The inﬂuence of stratospheric potential vorticity on ...rks/reprints/smy_scott_qjrms_2009.pdf ·...

QUARTERLY JOURNAL OF THE ROYAL METEOROLOGICAL SOCIETYQ. J. R. Meteorol. Soc. 135: 1673–1683 (2009)Published online 1 October 2009 in Wiley InterScience(www.interscience.wiley.com) DOI: 10.1002/qj.484

The influence of stratospheric potential vorticityon baroclinic instability

L. A. Smy* and R. K. ScottUniversity of St Andrews, UK

ABSTRACT: This article examines the dynamical coupling between the stratosphere and troposphere by considering theeffect of direct perturbations to stratospheric potential vorticity on the evolution of midlatitude baroclinic instability in asimple extension of an Eady model. A simulation in which stratospheric potential vorticity is exactly zero is used as a controlcase, and both zonally symmetric and asymmetric perturbations to the stratospheric potential vorticity are then considered,the former representative of a strong polar vortex, the latter representative of the stratospheric state following a majorsudden warming. Both types of stratospheric perturbation result in significant changes to the synoptic-scale evolution ofsurface temperature, as well as to zonally and globally averaged tropospheric quantities. In the case of a zonally symmetricperturbation, the linear growth rate of all unstable modes decreases with increasing perturbation amplitude. Initial growthrates in cases with significant asymmetric perturbations are also weaker than those of the control case, but final eddykinetic energy values are much larger due to the growth of low zonal wavenumbers triggered by the initial stratosphericperturbation. A comparison of the zonally symmetric and asymmetric perturbations gives some insight into the possibleinfluence of pre- or post-sudden-warming conditions on tropospheric evolution. Copyright c© 2009 Royal MeteorologicalSociety

KEY WORDS stratosphere-troposphere dynamical coupling; Eady model; tropospheric evolution

Received 22 December 2008; Revised 7 May 2009; Accepted 3 July 2009

1. Introduction

Observations of correlations between zonally symmetricanomalies of zonal wind and geopotential height inthe stratosphere and troposphere (e.g. Kodera et al.,1990; Thompson & Wallace, 1998, 2000; Baldwin &Dunkerton, 1999) have prompted much recent researchinto the dynamical coupling between these two regionsof the atmosphere. These correlations are time-lagged andshow tropospheric anomalies that persist on sub-seasonaltime-scales, longer than the corresponding stratosphericanomalies. Consequently, it has been suggested that thestratosphere may have a dynamical influence on thetropospheric circulation, even to the extent that medium-range weather forecasts might be improved by improvingthe representation of the stratosphere in forecast models(Scaife & Knight, 2008).

Dynamical links between the stratosphere and tropo-sphere have been suggested to exist on both sub-seasonaland longer, interannual time-scales. In the latter case, forexample, the stratosphere has been shown to influencetropospheric circulation patterns in both comprehensivegeneral circulation models (Hartmann et al., 2000; Shin-dell et al., 1999, 2001) and simplified primitive equationmodels (Polvani & Kushner, 2002; Kushner & Polvani,

∗Correspondence to: L. A. Smy, University of St Andrews, UK.E-mail: [email protected]

2004), possibly indicating some sensitivity of the tropo-spheric circulation to the details of the stratospheric evo-lution. The stratosphere is now widely believed to playan important role in climate variability (WMO, 2007),although the dynamical processes involved are not wellunderstood.

On shorter time-scales, on the other hand, the stronglag correlations between the strength of the winterstratospheric polar vortex and sea-level pressure dis-tribution (Baldwin & Dunkerton, 1999, 2001; Thomp-son et al., 2002; Charlton et al., 2003) again pointto a dynamical coupling that remains, however, farfrom well-understood. While descending zonal windanomalies within the stratosphere can be explained interms of descending critical layers (Matsuno, 1971), thepersistence of tropospheric anomalies on longer time-scales requires the consideration of additional dynam-ical processes. For example, Song & Robinson (2004)have suggested that stratospheric sudden warming eventsmight couple with the troposphere through an eddy-feedback mechanism. Another recent study by Thompsonet al. (2006) suggested, on the other hand, that the purelybalanced response of the troposphere to changes in strato-spheric wave drag and thermal heating may be sufficientto explain longer tropospheric correlation time-scales.

A dynamical link between the stratosphere and tropo-sphere involving the modulation of baroclinic instabilityby stratospheric zonal wind anomalies has also been con-sidered recently (Wittman et al., 2004, 2007). In simple

Copyright c© 2009 Royal Meteorological Society

1674 L. A. SMY AND R. K. SCOTT

baroclinic life-cycle experiments using a general circula-tion model dynamical core, Wittman et al. (2004) foundthat the strength of the stratospheric zonal winds had aninfluence on the tropospheric evolution, both in terms ofthe synoptic-scale development and zonal mean quanti-ties, such as the surface geopotential. Building on thisand earlier work by Muller (1991) that examined lin-ear growth rates in a one-dimensional Eady-type model,Wittman et al. (2007) further examined the dependenceof growth rates on stratospheric shear in a variety ofsimple and more comprehensive models. In particular,they found that increasing the vertical shear above thetropopause (a representation of a strong stratosphericvortex) increased growth rates across a range of zonalwavenumbers.

In this article, we again investigate the dynamical linkbetween stratospheric anomalies and baroclinic instabil-ity, but here restrict attention to anomalies that consistpurely of a change to the stratospheric potential vorticity.Our approach is motivated by a recognition of potentialvorticity as the principle dynamical quantity governingthe slow, balanced motion of the stratosphere. A strongpolar vortex is an approximately zonally symmetric dis-tribution of high potential vorticity, while a weak polarvortex may be due either to a distribution of low potentialvorticity or, alternatively, to a zonally asymmetric distri-bution of potential vorticity. The latter scenario is theone typically observed during major stratospheric suddenwarming events, when strong planetary wave-breakingredistributes the stratospheric potential vorticity, eitherthrough a strong displacement of the vortex (in the case ofa wave-one warming) or through a vortex split (in the caseof a wave-two warming: Charlton & Polvani, 2007). Theapproach we adopt here allows us to examine in detailhow such a redistribution of the stratospheric potentialvorticity may affect the tropospheric evolution. Note alsothat we are interested in resultant changes to the nonlineartropospheric evolution during the course of a barocliniclife cycle rather than changes to the tropospheric circula-tion resulting from an instantaneous potential vorticityinversion. The effects of zonal mean perturbations tothe stratospheric potential vorticity on the troposphericcirculation as a direct result of potential vorticity inver-sion were considered by Ambaum & Hoskins (2002) andBlack (2002), but not their influence on baroclinic insta-bility.

For simplicity, and to avoid the difficulties involved inbalancing an asymmetric potential vorticity distribution,we use a quasi-geostrophic model on an f -plane. Themodel configuration is described fully in section 2. Baro-clinic instability is arranged in the troposphere throughan Eady-type distribution of potential temperature at thesurface and tropopause. The case in which the strato-spheric potential vorticity is exactly zero is treated as acontrol (discussed in section 3.1). Following this we con-sider the effect on the instability of zonally symmetricperturbations to the stratospheric potential vorticity (as arepresentation of a strong vortex, section 3.2) and asym-metric perturbations (as a representation of the vortexfollowing a sudden warming, section 3.3). Finally, we

briefly consider the sensitivity of our results to details ofthe tropospheric mean state (section 3.4).

2. Model description

The numerical model used is the contour-advectivesemi-Lagrangian (CASL) model developed originally byDritschel and Ambaum (1997) and extended to cylindri-cal geometry by Macaskill et al. (2003). It solves thequasi-geostrophic equations on an f -plane in a cylindri-cal domain in uniform rotation about the central axis.One advantage of this model is that it allows easy ini-tialization of both zonal and non-zonal potential vorticityanomalies and a relatively straightforward interpretationof dynamical processes.

In cylindrical coordinates (r, θ, z) the equations takethe form

Dq

Dt= ∂q

∂t+ u · ∇q = 0, (1)

q = 1

r

∂

∂r

(r∂ψ

∂r

)+ 1

r2

∂2ψ

∂θ2+ 1

ρ0

∂

∂z

(ρ0

f 20

N2

∂ψ

∂z

),

(2)

u =(

−1

r

∂ψ

∂θ,∂ψ

∂r

), (3)

together with an isothermal lower boundary conditionψz = 0 at z = 0. Here q is the (anomalous) quasi-geostrophic potential vorticity, ψ is the geostrophicstream function, and u = (u, v) is the horizontalgeostrophic velocity. The physical parameters are thebackground density ρ0 = ρs exp(−z/H), where H is avertical scale height and ρs is a surface reference density,the ‘midlatitude’ Coriolis parameter f0 = 2� sin 45◦,where � = 2π day−1 is the (planetary) rotation rate, andthe constant buoyancy frequency N . Numerically, wehave chosen N ≈ 0.018 s−1 and H ≈ 8800 m as beingrepresentative of the troposphere, giving a Rossby defor-mation radius LR = NH/f0 of approximately 1525 km.

We use an Eady-type approximation in which the inte-rior tropospheric potential vorticity is uniform and poten-tial vorticity anomalies are concentrated in thin layersnear the surface and the tropopause. The standard Eadymodel has a basic state consisting of a uniform verti-cal shear U = �z corresponding to a uniform latitudinalpotential temperature gradient � = −�y, together withlower and upper boundary conditions ψz = θ at z = 0 andz = H . Following Bretherton (1966), this can be recastin terms of the evolution of upper and lower potentialvorticity sheets with basic state

Q = �δ(z) − �δ(z − H), (4)

together with isothermal boundary conditions at z = 0and z = H . As discussed in Juckes (1994), replacing theupper boundary at z = H with a tropopause separatingregions with static stability Nt (tropospheric) and Ns(stratospheric) leads to the same expression but with a

Copyright c© 2009 Royal Meteorological Society Q. J. R. Meteorol. Soc. 135: 1673–1683 (2009)DOI: 10.1002/qj

INFLUENCE OF STRATOSPHERIC POTENTIAL VORTICITY 1675

factor proportional to (Nt − Ns)/NtNs multiplying thesheet at z = H (see Equation 3.11 in Juckes, 1994). Sincehere we are interested in the dynamics of a jet localizedin latitude, we generalize the Eady basic state to a non-uniform function of latitude. This motivates the followingdefinition for the total basic state potential vorticity:

q = θs(r)δ(z) − θt(r)δ(z − H) + qstrat, (5)

with θs and θt corresponding to � in Equation (4)and where qstrat is a basic-state stratospheric potentialvorticity, to be specified below.

The initial surface and tropopause potential temper-ature distributions are specified by a simple latitudinalprofile of the form

θs,t(r) = θs,t tanh{(rj − r)/w}, (6)

of width 2w = LR and centred at rj = 4LR (approxi-mately 30◦ latitude). Analogously to Equation (4), θs,tcorrespond to the pole–equator potential temperature dif-ferences at the surface and tropopause, with θs, θt <

0. The choice 2w = LR is reasonable, being similar tothe size of typical eddies developing in a baroclinic flow;the sensitivity of our results to this parameter is describedbriefly in section 3.4. Choosing values θs = −0.2 andθt = −0.6 gives a basic state comprising a subtropicaljet with a maximum u of about 35 ms−1 and a verticalshear amounting to a difference of about 50 ms−1 betweenthe surface and tropopause, as shown in Figure 1(a).

The distribution of potential vorticity in the winterstratosphere is dominated almost entirely by the polarvortex, and may be approximated most simply as a regionof high potential vorticity over the Pole and low potentialvorticity in midlatitudes. In our model we therefore defineqstrat by

qstrat(r, z) ={

Q if r < rv, z � 1.5H,

0 otherwise,(7)

where rv is the vortex radius at a given height and Q

is the jump in potential vorticity at the vortex edge. Herethe lowermost vortex is situated at a distance of 0.5H

above the tropopause, as a crude representation of theweaker ‘sub-vortex region’ and to reduce the direct effectof the stratospheric potential vorticity on the troposphericwinds. It was found that the effect of the polar vortexon the tropospheric evolution is most pronounced whenthere is no sub-vortex region, and decreases slightly asthe region increases in depth. However, the sense of thestratospheric influence is unaffected by the depth of thesub-vortex region, and even the more detailed aspectsof the results show very little sensitivity to the depthof the region. Here, we include the sub-vortex region toillustrate that the influence of the polar vortex on theinstability is remote, and not due to local interactions inthe direct vicinity of the tropopause.

We consider three different forms of rv to representtypical polar vortex regimes. The simplest form corre-sponds to a zonally symmetric columnar vortex with

rv = a independent of height. Choosing a = 2LR gives apolar vortex edge situated near 60◦ latitude.

The other two forms correspond to a simple horizon-tal rearrangement of this profile, or deformation of thevortex edge, into a zonally asymmetric state. The firstof these takes the form of a simple horizontal displace-ment of the vortex from over the Pole, similar to a zonalwavenumber-one perturbation. The amplitude of the dis-placement used for the experiments reported in section3.3 is such that the vortex is displaced a distance LR;the dependence of the results on this displacement isalso considered. The second asymmetric state is givenby a zonal wavenumber-two perturbation of amplitude η

to the zonally symmetric state. In polar coordinates, thevortex boundary is displaced from rv = a (constant) torv = rv(θ), where

rv(θ) = α[a + η cos(2θ)], (8)

where the normalization factor α2 = a2/(a2 + η2/2) isincluded to ensure that the cross-sectional area of theperturbed vortex is the same as that of the zonally sym-metric vortex, regardless of η. A disturbance amplitudeη = 2 gives a vortex that is exactly split in a figure-of-eight pattern, while smaller values give a less perturbedvortex. For the results reported in section 3.3 we settledfor η = 1.

The complete tropospheric and stratospheric vorticitydistributions are shown in a perspective view in Figure 2.Note that we have added a weak (maximum displace-ment of 0.1LD) zonal wavenumber-six perturbation tothe surface and tropopause potential temperature fields toseed the baroclinic instability. This is the fastest-growingmode, and a single-wavenumber perturbation was chosento allow direct comparison of surface potential temper-ature fields among different cases during the nonlinearstages of the evolution. The control case, with qstrat = 0,is shown in Figure 2(a) and the cases with stratosphericperturbation – zonally symmetric, displaced vortex andsplit vortex – in Figure 2(b)–(d), respectively. The corre-sponding initial zonal velocity profiles for each case areshown in Figure 1.

The model equations are discretized using 80 layersin the vertical between z = 0 and z = 3H . This givesa vertical domain extending from the ground to approxi-mately the middle stratosphere. In this problem, the upperstratospheric potential vorticity has practically no impacton details of the tropospheric winds and the evolution ofthe baroclinic life cycle, so this truncation seems justi-fied. In the horizontal directions, the stream function andvelocity fields are calculated on a stretched grid of 128radial and 264 azimuthal points, although the potentialvorticity itself is first interpolated on to a grid four timesfiner for more accurate inversion. The lateral boundary islocated at a distance of 8LR from the Pole and has beenverified to have practically no effect on the troposphericevolution.

Finally, a few words are needed concerning the issueof numerical convergence. In the Eady model, the surface



Figure 1. Initial zonal wind profiles of the four main cases: (a) zero stratospheric potential vorticity (the control case), (b) zonally symmetric polarvortex, (c) asymmetric polar vortex (displaced) and (d) asymmetric polar vortex (split). The stratospheric potential vorticity jump in (b)–(d) is

Q = 0.4f0. The contour interval is 5 ms−1. Height is in units of H, radius is in units of LR.

(a) (b) (c) (d)

Figure 2. Perspective view of the potential vorticity contours in the four main cases: (a)–(d) as in Figure 1.

and tropopause dynamics exhibit a logarithmic singular-ity in the tangential velocity field as potential temperaturefronts develop (Juckes, 1994). Further, steep potentialtemperature gradients are a natural feature of the non-linear evolution during the instability. The logarithmicsingularity is here regularized by the fact that the ver-tical discretization is finite, but increasing the verticalresolution results in increasingly energetic flow at thesmallest horizontal scales. It was found by successivelydoubling both horizontal and vertical resolution togetherthat, although the smallest-scale features of the modelcontinue to exhibit differences at the highest resolutionperformed, large- and synoptic-scale features are essen-tially convergent. Further, bulk quantities like the eddykinetic energy are convergent below the resolution usedfor the main experiments reported below.

3. Results

3.1. Control

We first briefly describe the control case in which thestratospheric potential vorticity is exactly zero. The ini-tial zonal wind profile for this case is shown in Figure 1(a)and, as described above, comprises a baroclinically unsta-ble subtropical jet with a peak wind speed of approx-imately 35 ms−1 and a vertical shear amounting to adifference of about 50 ms−1 between the surface andtropopause.

The growth of the instability is such that significantnonlinearity develops by around days 6–8 of the evo-lution, with eddy kinetic energy peaking near day 10and then decreasing again in a decay stage, the clas-sic baroclinic life-cycle paradigm. The synoptic surfacepotential temperature distribution for the control case atdays 10, 12 and 14 of the life-cycle evolution is shownin Figure 3(a)–(c). Significant nonlinearity has devel-oped by day 10 with the usual wave-breaking, irreversiblemixing of potential vorticity and intensification of poten-tial temperature gradients. Because of the simplicity ofour model, only a very qualitative comparison of thisevolution with that observed in more sophisticated mod-els is possible. However, the evolution can be regardedas broadly similar to the LC1 life cycle of Thorncroftet al. (1993) with predominantly anticyclonic equator-ward wave-breaking during the saturation phase. Exami-nation of the zonal mean zonal velocity and wave fluxes(not shown) indicates a transfer of energy into a deeperbarotropic jet and an upward wave flux from the surfaceto the tropopause.

3.2. Zonally symmetric perturbation

To examine the effect of stratospheric potential vorticityon the tropospheric evolution, we next consider caseswith non-zero qstrat. We first consider the case of a zonallysymmetric perturbation to the stratospheric potentialvorticity. More realistic perturbations will follow in



(a) (b) (c)

(d) (e) (f)

Figure 3. Surface potential temperature at days 10, 12 and 14: (a)–(c) the control case Q = 0, and (d)–(f) with a zonally symmetric stratosphericanomaly of Q = 0.4f0.

section 3.3, but we note for now that a zonally symmetricperturbation can be regarded as a crude representation ofa strong polar vortex, and can therefore be contrastedwith the control case discussed above (which can beregarded as an extreme example of a weak vortex). Ourcomparison here is similar to the situation considered byWittman et al. (2004), who restricted attention to purelyzonally symmetric initial conditions, with or without astratospheric polar vortex.

The zonally symmetric perturbation to the stratosphericpotential vorticity is defined by Equation (7) with rv =a = 2LR. Here we focus on the case Q = 0.4f0, whichgives rise to a relatively strong polar vortex (in terms of u

near the vortex edge), but the intermediate case of Q =0.2f0 is also discussed briefly. As can be seen in Figure 1,the addition of the polar vortex in the stratosphere has adirect effect on the initial-state tropospheric zonal winddue to potential vorticity inversion. Note, however, thatalthough the tropospheric jet has increased to a maximumof 45 ms−1 (compared with 35 ms−1 in the control),the vertical shear between the surface and tropopauseis largely unchanged. In contrast, the addition of thepolar vortex results in a more significant change to thehorizontal shear throughout the troposphere, which has asubsequent effect on the tropospheric evolution.

Figure 3(d)–(f) shows the synoptic surface potentialtemperature evolution for the zonally symmetric pertur-bation with Q = 0.4f0 at days 10, 12 and 14. Again,

the baroclinic instability results in wave-breaking andirreversible mixing across the jet. However, in the per-turbed case increased horizontal shear throughout thetroposphere results in an eddy growth that is now moreconfined in latitude than before. This interpretation isconsistent with the results of Thorncroft et al. (1993),where the addition of barotropic shear to the initial meanzonal flow of an LC1 life cycle resulted in significantlydifferent nonlinear dynamics (the LC2 life cycle). Whileinterpretation of the evolution shown in Figure 3 in termsof LC1 and LC2 life cycles is difficult, owing to thesimplicity of our model, there are nevertheless clear dif-ferences between the two cases. For example, both thestrong equatorward wave-breaking of the control case andthe weaker, poleward wave-breaking are attenuated in thecase with Q = 0.4f0. At later times, days 12–14, thepoleward breaking in the control case results in strongtransport of low-latitude air into high latitudes, which ismuch weaker in the perturbation case.

To quantify the change in eddy growth with changes tothe stratospheric potential vorticity, we show in Figure 4the eddy kinetic energy as a function of time for thecase of exactly zero stratospheric potential vorticity (thecontrol, for which Q = 0) and two cases with increas-ing strength of stratospheric potential vorticity anomalies,Q = 0.2f0 and Q = 0.4f0. At early times there isa decrease in the growth rate of eddy kinetic energywith increasing Q, and, consequently, a reduction in



0

1

2

3

4

0 12 16

t (days)

EKE *104

4 8

Figure 4. Eddy kinetic energy as a function of time for the caseswith a zonally symmetric polar vortex with Q = 0 (bold solid line),

Q = 0.2f0 (dashed line) and Q = 0.4f0 (thin solid line).

the maximum value obtained around day 10. At firstsight, this dependence might appear contrary to the resultsof Wittman et al. (2007), who found using a primi-tive equation model that the growth rate of eddy kin-etic energy increased monotonically (at wavenumbersless than 7) with increasing vertical shear in the strato-sphere, their proxy for the polar vortex. In fact, thetwo sets of results appear to be consistent when wetake into account the details of the changes to the ini-tial shear caused by the stratospheric perturbations ineach case. In Wittman et al., the addition of verticalshear in the stratosphere appears to enhance the eddygrowth rate in a similar way as increasing the verticalshear in the troposphere in the traditional Eady model.In our model, on the other hand, the addition of thestratospheric potential vorticity anomaly has practicallyno effect on the vertical shear in the troposphere or any-where near the subtropical jet (Figure 1(a) and (b)). Asdiscussed above, however, it does increase the horizon-tal shear throughout the troposphere, with a resultantchange in the character of the life cycle, consistent withThorncroft et al. (1993). Of course, other differencesbetween the two modelling frameworks (e.g. cylindri-cal versus spherical geometry, quasi-geostrophic versusprimitive equations, latitudinal offset between the polarvortex and the subtropical jet) may also alter details ofthe evolution. However, the dependence of the growthrates on the tropospheric shear would appear to beconsistent.

Another useful measure of eddy growth, particularlysuited to the contour representation used here, is the waveactivity A, defined as

A(z, t) = 1

4ρ0(z)

∑k

qk

∮ k(z)

(r2 − r2

e

)2dθ, (9)

0

1

2

3

4

0 12 16

t (days)

wave activity

4 8

Figure 5. Combined surface and tropopause wave activity as a functionof time for the cases Q = 0 (bold solid line), Q = 0.2f0 (dashedline) and Q = 0.4f0 (thin solid line). Values have been normalized by

the initial angular impulse of the case Q = 0.

where the sum is over all contours k in a given verticallevel, where re is the radius of the undisturbed circularcontour enclosing the same area as k and qk is thevorticity jump on the kth contour. This is a nonlinearpseudo-momentum-based wave activity, second-order indisturbance amplitude, satisfying an exact conservationrelation (see Dritschel, 1988 and Dritschel & Saravanan,1994 for more details). The evolution of total waveactivity contained in the surface and tropopause potentialtemperature fields is shown in Figure 5 for the three casesQ = 0, 0.2f0, 0.4f0. Like the eddy kinetic energy,the wave activity increases due to the instability, andagain with weaker growth at higher values of Q. Atlate times the difference between the cases is even moremarked, with continued increase in the wave activity inthe case Q = 0 but not in the other cases. We noteincidentally that these differences in the troposphericwave activity are entirely due to the evolution of thebasic instability: the amount of wave activity ‘leaking’from the troposphere into the polar vortex is negligible,with values of the stratospheric wave activity remainingaround five orders of magnitude less than troposphericvalues throughout the evolution.

As in Wittman et al. (2004), it is also possible toconsider the surface geopotential height difference asa crude measure of the extent to which the instabilityprojects on to the Arctic Oscillation. Figure 6 showsthe difference between the surface geopotential heightanomaly at day 12 and day 0. The magnitude of thedipole structure resulting from the instability becomesweaker for larger values of Q, consistent with thereduction of eddy growth rates discussed above. Againthe difference in dependence on Q from that foundin Wittman et al. (2004) can be understood in termsof changes to the tropospheric shear induced by thestratospheric perturbation.



-5

-4

-3

-2

-1

0

1

2

3

4

5

0radius

surface geopotentialdifference

2 4 6 8

Figure 6. Difference between the surface geopotential at day 12 and day0 for the cases Q = 0 (bold solid line), Q = 0.2f0 (dashed line) and

Q = 0.4f0 (thin solid line).

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0

wavenumber

growth rate

1 2 3 4 5 6 7 8

Figure 7. The linear growth rate of eddy kinetic energy with differentinitial wavenumber perturbations in the troposphere: triangles corre-spond to the control case, Q = 0, and squares correspond to a zonally

symmetric stratospheric polar vortex with Q = 0.4f0.

Finally, we examine the effect of the stratosphericperturbation on the growth rate of other wavenum-ber disturbances to the surface and tropopause basicstate. Here we consider the difference between the twocases Q = 0 and Q = 0.4f0 and calculate the lin-ear growth rates of different wavenumbers by integrat-ing the model at early times only. One reason fordoing so is to verify that the addition of a polar vor-tex indeed reduces the growth rate at all wavenumbersrather than simply shifting the wavenumber of the fastest-growing mode. As seen in Figure 7, this is indeed thecase, with a modest reduction in growth rate across allwavenumbers.

3.3. Asymmetric perturbations

We now consider the effect of asymmetric perturbationsto the stratospheric potential vorticity on the troposphericevolution. Again, these perturbations are representativeof the shape of the polar vortex following a stratosphericsudden warming. Here, we focus on the displaced vortexcase (a wave-one warming) but note that very similarresults were also obtained in the case of a split vortex.

As before, the stratospheric perturbation has an instan-taneous effect through potential vorticity inversion on thetropospheric basic state, as shown in Figure 1(c) and (d)for the displaced and split-vortex cases with Q = 0.4f0.However, in comparison with Figure 1(b), we see that thebiggest differences in the winds between the symmet-ric and asymmetric vortex cases are in the stratosphere:because the asymmetric perturbation results in a lati-tudinally distributed zonal mean stratospheric potentialvorticity, the stratospheric jet is also broader and weakerthan in the case with a zonally symmetric vortex. In thetroposphere, on the other hand, the winds are very sim-ilar in all cases (symmetric, displaced vortex and splitvortex), in almost all respects, including the maximum ofthe subtropical jet and the vertical and horizontal shear.Thus the instantaneous effect of a rearrangement of thestratospheric potential vorticity on the tropospheric zonalflow is very small.

Figure 8 shows the synoptic surface potential tem-perature evolution at days 10, 12 and 14 for the caseQ = 0.4f0 and a vortex displacement of distance LR.What is immediately apparent is the strong departure fromsix-fold symmetry, which results from the interaction ofthe fastest-growing wave-six mode in the tropospherewith the growth of wave one initiated by the displacedpolar vortex. Although the growth rate of wave one ismuch smaller than that of wave six (Figure 7), the wave-one perturbation induced by the displaced polar vortexis larger than the initial wave-six tropospheric perturba-tion, with the result that significant growth of wave oneoccurs. This growth is naturally smaller for smaller ini-tial displacements of the polar vortex; however, it wasfound that even relatively small displacements (down toa distance of 0.2LR) were sufficient to cause significantdevelopment of a tropospheric wave one by day 16 (withgradually later development at smaller displacement val-ues). Thus a simple displacement of the polar vortex mayhave a significant effect on the tropospheric evolution.

The growth of eddy kinetic energy for the control case(Q = 0) and the two cases Q = 0.2f0 and Q =0.4f0 (with a vortex displacement of LR) is shown inFigure 9. The contribution of the initial stratospheric polarvortex displacement is evident in the eddy kinetic energyat t = 0. Despite these larger initial values, however,growth rates during the development of the instability aresmaller at larger values of Q, similar to the case of azonally symmetric stratospheric perturbation. Essentially,the eddy growth at early times is again dominated bythe fastest-growing mode, despite the presence of thelarge wave-one perturbation. This behaviour changesdramatically at later times, however, when the growth



(a) (b) (c)

Figure 8. Surface potential temperature at days 10, 12 and 14 for the case of a displaced polar vortex (centred at r = LR) with Q = 0.4f0.

0

1

2

3

4

0 12 16

t (days)

EKE *104

4 8

Figure 9. Eddy kinetic energy as a function of time for a displacedpolar vortex (centred at r = LR) with Q = 0 (bold solid line, sameas the control case), Q = 0.2f0 (dashed line) and Q = 0.4f0 (thin

solid line).

of the wave-one perturbation eventually overtakes that ofthe wave-six one: whereas saturation of wave six occursaround day 10, saturation of wave one does not occuruntil around day 16 or later, at significantly higher valuesof eddy kinetic energy. The evolution for the case of thesplit vortex (not shown) is qualitatively very similar tothat of the displaced vortex, with large eddy growth dueto the development of wave two dominating at late times.Eddy kinetic energy plots for this case are similar to thoseshown in Figure 9, while the surface potential temperaturefields show the late time development of wave four dueto the interaction of the wave-six and wave-two initialperturbations.

Instead of comparing the tropospheric evolution at dif-ferent Q, it is perhaps more instructive to consider thedifferences between the zonally symmetric and displacedpolar vortex cases, both with Q = 0.4f0, the poten-tial vorticity in the displaced vortex case being simply aredistribution of that in the symmetric case. The surface

-5

-4

-3

-2

-1

0

1

2

3

4

5

0

radius

surface geopotentialdifference

2 4 6 8

Figure 10. Difference between the surface geopotential height at day12 and day 0 for the displaced polar vortex (centred at r = LR) withQ = 0 (bold solid line), Q = 0.2f0 (dashed line) and Q = 0.4f0

(thin solid line).

potential temperature fields (compare Figure 3(d)–(f) andFigure 8) are significantly different between the two casesalready at day 10, but still more so at late times, whenmuch more vigorous mixing is found across a wider lati-tudinal region in the displaced vortex case. Similarly, theeddy kinetic energy shows significant differences at latetimes (compare the thin solid lines in Figures 4 and 9),although growth rates at early times are similar.

The more vigorous mixing across latitude has aninfluence on the zonal mean surface geopotential heightanomalies considered above. Figure 10 again showsthe difference between the surface geopotential heightanomaly at day 12 and day 0, for Q = 0, 0.2f0, 0.4f0but for the displaced vortex cases. As in the case ofthe zonally symmetric vortex, large Q again resultsin a less pronounced dipolar structure in midlatitudes.Comparing the thin solid line in this figure with thatin Figure 6 indicates that the redistribution of potentialvorticity in the stratosphere has also had an effect on



the resulting surface geopotential height following thebaroclinic development.

Finally, it is again useful to consider the evolution ofwave activity in these cases. At t = 0, the stratosphericwave activity (not shown) due to the displaced vortexis around four times that due to the initial troposphericwave-six perturbation. This stratospheric wave activitydecreases until around day 8 due to downward wave prop-agation (Scott & Dritschel, 2005) and is the source forthe subsequent growth of wave one in the troposphere.However, the initial stratospheric wave activity is muchsmaller than the subsequent difference in the troposphericwave activity between the zonally symmetric and dis-placed vortex cases, which can only be accounted forby the unstable growth of wave one, and not by simpledownward propagation of waves from the stratosphere.Final (day 16) values of stratospheric wave activity arearound four times larger than initial ones, indicating thatsome of the wave-one development in the troposphereis subsequently able to propagate upwards on the polarvortex edge (in contrast to wave six, which, as discussedabove, is trapped in the troposphere).

3.4. Influence of the basic state

To verify that the above results are not sensitive todetails of the tropospheric initial conditions, we haveperformed a number of variations with different values forthe surface and tropopause θ and different forms of thelatitudinal profiles, as well as different relative positionsof the polar vortex and subtropical jet. In all cases theresults are broadly similar to those reported above, withincreasing stratospheric potential vorticity perturbationresulting in weaker eddy growth in the troposphere,related to changes in horizontal shear. In this section webriefly describe a few a these variations.

In one variation we consider a surface potential temper-ature distribution that has a broader latitudinal structure,with 2w = 4LR for the surface distribution (and 2w = LRat the tropopause as before). This gives a slightly morerealistic zonal wind profile, as shown in Figure 12(a)and (b) for the cases of no polar vortex and a zon-ally symmetric polar vortex with Q = 0.4f0. The zonalwinds are now increasing monotonically at all heightsbetween the pole and the jet latitude, and decreasingequatorward of the jet, and, in particular, there is nowa surface shear that is cyclonic poleward of the jet lati-tude, closer to the observed zonal wind profile (comparethis with Figure 1(a), where the shear just poleward ofthe jet location is anticyclonic). This could be impor-tant, for instance in determining the direction of synop-tic wave-breaking during the evolution of the life cycle,where, for example, the poleward wave-breaking in Fig-ure 3 appears initially anticyclonic. We emphasize, how-ever, that here we are less interested in the details ofthe synoptic development than on the influence of thepolar vortex on the overall growth rate of the instabil-ity.

Figure 11 shows the surface potential vorticity atday 14 for three cases: (a) with Q = 0, similar to

the control case discussed above; (b) with Q = 0.4f0and a zonally symmetric vortex; (c) with Q = 0.4f0and a vortex that has been displaced horizontally by adistance LR. The corresponding figures from the previouscases are the right-hand panels in Figures 3 and 8.The increase in cyclonic meridional shear poleward ofthe jet has resulted in slower growth of the instabilityacross all cases, although the poleward breaking remainsinitially anticyclonic. However, and more importantly,the difference between cases remains much the sameas before. In terms of both eddy kinetic energy andwave activity (not shown), the largest growth is again forthe case Q = 0, while in the cases with Q = 0.4f0the case of the displaced vortex again exhibits largereddy growth at late times than the case of the zonallysymmetric vortex.

In a second variation we considered the effect of simplyadding a r-dependent barotropic shear to the initial state,similar to the LC2 life cycle case considered in Thorncroftet al. (1993). The resulting initial zonal wind profilesare broadly similar to those shown in Figure 12, as isthe subsequent evolution (not shown). In particular, weagain found a robust decrease in the growth rate of theinstability when the polar vortex was added, and a latetime increase in eddy kinetic energy in the displacedvortex case due to the growth of low wavenumbers.

Finally, we considered a basic state characterized bytwo surface temperature fronts located poleward andequatorward of jet latitude. Such a distribution arisesnaturally as a result of eddy mixing of surface temperaturedue to baroclinic waves, and can be considered to rep-resent the statistically stationary state of the atmosphere.This state would also be obtained by zonally averagingthe final temperature distributions in the above calcula-tions. For completeness, therefore, we repeated the mainseries of experiments using this tropospheric basic state,although it could be argued that such a state is not themost appropriate choice of initial conditions in the life-cycle approach. In fact, we found that the details of thissurface-temperature basic state made very little differenceto the influence of the polar vortex on the instability, withan increase in polar vortex strength again resulting in adecrease in growth rate. Overall, it appears therefore thatthe influence of the polar vortex on the evolution is insen-sitive to details of the initial basic state, at least withinthe limitations of our restricted model.

4. Discussion

To summarize our results, changes to the stratosphericpotential vorticity have a significant impact on the devel-opment of baroclinic instability in an Eady-like model.The dependence is such that increasing the strength ofthe polar vortex tends to decrease the eddy growth in thetroposphere. This is found not just in the zonally sym-metric cases, comparing zonally symmetric stratosphericperturbations of different potential vorticity magnitudes,but also in cases of zonally asymmetric disturbances to a



(a) (b) (c)

Figure 11. Surface potential temperature at day 14 for the case with broad surface-temperature initial condition: (a) Q = 0 (the control); (b) azonally symmetric vortex with Q = 0.4f0; (c) a displaced polar vortex (centred at r = LR) with Q = 0.4f0.

0

H_t

2H

3H

0

radius (r)

heig

ht (

z)

0

radius (r)

(a) (b)

2 4 6 2 4 6

Figure 12. Initial zonal wind profiles for the case with broad surface-temperature initial condition: (a) Q = 0 (the control); (b) a zonallysymmetric vortex with Q = 0.4f0. The contour interval is 5 ms−1.

polar vortex of given potential vorticity. The latter sce-nario extends previous work that considered only zonallysymmetric stratospheric perturbations. In particular, wefound that there is a large difference in the troposphericevolution between cases representing a strong vortex andcases representing the vortex following either a wave-one or wave-two sudden warming. Differences in thetropospheric evolution include the growth of eddy kin-etic energy and wave activity, as well as synoptic-scaledetails of the wave-breaking and the latitudinal extent ofmixing within the troposphere.

Our study differs fundamentally in philosophy fromthe related work of Wittman et al. (2004, 2007), in whichperturbations were made to the stratospheric zonal winds.It is of course true that by perturbing the stratosphericpotential vorticity, as is done here, one is also perturb-ing the tropospheric zonal flow. However, because ofthe fundamental nature of the potential vorticity (Hoskinset al., 1985) it is perhaps justified to consider such strato-spheric potential vorticity perturbations as dynamical per-turbations to the stratosphere only. In our case, theseperturbations have been carefully isolated from the tro-posphere by including a ‘sub-vortex’ region between thetroposphere and lowermost polar vortex, in which the

potential vorticity is unperturbed. Moreover, actual dif-ferences in the initial tropospheric zonal winds betweenzonally symmetric and asymmetric perturbation casesare very slight (compare Figure 1(b)–(d)). Finally, thiskind of perturbation is arguably closer to the situationresulting from a stratospheric sudden warming. One ofthe important results of the present article is that thepotential vorticity perturbations made here result in sig-nificantly larger differences to the tropospheric evolutionthan obtained by perturbations to the stratospheric windsalone.

One significant difference between our results andthose of Wittman et al., is the sense in which a strato-spheric perturbation affects the growth of the instability.Wittman et al., found an increase in eddy growth rateswith increasing stratospheric shear, whereas we find adecrease in growth rates with increasing stratosphericpotential vorticity. The results are not inconsistent whenfull account is taken of changes to the tropospheric shearresulting from the stratospheric potential vorticity per-turbation in our case, which tends to leave the verticalshear unchanged but increases the horizontal shear. Thedecrease in growth rates we observed may therefore beattributed to a change in the nature of the baroclinic devel-opment similar to that found by Thorncroft et al. (1993).



One conclusion that may be drawn from both Wittmanet al., and the present work is that the tropospheric evolu-tion depends rather sensitively on the stratospheric statethrough details of the shear in the troposphere and nearthe subtropical jet.

Acknowledgements

The authors wish to thank Andrew Charlton, DavidDritschel, and Lorenzo Polvani for useful comments anddiscussions during the preparation of this manuscript.Financial support was provided through an EPSRC CASEstudentship with the UK Meteorological Office.

References

Ambaum MHP, Hoskins BJ. 2002. The NAO troposphere–stratosphereconnection. J. Climate 15: 1969–1978.

Baldwin MP, Dunkerton TJ. 1999. Propagation of the Arctic oscillationfrom the stratosphere to the troposphere. J. Geophys. Res. 104:30 937–30 946.

Baldwin MP, Dunkerton TJ. 2001. Stratospheric harbingers ofanomalous weather regimes. Science 294: 581–584.

Black RX. 2002. Stratospheric forcing of surface climate in the Arcticoscillation. J. Climate 15: 268–277.

Bretherton FP. 1966. Critical layer instability in baroclinic flows. Q.J. R. Meteorol. Soc. 92: 325–334.

Charlton AJ, Polvani LM. 2007. A new look at stratospheric suddenwarmings. Part I. Climatology and modeling benchmarks. J. Climate20: 449–469.

Charlton AJ, O’Neill A, Stephenson DB, Lahoz WA, Baldwin MP.2003. Can knowledge of the state of the stratosphere be used toimprove statistical forecasts of the troposphere? Q. J. R. Meteorol.Soc. 129: 3205–3224.

Dritschel DG. 1988. Contour surgery: A topological reconnectionscheme for extended integrations using contour dynamics. J. Comp.Phys. 77: 240–266.

Dritschel DG, Ambaum MHP. 1997. A contour-advective semi-Lagrangian numerical algorithm for simulating fine-scale conserva-tive dynamical fields. Q. J. R. Meteorol. Soc. 123: 1097–1130.

Dritschel DG, Saravanan R. 1994. Three-dimensional quasi-geostrophic contour dynamics, with an application to stratosphericvortex dynamics. Q. J. R. Meteorol. Soc. 120: 1267–1297.

Hartmann DL, Wallace JM, Limpasuvan V, Thompson DWJ,Holton JR. 2000. Can ozone depletion and global warming interactto produce rapid climate change? Proc. Nat. Acad. Sci. (Wash. DC)97: 1412–1417.

Hoskins BJ, McIntyre ME, Robertson AW. 1985. On the use andsignificance of isentropic potential vorticity maps. Q. J. R. Meteorol.Soc. 111: 877–946.

Juckes MN. 1994. Quasigeostrophic dynamics of the tropopause.J. Atmos. Sci. 51: 2756–2768.

Kodera K, Yamazaki K, Chiba M, Shibata K. 1990. Downwardpropagation of upper stratospheric mean zonal wind perturbationto the troposphere. Geophys. Res. Lett. 17: 1263–1266.

Kushner PJ, Polvani LM. 2004. Stratosphere–troposphere couplingin a relatively simple AGCM: The role of eddies. J. Climate 17:629–639.

Macaskill C, Padden WEP, Dritschel DG. 2003. The CASL algorithmfor quasi-geostrophic flow in a cylinder. J. Comp. Phys. 188:232–251.

Matsuno T. 1971. A dynamical model of the stratospheric suddenwarming. J. Atmos. Sci. 28: 1479–1494.

Muller JC. 1991. Baroclinic instability in a two-layer, vertically semi-infinite domain. Tellus A 43: 275–284.

Polvani LM, Kushner PJ. 2002. Tropospheric response to stratosphericperturbations in a relatively simple general circulation model.Geophys. Res. Lett. 29: 1114. DOI: 10.1029/2001GLO14284

Scaife AA, Knight JR. 2008. Ensemble simulations of the coldEuropean winter of 2005–2006. Q. J. R. Meteorol. Soc. 134:1647–1659.

Scott RK, Dritschel DG. 2005. Downward wave propagation on thepolar vortex. J. Atmos. Sci. 62: 3382–3395.

Shindell DT, Rind D, Balachandran N, Lean J, Lonergan P. 1999. Solarcycle variability, ozone, and climate. Science 284: 305–308.

Shindell DT, Schmidt GA, Mann ME, Rind D, Waple A. 2001. Solarforcing of regional climate change during the Maunder minimum.Science 294: 2149–2152.

Song Y, Robinson WA. 2004. Dynamical mechanisms for stratosphericinfluences on the troposphere. J. Atmos. Sci. 61: 1711–1725.

Thompson DWJ, Wallace JM. 1998. The Arctic oscillation signature inthe wintertime geopotential height and temperature fields. Geophys.Res. Lett. 25: 1297–1300.

Thompson DWJ, Wallace JM. 2000. Annular modes in the extratropicalcirculation. Part 1: Month-to-month variability. J. Climate 13:1000–1016.

Thompson DWJ, Baldwin MP, Wallace JM. 2002. Stratosphericconnection to northern hemisphere wintertime weather: Implicationsfor prediction. J. Climate 15: 1421–1428.

Thompson DWJ, Furtado JC, Shepherd TG. 2006. On the troposphericresponse to anomalous stratospheric wave drag and radiative heating.J. Atmos. Sci. 63: 2616–2629.

Thorncroft CD, Hoskins BJ, McIntyre ME. 1993. Two paradigms ofbaroclinic-wave life-cycle behaviour. Q. J. R. Meteorol. Soc. 119:17–55.

Wittman MAH, Polvani LM, Scott RK, Charlton AJ. 2004.Stratospheric influence on baroclinic lifecycles and its connectionto the Arctic oscillation. Geophys. Res. Lett. 31: L16113.

Wittman MAH, Charlton AJ, Polvani LM. 2007. The effect of lowerstratospheric shear on baroclinic instability. J. Atmos. Sci. 64:479–496.

WMO. 2007. ‘Scientific assessment of ozone depletion: 2006’, WMOGlobal Ozone Research and Monitoring Project Report No. 50.World Meteorol. Org.: Geneva.


The inﬂuence of stratospheric potential vorticity on ...rks/reprints/smy_scott_qjrms_2009.pdf ·...

Documents

Transcript of The inﬂuence of stratospheric potential vorticity on ...rks/reprints/smy_scott_qjrms_2009.pdf ·...