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Finite-amplitude Lagrangian-mean wave activity diagnostics1

applied to the baroclinic eddy life-cycle2

Abraham Solomon ∗

Center for Ocean-Land-Atmosphere Studies, Calverton, Maryland

3

Gang Chen

Cornell University, Ithaca, New York

4

Jian Lu

Center for Ocean-Land-Atmosphere Studies, Calverton, Maryland

5

∗Corresponding author address: Abraham Solomon, Center for Ocean-Land-Atmosphere Studies, 4041

Powder Mill Road, Suite 302, Calverton, MD 20705-3106.

E-mail: solomon@cola.iges.org

1

ABSTRACT6

Lagrangian-mean wave activity diagnostics are applied to the non-linear baroclinic eddy7

life-cycle in a simple general circulation model of the atmosphere. The growth of these8

instabilities through baroclinic conversion of potential temperature gradients and the sub-9

sequent barotropic decay can exhibit two distinct life-cycles. One life-cycle results in equa-10

torward propagation of the growing eddy, anti-cyclonic wave-breaking and a poleward shift11

of the mean jet. The second life-cycle is distinguished by limited equatorward propagation12

and cyclonic wave-breaking on the poleward flank of the jet. Using a conservative finite-13

amplitude, Lagrangian-mean wave activity (negative pseudomomentum) to quantify wave14

growth and propagation, reveals more details about the life-cycles than could be discerned15

from eddy-kinetic energy (EKE) or other Eulerian metrics. It is shown that the distribution16

of pseudomomentum relative to the latitude of the axis of the jet can be used to provide17

a clear distinction between the two life-cycles at an early stage in their development and18

hence a prediction for the subsequent shift of the jet. This suggests that the distribution of19

pseudomomentum may provide some predictability for the atmospheric annular modes.20

1

1. Introduction21

The breaking of atmospheric waves has long been recognized as an important and com-22

plex process, crucial to the advancement in our understanding of the short term weather23

forecast, seasonal and intra-seasonal predictability and the general circulation of the at-24

mosphere (McIntyre and Palmer 1983). Simply stated, atmospheric wave-breaking is the25

irreversible deformation of material surfaces. In the upper troposphere, waves radiated by26

occluding tropospheric depressions, can break as they propagate upwards and equatorwards,27

redistributing angular momentum and heat. Many papers have been written about this28

phenomenon, employing a wide variety of diagnostics, modeling efforts and insightful obser-29

vations. Recent advances in the Lagrangian theory of wave-mean flow interaction Nakamura30

and Solomon (2010) provide a new perspective on this classic problem. Using diagnostics31

based on the distribution of material surfaces, wave-breaking can be more readily quantified.32

When a baroclinic jet becomes unstable its subsequent evolution may follow two very33

distinct paths, first noted by Simmons and Hoskins (1980) and subsequently dubbed Life-34

Cycle 1 (LC1) and Life-Cycle 2 (LC2) in Thorncroft et al. (1993). The dichotomy between the35

life-cycles has profound consequences for the evolution of the zonal mean wind. One of the36

most remarkable things about this instability is that identical perturbations, growing on two37

nearly indistinguishable, zonally symmetric background states, may evolve along these very38

distinct paths (Thorncroft et al. 1993). Hartmann and Zuercher (1998) demonstrated that a39

slight change in background shear could induce an abrupt transition from LC1 to LC2. This40

investigation will examine the results of a set of experiments, which explore a parameter space41

spanning this transition, using the combined diagnostic tools of a Lagrangian-mean, finite-42

amplitude wave activity and the Eliassen-Palm (EP) flux. Previous studies have either used43

EP-flux and linear wave theory (Hartmann and Zuercher 1998; Limpasuvan and Hartmann44

2000), or finite-amplitude diagnostics whose flux cannot be directly related to the tendency45

of the mean flow (Magnusdottir and Haynes 1996; Esler and Haynes 1999). Because this46

new Lagrangian framework is compatible with the familiar transformed Eulerian-mean, it47

2

provides a simple way to augment a conventional analysis of wave-mean flow interaction.48

The life-cycles have been characterized in various different ways. Simmons and Hoskins49

(1980) recognized that LC2 exhibits much slower decay of eddy kinetic energy than LC1.50

This difference in persistence was also noted by Thorncroft et al. (1993) when looking at51

wave activity. There is also a pronounced difference in the direction of wave propagation be-52

tween the life-cycles. Hartmann and Zuercher (1998) showed that the enhanced equatorward53

propagation exhibited in LC1 can be attributed to differences in the refractive indices asso-54

ciated with the zonal mean background state. Magnusdottir and Haynes (1996) employed55

a finite-amplitude wave activity and noted that the meridional excursion exhibited during56

LC1 was partly non-linear advection and not completely a wave propagation phenomenon.57

These differences in the propagation of waves and the dissipation of wave activity result in a58

poleward shift of the jet in LC1 relative to LC2, as noted by many authors. This jet shift is59

associated with a meridional dipole in angular momentum, which has a barotropic structure60

and projects onto the atmospheric annular mode (Limpasuvan and Hartmann 2000). This61

paper will complement these observations with a demonstration that such a dipole is exhib-62

ited in the distribution of angular pseudomomentum as well, a result which is not evident63

from the distribution of EKE.64

The annular modes represent the primary pattern of atmospheric variability in the ex-65

tratropics, whose phase has major influences on regional climate. There is no consensus on66

the fundamental, underlying cause for the variability of these pressure and wind patterns,67

however improved predictability would be invaluable for medium-range weather forecasting68

and projections of climate change. Several authors have drawn a connection between wave-69

breaking and the variability of the annular modes. Esler and Haynes (1999) constructed70

a wave-breaking index based on the meridional flux of wave activity and used it to relate71

variability of the waves to the state of the jet. Riviere and Orlanski (2007) relate life-cycles72

and wave-breaking to the structure of the storm-track and the North Atlantic Oscillation, a73

pattern closely related to the annular mode. Another recent study by Strong and Magnus-74

3

dottir (2008) also developed an index of wave-breaking and showed that it provided some75

skill for predicting the phase of the annular mode. In this paper, a new index based on76

the distribution of wave activity is introduced. This index is shown to clearly distinguish77

the two life-cycles across a suite of experiments, without the need to calculate any fluxes of78

momentum.79

The next section reviews the diagnostics used to evaluate the experiments in this study.80

The third section describes the simplified atmospheric model employed to carry out the81

simulations. The fourth section will detail one pair of experiments which exemplify this82

transition. The fifth section explores the parameter space spanned by zonal wavenumber83

and barotropic shear in order to determine any systematic variations in the life-cycle and84

develop a method for objectively distinguishing them. A discussion of the predictability of jet85

shifts using Lagrangian-mean diagnostics and comparison with more conventional Eulerian86

alternatives follows.87

2. Diagnostics88

This study will employ the finite-amplitude Lagrangian-mean wave activity (negative89

pseudomomentum) introduced in Nakamura and Solomon (2010), which employs the poten-90

tial vorticity equivalent latitude (φeq) as a meridional coordinate (Butchart and Remsberg91

1986). The wave activity at time t is defined as92

A(φ, z, t) =1

2πa cosφ

∫Q(φeq)≤q(φ′)

qdS ′ −∫

φ≤φ′

qdS ′

(1)

where q is the potential vorticity, the area element dS ′ = a2 cosφ′dλdφ′, a = 6, 378km the93

radius of the Earth and Q(φeq) is the Lagrangian-mean potential vorticity. The equivalent94

latitude (φeq) is defined such that the area covered by the two integrals are equal. That is95

4

to say96 ∫Q(φeq)≤q(φ′)

dS ′ =

∫φ≤φ′

dS ′, (2)

which specifies the value of Q as a function of φeq. The quasi-geostrophic (QG) potential97

vorticity (PV) is98

q = f + k · (∇× v) +f

ρ0

∂

∂z

(ρ0(θ − θ0)∂θ0/∂z

)(3)

where f is the Coriolis parameter, k the unit upward pointing normal vector, the geostrophic99

winds v, θ(λ, φ, z, t) is potential temperature, θ0(z, t) is the layer mean potential temperature100

and ρ0 ∝ exp(−z/H).101

Consider the conservation of PV, using the non-divergent, geostrophic wind v102

∂q

∂t= −v · ∇q = −∇ · (vq). (4)

If q is conserved, the first term in the definition of A (Eq. 1) is constant following the motion103

of a parcel of air, and the tendency of wave activity is due only to the flux of PV across the104

fixed, bounding equivalent latitude105

∂

∂t(A cosφ) = − 1

2πa

∫φ≤φ′

∂q

∂tdS = − cosφvq (5)

here bars denote zonal means and tildes denote deviations from zonal means. Assuming that106

the waves are geostrophic and ignoring the divergence associated with the gradient in f , it107

can be shown that108

cosφvq =1

ρ0∇ · F, (6)

where F is the QG EP-flux109

F = (F φ, F z) = ρ cosφ[−uv, f vθ/(∂θ/∂z)] (7)

and ∇ · F = (a cosφ)−1∂(cosφF φ)/∂φ+ ∂F z/∂z. Thus Eq. 5 is approximately110

∂

∂t(A cosφ) = − 1

ρ0∇ · F (8)

5

as demonstrated in Nakamura and Solomon (2010). Furthermore, the QG transformed111

Eulerian-mean zonal momentum equation on the sphere as derived in Pfeffer (1987) is112

∂

∂t(u cosφ) =

1

ρ0∇ · F + fvR cosφ. (9)

So that the Coriolis torque associated with the residual circulation, vR(φ, z, t), balances the113

tendencies of wave activity and the zonal mean wind114

∂

∂t(u+ A) = fvR. (10)

In practice the three terms in Eq. 10 tend to be of comparable magnitude, the nature of115

the balance being determined by the aspect ratio of the wave forcing (Pfeffer 1987). For116

tall, narrow forcing, in the absence of irreversible processes, the right hand side tends to be117

negligible, hence Eq. 10 is referred to as a non-acceleration theorem (Nakamura and Zhu118

2010a).119

3. Model120

The model used in this set of experiments is an atmosphere only model, which employs121

the GFDL spectral dynamical core on thirty vertical sigma levels specified as122

σk ≡ p(k)/ps = exp(γ(30− k)/30), for k = 0, . . . , 30 (11)

where p(k) is pressure on the kth level, ps the surface pressure and γ = log(2 × 10−2).123

There is no external heating or dissipation during the baroclinic eddy life-cycle, except for124

hyperdiffusion to damp the smallest scales. The initial condition is expressed in terms of125

pseudoheight, which is a linear function of log pressure, z = −H log(p/p0), where the scale126

height H = 7.5km and reference pressure p0 = 1e5Pa. The initial zonal wind is specified to127

resemble the experiments in Hartmann and Zuercher (1998)128

u(λ, φ, z) = u0(φ, z) + us(φ) + um(λ, φ, z) (12)

6

where u0 is a baroclinic jet, us a barotropic jet and um the velocity anomaly associated with129

the normal mode of zonal wavenumber m scaled to a standard amplitude for each case. The130

specified profile for the baroclinic jet is131

u0(φ, z) = U0 sin3(π sin2 φ)(z/zT exp(−z2/z2T − 1

2)), φ > 0 (13)

here the parameters are U0 = 45m s−1 and zT = 13km. The barotropic component of the132

initial wind field is133

us(φ) = Us

{exp

[−(φ− φe

∆φ

)2]− exp

[−(φ− φp

∆φ

)2]}

(14)

where φe = 20◦, φp = 50◦ and ∆φ = 12.5◦. The barotropic shear parameter Us is sys-134

tematically varied amongst the experiments in order to explore its effect on the evolution135

of the normal mode perturbation. The temperature and surface pressure are derived from136

geostrophic and hydrostatic balance such that the temperature is in thermal wind balance137

with the initial zonal wind, following the method described in Polvani and Esler (2007). The138

lower boundary of the model is specified as a geopotential surface Φ(φ, ps) = 0, which allows139

the surface pressure to vary in time. Geopotential height is calculated as140

Φ(φ, p) =

p0∫p

RTr(p′)

p′dp′ −

φ∫0

(af + u tanφ′)udφ′ (15)

where the reference temperature at the equator is141

Tr(p) = Ts +Γ0

(z−αT + z−α)1/α(16)

for Ts = 300K, Γ0 = −6.5K km−1 and α = 10.142

4. One Transition Case143

In this section one pair of experiments which clearly depict the transition from LC1 to144

LC2 will be examined in detail. The initial states for these two experiments are very similar.145

7

Both have a small zonal wavenumber 6 perturbation to the zonally symmetric background146

flow. The only difference between the two is the value chosen for the barotropic shear147

parameter Us. The LC1 case, which has slightly lower shear Us = 9.5m s−1, is associated148

with anti-cyclonic wave-breaking. Anti-cyclonic wave breaking refers to the NE-SW tilting149

of troughs on the equatorward flank of the jet, as illustrated with maps of Θ on PV surfaces150

in Thorncroft et al. (1993) (Fig. 7). Whereas, the LC2 case, in which Us = 10m s−1, exhibits151

so-called cyclonic wave breaking. Cyclonic breaking involves the cutting-off of nearly circular152

vortices on the poleward side of the jet, which become homogenized in their interiors and153

tend to be long-lived (Thorncroft et al. (1993) Fig. 9). As we see in Fig. 1 the two cases154

have nearly identical initial zonal mean wind distributions (left column), differing from each155

other by less than 1m s−1 within the domain (bottom-left). However, this subtle difference156

in initial conditions results in very different zonal mean wind fields on day 30 (center panel),157

at the end of the simulation. The evolution of the high shear case (LC2), is best described158

as a sharpening of the jet with a slight equatorward shift (top-right). The low shear case159

(LC1) results in a significant poleward shift of the jet, the jet also narrows in latitudinal160

extent and intensifies (middle-right). The difference between the two final states may be161

characterized as a dipole, with a node at 55◦, whose structure is fairly uniform below 15km162

(bottom-middle). These are basic features of the life-cycles, which have been discussed in163

previous papers and here will be explained in the context of wave activity.164

In the past decade a number of studies have investigated the vacillation of the extra-165

tropical eddy-driven jet as an equivalent barotropic process. Limpasuvan and Hartmann166

(2000) showed that the annular modes, the principal mode of variability in the extra-tropical167

circulation, exhibit largely barotropic structure in both the height field and zonal winds. This168

variability is manifest as a meridional dipole structure associated with tropospheric wave-169

breaking, with centers of action located between 30-40N and 50-60N. Subsequent work by170

Lorenz and Hartmann (2001) and Vallis et al. (2004) underscored the equivalent barotropic171

nature of the annular modes and the potential for understanding their variability with172

8

stochastic models driven by eddy forcing.173

To illustrate this perspective and draw a connection between these life-cycle experiments174

and the detailed analysis of observations and models undertaken by previous investigators,175

we consider the barotropic angular velocity in Fig. 2 (left panels). The barotropic angular176

velocity is simply a pressure weighted vertical average of the zonal wind177

[u](φ, t) cos(φ) =1

ps

ps∫0

u(p, φ, t) cos(φ)dp. (17)

Results from the LC2 experiment (top) show the slight equatorward shift of the mean west-178

erlies, with the development of zonal mean easterlies on the poleward flank. LC1 (middle)179

shows a pronounced poleward shift around day 12 and progressive strengthening and sharp-180

ening of the jet between days 15 and 20. The difference between the mean winds in these two181

life-cycles (bottom) exhibits a dipole structure reminiscent of the difference between extreme182

states of observed annular modes.183

Recent investigations into the annular modes have emphasized the role of wave-breaking.184

It now appears clear that the high phase of the annular mode is associated with anti-cyclonic185

breaking of synoptic scale waves (Benedict et al. 2004; Riviere and Orlanski 2007; Strong186

and Magnusdottir 2008). Employing the Lagrangian-mean wave activity to visualize the role187

of waves in these experiments provides a new perspective on the evolution of the circulation188

in this wave-breaking process. The barotropic wave activity [A](φeq, t) cos(φeq) is plotted in189

the right panels of Fig. 2. LC2 (top) and LC1 (middle) have similar distributions of angular190

pseudomomentum prior to day 15, with a localized maximum centered around 50N. LC1191

shows enhanced equatorward propagation between days 15 and 20, then steady decay of192

[A] at all latitudes as the simulation progresses. In contrast, LC2 exhibits a brief period of193

transience in wave activity around day 15, followed by a poleward shift of the maximum in [A]194

and steady increase until day 20. This new quasi-equilibrium state of wave activity persists195

for the duration of the experiment. These differences are comparable to the distinction196

between LC1 and LC2 made by Simmons and Hoskins (1980), which emphasized the fact197

9

that both cases attain similar maximum levels of eddy-kinetic energy, but in the LC2 case198

the EKE dissipates much more slowly. However, the meridional propagation of the eddies199

is not as clear from EKE, which exhibits only a single maximum that tends to follow the200

latitude of maximum winds (not shown). Using this Lagrangian-mean finite-amplitude wave201

activity, the meridional dipole in angular pseudomomentum mirroring that of velocity (lower-202

right), can be discerned in the difference between the two life-cycles. These opposing dipoles203

illustrate the exchange of pseudomomentum for angular momentum underlying this wave-204

mean flow interaction in a way that cannot be reproduced from linearized, Eulerian metrics205

that do not possess an exact non-acceleration theorem (Solomon and Nakamura 2012).206

We have seen that the distinction between the two cases is clearly depicted by the tem-207

poral evolution of wave activity. In Fig. 3 the time average distribution of wave activity is208

shown on the left. The two cases have similar distributions of mean wave activity, however209

it is apparent that the LC2 case has larger time averaged wave activity and the LC1 case210

has broader meridional extent. In the panels on the right, the temporal evolution of wave211

activity for the two cases are illustrated by concatenating thirty snapshots of wave activ-212

ity in the meridional plane. This space-time diagram illustrates that the growth of wave213

activity between days 6 and 12 is very comparable between the two cases, attaining nearly214

identical maximum values in the upper-troposphere. However, the subsequent evolution fol-215

lows extremely different paths. In the LC1 case (bottom-right) the waves begin to decay216

around day 12 and continue to diminish at all vertical levels as the simulation proceeds. In217

contrast, the LC2 case (top-right) shows some transience in the mid-troposphere around day218

12, but then the tropospheric waves grow again until they have equilibrated around day 24.219

Additionally, the secondary maximum of wave activity in the stratosphere, which emerges220

in both simulations around day 12, grows and descends in LC2 while it has all but vanished221

by day 20 in LC1. This figure indicates that the crucial difference in the evolution of these222

simulations must manifest itself between days 10 and 15, so in this interval we will examine223

the dynamics more closely, using the combined diagnostic power of A and the EP-flux.224

10

Fig. 4 shows five snapshots of wave activity between days 6 and 18 of each experiment.225

In the top row A is plotted in the meridional plane for the LC2 case with the EP-flux226

vectors overlain. Plotted beneath each snapshot is the near surface potential temperature227

gradient, which represents the wave source. On day 6 the waves are still negligibly small, but228

a region of eddy heat flux between 40N and 60N coincides with the latitudes of maximum229

magnitude temperature gradient. By day 9 a measurable density of pseudomomentum has230

accumulated between 5km and 10km, which continues to expand on day 12. The temperature231

gradient has developed a bimodal structure although upward flux persists throughout the232

extratropics with divergence aloft. On day 15 the maximum in wave activity has shifted233

toward the pole and the EP-flux is now poleward everywhere except on the equatorward234

flank of the secondary maximum in wave activity developing between 20km and 25km. Day235

18 depicts very strong wave activity north of 40N in both the troposphere and stratosphere.236

The orientation of this locus for wave activity, directly above a region of enhanced meridional237

temperature gradient and upward EP-flux, contributes to the persistence of waves in the LC2238

case.239

Corresponding snapshots for the LC1 case are plotted in the middle panels and the240

difference between the two cases is displayed in the lower panels. Until day 12 there is241

little difference visible between the two cases in either wave activity or surface temperature242

gradients. On day 12, although the distributions of wave activity are still nearly identical,243

there is a greater heat flux at high latitudes in the LC2 case. By day 15 a perceptible244

difference in the density of pseudomomentum in the troposphere near 60N has emerged.245

Furthermore, the EP-flux on day 15 in the LC1 case is everywhere equatorward, opposite246

to that of the LC2 case. On day 18 the wave activity in the LC1 case has diminished247

dramatically, strong equatorward fluxes below 15km continue to erode the remaining wave248

activity at high latitudes. At this point the difference in wave activity density between the249

two cases has also become visible in the stratosphere and a sizable discrepancy in surface250

temperature gradients has emerged.251

11

On day 15 the two life-cycles exhibit substantial differences in the distribution and ten-252

dency of momentum, which can be estimated by a variety of methods that we compare253

here. In the top row of Fig. 5 the tendency of wave activity is estimated by taking the254

difference between its density on days 16 and 14. Values from the LC2 experiment (left-top)255

show a dipole structure in the vertical with waves decaying below 15km and growing above,256

consistent with the vertical propagation of waves at this stage of the life-cycle discussed by257

Thorncroft et al. (1993). The tendency of wave activity in the LC1 case (center-top) displays258

largely similar structure to the high shear case, however the magnitudes are very different.259

To illustrate more clearly the discrepancy in their structure, the difference between their260

tendencies is plotted on the right. We find an increasing deficit of wave activity in the LC1261

case, relative to the LC2 case, throughout the high latitudes of the free atmosphere. The262

second row of Fig. 5 shows the instantaneous zonal mean poleward flux of PV, which in263

theory should be the negative of the wave activity tendency. The agreement in the spatial264

structure of these two quantities reassures us that our understanding of the quantity A is265

valid. In addition to these basic confirmations, we can also see a meridional dipole structure266

in the difference between these instantaneous tendencies on day 15.267

In the third row of Fig. 5 we see the EP-flux divergence, which agrees well with the268

PV flux. Overlain are the EP-flux vectors, which help to interpret their divergence. In the269

LC2 case, poleward flux at 35N diverges strongly both upward and downward suggesting270

reflection of waves near 10km. In the LC1 case, the pattern is reversed with equatorward271

flux at 60N, diverging in the vertical again around 10km. It is worth noting here that272

these three measures of the tendency of momentum rely on very different assumptions and273

approximations, yet they exhibit reasonable, quantitative agreement. In particular, the fact274

that the quasi-geostrophic EP-flux convergence matches the tendency of A, underscores the275

point that this Lagrangian framework is consistent with the familiar, flux based, Eulerian,276

linearized diagnosis. The lower panels show the Coriolis torque associated with the residual277

circulation. Note that this term is the same order of magnitude as the other tendency terms278

12

and typically opposes them. Both cases indicate poleward circulation at upper levels and279

an equatorward branch near the surface. The difference between the two shows a stronger280

circulation near the surface north of 40N in the LC1 case, which contributes to the difference281

in surface potential temperature gradients.282

Previous investigations have employed linear wave theory and refractive indices to dis-283

tinguish the evolution of the two life-cycles (e.g. Hartmann and Zuercher (1998)). Planetary284

waves tend to propagate toward regions of high refractive index, n2, and can be reflected285

where the index vanishes (Karoly and Hoskins 1982)286

n2(φ, z, t) =∂q(φ, z)/∂φ

a(u− cp)+N2m2 (18)

where the Brunt-Vaisala frequency N2 = (g/θ0)∂θ0/∂z and cp is the phase velocity of the287

wave. The dominant term in the refractive index is proportional to the meridional gradient288

of zonal-mean PV, so in regions where the PV gradient vanishes or becomes negative, wave289

propagation is suppressed according to linear theory. Great insights into wave-mean flow290

interaction have been developed within the framework of refraction and critical layer theory291

(e.g. Killworth and McIntyre (1985)), however linear theory cannot address all aspects of292

the baroclinic life-cycle once wave-breaking and mixing begin to significantly modify the293

mean state (Hartmann and Zuercher 1998). To investigate the changing PV gradients in294

the evolution of the life-cycles we will consider both the zonal-mean and Lagrangian-mean295

PV in Fig. 6. By differentiating Eq. 1 twice with respect to latitude (enforcing φeq = φ),296

it can be shown that the two mean-PV gradients differ due to a contribution from the wave297

activity298

∂q

∂µ=∂Q

∂µ+∂2A cos(φ)

∂µ2(19)

where µ = sin(φ), as derived in Nakamura and Zhu (2010a). The zonal-mean PV gradient is299

not a conserved quantity, because of the contribution from the curvature of the wave activity300

profile it can be modified dramatically in the presence of waves. The Lagrangian-mean PV301

is conserved for adiabatic, frictionless flow, so its gradient tends to evolve more slowly and302

systematically, responding only to irreversible mixing and other non-conservative processes.303

13

Fig. 6 shows snapshots of the meridional gradients of the two mean PV fields for the initial304

and final day of each experiment. On day 1 (left) the fields all appear indistinguishable, since305

the initial conditions for each experiment are very similar and essentially zonally symmetric.306

We see a localized maximum of the PV gradient near 45N at approximately 8km, close to307

the axis of jet. This distribution will tend to focus waves into the jet, making the jet a strong308

waveguide for the growing eddies. The final distribution of PV in each case shows reduction309

in the meridional extent of the region of strong gradients, associated with sharpening of310

the jets. The Lagrangian-mean PV increases monotonically with equivalent latitude by311

construction and indicates sharper gradients than the zonal-mean. The Eulerian, zonal-312

mean PV gradients are reversed on the flanks of the jet by the end of the simulation and313

have somewhat smaller maximum value.314

The set of panels farthest right in Fig. 6 show the difference between the initial and315

final distributions of PV gradients (day 30 - day 1). Here we see that the two life-cycles316

result in oppositely signed dipole anomalies in the upper troposphere, with a node near the317

initial jet axis. These patterns reflect the fact that the region of enhanced PV gradient is318

closely associated with the jet, which shifts oppositely in latitude over the course of the two319

life-cycles. The two averages portray fairly consistent changes, although the dipole patterns320

are narrower in equivalent latitude, due to the sharper final Qφeq distributions. The largest321

discrepancies are seen at high latitudes in the LC2 experiment, where waves continue to322

contribute to the zonal-mean PV gradients on day 30. This dipole pattern in PV gradient323

depicts a change in the curvature of the mean PV near the initial axis of the jet. An alternate324

metric for the shift of the jet can then be calculated both from the Lagrangian and zonal325

means.326

∆PVφφ(t) =1

a2

10∫5

∂2

∂φ2[PV (z, t)− PV (z, 0)] |45Ndz (20)

The panel at the bottom of Fig. 6 shows ∆PVφφ, the change in the mean tropospheric327

curvature of PV at the initial jet axis as a function of time, for both experiments. The328

Lagrangian-mean PV (solid curves) indicates enhanced curvature in both cases until day 13,329

14

when the two life-cycles diverge from one another. After day 13 the PV gradient on the pole-330

ward flank of the jet begin to decrease in LC2, ultimately resulting in the observed dipole and331

decreased curvature relative to the initial state. In contrast, the LC1 change in PV curvature332

remains positive, rising to its peak value after day 22. The corresponding diagnosis using q333

is plotted in dashed curves, indicating similar conclusions about the temporal development334

but smaller signal in the LC1 case. In the next section we will employ ∆PVφφ to characterize335

the jet shift resulting from the two life-cycles and utilize the association between maximum336

Q(φeq) gradients and the jet axis to define a wave index.337

5. The effects of increasing barotropic shear and vary-338

ing zonal wavenumber339

In Fig. 7 the time evolution of the global mean wave activity is plotted for three selected340

wavenumbers at a range of barotropic shears. Within this range of values for Us (0−12m s−1)341

we see a clear progression in the behavior at these wavenumbers (m = 5, 6, 7). These plots342

are similar to those of Hartmann and Zuercher (1998), who plotted eddy-kinetic energy343

for a wavenumber 6 disturbance growing in a similar background flow and calculated the344

relevant energy conversion terms in order to diagnose the evolution of the flow. The initial345

stage of exponential growth is due to baroclinic conversion through eddy-heat flux and is346

very similar for both life-cycles. The subsequent barotropic decay phase characterized by347

large eddy-momentum fluxes is very different in the two life-cycles. LC2 exhibits much less348

barotropic conversion during this second phase of the life-cycle (Hartmann and Zuercher349

1998).350

On the left, timeseries for wavenumber five, clearly depict the conventional LC1 progres-351

sion. The initial period of exponential growth is clear, followed by a wave breaking instability,352

which results in a dissipation of wave activity until a new quasi-steady state is achieved. This353

dissipation of wave activity is associated with an exchange of pseudomomentum for angular354

15

momentum, resulting in the observed changes in the zonal mean wind (Fig. 1). The growth355

of the instability is delayed with increasing shear (redder colored curves indicate higher Us).356

Also the maximum global mean wave activity attained generally increases with greater shear.357

The tendency of decreasing growth rate with rising shear, observed at m = 5 persists for358

all higher wavenumbers examined in this study. The maximum global mean wave activity359

attained is a complicated story, which will not be addressed in this study.360

The cases examined in detail were wavenumber six perturbations, which are plotted in the361

middle. Here we see the transition from LC1 to LC2 as the Us = 9.5m s−1(yellow) diverges362

from the Us = 10m s−1(orange) after day 13. The LC2 cases, initialized with Us ≥ 10m s−1,363

exhibit less dissipation of wave activity after the peak of the life-cycle than the experiments364

with slightly lower barotropic shear. Even the very low shear cases (bluer colors), which365

attain only half the peak amplitude of the LC2 cases, exhibit greater dissipation in the366

immediate aftermath of their exponential growth phase. Finally, the m = 7 case is plotted367

in the right panel. The two experiments with lowest initial barotropic shear still exhibit368

LC1. However, all m = 7 cases with Us ≥ 4m s−1 develop as LC2. These m = 7 cases369

attain lower maximum amplitudes than the corresponding longer wavelength perturbations370

and peak earlier, with their exponential growth phase ending prior to day 10 for almost371

all cases. The LC2 experiments can actually exhibit increasing pseudomomentum after the372

period of normal mode growth, as seen in these m = 7 timeseries, although that growth373

is not systematic nor a large fraction of the total wave activity. All simulations for higher374

wavenumbers exhibited the LC2, with smaller peak values of A and little or no dissipation.375

In order to objectively distinguish the two life-cycles, for every one of these simulated376

cases, we cannot simply rely on the magnitude of wave activity alone. The maximum am-377

plitude of the waves decreases dramatically with increasing wavenumber, so the temporal378

and spatial-domain averaged wave activity convolves the issues of persistence and peak am-379

plitude. The fundamental distinction between the LC1 and LC2 cases is the extent of380

wave-breaking on the equatorward flank of the jet, as illustrated by Thorncroft et al. (1993).381

16

In LC1 filaments of PV are sheared out to the south of the jet where they dissipate over time.382

The LC2 case is distinguished by a general absence of this equatorward wave propagation383

and subsequent breaking. So we can define an index which quantifies the fraction of wave384

activity found on the equatorward side of the jet axis385

WI(t) =

zT∫0

J(z,t)∫0

A(φeq, z, t)dφeqdz∫∫Adφeqdz

(21)

where the axis of the jet is defined as J(z, t) = maxφeq{∂Q/∂φeq}, the equivalent latitude of386

maximum PV gradient. This index is similar to a quantity considered by Magnusdottir and387

Haynes (1996), however they employed a fixed latitude to distinguish the two flanks of the388

jet.389

In Fig. 8 both WI and the domain averaged wave activity are plotted for each case ex-390

amined. In the left column we see the time average of each diagnostic, plotted with a distinct391

color for each case. The mean wave activity (top-left) does not vary consistently for all cases,392

though a general tendency toward lower values with increasing wavenumber is apparent. A393

related parameter space, spanned by eddy kinetic energy and frequency, was explored in394

Riviere and Orlanski (2007), who found that long waves break only anti-cyclonically, while395

shorter wavelengths may follow either life-cycle. WI (bottom-left) clearly distinguishes the396

two life-cycles, with LC1 in red hues and LC2 in bluer shades. An oblique separatrix be-397

tween the two life-cycles can now be discerned. This parameter-space is partitioned into an398

LC1 region spanned by low zonal wavenumbers and low values of barotropic shear, and an399

LC2 region representing the high shear or high wavenumber cases. Transitions from LC1 to400

LC2 with increasing shear are seen for both m = 6 and m = 7, with higher wavenumbers401

exhibiting only LC2. It is possible that m = 5 would exhibit a transition to LC2 if even402

higher values of initial barotropic shear were applied, but no such behavior was observed for403

the shear range examined.404

The panels on the right show timeseries of these two metrics for each experiment. Here the405

LC2 cases have been highlighted in colors by wavenumber and the LC1 cases are plotted in406

17

black. In the top panel the global mean wave activity illustrates that the LC2 cases cluster407

tightly according to wavenumber and clearly lack the strong dissipation seen in the LC1408

cases. However, the range of mean wave activities for the LC2 cases cannot be distinguished409

from that of the LC1 cases. WI (bottom) more clearly distinguishes the two cases, with410

essentially distinct ranges of values after day 10. The wavenumber 5 cases do not break411

until somewhat later, rising to values of WI ≥ 0.3 between days 10 and 15, but aside from412

those few cases the black curves are clearly separated from the the colored curves by day 10.413

For all the LC2 cases WI diminishes, nearly monotonically, throughout the course of the414

experiment, asymptotically approaching values below 0.2. Hence the tendency of WI during415

the eddy growth phase (days 6-10 or so depending on wavenumber and background shear)416

has opposite signs for the two life-cycles. This fact can be used to predict the subsequent417

evolution of the instability and the consequent changes in mean wind. Similar indices of418

wave-propagation and breaking have been constructed in Esler and Haynes (1999); Strong419

and Magnusdottir (2008), which proved to be useful predictors for the phase of the observed420

annular mode.421

6. Discussion422

The ultimate goal of this investigation was to provide a method for predicting the type of423

life-cycle resulting from a given initial condition. This goal was not attained and questions424

still remain about the critical factor leading to symmetry breaking in the evolution of this425

instability. However, as observed by many previous authors (e.g. Thorncroft et al. (1993);426

Hartmann and Zuercher (1998)), the life-cycles clearly distinguish themselves during the427

growth and propagation phase, prior to attaining their maximum amplitude and breaking.428

The fact that this occurs before any significant redistribution of angular momentum or shift429

of the jet implies that there is some predictive potential to be derived from the distribution430

of wave activity. The figure below illustrates the shift of the jet after the onset of wave-431

18

breaking, measured by ∆PVφφ, as a function of the tendency of WI a week or more earlier432

in the life-cycle. Here the tendency of WI is simply calculated as a finite difference, two days433

before the maximum in wave activity is reached. Because of the difference in growth rates434

between the long waves (m = 5, 6) and the shorter waves (m = 7, 8, 9), these characteristic435

metrics for the life-cycles had to be calculated during different intervals. For the short waves,436

which grow more rapidly, the tendency of WI was calculated on or prior to day 6 and the PV437

curvature on day 14. For the long waves WIt was calculated before day 12 and ∆PVφφ on438

day 18. Although the timing of the life-cycles differ with wavenumber, they all suggest about439

a week of lag between the propagation of waves relative to the jet axis and the ultimate shift440

of the jet.441

In Fig. 9, several clusters of points can be distinguished, the filled circles calculated using442

the Lagrangian diagnostics and open circles based on a Eulerian metric for comparison. The443

filled green circles, reflecting LC1 experiments, all plot in the first quadrant. This implies444

that the equatorward propagation of wave activity early in the life-cycle invariably precedes a445

poleward shift of the jet by about a week. The filled blue circles, LC2 experiments, generally446

plot in the third quadrant. This suggests that the negative tendency of WI is a strong447

indicator of a coming equatorward shift of the jet. The two marginal cases (points in the448

2nd and 4th quadrants) are m = 7, which fall somewhere between long and short baroclinic449

waves, so the timing of their life-cycles did not conform to these conventions, although their450

overall behavior still supports the broader conclusions. Plotted in black are the pair of451

m = 6, transition cases examined in detail. The two cases depict small magnitude and452

differing signs of WI tendency on day 11 and opposite shifts of the jet by day 18. The fact453

that the wave propagation is so weak in this pair of transition cases when compared with the454

tendency of WI for cases that are not as close to the abrupt transition between life-cycles,455

suggests that a threshold in wave-propagation distinguishes the two life-cycles.456

The predictive diagnostic WIt required several pieces of Lagrangian data. The distri-457

bution of wave activity in the equivalent latitude meridional plane is a Lagrangian-mean458

19

diagnostic, which is more robust than its Eulerian counterparts (eddy-kinetic energy, lin-459

ear zonal-mean pseudomomentum) (Solomon and Nakamura 2012). The jet latitude J(z, t)460

has been defined in terms of the maximum gradient of Lagrangian-mean PV, primarily for461

the practical purpose of determining the location of the jet in terms of equivalent latitude.462

Also, the eddy propagation phase was determined based on the timing of maximum of wave463

activity for each life-cycle. To determine how important this Lagrangian information is we464

considered an Eulerian diagnostic (open circles), plotted for comparison with WIt. This465

metric employs eddy-kinetic energy (EKE) in place of wave activity, jet latitude defined as466

the maximum of the zonal mean wind and timing based on the maximum of mean EKE467

during the life-cycle. Linear zonal-mean pseudomomentum is unsuitable for this type of468

analysis because of the vanishing q gradient, which makes it impossible to calculate domain469

averages since the linear pseudomomentum becomes infinite. From Fig. 9 we can see that470

the Eulerian diagnostic is not able to distinguish the two life-cycles, with the majority of471

LC2 cases exhibiting increasing EKE south of the latitude of maximum winds (open circles472

in the second quadrant). The timing of maximum EKE is fairly consistent with wave activ-473

ity, although A tends to develop more smoothly and shows less dissipation in the LC2 cases474

(not shown). The distribution of eddies relative to the maximum PV gradient seems to be475

a more reliable discriminant between the two life-cycles than the distribution with respect476

to the maximum winds. This supports the perspective that it is the mixing of PV by eddies477

and resultant modification of the PV gradient which is more fundamental to the dichotomy478

between life-cycles than the flux of momentum.479

7. Conclusions480

The use of Lagrangian-mean wave activity provides insights into the evolution of wave481

activity in these life-cycle experiments. The use of a conservative diagnostic allows easier482

quantification of the spatio-temporal evolution of wave activity density than conventional483

20

Eulerian diagnostics. Measures like eddy kinetic energy only provide robust information484

in the domain average. Lagrangian-mean finite-amplitude wave activity can illustrate the485

specific regions of the domain whose circulation is being modified by the presence of waves486

as well as reproduce prior conclusions about the temporal evolution of domain average pseu-487

domomentum. Domain average A clearly depicts the initial period of exponential growth488

common to both life-cycles and the subsequent dichotomy in trajectories, with dissipation in489

LC1 and persistence in LC2. The spatial distribution explains this difference, underscoring490

the equatorward breaking of waves in LC1 and the consequent deceleration on the south491

flank of the jet, resulting in the apparent poleward shift of the jet. In contrast LC2 evolu-492

tion involves little or no equatorward wave-breaking, hence no significant redistribution of493

angular momentum toward higher latitudes.494

The index of refraction has been used in previous studies to illustrate the role of critical495

lines and wave reflection in the evolution of life-cycle experiments Thorncroft et al. (1993);496

Hartmann and Zuercher (1998); Limpasuvan and Hartmann (2000). Such indices rely heavily497

on the gradient of zonal-mean potential vorticity, which provides the restoring force that498

permits Rossby-wave propagation. In the presence of finite-amplitude waves the gradients499

of PV can become obscured by zonal averaging, making it difficult to interpret the index500

of refraction and impossible to compute the linear pseudomomentum. In this study we501

considered the meridional gradient of Lagrangian-mean potential vorticity, which can only502

be modified through irreversible processes such as mixing. The Lagrangian-mean PV depicts503

the progressive evolution of a localized maximum in the PV gradient, which is associated504

with the curvature of the evolving jet. LC1 results in significant mixing on the equatorward505

flank of the jet, increasing the meridional PV gradient and sharpening the jet, whose mean506

latitude shifts toward the pole. Whereas, LC2 produces more mixing on the poleward flank507

of the jet, this also enhances the maximum PV gradient, concentrating it to the south of the508

original jet axis and diminishing the mean gradient to the north. This provides a context509

for thinking about baroclinic eddy life-cycles in terms mixing and PV staircases (Dritschel510

21

and McIntyre 2008; Nakamura and Zhu 2010b).511

Perhaps the most useful outcome of this study is a simple wave index, which utilizes512

both Lagrangian-mean wave activity and a Lagrangian perspective of jet location, thereby513

providing a clear distinction between the two life-cycles. We have demonstrated that, in a514

broad suite of simulations, the tendency of this index can be used to predict the subsequent515

evolution of the instability and hence the shift of the jet. The use of a Lagrangian perspective516

of jet axis location, via the maximum PV gradient in equivalent latitude, turned out to be517

a crucial piece of this construction and makes this diagnostic more applicable to the range518

of dynamics which might be encountered in observations than a fixed Eulerian prescription519

of jet latitude (e.g. Magnusdottir and Haynes (1996)). A future study will apply this wave520

index to observational reanalysis in order to advance our understanding of the role which521

wave-breaking plays in annular mode variability.522

Acknowledgments.523

Thanks to the anonymous reviewers whose feedback contributed to the revision of this524

paper. The lead author is supported by NSF grant: ATM-0947837. Dr. Gang Chen is525

supported by NSF grant: AGS-1064079. Dr. Jian Lu is supported by NSF grant: AGS-526

1064045.527

22

528

REFERENCES529

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oscillation. J. Atmos. Sci., 61, 121–144.531

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diagnostic for tracer transport on an isentropic surface. J. Atmos. Sci., 43, 1319–1339.533

Dritschel, D. G. and M. E. McIntyre, 2008: Multiple jets as pv staircases: the Phillips effect534

and the resilience of eddy-transport barriers. J. Atmos. Sci., 65, 855–874.535

Esler, J. G. and P. H. Haynes, 1999: Baroclinic wave breaking and the internal variability536

of the tropospheric circulation. J. Atmos. Sci., 56, 4014–4031.537

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shear. J. Atmos. Sci., 55 (3), 297–313.539

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variability. J. Climate, 13, 4414–4429.545

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wave life cycles. J. Atmos. Sci., 53, 2317–2353.549

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McIntyre, M. E. and T. N. Palmer, 1983: Breaking planetary waves in the stratosphere.550

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ments in the atmospheric general circulation. Part 1: Quasigeostophic theory and analysis.553

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25

List of Figures576

1 Zonal mean wind, u in units of m s−1, plotted in the northern hemisphere,577

meridional plane for the LC1 (middle row) and LC2 (top row) cases. The578

left column illustrates the initial state (Day 1) for each experiment and their579

difference (bottom row). The center column shows the final state (Day 30).580

The right column shows the net evolution of each case respectively. 28581

2 Barotropic angular velocity (left), [u] cos(φ) in units of m s−1, and barotropic582

angular pseudomomentum (right), [A] cos(φeq) in units of m s−1, plotted for583

both experiments: LC2 (top), LC1 (middle) and their difference (bottom).584

Note that the differences are plotted with distinct colorbars. 29585

3 The time mean wave activity (left column) is plotted in the northern hemi-586

sphere, meridional plane for the LC1 (bottom row) and LC2 (top row) cases.587

In the right panel all thirty snapshots of A are concatenated to illustrate its588

temporal evolution. 30589

4 The top row shows five snapshots of wave activity [m s−1], in the meridional590

plane, for the LC2 case on days 6, 9, 12, 15 and 18 of the simulation. The EP-591

flux vectors are overlain in white and the zonal mean potential temperature592

gradient [K deg−1] on the 723mb surface is plotted below in red. Center593

panels show the same quantities for the LC1 case. The bottom row shows the594

difference between the two cases (LC2-LC1) for each day. 31595

5 The tendency of wave activity on day 15 [ms−2] (same colorbar and units for596

each row), is plotted in the meridional plane (top), for the LC2 case (left), LC1597

case (center) and their difference (LC2-LC1) (right). The second row is the598

same as the top but for the zonal mean poleward flux of eddy PV. The third599

row shows the EP-flux convergence in colors with the flux vectors overlain in600

white. The bottom row show the Coriolis torque due to the residual circulation. 32601

26

6 Contours of the meridional gradients of PV, both the zonal-mean (lower) and602

Lagrangian-mean (upper) are displayed for day 1 (left), day 30 (middle) and603

their difference (day 30-day 1). For both snapshots of the LC1 experiment604

(left) and LC2 experiment (right), the PV gradients are all plotted with the605

same colorbar in units of m−1s−1. Below are timeseries of the ∆PVφφ, the606

change in the curvature of PV at the initial jet axis, Lagrangian-mean (solid)607

and zonal-mean (dashed), for the entire thirty days of each experiment. 33608

7 Each panel depicts the time evolution of the global mean wave activity for609

a single zonal wavenumber: left m=5, middle m=6, right m=7. The ten610

curves represent different values of the barotropic shear parameter Us between611

0− 12m s−1 (redder colors indicate higher shear). 34612

8 Top row shows the time-mean, global-mean wave activity with units of m s−1613

for each case on the left and the time evolution for all cases. The colored614

curves are associated with LC2 and the black curves are LC1. Bottom row615

same as the top, but for WI the wave-breaking index. 35616

9 The change in the PV curvature at the initial jet axis ∆PVφφ [m−2s−1] are plot-617

ted as a function of the tendency of the wave-breaking index during the eddy618

growth phase WIt [day−1]. Filled circles are calculated from the Lagrangian-619

mean PV, open circles reflect zonal-mean PV gradients. Green symbols are620

LC1 experiments, blue LC2 and the pair of cases examined in detail are plot-621

ted in black. 36622

27

LC2

z(km

)

Ubar DAY 1

20 40 60 80

5

10

15

20

25

Ubar DAY 30

20 40 60 80

5

10

15

20

25

DIFF(DAY 30 DAY 1)

20 40 60 80

5

10

15

20

25

20

0

20

40

LC1

z(km

)

20 40 60 80

5

10

15

20

25

20 40 60 80

5

10

15

20

25

LAT

20 40 60 80

5

10

15

20

25

20

0

20

40

LAT

DIF

F(LC

2LC

1)z(

km)

20 40 60 80

5

10

15

20

25

LAT

20 40 60 80

5

10

15

20

25

20

0

20

40

Fig. 1. Zonal mean wind, u in units of m s−1, plotted in the northern hemisphere, meridionalplane for the LC1 (middle row) and LC2 (top row) cases. The left column illustrates theinitial state (Day 1) for each experiment and their difference (bottom row). The centercolumn shows the final state (Day 30). The right column shows the net evolution of eachcase respectively.

28

DAYS

LC2

LAT

Angular Velocity Ucos( )

5 10 15 20 25 30

102030405060

20

0

20

40

DAYSEQ

. LAT

.

Angular Pseudomomentum Acos( eq)

5 10 15 20 25 30

10

20

30

40

50

60

0

10

20

30

40

DAYS

LC1

LAT

5 10 15 20 25 30

102030405060

20

0

20

40

DAYS

EQ. L

AT.

5 10 15 20 25 30

10

20

30

40

50

60

0

10

20

30

40

DAYS

DIF

F(LC

2LC

1)LA

T

5 10 15 20 25 30

102030405060

30

20

10

0

10

20

30

DAYS

EQ. L

AT.

5 10 15 20 25 30

10

20

30

40

50

60

30

20

10

0

10

20

30

Fig. 2. Barotropic angular velocity (left), [u] cos(φ) in units of m s−1, and barotropic angularpseudomomentum (right), [A] cos(φeq) in units of m s−1, plotted for both experiments: LC2(top), LC1 (middle) and their difference (bottom). Note that the differences are plottedwith distinct colorbars.

29

EQ. LAT.

LC2

z(km

)

TIME MEAN A

20 40 60

5

10

15

20

25

0

10

20

30

40

50

60

DAY

z(km

)

Space Time Plot of A

1 5 10 15 20 25 30

5

10

15

20

25

0

10

20

30

40

50

60

70

80

EQ. LAT.

LC1

z(km

)

20 40 60

5

10

15

20

25

0

10

20

30

40

50

60

DAY

z(km

)

1 5 10 15 20 25 30

5

10

15

20

25

0

10

20

30

40

50

60

70

80

Fig. 3. The time mean wave activity (left column) is plotted in the northern hemisphere,meridional plane for the LC1 (bottom row) and LC2 (top row) cases. In the right panel allthirty snapshots of A are concatenated to illustrate its temporal evolution.

30

LC2

z(km

)

DAY 6

5

10

15

20

25

20 40 602010

0

LAT

d/d

LC1

z(km

)

5

10

15

20

25

20 40 602010

0

LAT

d/d

DIF

F(LC

2LC

1)z(

km)

5

10

15

20

25

20 40 60404

LAT

d/d

DAY 9

20 40 60LAT

20 40 60LAT

20 40 60LAT

DAY 12

20 40 60LAT

20 40 60LAT

20 40 60LAT

DAY 15

20 40 60LAT

20 40 60LAT

20 40 60LAT

DAY 18

0

20

40

60

80

20 40 60LAT

0

20

40

60

80

20 40 60LAT

0

20

40

60

80

20 40 60LAT

Fig. 4. The top row shows five snapshots of wave activity [m s−1], in the meridional plane,for the LC2 case on days 6, 9, 12, 15 and 18 of the simulation. The EP-flux vectors areoverlain in white and the zonal mean potential temperature gradient [K deg−1] on the 723mbsurface is plotted below in red. Center panels show the same quantities for the LC1 case.The bottom row shows the difference between the two cases (LC2-LC1) for each day.

31

EQ. LAT.

dA/d

tz(

km)

LC2

20 40 605

10152025

LAT

vqz(

km)

20 40 605

10152025

LAT

EPFl

uxz(

km)

20 40 605

10152025

LAT

fvR

z(km

)

20 40 605

10152025

EQ. LAT.

LC1

20 40 605

10152025

LAT20 40 60

510152025

LAT20 40 60

510152025

LAT20 40 60

510152025

EQ. LAT.

DIFF(LC2 LC1)

20 40 605

10152025

2

0

2x 10 4

LAT

20 40 605

10152025

2

0

2x 10 4

LAT

20 40 605

10152025

2

0

2x 10 4

LAT.

20 40 605

10152025

2

0

2x 10 4

Fig. 5. The tendency of wave activity on day 15 [ms−2] (same colorbar and units for eachrow), is plotted in the meridional plane (top), for the LC2 case (left), LC1 case (center) andtheir difference (LC2-LC1) (right). The second row is the same as the top but for the zonalmean poleward flux of eddy PV. The third row shows the EP-flux convergence in colorswith the flux vectors overlain in white. The bottom row show the Coriolis torque due to theresidual circulation.

32

EQ. LAT.

Q z(km

)

DAY 1LC1

20 40 60

5

10

15

EQ. LAT.

DAY 1LC2

20 40 60

5

10

15

LAT.

qz(

km)

20 40 60

5

10

15

LAT.20 40 60

5

10

15

EQ. LAT.

DAY30LC1

20 40 60

5

10

15

EQ. LAT.

DAY30LC2

20 40 60

5

10

15

LAT.20 40 60

5

10

15

LAT.20 40 60

5

10

15

EQ. LAT.

DIFF.LC1

20 40 60

5

10

15

EQ. LAT.

DIFF.LC2

20 40 60

5

10

15

1.5

1

0.5

0

0.5

1

1.5

x 10 10

LAT.20 40 60

5

10

15

LAT.

20 40 60

5

10

15

1.5

1

0.5

0

0.5

1

1.5

x 10 10

5 10 15 20 25 301

0.5

0

0.5

1x 10 16

DAY

PV

LC1LC2

Fig. 6. Contours of the meridional gradients of PV, both the zonal-mean (lower) andLagrangian-mean (upper) are displayed for day 1 (left), day 30 (middle) and their difference(day 30-day 1). For both snapshots of the LC1 experiment (left) and LC2 experiment(right), the PV gradients are all plotted with the same colorbar in units of m−1s−1. Below aretimeseries of the ∆PVφφ, the change in the curvature of PV at the initial jet axis, Lagrangian-mean (solid) and zonal-mean (dashed), for the entire thirty days of each experiment.

33

5 10 15 20 25 30

1

2

3

4

5

6

7

8

9

10

DAYS

Mea

n A

m=5

5 10 15 20 25 30

1

2

3

4

5

6

7

8

9

10

DAYS

Mea

n A

m=6

5 10 15 20 25 30

1

2

3

4

5

6

7

8

9

10

DAYS

Mea

n A

m=7

0246899.5101112

Fig. 7. Each panel depicts the time evolution of the global mean wave activity for a singlezonal wavenumber: left m=5, middle m=6, right m=7. The ten curves represent differentvalues of the barotropic shear parameter Us between 0 − 12m s−1 (redder colors indicatehigher shear).

34

5 6 7 8 9024689

9.5101112

Wavenumber

U s

Mean A

1

2

3

4

5

6

5 10 15 20 25 300

2

4

6

8

10

DAY

Mea

n A

m=6m=7m=8m=9

5 6 7 8 9024689

9.5101112

Wavenumber

U s

WI

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

5 10 15 20 25 300

0.2

0.4

0.6

0.8

1

DAY

Wav

e In

dex

Fig. 8. Top row shows the time-mean, global-mean wave activity with units of m s−1 foreach case on the left and the time evolution for all cases. The colored curves are associatedwith LC2 and the black curves are LC1. Bottom row same as the top, but for WI thewave-breaking index.

35

0.1 0.05 0 0.05 0.1 0.151.5

1

0.5

0

0.5

1

1.5x 10 16

WIt

PV

LC1LC2

Fig. 9. The change in the PV curvature at the initial jet axis ∆PVφφ [m−2s−1] are plottedas a function of the tendency of the wave-breaking index during the eddy growth phaseWIt [day−1]. Filled circles are calculated from the Lagrangian-mean PV, open circles reflectzonal-mean PV gradients. Green symbols are LC1 experiments, blue LC2 and the pair ofcases examined in detail are plotted in black.

36