stability of an equatorial jet and the formation of ... · 1.1 figures 1.1 to 1.3 illustrate the...

STABILITY OF AN EQUATORIAL JET

AND THE FORMATION OF CYCLONE TWINS

by

Servando Marco Augusto De la Cruz-Heredia

A thesis subm itted in conformity with the requirements for the degree of Doctor of Philosophy

Graduate Department of Physics University of Toronto

©Copyright by Servando Marco Augusto De la Cruz-Heredia (2000)

Reproduced with perm ission o f the copyright owner. Further reproduction prohibited w ithout permission.

STABILITY OF AN EQUATORIAL JET

AND THE FORMATION OF CYCLONE TWINS

Doctor of Philosophy (2000)

Servando Marco Augusto De la Cruz-Heredia

Department of Physics

University of Toronto

Abstract

It has been observed th a t equatorial cyclone twins (ECTs), which may be a triggering

factor for El Nino events, are usually preceded by powerful jet-like westerly wind bursts.

These bell-shaped profiles are usually modelled (in a rather arbitrary manner) as being

Gaussian or sinusoidal in shape. These types of profiles are generally barotropically unstable

to infinitesimal perturbations in the most common linearized geophysical models, be they

quasigeostrophic, shallow water, or Boussinesq, but the structures of the instabilities are

very different from what would be required to trigger an ECT. A more generalized wind

flow, however, can lead to growing modes with symmetric lows across the equator that are

similar to ECTs. The instabilities may be understood in terms of the wave resonance theory,

in which neutral waves (which may be induced by shear, background vorticity, gravity, or

some other restoring mechanism) can phase-lock and grow in order to destabilize the flow.

ii


Acknowledgements

Thank you Kent for the academic guidance and support, Hiro for the discussions,

CONACYT for the financial support, my father Servando for the physics, and above all my

mom M arta Irene for her unconditional love and encouragement. This thesis is dedicated

to her.

iii


Contents

1 Introduction 1

1.1 Equatorial Cyclone T w i n s ........................................................................................ 1

1.2 The linear stability p ro b le m ..................................................................................... 15

1.3 Wave reso n an c e ................................................................................................................ 17

2 Overview of the models 20

2.1 Geophysical m o d e ls ......................................................................................................... 20

2.2 The Boussinesq m o d e l ...................................................................................................20

2.3 The Shallow-water m odel............................................................................................... 22

2.4 The Quasigeostrophic m o d e l ......................................................................................... 23

2.5 The Geostrophic momentum m o d e l ............................................................................ 25

3 Stability in the quasigeostrophic model: resonance in piecewise linear

flows 28

3.1 The one-dimensional potential vorticity eq u a tio n .....................................................28

3.2 Some simple, stable flows on an /-p lane .................................................................. 30

3.2.1 Constant and Couette f lo w s ............................................................................. 30

3.2.2 A piecewise linear flow with a discontinuity in u<jyy ..................................32

3.3 Unstable waves in /-p lane QG th e o ry ......................................................................... 33

3.4 Wave-wave interaction: two wave resonance ............................................................40

iv


3.5 Three and four wave resonance .................................................................................... 46

4 Stability in the quasigeostrophic model: resonance in numerically dis

cretized flows 53

4.1 Numerical approach to solving smooth p ro files .......................................................... 53

4.2 The hyperbolic tangent p ro file ....................................................................................... 54

4.3 The Gaussian j e t ...............................................................................................................65

4.3.1 The over-reflection m echanism ..................................................................... 66

4.3.2 The wave-resonance m ech an ism .................................................................. 67

4.4 The /3 e f fe c t .......................................................................................................................73

4.4.1 Rossby waves and stability c r i t e r i a ............................................................76

4.5 The Hyper-Gaussian je t on a /3 -p lane .........................................................................80

5 The Hyper-Gaussian jet on a /3-plane: shallow-water and Boussinesq mod

els 88

5.1 Equatorial flows and the role of gravity w a v e s .........................................................88

5.2 Neutral waves in shallow-water theory ......................................................................89

5.2.1 Neutral modes on a shallow-water equatorial /3 -p la n e ................................90

5.3 Stability of an equatorial Hyper-Gaussian jet in sh a llo w -w a te r...........................94

5.4 The Boussinesq m o d e l ................................................................................................... 99

5.5 Neutral waves supported by the Boussinesq m o d e l................................................ 101

5.6 Rossby wave-Kelvin wave coupling ...........................................................................103

5.7 Stability of an equatorial Hyper-Gaussian j e t .......................................................... 117

6 Summary, conclusions and future work 126

A The Quasigeostrophic numerical eigenvalue problem 134

B The Shallow-Water numerical eigenvalue problem 137

v


C The Boussinesq numerical eigenvalue problem 143

vi


List o f Tables

3.1 Conditions for instability (Nondivergent, barotropic QG model) ....................... 39

vii


List o f Figures

1.1 Figures 1.1 to 1.3 illustrate the formation of equatorial cyclone twins. The

sequence shows the genesis of Typhoon Lola and Tropical Cyclone Namu

in 1986. Infrared images from May 14 (this figure), 17 and 19 are shown.

Underneath the clouds of this image the westerly wind bursts (WWBs) have

nearly peaked................................................................................................................... 2

1.2 On May 17 the cyclones reach tropical storm intensity while the WWBs

begin to wane................................................................................................................... 3

1.3 By May 19 the two well-defined cyclones are apparent.......................................... 4

1.4 A “composite” of ECT evolution (from Lander, 1990). Figures 1.1 to 1.3

correspond to stages 2, 3 and 4 shown here. Pressure troughs are indicated

with dashed lines, “C” shows the locations of subdepression strength vortices

while the zonal wind flow is represented by the wind barbs (1 barb = 5 m s-1). 5

1.5 Solution of the shallow-water equations given a symmetric heat source cen

tred a t the equator (from Gill, 1979). Figure (a) shows contours of vertical

velocity and the horizontal velocity field, while the contours in (b) represent

the perturbation pressure which is everywhere negative. The heating region

resembles the central elliptical contours in (a)......................................................... 8

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1.6 The top figure shows surface level pressure contours (in mb) for 00Z, May 10,

1986, while the bottom figure shows the wind field magnitude and direction

in m s_1. The westerlies have commenced and continue to intensify a t this

point. The wind fields in this and the three subsequent figures are shown a t

the 850 mb level (approximately a t 1500 m above sea level)................................ 9

1.7 Same as Fig. 1.6 but for May 14 a t 06Z. Note the strong westerly wind burst

located about the equator................................................................................................. 10

1.8 By 12Z on May 19 the cyclones have matured and are well-defined, while the

equatorial winds have already waned considerably......................................................11

1.9 Increase of the cyclones’ perturbation kinetic energy (in arbitrary units) be

tween May 10 and May 21, 1986. The e-folding time is approximately 70

hours.......................................................................................................................................12

1.10 Daily averaged zonal wind speed during May 1986................................................. 13

1.11 Equatorial cross-section of May 14 (Fig. 1.7) a t 155°E. Vertical axis is pres

sure in mb. Contours indicate the zonal wind speed in m s_I (westerlies are

positive).................................................................................................................................14

3.1 Stability of Rayleigh’s broken-line shear layer profile. P lot (a) shows the

growth rate while (b) indicates the phase speeds (the thick line represent the

phase speed of the unstable mode). The curves drawn with “+ ” and "

signs show the phase speeds of the neutral vorticity modes localized about

the “kinks” at y = ±d. Plot (c) shows the structure ('P field at t = 0) of

the fastest growing mode (dashed contours are negative), together with the

initial, zonally averaged Reynolds stress R and vorticity £ = ( ^ IX + '£yy)x-

The horizontal dotted lines indicate the position of the kinks. A schematic

of the flow ua(y) is shown to the far right of (c)......................................................... 35

ix


3.2 Instability due to the interaction between two phase-locked (or “resonant”)

waves (thick horizontal arrows denote the direction of propagation of a free

vorticity wave). Fluid displaced from the middle layer has anticyclonic

(clockwise) vorticity while the vorticity of fluid advected from the lateral

layers into the middle one is cyclonic (relative to the background). Together

with the vertical displacement of the interfaces [the vertical arrows denote

the positions of Vig maxima as given in (3.17)], the combined effect results

in the growth of the disturbance. As the kinks become further apart (for

a given wavelength) the interaction becomes very weak and eventually the

instability vanishes..............................................................................................................41

3.3 Difference energy of the two-kink profile (dotted line). E D [as evaluated

from equation (3.27)] vanishes in the region where the mode is unstable (for

k < 0.639 , cf. Fig. 3.1). The circles represent the sum of the difference

energies of the waves at each kink, E d+ + E D_ [equation (3.28)]...........................45

3.4 Stability analysis of a “triangular je t” . (a) Three neutral waves exist locally

about each kink, although the profile as a whole is unstable, as seen by the

growth rate curve (b). Plot (c) shows the phase speeds of both the three-

kiuk profile and the individual neutral modes: “o,” generated by

the isolated kinks a t d = —1,0 ,1 , respectively. Plots (d) and (e) show the

structures of the unstable and stable modes a t k = 1.23, respectively. . . . . 47

3.5 A “staircase” profile with four kinks. At higher k there are two modes of

equal growth rates and opposite phase speeds, while at the smaller wavenuru

bers the unstable modes possess the same phase speed b u t different growth

rates. The dashed and solid vertical lines indicate the positions (k = 0.70

and k = 0.045, respectively) at which the structures shown in Fig. 3.6 are

plotted....................................................................................................................................48

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3.6 Plots (a) and (b) show the structures of the unstable modes generated by

the “staircase” profile a t k = 0.70. The mode shown in (a) propagates

towards the right with c = 0.42, while the other has the same growth rate

bu t c = —0.42. At a smaller wavenumber (k = 0.045) the phase speeds are

the same (c = 0), but the growth rates differ, with the most unstable mode

being shown in plot (c). Note the vertical scaling in (c) and (d) (blow-ups of

the mid-region are shown beneath each plot), and the fact th a t the Reynolds

stress becomes negative a t the middle of the channel................................................49

3.7 Structure of the unstable mode with c > 0 (a), and c < 0 (b), when k = 0.20. 50

3.S Multiple wave resonance in a three-layer How. Note the progression from two

isolated instances of Kelvin-Helmholtz instability a t each interface towards

the large scale instability in the final stages (from Sakai, 1997).......................... 52

4.1 Stability of a barotropic flow u0(y) according to the stability criteria in Ta

ble 3.1. The arrows indicate the intrinsic phase speeds of the shear modes.

Flow (a) is stable because u0yy < 0 for all y. and hence fails Rayleigh’s nec

essary criteria for instability. Similarly, u0yy > 0 in (6) and the flow is again

stable. In (c) there is an inflection point present (dashed line) but the flow

does not satisfy FjOrtfort’s criteria. The flow shown in (d) might be unstable

since there is an inflection point uoyy(!/c) = 0 and uayy{u - u{yc)) < 0 (from

Drazin and Reid, 1991)......................................................................................................55

4.2 Stability diagram for the tanh(at/) profile as a function of the shear (a) and

the channel width (26) when Uo = 1. Positive regions are unstable according

to equation (4.1)..................................................................................................................56

xi


4.3 Plot (a) shows the shape of the profile uo = tanh(y) (solid line) and its

second derivative (dashed line). The circles indicate the positions of the

sample points used by the numerical solver. The growth rate as a function

of wavenumber k is shown in (b) (for clarity, only positive growth rates have

been plotted), while the phase speeds between cT = —0.2 and 0.2 are shown

in (c) (see text for explanation of the markers). The darker dots represent

the phase speeds of the unstable modes. P lot (d) shows the structure of the

fastest-growing mode, together with the barotropic energy transfer and the

vorticity profile. The number of grid points was set to n = 400, and the

channel width to 20. All perturbation fields are assumed to vanish at the

walls........................................................................................................................................57

4.4 Same as Fig. 4.3 but using n = 14 grid points to solve the eigenvalue problem

(4.2). The markers “+ ” and indicate the location (in plot (a)) and

dispersion relationships (plot (c)) of the neutral waves generated by the kinks

as if they were isolated.......................................................................................................59

4.5 Same as figure (4.3) but using n = 50 (top four plots) and n = 51 (bottom

four)........................................................................................................................................61

4.6 The reason monotonic flows require two more inflections points for new insta

bilities to appear (as stated by the Balrnforth-Morrison criteria) is illustrated

above. While the flow pictured in (a) may be unstable, the number of un

stable modes will not change by adding an inflection point (b) since the

Doppler-shifting works against any new resonances taking place. Two more

inflection points (c) are thus necessary for new unstable modes to appear. . . 62

xii


4 .7 S ta b ility o f a G au ssian profile u o ( y ) = Uoe . P lo t (a) show s th e shap e

of the je t and details of the numerical sampling near the inflection points, (b)

shows the growth rate curves, (c) the phase speeds, (d) and (e) the structure

of the two most unstable modes a t k = 9.538 and k = 6.024, respectively.

Horizontal dotted lines in the Reynolds and vorticity diagrams indicate the

location of the inflection points........................................................................................64

4.8 The top figure shows the required wave geometry for over-reflection to take

place (after Proehl, 1996). The lower figure shows this geometry applied to

a barotropic Gaussian je t.................................................................................................. 66

4.9 Diagram showing the resonant waves generated by a Gaussian jet. The

crosses indicate the inflection points.............................................................................. 68

4.10 A piecewise linear “jet,” with the structure of the most unstable mode at

k = 0.837 shown in (d), and th a t of the peak secondary instability a t k =

0.784 in (e)............................................................................................................................69

4.11 Same as Fig. 4.10, but for a wider je t (structures shown a t k = 0.799)............. 70

4.12 A Hyper-Gaussian profile. Plot (d) shows the structure of the most unstable

mode a t k = 2.452. The secondary mode in (e) peaks a t k = 2.492.................. 72

4.13 Stability of a cosine jet. The structure of the only mode present is some

what similar to that of the most unstable Hyper-Gaussian mode shown in

Fig. 4.12(d), although only one maxima is present along the y direction. . . . 74

4.14 Same as Fig. 4.7 but for the lower half of the Gaussian je t .......................................75

4.15 Dispersion relationship cf the first ten Rossby waves in the absence of back

ground shear. The nondimensional width of the channel is 2b = 4, while

0 = 0.375, which represents a m id-latitude value of 1.5 x 10- n (m s)_1. . . . 78

xiii


4.16 Dispersion relation of a weak, westerly Hyper-Gaussian je t on a mid-latitude

/3-plane. When the shear is small (Uo = 0.01) the flow is stable and the

Rossby modes (of negative phase speeds) are only slighted affected.................. 81

4.17 A stronger Hyper-Gaussian je t (Uo = 0.1) leads to unstable modes, although

the lower Rossby modes can still be discerned. Close inspection at the the

lower critical wavenumber, however, reveals th a t the slower instability does

not bifurcate into a shear-deformed Rossby mode...................................................... 82

4.18 Eastward Hyper-Gaussian je t on an /-plane (to be compared with the /3-plane result

of Fig. 4.19). The scales used are the same as in Fig. 4.15. Each mode is

unstable for 0 < fc2 < k% where kc is the corresponding critical wavenumber. 83

4.19 Eastward Hyper-Gaussian je t on a /3-plane. Although the symmetric (and

most unstable) mode now presents a second critical wavenumber a t k =

0.668, it does not bifurcate into a Rossby mode..........................................................85

4.20 Structures of the unstable modes due to of a Hyper-Gaussian je t on a /3-

plane. The most unstable mode is pictured a t the top when k = 2.53. The

secondary mode peaks at k = 2.46 and is pictured a t the bottom ......................... 86

5.1 Neutral modes supported by the shallow-water model a t the equator (numer

ical solution). Note the appearance of boundary Kelvin waves propagating

westward with c = - 1 . These are due to the artificial walls required to

bound the numerical problem.......................................................................................... 92

5.2 Eastward Hyper-Gaussian je t on an equatorial /3-plane. While not all modes

have converged (see Fig. 5.3), it is useful to compare with the QG midlatitude

result shown in Fig. 4.19, for which the unstable modes present essentially

the same characteristics.....................................................................................................95

5.3 Same as the previous figure but using a high-resolution sparse calculation.

The number of interior grid points in this case was set to n = 10000................ 97

xiv


5.4 Structure of the most unstable (top) and the secondary (bottom ) modes at

k = 2.46 and k = 2.25, respectively. The contours represent isolines of hi,

while the plots labelled R and £ show the Reynolds stress and the vorticity. . 98

5.5 On a northern /-p lane (say) the most unstable unstable mode (pictured on

the top left) is symmetric and results from vorticity wave coupling, as does

the secondary mode shown to its right. The dynamics of the modes remain

unchanged when the flow is shifted to an equatorial /3-plane, and hence in

order to conserve the sense of rotation within the disturbance the signs of

the pressure perturbations must switch when crossing the equator (since

the sign of / changes). The most unstable mode is thus the antisymmetric

perturbation shown in the bottom left diagram........................................................ 100

5.6 Neutral modes supported by the nonhydrostatic Boussinesq model in an

equatorial channel 30km (top) and 10km (bottom) deep. Classification of

the modes is as in Fig. 5.1.............................................................................................. 104

5.7 A bounded piecewise linear profile with a shear discontinuity. The shears in

regions 1 and 2 are u ly and uly, respectively............................................................105

5.8 Nondimensional Boussinesq eigenvalue spectrum versus zonal wavenumber

when m = 10 k. Solid and dashed lines show the imaginary and real parts

of the frequency, respectively. Note the two regions of instability centred

approximately at k = 0.360 and k = 0.403................................................................ 108

5.9 The perturbation pressure (contours) and horizontal velocity (arrows) are

plotted for each of the unstable modes. The Reynolds stresses ( - (uv)xz (e,/e))

are shown to the right of the fields. The walls are located a t y — ± 1 .................109

5.10 Vertical perturbation velocities a t k = 0.360 (top), and k = 0.403 (bottom).

Note that in the top figure the flow a t midchannel is horizontally nondivergent.110


5.11 Nondiraensional eigenvalue spectrum versus zonal wavenumber when m —

10 fc, GM case. Solid and dashed lines show the imaginary and real parts of

the frequency. The values used were b = 1, ei = 0.2 and t i = 0.4. Compare

with the result from the Boussinesq model in figure (5.8)........................................113

5.12 “Decomposition” of the velocity profile shown in Fig. 5.7 into three simpler

flows......................................................................................................................................114

5.13 Frequency matching contours: (a) Kelvin wave at y = — b with the Rossby

wave at the kink (labeled K-R). (b) Both Kelvin waves (K-K). (c) Rossby and

Kelvin wave a t y = +b (R-K). The values chosen for the various parameters

are b = 1, ei = 0.2, and t i = 0.4. The dashed lines are the k-m ratios shown

in Fig. 5.11 and Fig. 5.15. They represent, from left to right, m — 10/;,

m = 5 k and m = 2 k ........................................................................................................ 115

5.14 Phase speeds of the three modes generated by the flows shown in Fig. 5.12.

The solid line is the Kelvin wave localized near y = 6, the dashed line shows

the dispersion curve of the Kelvin wave at y = - b, and the dotted line is

the phase speed of the kink-generated Rossby wave as a function of k. Here

b = 1, ei = 0.2 and t i = 0.4............................................................................................ 116

5.15 Comparison of the growth rates between the Boussinesq model (solid line),

and the GM model (dashed line) when m = 5 k (top) and m = 2 k (bottom ).

See also Fig. 5.13...............................................................................................................118

5.16 Growth rate contours in (k, m) space of an equatorial, Hyper-Gaussian je t

(Boussinesq model). Shading highlights the unstable regions. The top figure

shows the contours of the most unstable modes, while the bottom figure

shows the growth surface which lies underneath........................................................120

5.17 Growth rate and phase speed curves of the unstable modes generated by a

Hyper-Gaussian jet in the Boussinesq model when m = 0.......................................122

xvi


5.IS Structures of the two unstable modes a t m = 0. The instability pictured

a t the top peaks a t k = 2.42 and has a growth rate of Sj(o-) = 0.55. The

perturbation pictured on the bottom plot corresponds to the secondary mode.

It was calculated a t k = 2.38 and possesses a nondimensional growth rate

of a = 0.46. Dotted lines indicate the position of the inflection points at

±516 km from the equator. For visualization purposes only the central part

of the channel is shown, since the total width is 14 (in nondimensional units). 123

5.19 Same as Fig. 5.17 but with m = ir................................................................................ 124

5.20 Structures of the two unstable modes a t m = tr. The most unstable mode

(top) peaks a t k = 2.40 and has a growth rate of cr = 0.44. The secondary

instability pictured on the bottom plot was calculated a t k = 2.19 and pos

sesses a growth rate of cr = 0.29.....................................................................................125

B .l Numerical procedure used to determine the stability of an eastward Hyper-

Gaussian je t on an equatorial /3-plane using the shallow-water model. The

squares represent the coarse guesses (using 100 grid points) obtained by

solving the nonsparse eigenvalue problem (B.2) using a QR algorithm. The

three most unstable modes (tagged ) are saved and used as seeds for the

sparse solver. The results, using 10000 grid points, are tagged here as “0 ” .

These latter values are then chosen to begin a dense trace along k, using a

result a t point fc; as a guess for point i. The trace is also performed using

10000 grid points and the resulting curve is plotted above with dots . . . 141

xvii


Chapter 1

Introduction

1.1 Equatorial Cyclone Twins

Figures 1.1 to 1.3 show the remarkable formation of a cyclone pair which took place

during the month of May in 1986. The sequence of infrared satellite images begins on May

14 with the presence of an unorganized convective cloud complex along the equator between

150°E and 170°E. In the lower troposphere powerful westerly winds bursts (WWBs)1 blow

during these periods with an average zonal speed of 10 m s-1 in a jet-like fashion, with a

peak speed of nearly 25 m s-1 with respect to the background flow. A cyclonic circulation

forms in each hemisphere between 5 and 8 degrees to the north and south of the equator.

The wind bursts continue to intensify during the next couple of days until on the 16 and 17

the cyclones reach tropical storm intensity. By then the equatorial convection has collapsed

while the equatorial westerlies weaken as the cyclones mature. The westerlies continue to

wane and by the 19 the twin cyclones are well-defined and slowly begin to travel westward.

At this time the southern cyclone begins to abate, and by the 21 it has nearly vanished.

The northern storm then continues its journey westward and towards the pole.

1 Westerlies flow eastwards, while easterlies flow from east to west.

1


Chapter 1. Introduction 2

Figure 1.1: Figures 1.1 to 1.3 illustrate the formation of equatorial cyclone twins. The se

quence shows the genesis of Typhoon Lola and Tropical Cyclone Namu in 1986. Infrared

images from May 14 (this figure), 17 and 19 are shown. Underneath the clouds of this

image the westerly wind bursts (WWBs) have nearly peaked.


Chapter 1. Introduction

Figure 1.2: On May 17 the cyclones reach tropical storm intensity while the WWBs begin

to wane.



Figure 1.3: By May 19 the two well-defined cyclones are apparent.



STAGE 1

VV'f;-._c

STAGE 2

-EQ.

STAGE 3

- v * t'- - ~ V EQ.

STAGE 4

9 ' : / /

-EQ.

Figure 1.4: A “composite” of ECT evolution (from Lander, 1990). Figures 1.1 to 1.3 corre

spond to stages 2, 3 and 4 shown here. Pressure troughs are indicated with dashed lines,

“C” shows the locations of subdepression strength vortices while the zonal wind flow is

represented by the wind barbs (1 barb = 5 m s_1).



Although the previous paragraph specifically details the cyclogenesis of Typhoon Lola

and Tropical Cyclone Namu in 1986, the above evolution, pu t forth by Lander (1990), also

describes a composite of the formation of various equatorial cyclone twins (ECTs) (see

Fig. 1.4). The occurrence of storm pairs varies depending to the definition, but according

to the one by Lander2, their frequency in the Western Pacific ocean is approximately once

every 2 to 5 years (Nieto Ferreira et al., 1996). The suspicion th a t ECTs may be a by

product of the WWBs has been present in the literature (Lander, 1990; Hartten, 1996),

but so far there do not seem to be any thorough studies regarding this particular link. The

importance of WWBs and ECTs has been highlighted in the past few years due to their

apparent relation (possibly causal) to the occurrence of El Nino. The El Nino phenomenon

has been extensively described in the literature (Philander, 1983; Ramage, 1986), and has

gained recognition due to its widespread effects (droughts in Asia and the collapse of the

Peruvian fishery, among many others) (Wyrtki, 1975). A “typical” El Nino event takes

place every 2 to 7 years and is characterized by a large tongue of particularly warm water

propagating westward from the coasts of Ecuador and Peru. The anomalous temperature

rise exists for about a year accompanied by a weakening (and sometimes reversal) of the

easterly trade winds in the western Pacific (Cane et al, 1986). These trade winds, called the

Walker Circulation, form a zonal cell consisting of low altitude easterlies which rise in the

North Australian-Indonesian Low Pressure Zone and become westerlies a t high altitudes,

sinking later upon reaching the South-east Pacific High Pressure Zone. The disruption of

the Walker Circulation is actually a precursor to an El Nino (although it does not necessarily

imply th a t one will take place), and it has been suggested th a t the ECTs resulting from the

21) Twin cyclones form nearly simultaneously.

2) They form a t about ±5 degrees about the equator.

3) They form along the same longitude

4) Their wind, pressure, and cloud patterns are nearly symmetrical with respect to

the equator.



ensuing WWBs may trigger the event (Keen, 1982; Ramage, 1986). ECTs have preceded

most El Nino events (Ramage, 1986), including those th a t took place in 1986-87 (shown in

Figs. 1.1 to 1.3 and Figs. 1.6 to 1.8 below), and 1991-92 (Nieto Ferreira et al., 1996).

Studies regarding the formation of ECTs have mostly focused on the effects of heat

ing. Gill (1979) used a linearized shallow-water model to study how a resting atmosphere

responds to a heat source located at the equator. In order to model convection in the ide

alized system mass was extracted from the flow, and analytical time-independent solutions

were found. When the heat source is symmetric with respect to the equator two lows form

to the north and south of the forcing, as shown in Fig. 1.5. Nieto Ferreira et al. (1996)

later solved a similar problem using a nonlinear shallow-water model on a sphere assuming

a time-dependent heat forcing (with similar spatial structure), and again symmetric lows

appeared a t ±9° of latitude. The resulting cyclones, however, were rather weak (peak winds

of less than 9 m s-1), and the heat sources invariably generate an ECT [as opposed to the

more common single-cyclone events (Hartten, 1996)]. As with Gill’s study, no background

WWBs were present.

Figure 1.6 shows the surface pressure and 850 mb horizontal wind fields from the NCEP

reanalysis on May 10,1986. A t this point the westerly wind bursts have already commenced

and four days later the strong gusts are still apparent (Fig. 1.7). By the 19 the cyclone pair

is clearly discernible while the westerlies have begun to weaken considerably. The growth

of the storms can be clearly seen in Fig. 1.9, which shows the change in the eddy kinetic

energy density over a box which surrounds each cyclone. This growth is roughly exponential

(with an e-folding time of approximately 70 hours), and although the central pressures dip

only to about one thousand millibars (approximately 12 mb below the surrounding surface

pressures), the peak wind speeds rise above 20 m s-1 , the threshold of hurricane intensity

(Saffir-Simpson hurricane scale, Beven 1999).



( a )

■-10

(b)

Figure 1.5: Solution of the shallow-water equations given a symmetric heat source centred

a t the equator (from Gill, 1979). Figure (a) shows contours of vertical velocity and the

horizontal velocity field, while the contours in (b) represent the perturbation pressure

which is everywhere negative. The heating region resembles the central elliptical contours

in (a).



1014

Figure 1.6: The top figure shows surface level pressure contours (in mb) for 00Z, May 10,

1986, while the bottom figure shows the wind field magnitude and direction in m s-1.

The westerlies have commenced and continue to intensify a t this point. The wind fields

in this and the three subsequent figures are shown a t the 850 mb level (approximately

a t 1500 m above sea level).



Figure 1.7: Same as Fig. 1.6 but for May 14 a t 06Z. Note the strong westerly wind burst

located about the equator.



1004

Figure 1.8: By 12Z on May 19 the cyclones have matured and are well-defined, while the

equatorial winds have already waned considerably.


Eddy

ki

netic

en

ergy

(a

rbitr

ary

units

)


day

— Northern cyclone EKE— Sou thern cyclone EKE300

250

200

150

100

Figure 1.9: Increase of the cyclones' perturbation kinetic energy' (in arbitrary units) between

May 10 and May 21, 1986. The e-folding time is approximately 70 hours.



Figure 1.10 shows the averaged3 zonal wind speeds from the NCEP reanalysis during

the month of May. At the sta rt of the month the usual easterlies are present, but by

May 9 the flow reverses and on May 16 the average wind speed is 6 m s ' 1. Typhoon

Lola reaches tropical storm intensity a t th a t point, followed by the appearance of Tropical

Cyclone Namu the next day. Afterwards the winds quickly wane so th a t by the end of

the month the easterlies have resumed. Figure 1.11 shows the structure (latitude/height

i

oNto><

Date (May)

Figure 1.10: Daily averaged zonal wind speed during May 1986.

cross-section) of the WWB in Fig. 1.7. It has the form of a localized je t about the equator,

confined to the lower troposphere. The zonal wind speed at its core is approximately

20 m s " 1 towards the east, the je t being embedded in a westward flow which possesses a

speed between 10 and 6 m s-1 . It is the purpose of this thesis to study the possible role of

3Spatial average is taken within a “corridor” containing the je t, located between 5°N and 5°S, 120°E

and 170°E, and from 300 to 1000 mb.



20S 15S 10S 5S EQ 5N 10N 15N 20N

Figure 1.11: Equatorial cross-section of May 14 (Fig. 1.7) a t 155°E. Vertical axis is pressure

in mb. Contours indicate the zonal wind speed in m s_1 (westerlies are positive).



westerly wind bursts in the cyclogenesis of equatorial cyclone twins. As can be seen from

Fig. 1.11 the shape of the WWB suggests modelling the burst as a bell-shaped je t confined

to the troposphere. We shall thus take into account the finite depth of the flow in order

to analyze the effect it has on the stability of the profile. The focus will be concentrated

on barotropic jets (background flows whose velocity is solely dependent on latitude) with

varying degrees of curvature in the vicinity of the je t maximum.

In order to attem pt to understand the mechanism behind the initial formation of a pair

a linear analysis is carried out. Although finite in depth the je t will be assumed to be

zonally infinite and stationary (the average strength of the je t increases by about 15% in

the three days previous to the appearance of the cyclones). Emphasis will be placed on the

dynamics from a “wave resonance” perspective, as explained in Section 1.3.

1.2 The linear stability problem

In order to elucidate the manner in which a given flow may become unstable4 a series of

assumptions are made, starting with the choice of a suitable model of the flow's behaviour

(which simplifies the problem at the expense of eliminating some physical processes). If

one further assumes th a t the fluctuations which initially take place are small compared to a

time-independent basic state, the model can then be linearized and the resulting equations

become even simpler. The problem is now reduced to determining whether a small wave

like disturbance will grow in time (i.e. become unstable) given a certain basic flow which

resembles one found in nature.

Unfortunately the stability of even simple flows in idealized fluids usually poses ana

lytically intractable problems. In these cases the most common approach is to solve the

problem numerically, a procedure which often involves spatial discretization of the flow

4 We shall confine our attention to normal modes.



fields. An example of this methodology is given in Appendix A. Briefly, the basic flow

and the perturbation’s as-yet-to-be-determined structure are sampled by a certain num

ber of grid points. This leads to an eigenvalue problem which, if sta ted in m atrix form5,

can be solved by various methods, such as the application of certain transformations (e.g.

“QR m ethod”), or through iterative techniques such as the “power method” . Each have

their own particular weaknesses and strengths: transformation methods are usually ro

bust numerically and provide the full eigensolution a t the expense of high computational

and memory consumption (which can severely limit the resolution, for example, Moore

and Peltier (1987), Yamazaki and Peltier (2000b)), while iterative procedures tend to be

resource efficient but only provide partial solutions and convergence is more uncertain.

Since it is not clear, a ■priori, how many grid points will be required in order for the

solution to converge, a combination of both methods is sometimes useful. “Convergence”

in this la tte r context does not refer to the result of a given numerical procedure, but to the

determination of the “true” dynamics of the perturbed flow. The solution has converged

when an increase in the resolution (i.e. number of grid points) no longer changes the

result. This usually implies solving the eigenvalue problem, doubling (say) the number of

sample points, and repeating the procedure until there are no noticeable differences between

two subsequent iterations. O ther factors, such as the necessity of probing wider domains

for infinite flows (which must be artificially bounded in order to be solved numerically),

also affect convergence. We’ll call this situation “physical convergence” , to distinguish it

from the “numerical convergence” mentioned in the previous paragraph. The underlying

physics, unfortunately, is usually obscured by the above numerical details. It is here the

wave resonance theory may be of most use, as seen below.

5Solving the differential equation can be achieved in other ways, e.g. by means of a Runge-Kutta

algorithm, o r by spectral methods which also require solving an eigenvalue problem (Moore and Peltier,

1990).



1.3 W ave resonance

The idea of instability as the product of the “resonance of waves” has been mostly

developed in the last twenty-five years or so (Yih, 1974; Cairns, 1979; Ripa, 1982; Hoskins

et al., 1985; Hayashi and Young, 1987; Fabrikant and Stepanyats 1998), although the notion

was present as far back as Taylor (1931). The basic idea is as follows: we consider th a t the

energy of a perturbed flow E p is given by the sum of the energy of an undisturbed state

Eu plus the energy of any disturbances th a t may be present E d s o th a t E p = Ep + E D. If

there are no external forces (dissipative or otherwise), then conservation of energy requires

that modes with nonzero E d must be stable, otherwise they cannot be excited without

changing the to tal energy of the system. Hence only disturbances for which E d = 0 can

amplify spontaneously6. When waves having difference energies of the same magnitude but

opposite signs interact while remaining stationary with respect to each other (phase-locking

through Doppler shifting), the possibility of instability arises since the resulting difference

energy is zero.

The above theory has been applied to various hydrodynamic problems with notable suc

cess. Satomura (1981a), for example, analyzed the stability of a nonrotating, shallow-water

Couette flow, and noted th a t gravity modes could somehow “mix” in order to produce

unstable modes. Satomura, however, did not seem to be formally aware of the wave reso

nance idea until a subsequent note (Satomura, 1981b). Hoskins et al. (1985) offered a clear

picture of Rossby wave locking when discussing shear instabilities. Some time later Sakai

(1989) applied the theory to a two-layer model on an /-p lane. Vertical shear in this case

provided the necessary Doppler shift for Kelvin and Rossby waves to phase-lock and create

an unstable mode. Kobayashi and Sakai (1993) studied the barotropic stability of zonal

6The quantity E d has been given various names in the literature, the most common ones being “pseu

doenergy,” “wave energy,” “disturbance energy” and “difference energy,” among others. We shall use the

latter (Iga, 1993).

Reproduced with perm ission o f the copyright owner. Further reproduction prohibited w ithout perm ission.


and meridional flows confined near a wall on a /3-plane. They attribu ted the instability of

the lower half of a sech2(y) je t to the coalescing of the two continuous modes nearest to

the inflection point (the continuum having been discretized by finite differentiation). Iga

(1993) later reanalyzed Orlanski’s problem and described the resulting instabilities in terms

of resonances between various Rossby and gravity waves. He also noted th a t the resonance

Sakai (1989) described was actually between a Rossby wave and a mixed Rossby-gravity

wave. Furthermore, Iga seems to have encountered, bu t not fully explained, an instance

in which four modes could resonate in order to generate a single instability. Baines and

Mitsudera (1994) presented a clear dynamical picture, similar to Batchelor’s description

of the Kelvin-Helmholtz instability (Batchelor, 1967), regarding the stability of a piece-

wise linear profile with two discontinuities in the background vorticity field (Gill, 1982).

Kushner et al. (1998) studied a horizontally bounded Couette flow comparing various fluid

dynamics models (semigeostrophic and the full Euler equations amongst them), and found

weak Kelvin wave-Kelvin wave coupling.

One advantage of the wave resonance formulation is th a t it is possible to gain insight into

the instabilities th a t may arise in a given flow by simply analyzing the neutral modes which

its various geometric components (walls, velocity discontinuities, etc.) can individually

support. These depend on the model with which the flow is being analyzed. A wall,

for example, does not contribute any extra dynamics in quasigeostrophic theory, but can

support the presence of boundary-trapped Kelvin waves in models which capture those

waves (Allen et al, 1997). If the Doppler shift provided by the background flow causes the

frequencies of two counterpropagating waves to match, these might then phase-lock and lead

to an unstable mode. Since it is usually simpler (and sometimes even analytically possible)

to obtain localized solutions of the neutral modes, one can then determine the approximate

location of unstable regions in wavenumber space by superimposing the dispersion curves.

Furthermore, simply adding the neutral fields can also provide a crude idea of what the



resulting unstable disturbance will look like, and allows one to understand the dynamical

processes responsible for the instability.

From this perspective the progression from the more complex models (e.g. Boussinesq)

towards the simpler ones (e.g. quasigeostrophic) simply involves the subsequent filtering of

neutral modes which in turn limits the number and nature of the wave resonances which

can take place. In this thesis we will systematically analyze the linear stability of simple

one-dimensional flows which vary solely in the meridional direction, i.e. the basic velocity

field is purely barotropic. Chapter 2 provides a brief overview of the models which will be

used. Chapters 3 and 4 deal with the stability of quasigeostrophic and barotropic flows.

The former chapter focuses on piecewise linear flows which provide a simplified picture of

the dynamics behind the wave resonance mechanism, while the la tter extends the concept

to smooth, numerically discretized flows. In Chapter 5 the stability of models which can

support gravity waves is studied. The shallow-water barotropic je t is analyzed and, since

no assumptions are made with respect to the magnitude of the Coriolis parameter, the flow

can now be shifted towards the equator. The second half of the chapter continues with

a further generalization provided by the Boussinesq equations, and focuses on the effect

of changing the depth of the domain within which the flow takes place. Rossby wave-

Kelvin wave resonance is analyzed in detail. A summary and final conclusions are given in

Chapter 6.


Chapter 2

Overview of the m odels

2.1 G eophysical models

The first step towards solving the stability problem requires modelling the flow in a

manner which simplifies the governing equations while retaining the dynamical aspects

relevant to the problem at hand. Sound waves, for example, are not expected to play a

role in the relatively slow, large-scale motions which interest us, and hence we can dispense

of this extra complexity by assuming the flow to be Boussinesq. Subsequent assumptions

then lead to the various models described in this chapter.

2.2 The Boussinesq m odel

We begin by considering a Boussinesq, inviscid and adiabatic flow on a /3-plane, that

is, a flat, ro tating frame of reference such that the planetary vorticity / = 2Qsin(0) varies

linearly with latitude (£2 is the earth’s angular velocity). The momentum equations are:

Du 1 dp ( .m - f v = -=P e-x’ (2' la)

20


Chapter 2. Overview of the models 21

% + f u = - i f , (2.1b)Dt pay

(2-ic)Dt p o z p

while mass conservation for an incompressible flow:

S +S +£=°’ ^requires th a t

~ = ( 2 -l e )Dt

where (x, y, 2) represents the zonal, meridional and vertical directions, the respective ve

locity components being (u , v, w). Time, pressure, and gravity are denoted by t, p = p(z) +

p ( x ,y , z , t ) , and g, while the Coriolis param eter a t latitude y takes the form / ( y ) — f t + fa y

on a coordinate system centred about yi . In accordance to the Boussinesq approximation

the density p = p + p(x , y, z, t) is assumed constant except when it gives rise to buoyancy

forces in the vertical momentum equation.

The above system can be linearized by assuming th a t time-dependent motions are depar

tures from a steady background flow, and as such we can expand the fields in the following

manner:

y7 = yro(y,z) + eJ7l ( x , y }z , t ) , (2.2)

where e < 1 and l^ol ~ |^ i |- By substituting the fields in the form (2.2) into system

(2.1) and assuming a basic flow ( u o , V o , W q ) = (u o ( y ) , 0 , 0) we obtain, a t 0 ( t ) , the following

linearized equations

dudt

1 dui ( du0\ 1 dpi . .= (2'3a)



(2.3b)

(2.3c)

du\ dv\ dwid x dy dz

(2.3d)

(2.3e)

where N is the Brunt-Vaisala frequency:

N 2 = _ 9 ^ P o ~ p dz

2.3 The Shallow-water m odel

By assuming th a t the fluid is in hydrostatic balance and th a t the density is constant we

immediately obtain, from (2.1c):

If we consider the undisturbed depth of the fluid layer to be h{x, y, t) integration of this

last equation yields

Substituting (2.5) into (2.1a) and (2.1b), and assuming th a t the horizontal velocities

u and v do not depend on depth (Pedlosky, 1987) results in the horizontal momentum

equations:

P = ~P9Z + pah- (2.5)

(2.6a)



dv dv dv . dh ,0 . . .— + u — + v — + f v = - j t - (2-6b)dz oy dy

Since u and v are independent of z equation (2.Id) can be immediately integrated (over

a flat bottom surface) to yield

— - ( ! +s ) ’ <">which, combined with the vertical speed of the free surface

dh dh dh

Y t + U d i + VY

gives:

. Ull UU Utl ,rt“ L=/l = -57 + “ 5Z + t' ^ ( )

dh , 8{uh) , d(vh) _ n ^¥ + ~ a ^ + _aT ~°- ( }

The shallow-water equations can then be linearized by employing an expansion similar

to (2.2) but with no vertical variation, resulting in the following system a t 0(e):

dudt

i, i du i ( r dun\ dh\ \t +u^ - { f - W Vl = - g^ (2-10a)

(2.10b)dvi dvi dhi_ + Uo_ + M = _9_ ,

a/ii , a/ii dui , a(«i/i0) ,o in „,~dt ° Y ^ 0 &T “ a^ - " (2' 10c)

2.4 The Quasigeostrophic model

We can readily render system (2.1) dimensionless by means of the following scaling:

p + W i U L ) / ( g H ) ] p

1-4 L x 1 1-4 (L/U) t / - +

1-4 Ly (u,v) 1-4 U (u ,v ) p 1-4

^ H z w M- (UH/L) w p 1-4



If we define the Rossby number Ro = U/{feL) the non-dimensional system becomes:

RoTTt-!v = - \ % (2'Ua)

R oi r t + f u = - W y (2' Ub)

Dt

? + ? + ? = 0’ <2-lld)o x oy oz

g = 0. (2.1Ie)

Expanding the fields in a power series of the Rossby number,

T = F g + R o T a + 0 ( R o 2), (2.12)

shows th a t a t 0 (1 ) the flow is geostrophic and hydrostatic:

’■ - i - » . - t ’ » - & ■

and hence the horizontal geostrophic velocity is nondivergent, allowing us to substitute pg

for the streamfunction Expanding in a Rossby series and substracting 3(2.11b)/cte from

3(2.11a) /d y (using (2.l id ) , (2.13), and hence the fact th a t wg vanishes at first order) yields,

a t 0 ( R o ),

V3t d x d y d y d x ) \ d x 2 dy2 dz ’ '

where the last term on the lhs can be written in terms of $ by substituting an expansion

for p of the form (2.12) into (2.11e) so th a t a t 0{Ro) we have:

( d 3 $ 3 d<$ d \ d 2<S> „ „( a t + dx dy dy d x ) d z2 + Wa ~ ’ ^


Chapter 2. Overview o f the models 25

where B = ( ( H N o ) / f e L ) 2 is a stratification parameter called the Burger number (N o = N

is assumed constant). Equation (2.14) thus becomes

d d * d d $ d \ ( d 2<$ d2<£ , , a \ _ nd t + ^ T y - ^ d i ) { d * + W d * + P ty ) - ° ’ (2'16)

which is the quasigeostrophic potential vorticity equation. Linearizing about a purely

barotropic zonal flow u0(y) finally yields:

( d W , - . A V a f , ( . d?u0\ _

The quantity within the second parenthesis is the perturbation potential vorticity (later

labelled “Q ”), while the rightmost parenthesis contains the meridional gradient of the

potential vorticity of the basic sta te IIy. Note that the nondivergent limit can be obtained

by letting B -+ oo, and the above reduces to the barotropic equation. Although this latter

case is valid a t low latitudes the vertical structure of the flow no longer plays a role in the

dynamics, an aspect which will be of importance later on when dealing with flows of finite

depth.

2.5 The Geostrophic m om entum m odel

Let us return momentarily to our original nonlinear, dimensional Boussinesq system

(2.1). Splitting u and v into a sum of geostrophic and ageostrophic components (but not

assuming th a t the la tter are necessarily smaller), we have th a t the geostrophic velocities

are

P -18 .)

" • — m (2I8b)



which upon substitution into equations (2.1a) and (2.1b) give:

^ - f v + f v g = 0, (2.19a)

^ t + f u - f u 9 = 0. (2.19b)

The ageostrophic parts are thus (making V = ( 1 / /) D/Dt):

ua = - V v , va = Vu ,

and hence solving for v and u in (2.19) yields

v = vg + V u = vg + Vug - V 2 v , (2.20a)

u = ug — V v = ug — Vvg — V 2u. (2.20b)

The geostrophic momentum approximation assumes th a t (Hoskins, 1975)

V 2u <S u, V 2v v,

so th a t the terms which are differentiated twice are ignored, and thus the resulting horizontal

momentum equations become

Assuming the flow to be hydrostatic we obtain, upon linearizing, the geostrophic momentum

model:

dulgdt

duig ( du0\ 1 dpi


Chapter 2. Overview of the models_____________________ 27

d v \ g d v \ g . 1 d p I /n n n L \_ £ + tl0_ + /tI1 = _ _ _ (2.22b)

^ = - p ig , (2-22c)dz

^ i + ^ + 5 l = 0, (2.22d)o x oy oz

dpi■ + U0~ - - ^ N 2WX = 0, (2.22e)

where

dt dx g

1dpi

p d y ’

1 9pip d x '

(2.22f)

(2.22g)


Chapter 3

Stability in the quasigeostrophic

model: resonance in piecewise linear

flows

3.1 The one-dim ensional potential vorticity equation

We shall first analyze the neutral modes which the quasigeostrophic (QG) model can

sustain. Unlike stable (unstable) modes whose amplitude decreases (increases) with time,

neutral waves oscillate without growth or decay. In the derivation of the QG vorticity

equation (2.16) some types of waves have already been eliminated due to the simplifications

made, such as the filtering of sound waves by the incompressibility assumption. Gravity

waves (whose restoring force is provided by g) have also been filtered through the time

scaling which eliminates the higher frequency modes.

We begin by considering a perturbation streamfunction which has the form1 (dropping

Ut is understood th a t physical quantities will be obtained by taking the real part of the fields.

28


Chapter 3. Stability in the quasigeostrophic model: resonance in piecewise

where c = a / k and a y subscript denotes differentiation with respect to th a t direction.

Recall th a t these equations are non-dimensional and th a t Uo = Uoiy)- For the above to

must be constant. This leads to the following eigenvalue problem for the vertical structure

function 4>:

where the boundary conditions are obtained by requiring th a t wa vanish at the horizontal

walls which bound the flow vertically. Note that on an /-p lane all rotational and stratifica

tion effects are encompassed by the Burger number. If the flow is nonrotating or infinitely

stratified (so th a t no vertical motion is possible), then m = 0 and the perturbation is purely

barotropic2.

The potential vorticity equation governing the flow becomes

where we have defined the “equivalent” wavenumber y 2 = k2 + m2. Since m does not

appear anywhere else in the equation it is clear th a t the vertical structure can be taken

background shear. The perturbation has a baroclinic structure determined by (3.3) and thus becomes a

purely barotropic mode when m — 0 (Pedlosky, 1987).

linear flows 29

the subscript):

(3.1)

so th a t substitution into equation (2.17) yields

hold for all y a t a given zonal wavenumber k each of the grouped terms over the braces

= ~ m 2/p{z), (3.3)

(3.4)

2The basic flow uQ(y) is barotropic since instability can take place solely as a result of the horizontal


Chapter 3. Stability in the quasigeostrophic model: resonance in piecewiselinear flows 30

into account by simply solving the purely barotropic case (i.e. the nondivergent limit for

which to = 0), but using p instead as an equivalent zonal wavenumber. The solutions of

equation (3.4) with the appropriate boundary conditions provide the meridional structure

of perturbations which propagate according to (3.1). All other perturbation fields can be

then calculated thereof.

3.2 Some simple, stable flows on an /-p lan e

In general it is not possible to analytically solve equation (3.4) for an arbitrary shear

flow, but we shall now look at two simple cases for which fS = 0. The vorticity equation

can then be written as follows:

(u0 - c) (ipyy - /i2V>) - Uoyyip - 0. (3.5)

3.2.1 C onstant and C ouette flows

We consider first the simplest case, th a t of a constant zonal flow independent of y. We

have, for this case:

(u0 - c) (iiyy - n2i/S) = 0. (3.6)

Except for the singular mode which results when c = uo (see below), there are no solutions

to the differential equation which satisfy the boundary conditions, and hence no other

modes are present.

For the case of Couette flow, i.e. constant shear uo{y) = Uqyy, we obtain the following:

(u0yy - c) (tpyy - y?ip) = 0. (3.7)

As before, there are no solutions which satisfy the boundary conditions unless uoyy — c = 0


Chapter 3. Stability in the quasigeostrophic model: resonance in piecewiselinear flows ____________________________________ 31

(i.e. a t a critical point), in which case the above becomes

ipyy - y 2ip = 6 ( y - c/uoy) = S(y — d), (3.8)

which has solutions of the form

4>(y) = j Q(y,y')S{y'- d ) d y '= Q{y,d), (3.9)f-b

where Q is the Green’s function

\Qi (y ,d) - b < y < dG(y,c) = \ (3.10)

(ff2(y.c') d < y < b ,

which satisfies

Qyy - = 0, y # c', (3.11)

and which must vanish a t the boundaries. Together with the jum p conditions (Arfken,

1985):

lim Gx(y,d) = lim S2(!/>c')>y->c_ y-*c’+

lim 4-G i(y ,d ) - lim ^ -Q i(y ,d ) = - 1 , y-*c:+ ay y-*c'_ dy

we find that:

! smh(fi[b — y])sinh(fi[b + c / u 0y])/(fi.sin\i(2iib)), c /u 0y < y < b

sinh(/i[i> + y]) sinh(/i[6 - c/noJ)/(/rsinh(2/r6)), —b < y < c /u 0y.

Case (1960) was the first to analyze this continuum of modes in a Couette flow by solving

the initial-value problem. Tnese singular perturbations are neutrally stable3, and their

phase speeds lie between the minimum and maximum values of the background flow speed.Solving the initial-value problem as Case did actually shows th a t the perturbations vanish algebraically

as 1 /i or faster as t oo. The mode continuum, however, completes the normal mode solution.


Chapter 3. Stability in the quasigeostrophic model: resonance in piecewise linear flows________________________________________________________________32

3.2.2 A p iecew ise linear flow w ith a d iscontinu ity in uo,jy

The third case involves a piecewise linear profile of the form

Uoj, V, 0 < y < b ,u 0(y) = U 0 + <

~ b < y < 0,

where denotes the constant shear in each region. In addition to the homogeneous

boundary conditions a t ±6 we require th a t the geostrophic and ageostrophic components

of the pressure and meridional velocity be continuous, so that

[*]»=o = 0 (3.12)

and4

' d'S ( d d \ 3® 9® duay dx + + U° d x ) dy dx dy

Here [C]y=yo = C(yo+0) — C(i/0 - 0 ) denotes the “jum p” of the bracket contents C a t y = j/o-

Recall also th a t we have assumed zero p. The streamfunction which is a solution of the

potential vorticity equation and which satisfies the above continuity conditions is

4 , ( y ) = - L ( e M H i i l ) _ e - * ( M y l > ) _ ( 3 1 4 )

where the dispersion relationship

a = U„fc - ~ f a i - O k (3 15j2/i (e"6 + e -"1) ' ' '

= 0. (3.13)y = 0

4The first continuity condition may be strengthened by requiring th a t the normal displacement

%){x, y, z, t) of any point across the interface be unique (Chandrasekhar, 1981). Since

ihty = i ^ ( n ) c o s ( \ / I f n 2 )e , | i l " 'Tl) = d%)/dt + u0 d Z ) /d x = ik (u 0 - c)2) =

ifc(uo - c)tj(y) cos(y /B m z) ,

we obtain:

[?]y=0 = - c)}y=Q = 0.



must be obeyed. This is a type of vorticity mode located about the “kink” , and therefore the

presence of walls has little effect on the perturbation (note th a t there is no quantization).

W hen b oo, we simply obtain:

If u~ > u* the wave propagates sinusoidally to the left. The meridional velocity of the

interface is

which is 1/4 of a wavelength ahead of the amplitude of the displacement.

3.3 U nstable waves in /-p lan e QG theory

In general the presence of a nonuniform background flow u0(y) leads to the possibility of

instability, th a t is, the solution of the potential vorticity equation (3.4) may yield complex

eigenvalues which in turn give complex frequencies of the form a = crr 4- icq. Upon substi

tution into equation (3.1) the perturbations with positive (negative) cq will grow (decay)

exponentially in time. If <q = 0 the flow is said to be neutrally stable.

The above condition must be used when the basic flow velocity is discontinuous a t y = 0, but otherwise it

is equivalent to (3.12). Regarding the second condition, one can write (3.5) as (Drazin and Reid, 1991):

((no - c)'Pi - - /I2 (uo - C) Ip = 0,

so th a t integrating between - e and +e and then letting e -* 0 yields

t ( u 0 - c ) V > s - t l 0 j , V ’ ] ! < = 0 = 0

which is equivalent to (3.13) (with >3 = 0).

tp(y) — e (3.16a)

(3.16b)

= k e - ^ cos( y / B m z )dx

(3.17)



As an example consider the following piecewise linear profile:

■Uoyd, d < y < b

u o{y) = ' uqyy, —d < y < d (3.18)

—UQyd, —b < y < —d,

where u 0y is the constant shear of the flow between —d and d. Solving the vorticity equa

tion and applying the appropriate continuity and boundary conditions yields a dispersion

relationship for the resulting mode

ff2 = < fc2 + e4M + 2e- ^ i _ 2e-2 - e- 4"A‘ + e"4"*2 -4/i2e4 d (1 — e~Atlb) V

4 /id [e4"1' - /ide4*"' - e~2t,A' - e ' 2 + e + / id e -4" ^ ] ) ,

where A n = b — nd. In order to simplify the result we shall assume th a t the channel width

is infinite, so th a t when An —» oo the above becomes

n UL k 2a 0 y *

4 / i 2■ ( ( 1 - 2 / i d ) 2 - e - W ) ,

and the streamfunction is

where

ip{y) = A e ~ ^ +d' +

u0y(k/ij.)e

(3.19)

(3.20)

(3 .2 1 )B 2 (a + uoykd) — uoy(k/t i)’

As can be seen in Fig. 3.1(a), when /id < 0.639 the frequency is purely imaginary, and the

mode is unstable. Note th a t the growth rate is given by

£7j = kd{jl) = ( f i 2 - m 2) ‘ /2 Cj(/Li), (3.22)



0.3

0.44o3

-0 .4- 0.2

-CU-0 .3 . 0 .4 0.6 0.80.2 0.4 0.6 1 0.2

+d

- d

kx

Figure 3.1: Stability of Rayleigh’s broken-line shear layer profile. P lot (a) shows the growth

rate while (b) indicates the phase speeds (the thick line represent the phase speed of the

unstable mode). The curves drawn with “+ ” and signs show the phase speeds of

the neutral vorticity modes localized about the “kinks” a t y = ±d. Plot (c) shows

the structure ('F field at t = 0) of the fastest growing mode (dashed contours are

negative), together with the initial, zonally averaged Reynolds stress R and vorticity

£ = ( ^ XI + ^yy)x . The horizontal dotted lines indicate the position of the kinks. A

schematic of the flow u0(y) is shown to the far right of (c).



and hence the maximum growth rate occurs when m = 0, i.e. when the perturbation is

purely barotropic5.

Rayleigh (1896) was the first to analyze this problem in the nondivergent case. Clearly

the existence of the instability depends on the presence of the shear Uoy within the flow.

The reason for this can be seen by studying the energy transfers which are taking place, i.e.

multiplying the linearized potential vorticity equation (2.17) by pvt and integrating over

over z and y (recalling th a t the normal velocities vanish at the boundaries) yields:

where ( )x represents the zonal average along a latitude circle. T he quantity in left paren

thesis is the total perturbation energy E\y, and hence the lhs of the above equation is the

rate a t which the total energy of the disturbance (per unit zonal length) changes in time,

while the rhs represents the source of energy, which depends critically on the horizontal

shear of the mean flow (the term -within parentheses on the right is called the Reynolds

stress). Insight with respect to the importance of the shear can be further gained from

Howard’s semicircle theorem (Pedloslcy, 1987), which provides bounds for the phase speed

and growth rate of a disturbance in a two-dimensional flow uo(y,z):

min(uo) < c;- < max(n0),

ci — ^ (max(uo) — min(u0))2 .

5Note tha t, although mathematically equivalent to solving the divergent problem with the “composite wavenumber” p , the nondivergent version of the potential vorticity equation (3.4) becomes the 2D Euler

equation (Landau and Lifshitz, 1987), and hence the applicability of the theory presented here extends

beyond the confines of the geophysical flows we are analyzing.



Again, if there is no shear present (either horizontal or vertical), then max(uo) = min(iio)

and the flow is stable. Clearly the presence of horizontal shear is a requirement for

barotropic instability, albeit a rather weak one and, as the Couette flow demonstrates,

it is not a sufficient condition. Rayleigh (1880), however, obtained a stronger criteria by

substituting complex solutions of the form c = c, + iCi and iji = ijir + ifa into equation (3.5)

and finding the conditions under which c; ^ 0, namely, th a t Uoyy change sign at some point

yc w ithin the domain of the flow. Taylor (1915) provided a physical explanation for this in

terms of the rate of change of s-momentum of a vertical slab of fluid (parallel to the x-z

plane, of thickness 6y),

where 2)o is the original position of the fluid particles within the layer (before the distur

bance). Instability takes place if the quantity inside the parentheses is positive, but if u0yy is

everywhere positive (negative) then all layers gain (lose) momentum, which cannot happen

if the flow is inviscid. Thus u0yv must change sign for the flow to be unstable. Fjprtoft (1950)

strengthened this condition (Drazin and Reid, 1991) by showing th a t uoyy(u - u(yc)) < 0

somewhere within the flow was a necessary condition for instability. Kuo (1949) made the

generalization of Rayleigh’s criteria to a /3-plane and concluded th a t in order for the flow to

be unstable ^ — u0yy must change sign somewhere in the domain (or, conversely, a sufficient

condition for stability is th a t the sign remain unchanged). Although Kuo claimed that this

condition was also sufficient for instability, it was later shown th a t this was not the case

(Howard and Drazin, 1964). Pedlosky (1964) generalized Fjortfort’s criteria to a /3-plane,

and later Ripa (1982) obtained the more general stability conditions for a one-layer model

on a /3-plane which encompassed all previous necessary conditions in the nondivergent limit

(q.v. Section 5.2.1).

A necessary and sufficient condition for instability was obtained by Rosenbluth and

Simon (1964) for monotonic flows with a single inflection point a t yc and zero /3. Writing

;PoUoyySy



the vorticity equation (3.5) in terms of the meridional displacement 1) (see footnote 4) gives

Noting th a t rj must vanish a t the walls we have that for p = 0 the eigenvalue problem

becomes

decreases. Thus, if the flow is stable when p = 0, it will be so for all p, and hence the

c traces a semicircle over the upper half of the complex plane. Using this methodology the

following necessary and sufficient condition for instability is obtained6:

where V denotes the Cauchy principal value.

Generalizations of the stability criterion (3.25) had not been forthcoming until recently,

when an extension to monotonic flows with more than a single inflection point was derived

by Balmforth and Morrison (1999). Using again the Nyquist m ethod they arrived a t the

following necessary and sufficient condition for instability:

6There is a typo in the original paper, where the exponent of u0w is incorrectly stated as 3 instead of 2.

Furthermore, the condition remains valid for walls located arbitrarily a t 6_ and 6+ (6+ > 6_), as long as

the flow is monotonic and has a single inflection point between them.

Lin (1955) has shown th a t if the flow in question is unstable then C; will increase as p2

importance of analyzing the above case. By Nyquist’s theorem the number of zeros of the

gain function is determined by the number of times a plot of ®(c) encircles the origin when

uov(y) M v c ) - M y ) )l

^yy{y)i>(y,Vo)

where xjj(y, yc) is the solution of the Fredholm problem

■^yy(y')xl>{y' ,yc)dy' ■



Table 3.1: Conditions for instability (Nondivergent, barotropic QG model)

Condition Type 0 Profile [0 = 0)

uo,jy changes sign N A rbitrary

uosy{u — n(f/c)) < 0 somewhere N Arbitrary

0 - uoy,j changes sign N A rbitrary (0 ^ 0)

ua(0 - uoyy) > 0 somewhere N A rbitrary (0 ± 0)

N, S uoy > 06UOy(Uo(yC)-U0) |6_ Uqy(uO-Uo(yc))

r 1 UOyyWtKy,Vc)j..^ ,J - i M ’j)-Mvc) ” N, S U 0 y > 0

°N = necessary, S = sufficient ^Single inflection point



Unfortunately the above calculation makes the evaluation of the stability criteria somewhat

more involved than in the previous case. However, by analyzing the number of times a locus

encircles the origin in the Nyquist plot (which determines whether there is instability or

not), the authors note that, given N inflection points in the profile, there can be a t most

(iV + l ) /2 unstable modes at a given wavenumber k. Two more inflection points are thus

required if a new instability is to appear. The stability criteria mentioned so far have been

summarized in Table 3.1.

3.4 Wave-wave interaction: two wave resonance

From the criteria mentioned in the previous section it is clear th a t shear (and, in par

ticular, a t least one inflection point within the flow) is critically im portant for instability

to be able to take place in an inviscid fluid. The mechanism by which the perturbation

sta rts to grow to destabilize the flow, however, has been subject to extensive discussion in

the literature. One of the foremost theories on the problem has been the idea of instability

as the product of the “resonance” between waves, as explained below.

We shall first analyze flows which take place on an /-p lane so as to eliminate any 0-

Rossby waves. The vorticity modes which result will thus be a product of the background

shear gradient, easily isolated when said gradients are discontinuous (with otherwise linear

shear). We consider a piecewise linear profile with two kinks described by equation (3.18).

The structure of the instability is shown in Fig. 3.1c. Two vortices appear within half a

wavelength, each centred a t the latitude of a kink, and shifted with respect to one another

so th a t the streamlines tilt opposite to the inclination of the shear. The Reynolds stress,

labelled R in the figure, is constant within the shear region, while the vorticity plot £ clearly

shorvs two peaks, each located a t a kink in the background velocity field.

The mechanism by which the instability develops can be understood in terms of vorticity



Figure 3.2: Instability due to the interaction between two phase-locked (or “resonant” )

waves (thick horizontal arrows denote the direction of propagation of a free vorticity

wave). Fluid displaced from the middle layer has anticyclonic (clockwise) vorticity while

the vorticity of fluid advected from the lateral layers into the middle one is cyclonic

(relative to the background). Together with the vertical displacement of the interfaces

[the vertical arrows denote the positions of ui9 maxima as given in (3.17)], the combined

effect results in the growth of the disturbance. As the kinks become further apart (for

a given wavelength) the interaction becomes very weak and eventually the instability

vanishes.



advection, and has been successively clarified by Hoskins et al. (1985), Baines and Mitsudera

(1994), and Heifetz et al. (1999). The instability arises due to the combined effect of the

two sinusoidal waves which counter-propagate along the shear discontinuities according

to equation (3.19). This superposition of waves can be clearly seen by comparing the

neutral mode from equation (3.16a) with the unstable solution (3.20). W hen the modes

become stationary with respect to one another (due to the Doppler shift), the meridional

displacement [given by equation (3.17)] and the advected vorticity act in tandem to make

the disturbance grow7. The velocity matching is highlighted in Fig. 3.1(b) which shows the

phase speeds of the unstable mode and the neutral waves. Figure 3.2 schematically details

the mechanism of the instability.

Integrating the energy equation (3.23) by parts over y and noting th a t v lg vanishes at

the boundaries gives the rate of change of the wave energy E\V'.

We can write the linearized potential vorticity equation (2.17) in the following manner:

3 — ■£Where Q is the perturbation potential vorticity and FI is the potential vorticity of the basic

state. Since

P (*xQ)x = P <«x (^xx + - m2* ) )x

we can thus obtain an equation in terms of the potential vorticity

7Hoskins et al. (1985) elegantly summarized the instability mechanism in the following sentence: “The

induced velocity field o f each Bossby wave keeps the other in step, and makes the other grow



Substituting the meridional displacement 2) into the potential vorticity equation (3.26)

yields the following relationship

Ed

The quantity within the parentheses is the difference energy' which must vanish if the

mode is to be unstable8 (Hayashi and Young, 1987). Since E w is positive definitive this

implies th a t

9n „u o w - > 0 dy

somewhere in the domain is a necessary condition for instability (see Table 3.1).

Consider the case of the shear profile with two kinks as given in equation (3.18) when

b oo (an infinitely wide channel). For simplicity we shall set the flow to be nondivergent.

If we assume th a t the kinks are far apart, solving locally yields a solution like th a t of the

8Equation (3.27) simply provides an expression for the difference energy in terms of the streamfunction

and particle displacements. Although consistent with the fact th a t E d m ust vanish for the mode to become

unstable, the requirement itself is obtained by the condition th a t the total energy of the system remain

constant i.e. E p - E u = 0.

from where

and thus

(3.27)


Chapter 3. Stability in the quasigeostrophic model: resonance in piecewiselinear flows___________________________________________ 44

isolated kink [equation (3.16a)], so th a t the difference energy for each mode becomes:

+ \ (pH) J <2)±)x u0{y) u0yy(y) dy,

where

_ e - k \y ^ d \ g i(kx -c rt)

= ±U0yd =F

and thus,

E d ± = ( j e 2kly* dl + y e “ l9=F‘' 1) d y + ^ Y ( ^ ) ^ u °y a > = ± i

= ^ f c ( l - 2 M ) ,

so th a t upon “superposition” of the modes the total difference energy is

E d+ + E d_ = pH k (1 - 2kd) (3.28)

which vanishes when

k d =2

This is also the phase-speed matching condition where c+ = c_, which lies near the growth

maximum of the a-k plot [Fig. 3.1(b)], A plot of the difference energy of the two-kink

profile is shown in Fig. 3.3, which shows that E d vanishes in the region where instability

takes place. Although simply adding the difference energies of each modes is but a crude

approximation of the to tal difference energy, the quantity E d+ + E d. is zero at k ss 0.5,



near the growth rate maxima. This type of analysis thus allows to determine where in

wavenumber space instability may arise and also elucidate the mechanism (e.g. vorticity

advection between phase-locked vorticity waves) behind the instability.

0.2

0.1

- 0.1

Q

- 0.2

-0 .3

- 0 .4

-0 .50 .70.1 0 .3 0 .4 0.5

k0.60.2

Figure 3.3: Difference energy of the two-kink profile (dotted line). E d [as evaluated from

equation (3.27)] vanishes in the region where the mode is unstable (for k < 0.639 , cf.

Fig. 3.1). The circles represent the sum of the difference energies of the waves a t each

kink, Ep+ + E d- [equation (3.28)].



3.5 Three and four wave resonance

In the light of the “wave resonance” interpretation the structure of the double kink

profile in Fig. 3.1(c) can be attributed to the phase-locking of two vorticity waves, each

identified by the localized train of pressure variations a t y = ±1. In this context it is clear

th a t the presence of walls (which in QG theory do not support any waves) has no further

effect except th a t of inhibiting instability. A flow of the form

is stable no m atter what the values of the constants ujf are, either because the waves’ phase

speeds cannot possibly match if the shear slopes are of the same sign, or because the total

difference energy cannot vanish when the slopes have opposite signs.

Figure 3.4 shows the “triangular je t” first analyzed by Rayleigh (1880) and discussed

by Hayashi and Young (1987) as a simple example in order to illustrate the wave resonance

mechanism. Due to the symmetry of the profile the dynamics in this case are actually more

involved than what was suggested in the original analysis. The waves located at y = ±1

propagate in the same direction and possess identical Doppler shifts relative to the middle

mode, so th a t their (isolated) dispersion curves (tagged “+ ” and “—” ) exactly overlap in

plot (c). The phase speeds of all three waves m atch a t k = 1.50, near the wavenumber of

maximum growth rate (k = 1.23). The resulting instability is thus the simultaneous reso

nance of all three modes, as can be seen from the structure of the perturbation in plot (d).

The mode is symmetric about the y = 0 latitude, and resembles two overlapping instances

of the shear layer profile in Fig. 3.1, albeit shifted apart in the y direction. The previous

statement also applies to the Reynolds stress R, which now remains constant throughout

the entire — 1 < y < 1 region. The vorticity profile shows three peaks, one a t each kink,

{v - <0 d < y

— d < y < d



(a)

(c)

(b)0.4

0.2

- 0.2

-0.42.50.5

k(d )

0.5

-0.5

0.5 2.5k

kx

5

•5

Figure 3.4: Stability analysis of a “triangular je t” , (a) Three neutral waves exist locally

about each kink, although the profile as a whole is unstable, as seen by the growth

rate curve (b). P lot (c) shows the phase speeds of both the three-kink profile and the

individual neutral modes: “o,” “+ ,” generated by the isolated kinks at d = —1,0,1,

respectively. Plots (d) and (e) show the structures of the unstable and stable modes at

k — 1.23, respectively.



where the neutral modes would be localized. This overall behaviour is to be expected since

the triangular profile itself can be viewed as the superposition of two meridionally shifted

shear layer flows, such as the one described by Eq. (3.18). The remaining neutral mode in

plot (e) is a stable combination of the modes at y = ±1 . These resonances can be easily

split by destroying the symmetry, where changing the value of the northern shear creates

two distinct modes generated by the pairings ( - , o) and (+ , o).

(c)

Figure 3.5: A “staircase” profile with four kinks. At higher k there are two modes of equal

growth rates and opposite phase speeds, while a t the smaller wavenumbers the unstable

modes possess the same phase speed but different growth rates. The dashed and solid

vertical lines indicate the positions (k = 0.70 and k = 0.045, respectively) a t which the

structures shown in Fig. 3.6 are plotted.


Chapter 3. Stability in the quasigeostrophic model: resonance in piecewiselinear flows

- 101-10 kx

-6 0-6 0

55i — I

Figure 3.6: Plots (a) and (b) show the structures of the unstable modes generated by the

“staircase” profile a t k = 0.70. The mode shown in (a) propagates towards the right

with c = 0.42, while the other has the same growth rate but c = -0 .42 . At a smaller

wavenumber (k = 0.045) the phase speeds are the same (c = 0), but the growth rates

differ, with the most unstable mode being shown in plot (c). Note the vertical scaling in

(c) and (d) (blow-ups of the mid-region are shown beneath each plot), and the fact that

the Reynolds stress becomes negative at the middle of the channel.



As the number of kinks increases so does the complexity of the interactions. Consider

a “staircase je t” such as the one pictured in plot (a) of Fig. 3.5. Two distinct regimes can

be seen from both the growth rate curve pictured in plot (b), and the phase speeds shown

in (c). W hen k > 0.078 there are two unstable modes which possess the same growth rates

but phase speeds of opposite sign, while at smaller wavenumbers the two growth rate curves

“unfold” and the phase speeds of both unstable modes become equal. P lot (a) in Fig. 3.6

shows the structure of the mode propagating towards the right a t k = 0.702. Although the

vorticity is somewhat affected by the presence of the lower shear layer, the streamfunction

is very similar to that of a profile with two kinks such as the one in Fig. 3.1, with the

Reynolds stress constant in the 1 < y < 2 shear layer. We a ttribu te the appearance of

this mode to the resonance of the neutral waves which counterpropagate about the kinks

a t y = 1,2, as can be seen from the crossing of the corresponding phase speed curves at

k = 0.9 and c = 0.6 in Fig. 3.5(c). Similarly, the unstable mode which propagates towards

the left, shown in Fig. 3.6(b), is mostly confined to the —2 < y < - 1 band, and is due to

the resonance between the neutral waves localized a t y = —1, —2.

-10 -10

Figure 3.7: Structure of the unstable mode with c > 0 (a), and c < 0 (b), when k = 0.20.

Plots (c) and (d) in Fig. 3.6 show the structures of the modes at k = 0.045. Note that

the meridional extent has been increased in the figures since the perturbations are no longer



confined to narrow bands about each pair of kinks. Regions in which the Reynolds stress

is negative appear when — 1 < y < 1, and correspond to the sections of the disturbance for

which the streamlines have a positive slope (i.e. the same tilt as the background flow). This

“energy' sink” is most notable in the weaker instability pictured in plot (d). In these cases the

resonances are not so clear-cut since Reynolds stresses and vorticity spikes simultaneously

appear in the regions - 2 < y < — 1 and 1 < y < 2. Figure 3.7 shows an intermediate step

between small and large wavenumbers (k = 0.20). Consider the mode in plot (a) which

propagates towards the right. Here a local maximum appears about y = — 2 where the

southernmost kink is located, and becomes more prominent as k decreases. The overall

effect is th a t the instability begins to resemble th a t of a single shear layer with kinks a t

y = ±2. The phase speeds of the “outer” neutral modes tagged “A” and “V” cross at

k ~ 0.4, and hence the long-wave unstable modes can be attribu ted to the resonances

between these two waves modified by the presence of the internal vorticity discontinuities.

Figure 3.8 shows an example of how multiple wave resonance might look like in a three-

layer flow. While the disturbances here are of finite amplitude and mixing is clearly taking

place, it can nonetheless aid in visualizing the mechanism behind three wave resonance. A

container holding three liquid layers of slightly different densities is tilted and a vortex sheet

is created a t each of the two interfaces. The mechanism behind the localized instabilities is

essentially the limit of vanishing d in Fig. 3.1 (Baines and Mitsudera, 1994). Although the

density is not the same for all the layers, the mechanism by which the instability takes place

is attributable to the shear (Drazin and Reid, 1991). The two small-scale disturbances grow

and propagate locally in opposite directions a t similar speeds, reflecting the conditions at

k = 0.70 in Fig. 3.5(c). In the final stages the two disturbances seem to phase-lock to create

the large-scale mode as in part (c) of Fig. 3.5 when k < 0.05.



f t l f i t T " * "

Figure 3.8: Multiple wave resonance in a three-layer flow. Note the progression from two

isolated instances of Kelvin-Helmholtz instability at each interface towards the large

scale instability in the final stages (from Sakai, 1997).


Chapter 4

Stability in the quasigeostrophic

model: resonance in num erically

discretized flows

4.1 Num erical approach to solving sm ooth profiles

Piecewise linear barotropic profiles are highly idealized, and are not really intended to

represent actual physical flows. Atmospheric and oceanic currents can be modelled more

appropriately by the use of smooth profiles, such as a cosine je t (Yanai and Nitta, 1968),

or a Gaussian je t (Marinone and Ripa, 1984). While still unrealistic, such flows can begin

to approach what is observed in Nature. Even simple shear flows u0{y), however, lead to

analytically intractable problems when substituted into the QG potential vorticity equation

(3.4), a situation which worsens considerably with the more sophisticated models. Ways

of numerically solving the equations of motion have been discussed in Section 1.2, and

detailed descriptions of one particular methodology (via m atrix methods) is given in the

appendixes. The underlying physics, however, is usually obscured by the numerical details.

53


Chapter 4. Stability in the quasigeostrophic model: resonance in numericallydiscretized flows 54

It is here where the wave resonance theory may be of most use. By simply being aware

of the neutral waves which the model can support (in the absence of any basic flow) one

may conjecture the possible resonances th a t could take place, and even determine the most

likely regions (e.g. in wavenumber space) in which instability may take place.

Consider, for example, the simple /-p lane quasigeostrophic flows shown in Fig. 4.1. Dis

cretization of these flows will generate shear modes propagating in the directions indicated

by the arrows. If there is no inflection point present [Fig. 4.1(a) and Fig. 4.1(b)] then no

resonance can take place and the flow is stable in accordance with Rayleigh’s criteria (see

Table 3.1). Flow (c) presents an inflection point, but the Doppler shift acts against the

phase-locking of the counterpropagating modes. Rayleigh’s criteria is satisfied but Fjortoft’s

is not and hence the flow is stable. Only for flow (d) can resonance take place, but it’s

not guaranteed by any of the simple criteria mentioned above — the Rosenbluth-Simon

condition (3.25) must be evaluated.

4.2 The hyperbolic tangent profile

Consider the following smooth profile:

u0(y) = Uo tanh(aj/), —b < y < b .

This flow satisfies Rayleigh’s inflection point criteria (since uaw changes sign at y = 0), but

th a t in itself is no guarantee of instability. Applying the stronger stability criterion (3.25)

to this flow yields the following condition for instability

4a6 + e2ab - e~2ab______________ 2____________2aUo aUj] tanh(a&) ( l — tanh2(ai>))

A contour plot of the lhs of the above equation is shown in Fig. 4.2. As the shear becomes

more pronounced (large values of a), the above condition is satisfied for very narrow chan-



(a) (b)

( d )

M v )

(c)

Figure 4.1: Stability of a barotropic flow u0(y) according to the stability criteria in Table 3.1.

The arrows indicate the intrinsic phase speeds of the shear modes. Flow (a) is stable

because uoyy < 0 for all y, and hence fails Rayleigh’s necessary criteria for instability.

Similarly, Uom > 0 in (6) and the flow is again stable. In (c) there is an inflection

point present (dashed line) but the flow does not satisfy F jortfort’s criteria. The flow

shown in (d) might be unstable since there is an inflection point Hora(j/c) = 0 and

UQyy{u — u(yc)) < 0 (from Drazin and Reid, 1991).



2 .5

j-20 .5

a-10

-5 0

32 2.51.510.5

Figure 4.2: Stability diagram for the tanh(oy) profile as a function of the shear (a) and the

channel width (26) when Uo = 1. Positive regions are unstable according to equation

(4.1).



nels. This is to be expected as in the limit of infinite a the flow becomes a vortex sheet for

which a = ± i k , i.e. it is always unstable independent of the width of the domain.

(c)0.2

k x R fi.

Figure 4.3: P lot (a) shows the shape of the profile uo = tanh(y) (solid line) and its second

derivative (dashed line). The circles indicate the positions of the sample points used by

the numerical solver. The growth rate as a function of wavenumber k is shown in (b) (for

clarity, only positive growth rates have been plotted), while the phase speeds between

cr = —0.2 and 0.2 are shown in (c) (see text for explanation of the markers). The darker

dots represent the phase speeds of the unstable modes. P lo t (d) shows the structure of

the fastest-growing mode, together with the barotropic energy transfer and the vorticity

profile. The number of grid points was set to n = 400, and the channel width to 20. All

perturbation fields are assumed to vanish a t the walls.



The hyperbolic tangent profile has been studied in detail by Michalke (1964). Although

analytical solutions to this problem have not been found, it is possible to numerically solve

the vorticity equation in order to study the stability of the flow. Following Appendix A

we can write equation (3.5) in m atrix form (for simplicity we shall consider the purely

barotropic case):

[u0 (fc21 - dL2)) + Dg>u0] ip = c (fc2 X - dL2)) ip. (4.2)

By the stability criteria (3.25) an unbounded flow of the form Uo = tanh(y) should be

unstable. Although infinite domains are not numerically feasible1, the walls may be located

far enough so as not to affect the stability of the flow. The dispersion relationship and the

structure of the fastest growing mode is shown in Fig. 4.3. A plot of the phase speeds is also

shown in part (c) of the figure. In this case setting the walls a t b = ±10 gave the same result

as b = ±20, and thus the domain was considered wide enough to be “infinite” . Increasing

the meridional resolution from 400 to 800 grid points did not lead to any noticeable changes,

and hence we assume th a t the problem has “physically converged” . The maximum growth

rate occurs a t fc = 0.44, and the longwave cutoff takes place a t fc = 1 (this cutoff can be

determined analytically by solving the neutral case for which c* = 0). The phase speeds

are symmetric about cr = 0 and vary from - 1 to 1 (the region has been trimmed in the

p lo t). The markers “+ ” and ” trace the dispersion relationships of the waves generated

by the two kinks adjacent to the inflection point on either side (see below).

In order to understand what the numerical solver is doing we begin by analyzing the

results obtained when the resolution is very low. Figure 4.4 is identical to Fig. 4.3 except

th a t in this case the number of interior grid points has been set to n = 14. In addition,

two curves (“+ ” and “—”) have been plotted in part (c). These represent the dispersion

relationships of the isolated waves generated by the potential vorticity discontinuities (the

XA change of variables may sometimes be introduced in order to bound the region (Michalke, 1964).



-0.5 0 0.5 1 1.5•1.5 •1«o(y)

1.50 0.5 1

Figure 4.4: Same as Fig. 4.3 but using n = 14 grid points to solve the eigenvalue problem

(4.2). The markers “+ " and " indicate the location (in plot (a)) and dispersion

relationships (plot (c)) of the neutral waves generated by the kinks as if they were

isolated.



kinks) which are created due to the discretization. The kink generating each wave can be

identified by comparing the markers in subplots (a) and (c).

There are various features to note about this last result. Even at this low resolution

the unstable mode has been fairly well captured. The shortwave cutoff is now located at

around fc ~ O.S instead of k = 1, although the most unstable wavenumber is located a t

the appropriate wavelength (fc = 0.44). Its structure is shown in subplot (c) and discussed

below. Comparing the neutral dispersion relations to the full eigensolution it is possible

to identify the modes which resonate. We thus note th a t the instability is caused by

the resonance between the (modified) “+ ” and ” waves generated on each side of the

inflection point. Although the dispersion curves of all the kink-generated modes intersect

a t sufficiently small fc when taken in isolation, it is the two modes which are nearest to the

inflection point which most closely follow the curves th a t actually coalesce. Twelve more

eigenvalue curves are clustered near c = ±1, corresponding to the remaining grid points

within the flow (whose neutral dispersion curves are not shown). The structure of the most

unstable mode is shown in Fig. 4.4(d), together with the barotropic energy' transfer and the

vorticity profile. Note th a t the streamfunction resembles the result shown in figure (3.1c)

for the two-kink piecewise linear profile.

Comparison with the result in Fig. 4.3 suggests that in spite of the increase in resolution

the mechanism by which the instability takes place does not change, i.e. the vorticity waves

generated by the kinks nearest to the inflection point resonate and begin to grow. It is,

in essence, a modified version of the two-kink profile. The influence of the other kinks,

however, is also an integral part of the resulting dynamics, since they all contribute to

the to tal difference energy'. The overall effect seems to be th a t of having two “composite”

waves, each grouped together by the sign of the curvature of the flow. This also explains

why a change in sign in uoyy is required for the flow to be unstable: the curvature of the

flow determines the propagation direction of the waves which must reverse a t some point in



-0.S 0 O.S 1.5•1.5

(b )

R (i,k x

(b )

k x R

Figure 4.5: Same as figure (4.3) but using n = 50 (top four plots) and n = 51 (bottom

four).



order for resonance to take place. The above arguments are very similar to those given by

Baines and Mitsudera (1994) for arbitrary profiles, although the view presented here stems

from a numerical standpoint. Furthermore, they help obtaining an intuitive understanding

of the requirement that, given a monotonic flow with an inflection point, two more inflection

points are necessary if a new instability is to appear (see Section 3.3). This stems from the

fact th a t any neutral modes arising from a single inflection point will be Doppler-shifted

against any possible phase-locking. At least one more inflection point is required if new

resonances are to take place, as seen diagrammatically in Fig. 4.6.

Figure 4.6: The reason monotonic flows require two more inflections points for new insta

bilities to appear (as stated by the Balmforth-Morrison criteria) is illustrated above.

While the flow pictured in (a) may be unstable, the number of unstable modes will not

change by adding an inflection point (b) since the Doppler-shifting works against any

new resonances taking place. Two more inflection points (c) are thus necessary for new

unstable modes to appear.



One obvious difference between low and high resolution results is the shape of the

vorticity profile. The double “hump” obtained when n — 400 is absent since the dip at

y = 0 is missing. This is an im portant feature since the vanishing of the vorticity at

the inflection point in this case can be easily shown analytically. The relation between

the vorticity 5 and the streamfunction for the nondivergent flow is readily obtained as

follows:

„ = 9 ^ #■£= " d x2 + dy2

= - (fc2^ - ipyy) ei{kx- ° l).

Since E = £(y) e'!*1-"*) We find that, using (3.5),

f = - (/c2^ -

=u0 - c

so th a t the vorticity of the unstable mode vanishes a t the inflection point (since uo —c yf 0).

Given an even number of grid points it is seen th a t the values of £ nearest to the inflection

point slowly begin to decrease as n grows. If the number of grid points is odd, so th a t the

inflection point is sampled in the discretization, the vorticity vanishes a t it should a t y = 0,

but the shortwave cutoff is notably underestimated (see Fig. 4.5). In order to explain this

behaviour we note th a t the difference energy of the mode at y = 0 (marked with circles ( “o”)

in the plot) is zero, and hence does not actually participate in the resonance. The resonating

modes, however, are further apart, and the net effect is similar to th a t of decreasing the

resolution. When the number of grid points is sufficiently high, however, both “even” and

“odd” samplings give the same result, i.e. the problem physically converges.



0.5

-0.5,

(a)

x ^2

X 1.5

S 1a

0.5

00.5Mv)

(c )

fc(d)

0.64i

0.62

0.58'

fc

0.5

-0 .5

0.5

-0.5

Figure 4.7: Stability of a Gaussian profile wofe) = Uoe~(y/T'>2. P lot (a) shows the shape

of the je t and details of the numerical sampling near the inflection points, (b) shows

the growth rate curves, (c) the phase speeds, (d) and (e) the structure of the two most

unstable modes at k = 9.538 and k = 6.024, respectively. Horizontal dotted lines in the

Reynolds and vorticity diagrams indicate the location of the inflection points.



4.3 T he Gaussian jet

Bell-shaped jets are a common feature in the atmosphere (e.g. the strong “west wind

bursts” which occasionally take place at the equator, Lander 1990), and ocean (e.g. the

Equatorial Undercurrent, Pedlosky 1996), and thus the dynamics of barotropic “cosine,”

“hyperbolic secant,” and “Gaussian” jets have been the subject of many studies regarding

the stability of geophysical flows (Lipps 1963; Yanai and N itta 1968; Kuo 1973; Marinone

and Ripa 1984). As with the hyperbolic tangent profile, numerical solvers are required in

order to find any unstable modes. Unlike the hyperbolic tangent case, these jets are non

monotonic and possess two inflection points, which complicates the underlying dynamics.

Figure 4.7 shows the growth rate, phase speed diagram, and structures of the most unstable

modes for a Gaussian je t of the form

M v ) = Uae ~ ^ \ (4.3)

with Uo = 1 and r = 0.1, as shown in subplot (a). The smaller in-plots are blow-ups of

the profile showing the positions of the sampled points with respect to the location of the

inflection point (marked by the horizontal line). The domain extends from y — - 1 /2 to

y = 1/2 and resolution is n = 800. Plots (b) and (c) show growth rate and phase speed.

The latter plot also shows the dispersion relationships of the modes generated by the closest

kinks to the inflection points (see subplot (a) for marker correspondence). Only two curves

appear because curves on opposite sides of the mid-jet exactly overlap. Note th a t the phase

speed of the most unstable disturbance decreases as fc becomes smaller, while the opposite

behaviour is observed for the weaker mode. Plots (d) and (e) show the structures of the

modes a t each peak (fc = 9.538 and fc = 6.024, respectively).

A review of the possible mechanisms behind the appearance of these modes is given

below.



4.3.1 T h e over-reflection m echanism

r e f l e c t o rI I I I I

i n c i d e n t

T r a n s m it te d

r e f l e c t e d

d e c a yy ry c

I I I I I I I I I

Figure 4.8: The top figure shows the required wave geometry for over-reflection to take

place (after Proehl, 1996). The lower figure shows this geometry applied to a barotropic

Gaussian jet.

Proehl (1996) explained the instability mechanism of the barotropic Gaussian je t based

on the idea of “wave over-reflection” . This theory has been developed since the 1970s by

Lindzen (1988). Briefly, waves impinging on a region of wave decay (II) (see Fig. 4.8) are

partly transm itted into region (III) (if region (II) is thin enough), and partly reflected back

into (I). Over-reflection takes place when the amplitude of the reflected wave is larger than

that of the incident wave, i.e. the reflection coefficient is greater than one. If the reflected

waves are again bounced back towards region (II) a quantization condition is created (since



the wave must satisfy the appropriate boundary conditions), and the continuously over

reflected wave would lead to an unstable mode. The growth rate can be then estimated

using the “laser formula”

In R° ~ 2t ’

where r is the time the over-reflected wave takes to traverse region (I), and R is the reflection

coefficient.

For the case of the Gaussian je t the equivalent regions required for the over-reflection

geometry are also labelled (I), (II) and (III) in the bottom diagram of Fig. 4.8. A disturbance

propagating in regions (I) is over-reflected in regions (II) and grows as the process repeats

itself (Proehl, 1996).

The over-reflection theory, however, has encountered some difficulties. Smyth and

Peltier (1989), for example, have noted that the over-reflection mechanism fails near the

Kelvin-Helmholtz/Holmboe instability transition in a stably stratified Boussinesq shear

layer. Takehiro and Hayashi (1992) subsequently found the theory unsuitable to explain

shear instabilities in a shallow-water model, as the over-reflection mechanism cannot be ap

plied to “divergent waves” (as opposed to “vorticity modes” ). As will be seen in Chapter 5

the wave resonance theory can provide insight into the stability of models which support

divergent waves.

4.3.2 T he wave-resonance m echanism

In the Gaussian je t there are three distinct regions in which waves are grouped by their

direction of propagation. Modes located where v.QylJ > 0 propagate towards the right while

those in the central region where u0yy < 0 have a negative phase speed c. These are shown

schematically in Fig. 4.9.

Recalling the hyperbolic tangent profile it is clear th a t resonances can take place near the


Chapter 4. Stability in the quasigeostrophic model: resonance in numericallydiscretized flows &

Figure 4.9: Diagram showing the resonant waves generated by a Gaussian jet. The crosses

indicate the inflection points.

inflection points. From Fig. 4.7 it can be seen th a t both instabilities possess the same phase

speed (c ~ 0.607) a t their respective critical wavenumbers, which is equal to the background

velocity at the inflection points. This value also lies between the phase speeds of the four

neutral modes closest to the inflection points as shown in plot (c), suggesting a resonance

between “A” and “V” modes in each half of the flow. The reason behind the appearance of

two unstable modes with such dissimilar structures can be better understood by using the

crude piecewise linear analogue of the Gaussian je t shown in Fig. 4.10. Again, two unstable

modes appear, each qualitatively similar to its “counterpart” in the Gaussian jet. The

most rapidly growing perturbation shown in plot (d) can be attribu ted to the simultaneous

resonance of the modes in each half of the jet, i.e. two instances of the same instability

which takes place in Fig. 3.1 with the maxima on each side of the channel in phase. The

secondary mode in plot (e), on the other hand, results from the same simultaneous resonance

bu t when the maxima are out of phase. When the mid-region between the kinks vanishes



(b)0.3

0.2

0.1

- 0.1

- 0.2

-0.3,0.5

k(d)

-10

10

m

-10kx

Figure 4.10: A piecewise linear “jet,” with the structure of the most unstable mode at

fc = 0.837 shown in (d), and that of the peak secondary instability a t fc = 0.784 in (e).


Chapter 4. Stability in the quasigeostrophic model: resonance in numericallydiscretized flows___________________________________________________________70

we obtain the “triangular je t” (cf. Fig. 3.4), and the secondary mode disappears as the

decoupling effect of this region is no longer present.

(a) (b)

642

a 0 -2 - 4 -6

lAAAAAAAAAA

£

1.5

1

0.5

0

-0 .5

-1

-1 .5

M v )

(o)

0.3

0.2

- 0.1

- 0.2

-0.3,1.50.5

fc(d )

10

0e -

^

l l f lzzr

* -1 00.5 1fc 1.5 kx R fi.

10 M

-10kx R tig

Figure 4.11: Same as Fig. 4.10, but for a wider je t (structures shown a t k — 0.799).

Consider the effect of making the mid-region even wider as shown in Fig. 4.11. The shear



layers are now sufficiently far apart so th a t the structure of the two unstable modes (which

have the same growth rates) are essentially two independent instances of the instability

shown in Fig. 3.1, one with the meridional extrema in phase and the other with them out

of phase. The primary mode in the narrow case thus “splits,” resulting in the disturbance

shown in Fig. 4.11(d). The secondary mode remains fairly unchanged, with the wave

trains in each half of the perturbation pictured in Fig. 4.10(e) separating further to yield

Fig. 4.11(e).

Using a smooth, wide je t mimics this behaviour, as can be seen from Fig. 4.12 which

shows a Hyper-Gaussian profile of the form

ua(y) = U0e - ^ \ (4.4)

where now Uq = 1 and r = 0.555. Although the meridional e-folding distance of the

je t remains the same, the separation between the inflection points increases due to the

“flattening” of the peak. The resulting effect is th a t the instabilities sta rt becoming more

localized about the inflection points, as shown in plot (c), where now two maxima are

apparent across the width of the channel.

One clearly cannot take the analogy between a smooth flow and a piecewise linear

profile too far. An analysis similar to the one above can be done on the cosine je t (see, for

example, Yanai and N itta 1968) as is shown in Fig. 4.13. Qualitatively, the single unstable

mode which this je t supports is similar to the most unstable mode of the Gaussian je t [(see

Fig. 4.13(c) and Fig. 4.10(c)]. The secondary mode, however, is absent, as the comparison

between the stability of Gaussian and cosine jets shows. The idea of wave resonance may

help in elucidating how and where (in wavenumber space) an unstable mode develops, but

it remains unclear how well it can actually predict whether a certain flow will be unstable,

and if so how many instabilities there will be (advances towards obtaining such conditions,

however, have been made by Baines and Mitsudera, 1994). So far there are no criteria

th a t can provide sufficient conditions for the instability of a symmetric jet. Necessary and



Figure 4.12: A Hyper-Gaussian profile. Plot (d) shows the structure of the most unstable

mode a t k = 2.452. The secondary mode in (e) peaks at k = 2.492.



sufficient conditions for certain monotonic je ts do exist, however, as seen in section 3.4.

Applying the Rosenbluth and Simon stability criterion (3.25) proves th a t half a cosine

je t is stable, while each half of the Gaussian (and the Hyper-Gaussian) je t satisfies the

instability condition, and can readily be solved numerically as shown in Fig. 4.14. The

growth rate curve in plot (b) is identical to the one of the secondary mode in Fig. 4.7, and

again the instability is due to the resonance between the two modes ( “A” and “V”) nearest

to the inflection point, as shown in plot (c). The structure of the most unstable wave is

drawn in plot (d), and is just the lower half of the perturbation pictured in Fig. 4.7(d).

This suggests th a t the criteria of Rosenbluth and Simon may yield a sufficient (but not

necessary) condition regarding the stability of symmetric jets.

In essence then, the discretized Gaussian je t generates three “families” of waves, each

grouped by the sign of the curvature of the flow, and distinguished by their intrinsic phase

speeds. Numerically only four modes actually phase-lock, although the instability depends

on the structure of the flow as a whole (since the to tal difference energy must vanish). Each

instability is the product of a different resonance between the waves shown in Fig. 4.9.

The weaker one is due to the simultaneous resonance of the waves localized about each

inflection point, and can be readily “split” into its components as shown in Fig. 4.14.

More importantly, the necessary and sufficient stability condition (3.25) in single-inflection

monotonic flows might be applicable to symmetric jets as a sufficient condition. Both

instabilities can be modelled in a qualitative manner using a simple piecewise je t such as

the one shown in Fig 4.10(a).

4.4 The (3 effect

So far we have focused our attention on rotating flows which take place on a plane. In

actuality the motions which concern us, albeit essentially horizontal, take place on a curved


Chapter 4. Stability in the quasigeostrophic model: resonance in numericallydiscretized flows 7-

Figure 4.13: Stability of a cosine jet. The structure of the only mode present is some

what similar to th a t of the most unstable Hyper-Gaussian mode shown in Fig. 4.12(d),

although only one maxima is present along the y direction.



(a) (b)

-0 .5 ,

(C)

0.5

25k

0.64

0.62

0 .6;

0.5820 25

k

-0 .5 kx R tly

Figure 4.14: Same as Fig. 4.7 but for the lower half of the Gaussian jet.



earth, and hence a spherical geometry is more appropriate. Small meridional motions2

about a central latitude 9i, however, may be assumed to take place on a flat surface for

which the Coriolis param eter / varies linearly with y, and thus the ambient vorticity on a

/3-plane becomes:

One well-known consequence due to the background vorticity gradient is its stabilizing

effect, as will be seen by the generalization of the semicircle theorem discussed in the follow

ing section. On the other hand it has also been shown, in light of the wave resonance theory,

th a t waves generated by the presence of /? can produce new unstable modes (Kobayashi and

Sakai, 1993). In this chapter we shall analyze what effect the ambient vorticity gradient

has on the jets which currently concern us.

4.4.1 R ossby waves and stability criteria

We consider flows such th a t u0 is constant. The current is assumed to be bounded by

rigid walls a t y = ±6, and hence the no normal-flow condition is still i/>(±6) = 0 . If /3 is

constant and nonzero we have

f ( y ) = fe + Pe y

where

f t = 2fisin(6>e),

01 = cos(0<)-

(4.5)

which admits the solution:

(4.6)

20 f scales smaller than the radius of the earth tq.


C h a p te r 4. S ta b il i ty in th e q u as ig eo stro p h ic m o d e l: re so n a n c e in n u m erica lly d isc re tiz e d flows_______________________________________________________ 77

where $ = m r /2 ,I2 = (rar)2/(26)2 (n = 1 ,2 ,3 . . . ) , and where the dispersion relationship

. 13k .a = u°k - 2 . nH5 ( ^M T 4(l2

must hold. These are divergent Rossby waves which appear due to the background potential

vorticity gradient caused by the /3-effect. Note th a t the walls provide the quantization

condition, so th a t if 6 —> oo the dispersion relationship becomes:

a = u0k - (4.8)

and we are left with a single Rossby wave.

W hen the background flow is nonzero the problem is usually analytically intractable.

An exception is the Couette flow on a /3-plane, for which the vorticity equation

= (4'9)

can be written in the form of W hittaker’s Equation by making

3(y) = (2tt/uoy)(u0yy - c ) ,

so th a t (Howard and Drazin, 1964):

'Pa + ( j - $ = °> (4 1 °)

where re = /3/(2/iUos). A solution can then be obtained in terms of confluent hypergeo

metric functions (Kummer’s functions):

•0 (3) = A e -(j/2 ,3 M (1 - re, 2,3) + B e~(j/2)3 U (1 - re, 2 ,3) , (4-11)

where A and B are constants (Abramowitz and Stegun, 1972). The streamfunction must

satisfy the homogeneous boundary conditions a t ± 6, which implies th a t the determinant

c-(J(+«/2) J(+6) M ( l — re, 2, 3(+4)) e- (3(+e/2) J(+|>) u ( 1 - re, 2, 3(+t))(4.12)

e -(j(-6 )/2 ) 3( 6) M (1 - re, 2 , 3 M ) ) e - b ( - n / 2> 3 ,_ t) I f (1 - re, 2 , 3 (_ 6))



- 0.1

- 0.2

-0 .3

-0 .4

-0 .5

- 0 .(

- 0 .7

k

Figure 4.15: Dispersion relationship of the first ten Rossby waves in the absence of back

ground shear. The nondimensional width of the channel is 2b = 4, while 0 = 0.375,

which represents a m id-latitude value of 1.5 x 10“ n (m s ) " 1.



must vanish. At a given wavenumber the number of deformed Rossby modes decreases

(starting with the waves of highest mode number) as the shear becomes stronger. In

the numerical eigenvalue problem this translates as singular modes from the discretized

continuum appearing a t the expense of vanishing Rossby waves.

Howard’s semicircle theorem may be extended to quasigeostrophic flows on a /3-plane

(Pedlosky, 1987) in order to obtain the following bounds on the complex value of a mode’s

phase speed:

change sign in order for instability to take place3. Thus a barotropic flow with weak enough

shear will be stable. Furthermore, Kuo also noted th a t the phase speed of an unstable mode

is constrained by the background flow velocity to be C/max > c > Umi„ (Kuo, 1949).

From the wave resonance perspective, the range of phase speeds follows from the fact

th a t there are no shear modes capable of resonating outside the range of uo(y). Although

Rossby modes are present they all possess intrinsic phase speeds of the same sign (they

propagate westward), and remain neutral as a result. Rossby modes, however, are capable

of phase-locking with shear modes within the continuum, resulting in new instabilities. This

N onetheless Pedlosky (1987) has noted tha t “the presence of the /3-effect can be expected to introduce the

possibility o f new, unstable modes in the presence of shear, although alone it provides a stabilizing restoring

mechanism as manifested by the Rossby wave.’’.

min(uo) —2 ( / i 2 + tt2/ 4 )

< Cr < max(uo),

c? < - (max(uo) - min(u0))2 + (max(tto) - min(uo))

for a 2D flow Uo(y, z). In spite of the radius increase, /? is a stabilizing param eter according

to Kuo’s theorem, which requires th a t for a barotropic flow

d2uo(y)dy2

(4.14)



explains the asymmetry of eastward jets with respect to their westward counterparts (Ped

losky, 19S7), since in the latter case the Rossby waves lie embedded within the continuum

of shear modes, yielding resonances between modified Rossby and shear modes (Kobayashi

and Sakai, 1993). Although the stability of eastward jets is also affected, no resonances can

take place with westward-propagating modes (associated with the Rossby modes), since the

Doppler shift due to the background flow acts against the phase-locking.

4.5 The Hyper-Gaussian jet on a /3-plane

W hen (S is nonzero the continuum modes are modified and modes with negative phase

speeds appear. The la tter can be identified as shear-deformed Rossby modes, as can be seen

from the phase speed diagrams in Fig. 4.16 and Fig. 4.17. When the Hyper-Gaussian je t

is very weak there is no instability present and the discretized continuum appears between

c = 0 and c = 0.01, which corresponds to the velocity range of the background flow.

The waves with negative phase speeds correspond to the (slightly) shear-deformed Rossby

modes. As the background flow becomes stronger (Fig. 4.17) the two unstable modes appear

and the dispersion curves of the Rossby modes are further shifted and deformed. A longwave

cut-off appears for the more unstable mode (which is the anti-symmetric perturbation), but

close inspection of the bifurcations a t both long and short critical wavenumbers shows that

the resonating waves have phase speeds larger than zero, and hence belong to the shear

mode discretized continuum.

We thus again consider the Hyper-Gaussian westerly je t given by equation (4.4) and

analyzed in Section 4.3 in the absence of /3. The scales for L and U are 106 m and 40 m s- ’ ,

respectively, and the e-folding meridional width of the je t is 555 km. The flow is constrained

by a channel 3000 km across, which is wide enough to avoid any wall effects. Given the

above scaling the dimensional growth rates (in seconds) can be obtained by multiplying the



-0.05

- 0.1

-0.15u

S '

- 0.2

-0.25

-0.3

0 3 41 2 5 7 86k

Figure 4.16: Dispersion relation of a weak, westerly Hyper-Gaussian je t on a mid-latitude

/3-plane. W hen the shear is small ([/0 = 0.01) the flow is stable and the Rossby modes

(of negative phase speeds) are only slighted affected.



Figure 4.17: A stronger Hyper-Gaussian je t (t/o = 0.1) leads to unstable modes, although

the lower Rossby modes can still be discerned. Close inspection at the the lower critical

wavenumber, however, reveals th a t the slower instability does not bifurcate into a shear-

deformed Rossby mode.



0.9

0.8

0.7

0.6

0-5

0.4

0.3

0.2

0.1

0 2 4 6 8

0.8

0.6u

S0.4

0.2

0 2 4 6 8

Figure 4.18: Eastward Hyper-Gaussian je t on an /-p lane (to be compared with the

/3-plane result of Fig. 4.19). The scales used are the same as in Fig. 4.15. Each mode is

unstable for 0 < k 2 < k l where kc is the corresponding critical wavenumber.



values from the figures by (L / U ).

For comparison, Fig. 4.18 shows the growth rate curves and their corresponding phase

speeds obtained for the je t on an /-p lane, while Fig. 4.19 shows the same plots when

0 = 0.375 (a mid-latitude /3-plane).

On the /3-plane the neutral waves which coalesce a t the higher wavenumbers correspond

to the discretized modes from the continuum located nearest to the effective inflection point

where /3 — u0yy vanishes. The symmetric mode, however, presents now a long-wave cutoff at

k = 0.668 due to the effect of /3. Careful examination near this critical wavenumber reveals

th a t the unstable mode and its complex conjugate are again connected to two neutral modes

from the discretized continuum, so th a t the modified Rossby modes (of negative phase

speeds) are not directly participating in the resonance. Furthermore, the structure of the

symmetric instability a t various wavenumbers does not differ much from the /-plane case

(this also holds true for the anti-symmetric mode). Figure 4.20 shows the structure of the

symmetric and anti-symmetric modes at their respective peaks. Although the effect of 0

in this case causes the longwave cutoff, the nature of both instabilities remains essentially

the same.

Flipping the je t so th a t it flows westward reveals little difference with respect to its

westward /-p lane counterpart in any respect. The westward phase speeds of the unstable

modes become slightly greater but there is no longwave cutoff nor is there any notable

change in the growth rate curves. This seems to be due to the fact th a t the strength of the

je t essentially “swamps” the shear-deformed Rossby waves (i.e. the Rossby waves disappear

and singular modes appear), which in the case of no background flow possess phase speeds

up to approximately c = —0.34 (i.e. smaller in magnitude than the je t minimum of -1).

The effect of 0 on the Gaussian je t is similar to that on the Hyper-Gaussian profile —

the most unstable mode presents a longwave cutoff but the structures and growth rates of

the unstable disturbances remain essentially unchanged. This holds true for both eastward



0.8

0.6

0.4

0.2

0.5

-0.5

82 64k

Figure 4.19: Eastward Hyper-Gaussian je t on a /3-plane. Although the symmetric (and

most unstable) mode now presents a second critical wavenumber a t k = 0.668, it does

not bifurcate into a Rossby mode.



1.5

mi '"i

1.5

1.5

Figure 4.20: Structures of the unstable modes due to of a Hyper-Gaussian je t on a /3-plane.

The most unstable mode is pictured a t the top when k = 2.53. The secondary mode

peaks at k = 2.46 and is pictured a t the bottom.


Chapter 4. Stability in the quasigeostrophic model: resonance in numericallydiscretized f l o w s ___________________________________________________ 87

and westward flows.


Chapter 5

The H yper-G aussian jet on a /3-plane:

shallow-water and Boussinesq models

5.1 Equatorial flows and the role o f gravity waves

Unstable modes in barotropic QG flows result from the resonance of the various vorticity

waves which a given geometry can support through shear in the basic flow and background

vorticity. The relatively simple dynamics of the QG formulation, however, comes a t a cost,

such as the fact that the regions in which this model can be applied are constrained, through

the smallness of the Rossby number, to latitudes far away from the equator. Although the

barotropic equation can be applied a t low latitudes, it makes no provision for flows of finite

depth. Furthermore, they both fail to capture many dynamical features which can take

place even at midlatitudes. This is of course due to the many simplifications made in the

derivation from the more general Boussinesq equations (2.1). In view of the wave resonance

theory it is clear th a t by successively filtering out certain types of waves some resonances

are no longer possible, and hence unstable modes which are captured by more complex

models do not show up in corresponding QG flows. In this chapter we shall analyze the

88


Chapter 5. The Hyper-Gaussian jet on a /J-plane: shallow-water andBoussinesq models_______________________________________________ 89

stability of the shallow-water and Boussinesq systems which can sustain gravity waves and

also allow for the modelling of flows at any latitude (in particular, a t the equator).

Many studies of equatorial flows have focused on the dynamics of the Equatorial Un

dercurrent (EUC), a powerful eastward flow which consists of a thin ribbon of water ap

proximately 200 km wide located between 100 and 200 m deep, reaching speeds of up to

1.5 m s_1 under the ocean’s surface (Pedlosky, 1996). Marinone and Ripa (1984), for exam

ple, analyzed the stability of various Gaussian jets using a one-layer, shallow-water model.

The westerly wind bursts which concern us, however, are much wider jets, and hence we

will focus our attention in this chapter on the stability of an atmospheric Hyper-Gaussian

profile.

5.2 N eutral waves in shallow-water theory

We shall now analyze the neutral modes which the shallow-water (SW) model is capable

of capturing. We begin by nondimensionalizing the linearized SW system (2.10) as follows:

0x ,y ) L{x,y ) (u0,u i , v , ) i-> U(u0, u lt Vi)

so th a t it becomes

(5.1a)

(5.1b)

(5.1c)


Chapter 5. The Hyper-Gaussian jet on a 3-plane: shallow-water and Boussinesq models_________________________________________________________ 90

where F = U / \ /g H is the Froude number. Substituting solutions o f the form

(5-2)

for all perturbation fields we obtain:

i (ku0 - o) ui - ~ ^ j vi + hi = 0 (5.3a)

i (ku0 - a ) i i i + f i i + ( J k j = 0 (5'3b)

i (kuo — a ) h \ + ikhoiii -I— = 0. (5.3c)ay

Consider the case where ho is constant and hence uo = 0. Dropping hats and eliminating

hi (see Appendix B) from the above yields,

a2vi + icrfui + (kh0Ui - ih0Viv)y = 0 (5.4a)

a2ui - i a f v i - (khoiii - ihoViy) = 0. (5.4b)

Solving for Vi then gives the governing equation for the perturbations with nonzero merid

ional velocity component:

v iyy + (ct2 - / 2) - ^ f y ~ ^ v i = (5-5)

5.2.1 N eutral m odes on a shallow-water equatorial /3-plane

For simplicity we shall assume th a t a flow in the presence of a varying Coriolis parameter

takes place on the equatorial /3-plane. Substituting / = f3y into equation (5.5) and solving

yields solutions of the form

F 2/}'ho

d/4)y , (5-6)


Chapter 5. The Hyper-Gaussian jet on a 5-plane: shallow-water andBoussinesq models___________________________________________ 91

where is the n-th Hermite polynomial. The dispersion relationship for these waves is

fc2 + (2n + 1) ( ^ 0 ) ~ W = 0- (5-7)

where n = 1 ,2 ,___ Note th a t quantization occurs in spite of the fact th a t there are no

lateral walls present — the meridional gradient of the Coriolis param eter constrains the

waves within a band about the equator. For low-frequency waves the above reduces to:

kp

fc2 + (2n + 1 ) ( 0 F / y / h ^ ’(5.8)

which corresponds to the westward-propagating Rossby modes, while for large values of a

we have

( P F \ \ 1/2+ (2n + l) £ = , (5.9)

which are the fast gravity waves. When n is zero we obtain the Yanai wave, which behaves

like a gravity wave when k is small and like a Rossby wave at large wavenumbers. Its

frequency lies between slow and fast waves, and when k = 0 it is equal to

P s fh o F '

The full dispersion relation can be written in the following manner:

, i ft

(5.10)

k 2 a ± 2 V<r2

There is also a solution for which «i = 0 throughout the domain, in which case the

dispersion relation and wave structure for the equatorially trapped mode are

a = yfh0, (5.12)

Ul „ e-{msM-0)y\ (513)


Chapter 5. The Hyper-Gaussian jet on a /3-plane: shallow-water andBoussinesq models_______________________________ 92

Yanai

GravityEquatorial Kelvin

0.5

Rossby

-0 .5

Yanai Boundary Kelvin Gravity

-1 .54 .52 .5

k3.50.5 1.5

Figure 5.1: Neutral modes supported by the shallow-water model a t the equator (numerical

solution). Note the appearance of boundary Kelvin waves propagating westward with

c = - 1 . These are due to the artificial walls required to bound the numerical problem.


Chapter 5. The Hyper-Gaussian je t on a /3-plane: shallow-water andBoussinesq models__________________________________________ _ _ _ _ _ _______93

This is an eastward propagating Kelvin wave. A diagram showing the entire spectrum is

given in Fig. 5.1. This plot was made by solving the numerical eigenvalue problem derived

in Appendix B using 50 grid points. The basic depth ho was assumed to be equal to one,

while the scaled /3 a t the equator was set to 0.912. The Froude number was F = 1 and

the width of the channel (required in order to bound the eigenvalue problem) was assumed

to be 12, or approximately 8 times the e-folding half-width of the unperturbed equatorial

Kelvin wave. The numerical solution is in complete agreement with the dispersion relations

obtained directly by plotting equations (5.11) and (5.12).

Stability conditions for the shallow-water model on a /3-plane were obtained by Ripa

(1982). Given a zonal je t of shape n0(y) the flow is stable if

[ a - u 0(i/)]<3s (y) > o and [a - u0{ y ) f < grH(y) for all y (5.14)

for some value of a , where

Q { y ) = m ^ M , (5.i5)

and H[y) is the layer depth of the basic state. When the flow is nondivergent the latter

condition is trivially satisfied since gr - t oo, and the first condition simply reduces to

F jortoft’s theorem when a is equal to the flow velocity at the inflection point. The wave

resonance interpretation is then the one given in Section 4.1 and illustrated in Fig. 4.1.

W hen the flow is divergent the second condition implies th a t if the speed range of the

profile is less than the minimum phase speed of the long gravity waves the flow will be

stable (the a guaranteeing th a t if the flow is stable in one inertial frame of reference it must

be stable in all inertial frames of reference). In other words, the waves’ phase speeds must

match if instability is to take place.


Chapter 5. The Hyper-Gaussian jet on a /3-plane: shallow-water andBoussinesq models________________________________________________________ 9^

5.3 Stability o f an equatorial H yper-G aussian jet in

shallow-water

The background flow is again chosen to be a Hyper-Gaussian je t identical to the one

used in the previous chapter a t midlatitudes. The basic zonal flow and the pressure field

are assumed to be in geostrophic balance, and so the background basic height field and

resulting velocity profile are:

where again U0 = 1 and r = 0.555. The Froude number F was calculated to be equal to

0.08 by assuming th a t the atmospheric depth scale is 10 km and setting the scale velocity

to be 25 m s_1. This is clearly an over-simplification, but it serves to understand what

effect, if any, shallow-water gravity waves might have on the above je t if the atmosphere

is treated simply as a thin, homogeneous layer of fluid. Furthermore, the Froude number

we are considering is in itself an upper limit to what one could reasonably expect for the

flow we are analyzing since we are using the peak velocity Uo to determine it. On the

other hand, we note th a t when the Froude number vanishes the eigenvalue equation (B.l)

is reduced to th a t of the barotropic case (q.v. Chapter 4). This is to be expected since it is

equivalent to making the phase speed of the gravity waves infinite, and hence render them

incapable of resonating with any other neutral mode.

Thus, as the Froude number decreases the phase speed of the gravity waves becomes

larger and hence the possibility that these may be Doppler-shifted enough to resonate

with the shear-generated waves, the slow shear-deformed Rossby waves, or the westward

propagating gravity waves is negligible unless the background flow is very strong. In effect,

the je t we are analyzing seems too weak to do so, as can be seen from the growth rate curve

(5.16)

u0 = U0e - ^ (5.17)


Chapter 5. The Hyper-Gaussian je t on a /3-plane: shallow-water andBoussinesq models_________________________________________________ 95

0.7

0.6

0.4

0.3

0.2

0.1

2 6 840

0.8

0.6

0.4

0.2

- 0.2

-0.4

2 6 80 4k

Figure 5.2: Eastward Hyper-Gaussian je t on an equatorial /3-plane. W hile not all modes

have converged (see Fig. 5.3), it is useful to compare with the QG midlatitude result

shown in Fig. 4.19, for which the unstable modes present essentially the same character

istics.


Chapter 5. The Hyper-Gaussian jet on a 5-plane: shallow-water andBoussinesq models_____________________________________________ 96

shown in Fig. 5.2. Again there are two main unstable modes, both having a short-wave

cut-off a t k r : 5 w ith the most unstable mode possessing a second critical wavenumber at

fc s« 1. The dispersion relationship shows again the discretized continuum between c = 0

and c = 1, together with the slow westward-propagating shear-deformed Rossby waves. The

high-speed gravity waves do not participate in either instability, and both show essentially

the same behaviour as their QG counterparts, as can be seen by comparing Fig. 5.2 with

Fig. 4.19.

In order to verify the above result (which was obtained using 200 grid points along

y), and to check whether the weaker modes are spurious or not, we carry out the same

calculation a t a high resolution using the sparse techniques described in Appendix B. The

curves, shown in Fig. 5.3, indicate that none of the weaker modes are in fact real, and

furthermore show th a t the slight bulging of the growth rate curves a t high wavenumber in

Fig. 5.2 is a product of the low resolution.

Figure 5.4 shows the structures of the two modes. They are similar to the barotropic

modes (Fig. 4.20) save for one im portant difference: the anti-symmetric mode is now the

most unstable perturbation while the meridionally symmetric mode becomes the weaker

instability. The reason for this behaviour is due to the change in sign of the Coriolis pa

rameter across the equator. At midlatitudes the unstable modes consist of two eddies which

arise as a result of the vorticity wave coupling. At the equator the dynamics are very sim

ilar since the gravity waves are too fast (due to the smallness of the Froude number) to

resonate with the slower shear-generated modes. Again two cross-jet eddies form with each

instability, and both modes possess essentially the same wind fields as they did at m idlati

tudes, th a t is, the most unstable mode has two co-rotating counter-clockwise eddies while

the secondary mode consists of counter-rotating vortices. On the southern hemisphere,

however, a counter-clockwise vortex implies a pressure high, while clockwise flow requires

a pressure dip, which is the result shown in Fig. 5.4. Since the Froude number is so small


Chapter 5. The Hyper-Gaussian jet on a /?-plane: shallow-water andBoussinesq models _____________________________ _ _ __________________ 97

0.9

0.7

0.6

^ 0.5&

0.4

0.3

0.2

0.1

1.2

0.4

0.2

- 0.2

-0.4

6 7 80 2 3 4 51

k

Figure 5.3: Same as the previous figure but using a high-resolution sparse calculation. The

number of interior grid points in this case was set to n — 10000.


Chapter 5. The Hyper-Gaussian jet on a /3-plane: shallow-water andBoussinesq models 98

- 3

3

- 3

Figure 5.4: Structure of the most unstable (top) and the secondary (bottom) modes at

k = 2.46 and k = 2.25, respectively. The contours represent isolines of hi, while the

plots labelled R and £ show the Reynolds stress and the vorticity.


Chapter 5. The Hyper-Gaussian jet on a /?-plane: shallow-water andBoussinesq models_____________________________________ 99

the perturbation height becomes:

kl F 2r 2 - fc2/i0(r{vih0)y - % i ) (5.18)

~ -T JT - (r(Ul/l0)y - k h o T j V l ) , (5.19)

whereupon substitution into (B.4) gives

[ k h 0r j v i + ( r(u1/io)y - k h 0T ] V i ) ] , (5.20)

and since ho » 1 the relationship between meridional and zonal winds reduces to the one

found in QG theory:

Thus the corresponding wind fields of each unstable mode remain unchanged and it is

the pressure which adjusts to the change of sign in / . Clearly care must be taken when

evaluating the pressure field: the values of the nondimensional wind fields are of 0(1),

while hi is approximately of (?(10~3). Using 4000 grid points the residuals of the numerical

solution were of <9(10-2), and it required 10000 grid points to bring them down to O (10-4).

5.4 The Boussinesq m odel

The derivation of the shallow-water equations is based on the assumption th a t the

density is constant and the flow is hydrostatic. As a consequence the resulting fields are

independent of 2 and there is no variation with depth, in contrast w ith the Boussinesq model

presented in Section 2.2. As with the quasigeostrophic model, the vertical structure in the

Boussinesq model of a barotropic instability is specified by the solution of the baroclinic

eigenvalue problem (q.v. Section 3.1). Unlike the QG case, however, the dependence

of the barotropic mode on the vertical wavenumber m is more involved, and cannot be

i k u i ~ — V \ y . (5.21)


Chapter 5. The Hyper-Gaussian jet on a 3-plane: shallow-water andBoussinesq models__________________________________________ 100

/-p lane

G GG G

/3-plane

G OEq

G G

Figure 5.5: On a northern /-plane (say) the most unstable unstable mode (pictured on the

top left) is symmetric and results from vorticity wave coupling, as does the secondary

mode shown to its right. The dynamics of the modes remain unchanged when the

flow is shifted to an equatorial /?-plane, and hence in order to conserve the sense of

rotation within the disturbance the signs of the pressure perturbations must switch

when crossing the equator (since the sign of / changes). The most unstable mode is thus

the antisymmetric perturbation shown in the bottom left diagram.


Chapter 5. The Hyper-Gaussian jet on a /?-plane: shallow-water andBoussinesq models 101

simply taken into account by defining an “equivalent” wavenumber. Since the value of m

is interpreted in terms of the depth of the flow (which is confined between the ground and

the tropopause), we can therefore analyze the behaviour of an equatorial barotropic je t in

a finite depth domain.

5.5 N eutral waves supported by the Boussinesq m odel

The linearized Boussinesq system has already been derived in Chapter 2, equation (2.3).

Using separation of variables and assuming solutions of the form

# r , y , z , t ) = 4>[y) ( ^ ) ei<tl" ‘)-

for ui*Vi,pi, and

ip (x ,y ,z , t ) = <£(y)¥3(z)e’(fcc- '7i>

for wi and pi results in two eigenvalue problems in y and z. The former can be written as

7 % + 2waisP l y y - P l y f p + u 2 - P i

' 2 _ k f \ + ikfuiUy _ f v + u= 0, (5.22)

u(aJ2 + f y ) N $ - w 2 \

where u = k u 0—cr and r/ = uoy —f . In order to keep the problem separable the stratification

N is assumed to be constant.

The vertical wavenumber m is the eigenvalue resulting from the z-component of the

separation of variables

= —rr?tp where tp = 0 a t z = 0, H, (5.23)

which, as previously mentioned, describes the dynamics of the baroclinic perturbation

within the barotropic flow (Pedlosky, 1987). The solution to equation (5.23) is:

ip = <po sin(mz) m = ~ - n = 0 ,1 ,2 ___ (5.24)H


Chapter 5. The Hyper-Gaussian jet on a /3-plane: shallow-water andBoussinesq models _________________________________________ ___ 102

We will consider the fundamental vertical mode, n = 1, so th a t m = n j H is solely a

function of the domain’s height.

For our purposes it is more convenient to cast the problem in term s of the meridional

perturbation velocity iq ,

| k2 (u 2 - N$) + m 2u 2j jfcuu)! + / u l!(j -

| ( u j 2 - iVg) + m 2uj2j (m2/w ) {uiviy — krjvi) +

k (w2 - No) (uviyy - hrjyVi + t / % ) ] -

2km2ULjyNo (uviy — krjVij = 0, (5.25)

from where the other fields can be readablv derived as follows:

Pi = r- u v i y ) (ui2 - JVq)

■■

(it/2 — No) k 2 + m 2u 2

muip!u 2 - N 2 ’

Pi :_ f i m \ N%pi

\ g J u 2 - N o ’

i = (£) ^ + i '-i m w i ) .

In the absence of any background flow equation (5.25) reduces to

V l w +

2 o2 — f 2 k N 2 - a 2 ~ a Jyf y - k 2 • 0.

which, on the equatorial /3-plane / = /3y, has solutions of the form

m ~ e(7/2),K„ (7 ) ,

where

(5.26)

(5.27)

(5.28)

(5.29)

(5.30)

(5.31)

_ f m 2(py


Chapter 5. The Hyper-Gaussian jet on a /3-plane: shallow-water andBoussinesq models______________________________________ 103

and the resulting dispersion relationship is:

, 1/2k = - A I 13 4m

2<r 2 V VnI (2n + l)/3 - (5.32)

The above equations describe all gravity and Rossby modes except for the equatorial

Kelvin wave. As before the la tter can be obtained by making Vi = 0 in equation (2.1)

which, together with the fact that u0 = 0, yields a wave of the form:

Ur ~ e~lm' /2)y\ (5.33)

with m* = m/(Nf3) and

a = + ( N /m ) k , (5.34)

where the positive sign is required if the perturbation is to remain bounded in y, i.e. the

Kelvin wave propagates eastward. The full dispersion relationship is shown in Fig. 5.6 for

two different depths set by the choice of the vertical wavenumber m . The plot was made

using the numerical algorithm described in Appendix C using a resolution of n = 50 grid

points, and the result agrees with direct calculation of equations (5.32) and (5.34).

5 .6 R o s s b y w a v e - K e l v i n w a v e c o u p l i n g

In order to understand how the finite depth can affect the nature of the resonances

between different types of waves we shall analyze a very simple profile consisting of a

piecewise linear flow which possesses a shear discontinuity (a “kink”) at the midpoint. This

Boussinesq flow is contained by a midlatitude channel of finite depth which has no vertical

shear, so th a t the profile can be specified as follows:

Iu\yy 0 < y < bL

u l tJy - bL < y < 0,


Chapter 5. The Hyper-Gaussian jet on a /3-plane: shallow-water andBoussinesq models_______________________________________________________ 104

k

Figure 5.6: Neutral modes supported by the nonhydrostatic Boussinesq model in an equa

torial channel 30km (top) and 10km (bottom) deep. Classification of the modes is as in

Fig. 5.1.



with constant |uj | < / in regions j = 1,2. The Brunt-Vaisala frequency N is constant,

and as explained in the previous section we consider the domain to be vertically bounded,

its depth also constant and equal to H. A diagram of the flow (viewed from above) is shown

in Fig. 5.7. Equation (5.22) can be readably nondimensionalized as follows1:

wall

Region 1

y = 0

Region 2

wall

Figure 5.7: A bounded piecewise linear profile with a shear discontinuity. The shears in

regions 1 and 2 are u j and u\y, respectively.

Pyy ~ Pyei]y + 2e2w u y

ey + t 2u 2 ~ Pkr}y - 2ekuoj,. o Uri + e2ui2)

• - w = 0 , (5.35)u (i2ui2 + erj) 1 - e2(//JVo)2w2_

where we have scaled x and y by 2L in the horizontal; z, 1 /m and H by 2L f /N o in the

vertical; the flow velocity uo by the mean cross-channel velocity difference u0 = 2Luoy, and

time t by 2 i /u o . We have also nondimensionalized the quantities y = (l/e)(ey - 1) and

u = k((e j /e)y—c). Here e = Uo/2Lf = Uoj/// represents the mean Rossby number, which in

turn can be expressed in terms of the “local” Rossby numbers th a t characterize each region

1In this section we will drop the subscript “1” from the perturbation pressure p and perturbation

meridional velocity v.


Chapter 5. The Hyper-Gaussian jet on a /3-plane: shallow-water andBoussinesq models ______________________________________________ 106

as e = (ei + £2)/2. The restriction on the magnitude of the shear requires th a t t j < 1. Note

that the above scaling is valid only off the equator.

The required boundary' condition at the walls is given by the fact th a t the meridional

velocity v must vanish, and hence

which, together with the appropriate continuity conditions for p and v across the kink,

define the Boussinesq eigenvalue problem.

Even for this simple profile, however, we cannot analytically solve equation (5.35). A

numerical solution is nonetheless readably available by reducing the Boussinesq system (2.1)

to a single equation in the meridional perturbation velocity:

where S = (f / N ), and asking for v to vanish at the walls. We can then recast the above

case. The frequency eigenvalue can be real or complex depending on whether the mode is

neutral or unstable (the imaginary part thus representing the growth rate). The shear is

discontinuous a t y = 0, but we can still approximate it by means of the following function

(MacKay and Moore, 1995):

twpy — kp = 0, (5.36)

|fc2 (t2S 2u 2 — l) + e2m2u;2j [us + eiufcuj —

|fc2 (e2S2u 2 — l) + e2m2u;2j (e2m 2u) (uiVy — ktjv) +

k (e2S 2w2 — l ) (kvy + euvyy - ekr)yv) j —

equation as a generalized eigenvalue problem similar to the one solved in the shallow-water

(5.38)

where

£j(y) = -«2 e -C y _i_ e - c ( c i / 6 3 ) v

■)]}(5.39)


Chapter 5. The Hyper-Gaussian jet on a /3-plane: shallow-water andBoussinesq models____________________________ 107

Here C is a curvature param eter which generates the profile shown in Fig. 5.7 as C oo.

For our calculations a value of C = 500 was found to be sufficiently high to reproduce the

physics of the piecewise linear flow. The parameters chosen for the calculations were as

follows: 4 = 1, t i = 0.2, and t 2 = 0.4. Sensitivity analyses indicated th a t a resolution of

100 grid points was sufficient.

In order to determine the stability as a function of height and the zonal wavenumber fc

we calculate the growth rates along m = a fc “cuts" where a is a constant ranging between 0

and 10 (recall th a t m = w/Ff). Figure (5.8) shows the nondimensional eigenvalue spectrum

obtained when a = 10. Two distinct regions of instability are present, one centred at

k ss 0.36 and another a t k as 0.403. A plot of the Reynolds stresses together with pressure

and horizontal velocity fields for the most unstable wave in each region of instability is

shown in Fig. 5.9.

One conspicuous difference between the two cases is the structure of the perturbations

along the indirection. At k = 0.360 the perturbation pressure has maxima at y = - 6 (—1

in this case), and at the middle of the channel where the kink is located, with relatively

little activity occurring in the northern region. The energetics show th a t the Reynolds

stresses are also localized in the southern half of the channel. The disturbance pressure

at k = 0.403, on the other hand, is distributed throughout the width of the channel. The

pressure and velocity fields in this case clearly show the perturbation maxima localized

near the boundaries. The horizontal Reynolds stresses also encompass the entire domain,

although the energy transfer varies due to the stronger shear when y < 0. Figure (5.10)

shows the vertical velocity fields. At fc = 0.360 the flow is mostly nondivergent except

near the south wall where w is large, while a t fc = 0.403 there is noticeable divergence at

both boundaries. Note th a t w w 0 at y = 0 in both cases. The dynamical mechanisms

pertaining to each mode thus appear to be quite different, and we now attem pt to gain a

better understanding of the physics involved.


Chapter 5. The Hyper-Gaussian jet on a /3-plane: shallow-water andBoussinesq models_______________________________________________ 108

0.02

- 0.02

- 0.04

- 0.06

- 0.08

- 0.1

- 0.12

- 0.14

- 0 .1 6 ---------------------1---------------------1---------------------1---------------------1---------------------1---------------------1---------------------1-------------------0.33 0.34 0.35 0.36 0.37 0.38 0.39 0.40 0.41

k

Figure 5.8: Nondimensional Boussinesq eigenvalue spectrum versus zonal wavenumber when

m = 10 k. Solid and dashed lines show the imaginary and real parts of the frequency,

respectively. Note the two regions of instability centred approximately a t k = 0.360 and

k = 0.403.


Chapter 5. The Hyper-Gaussian jet on a ,5-plane: shallow-water andBoussinesq models_________________________________ 109

a) Pressure field/horizontal velocity k - 0.360 Energetics

0.6

0.4

0b) Pressure field/horizontal velocity k = 0.403 Energetics

0.6

0.4

0.2

Figure 5.9: The perturbation pressure (contours) and horizontal velocity (arrows) are plot

ted for each of the unstable modes. The Reynolds stresses (— (uv)xz (e,/e)) are shown to

the right of the fields. The walls are located a t y ~ ±1.


Chapter 5. The Hyper-Gaussian jet on a (3-plane: shallow-water andBoussinesq models__________________________________ HO

a) Vertical velocity w k = 0.360

0.6

0.4

0.2

- 0.2

-0.4

- 0.6

kxb) Vertical velocity w k — 0.403

0.6

0.4

0.2

- 0.2

-0.4

- 0.6

-o.:

kx

Figure 5.10: Vertical perturbation velocities a t k = 0.360 (top), and k = 0.403 (bottom).

Note th a t in the top figure the flow at midchannel is horizontally nondivergent.



We first note th a t the flow shown in Fig. 5.7 has already been analyzed in Chapter 3 in

terms of the quasigeostrophic theory (see equations (3.15) and (3.16b)), which resulted in a

stable vorticity mode generated by the presence of the kink. I t is clear th a t QG theory will

not provide much insight into the instabilities arising in the Boussinesq case, and we thus

extend our analysis into the geostrophic momentum (GM) regime. The relevant system of

equations (2.22) has already been derived in Chapter 2.

As before, we can reduce the GM system to a single nondimensional equation in the

perturbation pressure

where -qy is a <5-function a t the kink. For each segment of the piecewise linear flow the above

equation simplifies to:

a t the walls, respectively. We can now analyze the stability of the profile shown in Fig. 5.7

using GM dynamics, and again assume solutions of the form

Pw ~Py{^)+p (eri (*2 + m2) + = ° ’ (5.40)

P y y - X2p = 0, A2 = fc2( l — e) + m2(l - e), (5.41)

where the continuity and boundary conditions on the pressure are:

= 0 (5.42)

a t the kink, and

eupy - kp = 0 (5.43)

p = A e ^ + B e - ^

p = C e ^ y + D e ~ ^ y

Region 1 (y > 0),

Region 2 (y < 0),

(5.44a)

(5.44b)



where A / = fc2( 1 - e,j + m 2(l - ej), j = 1,2. Applying the appropriate continuity and

boundary conditions leads to the following system of equations:

A + B - C - D = 0,

Ar)2 (k + eaXi) + Brj2 (k — ecrAi) — Cr)i (k + eaX2) — Drji (k — eaX2) = 0,

Aex' b [eAr (a - ( d /e )kb) + fc] - B e - ’" 6 [eA! (a - (ei/e)kb) — fc] = 0,

Ce~x*b [eA2 (or + (e2/e)kb) + fc] - Dex*b [eA2 (or + (e2/e)A:&) - fc] = 0,

where r]j = (1 /t )(ej — 1), j = 1,2. Non-trivial solutions of the above system lead to three

dispersion relationships. Figure 5.11 shows a plot of the growth rate versus the wavenumber

when m = 10 k. Comparison with Fig. 5.8 shows very good agreement between the GM

and Boussinesq solutions. I t therefore appears that GM theory manages to capture the

additional dynamics present in the Boussinesq model. We will now proceed to identify the

mechanism responsible for the instabilities shown in Fig. 5.8 and Fig. 5.11.

Unlike the Boussinesq case, it is possible to obtain analytical expressions for the various

modes which arise in the GM model due to the presence of the walls and the kink. We can

then interpret the instabilities which arise in terms of resonances between these modes, and

furthermore explain the reason why the QG model is incapable of capturing the instabilities.

We commence by analyzing the waves generated in the GM case by solving locally at

the walls and near the kink. We thus “decompose” the basic flow into the profiles shown

in Fig. 5.12.

Assuming a solutions of the form

p(y) - Ae1 11" -6) (5.45)

near the boundary at y = +b we obtain, using equation (5.43), the following dispersion


Chapter 5. The Hyper-Gaussian jet on a /3-plane: shallow-water andBoussinesq m odels___________________________________________ 113

0.02

- 0.02

- 0.04

- 0.06

- 0.08

- 0.1

- 0.12

- 0.14

. 0 .1 6 -------------------- 1---------------------1---------------------1---------------------1---------------------1---------------------1---------------------1--------------------0.33 0.34 0.35 0.36 0.37 0.38 0.39 0.40 0.41

k

Figure 5.11: Nondimensional eigenvalue spectrum versus zonal wavenumber when m = 10 k,

GM case. Solid and dashed lines show the imaginary and real parts of the frequency.

The values used were 6 = 1, ei = 0.2 and e2 = 0.4. Compare with the result from the

Boussinesq model in figure (5.8).


Chapter 5. The Hyper-Gaussian jet on a /3-plane: shallow-water andBoussinesq m odels ________________ 114

y =

y = b

Figure 5.12: “Decomposition” of the velocity profile shown in Fig. 5.7 into three simpler

flows.

relation for the mode trapped at the northern wall:

keAi

In a similar manner, let

t+ K t )'

p(y) ~ .4e|A2|Cy+6J

i frequency of thea t the opposite wall. In this case the frequency of the boundary wave is

^ , / 2\

(5.46)

(5.47)

(5.48)

We identify these two boundary trapped modes as counterpropagating coastal Kelvin waves

(Kushner et al, 1998). At the kink the structure is identical to th a t given in equation (5.44),

but with A s= D = 0, so that the pressure remains bounded as y -¥ ±oo. The continuity

condition (5.42) then gives a Rossby wave

(r}2-r}{)ke(7?iA2 4- *

which a t order one reduces to equation (3.16b), its QG equivalent.

(5.49)


Chapter 5. The Hyper-Gaussian jet on a /3-plane: shallow-water andBoussinesq models______________________________________________________ 115

In order for resonance to take place, the frequencies of the waves must match. Figure 5.13

shows the curves where the Doppler-shifted frequencies of the three neutral modes coincide

in k-m space. In this plot, each solid line represents the intersection of the cr-surfaces

generated by equations (5.46), (5.48) and (5.49) as functions of k and m. Note th a t the

m = 10 k dashed line cuts a Kelvin wave-Rossby wave matching curve at k « 0.36, and the

Kelvin wave-Kelvin wave matching condition a t k ~ 0.4.

4.5

R-K3.5

,K-K

K-R2.5

0.5

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Figure 5.13: Frequency matching contours: (a) Kelvin wave a t y = - b with the Rossby

wave at the kink (labeled K-R). (b) Both Kelvin waves (K-K). (c) Rossby and Kelvin

wave at y = +b (R-K). The values chosen for the various param eters are b = 1, ei = 0.2,

and t 2 = 0.4. The dashed lines are the k-m ratios shown in Fig. 5.11 and Fig. 5.15. They

represent, from left to right, m = 10 k, m = 5 k and m = 2 k.


Chapter 5. The Hyper-Gaussian jet on a /3-plane: shallow-water andBoussinesq models___________________________________________ ______

R-K- 0.1

K-R- 0.2

" K-K-0.3

-0.4

-0.50.50.450.40.35

Figure 5.14: Phase speeds of the three modes generated by the flows shown in Fig. 5.12. The

solid line is the Kelvin wave localized near y = b, the dashed line shows the dispersion

curve of the Kelvin wave a t y = —b, and the dotted line is the phase speed of the

kink-generated Rossby wave as a function of k. Here b = 1, ei — 0.2 and tz = 0.4.

The results obtained so far can thus be interpreted as resonances between the various

waves generated by the flow. By comparing Fig. 5.8, Fig. 5.11, and Fig. 5.13, we find that

the instabilities are indeed centred where the frequencies are equal, and the interpretation

of the eigenfunctions becomes clear. At k = 0.360 the instability arises due to the phase-

locking of the Kelvin wave at y — —b with the Rossby wave localized near the kink at y = 0.

This is consistent with the pressure field shown in Fig. 5.9(a). The second growing mode

[Fig. 5.9(b)], on the other hand, is generated by the resonance between the Kelvin waves

trapped a t each boundary. This last instability has been analyzed in detail by Kushner

et al. (1998), and arises due to the fact th a t the flow on the whole is anticyclonic, which

supplies the appropriate Doppler shifts required for the coupling to take place (since the


Chapter 5. The Hyper-Gaussian jet on a /3-plane: shallow-water andBoussinesq models___________________________________________________ 117

waves propagate with the wall to their right). We note, however, th a t when k & 0.47 a third

matching condition between the Rossby wave and the Kelvin wave a t y = +b is present,

and yet no instability was found in this region. The reason is th a t although the frequencies

coincide, the waves’ difference energy is of the same sign. This fact is confirmed by noting

th a t the sign of the difference energy is opposite to th a t of c_1dc/dl:, where c is the phase

speed (Hayashi and Young, 1987). Figure 5.14 clearly shows th a t both the Rossby wave

and the northern Kelvin wave have positive difference energy when k s» 0.47.

Kushner et al. (1998) have noted th a t unless -C f * / N ’, an unphysical “mirage

wave” develops in the GM model. This boundary-trapped wave is a product of the formula

tion itself, and leads to an unrealistic mode in which the particles remain motionless in the

presence of a nonvanishing perturbation pressure. Whereas in the Boussinesq model the

flow becomes stable as |m*/fc*| ->• 0, in GM theory both instabilities are present and con

tinue to grow. Figure 5.15 shows how the growth rates behave in both the Boussinesq and

GM models as a decreases. The Boussinesq instabilities become weaker and shift towards

higher wavenumbers relative to the GM modes. Although both GM instabilities become

unphysical, the mirage wave-mirage wave resonance only involves spurious modes. The

Kelvin wave-Rossby wave (K-R) mode, on the other hand, subsists even as one approaches

the nondivergent lim it (which can be obtained by making m = 0 in (3.16b)), and thus

interaction between “real” and “mirage” modes takes place.

5.7 Stability o f an equatorial H yper-G aussian jet

We now analyze the stability of the Hyper-Gaussian je t described in Section 5.3 for a

nonhydrostatic Boussinesq flow, and again consider a je t of the form

u0 = U0e - ^ \ (5.50)

The scaling chosen is appropriate for the equator and similar to the one used previously


Chapter 5. The Hyper-Gaussian jet on a /3-plane: shallow-water andBoussinesq models___________________________________ _____________ 118

0.04

£oO -0.04

0.65 0.7 0.75 0.8 0.85k

0.08

0.04u

JSI uo!~) -0.04

-0.08

1.5 1.6 1.7 1.8 1.9k

2.0

Figure 5.15: Comparison of the growth rates between the Boussinesq model (solid line),

and the GM model (dashed line) when m = 5 k (top) and m = 2 k (bottom). See also

Fig. 5.13.



for the shallow-water model, to wit:

( x ,y , k ~ l) L (x , y , k ~ l) ( z ,m ~ l) >-> H ( z , m ~ l )

(uo.t»i,Ui) i-t U{u0. u h Vi) ( t , / _1,<7_1) r-t {L/U)(t, , a ~ l )

P * ( U / I ? ) 0

so th a t the nondimensional iq equation becomes:

|fc2 (s2w2 — l) + e2m2u 2j Jfcojui + f v iy| —

|fc2 (s2 ai2 — l ) + e2m 2o;2j £ (e2m 2/w ) {oJViy — kyvi) +

k (s2 w2 - l ) (u>vlyy - krjyVi + k f v i y) ] -

2ke2m 2uiuy (wviy - k r \ v = 0, (5.51)

where s = (U / L ) / N and e = (U /H ) /N . Using the following scales:

U = 25m s_1,

L = 106m,

H = 104m,

N — 0.01s-1 ,

results in s = 0.0025 and e = 0.25. The background flow is scaled in a similar manner and

the r param eter is again chosen to be equal to 0.555, so th a t the to tal e-folding width of

the Hyper-Gaussian je t is 1110 km. Lateral walls are imposed far from the core of the flow

at y = ±7000 km where the meridional boundary condition is Vi = 0. The resulting eigen

value problem is similar to the one in the previous chapter and is derived in Appendix C.

Figure 5.16 shows the contours which represent the growth rates in (k, m) space. The


Chapter 5. The Hyper-Gaussian jet on a ,3-plane: shallow-water andBoussinesq models_______________________________________________________ 120

m8

m

Figure 5.16: Growth rate contours in (k, m) space of an equatorial, Hyper-Gaussian jet

(Boussinesq model). Shading highlights the unstable regions. The top figure shows the

contours of the most unstable modes, while the bottom figure shows the growth surface

which lies underneath.


Chapter 5. The Hyper-Gaussian jet on a >9-plane: shallow-water andBoussinesq models_______________________________________________________ 121

top figure shows the isolines of the most unstable perturbation (which correspond to two

distinct modes, as seen below) while the bottom plot shows the secondary instabilities.

Figure 5.17 shows the growth and phase speed curves corresponding to a domain of

infinite depth for which m = 0. When m vanishes the dynamics and thermodynamics are

decoupled in the original Boussinesq system (2.1). The la tter simply yields buoyancy waves

of intrinsic frequency iV, while the dynamical portion reduces to the nondivergent form of

the shallow-water equations. Since the instabilities analyzed so far for the Hyper-Gaussian

flow are the result of vorticity wave coupling it is not surprising th a t in this lim it the results

are essentially the same as those in Section 5.3, as seen in Fig. 5.18.

As the domain becomes shallower the peaks shift slightly towards smaller wavenumbers

and the growth rates decrease. Figure 5.19 shows the stability of a je t confined to a channel

which is 10 km deep (m = it). As before the fastest-growing mode is the antisymmetric

perturbation which peaks a t fc = 2.40 (A = 2614 km) with a growth rate of S (o j = 0.44

(e-folding time of 25 h). The symmetric mode achieves its maximum value when k = 2.19

(A = 2869 km) where 3 (o j = 0.29 (e-folding time of 38 h). The pressure maxima and

minima in both cases are located near the inflection points, at about 516 km to the north

and south of the equator. The shape of the dispersion relationship and structure of the

functions (Fig. 5.20) suggest th a t the mechanism behind the instabilities are the same

as before, namely, shear wave resonance about the inflection points. This conclusion is

corroborated by the Reynolds stresses and the vorticity patterns.

In spite of the results obtained in Section 5.6, which suggest th a t Rossby waves and

Kelvin waves may resonate, no Rossby wave-gravity wave coupling was found for an equa

torial Hyper-Gaussian je t given any reasonable physical geometry. Instabilities of this type

— in which shear-generated vorticity modes phase lock with some type of gravity wave —

were found only for very shallow (less than 6 km deep), narrow channels (approximately

twice the e-folding width of the jet), which are clearly unrealistic.


Chapter 5. The Hyper-Gaussian jet on a /3-plane: shallow-water andBoussinesq models______________________________________ 122

0.6

0.5

0.4

0.2

0.1

1

0.6

0.2

- 0 .2,

Figure 5.17: Growth rate and phase speed curves of the unstable modes generated by a

Hyper-Gaussian je t in the Boussinesq model when m — 0.


Chapter 5. The Hyper-Gaussian jet on a /3-plane: shallow-water andBoussinesq models_______________________________________________ 123

- 1

- 2

- 3

3

- 1

- 3

Figure 5.18: Structures of the two unstable modes at m = 0. The instability pictured at

the top peaks a t k = 2.42 and has a growth rate of S(cr) = 0.55. The perturbation

pictured on the bottom plot corresponds to the secondary mode. I t was calculated at

k = 2.38 and possesses a nondimensional growth rate of a = 0.46. D otted lines indicate

the position of the inflection points a t ±516 km from the equator. For visualization

purposes only the central part of the channel is shown, since the to tal width is 14 (in

nondimensional units).


Chapter 5. The Hyper-Gaussian jet on a /3-plane: shallow-water andBoussinesq models___________________________________________

0.5

0.4

■3. 0.3

0.2

0.1

1

0.6

^ 0.4 £

0.2

- 0.2,

Figure 5.19: Same as Fig. 5.17 but with m = tt.

124


Chapter 5. The Hyper-Gaussian jet on a /3-plane: shallow-water andBoussinesq models____________________________________ 125

3

;'/////,

Figure 5.20: Structures of the two unstable modes a t m = it. The most unstable mode

(top) peaks at k = 2.40 and has a growth rate of cr = 0.44. The secondary instability

pictured on the bottom plot was calculated a t k = 2.19 and possesses a growth rate of

a = 0.29.


Chapter 6

Summary, conclusions and future

work

In Section 5.1 it was noted th a t Marinone and Ripa (19S4) have previously analyzed the

stability of an equatorial Gaussian je t when studying the Equatorial Undercurrent using

the shallow-water equations. An im portant result from this work was the notion th a t some

of the resulting instabilities (those pertaining to wide1 easterly jets) could be linked with

“negative energy perturbations” , i.e. growing waves in which the net energy transfer is

from the perturbations and into the mean flow. Furthermore, it was found th a t narrow jets

were also unstable and th a t the perturbations in these cases were not unlike those which

appear in the “classic” barotropic problem, where kinetic energy flows from the mean flow

into a nondivergent disturbance (Yanai and Nitta, 1968). The idea of instability resulting

from the transfer of energy from a background flow into the perturbation no longer held

true in general.

The subsequent literature thus elaborated on the nature of the mechanism behind the

*The terms “wide” and “narrow” used here compare the e-folding w idth of the Gaussian je t with the

e-folding width of an equatorial Kelvin wave in the ocean.

126


Chapter 6. Summary, conclusions and future work 127

instability of shear flows and negative energy waves. Lindzen (1988) pu t forth the over

reflection theory as a means to explain instability — briefly discussed in Section 4.3.1 — but

its validity has been debated in the literature (e.g. Smyth and Peltier, 1989). Concurrently

the wave resonance theory began to gain strength when Hayashi and Young (1987) analyzed

the stability of shallow-water shear flows on an equatorial /?-plane. The idea th a t instability

was the product of the resonance between waves with difference energies of opposite signs

solidified, and later Takehiro and Hayashi (1992) described the over-reflection mechanism

itself in terms of wave-wave interactions.

A steady stream of authors have applied the wave-resonance theory in a series of QG and

shallow-water models, studying both baroclinic (usually two-layer models, e.g. Sakai, 1989)

and barotropic (in which u0 = no (y) e.g. Hayashi and Young, 1987; De la Cruz-Heredia and

Moore, 1999) flows. The reanalysis by Iga (1993) of Orlanski’s problem recently studied the

stability of a frontal surface between two fluid layers from the wave resonance perspective.

We now apply these ideas to a phenomenon th a t is similar to the one which concerned

Marinone and Ripa (1984): the stability of an equatorial “bell-shaped” jet. Chapter 1

explains the nature of the equatorial jets which concern us, namely, the strong westerly

wind bursts (WWBs) which seem to play an im portant part in the formation of equatorial

cyclone twins (ECTs). Aside from the obvious differences in scales we consider another,

more subtle aspect of the profile which takes into account a variation in the shape. This

slight modification is found to lead to im portant differences with respect to the ubiquitous

Gaussian je t so often studied in the literature. Furthermore the finite depth of the flow

is also taken into account in order to investigate the effect of having a vertically bounded

domain. Chapter 2 details the models which were used to analyze the flow.

We begin by studying some simple flows in Chapter 3, which deals with the stability of

piecewise linear profiles when the flow is quasigeostrophic (QG). G ravity modes are filtered

out a priori, and only vorticity modes are capable of resonating. The depth of the flow,



set through the vertical wavenumber m , is of little consequence regarding the nature of the

instabilities, and in fact enters the potential vorticity equation in a trivial manner (through

an “equivalent” wavenumber which we labelled //). Instability is solely the result of phase

locking between waves resulting from the discontinuities in the meridional gradient of the

shear. Wave resonance analysis on piecewise linear flows with more than two discontinu

ities in the curvature of the basic velocity field (i.e. “kinks” ) shows th a t resonance can

take place between more than two localized waves. Depending on the wavenumber (and

hence on the meridional extent of the “free” modes), a “compound” instability resulting

from two simultaneous, localized resonances can be obtained. As the wavelength increases

a large scale resonance may take place, but the small-scale effects remain, complicating

the overall dynamics. The resulting instability in these cases could be attributed to the

resonance between four modes. Examples from the literature seem to indicate th a t this

is a rather common occurrence. The transition between Kelvin-Helmholtz and Holmboe

instability studied by Smyth and Peltier (1989), for example, shows a transition between

unstable modes of identical growth rates and opposite phase speeds towards unstable modes

of different growth rates and equal phase speeds a t smaller wavenumbers, including the ap

pearance of an energy sink in mid-flow, similar to what is seen for the staircase profile

(Fig. 3.5). Smyth and Peltier analyzed a hyperbolic-tangent profile w ith varying B runt-

Vaisala frequency in the vertical using a Boussinesq model which can support gravity waves.

It was later shown by Baines and Mitsudera (1994) that Holmboe instability can be ex

plained in terms of resonance between vorticity and gravity waves. The QG model does not

support gravity waves, but in this case it is the additional kinks which provide the waves

necessary to create the two independent instabilities at the larger wavenumbers. Other

examples of resonance between unstable modes can be found in Iga (1993) (see his Fig

ure 6), and Nakamura (1988) [see his Figure 7(b)]. The former seems to be a four wave

resonance between two Rossby waves and two mixed Rossby-gravity waves (R i and Mo



in Iga’s notation). The latter is apparently the resonance between two boundary-trapped

waves and two inertia-gravity modes. In Nakamura’s case the four wave mixing ceases at

the lower wavenumbers due to the disappearance of the inertia critical levels (which “trap”

the inertia-gravity waves), simply leaving the Eady mode (due to the resonance between

the boundary waves).

All the profiles in Chapter 3 featured some sort of discontinuity in the gradient of the

background shear, which facilitates the identification of the resonating waves (since they

are generated by, and localized about, the discontinuities), and their relative Doppler shifts.

Kobayashi and Sakai (1993) noted th a t when the profiles are smooth the instability is the

result of the resonance between the two modes of the continuum closest to the “effective”

inflection point (i.e. where /? - u0yy vanishes). Barotropic instability in a smooth boundary

current was later studied by Katsum ata (1997) and was similarly explained in terms of

resonances between various Rossby waves whose direction of propagation depend on the

sign of the background vorticity gradient.

These findings agree with the results in Chapter 4, where the stability of a hyperbolic-

tangent flow was studied as a function of the number of grid points. The finite difference

method is widely used in order to obtain numerical solutions of hydrodynamic problems

which are analytically intractable, and is particularly useful when the sparseness of the

resulting eigensystem is exploited (Proehl, 1996; De la Cruz-Heredia and Moore, 1997).

Although certainly not the only method available (and sometimes not even the most ap

propriate), it nonetheless provides a convenient way of visualizing how the numerical solvers

discern instability from the wave resonance perspective.

In the case of the hyperbolic-tangent profile there is a continuum of counterpropagating

vorticity waves generated by the curvature of the flow at either side of the inflection point.

When discretized, a finite number of neutral modes results, and the flow is unstable due

to the resonance of the waves nearest to the inflection point. In a sense the flow is the



counterpart to Rayleigh’s broken-line shear layer profile, where in the latter case the two

kinks generate the resonating vorticity waves. Although analyzing piecewise linear “ana

logues” of smooth profiles may help understand the dynamics behind an unstable mode,

wave resonance is unable to predict whether two modes will indeed phase-lock or not. The

“half-wave” sine profile, for example, is qualitatively very similar to a hyperbolic-tangent

flow, bu t is stable.

The above procedure is nonetheless useful in understanding the instabilities arising

in Gaussian and Hyper-Gaussian jets, which result in growing modes whose structures are

mostly localized about each inflection point, resulting in a symmetric and an anti-symmetric

perturbation with nearly the same growth rates. For “wide” jets, such as the Hyper-

Gaussian profile, the individual resonances become better defined, with the symmetric

mode “splitting” into two, in-phase trains of highs and lows a t opposite latitudes with

respect to the centre of the jet.

The stability of the Hyper-Gaussian profile on a /?-plane does not differ much from the

/-p lane results, no m atter what the direction of the je t is. This is due to the scales used,

which were chosen according to the values set by an equatorial W WB. The effect of the

background vorticity gradient was small compared to th a t of the curvature of the shear,

and the instabilities themselves remained essentially unchanged except for the long-wave

cutoff of the symmetric mode.

The QG approximation does not allow us to model flows a t the equator unless the depth-

independent barotropic equation is used. Furthermore, neither model supports Kelvin or

gravity waves. In order to bypass the latter lim itation the shallow-water (SW) equations

provide a dynamical framework which, aside from being able to deal with the vanishing

of the Coriolis parameter, also support dynamics which are not present in the QG regime.

New types of waves which were previously filtered out at O(Ro) are now present in the

form of the Yanai, Kelvin and gravity waves, as seen in Chapter 5.



Using scales similar to that of a WWB, however, shows th a t these waves have essentially

no effect on the shear-generated modes which appear in the QG case. This is due to the fact

th a t the Froude number is small and hence the phase speeds of the gravity modes are well

beyond those of the shear waves, so th a t no resonance can take place given the relatively

small Doppler shifts. There is, nonetheless, an im portant effect due to the location o f the

je t a t the equator in which the symmetric and anti-symmetric modes are “interchanged,”

although the growth rate curves remain essentially the same. The reason behind this be

haviour can be understood in terms of the perturbation’s wind fields. On an /-p lane located

in the northern hemisphere, say, the symmetric mode consists of two cyclones which imply

the existence of symmetric pressure lows. At the equator the wind fields remain unchanged,

and in order to maintain the sense of rotation in the southern hemisphere the sign of the

pressure field must switch. The same argument applies to the anti-symmetric mode and

hence the exchange of the mode structures. The symmetric mode is no longer the most

unstable, and the dominance of the anti-symmetric instability suggests th a t the appearance

of single cyclones should be most common as the result of a WWB. This has indeed been

observed to be the case (Hartten, 1996), as nearly 67% of the observed WWBs in the 1980s

have preceded the formation of individual cyclones. The mechanism which determines the

final structure cannot be obtained from the above analysis, although heating may certainly

play a role (see below). Another factor is the initial effect of non-modal disturbances which

can shift dominance between modes during the early stages of growth. Farrell (1984) has

noted th a t the growth rate of non-modal disturbances (whose structures change with time)

can initially be much larger than th a t of the normal modes, with the latter dominating at

greater times. In the interim, however, the non-modal disturbances may interact with the

modal waves in a manner which ultimately favours one normal mode over the other.

Using a nonhydrostatic Boussinesq model the effects of channel depth and gravity waves

can be readably incorporated. The former in particular is of special relevance since it is


Chapter 6. Summary, conclusions and future work 1 3 2

shown th a t resonances between shear and gravity modes can take place by modifying the

depth of the channel through the vertical wavenumber m (which is inversely proportional

to the height once the fundamental mode is chosen). A Hyper-Gaussian je t similar to a

10 km deep WWB presents again a symmetric and an anti-symmetric mode. The latter,

which has an e-folding time of 25 hours, peaks when the wavelength is 2614 km. The

weaker symmetric mode grows with an e-folding time of 38 hours and peaks a t A = 2869

km. These are slightly weaker than their purely barotropic counterparts (w'hen m = 0), but

the dynamics behind them is essentially the same as in the mid-latitude case. The pressure

extrema are about 10 degrees apart from each other, a t approximately 500 km on either

side of the equator. This is in good agreement with the ±5° latitude at which ECTs form,

and it is encouraging th a t this value is obtained for a je t of realistic dimensions (based on

actual WWBs events) within a vertically bounded domain. This result also improves upon

the heat-forced models described in Section 1.1, which lead to a much larger separation

between the storms and do not seem to explain the dominance of single cyclone formation.

We conclude from the above results th a t gravity waves do not play a noticeable role

in the generation of the cylones when considering a barotropic background flow of WWB

dimensions, essentially reducing the dynamics to th a t of a 2D plane flow (the nondivergence

in the shallow-water model, for example, being established through the smallness of the

Froude number). Although /? introduces a longwave cutoff to the most unstable mode,

the peak growth rates and their position in wavenumber space are only affected to a small

degree.

Studies on the formation of equatorial cyclone twins (Gill, 1979; Nieto Ferreira et al.,

1996) have focused on the effects of heat sources straddled on the equator. Convection plays

a vital role in tropical cyclogenesis (Holton, 1992), and the results from these studies show

how such forcing can lead to symmetric, cross-equatorial lows. Strong convective events

which are observed in the equatorial regions, however, do not necessarily lead to ECTs, and



hence other processes are certainly a t play. A good candidate and one of the most notable

events which precedes ECT formation is the appearance of strong westerly wind bursts.

These powerful jets are capable of generating symmetric cross-equatorial lows, aiding in

the organization of the deep equatorial convection (Keen, 1982) which can eventually lead

to the striking twin hurricanes shown in Fig. 1.3. Clearly it remains necessary to analyze

the combined effects of both processes in order to better understand how equatorial cyclone

twins — and tropical cyclogenesis in general — takes place.


A ppendix A

The Quasigeostrophic numerical

eigenvalue problem

The QG potential vorticity equation (3.4) can be readably written in m atrix form

[u0 (/j2I - D (2)) - ( / 3 I - D (2)u 0)] V> = c (/r21 — D (2)) i/>, (A.l)

or, more compactly,

A ip = cBt/i, (A.2)

and is thus solved by seeking the eigenvalue c and eigenvector

ip = (i/>(!/[11),t/>(!/121),--- ,4>{y[n]))T

which satisfy the eigensystem for a given profile u(y) as a function of the wavenumber // (or

k for the nondivergent case). The value j/M represents the latitude a t grid point i, I is the

identity matrix, and the n-th derivative operator. Five-point central difference formu

las were used in all cases (Gerald and Wheatley, 1994) except near the boundaries, where

forward and backward differences were used1. The errors for these numerical derivatives are

1Using three-point formulas did not seem to affect the results in any notable manner, but the five-point

formulas were preferred for their higher accuracy in spite of the slight increase in memory requirements.

134


Appendix A. The Quasigeostrophic numerical eigenvalue problem 135

of 0 {{A y )4), where A y is the distance between two grid points. W hen no boundaries are

present artificial walls are placed far away from the region of interest so as not to modify

the dynamics. If doubling the width of the channel did not change the results it was as

sumed th a t the boundaries were “far enough” so as to consider the domain to be effectively

“infinite” .

The five point finite-difference second derivative operators applied to fields which vanish

a t the boundaries (or a t infinity) are:

/

d L1' =12A y

0

12(Ay)2

0 18 - 6 1 o N0 8 - 1

°OO1 ••

8 - 1

1 - 8 0 8 - 1

1 - 8 0 8

-1 6 -1 8 10 J-2 0 6 4 - 1

16 -3 0 16 - 1

- 1 16 -3 0 16 - 1

1 16 -3 0 16

0

0

-1

- 1 16 -3 0 16

- 1 4 6 -2 0 j

For fields whose boundary values are unknown five-point one-sided differences were used


Appendix A. The Quasigeostrophic numerical eigenvalue problem 136

instead:

DL1' =12A y

-2 5 48 -3 6 16 - 3

- 3 -1 0 18 - 6 1

1 - 8 0 8 - 1

1 - 8 0 8

- 1 6 -1 8 10

0 3 -1 6 36 -4 8

0

1 35

11

-1

d L2) = 12(A y f

V

-104 114

- 2 0

16

6

-3 0

-1

- 1

11

—56

4

16

16

4

11

- 1

- 1

-3 0

6

0

-56 114

16 - 1

- 2 0 11

-1 0 4 35 j

The generalized eigenvalue problem Ai/> = c B ip was first converted into the standard form

(B_1A)V> = crf> and then solved using the M A TLA B subroutine eig. A 400 grid point

resolution gave essentially the same results as an 800 point grid and was deemed sufficient

for all calculations.


A ppendix B

The Shallow-W ater numerical

eigenvalue problem

The nondimensional shallow-water system (5.3) can be readably written as a fifth-order

generalized eigenvalue problem of the form

A 5cr5t)i + A4(T4« i + A 3a 3Ui + A 2cr2Ui + A i ODi + A 0Ui = 0, (B.l)

where the matrices A 0 - A 5 are of order n (the number of interior grid points), and depend

on the wavenumber k and the various parameters of the basic sta te including the geometry of

the background flow i.e. F, f , u0(y) and ho(y). When derivatives are required the operators

described in Appendix A are used. The boundary condition is simply Vi = 0 at the lateral

walls.

The above system can be reduced to a first-order generalized eigenvalue problem by

137


Appendix B . The Shallow-Water numerical eigenvalue problem 138

means of the following m atrix composition

( a„ A-i A2 A3 a4M

' 0 0 0 0 -As^M0 Ao 0 0 0 av 1 Ao 0 0 0 0 av 1

0 0 Ao 0 0 a2Vx = a 0 Ao 0 0 0 a2v x0 0 0 Ao 0 cr3v 1 0 0 Ao 0 0 a3ui

v° 0 0 0 Ao j 1 ° 0 0 Ao 0 } [ a ^ J

or, more compactly,

A v e = a B v e (B.2)

The solution of the above eigenvalue problem yields the meridional velocity eigenvector,

and perturbation height and the zonal velocity can then be calculated as follows:

iF 2hi = F 2p2 _ jr hQ (r («ifto)s - k h ^ v x ) , (B.3)

ui = p (v v i + j $ h i j , (B.4)

where T = ku 0 — a and 7} = Uoy — f .

The matrices A and B in equation (B.2) are of order 5n but are very sparse (each

submatrix having at most five diagonals resulting from the finite differences operators). In

order to fully exploit this fact the following reordering of the eigenvector is first carried out

so as to reduce the bandwidth of A and B:

v e = (fi*', cruj"*, <r2uj"l, a3v cr4u["')T.

The eigenvalue problem is then solved as follows: given an initial guess ao = a — A<r0

to the eigenvalue we seek, it is possible to rewrite equation (B.2) as

(A — oqB) v e = AooBue,


Appendix B. The Shallow-Water numerical eigenvalue problem 139

which can be rewritten in the following way

LU«e = AcrBu,,,

where L and U are the lower and upper triangular matrices resulting from a sparse LU-

decomposition of ( A - c t o B ) . This transformation is highly efficient and retains most of both

the sparsity and the narrow band structure of the original system thanks to the reordering

described above.

We then proceed to solve the problem using a generalized inverse iteration method

(Kerner, 1986; De la Cruz-Heredia and Moore, 1997; Yamazaki and Peltier, 2000a) for

i = 0 ,1 ,2 , • • • in the following manner:

r 'B y '

y (i+i) = u _1(L_1 (A<r(i+1) B y ;))

r (>+i) = (((A<r(i+1)) V B * ) (L*)-1) ( U * ) '1

where r° = (y0)*. In the above procedure the asterisk denotes the Hermitian conjugate.

The vectors y and r are normalized a t each step to 1 + Oi and the iterations are interrupted

if convergence is achieved, namely when

|y0+l) _ y(i)||v «+i) | < T 0 L ’ < TOL, and

5 (Acr(*+P)< TOL,

where <5h+1> = A ct('+1) - AcM. The tolerance TOL was chosen to be O(10-8).

Both the initial eigenvalue (cro) and eigenvector (y°) guesses are obtained by first solv

ing the nonsparse eigensystem (B.2) a t a low resolution of 100 grid points. This is accom

plished by means of a straightforward QR transformation. The calculation is performed

over a coarse one-dimensional mesh in wavenumber space, so th a t the fc region of interest

is thoroughly but not densely sampled. Once the most unstable modes have been detected

the sparse solver described above is used. The resolution in this la tter case — 10000 grid



p o in ts resulting in m atrices o f o rder 50000 — is qu ite high, b u t th e sparse sto rage scheme

allows us to solve th e problem on a m edium -sized w orkstation . T h e th ird and final step

consists o f using th e results from th e sparse solver as s ta r tin g po in ts to trace growth ra te

curves in wavenum ber space, increasing the sam pling ra te a long fc by 20 tim es a t th e highest

resolution.

Below are the m atrices p e rta in ing to th e shallow -w ater eigensystem (B .l) . T hey are

shown in M A T L A B n o ta tion , where “ . . . ” denotes a line con tinua tion . Q uan tities which

have been differentiated w ith respect to y are denoted as “Xy” , w here “X” is th e appropria te

field. T he identity and finite difference operato rs are “I ” and “DM” , respectively, where in

the la t te r case “N” can be “1” or “2” depending on the o rder of th e derivative.

A0 = -F~2*k"3*f~2*H*U - k ‘ 5*H‘ 2*U + . . .

F‘ 4 * k " 3 * f‘ 2*U*3 + 2*F“2*k‘ 5*H*U“3 - . . .

F“4*k-5*U "5 + k~3*H~2*fy - . . .

F '2 * k ‘ 3*H *iT2*fy - k '3*f*H *H y - . . .

k “3*U*Hy"2 + 3*F~2*k“3*f*H*U*Uy - . . .

F"4*k‘ 3*f*U “3*Uy + k~3*H*Hy*Uy + . . .

2*F“2 *k“3*U‘ 2*Hy*Uy - . . .

2*F‘ 2*k“3*H*U*Uy“2 + k ‘ 3*H*U*Hy*Dl - . . .

2*F~2*k~3*U“3*Hy*Dl + . . .

2*F“ 2*k~3*H*U"2*Uy*Dl + k"3*H*U*Hyy - . . .

F‘ 2*k"3*U“3*Hyy - k '3*H '2*U yy + . . .

F*2*k"3*H*U"2*Uyy + k ‘ 3*H'2*U*D2 - . . .

F~2*k"3*H*U“3*D2;

A1 = F‘ 2*k‘ 2*f~2*H + k '4 * H ‘ 2 - . . .

3*F“4 *k“2*f "2*U“2 - 6*F"2*k'4*H*U~2 + . . .



0.6

0.5

0.4

0.2

k

Figure B .l: Numerical procedure used to determine the stability of an eastward Hyper-

Gaussian je t on an equatorial /?-plane using the shallow-water model. The squares rep

resent the coarse guesses (using 100 grid points) obtained by solving the nonsparse eigen

value problem (B.2) using a QR algorithm. The three most unstable modes (tagged )

are saved and used as seeds for the sparse solver. The results, using 10000 grid points,

are tagged here as “0 ” • These la tter values are then chosen to begin a dense trace along

k, using a result a t point fc; as a guess for point ki±i. The trace is also performed using

10000 grid points and the resulting curve is plotted above with dots



5 * F '4 * k ‘ 4*U"4 + 2*F*2*k‘ 2*H*U*fy + . . .

k “2*Hy"2 - 3*F ‘ 2*k‘ 2*f*H*Uy + . . .

3*F‘ 4*k‘ 2*f*U ‘ 2*Uy

4*F"2*k"2*U*Hy*Uy + . . .

2*F‘ 2*k‘ 2*H*Uy‘ 2 - k ‘ 2*H*Hy*Dl + . . .

6*F‘ 2*k'2*U"2*Hy*Dl

4*F“2*k~2*H*U*Uy*Dl - k ‘ 2*H*Hyy + . . .

3*F"2*k‘ 2*U'2*Hyy - . . .

2*F‘ 2*k"2*H*U*Uyy - k"2*H‘ 2*D2 + . . .

3*F‘ 2*k~2*H*U‘ 2*D2;

A2 = 3*F‘ 4 * k * f‘ 2*U + 6*F"2*k‘ 3*H*U -

10*F"4*k“3*U“3 - F~2*k*H*fy - . . .

3*F"4*k*f*U*Uy + 2*F‘ 2*k*Hy*Uy - . . .

6*F"2*k»U*Hy*Dl + 2*F"2*k*H*Uy*Dl - . . .

3*F~2*k*U*Hyy + F"2*k*H*Uyy - . . .

3*F*2*k*H*U*D2;

A3 = -F * 4 * f* 2 - 2*F‘ 2*k~2*H + . . .

10*F"4*k"2*U '2 + F‘ 4*f*Uy + . . .

2*F‘ 2*Hy*Dl + F"2*Hyy + F‘ 2*H*D2;

A4 = -5 * F “4*k*U;

A5 = F ~ 4 * I;


A ppendix C

The Boussinesq numerical eigenvalue

problem

Mathematically the Boussinesq eigenvalue problem is equivalent to the shallow-water

system, i.e. equation (5.51) can also be written as a fifth-order generalized eigenvalue

problem of the form

A 5<75Ui + A 4o-4i;i + A ^ t ) ! + A 2(T2Ui + AiffUr + A 0Vi = 0, (C.l)

where the matrices Ao - A 5 are of order n (the number of interior grid points), and depend

on both wavenumbers k and m (or, more appropriately, the depth of the domain). The

same methodology given in Appendix B is also applied in this case. The nondimensional

matrices are calculated as follows:

AO = k ‘ 6*U + e ‘ 2*k‘ 4*nT2*f"2*U - . . .

2 * e ‘ 2*k"6*m‘ 2*U*3 - 2*k‘ 8 * s ‘ 2*NH*U'3 - . . .

e '4 * k ‘ 4 * n r4 * f“2*U"3 - . . .

e~2*k~6*m‘ 2*s~2*NH*f~2*U~3 + e"4*k~6*m~4*U‘ 5 + . . .

2 * e ‘ 2*k-8*m‘ 2*s'2*NH*U‘ 5 + k ‘ 10*s‘ 4*NH*U'5 - . . .

143


Appendix C. The Boussinesq numerical eigenvalue problem 144

k "4 * fy + e~2*k"4*nT2*U~2*fy + . . .

2*k“ 6*s'2*NH*U“2 * fy - . . .

e ‘ 2*k‘ 6*m“2*s"2*NH*U"4*fy - . . .

k ‘ 8 * s ‘ 4*N H *ir4*fy - . . .

3*e"2*k"4*nT2*f*U*Uy + . . .

e ‘ 4*k~4*m"4*f*U‘ 3*Uy + . . .

e ‘ 2*k"6*m‘ 2 * s ‘ 2*NH*f*U~3*Uy + . . .

2 * e"2*k‘ 4*m‘ 2*U*Uy‘ 2 - . . .

2*e“2*k“4*m"2*U~2*Uy*Dl + k~4*Uyy - . . .

e"2*k"4*m “2*U*2*Uyy - 2*k"6*s~2*NH*U‘ 2*Uyy + . . .

e _2*k“6*m "2*s‘ 2*NH*U"4*Uyy + . . .

k ‘ 8*s-4*NH*U‘ 4*Uyy - k ‘ 4*U*D2 + . . .

e"2*k"4*m"2*U“3*D2 + 2*k‘ 6*s'2*NH*U‘ 3*D2 - . . .

e ‘ 2*k '6*m ‘ 2 * s ‘ 2*NH*U"5*D2 - k ‘ 8 * s ‘ 4*NH*U‘ 5*D2;

A1 = -k * 5 * I - e p ‘ 2*k"3*m“2 * f“2 + . . .

6*ep~2*k"5*m~2*U‘ 2 + 6*k~7*s'2*NH*U“2 + . . .

3*ep"4*k~3*m ~4*f“2*U‘ 2 + . . .

3 *ep ‘ 2*k‘ 5*m '2*s‘ 2*NH*f‘ 2*U‘ 2 - . . .

5*ep"4*k"6*m “4*U"4 - 10*ep"2*k"7*m “2*s~2*NH*U~4 - . . .

5*k"9*s"4*NH*U‘ 4 - 2*ep‘ 2*k"3*m‘ 2*U*fy - . . .

4*k"5*s"2*NH*U*fy + . . .

4*ep"2*k"5*m ‘ 2 * s“2*NH*U‘ 3 * fy + . . .

4 * k "7 * s‘ 4*NH*U"3*fy + . . .

3*ep“ 2*k"3*m"2*f*Uy -

3*ep~4*k"3*m "4*f*U '2*Uy - . . .



3 *ep '‘2*k"5*m~2*s‘ 2*NH*f*iT2*Uy - . . .

2*ep~2*k"3*m“2*Uy~2 + 4*ep‘ 2*k‘ 3*nT2*U*Uy*Dl + . . .

2*ep‘ 2*k'3*m ‘ 2*U*Uyy + . . .

4*k~5*s~2*NH*U*Uyy - . . .

4*ep‘ 2*k‘ 5*m '2*s‘ 2*NH*U"3*Uyy - . . .

4 * k "7 * s ‘ 4*NH*U'3*Uyy + k '3*D 2 - . . .

3 * e p '2 * k ‘ 3*m-2*U"2*D2 - 6*k‘ 5*s'2*NH*U‘ 2*D2 + . . .

5*ep~2*k“5*nT2*s"2*NH*ir4*D2 + 5*k‘ 7 * s ‘ 4*NH*U‘ 4*D2;

A2 = -6 *ep"2*k '4*m '2*U - 6*k‘ 6*s~2*NH*U - . . .

3 *ep“4*k~2*nT 4*f“2*U - . . .

3*ep '2*k"4*ra‘ 2*s"2*NH*f‘ 2*U + . . .

10*ep"4*k‘ 4*m‘ 4*U‘ 3 + 20*ep"2*k“6*m"2*s“2*NH*U‘ 3 + . . .

10*k‘ 8*s"4*NH*U"3 + e p ‘ 2*k‘ 2*m '2*fy + . . .

2*k“4 * s ‘ 2*NH*fy - . . .

6*ep'2*k"4*m "2*s"2*NH*U“2 * fy -

6*k"6*s"4*NH*U"2*fy + . . .

3 *ep“4*k"2*m"4*f*U*Uy +

3 * ep"2*k“4*m"2*s"2*NH*f*U*Uy - . . .

2*ep~2*k“2*nT2*Uy*Dl - ep"2*k"2*m “2*Uyy - . . .

2*k“4 * s “2*NH*Uyy + . . .

6*ep~2*k‘ 4*m‘ 2 * s ‘ 2*NH*IT2*Uyy + . . .

6*k“6 * s ‘ 4*NH*U‘ 2*Uyy + 3*ep‘ 2*k“2*m“2*U*D2 + . . .

6 * k '4 * s ‘ 2*NH*U*D2 - 1 0 * ep '2 * k ‘ 4*nT2*s‘ 2*NH*U"3*D2 - . . .

10*k-6*s~4*NH*U-3*D2;



A3 = 2*ep"2*k"3*nT 2*I + 2*k"5*s~2*NH*I + ep*4*k*nT4*f~2 + . . .

ep “2*k"3*m "2*s“2*NH*f*2 - 10*ep“4*k“3*m“4*U '2 - . . .

20*ep‘ 2*k‘ 5*nT2*s"2*NH*U‘ 2 - 1 0 * k '7 * s ‘ 4*NH*U‘ 2 + . . .

4*ep- 2*k"3*nr2*s~2*NH*U*fy +

4*k"5*s'4*N H *U *fy - ep~4*k*m"4*f*Uy - . . .

e p “2*k~3*m"2*s"2*NH*f*Uy -

4*ep“ 2*k"3*m‘ 2 * s ‘ 2*KH*TJ*Uyy - . . .

4*k"5*s"4*NH*U*Uyy - e p “2*k»m“2*D2 - . . .

2*k‘ 3 * s ‘ 2*NH*D2 + 10*ep‘ 2*k‘ 3*m‘ 2 * s ‘ 2*NH*U‘ 2*D2 + . . .

10*k~5*s‘ 4*NH*U‘ 2*D2;

A4 = 5*ep“4*k“2*m"4*U + 10*ep"2*k“4*m~2*s"2*NH*U + . . .

5*k"6*s~4*NH*U - e p “2*k"2*m"2*s"2*NH*fy - . . .

k “4*s'4*N H *fy + e p “2*k~2*m*2*s"2*NH*Uyy +

k ‘ 4*s~4*NH*Uyy - 5*ep"2*k"2*m~2*s"2*NH*U*D2 - . . .

5*k"4*s"4*NII*U*D2;

A5 = -ep*4*k*m *4»I - 2*ep"2*k"3*m “2 * s“2*NH*I - . . .

k ‘ 5 * s ‘ 4*NH*I + e p “2*k*m‘ 2 * s ‘ 2*NH*D2 + k "3 * s ‘ 4*NH*D2;

where “ep” represents th e param eter (Section 5.6). C oarse guesses were m ade solving

th e full system w ith 100 in terio r grid poin ts which were la te r refined using the sparse

techniques explained in A ppendix B. T he m axim um num ber of grid poin ts employed was

10000, which satisfied th e k inetic and po ten tia l energy balances:

^ ( “ ? + v\ + ™\)xz = - = (P\wi)xs - ( u i t i i ) „ u0y - i ,

2N 2p2 dt ^ xz = f


Appendix C. The Boussinesq numerical eigenvalue problem

to within 0.01%.

147


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stability of an equatorial jet and the formation of ... · 1.1 figures 1.1 to 1.3 illustrate the...

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