The Effects of Variations in Jet Width on the Growth of Baroclinic...

1 JANUARY 2004 23H A R N I K A N D C H A N G

q 2004 American Meteorological Society

The Effects of Variations in Jet Width on the Growth of Baroclinic Waves:Implications for Midwinter Pacific Storm Track Variability

NILI HARNIK

Lamont-Doherty Earth Observatory, Columbia University, Palisades, New York

EDMUND K. M. CHANG

ITPA/MSRC, State University of New York at Stony Brook, Stony Brook, New York

(Manuscript received 3 February 2003, in final form 1 July 2003)

ABSTRACT

The effects of variations in jet width on the downstream growth of baroclinic waves are studied, using asimple quasigeostrophic model with a vertically varying basic state and variable channel width, as well as asimplified primitive equation model with a basic state that varies in latitude and height. This study is motivatedby observations that in midwinter in the Pacific the storm track is weaker and the jet is narrower during yearswhen the jet is strong.

The linear models are able to reproduce the observed decrease of spatial growth rate with shear, if the narrowingof the jet is accounted for by assuming it decreases the meridional wavelength of the perturbations, whichhampers their growth. A common suggestion has been that perturbations are weaker when the jet is strongbecause they move faster out of the unstable storm track region. The authors find that one needs to take intoaccount that the jet narrows when it strengthens; otherwise, the increase of growth rate is strong enough tocounteract the effect of increased advection speed.

It is also found that, when the model basic state is Eady-like (small or zero meridional potential vorticitygradients in the troposphere), the short-wave cutoff for instability moves to large-scale waves as shear is increased,due to the accompanying increase in meridional wavenumber. This results in a transition from a regime whereupper-level perturbations spin up a surface circulation very rapidly, and normal-mode growth ensues, to a regimewhere the initial perturbations take a very long time to excite growth. Since waves slow down when a surfaceperturbation develops, this can explain the observations that the storm track perturbations are more ‘‘upperlevel’’ during strong jet years and their group velocities increase faster than linearly with shear.

1. Introduction and motivation

In this paper we study the effects of jet width on thegrowth of baroclinic waves in a storm track setup, withthe goal of understanding the observation that the mid-winter storm track is stronger during years when the jetis weak. Nakamura (1992) was the first to show that theintensity of the storm track is positively correlated tojet strength for jet wind speeds below 45 m s21 (relevantto the Atlantic storm track) and is negatively correlatedat higher jet speeds (relevant to the Pacific). This ismanifest most clearly as a midwinter minimum of thePacific storm track, but is also evident on interannualtime scales (Zhang and Held 1999; Chang 2001; Nak-amura et al. 2002). While Nakamura’s study mostly em-

Corresponding author address: Dr. Nili Harnik, Lamont-DohertyEarth Observatory, Columbia University, 61 Route 9W, Palisades,NY 10964.E-mail: [email protected]

phasized the seasonal time scales, Zhang and Held(1999) pointed out the distinction between interannualand seasonal time scales but hypothesized that the causefor a negative correlation is similar for both time scales.Chang (2001), on the other hand, found evidence thatdifferent mechanisms cause the relation for the two timescales, with changes in diabatic heating contributingstrongly on seasonal time scales, and changes in eddystructure and evolution dominating the interannual timescales. In this study we concentrate on understandingthe interannual-time-scale variability, by studying dy-namical effects that might cause a negative correlationbetween storm track and jet strength. We will discussthe relevance of our results to the seasonal time scalein section 5.

Figure 1 shows the January mean jet and strom trackvariance, averaged over the seven years with strongestand weakest jets. It is clear that the spatial growth ratein the downstream direction in the storm track is stron-ger during the weak jet years. The cause for this relation

24 VOLUME 61J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S

FIG. 1. (top) The storm track as represented by a 24-h difference-filtered meridional wind (m 2 s22) averaged overthe seven weakest (left) and strongest (right) jet years. (bottom) Same as top, only 300-hPa zonal wind (m s 21).

is not obvious since we expect eddies to grow faster,rather than slower, when the jet is strong. One simpleexplanation (suggested by Nakamura 1992) is that ed-dies move more quickly out of the storm track regionwhen the jet is strong, allowing them less time to am-plify. This increase in propagation speed, however,needs to overcome the increase in growth rate. Chang(2001) compared the energy budget of the storm tracksfor Januaries with weak and strong jets, using GCMsimulations and observations. He found that on inter-annual time scales dynamical effects were the most im-portant for explaining the observed variability, with ed-dies being more top-trapped near the tropopause, prop-agating with a much higher group velocity, and beingless efficient in tapping the surface baroclinicity whenthe jet is strong. This suggests that, when the jet isstrong, eddies move faster out of the unstable region,as expected, but they also grow less efficiently, resultingin a weaker storm track. The changes in group velocityand growth rate seem to be related to the changes inthe vertical structure of the eddies, but the exact mech-anism is still unclear. In particular, as Chang (2001)

pointed out, the simplest model for baroclinic instability,the Eady (1949) model, does not predict these changes.

This suggests that a linear model of baroclinic growth,which relates eddy growth to the shear in the center ofthe jet, may not be sufficient to explain the observedrelation. It is possible, however, that a linear model thatincorporates the meridional jet structure and variabilitywill be able to reproduce behavior like the observed.To examine this we calculate the January mean zonalwind averaged over the Pacific jet region (1208–1508E,1

referred to as UPAC), using National Centers for Envi-ronmental Prediction–National Centers for AtmosphericResearch (NECP–NCAR) reanalysis for 1949–2001.The top plot in Fig. 2 shows the 300-hPa wind profiles,centered about the jet maximum, for the strongest andweakest jet profiles, as well as the average over thestrongest, middle, and weakest third of the years. Thejet strength varies roughly between 50–76 m s21, and

1 The overall results are not sensitive to the range of longitude overwhich we average.


FIG. 2. (top) Latitudinal profile (centered around the jet maximum)of the strongest and weakest jets (thin solid and dashed lines, re-spectively), and the average over the strongest, middle, and weakestone-third of the jet profiles (thick dashed–dotted, solid and dashedlines, respectively). (middle and bottom) The zonal wind over thePacific, averaged over the five strongest and weakest jet years (usedin the primitive equation linear model runs, see text for details).Negative values dashed; contour interval is 5 m s21.

there is a clear tendency of the jet to be narrower whenit is strong.

The reason for the stronger jets being narrower is notclear, but there is some evidence that this is not due tofeedback forcing from the local storm track eddies butrather forced externally. Nakamura et al. (2002) sug-gested that the changes in stationary wave could beassociated with a wave train that appears to originatefrom the North Atlantic reaching into the far east, whileYin (2002) suggests that this could be the response toforcing by tropical convection based on an idealizedGCM study. This is also consistent with the suggestion

of Lee and Kim (2003) that the Pacific jet is externallydriven (as opposed to the Atlantic jet which is eddydriven). In this study, we will assume that the observedjet strength–width relation is externally set and use thisrelation to constrain eddy evolution.

Regardless of the causes, we expect the variations injet structure to affect wave growth. For example, theEady model predicts that the growth rate of the mostunstable normal mode decreases as the meridionalwavenumber increases. One simple model of the stormtracks (Chang 2001) is of upper-level wave packets en-tering the storm track region from upstream and prop-agating along the jet as they amplify. This implies theexistence of a waveguide that prevents the eddies frompropagating away from the jet region in the meridionaldirection. This view, which has some support in theliterature (e.g., Lee and Anderson 1996; Nakamura andSampe 2002), implies that the meridional wavenumberof the perturbations is set by the waveguide, which isdetermined by the jet structure. Ioannou and Lindzen(1986) showed, using the WKB approximation, thatgrowth of baroclinically unstable waves on a jet can beapproximated to first order by waves growing on a ver-tically varying basic state taken from the jet center, withthe meridional wavelength set by the meridional struc-ture of the jet (a rough approximation is to have half ameridional wavelength equal to a half-width of the ob-served jet). We used a 3D primitive equation model tocalculate the growth rate and structure of the most un-stable modes on 2D (latitude–height) basic-state jets thatwe strengthen and narrow as observed, and the results(see section 3) support this interpretation. While this isnot the only way in which meridional wind variationscan affect wave growth, it is a straightforward directeffect on growth rate, which can easily be incorporatedin a simple model.

In this paper we examine the effects of shear and me-ridional jet structure on the linear growth and propagationof waves in the context of storm tracks. We start bypresenting a simple model of the storm tracks that weuse as a framework for our model studies (section 2).We then present results from primitive equation and qua-sigeostrophic model runs in sections 3 and 4. In section5 we discuss the relevance of our idealized calculationsto the storm tracks and we conclude in section 6.

2. A simple model of spatial growth rate

Spatial, rather than temporal growth, is the relevantquantity for storm tracks that are characterized by thetime-mean eddy variance. Assuming storm tracks areseeded by upper-level waves coming into the storm trackregion from upstream (see section 5 for references), westudy the spatial growth and evolution of an upper-levelwave packet, which propagates along the jet stream andamplifies by inducing a surface perturbation that rein-forces the initial upper-level wave (Hoskins et al. 1985).


TABLE 1. The zonal wavenumber, growth rate, group velocity, andimplied spatial growth rate (the temporal growth rate divided by thegroup velocity) of the most unstable linear modes for the differentruns of the 3D primitive equation model (see text).

Jet speed FrictionWave-number

Growthrate

Groupvelocity

Impliedspatial

growth rate

Strong WeakMediumStrong

776

0.470.360.20

16.316.016.3

0.340.260.14

Weak WeakMediumStrong

776

0.390.290.13

11.711.311.5

0.390.300.13

Weak*1.3 WeakMediumStrong

776

0.550.430.28

15.915.114.1

0.400.330.23

In section 5 we discuss the relevance of our results toreal storm tracks.

We obtain the spatial growth rate using an equation forthe eddy energy E along the storm track (x direction). Thisderivation, which follows the heuristic formulation putforth by Swanson et al. (1997), is presented in Chang(2001) and is repeated here for clarity of the discussion:

]E ]E1 = · F ø C 5 source 1 sink, (1)g]t ]x

where F is the flux of eddy energy, which equals theeddy energy times a group velocity Cg, and we assumeCg is constant in x in our model.

In steady state, the eddy energy flux divergence bal-ances the sources and sinks. We assume the source termcan be expressed as a growth rate s times eddy energyand use baroclinic instability theory to estimate s. Forsimplicity, we also assume damping is proportional toeddy energy (we relax this assumption later on):

source ø sE, (2)

sink ø 2gE. (3)

Substituting into Eq. (1) and taking a time average,we get an expression for the spatial growth of eddyenergy (Sgr):

1 ]E s 2 gS [ ø . (4)gr E ]x Cg

There are two competing effects of group velocityand growth rate. When the jet strengthens, the eddiesmove faster out of the unstable region, which tends toweaken Sgr. On the other hand, when the jet is stronger,the local growth rate increases; causing Sgr to increase.Equation (4) suggests that it is the ratio of s and cg thatmatters. Which effect dominates depends on which ofthe two varies more with jet strength.

3. Stability analysis of Pacific jet profiles using aprimitive equation model

The model used is based on the dynamic core of theGeophysical Fluid Dynamics Laboratory (GFDL) cli-mate model (see Held and Suarez 1994). The integra-tions are performed with 10 evenly distributed sigmalevels, at a horizontal resolution of T42 (64 Gaussiangrid points in the meridional direction). The basic-statetemperature and wind are a zonal mean across the Pacificstorm track entrance region (1208E–1808), taken fromthe NCEP–NCAR reanalysis data for different years.The resulting basic states (shown in Fig. 2, middle andbottom panels), which are 2D, represent strong and weakjet years in the Pacific.2 We see that, indeed, the jet isstronger but narrower during the strong jet years. Note

2 The strong jet years included in the composite are 1970, 1977,1981, 1985, and 1988, while the weak jet years are 1971, 1972, 1973,1989, and 1993.

that the maximum speed of the strong jet is 1.3 timesthat of the weak jet.

The model is dry, with a 30-day Newtonian coolingin the free atmosphere, reaching 2 days in the lowestsigma level. As discussed in Valdes and Hoskins (1988),the growth rate and structure of the unstable modes arenot very sensitive to the Newtonian damping. Surfacefriction is represented by Rayleigh friction with a damp-ing time scale of 2, 1, and 0.5 days for the weak, me-dium, and strong friction experiments, respectively. Asin previous work (e.g., Farrell 1985; Valdes and Hoskins1988), the growth rates are very sensitive to the strengthof surface friction.

For each basic state and zonal wavenumber, a smallperturbation is introduced and the model integrated.Nonlinearity is suppressed by rescaling the perturbationonce its amplitude (in terms of vorticity perturbation)reaches 1% that of the basic state. The experiment isrun until the growth rate and wave structure becomesteady.

The zonal wavenumber, growth rate, group velocity,and implied spatial growth rate (the temporal growthrate divided by the group velocity) for the different casesare tabulated in Table 1. Here, the group velocity isestimated by finite differencing the wave frequency bythe zonal wavenumber between the two adjacent fastestgrowing normal modes. Comparing the strong and weakjet cases, we see that for all three friction values, thetemporal growth rate is higher for the strong jet, but forthe weak and medium friction cases, the increase ingrowth rate is less than that implied by the change inthe jet speed (30%). The group velocity for all cases ishigher for the strong jet runs by more than 30%. Theimplied spatial growth rate is therefore higher for theweak jets except for the strong friction runs (especiallythe weak jet) for which the growth rate is strongly sup-pressed by surface friction.

The structure of the most unstable mode for both jetsand medium friction are shown in Fig. 3. Shown arerms y at 250 hPa (normalized to unit amplitude at the


FIG. 3. (a) The latitude variation of the meridional wind anomalyat 250 hPa and (b) heat flux at 750 hPa of the most unstable normalmodes of the primitive equation model runs, and (c) the observed700-hPa Jan heat fluxes for the strong (solid with open circles) andweak (dashed dots) jets of Fig. 2. In (a) and (b) the meridional windsand heat fluxes are normalized such that the maximum 250-hPa yequals 1 m s21 for both cases. In (c) the heat flux for the strong jetis multiplied by 1.4 to facilitate the comparison of meridional struc-ture.

peak, plot a) and poleward heat flux at 750 hPa (alsonormalized, plot b). Clearly, the most unstable mode forthe narrower jet has a significantly narrower structure.In plot c, we show the meridional profile of the observedbandpass-filtered 700-hPa poleward heat flux averagedbetween 1208E and 1808, taken from the strong jet andweak jet years. The results show that the observed heatflux is indeed narrower during the strong jet years. How-ever, the correlation between the meridional width of700-hPa heat flux and jet strength is quite weak (20.25,which is only significant at the 90% level but not at the95% level) for the entire 51 years of reanalysis data.

To show that this behavior of the spatial growth rateis indeed due to the change in jet profile, we generatea new jet profile by increasing the strength of the weakjet by 1.3 times while maintaining its latitudinal profileso that its structure is of the weak jet, while its maximumstrength is of the strong jet. The results for this case arealso shown in Table 1. Compared to the weak jet case,we see that the temporal growth rate increases by morethan 30%, which is the amount by which the jet strengthincreased. This is due to the effect of surface friction,as expected from Eq. (4) based on the arguments inChang (2001). The resulting spatial growth rate is largerfor the weak times 1.3 jet than for the weak jet, eventhough the group velocity has increased. This showsthat simply increasing the advection out of the regionis not sufficient to suppress spatial growth. We musttake into account both the effects of an increase in thegroup velocity due to the increase in jet speed, as wellas the reduction in increase of temporal growth rate dueto the narrowing of jet, in order to explain the decreasein spatial growth rate for the strong jet case.

These model runs suggest that the variations in me-ridional confinement may explain the observed reduc-tion in the rate at which temporal growth rate increaseswith shear. In the next section we use a much simplerand numerically cheaper model to examine in detail howthe jet structure affects temporal and spatial growth rate,group velocity, and vertical wave structure for varioustypes of basic-state and model parameters.

4. Linear quasigeostrophic model results withvariations in jet width taken into account

We use a simple linear quasigeostrophic b-plane mod-el of baroclinic instability, with different jet profiles, toperform two kinds of calculations. The first is an ei-genvalue calculation for the vertical structure, growthrate, and phase speed of the corresponding baroclini-cally unstable normal modes for a range of zonal wave-numbers. The second is a time-dependent integration ofour model, starting from an upper-level wave packet,centered around a specified zonal wavenumber ko. Weevaluate the spatial growth rate from Sgr 5 s*/Cg, wheres* is the effective growth rate of perturbations in themodel, including the effect of damping. The basic-stateand model setup is similar to Harnik and Lindzen (1998)and the numerical solution method is similar to Farrell(1982), where we decompose the equations into zonalFourier components, and for each zonal wavenumberwe solve the vertical eigenvalue problem, project theinitial perturbation on the eigenvectors, and sum theircontributions to obtain the time-dependent solution.

We incorporate the observed relation that the jet nar-rows as it strengthens into our model by assuming thejet width determines the meridional structure of the ed-dies and specify the meridional wavenumber accord-ingly. The validity of this approach is supported by the


FIG. 4. The relation between the jet strength (m s21) at modeltropopause and at 300 hPa for observations and the jet half-width (in8lat) for model (line with solid circles) and observed (diamonds).

results of Ioannou and Lindzen (1986), as well as theresults of section 3.

The model basic-state wind and temperature have atroposphere and stratosphere, and the density decreasesexponentially with height. Our channel is centered at408 latitude. The top boundary is a lid at around 22 km,and we have a sponge layer that starts at around 15 km,to absorb any wave activity that propagates to theseheights.3 We have 70 grid points in the vertical. Thereare various ways to specify the tropospheric shear,which differ in the resulting meridional gradient of po-tential vorticity (PV). The simplest is to specify a con-stant shear with height, and let the PV gradient increasewith shear [a Charney model (Charney 1947)]. A dif-ferent approach is to hold the meridional PV gradientfixed, while we increase shear by adding vertical cur-vature to the wind profile [an Eady-like model (see Lind-zen 1994; Harnik and Lindzen 1998)]. The value ofmeridional PV gradients makes a big difference for thegrowth of short waves. In the Charney model, baroclinicgrowth occurs through an interaction between surfaceedge waves and internal Rossby waves. In the modifiedEady model [which differs from the Eady model bybeing on a b plane (Lindzen 1994)] there are no internaltropospheric waves because the meridional gradients ofPV are zero. Instead, the model supports tropopausewaves, which interact with the surface waves and grow,provided these waves are deep enough. Since the ver-tical scale is proportional to the horizontal scale of thewaves, short waves are stable. As a result, one of themain differences between the Eady and Charney modelsis that short waves are neutral in the former and unstablein the latter. This is relevant to our calculations becausethe decrease of meridional wavelength with shear push-es the model to the short-wave regime.

Unless otherwise specified, in all the following runswe increase the meridional wavenumber as we increaseshear. Figure 4 shows the relation between meridionalwavenumber and tropopause wind in our model, plottedwith the observed relation between the maximum jetspeed, using 300-mb zonal winds averaged over thePacific (1208–1508E), and the distance between the lat-itudes at which the winds reach half their peak value.We use a simple linear fit to the observed jet strength–jet width relation, and the rough approximation that halfa meridional wavelength equals the half-width of theobserved jet, so that the meridional width in our modelvaries between a meridional half-width of 368 (equiv-alent to a zonal wavenumber of 3.8 in units of numberof waves that fit into a latitude circle) for a jet of around22 m s21, to 158 (zonal wavenumber 9.4) for a jet of77 m s21. Observations lie along the strongest three orfour jet profiles. An alternative to the linear fit is toassume that the meridional temperature difference be-

3 Since we are interested mostly in medium and short waves, whichdo not propagate up to the stratosphere, the waves hardly feel thesponge layer and lid.

tween the northern and southern boundaries is fixed(corresponding roughly to fixed zonal transport due tothe thermal wind relationship). This fits the observationsjust as well, with the main difference being over theweak jet regime. The results presented below are notsensitive to the specific jet strength–width relation used,provided it roughly captures the observed relation.

a. Charney/Green-type basic state

First, we examine results based on the Charney pro-files, with no damping. Figures 5a,b show the basic statefor a set of runs in which the wind is specified to havea constant shear in the troposphere and zero shear inthe stratosphere. The tropospheric wind (below about 9km) increases from 3 m s21 at the surface with verticalshear between 2.5–8.6 m s21 km21 (runs 1–6, respec-tively). Also shown (Fig. 5b) is the Brunt–Vaisala fre-quency, which is the same for all runs, and the corre-sponding profiles of meridional PV gradients, whichincrease with shear in the troposphere between 2.4–5.8b (for runs 1–6, respectively).

Figure 5c shows the normal-mode growth rate for thesix basic states. We see that, for the most unstable wave-numbers (zonal wavenumber , 11), the growth rateincreases with jet strength for runs 1–5 and decreases


FIG. 5. (a) Zonal mean wind (m s21), (b) the meridional gradient of PV (units of b), (c) thenormal-mode growth rates (day21), and (d) the corresponding spatial growth rate (1026 m21) asa function of zonal wavenumber and nondimensional shear (used to number the runs), for the sixbasic states used in the model run. The nondimensional shear (run number) is marked in (a)–(c);thick line is run 6; thick dashed is run 5. Also shown in (b) is the Brunt–Vaisala frequency (1024

s22).

between runs 5 and 6. Dividing by the group velocity(which increases slower than linearly with shear), weget a spatial growth rate that increases, then decreaseswith shear, with the transition occurring at runs 3 or 4(Fig. 5d). Our model therefore seems to account notonly for the decrease of storm track strength with shear,but also for the fact that the decrease is only observedfor large shears.

To understand this transition, we calculate the growthrate for a range of meridional wavenumbers and shear

for zonal wavenumber 8 (results for other zonal wave-numbers are similar). The results are shown in Fig. 6.If the meridional wavenumber is kept constant as theshear is increased (moving up from curve 1 to 2 to 3,etc., in Fig. 6), the growth rate increases with increasingshear, as expected, but the increase is fastest for smallermeridional wavenumbers. If we keep the shear the samebut increase the meridional wavenumber (stay on thesame curve but migrate to the right), the growth rategenerally decreases with increasing meridional wave-


FIG. 6. The normal-mode growth rate (day21) as a function ofmeridional wavenumber for zonal wavenumber 8. Curves are for thedifferent basic states of Fig. 5 (numbered). The meridional wave-numbers used for each run and the corresponding growth rate aremarked by the circles.

number,4 with the decrease fastest for large shear andlarge wavenumbers. Combining the two effects, as theshear increases and jet width decreases, we move fromone curve to the next while shifting toward the right(following the trajectory of the circles shown in Fig.6). We see that the shear effect wins for small shearsand wavenumbers and the wavenumber effects wins forlarge shears and wavenumbers, resulting in the two re-gimes.

This behavior also holds for the time-dependent mod-el. We start with an upper tropospheric wave packetcentered around zonal wavenumber 8.5 (using otherwavenumbers yields similar results) and let it evolve oneach of the six basic states. Figure 7a shows the initialtropopause wave packet geopotential height, and Figs.7b–f show the longitude–time evolution of the wave-packet envelope at 9 km for runs 4 and 6 and the time–height structure of the envelope at the center of the wavepacket for runs 2, 4, and 6. We calculate the envelopeby interpolating between maxima of the absolute valueof the wave geopotential height field. It is clear, bylooking at a specific envelope value for each of the runs(e.g., the dashed lines), that the temporal and spatialgrowth rates are largest for intermediate shear values(the dashed line appears earlier and at a smaller lon-gitude in run 4, compared to runs 2 and 6).

We see two stages in the evolution of the wave packet.During the initial stage the vertical structure of the wavepacket changes as it spins up a surface circulation (e.g.,days 0–3 of run 2, Fig. 7b), while in the second stage

4 For the very long waves (planetary scales), the growth rate in-creases with wavenumber. Given the horizontal scales that are rele-vant to the storm tracks, this is not relevant here.

the vertical structure is fixed in time with the amplitudeincreasing exponentially in time (e.g., after day 4 of run2), as is the case with normal modes. Estimating agrowth rate and group velocity from the amplitude andlongitude of the center of the wave packet, respectively,we find that the group velocity and growth rate alsovary in the initial stage, reaching a constant value inthe second stage, which we verify to be equal to thevalues obtained from the normal-mode calculation forzonal wavenumber 8.5 (Fig. 5c). We therefore refer totwo stages as the nonmodal and normal-mode growthstages, respectively. We find that, as shear is increased,it takes the waves less time to reach the normal-modestage.

Figure 8a shows the group velocity divided by shear,for each of the six runs, estimated from the slope of alinear fit to the longitude of the tropopause wave packetcenter for different time periods, representing the initialnonmodal stage (days 0–3.3) and the normal-mode stage(days 18.6–25). We also use days 0–7 to represent acharacteristic storm track value, which may comprise amix of both stages. We see that the group velocity islarger in the initial stage (thin dashed) compared to thenormal-mode stage (solid) for runs 1–3. For runs 4–6,the normal-mode stage is reached in less than 3 days.We also see that, contrary to observations, the ratio ofgroup velocity to shear decreases with shear at all stagesof wave growth (the group velocity itself increases withshear).

b. Eady-like basic state

The vertical structure of the wave packets shownabove is largest at the surface, unlike the observed wavepackets, which have a peak at the tropopause with asecondary peak near the surface. This is expected basedon the normal modes of the Charney model. Waves inthe Eady model, on the other hand, do have a peak atthe tropopause.5

Figures 9a,b show the zonal mean wind and merid-ional PV gradients for the corresponding set of runswith the Eady-like basic states. The Brunt–Vaisala fre-quency is the same as in Fig. 5b. The wind is specifiedso that the mean tropospheric shears are the same as inFig. 5a and the shear is zero above about 10 km. Thechange in shear with height, along with the increase inthe Brunt–Vaisala frequency from tropospheric tostratospheric values, result in having a peak of PV gra-dients at the tropopause. Note that a relatively smalldifference in the wind profiles in Figs. 9 and 5 resultsin a significant difference in the PV gradient in thetroposphere. The PV gradients at the tropopause are alsomuch larger due to the sharper change in vertical wind

5 In the classical Eady model, there is complete symmetry aboutthe midtroposphere, but in the modified Eady model (b ± 0, but qy

5 0) waves are larger at the tropopause.


FIG. 7. (a) Longitude–height structure of the initial wave packet geopotential height. (b), (d),(f ) Time–height plots of the amplitude of the wave packet envelope at wave packet center forruns 2, 4, and 6 of the Charney-type model. (c), (e) Longitude–time plots at the tropopause ofwave packet amplitude for runs 4 and 6. Line types are the same in plots (b)–(f ).

shear with height near the tropopause. The meridionalwavenumber is varied with shear as in Fig. 4.

The corresponding normal-mode growth rates for thecase with no damping (Fig. 9c) increase, then decreasewith shear, as in the Charney basic states, but there isan additional effect on growth rate of shifting the short-wave cutoff to smaller zonal wavenumbers (from zonalwavenumber 10 in run 1 to zonal wavenumber 4 in run6). In our runs, this can have a very large effect on theevolution of initial upper-level wave packets of zonalwavenumbers that are close to the short-wave cutoff.The implied spatial growth rate is shown in Fig. 9d.Again, we see that for fixed zonal wavenumber, the

spatial growth rate first increases with shear but thendecreases as the jet gets too narrow, similar to the Char-ney case, but for this profile, since short waves are sta-bilized, the maximum spatial growth rate occurs whenk 5 8 instead of at much shorter waves.

Figure 10 shows the envelope structure for runs 2, 4,and 5 of Fig. 9 starting from the same initial upper-levelwavepacket perturbation of Fig. 7a. The increase, thendecrease, of growth rate with shear is evident. The maindifference compared to the results from previous runsis the short-wave cutoff shifting enough toward the longwaves so that the wavenumbers in the initial wavepacketare stable. In our case the initial wave packet is essen-


FIG. 8. The group velocity divided by shear, estimated from days 0 to 3.3 (thin dashed), 0 to 7 (thickdashed), and 18.6 to 25 (solid) of the time-dependent model solutions, for the six runs of the (a) Charneyand (b) Eady models. Results are in nondimensional model units.

tially stable in run 6 and marginally unstable in run 5.As a result, the vertical structure of the wave packetremains an upper-level wave packet for a relatively longtime.6 The persistence of upper-level waves occurs be-cause the model supports normal modes with such astructure—in the stable regime of the Eady model, thenormal modes are upper level7 (see Rivest et al. 1992).

When the model is stable, the differences betweenthe initial nonmodal stage and the normal-mode stageare very small. In the unstable runs, on the other hand,the initial wave packets have a more upper-level verticalstructure and a faster group speed than the normal modeswhich are slowed down by the induced surface com-ponent. As a result, the transition of the model from anunstable to a stable regime as shear is increased involveslarge changes in the wave growth rate and group ve-locity. This is clear in Fig. 8b, which shows the groupvelocity divided by shear, for the initial and normal-mode stages. The two are similar for run 6, while forruns 1–5 the group velocity is larger in the initial stage.Looking at the 0–7-day estimate (thick dashed line) wesee that the group velocity increases faster than linearlywith shear between runs 4–6, similar to the observedbehavior in the Pacific storm track entrance (Chang2001). We note that, since in observations we average

6 Even though the normal modes may eventually emerge, they willnot be relevant for storm tracks if the perturbations leave the unstableregion (which has a finite longitudinal extent) fast enough.

7 There is actually a pair of normal modes for each wavenumber—a surface edge wave and a tropopause wave.

over many eddies, we expect the transition from thestable to unstable regime to appear to be less abruptthan in our model.

Our results suggest that the transition of the Eady-like model to a state where upper-level wave packetsare essentially neutral yields wave behavior that is closerto the observed, compared with the Charney-like basicstate. This, however, might not hold under more realisticconditions. The neutrality of short waves in the Eadymodel depends on the meridional gradients of PV beingexactly zero, but in reality we expect some gradients.Observations suggest that the PV gradients on isen-tropes are somewhere in between the Charney and Eadymodels, with gradients being on the order of b in thetroposphere (e.g., Kirk-Davidoff and Lindzen 2000).Repeating our runs, however, with qy 5 b instead ofzero in the troposphere, we find that the main resultsare the same. The short waves are not neutral but theygrow very slowly (much slower than the medium-scalewaves), and the transition to the short-wave regime oc-curs at larger zonal wavenumbers for larger shears. Asin the Eady-like runs, for strong shears, the normal-mode stage is reached at a later time and the wavestructure during the initial stage is much more upperlevel. The similarity to the Eady-type model is not sur-prising because when shear is constant with height (asin the Charney model), y is much larger than b forqstrong jets (Fig. 5). Constraining the tropospheric PVgradients to be b results, for strong jets, in a very largepeak of PV gradients at the tropopause. This shifts thedynamics of the normal modes to a tropopause–surface


FIG. 9. As in Fig. 5 [without the Brunt–Vaisala curve in (b)] but for the Eady-type run.

interaction (as in the Eady model) rather than a tropo-sphere–surface interaction as in the Charney model. Forthe observed type of behavior, we need the upper-levelwavepacket to persist for a sufficiently long time. Rivestand Farrell (1992) showed that the pair of neutral short-wave modes in the Eady model transform to a decayingtropopause wave and an unstable surface wave, when asmall y is added to the model. Moreover, for smallqvalues of y, the slightly decaying tropopause waves,qwhich they referred to as ‘‘quasi modes,’’ have a decayrate that is proportional to y, allowing them to be quiteqpersistent in a time-dependent model.

A relevant study by Zurita and Lindzen (2001) sug-gests that, in both the Charney and the y 5 b models,q

short waves can be neutralized if the PV gradients arewiped out in the vicinity of their critical surface. It isvery likely that such a neutralization will occur quiterapidly during the wave life cycle since wave–mean flowinteraction is strongest at the critical surface. Moreover,the analysis of Zurita and Lindzen (2001) suggests thisneutralization will happen at larger zonal wavenumbersin the strong/narrow jet runs. These neutralized waves,however, have the characteristics of lower-troposphericrather than upper-level waves; therefore we expect theEady-like model to better explain the observed groupvelocity and vertical structure dependence. Nonetheless,this neutralization mechanism can enhance the models’transition to a stable regime in the qy 5 b runs.


FIG. 10. The time evolution of wave packet envelope in the Eady-like model for runs 2, 4, and5 (top to bottom): (a), (c), (e) as in Fig. 7c; (b), (d), (f ) as in Fig. 7b.

In the appendix we describe the results of addingvarious forms of linear damping that are relevant to thestorm track entrance regions. We find that damping doesnot change the main results of this study.

We also verify that the variations in meridional wave-number are the key factor in yielding the observed typeof behavior in our model runs by repeating the calcu-lations for all basic states, keeping the meridional wave-number constant. We find that both the growth rate andthe spatial growth rate increase with shear for all wave-numbers. This suggests that the narrowing of the jet iscrucial for the weakening of growth rate with shear.Moreover, without it, we expect an increase of the stormtracks with shear. We have repeated our runs with var-

ious combinations of a different value for the meridionalwavenumber, midchannel latitude, top boundary con-dition, b, different profiles for the wind and Brunt–Vaisala frequency, and a constant density with height.The results suggest that the increase of storm trackstrength with shear (when the jet width is kept fixed)for the inviscid case is mostly due to the effect of beta,with other factors having little effect.

5. Discussion and the relevance to storm tracks

In the previous section we showed that the narrowingof the jet as it strengthens results in a decrease in tem-poral as well as spatial growth rate. In addition, we


showed that, if we have an Eady-type basic state, thenarrowing also causes the model to be stable to theinitial upper-level perturbation because the waves aretoo shallow to induce a surface component. As a result,the group velocity increases more than the increase ofshear would imply, and the perturbations become moreupper level for large and narrow jets. While this is sim-ilar to the observed relation, the relevance of our resultsto the real world is not obvious given the simplicity ofour model. In this section we discuss the relevance ofour idealized model, to the observed variability in thePacific storm-track entrance region, and the potentialrelevance to seasonal time scale variations and the At-lantic storm track.

There is considerable evidence that storm tracks areregions where upper-level wave packets coming fromupstream amplify as they propagate by inducing a cor-responding surface perturbation through baroclinic in-teractions (e.g., Wallace et al. 1988; Sanders 1988;Chang and Yu 1999; Whitaker and Barcilon 1992). Fur-thermore, Chang (2001) showed that, in the Pacificstorm track entrance the main balance is between baro-clinic growth and downstream propagation, with damp-ing playing a relatively minor role since the eddies arestill too small for barotropic shearing and other nonlin-ear processes to be important. Examination of daily PVmaps suggests that the PV contours tend to undergoreversible sinusoidal oscillations near the Pacific storm-track entrance region, suggesting that waves behave rel-atively linearly over there. Our model, therefore, is rel-evant to the storm-track entrance region. The degreeand way in which the strength of perturbations in thestorm-track entrance region affects the rest of the stormtrack is a separate issue, which is beyond the scope ofthis study. A related basic underlying assumption wemade is that the variability in jet structure is externallydriven. This is consistent with the observation of Chang(2001) that eddies contribute to the jet structure mostlyin the Pacific exit region.

One of the main simplifications we made is that thebasic state is constant in the zonal direction. This greatlyreduced the needed computation time, allowing us totest many possible mechanisms and verify that the re-sults are not sensitive to the choice of parameters. The-oretically, this should be a reasonable assumption forcases in which the zonal wavenumber is smaller thanthe zonal scale on which the basic state varies, and weexpect it to apply at least qualitatively to the real stormtracks, but the results need to be tested in a full 3Dmodel with a zonally varying basic state. There havebeen a few studies of the effects of the basic-state shearon storm-track eddy structure and growth in nonzonalbasic states (e.g., Niehaus 1980, 1981; Frederiksen1983; Pierrehumbert 1984; Cai and Mak 1990), butthese studies do not address the question of the observedvariability; hence it is hard to draw any relevant con-clusions from them.

Another basic assumption is that the jet acts as a

waveguide and the eddies are meridionally confined bythe jet, with the eddies being constrained to be narrowerwhen the jet is strong. As discussed in section 3, ob-served 700-hPa heat flux does show weak negative cor-relation between its width and the jet strength. A caveatis that similar relations are not observed for bandpass-filtered meridional velocity (or geopotential height) var-iance. Currently, we do not completely understand thisdiscrepancy between the heat flux and velocity variance.Our speculation is that during the strong jet years, sincethe spatial growth rate is small, the structure of theeddies are more strongly influenced by that of their up-stream seed (as in our model results discussed in section4); whereas during the weak jet years, since spatialgrowth is faster, the structure of the eddies more quicklytakes on the normal-mode structure. Since the Pacificstorm track is seeded by a northern and a southernbranch (e.g., Chang and Yu 1999; Hoskins and Hodges2002), it is conceivable that the average storm trackwidth (in terms of velocity variance) may be broaderduring the strong jet years because the storm track ed-dies reflect more the influence from the two meridionallyseparated branches. On the other hand, the heat flux isa measure of the local baroclinic eddy source and isexpected to reflect the modal structure more closely.Whether this is the case is an issue to be examinedfurther using a model that incorporates zonal variationsin the basic state.

As we point out in the introduction, our study seeksto explain the observed variability on interannual timescales. The negative correlation between storm track andjet strength, however, is also very prominent in the sea-sonal cycle (Nakamura 1992). From observations, wefind that stronger jets are narrower when looking at theinterannual variations of all winter months (monthlymeans of November–March) and also when looking atthe climatological seasonal cycle (the jet is narrowestand strongest during January–February and widest andweakest during November). The change in width, how-ever, is smaller in the seasonal cycle relative to inter-annual time scales. To check if this narrowing can ex-plain the climatological seasonal cycle, we investigatethe seasonal change in the spatial growth rate, by con-ducting stability studies based on monthly mean jet pro-files from September through May, using the 3D modeldescribed in section 3. The results, which are displayedin Table 2,8 show that toward midwinter, both the tem-poral growth rate and group velocity increase and aremaximum during January, with the growth rate effectwinning over the group velocity, so that the spatialgrowth rate is maximum during midwinter. While thejet is slightly narrower during January than during fallor spring, the seasonal change in jet width is muchsmaller than the interannual change in midwinter, hence

8 Here, we only show results corresponding to the medium frictioncase (damping timescale of 1 day), but results for the other cases aresimilar.


TABLE 2. The growth rate, group velocity, and implied spatialgrowth rate of the most unstable linear modes, for the 3D primitiveequation model runs, using basic states of various months and mediumfriction (see text). The most unstable zonal wavenumber for allmonths is 7.

Month Growth rate Group velocityImplied spatial

growth rate

SepOctNovDecJan

0.050.150.250.310.33

8.110.212.213.814.5

0.070.170.240.260.26

FebMarAprMay

0.280.240.140.04

13.212.010.7

9.7

0.250.230.150.05

the smaller effect on wave growth. This is consistentwith the results of Chang (2001), that seasonal changesin the effects of diabatic heating (which are not incor-porated in our study) are important in understanding themidwinter suppression. Note, however, that while thejet narrowing cannot account for the midwinter sup-pression in the climatology, it probably contributes toit during strong years (since the narrowing effects oninterannual variability are much stronger in midwintercompared to fall and spring). This is consistent with thefinding of Nakamura et al. (2002) that the midwintersuppression is much stronger during years when themidwinter jet is strong (and narrow).

The observed transition from a positive to negativecorrelation of the Northern Hemisphere midwinter stormtrack strength with shear occurs at jets of around 45 ms21, which is roughly the upper limit for jet strength inthe Atlantic (Nakamura 1992). Since our model alsomakes a transition from a positive to a negative cor-relation at roughly the same jet strengths, it is possiblethat the Atlantic storm track variations are governed bythe same dynamics as in the Pacific, but the two happento fall in these two separate regimes. An analysis of thejet strength versus width in the Atlantic entrance showsthat stronger jets are narrower, suggesting at least thatthis is possible. Given, however, that the waves in theAtlantic are much more nonlinear than in the westernPacific (e.g., in terms of the ratio between eddy andmean flow quantities), the linear arguments of this studyhave to be applied with caution.

Observations show the Atlantic jet has a larger baro-tropic component, and it varies much more in latitudecompared to the Pacific jet. Lee and Kim (2003) sug-gested these differences are a result of the Atlantic jetbeing eddy driven, while the Pacific jet is driven bytropical heating and it organizes the eddies. While thissuggests that the two storm tracks are in different dy-namical regimes, Lee and Kim (2003) also showed inan idealized model that the system makes a transitionfrom an eddy-driven regime to a subtropical jet regimewhen the tropical forcing is increased. It is thereforepossible that the Atlantic storm track will behave like

that in the Pacific if the tropical heating, and corre-spondingly the jet there, ever got stronger. Our studysuggests that such a change will entail changes in theinterannual variations of the storm track. On the otherhand, the observation that waves in the Atlantic stormtrack are much more nonlinear in nature, compared towaves over the Pacific entrance region, might be dueto a different reason, possibly that the Atlantic stormtrack is seeded by already large waves from the Pacific.In this case, our linear model will probably not be rel-evant at all.

6. Conclusions

In this work, we have shown that the observed in-terannual variations of the Pacific storm track in its en-trance region are not inconsistent with linear baroclinicinstability theory, as long as the meridional structure ofthe jet is taken into account. Noting that the midwinterPacific jet stream is narrower during years when it isstrong, we examine the effects of jet width on the var-iations of wave growth with shear, using a simple qua-sigeostrophic channel model of downstream baroclinicgrowth as well as a 3D primitive equation model witha zonal-mean basic state. The former is done by assum-ing that the jet sets the meridional width of the waves.The resulting model is able to reproduce the observedweakening of the storm track with jet strength. More-over, our model yields a negative correlation for jetsstronger than some value and a positive correlation forjets weaker than that. Nakamura (1992) found a negativecorrelation for jets stronger than 45 m s21 and a positivecorrelation below. The narrowing of the jet inhibits thelocal growth rate by increasing the meridional wave-number of the perturbations. At the same time, thestrengthening of the jet tends to increase the growthrate. The narrowing is most effective for large wave-numbers, resulting in the two correlation regimes. With-out the narrowing of the jet, the effect of the increasein group velocity with jet strength is not sufficient toovercome that due to the increase in local growth rate.Therefore, an increase in group velocity, which has beensuggested as a possible cause for the weakening duringstrong jet times (e.g., Nakamura 1992), cannot by itselfexplain the observations.

The narrowing of the jet as it strengthens can explainthe decrease in storm track strength, but not the ob-served increase of Cg/shear with jet strength, or the moreupper-level structure for stronger jets. These observa-tions can be explained, however, if we have an Eady-like basic state in which short waves are neutral or al-most neutral. The increase in meridional wavenumberas the jet narrows causes the short-wave cutoff to moveto longer zonal waves, rendering the model stable to thewavepacket perturbations coming in from upstream (as-suming the zonal wavenumber of the wavepackets doesnot change much with shear).

Understanding the relation between the storm tracks


and the basic state is central to understanding stormtrack variability. Besides the obvious effects on localweather, there is evidence that the storm tracks are im-portant for understanding the low-frequency variabilityof the atmosphere, as well as for the assessment of im-pact of possible climate changes (see, e.g., Robinson2000; Peng and Whitaker 1999; Held et al. 1989; Bran-stator 1992, 1995) because the storm tracks transportsignificant amounts of heat and momentum that act asforcings to the large-scale flow. Various observationalstudies suggest that the storm tracks have undergonemajor changes on decadal time scales (see Harnik andChang 2003, and references therein). On longer time-scales, storm tracks are a major source of precipitationin midlatitudes, which can be important in the growthof ice sheets during times when the ice extends to lowenough latitudes. Thus, understanding storm track var-iability may be an important step in achieving under-standing of climate variability.

Acknowledgments. Nili Harnik is supported by Na-tional Oceanic and Atmospheric Administration GrantUCSIO-CU-02165401-SCF and by a Lamont-DohertyEarth Observatory postdoctoral fellowship. Edmond K.M. Chang is supported by National Science Foundation(NSF) Grant ATM-0296076 at the State University ofNew York at Stony Brook. Some of the work on thispaper was conducted when the authors were at the Flor-ida State University, where they were supported by NSFGrant ATM-0003136. Nili Harnik would also like toacknowledge hospitality provided by Professor R. Lind-zen at MIT. We thank Pablo Zurita-Gotor and SukyoungLee for very helpful comments.

APPENDIX

The Effects of Surface Damping

In this section we examine whether the addition oflinear damping at or near the surface affects our results.Surface damping of momentum and temperature are themain sinks of wave activity in the storm-track entranceregions (see Chang 2001). Other damping processes,mainly upper-tropospheric nonlinear momentum damp-ing via wave breaking, are significant more downstream.We use various combinations of Newtonian damping ontemperature and Rayleigh damping on momentum, bothimposed on the lowest levels of the troposphere, as wellas Ekman damping. Running our model with the variousforms of damping and various basic states, we find thatdamping does not qualitatively change the main resultand that, if we take into account that the jet narrows asit strengthens, the growth rate increases, then decreases,with shear. Without taking the change in jet width intoaccount, we find that damping leads to an increase inSgr with shear, as Eq. (4) implies. Clearly this effect isovercome by the effects of the narrowing of the jet.Surface damping does, however, affect our results in

several ways, which we present using the runs withEkman damping.

Figure A1a shows the normal-mode growth rates forthe six runs of the Charney model (basic states of Fig.5), with a 1-day Ekman damping. We see that dampingis much more effective on short waves.A1 Besides anoverall reduction of growth rates, this results in an ef-fective ‘‘shortwave cutoff.’’ Surface damping also af-fects the vertical structure of the waves since it reduceswave amplitude mostly at the surface. The correspond-ing most unstable normal modes (not shown) are upperlevel, while the short-wave normal modes have a max-imum at the tropopause and the surface with the latterbeing slightly larger, but the tropopause-to-surface am-plitude ratio is larger than for the undamped runs. Basedon the vertical structures, it is clear that run 6 is entirelyin the short-wave regime. The ‘‘short-wave cutoff,’’ de-termined from looking at the growth rates and verticalstructures, shifts slightly to longer zonal wavenumbersas shear is increased in runs 1–5. The transition to run6 for which all zonal wavenumbers are ‘‘short’’ occursquite abruptly.A2

In the previous section we argue that the modifiedEady basic state is important for explaining the observedtype of variability with shear, namely that waves aremore upper level for strong shears, and their group prop-agation increases faster than linearly with shear. It seemspossible that adding surface damping to the Charneymodel can yield similar behavior. We therefore examinethe time evolution of an initial upper-level wave packetin the presence of damping. Figure A1c shows the ver-tical structure of wave packet envelope at its center(normalized to unit amplitude at the surface), averagedover the first week of integration, for each of the basicstates (see comment in fn A2). We see that for runs 1–5, the wave packets are more upper level for weakshears, but in run 6 the wave packet becomes more upperlevel than runs 3–5, suggestive of the observed behavior.At the same time, the group velocity increases slowerthan linear with shear (Fig. A2a) for runs 1–5, but itincreases faster than shear between runs 5–6.

Although adding surface damping to the Charneymodel makes its behavior closer to that observed, theEady-type basic state still seems more likely to explainthe observed behavior. The corresponding damped runs

A1 This explains differences between our results and Farrell (1985),who found that waves in the Charney model are stabilized by Ekmandamping. Farrell used a meridional wavelength of around 288 latitudefor a relatively weak jet (30 m s21 at the tropopause) compared toour runs (wavelength of around 388 latitude for a jet of 66 m s21).

A2 When we add damping, the numerically obtained short-wavenormal modes (all wavenumbers of run 6 fall under this category),are sensitive to the vertical resolution we use. This is due to theinability of the numerics to resolve the singularity that appears at thecritical surface, when the growth rate of the mode vanishes due tothe damping (see Farrell 1989). The vertical structure and growthrate estimated from the initial stage do converge, however, suggestingthat the spuriously unstable short and shallow modes do not contributesignificantly to the initial stages of evolution.


FIG. A1. (a), (b) The normal-mode growth rates (day21) for the Ekman damping runs of theCharney and Eady models, respectively. (c), (d) The corresponding vertical structures of the wavepacket envelopes at wave packet center, averaged over the first 7 days of the time-dependentmodel integrations. Thick dashed-dotted, dashed, and solid lines are for runs 1, 5, and 6, respec-tively.

for the modified Eady model (shown in Figs. A1b,d andFig. A2b) show a more robust behavior, with the shiftingof the short-wave cutoff to longer zonal wavenumbersbeing much more gradual, the ratio of tropopause tosurface amplitudes increasing with shear between runs4–6, and the group velocity increasing faster than lin-early with shear between runs 4–6 (using the 0–7-dayestimates, thick dashed line). In the Charney model, theshort-wave normal modes are essentially troposphere–surface waves with a reduced surface amplitude while

in the modified Eady model the normal modes are tro-popause–surface waves. We note that the model with

y 5 b and surface damping (not shown) behaves sim-qilarly to the modified Eady model because its modes aretropopause–surface waves.

Comparing the growth rates for the damped and un-damped modified Eady model (Figs. A1b and 9c), wesee that the effect of damping is relatively uniform onthe growth rate of the unstable waves, but it slightlydestabilizes the short waves [see Snyder and Lindzen


FIG. A2. As in Fig. 8 but for the corresponding damped model runs.

(1988) for an explanation]. On the other hand, dampingdelays the time at which the wave packets reach thenormal-mode stage because the normal modes, whichhave a surface component, grow slower. This is evidentfrom comparing Fig. 8 to Fig. A2, which shows thatthe week-based estimates of the group velocity are clos-er to the initial stage estimates in the damped comparedto the undamped models. The overall effect is to makethe transition to the almost-stable short-wave regime, asthe jet is strengthened and narrowed, more gradual thanin the inviscid case.

REFERENCES

Branstator, G., 1992: The maintenance of low-frequency atmosphericanomalies. J. Atmos. Sci., 49, 1924–1945.

——, 1995: Organization of storm track anomalies by recurring low-frequency circulation anomalies. J. Atmos. Sci., 52, 207–226.

Cai, M., and M. K. Mak, 1990: On the basic dynamics of regionalcyclogenesis. J. Atmos. Sci., 47, 1417–1442.

Chang, E. K. M., 2001: GCM and observational diagnoses of theseasonal and interannual variations of the Pacific storm trackduring the cool season. J. Atmos. Sci., 58, 1784–1800.

——, and D. B. Yu, 1999: Characteristics of wave packets in theupper troposphere. Part I: Northern Hemisphere winter. J. Atmos.Sci., 56, 1708–1728.

Charney, J. G., 1947: The dynamics of long waves in a baroclinicwesterly current. J. Meteor., 4, 135–163.

Eady, E. T., 1949: Long waves and cyclone waves. Tellus, 1, 33–52.Farrell, B. F., 1982: The initial growth of disturbances in a baroclinic

flow. J. Atmos. Sci., 39, 1663–1686.——, 1985: Transient growth of damped baroclinic waves. J. Atmos.

Sci., 42, 2718–2727.——, 1989: Unstable baroclinic modes damped by ekman dissipation.

J. Atmos. Sci., 46, 397–401.

Frederiksen, J. S., 1983: Disturbances and eddy fluxes in NorthernHemisphere flows: Instability of three-dimensional January andJuly flows. J. Atmos. Sci., 40, 836–855.

Harnik, N., and R. S. Lindzen, 1998: The effect of basic-state potentialvorticity gradients on the growth of baroclinic waves and theheight of the tropopause. J. Atmos. Sci., 55, 344–360.

——, and E. K. M. Chang, 2003: Storm track variations as seen inradiosonde observations and reanalysis data. J. Climate, 16,480–495.

Held, I. M., and M. J. Suarez, 1994: A proposal for the intercom-parison of the dynamical cores of atmospheric general circulationmodels. Bull. Amer. Meteor. Soc., 75, 1825–1830.

——, S. W. Lyons, and S. Nigam, 1989: Transients and the extra-tropical response to El Nino. J. Atmos. Sci., 46, 163–174.

Hoskins, B. J., and K. I. Hodges, 2002: New perspectives on theNorthern Hemisphere winter storm tracks. J. Atmos. Sci., 59,1041–1061.

——, M. E. Mcintyre, and A. W. Robertson, 1985: On the use andsignificance of isentropic potential vorticity maps. Quart. J. Roy.Meteor. Soc., 111, 877–946.

Ioannou, P., and R. S. Lindzen, 1986: Baroclinic instability in thepresence of barotropic jets. J. Atmos. Sci., 43, 2999–3014.

Kirk-Davidoff, D. B., and R. S. Lindzen, 2000: An energy balancemodel based on potential vorticity homogenization. J. Climate,13, 431–448.

Lee, S. Y., and J. L. Anderson, 1996: A simulation of atmosphericstorm tracks with a forced barotropic model. J. Atmos. Sci., 53,2113–2128.

——, and H. K. Kim, 2003: The dynamical relationship betweensubtropical and eddydriven jets. J. Atmos. Sci., 60, 1490–1503.

Lindzen, R. S., 1994: The eady problem for a basic state with zeroPV gradient but b ± 0. J. Atmos. Sci., 51, 3221–3226.

Nakamura, H., 1992: Midwinter suppression of baroclinic wave ac-tivity in the Pacific. J. Atmos. Sci., 49, 1629–1642.

——, and T. Sampe, 2002: Trapping of synoptic-scale disturbancesinto the North Pacific subtropical jet core in midwinter. Geophys.Res. Lett., 29, 1761, doi:10.1029/2002GL015535.

——, T. Izumi, and T. Sampe, 2002: Interannual and decadal mod-


ulations recently observed in the Pacific storm track activity andeast asian winter monsoon. J. Climate, 15, 1855–1874.

Niehaus, M. C. W., 1980: Instability of non-zonal baroclinic flows.J. Atmos. Sci., 37, 1447–1463.

——, 1981: Instability of non-zonal baroclinic flows: Multiple-scaleanalysis. J. Atmos. Sci., 38, 974–987.

Peng, S. L., and J. S. Whitaker, 1999: Mechanisms determining theatmospheric response to midlatitude SST anomalies. J. Climate,12, 1393–1408.

Pierrehumbert, R. T., 1984: Local and global baroclinic instability ofzonally varying flow. J. Atmos. Sci., 41, 2141–2162.

Rivest, C., and B. F. Farrell, 1992: Upper-tropospheric synoptic-scalewaves. Part 2: Maintenance and excitation of quasi modes. J.Atmos. Sci., 49, 2120–2138.

——, C. A. Davis, and B. F. Farrell, 1992: Upper-tropospheric syn-optic-scale waves. Part 1: Maintenance as eady normal modes.J. Atmos. Sci., 49, 2108–2119.

Robinson, W. A., 2000: Review of WETS—The workshop on extra-tropical SST anomalies. Bull. Amer. Meteor. Soc., 81, 567–577.

Sanders, F., 1988: Life-history of mobile troughs in the upper west-erlies. Mon. Wea. Rev., 116, 2629–2648.

Snyder, C. M., and R. S. Lindzen, 1988: Upper-level baroclinic in-stability. J. Atmos. Sci., 45, 2445–2459.

Swanson, K. L., P. J. Kushner, and I. M. Held, 1997: Dynamics ofbarotropic storm tracks. J. Atmos. Sci., 54, 791–810.

Valdes, P. J., and B. J. Hoskins, 1988: Baroclinic instability of thezonally averaged flow with boundary-layer damping. J. Atmos.Sci., 45, 1584–1593.

Wallace, J. M., G. H. Lim, and M. L. Blackmon, 1988: Relationshipbetween cyclone tracks, anticyclone tracks, and baroclinic waveguides. J. Atmos. Sci., 45, 439–42.

Whitaker, J. S., and A. Barcilon, 1992: Type B cyclogenesis in azonally varying flow. J. Atmos. Sci., 49, 1877–1892.

Yin, J. H., 2002: The peculiar behavior of baroclinic waves duringthe midwinter suppression of the Pacific storm track. Ph.D. the-sis, University of Washington, 121 pp.

Zhang, Y. Q., and I. M. Held, 1999: A linear stochastic model of aGCM’s midlatitude storm tracks. J. Atmos. Sci., 56, 3416–3435.

Zurita, P., and R. S. Lindzen, 2001: The equilibration of short Charneywaves: Implications for potential vorticity homogenization in theextratropical troposphere. J. Atmos. Sci., 58, 3443–3462.

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