2 Transport of heat, momentum and potential vorticity

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1 Transport of heat, momentum and potential vorticity. Conventional mean momentum equation We ll write the inviscid equation of onal motion (we ll here be using log-pressure coordinates with = H ln p=p, with H constant) as and the continuity equation du dt fv = x () r u = () where () = p=gh. We introduce the onal mean of any variable a as a(y; ; t) = L Z L a(x; y; ; t) dx ; (3) (where we assume periodic behavior in x such that a(l; y; ; t) = a(; y; ; t) note that this means that (a=x) = ) and then the eddy component is It follows that, by de nition, a (x; y; ; t) = a(x; y; ; t) a(y; ; t) : a = : (4) Note that the de nition of the eddy variable is contingent on the de nition of mean. Then the mean of (), invoking (), becomes u t + v u y + w u f v = r u u ; (5) thus introducing the concept of eddy momentum ux u u = u v ; u w (it has no onal component in our onal mean formulation). The way in which the eddies in uence the mean state is evidently equivalent to a body force r u u acting on the mean onal motion. This ux term clearly represents the advection, by the eddy component of the meridional velocity, of the eddy component of onal momentum. The onal momentum per unit mass is u; so the advective ux of momentum across a coordinate surface (of constant y and ) is uu, whence its onal mean is uu = uu + u u ; hence it is natural to regard u u as the eddy ux of momentum. It is important to realise, however, that this de nition of eddy momentum ux is a consequence of our de nition (3) of what is meant by mean, speci cally, that the average is taken at at constant pressure (constant in log-p coordinates).

2 One could easily de ne other means: since transport is fundamentally a Lagrangian process, one might try taking some kind of Lagragian mean, but we will not do that here. One can take a di erent integration path in (3) the way we have done it is to integrate at constant latitude and height (i.e.; constant pressure) by using a di erent coordinate system, in which case the whole problem can look rather di erent, as we are about to see.. Wave, mean ow interaction in isentropic coordinates In (x; y; ) coordinates, the inviscid onal equation of motion is u t + uu x + v u y + _ u fv = M x (6) and that of continuity t + x (u) + y (v) + _ = (7) where = g p= is the isentropic coordinate density, and M = c p T + g = + (8) is the Montgomery potential, with (p) = c p (p=p ) the Exner function and = g. (Note that is real height here, not log-pressure height.) It follows from (8), with some juggling and the assumption of hydrostatic balance, that Now our de nition of onal mean is M = : (9) a(y; ; t) = L Z L a(x; y; ; t) dx where the integral is now, of course, taken at constant, and the eddy term is a (x; y; ; t) = a(x; y; ; t) a(y; ; t) : If you are interested in the Lagrangian perspective, there are some additional notes on this on the class web page. The conceptually powerful generalied Lagrangian mean theory of wave, mean ow, interaction can be found in Andrews & McIntyre, J. Fluid Mech., 978 (though it is not for the faint hearted!). The derivation of the -coordinate equations is discussed in Appendix 3A in Andrews, Holton and Leovy [987] (at the end of Chapter 3).

3 It makes sense to de ne mean meridional velocities in a density-weighted sense: v = (v) ; _ = _ () since then the mean continuity equation has no eddy ux terms: t + y (v ) + _ = : The momentum equation (6), on multiplication by, use of (7), and onal averaging, gives t (u)+ y (v u)+ _ u f v = y (v) u _ u M x : () The rst two terms on the rhs are easily identifable as advection by the eddy mass uxes, but what about the third term? Since M=x =, it is an eddy term M x = M x and, in fact, the term can be written, using the de nition of and (8) M x = p M g x = g pm x g p M x : But M= = (p), so p M x = p p (p) = pd x dp x = Z p d x dp dp = ; hence In total, therefore, () becomes t (u)+ y (v u)+ _ u M x = g pm = x f v = y h(v) u i M p g x : _ M u p g x ; () 3

4 there is an eddy momentum ux whose y and components are given by (v) u ; _ M u p : (3) g x The additional, non-advective, term in (3) represents the form stress acting on the isentropic surface. If one has, e.g., surface topography (x; y) there is an inviscid stress on the earth s surface given by psin = p=x, where 3 is the angle of the local surface with respect to the horiontal; accordingly there is a form drag of p=x acting on the atmosphere above. Now, p M x = p M x = p d p dp x + g x = p d p + gp dp x x since we just established that the rst term is ero. Hence M p g x = p x : = gp x ; Since =x here is just the slope of the isentropic surface, we can therefore immediately identify this extra term as the form drag acting on an isentropic surface due to the presence of the eddies. So, in a di erent coordinate system, eddy transport of momentum can look quite di erent, physically as well as mathematically..3 QG momentum, heat, and PV transport.3. The quasigeostrophic equations on a -plane We consider motion with characteristic horiontal length scale L, height scale h, velocity scale U, time scale L=U on a -plane for which the Coriolis parameter is f = f + y. We make the assumptions that: (i) the Rossby number Ro = U=f L is small, (ii) L=f Ro (which, in a spherical context, means that L << a, where a is the Earth s radius), (iii) the isentropic slopes j x j=j j and j y j=j j are Ro (h=l), (otherwise vertical motions would not be small), and 3 Under the shallow atmosphere" approximation which all these equations assume, the slope is small, so =x = tan ' sin : 4

5 (iv) the static stability is, to leading order, a function of only [this stems from (iii)], so we write = () + ~, where ~ is O(Ro) smaller than. We then de ne a background geopotential, in hydrostatic balance with the background : = g c p T = H : Under these assumptions, we nd that the leading order equations give geostrophic balance, in which we write the geostrophic velocities [i:e:, the leading order terms in a Rossby number expansion of (u; v; w)] as u = y ; v = x ; w = ; (4) where is the geostrophic streamfunction, which we de ne to be Hydrostatic balance then gives us = f H f = f H [ = [ ()] =f. (5) ()] = f H ~. (6) At next order, we obtain our quasigeostrophic equations. The equations of motion are D g u yv f v a = G (x), D g v + yu + f u a = G (y) (7), where G represents frictional or other forces per unit mass, D g is the time derivative following the geostrophic ow D g t + u x + v y and (u a ; v a ; w a ) is the ageostrophic velocity (i:e:, the di erence between the actual velocity and the geostrophic one). Similarly, the thermodynamic equation becomes D g + w a ; = c p Q = () J, (8) where J is the diabatic heating rate per unit volume (and Q = J =c p is the heating rate expressed in units of K per unit time). From these, we can readily derive the equation for quasigeostrophic potential vorticity, q : D g q = X, (9) where q = f + y + v x u y + f! ~ ; = f + y + () 5

6 and where with X = G (y) x x + y G y (x) + f + N = H ; J ; f N, (), () being the square of the buoyancy frequency. Eq. (9) tells us that, for conservative ow (G =, J =, whence X = ), q is conserved following the geostrophic ow. When the ow is not conservative, X represents the local sources or sinks of q arising from viscous and/or diabatic e ects..3. PV uxes and the Eliassen-Palm theorem Consider perturbations to a steady, onally-uniform basic state U = U(y; ) ; = (y; ) ; = (y; ) where The basic state PV is: y = U ; H y = f U. Q(y; ) = f + y + yy + f N = f + y +. (3) Write = + (x; y; ; t) where is a small perturbation. Now, v = x and q = = xx + yy + f N therefore v q = x xx + x yy + f x N Simple manipulation yields the following identities: ; (4) : (5) 6

7 . x xx = h ( x) i = ; x. x yy = x y y = x y = x y y y ; y xy h y i x and x f N = f N x = f N x = f N x : f N x f N h( ) i x Therefore, from (5), where F (y) F = = F () v q = r F. (6)! x y u f = v x =N f v : (7) = F is known as the ELIASSEN-PALM ux. Note that the northward component of F is (minus) the northward eddy ux of onal momentum, while the vertical component is proportional to the northward eddy ux of heat, v T : Now, assume that the perturbations are small, such that the QGPV equation (9) can be linearied to give ( t + U x ) q + v Q y = X ; multiply by q and average: If we de ne q + v q Q y = v X : (8) t A = q =Q y and D = v X =Q y ; 7

8 then, using (6), A t + r F = D. (9) Eq. (9) is the ELIASSEN-PALM RELATION. It is a conservation law for onally-averaged wave activity whose density is A. Note that D! for conservative ow. The signi cance of this relation is that it gives us a measure of the ux of wave activity through wave propagation. For example, if the waves are conservative (D = ) then A must increase with time wherever F is convergent and decrease wherever it is divergent. Thus F is a meaningful measure of the propagation of wave activity from one place to another. This becomes most obvious for almost-plane waves (WKB theory) when, as we shall see, where c g is group velocity, in which case F = c g A A t + r c g A = D. However, note that F remains valid as a measure of the ux of wave activity even when WKB theory is not valid and we cannot even de ne group velocity. [N.B. there are some subtleties to these arguments if Q y changes sign anywhere, since A is then not positive de nite (and so increasing A does not necessarily mean increasing wave amplitude). Indeed, if this occurs, the basic state may be unstable the Charney-Stern necessary condition for instability can in fact be readily obtained from (9).].3.3 The Eliassen-Palm theorem For waves which are steady (A t = ), of small amplitude and conservative (D = ), the ux F is nondivergent. This is, through (6), the same thing as saying that the northward ux of quasigeostrophic potential vorticity vanishes under these conditions [a result we could of course have obtained from (8) without involving F]:.3.4 Potential vorticity transport and the nonacceleration theorem We now consider the problem of how eddies (in this case, quasigeostrophic eddies) impact the onal mean circulation. We return to eddies which may not be small and to the QGPV budget, which becomes, on onal averaging q t + (v q ) y = X. (3) Note that, unlike (5), there is no mean advection term in (3). This is because there is no advection by w in this quasigeostrophic case and v = x =. Similarly, the eddy ux contains no vertical component, because of QG scalings. 8

9 The dynamical in uence of the eddies on mean potential vorticity, therefore, is described entirely 4 by the northward ux v q. Given the equivalence between pv q and rf, we could make the same statement about r F which informs us immediately that F is telling us something about wave transport as well as propagation. Now, we know from the Eliassen-Palm theorem that if the waves are everywhere (I) of steady amplitude, (II) conservative, and (III) of small amplitude, (so that terms of O( 3 ) in the wave activity budget are negligible) then F is nondivergent and v q =. Under these conditions, therefore, the budget equation for onally-averaged QGPV is independent of the waves (if we assume that X is so independent). Now, q = f + ( ) therefore t = qt = X : If q t is independent of the waves, then so is t ; since is an elliptic operator, the solution of the above equation for t invokes boundary conditions. If, however, we invoke the further condition that (IV) the boundary conditions on t are independent of the waves then t is everywhere independent of the waves. Since u = y and = (f T =g), it then follows that, under conditions (I)-(IV), the tendencies of the mean geostrophic ow, mean temperature and mean QGPV are independent of the waves. This is known as the nonacceleration theorem and conditions (I)-(IV) are sometimes known as nonacceleration conditions..3.5 Zonal mean momentum and heat budgets: Conventional Eulerian-mean approach In a similar fashion, we can take the onal mean of the quasigeostrophic momentum and heat equations. The onal mean of the rst of (7) gives us u t f v a = G (x) (u v ) y, (3) 4 One should add the caveat that, in principle, the eddies could also in uence X (e.g., eddies could modify mean precipitation). 9

10 where we have used v = x = (the mean ageostrophic wind is not ero, however). Similarly, the heat equation (8) gives t + w a = () J (v ) y. (3) The mean ow equations are closed by the continuity equation v a;y + (w a) = (33) and the thermal wind equation f u = H ~ y. (34) (The two latter equations are linear and so the onal averaging is trivial). The evolution of the onal mean state in the presence of eddies is therefore speci ed by (3) - (34). In this case, the e ects of wave transport are manifested in two terms the convergence of the eddy uxes of momentum u v and heat v (strictly, the eddy ux of heat is c p v T, but we shall refer to v in this way for simplicity). Both these terms force the mean ow equations and it is important to note that the whole system is coupled, i:e:, the heat uxes can impact on the mean winds just as much as can the momentum uxes. Thermal wind balance requires this to be true. Consider, for example, an upward propagating wave with v 6= but u v =. The mean state could not respond with a changing mean temperature only; thermal wind balance demands a corresponding change in u. From (3) this could only be achieved through an ageostrophic meridional circulation, which would impact on both the momentum and heat budgets. Thus, the waves will not only drive u t and t, but also v a and w a (except in the unlikely case where the eddy forcing terms conspire not to disturb thermal wind balance). To put the same statements into mathematics, what (3) - (34) give us is a set of 4 equations in the 4 unknowns u t, t, v a and w a in terms of the two eddy driving terms. In general, when both the eddy ux terms are nonero, there is no simple way of saying which forcing achieves what response. Moreover, the central role of the potential vorticity ux obvious in the PV budget is not at all obvious here. Indeed, we have seen from the mean potential vorticity budget that under nonacceleration conditions u t and t, must be ero under these conditions. What must (and does) happen under such circumstances is that the eddies induce ageostrophic mean motions whose e ects in (3) and (3) exactly balance the eddy ux terms. All in all, this approach to the mean momentum and heat budgets is not giving us much insight into what is going on.

11 .4 Transformed Eulerian-mean theory The most important lesson to learn from the isentropic-coordinate analysis we did earlier is that the de nition of mean, and consequently of concepts like eddy uxes, is non-unique. We can take this message to heart by allowing ourselves some exibility in de ning what we mean by the mean meridional circulation. We begin by noting that, from (33), we may de ne an ageostrophic mean streamfunction a such that (v a ; w a ) = ( a ), a y. (35) Now, we look for a more revealing way of de ning mean circulation. To make things as simple as possible, we insist that our modi ed circulation be nondivergent also, so we write ( (v ; w ) = ), ; (36) y where the new streamfunction is If we substitute this into (3), we obtain = a + c : t + w = () J y (v ) c y. Noting that = (), it follows that if we make the choice c = v, so that the streamfunction of the so-called residual circulation is we obtain the thermodynamic equation = a v, (37) t + w = () J. (38) We have thus succeeded in deriving a mean heat equation in which there are no explicit eddy terms; heat is transported solely through the mean vertical residual" motion. It might be thought, of course, that the eddy terms are still there, implicit in w ; but this was also true of w a which, as noted earlier, is in general in uenced by the eddies. What we have done is to rede ne this in uence so as to put the mean heat budget into its simplest possible form.

12 is We now need to complete our transformed system of equations. The continuity equation v y + (w ) = (39) [we arranged this by (36)]. The thermal wind equation stays as before, i:e: : f u = H ~ y : (4) The momentum equation is less trivial. We need to replace v a using (36); the result is u t f v = G x + r F, (4) where F is the Eliassen-Palm ux. This transformation which is nothing more than a di erent way of writing the same equations makes the role of the eddies look rather di erent. We now have, in (38) - (4), a set of equations for v ; w ; u=t and =t in which the only term representing eddy forcing is a term r F, which appears as an e ective body force (per unit mass) in the mean momentum equation. [We could, for example, rede ne G x to absorb this term]. It is clear, therefore, that under nonacceleration conditions when rf = and the boundary conditions are independent of wave-dependent terms, v, w, u=t and =t are independent of the waves. When nonacceleration conditions are not satis ed, the transformed equations o er a more transparent approach to the problem simply because the single term represented by the e ective body force rf entirely summaries the eddy forcing of the mean state (there being no thermal eddy forcing to confuse the issue). In fact, this formulation gives us another interpretation of F: as an eddy ux of (negative, i:e:, easterly) momentum which, because of the properties we have just described, is a more reliable measure of the wave transport of momentum than u v alone. Moreover, we now see that the Eliassen-Palm ux gives us a uni ed picture of wave propagation (through its role in wave activity conservation) and transport (through its interpretation as a momentum ux). This perspective is conceptually very powerful, as we shall see. Moreover, through the relationship (6) we immediately recover the central role of PV uxes in the eddy forcing of the mean state..4. Relationship to isentropic mean perspective Finally, note again that, while the perspective which regards F as a momentum ux may seem to be the result of mere mathematical juggling, it is rather more than that. It might seem odd to have what looks like an eddy heat ux appearing in the vertical momentum ux in this formulation. But it can be shown (see relevant Problem Set) that, for small-amplitude eddies under adiabatic conditions:

13 . this heat ux term corresponds to the form stress term that appeared in the isentropic coordinate formulation, and. that the residual circulation corresponds to the density-weighted mean circulation is isentropic coordinates. [It is also, under these assumptions, directly related to the Lagrangian mean motion.] Thus, aside from the mathematical advantages of the TEM set of equations over the conventional set, the formaultion is also describing the physics rather well. [You might well ask: why not use the isentropic coordinate version? The answer is that, for wave problems, isentropic coordinates are a bit unwieldy.].4. Boundary conditions on the TEM equations Finally, a short but important note about boundary conditions on the TEM ow. At a at lower boundary, the boundary condition w = translates straightforwardly, under conventional averaging, to w =. For the TEM equations, however, (36) and (37) tells us that! w = v y which is nonero whenever there is a heat ux along the boundary. So, in this respect, the TEM framework is the more complicated (though it is actually telling us something about the dynamics near the boundary see Problem Set). As we shall see, on a lower boundary with topography, the opposite may be true. In general w is nonero on such a boundary. 3