4 The impact of large-scale waves on the stratosphere

4.1 Forcing of the mean state

As we have seen, once we know r � F from observations (or theory) we arein a position to address the waves� impact on the mean state, using ourquasigeostrophic TEM1 equations

@�u

@t� f��v� = �Gx +

1

�r � F , (1)

@��

@t+ �w�

@��

@z= (��)�1 �J , (2)

@v�@y

+1

�

@(�w�)

@z= 0 , (3)

and

f0@�u

@z= ���

H

@��

@y. (4)

The problem is then simply the mathematical one of calculating the solutionfor @�u=@t, @��=@t, �v� and �w�. Note that in our qualitative diagnosis of thestratospheric circulation �in the absence of eddies,�we used the second andthird of these equations as they are: we were (if unknowingly, at the time)deducing the residual circulation without any need to invoke the absence ofeddies. As before, we use the continuity equation to de�ne a streamfunction�� such that

�v� = �1

�

@(���)

@z; �w� =

@��@y

. (5)

4.1.1 The lower boundary condition

Before trying to understand the nature of a response to a given forcing, weneed to be clear about boundary conditions on ��. They are straightforwardas z !1, but a bit more subtle at the surface. Recall that

�w� = �w +@

@y

v0�0

��z

!:

1We�ll talk about non-QG later.

1

On a �at surface at z = 0 with nonzero heat �ux, �w(y; 0) = 0 and hence�w�(y; 0) is generally nonzero if the heat �ux is �nite; this is the case in (e.g.)the Eady and Charney problems. On the other hand, if we think of a simpletopographically forced Rossby wave with no mean horizontal temperaturegradient near the lower boundary, then �w(y; 0) 6= 0 in general: the kinematicboundary condition is

w = u � rh0Bwhere the boundary is at z = h0B and h

0B = 0, then, to O("

2),

�w = u0 � rh0B = r � u0h0B =@

@y

�v0h0B

�:

So in this case �w is not necessarily zero on the boundary if hB is nonzero.The explanation of this is as follows. If v0h0B is nonzero, there is a nonzerocorrelation between v0 and hB. As shown in Fig 4.1.1a, this means there is a

net northward (or southward) mass �ux in the �valleys�which is not compen-sated in the �hills�simply because the corresponding region is underground.If the �ow is adiabatic near the ground,

Dg�0 + w0��z = 0 .

2

Butw0 = DghB ,

whencehB = ��0=��z

and

v0h0B = �v0�0

��z

which is negative (/positive) for any northern (/southern) hemisphere upwardpropagating wave with Fz > 0. So v0 tends to be poleward in the valleys,as shown in the �gure. Assuming the topography is �nite in y, this mass�ux must be a function of y and there is therefore a mass convergence; theonly way this can be balanced is by a mean vertical motion (as shown in Fig4.1.1b) � which is precisely what our equation for �w is telling us.Our boundary condition on �w therefore becomes

�w = � @

@y

v0�0

��z

!:

The condition on �w�, therefore, is simply

�w� = 0

on z = 0. So the condition on the residual cirulation on an isothermaltopographic boundary is simpler than for the conventional mean2. (Onceagain, the simple condition on �w� in this steady, small-amplitude situationre�ects the Lagrangian problem� there can be no net mass �ux through theboundary, no matter what its shape is. Amaterial tube lying on the boundarymust remain on the boundary; therefore it can have no mean velocity normalto the boundary.)Note the relationship between the boundary condition on �w and the form

drag on the boundary. We saw earlier that f0v0h0B = �p0h0x for geostrophic�ow, and h0x is just the local slope of the boundary, and p0h0x is the form

2As we noted earlier, this is not a general statement. On a �at, baroclinic boundary �e:g:, in the Eady and Charney baroclinic instability problems � the opposite is true: �w iszero but �w� is not. In these cases, the residual and Lagrangian means behave di¤erentlynear the surface.

3

stress acting on the boundary. So the form drag acting on the atmosphericcolumn is

�p0h0x = �f0v0h0B = ��f0v0�0

��z= �Fz(y; 0) :

So there is a net (negative) momentum input to the atmosphere, manifestedby the surface form drag from the topography, which in the TEM formalismis manifested entirely in this problem by the EP �ux out of the boundary.In the conventional Eulerian mean case, it is the mean meridional �ow outof the boundary that carries the form drag into the atmosphere. The meanvertical angular momentum �ux (for our QG �-plane) is

Az = � �wf0y = ��f0y@

@y

�v0h0B

�= � @

@y

��f0y v0h0B

�+ �f0v0h0B :

The �rst term integrates to zero and so is irrelevant to the net transport.

4.1.2 The �nonacceleration�case

If our planetary wave is undissipated, we have r � F = 0 everywhere. Wehave already seen that the O(�2) problem for @u=@t, @�=@t, �v� and �w� isindependent of the waves in this limit, provided the boundary conditions arealso (and we have just seen that this is true). If �J and �Gx are zero, then oursolution is simply

@�u

@t=@��

@t= �v� = �w� = 0

everywhere. The conventional mean circulation is nonzero, however, being

�w = � @

@y

v0�0

��z

!

everywhere. Note that if we assume latitudinal symmetry, and therebyneglect u0v0, the vanishing of r � F implies that f0v0�0=��z is independent ofz; hence �w is a function of y only, and �v = 0. So, in conventional terms,the situation looks like Fig 4.1.2, with the heat budget being satis�ed by abalance between the �eddy heat �ux convergence�, �@

�v0�0�=@y, and mean

adiabatic heating and cooling, � �w��z = ��zh@�v0�0�=@yi=�z everywhere.

4

4.1.3 The steady problem

In the steady inviscid case (we assume that the only waves or �eddy friction�present are included in our r � F), we must have

�f��v� =1

�r � F .

So if we know r � F, we immediately know �v�� the response in terms of�v� is purely local. In order to reach a steady state, the Coriolis e¤ect ofthe residual circulation must exactly balance the wave forcing. This is oftenreferred as �wave pumping�� the wave drag forces the mean residual �owpoleward (because the angular momentum must decrease in the presence ofthe drag). We can then determine �w� from continuity, to �nd

@

@z(� �w�) = ��

@�v�@y

=1

f0

@

@y(r � F) .

Given � �w� ! 0 as z ! 1, this de�nes �w� everywhere. In fact, for a purelyupward propagating wave with Fy = 0, r � F = @Fz=@z and Fz must vanishat large z, we can integrate to get

� �w� =1

f0

@Fz@y

:

Note that �w� then depends only onr�F at and above the level of interest: thisis the concept of �downward control�of the meridional circulation. [Notealso that this solution does not satisfy our lower boundary condition � we

5

will get back to this in a moment]. Then the thermodynamic equation mustsatisfy

�J = �� �w���z =���zf0

@Fz@y

.

The mean state must be pushed out of radiative equilibrium just enough fordiabatic heating/cooling to balance the adiabatic cooling/heating associatedwith �w�. Note that we can calculate the diabatic heating from a knowledge ofr�F � evidence that in the quasigeostrophic regime it is the dynamics thatdrives the diabatic heating and not vice-versa. From a radiative calculationwe can than determine �T ; for our Newtonian cooling, it is just

��( �T � Te) = �J =�cp .

The overall picture is shown in Fig 4.1.3.

Consider now the thorny issue of the bottom boundary condition. Thisseems to be saying that there is something wrong with this solution� andof course there is. We have a nonzero form drag on the topography andhence a nonzero net stress on the atmosphere. The mean momentum of theatmosphere (given our assumptions) will therefore change with time � therecan be no truly steady state unless other process come to act. The only waya steady state can be achieved appears to be as follows:

6

1. The mean �ow near the ground changes at leading order such thatthe net stress vanishes (in our case, this can be done either by themean wind at the surface vanishing so that there is no form drag� inwhich case the wave will disappear too!� or the mean wind at low levelschanging such that vertical propagation is not permitted). The kindof steady circulation we have just discussed will, for a steady forcing,be established locally in the stratosphere quite quickly (in fact on atime scale � 1=� � 10 days). However, changing the mean �ow lowerdown will take much longer, on two counts � the meridional velocitieswill be weaker there because of the pressure factor, and, since @u=@tis proportional to �2, it will take a time proportional to ��2 (very longfor small �) to e¤ect a �nite change in the mean �ow. So, on a �nitetimescale (and the forcing only exists during the winter season) oursteady state may never be reached at low levels and the situation willbe as depicted in Fig 6.10.

2. Alternatively, surface friction will come into play. Even though thesee¤ects are weak by most standards, the required force required to bal-ance the topographic stress is not usually very large and can likely beachieved by a weak induced �ow at the surface.

Note that if we had forced our wave thermally, rather than by topography,there would be no net momentum input to the atmosphere and we would nothave run into this problem. Then Fz would be zero at the ground and wecould get a truly steady state, with the residual circulation being closedo¤ within the forcing region where F is divergent (in fact where the e¤ectiveforce on the mean �ow, ��1r�F, must exactly balance that in the dissipationregion. See Fig 4.1.3. This simple picture assumes that the region of wavedissipation lies directly above the forcing. If the waves were to propagatehorizontally as well as vertically, this would not be true and the considerationsnoted in (i) and (ii) above would become relevant to this case also.

4.1.4 The transient case (the �Kuo-Eliassen problem�)

We now take the opposite limit, and look at the initial transient response ofthe system (1)-(4) when the basic state is initially in radiative equilibrium.We �rst take @=@t of the thermal wind equation (4)

f0@2�u

@z@t= ���

H

@2��

@y@t

7

We then substitute from the momentum and heat equations (1) and (2) andsubstitute for �v� and �w� from (5) to get

N2@2��@y2

+ f 20@

@z

�1

�

@

@z(���)

�= f0

@

@z

�1

�r � F

�+

�

�H

@ �J@y

+ f0@ �Gx@z

. (6)

Given r�F from our wave solution, together with �J and �Gx, we can thensolve this for ��, given appropriate boundary conditions. In terms of �v and�w, we know that �v = 0 on the sidewalls y = 0; L so that we may de�ne � = 0there. Since

�� = �+v0�0

��z,

it follows (since v0 = 0 on the walls) that �� = 0 there also.Suppose, for the sake of getting a feel for the solutions, we assume that we

can write the eddy forcing as r � F = �B(z) sin `y. Note that B is negative� the wave acts as an easterly force on the mean state.We now need to solve our equation (6) for ��. We consider the wave

driving to be the only forcing of the meridional circulation, i:e:, �J = 0 and�Gx = 0 (so we have an inviscid system, initially in radiative equilibrium).

8

Our equation now becomes

N2@2�

@y2�+f 20

@

@z

�1

�

@

@z(���)

�= �f0

dB

dzsin `y :

Now, writing��(y; z) = X(z) sin `y ;

our equation becomes�d

dz� 1

H

�dX

dz� N2

f 20`2X = � 1

f0

dB

dz(z)

In general, the solution is not very straightforward. However, since the prob-lem is linear, we can use a Green function approach, writing

B(z) =

Z z

0

B(z0)�(z � z0)dz0 .

Then in general the solution is

X(z) =

Z z

0

G(z; z0)B(z0)dz0 ,

where the Green function G(z; z0) for the problem is the solution to�d

dz� 1

H

�dG

dz� N2

f 20`2G = � 1

f0

d

dz[�(z � z0)]:

Thus the Green function is the solution we would get in response to a �-function in B (and therefore in minus r�F, i:e:, in the convergence of F)such as is shown in Fig 4.1.4a. This in itself is an interesting case, apart fromits value in constructing the solution to the more general problem. Note thata �-function in B, and therefore in �r � F, at z = z0 corresponds to a stepfunction there in Fz, and hence total absorption of an upward propagatingwave at z = z0.Now, except at z = z0;

G � e�z ,

where

2�H = 1�r1 +

4H2

D2

9

withD = f0=N`

which is the �Rossby height�for the problem (i:e:, the length scale `�1 is theRossby radius based on the height D). Thus there are two solutions for X,one of which always grows with height and one always decays (so it is alwayseasy to decide which satis�es our boundedness constraint). For H >> D,� � �1=D; for H << D, � � �H=D2, +1=H (note that the negative rootfor �, decaying upward, has a smaller absolute value than the negative root;this asymmetry re�ects the e�z=H dependence of pressure).Above z0, then

G = A� e��(z�z0) ,

where �� is the negative, bounded root. To make things simple, let us assumethat �+z0 � 1. Then a solution

G = A+ e�+(z�z0)

in z < z0 will almost satisfy the boundary condition G = 0 on z = 0:We now need matching conditions across z0 to close the solution. If we

integrate our o:d:e once across z0 we get

lim�!0

Z z0+�

z0��

�d2G

dz2� 1

H

dG

dz� N2

f 20`2G

�dz = � 1

f0lim�!0

Z z0+�

z0��

@

@z(�[z � z0])dz

= 0 ;

whence

�

�dG

dz� 1

HG

�= 0

where � means the increment across z0. Integrating a second time, we getthe second condition

�G = � 1f0.

These two conditions give us

(�� � 1

H)A� = (�

+ � 1

H)A+ ,

andA� � A+ = �

1

f0.

10

Hence

f0A+ =1H� ��

�+ � �� > 0

and

f0A� = ��+ � 1

H

�+ � �� < 0 .

The resulting circulation (actually the mass streamfunction ���) is shown inFig 4.1.4.

The residual northward �ow corresponding to this Green�s function solu-tion is

�v� =

��(�� � 1

H)A� e

��(z�z0) sin `y ; z > z0�(�+ � 1

H)A+ e

�+(z�z0) sin `y ; z < z0

i:e:;

�v� =

(H

f0�D2e�

�(z�z0) sin `y , z > z0H

f0�D2e�

+(z�z0) sin `y ; z < z0

where

� =

�1 +

4H2

D2

� 12

.

Since eF0 < 0, �v� is negative (equatorward) both above and below z0, as seenon Fig 4.1.4. The return �ow is of course in a delta-function at z0; this ispositive (poleward) as shown on Fig 4.1.4.

11

The mean acceleration may now be determined from

@�u

@t= f0�v� +

1

�r � F

Above and below z0, r � F = 0 and so @u=@t = f0�v�; we can obtain @u=@ttrivially from the above relations for �v�. In particular, the acceleration iseverywhere negative. Note that, unlike �v�, the acceleration does not have adelta-function at z0, since the delta-function in f0�v� is balanced by that in��1r � F there. The temperature tendency follows simply from �w� :

@�

@t= � �w���z :

The situation we have is therefore as depicted in Fig 4.1.4:

1. A wave propagates up from z = 0, undissipated until it reaches z = z0,where all the wave activity is entirely dissipated, implying a �-functioneasterly force acting on the mean �ow.

2. This drives a residual circulation. At z = z0, this has a �-functionpoleward �ow, and therefore gives a positive Coriolis acceleration, thustending to oppose the r�F forcing there (shades of Lenz�law). The re-circulation extends a height j��j�1 � min(1; D) above and (�+)�1 �min(H;D) below, where the �ow is equatorward and thus induces anegative (easterly) acceleration. The vertical residual motion inducesadiabatic warming or cooling in the pattern shown.

3. The vertical pro�le of the mean acceleration in mid-channel. Theoverall e¤ect of the induced circulation is to smooth out the response(@u=@t) relative to the r � F forcing. This is what it must do to keepthe mean state in thermal wind balance. (and of course the circulationcan only redistribute mean momentum� it cannot create or destroy it).

For a more realistic case in which r � F has a smooth pro�le, we needto regard the above solution as a Green�s function, as we noted earlier. Thesolution to a smooth forcing is a convolution of the forcing with our Green�sfunction and so is smoothed out in the vertical. Note that if the vertical scaleof the forcing is much greater than j��j�1 and (�+)�1, the smoothing e¤ectof the residual circulation is ine¤ective.

12

4.1.5 More realistic solutions on the sphere

[Haynes et al., J. Atmos. Sci., 48, 651, 1991]On the sphere, the transient problem can be solved using a similar ap-

proach to the QG ��plane case. Just as sin functions form the natural basisfor calculating the meridional streamfunction a bounded �-plane, so on thesphere the natural basis functions are �Hough functions�� the eigenvectorsof the unforced, spherical equivalent of (6). The steady problem is alsobroadly similar; the steady angular momentum budget can be written

�v� � rM =�v�a

@M

@'+ �w�

@M

@z=1

�r � Fs (7)

where M = a2 cos2 ' + ua cos' is the angular momentum per unit massand Fs = F a cos' is the appropriate spherical geometry form of the EP�ux. So just as we could determine the QG solution by integrating thesteadu momentum budget down from in�nity, we can do the same here, butnow the integration is downward along the M contours. In the QG limit� and in practice, outside the tropics � M is dominated by the planetarycomponent and these contours are essentially vertical, as shown in Fig 1Some results are shown in Fig. 4.1.5. There is a local patch of nonzero

r�F between about 20oN and 60oN and 30�50 km altitude. The top �gureshows the mass streamfunction (��� in our notation) for the instantaneous,Kuo-Eliassen, response. Note that, even though this is the transient case,there is little response above the forcing. This is because, even in thetransient case, the mass circulation is mostly below the forcing, just becauseof density strati�cation (but this is not true of ��, nor of �v� or @�u=@t). Thecirculation extends in latitude beyond the forcing latitudes, but does notpenetrate deeply into the tropics.The long-term steady response is shown in the bottom frame of Fig. 4.1.5.

This is just as we would have predicted from the QG analysis: northwardresidual �ow is con�ned to the forcing region, and the compensating verticalmotion is con�ned to the same latitudes and lower altitudes. Since the eddyforcing is here presumed to be con�ned to extratropical latitudes, so then isthe meridional circulation excluded from the tropics, unlike what is deducedfrom observations, where upwelling is con�ned to the tropics, an importantfact for stratospheric chemistry. The middle �gure shows the response forforcing that varies with an annual cycle: because of the time-dependence, thelatitudinal con�nement is no longer evident, and there is � in �midwinter��

13

Figure 1: Angular momentum distributions [Haynes et al., 1991]

14

15

upwelling across the tropics, qualitatively in keeping with what observationsrequire. So one might be tempted to see this as an explanation for tropicalupwelling. But [Plumb & Eluszkiewicz, J. Atmos. Sci., 56, 868, 1999]one cannot explain the annual mean in this way: the solution shown in themiddle frame of 4.1.5 is periodic in time, so air downwells at the equator on�summer.� To explain what happens in the tropics, one needs the forcing toextend deep into the tropics. But note, from Fig. 1, that the M gradientsvanish in the tropics (the same thing as saying that �a vanishes there), andso (7) tells us that only very weak wave drag is needed there. In fact, Plumb& Eluszkiewicz showed in their 2D model that weak friction was enough topermit subtropical forcing to force upwelling right across the equator. Inreality, it seems there is enough penetration of large-scale waves to do thejob: models suggest that upwelling at the equatorial tropopause is controlledby synoptic scale eddies reaching the tropics.Note also that, just like the Held-Hou theory of the tropical Hadley circu-

lation, weakM gradients in the tropics make it possible for thermally drivencirculations to exist, independent of any eddy forcing. Thermally driven cir-culations in the tropical stratosphere have been simulated by Dunkerton (J.Atmos. Sci., 46, 2325, 1989) and Semeniuk & Shepherd (J. Atmos. Sci., 58,3097, 2001). However, while such circulations may be relevant in the upperstratosphere, they are much too weak in the lower stratosphere to explainobservations.The steady circulation of the bottom frame of Fig. 4.1.5 similarly does

not extend deep into high latitudes, whereas tracer observations and polarwarming require at least some descent there. Again, this could be in partbecause some r � F is located there (through thermal dissipation or pene-tration of wave breaking into the vortex). This is re�ected in calculationsof the meridional circulation in the stratosphere. Fig. 2 (we have seen thisbefore) shows, in the rightmost frame, calculated diabatic heating rate (Kday�1) on one day in the southern hemisphere at 1100K Notice how thestrongets cooling (and therefore strongest diabatic descent _�) is found notdeep within the vortex, but at its edge, in fact on the poleward edge of theanticyclone/�cat�s eye�where the stirring is greatest. Nevertheless, thereis some descent going on throughout the vortex. We�ll see later, in fact,that tracer observations within the polar vortex show mesospheric air de-scending down into the middle, and perhaps even lower, stratosphere duringthe course of a winter, implying that the dynamical forcing of the descentis partly e¤ected in the mesosphere. This suggests that gravity waves �

16

Figure 2: The southern upper stratosphere, 1992 Sep 09. Left: 10hPageoppotential; middle, 1100K tracer calculation for 10 days, ending on 09/09and initialized with latitude circles; right, diabatic heating rate (K day�1) at10hPa.

which must be primarily responsible for the mesospheric circulation � mayplay a role in the high-latitude circulation.

4.2 Variability of the stratospheric circulation

4.2.1 Observations

Whilw the climatological picture we have been dealing with thus far repre-sents the seasonal average situation, the stratosphere exhibits very strongweek-to-week and interannual variability. Fig. 3 shows the variability ofmean 30hPa �T (80 � 50oN), and amplitudes (dam) of zonal waves 1 and2 at 60oN over several winters. There is a high degree of variability in all 3quantities. Bursts of planetary wave activity (which may be of wave 1, wave2, or wave 1 followed by wave 2) are invariably associated with weakening oreven reversal of the high latitude temperature gradient. At such times, thehigh latitude mean zonal shear weakens or reverses; these events are knownas stratospheric warmings3. There are typically 2 or 3 such events through

3When the mean zonal �ow at 60o reverses all the way down to 10hPa, they are clas-si�ed as major warmings; otherwise they are referred to as minor. However, there is acontinuous distribution of such events [Coughlin & Gray, J. Atmos. Sci., 2009] and thereappears to be no fundamental di¤erence between the two.

17

Figure 3: [Labitzke, J. Geophys. Res., 1981]

18

a northern winter; a major warming occurs every 2 years or so. In thesouthern hemisphere, warmings occur (usually in spring) but only one majorwarming has been observed (in 2002).A particularly clean example of a major warming occurred in January

2009. Figs. 4.2.1 and 4.2.1 show the structure of 10hPa height and temper-

ature at its height on 2009 Jan 23. There is a clear maximum of temperatureover the polar cap (the warmest region of the hemisphere, in fact) and thepolar vortex has split into two in a dramatic wave 2 structure.Some years are very disturbed, others are not, resulting in large interan-

nual variability. Fig. 4.2.1 shows the frequency distribution of monthly mean30hPa polar temperature, revealing large degree of interannual variabilitythat is associated with such events during winter. (In the southern hemi-sphere, interannual variability is relatively weak in midwinter, and greatestin spring.) In February, the spread is more than 40oC.

4.2.2 Early theory

Warmings appear to be just large-amplitude wave breaking events in whichthe material ejected from the vortex comprises a large fraction of the vortexitself, but saying that alone does not explain very much. The �rst dynamicaldescription of warmings was by Matsuno [J. Atmos. Sci., 1971]. Given the

19

wave ampli�cation, the short term behavior (on times less than the radiativerelaxation time � ten days or more in the middle and lower stratosphere) ischaracterized by wave activity convergence, since

r � F = �@A@t

:

The response of the mean state is then as in the transient Kuo-Eliassenproblem, with mean subsidence over the polar cap in the middle and lowerstratosphere, with consequent adiabatic warming. (The reverse is observed inthe mesosphere, as we would expect if r �F is con�ned to the stratosphere.)Consistent with thermal wind balance, the vertical zonal wind shear willweaken or even reverse. In a large event, westerlies will become easterlies inthe upper stratosphere. Then the wave will break in the critical layer where�u reverses, allowing sustained convergence of wave activity after the transientgrowth has stopped. Then the winds will become even more easterly, leadingto descent of the zero wind line through the stratosphere.To some extent, then, the warmings seem to be a straightforward con-

sequence of the wave ampli�cation. Which just begs the question: why dothe waves amplify? Originially, it was thought that the �ux of wave activityinto the stratosphere is under tropospheric control, and that variability ofthis �ux simply re�ects tropospheric variability. However, modeling results

20

21

suggest otherwise.

4.2.3 Modeling

Truncated �-channel The simplest model of planetary wave - mean �owinteraction is that of Holton and Mass [J. Atmos. Sci., 33, 2218, 1976].This is a � � channel model, severly truncated so that both 0 and �u arerepresented by a single mode, sin ly, in latitude. The geopotential amplitudeof the wave is �xed at the lower boundary, and the mean state is relaxedtoward a speci�ed radiative equilibrium state by Newtonian relaxation.The model exhibits threshold behavior. Fig. 4.2.3 shows the evolution of

the mean zonal �ow in mid-channel just below and just above the thresholdforcing amplitude. With subcritical forcing, the system evolves to a steadystate, in which the wave�s impact on the mean �ow is weak. With super-critical forcing, however, the system exhibits sustained vacillations in whichthe mean �ow is much weaker than that corresponding to unforced equillib-rium. The evolution of wave and mean �ow in a strongly supercritical sateis shown in Fig. 4.2.3. The wave exhibits strong (and in this case nonperi-odic) variability, even though its amplitude is �xed at the lower boundary.

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Bursts of wave activity are accompanied by reductions � reversals, in thelower stratosphere � of the mean zonal �ow, qualitatively like what we seein observations.Results from these truncated models suggest that the variability is a con-

sequence of internal dynamics, rather than something that is externally im-posed. Investigation of the dynamics in the Holton-Mass model led to aconclusion that in the supercritical state, the steady state solution is un-stable, through a process of resonant self-tuning [Plumb, .J. Atmos. Sci.,38, 2514, 1981], in which the wave, mean-�ow interaction moves the meanstate towards a resonant con�guration for the wave: thus the wave ampli�es,increasing the shift towards resonance. But can such simple truncated mod-els be trusted? In order to get resonance, one needs a cavity to contain thewave. We saw that vertical propagation can be inhibited by approriate winddistributions; in addition, however, the Holton-Mass model allows unrealisticlatitudinal con�nement by its very design. So can this kind of behavior befound in a more realistic setting?It can, as shown in Fig. 4. Scott and Polvani used a global �dynamical

core� model which incorporates full dynamics (at T42 resolution, in thiscase) but a simpli�ed set of �physics� parameterizations. They forced awavenumber 2 wave in the troposphere by speci�ying a constant thermalforcing there. Strong damping was imposed in the troposphere to suppressbaroclinic instability and tropospheric variability. Nevertheless, as Fig. 4shows, the mean �ow exhibits persistent vacillations in the stratosphere.

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Figure 4: [Scott & Polvani, 31, L02115, doi:10.1029/2003GL017965, 2004]

Focusing in on two of these events, Fig. 5 shows the EP �ux divergencewithin the stratosphere, and the integrated �ux upward through 200hPa.The strong wave bursts coincide with the rapid reductions of mean �owevident in Fig. 5. The key message from these results is that even thoughthe waves are clearly forced from the troposphere (and hence, e.g., the EP�ux at the bottom of the stratosphere is upward) the magnitude of the �uxinto the stratosphere is, at least to some extent, under stratospheric control,and hence stratospheric variability is dynamically internal.This is turn has implications for tropospheric dynamics, as there are

indications that this variability has an impact all the way down to the surface.Fig 6 shows evolution of the �annular mode index�(the projection, at eachlevel, of the zonal �ow onto the mode of dominant variability at that level)composited with respect to weak vortex events (i.e., warmings) and to strongvortex events. What appears to be a statistically signi�cant and lingeringsignal is evident all the way to the surface (such that a stratospheric warmingevent corresponds to an equatorward shift of the surface winds).

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Figure 5: [Scott & Polvani, 31, L02115, doi:10.1029/2003GL017965, 2004]

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Composite of 18 Weak Vortex Events

-90 -60 -30 0 30 60 90Lag (Days)

hPa

A10

30

100

300

1000

km

0

10

20

30

Composite of 30 Strong Vortex Events

-90 -60 -30 0 30 60 90Lag (Days)

hPa

B10

30

100

300

1000

km

0

10

20

30

Figure 6: [Baldwin & Dunkerton, Science, 2001]

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