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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: [email protected]
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
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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