8 3D transport formulation

Transcription

1 8 3D transport formulation 8.1 The QG case We consider the average (x; y; z; t) of a 3D, QG system 1. The important distinction from the zonal mean case is that the mean now varies with x, or (more generally) there is no spatial direction that is preferred a priori. The average may be a time average, but it could also be something else (such as a low-pass spatial or temporal lter) so we will retain terms like a=t even though such terms would be zero for a time-average of a statistically steady system. The basic QG Boussinesq eqs are t + (u r) u + fk u a = X b t + (u r) b + w b a z = B r u = 0 (1) r u a = 0 f z = k rb u is the (horizontal) geostrophic velocity, u a the ageostrophic velocity, X is the applied frictional (or other) force, B is the buoyancy forcing/dissipation, and all else is standard notation. The mean of the momentum eqn is t + (u r) u+fk u a = X (u 0 r) u 0 (2) the eddy term can of course be written (u 0 r) u 0 = j i mean buoyancy budget is, similarly, u 0 j u0 i. The b t + (u r) b + w b a z = B r u 0 b 0 : (3) Note that since b=z is assumed constant in QG theory, there is no vertical advection eddy term De ning the residual (ageostrophic) circulation As for the zonal mean case, we begin by de ning a residual circulation with the aim of cleaning up the buoyancy budget. Ideally, we want to de ne a residual mean ageostrophic circulation such that w b a z = w b a z + x u0 b 0 + y v0 b 0 1 Apsects are discussed in Plumb, J. Atmos. Sci., 43, p1675 (1986) and 47, p1825 (1990). 1

2 and de ning horizontal components to preserve continuity. Since does not vary with x or y, we can achieve this with This de nition leaves (3) in the simple form u a = u a + r R ; (4) R = ku0 b 0 : (5) b t + (u r) b + w a b z = B : (6) Note, however, that (5) is not the most general form: eq. (6) is unchanged if we make the de nition R = ku0 b 0 + k ; (7) b=z is yet to be determined. (We will need this exibility later.) Transformed momentum budget With the de nition (4), the meam momentum eq. becomes t + (u r) u+fk u a = X (u 0 r) u 0 + fk r R u0 b 0 = X (u 0 r) u 0 fk z fr 0 M = M xx M yx M zx = X j M ji M xy M yy M zy 1 A = 0 B u 02 + f u 0 v 0 u 0 v 0 v 02 + f f v0 b 0 f u0 b 0 The i th component of the QGPV ux is v u 0 i q0 = u i x y + f b 0 z = j Q ji 1 C A : 0 Q = Q xx " = 1 2 Q yx Q zx Q xy Q yy Q zy 1 A = 0 B b02 u 02 + v 02 + f u 0 v 0 v 02 " " u 02 u 0 v 0 f u0 b 0 f v0 b 0! = " K + " P 1 C A 2

3 is the energy density (the sum of kinetic and potential energy densities). Now, if we choose = 1 " u f 02 v 02 = 1 f (" P " K ) ; (8) then j M ji = k ( j Q ji ) = k u 0 q 0. Therefore the transformed mean momentum equation becomes t + (u r) u+fk u a = k u 0 q 0 + X : (9) Then, in (6) and (9), we have two equations that appear to be analogous to the TEM zonal mean problem: there are no explicit eddy terms in the buoyancy budget, and the eddies appear in the momentum budget in the form of the PV ux (speci cally, as a force per unit mass equal in magnitude, and normal, to the PV ux). If the forcing of the mean momentum budget by the PV ux is what we want, (8) is what we have to do to achieve that. The mean PV budget is straightforward: q t + urq = r u0 q 0 + k:r X+f z B : (10) Note that no transformations are involved here: the QGPV is advected only by the geostrophic ow, and all our transformations are of the ageostrophic velocity. Note that the PV budget cares only about the divergent part of the PV ux, and yet it is the full ux that appears in the mean ow eq. (9). In principle, we could construct the whole geostrophic solution from the PV budget, and thence from a knowledge on the PV ux divergence alone. So why does the momentum equation appear to require more informaion? It turns out that the geostrophic (though not the ageostrophic) ow only really cares about the divergent part of the ux. We can easily see this by looking at the consequences of subtracting a purely rotational ux to the PV ux, to leave u 0 q 0 R = u0 q 0 when the forcing of the momentum eq becomes r k k u 0 q 0 = k u 0 q 0 R + r : The term r can then be absorbed into a rede nition of the ageostrophic circulation: de ning = 1 f (" P " K + ) : (11) Using this technique, we can remove any rotational part of the PV ux from the momentum equation. The important conclusion from all this is that, if we wish to parameterize the eddies, it is the divergence of the PV ux that we need to parameterize in order to be able to calculate the response of the mean geostrophic ow. Just as in the zonal mean case, we can then only determine the transformed ageostrophic ow: to get the untransformed ageostrophic ow (if we really need to know it, though it is hard to see why we would), we must also parameterize. 3

4 8.1.3 The eddy PV ux The eddy enstrophy equation is t e + u re + u0 q 0 rq = S 0 q 0 ; (12) e = 1 2 q02 is the eddy enstrophy and (for convenience) S 0 represents both the nonconservative sources and sinks of PV and the nonlinear term. In the zonal mean case, in steady state and if enstrophy is being dissipated (S 0 q 0 < 0), this led us to expect a downgradient ux of PV. Now, even in steady state, we have an additional term, which complicates things. (The non-zonally-averaged case is thus never steady, in a following-the- ow sense, whenever there are spatial variations of q 02, which will usually be the case.) Under some circumstances, however, one can make some progress 2. If the mean ow is steady and along the PV contours (which requires that the ow be almost conservative and that the impact of eddies is not too large) such that u rq = J ; q = 0 is the geostrophic mean streamfunction and J the Jacobian in (x; y) space, then = (q). Now, divide up the eddy ux into components associated with the nonconservative and advective terms: u 0 q 0 = u 0 q 0 N + u0 q 0 A and u 0 q 0 N rq = S0 q 0 ; u 0 q 0 A rq = u re = r (ue) = r ek r = r e d dq k rq = rq k r e d : dq Therefore, there is a ux u 0 q 0 = k r e d A dq which, under the stated assumptions, 2 Marshall and Shutts, J. Phys. Oceanogr., 11, p1677 (1981); Illari and Marshall, J. Atmos. Sci., 40, p2232 (1983). 4

5 1. satis es u 0 q 0 rq = u re A 2. is rotational, so that (following the procedures of Section 8.1.2) it does not enter into the geostrophic mean ow problem. Fig. 1 shows an example of this decomposition. The raw ux (a) is not particularly downgradient; once the rotational part (b) is subtracted, the remaining ux (c) is downgradient ever it has signi cant amplitude, except near the western boundary the assumption = (q) may not be very good. A second example, from the shallow water study of Petersen and Greatbatch (Atmosphere-Ocean, 39, p1, 2001), is shown in Fig. 2. They did not use the Marshall-Shutts method, but just removed the rotational part of the PV ux directly via a Helmholtz decomposition assuming no ux components through the boundaries. Their raw ux (a) is strongly upgradient in places, but the divergent part (b) is very much downgradient, even near the western boundary (and without a signi cant skew component, even though the method does not guarantee removal of the skew ux). So does this leave us? The QG mean ow problem requires input of the divergence of the eddy PV ux. But what we might hope to parameterize i.e., what us related to eddy enstrophy dissipation is the downgradient component. Even if we make the asumptions required to take the Marshall-Shutts approach, we still know nothing of the ux component along the PV contours. (Although, in practice, Fig. 1 suggests that the skew component of the ux with the advective part removed may be weak in the open ocean.) It is possible to transform the skew (along-contour component of the) ux into a rede nition of the geostrophic ow, but since the mean geostrophic ow is probably what we want to know, that would still leave us with the need to parameterize the extra component. At present, it seems we have to be pragmatic in tackling the parameterization problem: to be guided by what works in practice fo the problem at hand, rather than having a sound theoretical basis for the whole problem. In some circumstances (such as long, thin mean jets), approximations may be appropriate that will allow one to be more rigorous. 5

6 Figure 1: PV ux decomposition at the top level of a model of wind-driven gyres. [From Wardle (Ph. D. thesis, 1999).] Arrows show (a) eddy PV ux u 0 q 0 ; (b) u 0 q 0 ; (c) A u0 q 0 = N u0 q 0 u 0 q 0. Contours are q, shading is A the ux is downgradient. 6

7 Figure 2: Similar to Fig 1. (Peterson & Greatbatch, 2001.) 7

8 8.2 Non-QG: isopycnal coordinates The same situation arises when we consider the non-qg case for adiabatic eddies in isopycnal coordinates. The mean continuity equation is or if we de ne t + r (u) = r 0 u 0 ; t now k is normal to the isopycnals. The isopycnal momentum equation is + r (U) = 0 ; (13) U = u + 1 k r (14) t u a = rb + X ; B = M + 1 2u u is the Bernouilli function (M being the Montgomery potential). Since only the isopycnal components of this equation are valid, and u is along the isopycnals, only the k component of a = P is relevant, so we can write + k up = rb + X : (15) t Note that the mean of the second term is up = up ; by the de nition of mass-weighted mean, and that up = u P + ^u ^P ; ^a = a (15) becomes a is the mass-weighted eddy term, as before. Then the mean of t + k u a + r ~ B = k ^u ^P r" K + X (16) (since a = P ), B ~ = M u u is a pseudo-mean Bernouilli function, by which I mean the Bernouilli function of mean state variables: the actual mean of B is B = B + " K, " K is the eddy KE density, as before. Unlike the QG case, we cannot simply absorb the eddy KE term into the factor. To exploit the generality of (14), we can rewrite (16) as t + k U a + r B ~ = k ^u ^P 1 r 2 u0 u 0 P r + X 8

9 but setting = " K = P does not eliminate the KE term unless P is uniform (such as when a ' f and is uniform, as in QG). However, if we do so, we get t + k U a + r ~ B = k ^u ^P + " K P r P + X : The extra term is along there PV gradient, and therefore just adds to the component of the PV ux that is normal to the PV gradient. One might be able to make some more sense of things by thinking in terms of vorticity and divergence rather than velocity. k r (16) gives the mean vorticity (mass-weighted mean PV) budget a t + r u a = r ^u ^P + k r X (17) while r(16) gives the divergence eq D t k r u a + r 2 ~ B = k r ^u ^P r u0 u 0 + r X : (18) It may be depending on the situation that the eddy terms in the mean divergence equation may be neglected by a suitable choice of balance (invoking (14), perhaps, if necessary, to introduce more freedom into the choice) involving u rather than u. But that still leaves the two issues in (17): 1. the PV ux is the full (divergent part of) the ux, not just the downgradient part, and only the latter appears related to processes such as the enstrophy cascade and dissipation; 2. the advected quantity a is the raw mean absolute vorticity, while the advecting velocity is transformed, and so further eddy terms (those involve in the transformation of u ) are implicit in the vorticity inversion a! u. One might be tempted to get around this by de ning a pseudo-mean absolute vorticity ~ a = f + kr u 6= a. This introduces extra eddy terms, ~ a t + r u ~ a = r ^u ^P 00 t + r (u 00 ) + k r X (19) 00 = a ~ a = 1 k r u k:r u = k u 0 r ( 0 =) ; which is an eddy term (in the sense that it is O (eddy amplitude) 2 ). While the time derivative will vanish for statistically steady eddies, the advective 9

10 term remains in addition to the PV ux. While we are left with a sinlge eddy forcing term (the sum of rst 3 terms on the RHS of (19)) and a closed set of equations for the mean state, provided one also makes the transformation a! ~ a in (18) as well as u! u in the de nition of ~ B, thereby admitting new eddy terms into the RHS of (18), which may be negligible. Note that to keep the mean thickness eq. (13) consistent with this approach, we simply set = 0 in (14). 8.3 The GM parameterization We will discuss: Gent & McWilliams, J. Phys. Oceanogr., 20, p150 (1990); Gent at al., J. Phys. Oceanogr., 25, p463 (1995); Gent & McWilliams, J. Phys. Oceanogr., 26, p2539 (1996) D transport in (PV, ) coordinates We will discuss Kushner & Held, J. Atmos. Sci., 56, (1999). 10