atmospheric waves chapter 6 - university of california, davis

Atmospheric Waves Chapter 6

Paul A. Ullrich

[email protected]

Part 5: Mixed Rossby-Gravity Waves

Trapped waves decay exponentially rapidly away from a boundary (such as the equator or a coastline). Trapped waves include: •  Equatorial (and coastal) Kelvin waves •  Equatorial Rossby (ER) waves •  Mixed Rossby-Gravity (MGR) waves at the equator •  Equatorial inertia-gravity waves

Trapped Waves

Paul Ullrich Atmospheric Waves March 2014

•  Shallow water equations, no topography

•  Coriolis parameter at the equator is approximated by equatorial β-plane:

Equatorial Rossby and Mixed Rossby-Gravity Waves

f ⇡ �y ⇡✓2⌦

a

◆y

Du

Dt

� �yv +@�

@x

= 0

Dv

Dt

+ �yu+@�

@y

= 0

D�

Dt

+ �

✓@u

@x

+@v

@y

◆= 0 � = ghwith


Linearize shallow water equations about a state at rest with mean height H


Seek wave solutions of the form (allow amplitudes to vary in y):

u = u0, v = v0, h = H + h0

@u0

@t� �yv0 +

@�0

@x= 0 (1)

@v0

@t+ �yu0 +

@�0

@y= 0 (2)

@�0

@t+ gH

✓@u0

@x+

@v0

@y

◆= 0 (3) �0 = gh0with

(u

0, v

0,�

0) =

⇣u(y), v(y),

ˆ

�(y)

⌘exp(i(kx� ⌫t))


Yields the system


�i⌫u� �yv + ik� = 0 (1)

�i⌫v + �yu+@�

@y= 0 (2)

�i⌫�+ gH

✓iku+

@v

@y

◆= 0 (3)

u = i�y

⌫v +

k

⌫� (4)Solve (1) for :

u



Plug (4) into (2) and (3), rearrange terms

Solve (6) for : �

��2y2 � ⌫2

�v � ik�y�� i⌫

@�

@y= 0 (5)

�⌫2 � gHk2

��+ i⌫gH

✓@v

@y� k

⌫�yv

◆= 0 (6)


� = � i⌫gH

(⌫2 � gHk2)

✓@v

@y� k

⌫�yv

◆(7)

Plug (7) into (5) and rearrange terms. Yields second-order differential equation for amplitude function


More complicated equation than before since the coefficient in square brackets is not constant (depends on y) Before discussing the solution in detail, let’s first look at two asymptotic limits when either H→ ∞ or β=0

@2v

@y2+

✓⌫2

gH� k2 � k

⌫�

◆� �2y2

gH

�v = 0 (8)


Asymptotic limit H→∞


Means that the motion is non-divergent, consequence of the continuity equation (3):

Equation (8) becomes

✓@u0

@x+

@v0

@y

◆= � 1

gH

@�0

@t! 0 H ! 1

@2v

@y2+

�k2 � k

⌫�

�v = 0 (9)

for


Solutions for the amplitude function in (9) exists of the form: Provided that the frequency ν satisfies the Rossby wave dispersion relationship: It shows that for non-divergent barotropic flow, equatorial dynamics are in no way special Earth’s rotation enters in form of β, not f: Rossby wave




@2v

@y2+

�k2 � k

⌫�

�v = 0 (9)

v

v = v0 exp(i`y)

⌫ =��k

k2 + `2


All influence of rotation is eliminated. Due to linearity, solutions of (9) exists of the form

Equation (8) reduces to the shallow water gravity model Non-trivial solutions if frequency ν satisfies

Pure gravity wave response (eastward and westward)

Asymptotic limit β=0


v = v0 exp(i`y)

@2v

@y2+

⌫2

gH� k2

�v = 0 (10)

⌫ = ±pgH(k2 + `2)


In the general case with we seek solutions for the meridional distribution of with boundary condition


@2v

@y2+

✓⌫2

gH� k2 � k

⌫�

◆� �2y2

gH

�v = 0 (8)

v

v ! 0 |y| ! 1for

For small y the coefficient in square brackets is positive and solutions oscillate in y. For large y the coefficient is negative, solutions either grow or decay.

Only the decaying solution satisfies boundary condition.


Matsuno (1966) showed that decaying solutions only exist if the constant part of the coefficient in square brackets satisfies the relationship


pgH

�

✓⌫2

gH� k2 � k

⌫�

◆= 2n+ 1 n = 0, 1, 2, . . .

This is a cubic dispersion relation for frequency ν. It determines the allowed frequencies ν of equatorially trapped waves for the zonal wavenumber k and meridional mode number n.


Then the solution has the form Where v0 is a constant and designates the nth Hermite polynomial. The first few Hermite polynomials have the form Index n corresponds to the number of N-S roots in v.

In (8) replace y by the nondimensional meridional coordinate


⇠ =

"✓�pgH

◆1/2

y

#

v(⇠) = v0Hn(⇠) exp

✓�⇠2

2

◆

Hn(⇠)

H0(⇠) = 1, H1(⇠) = 2⇠, H2(⇠) = 4⇠2 � 2


Meridional (N-S) structure of the amplitude function:

N

S

0

n = 0 n = 1 n = 2

ξ )(v λ

n = 5 n = 10

Hermite Polynomials

v(⇠) = v0Hn(⇠) exp

✓�⇠2

2

◆n = 0, 1, 2, . . .

Hn(⇠) = (�1)

nexp(⇠2)

dn

d⇠n�exp

�⇠�2

��Hermite polynomials:

v(⇠)


In general, the three ν solutions of can be interpreted as an equatorially trapped

1)  Eastward moving inertia-gravity wave 2)  Westward moving inertia-gravity wave 3)  Westward moving equatorial Rossby wave

However, the case n = 0 must be treated separately: For n = 0 the meridional velocity perturbation has a Gaussian distribution centered at the equator


pgH

�

✓⌫2

gH� k2 � k

⌫�

◆= 2n+ 1 n = 0, 1, 2, . . .


For n = 0 (special case) the dispersion relation factors as The root (westward propagating gravity wave) is not permitted, violates an earlier assumptions when eliminating The two other roots are


✓⌫pgH

� k � �

⌫

◆✓⌫pgH

+ k

◆= 0

⌫ = �kp

gH�

⌫ = kp

gH

"1

2±

s

1 +4�

k2pgH

#


The positive root corresponds to an eastward moving equatorial inertia-gravity wave

The negative root corresponds to a westward moving wave, which resembles •  An equatorial (westward moving) inertia-gravity (IG)

wave for long zonal scale (k → 0)

•  An equatorial Rossby (ER) wave for zonal scales characteristic of synoptic-scale disturbances

This mode (n=0, westward moving) is therefore called Mixed Rossby-Gravity (MRG) wave.



H L

L H

Plane view of horizontal velocity and height perturbations in the (n=0) Mixed Rossby-Gravity (MRG) wave, propagates westward

Equator

Equatorial Mixed Rossby-Gravity Waves


atmospheric waves chapter 6 - university of california, davis

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