8 3D transport formulation

8.1 The QG case

We consider the average � (x; y; z; t) of a 3D, QG system1 . The importantdistinction 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. Theaverage may be a time average, but it could also be something else (such asa low-pass spatial or temporal �lter) so we will retain terms like @�a=@t eventhough such terms would be zero for a time-average of a statistically steadysystem.The basic QG Boussinesq eqs are

@u

@t+ (u � r)u+ fk� ua = X

@b

@t+ (u � r) b+ wa

@b

@z= B

r � u = 0 (1)

r � ua = 0

f@u

@z= k�rb

where u is the (horizontal) geostrophic velocity, ua the ageostrophic velocity, Xis 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

@�u

@t+ (�u � r) �u+fk� �ua = �X�(u0 � r)u0 (2)

where the eddy term can of course be written�(u0 � r)u0

�i= @j

�u0ju

0i

�. The

mean buoyancy budget is, similarly,

@�b

@t+ (�u � r)�b+ �wa

@�b

@z= �B �r �

�u0b0

�: (3)

Note that since @b=@z is assumed constant in QG theory, there is no verticaladvection eddy term.

8.1.1 De�ning the residual (ageostrophic) circulation

As for the zonal mean case, we begin by de�ning a residual circulation with theaim of �cleaning up�the buoyancy budget. Ideally, we want to de�ne a residualmean ageostrophic circulation such that

�wa�@�b

@z= �wa

@�b

@z+

@

@x

�u0b0�+

@

@y

�v0b0�

1Apsects are discussed in Plumb, J. Atmos. Sci., 43, p1675 (1986) and 47, p1825 (1990).

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and de�ning horizontal components to preserve continuity. Since @�b=@z doesnot vary with x or y, we can achieve this with

�ua� = �ua +r�R ; (4)

where

R =k�u0b0@�b=@z

: (5)

This de�nition leaves (3) in the simple form

@�b

@t+ (�u � r)�b+ �wa�

@�b

@z= �B : (6)

Note, however, that (5) is not the most general form: eq. (6) is unchangedif we make the de�nition

R =k�u0b0@�b=@z

+ k� ; (7)

where � is yet to be determined. (We will need this �exibility later.)

8.1.2 Transformed momentum budget

With the de�nition (4), the meam momentum eq. becomes

@�u

@t+ (�u � r) �u+fk� �ua� = �X�(u0 � r)u0 + fk�r�R

= �X�(u0 � r)u0 � fk� @

@z

�u0b0

@�b=@z

�� fr�

= �X� @jMji

where

M =

0@ Mxx Mxy

Myx Myy

Mzx Mzy

1A =

0B@ u02 + f� u0v0

u0v0 v02 + f�

�f v0b0

@�b=@zf u0b0

@�b=@z

1CA :

The ith component of the QGPV �ux is

u0iq0 = u0i

�@v0

@x� @u0

@y+ f

@

@z

�b0

@�b=@z

��= @jQji

where

Q =

0@ Qxx QxyQyx QyyQzx Qzy

1A =

0B@ u0v0 v02 � ""� u02 �u0v0f u0b0

@�b=@zf v0b0

@�b=@z

1CAwhere

" =1

2

u02 + v02 + f

b02

@�b=@z

!= "K + "P

2

is the energy density (the sum of kinetic and potential energy densities).Now, if we choose

� =1

f

�"� u02 � v02

�=1

f("P � "K) ; (8)

then @jMji = k� (@jQji) = k� u0q0. Therefore the transformed mean momen-tum equation becomes

@�u

@t+ (�u � r) �u+fk� �ua� = �k� u0q0 + �X : (9)

Then, in (6) and (9), we have two equations that appear to be analogous to theTEM zonal mean problem: there are no explicit eddy terms in the buoyancybudget, 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, tothe PV �ux). If the forcing of the mean momentum budget by the PV �ux iswhat we want, (8) is what we have to do to achieve that.The mean PV budget is straightforward:

@�q

@t+ �u�r�q = �r � u0q0 + k:r� �X+f

@

@z

�B

@�b=@z

�: (10)

Note that no transformations are involved here: the QGPV is advected only bythe 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, wecould construct the whole geostrophic solution from the PV budget, and thencefrom a knowledge on the PV �ux divergence alone. So why does the momentumequation appear to require more informaion? It turns out that the geostrophic(though not the ageostrophic) �ow only really cares about the divergent part ofthe �ux. We can easily see this by looking at the consequences of subtracting apurely rotational �ux to the PV �ux, to leave�

u0q0�R= u0q0 �r� k�

when the forcing of the momentum eq becomes

�k� u0q0 = �k��u0q0

�R+r� :

The term r� can then be absorbed into a rede�nition of the ageostrophic cir-culation: de�ning

� =1

f("P � "K + �) : (11)

Using this technique, we can remove any rotational part of the PV �ux fromthe 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 orderto be able to calculate the response of the mean geostrophic �ow. Just as inthe 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 �.

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8.1.3 The eddy PV �ux

The eddy enstrophy equation is

@

@te+ �u � re+ u0q0 � r�q = S0q0 ; (12)

where e = 12q02 is the eddy enstrophy and (for convenience) S0 represents both

the nonconservative sources and sinks of PV and the nonlinear term. In thezonal mean case, in steady state and if enstrophy is being dissipated (S0q0 < 0),this led us to expect a downgradient �ux of PV. Now, even in steady state, wehave an additional term, which complicates things. (The non-zonally-averagedcase is thus never �steady�, in a following-the-�ow sense, whenever there arespatial variations of q02, which will usually be the case.)Under some circumstances, however, one can make some progress2 . If the

mean �ow is steady and along the PV contours (which requires that the �ow bealmost conservative and that the impact of eddies is not too large) such that

�u � r�q = J�� ; �q�= 0

where � is the geostrophic mean streamfunction and J the Jacobian in (x; y)space, then � = � (�q). Now, divide up the eddy �ux into components associatedwith the nonconservative and advective terms:

u0q0 =�u0q0

�N+�u0q0

�A

where �u0q0

�N� r�q = S0q0 ;

and �u0q0

�A� r�q = ��u � re

= �r � (�ue)= �r �

�ek�r �

�= �r �

�ed �

d�qk�r�q

�= r�q � k�r

�ed �

d�q

�:

Therefore, there is a �ux

�u0q0

�A= k�r

�ed �

d�q

�which, under the stated assumptions,

2Marshall and Shutts, J. Phys. Oceanogr., 11, p1677 (1981); Illari and Marshall, J. Atmos.Sci., 40, p2232 (1983).

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1. satis�es�u0q0

�A� r�q = ��u � re

2. is rotational, so that (following the procedures of Section 8.1.2) it doesnot enter into the geostrophic mean �ow problem.

Fig. 1 shows an example of this decomposition. The �raw��ux (a) is notparticularly downgradient; once the rotational part (b) is subtracted, the re-maining �ux (c) is downgradient where ever it has signi�cant amplitude, exceptnear the western boundary where the assumption � = � (�q) may not be verygood.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 theMarshall-Shutts method, but just removed the rotational part of the PV �uxdirectly via a Helmholtz decomposition assuming no �ux components throughthe boundaries. Their �raw��ux (a) is strongly upgradient in places, but thedivergent part (b) is very much downgradient, even near the western boundary(and without a signi�cant skew component, even though the method does notguarantee removal of the skew �ux).So where does this leave us? The QGmean �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 theadvective part removed may be weak in the open ocean.) It is possible totransform the skew (along-contour component of the) �ux into a rede�nitionof the geostrophic �ow, but since the mean geostrophic �ow is probably whatwe want to know, that would still leave us with the need to parameterize theextra component. At present, it seems we have to be pragmatic in tacklingthe parameterization problem: to be guided by what works in practice fo theproblem at hand, rather than having a sound theoretical basis for the wholeproblem. In some circumstances (such as long, thin mean jets), approximationsmay be appropriate that will allow one to be more rigorous.

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Figure 1: PV �ux decomposition at the top level of a model of wind-drivengyres. [From Wardle (Ph. D. thesis, 1999).] Arrows show (a) eddy PV �uxu0q0; (b)

�u0q0

�A; (c)

�u0q0

�N= u0q0 �

�u0q0

�A. Contours are �q, shading is

where the �ux is downgradient.

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

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8.2 Non-QG: isopycnal coordinates

The same situation arises when we consider the non-QG case for adiabatic eddiesin isopycnal coordinates. The mean continuity equation is

@��

@t+r � (���u) = �r �

��0u0

�;

or@��

@t+r � (��U) = 0 ; (13)

if we de�neU = �u� + ���1k�r� (14)

where now k is normal to the isopycnals.The isopycnal momentum equation is

@u

@t� u� �a = �rB +X ;

where B = M + 12u � u is the Bernouilli function (M being the Montgomery

potential). Since only the isopycnal components of this equation are valid, andu is along the isopycnals, only the k component of �a = �P is relevant, so wecan write

@u

@t+ k� u�P = �rB +X : (15)

Note that the mean of the second term is

u�P = �� uP�;

by the de�nition of mass-weighted mean, and that

uP�= �u� �P � + uP

�;

where a = a� �a� is the mass-weighted eddy term, as before. Then the mean of(15) becomes

@�u

@t+ k� �u���a +r ~B = ���k� uP

��r"K + �X (16)

(since ��a = �� �P �), where ~B =M + 12�u � �u is a pseudo-mean Bernouilli function,

by which I mean the Bernouilli function of mean state variables: the actualmean of B is �B = �B + "K , where "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

@�u

@t+ k�U��a +r ~B = ���k� uP

��r

�1

2u0�u0

�� �P �r�+ �X

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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

@�u

@t+ k�U��a +r ~B = ���k� uP

�+"K�P �r �P � + �X :

The extra term is along there PV gradient, and therefore just adds to thecomponent 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 meanvorticity (mass-weighted mean PV) budget

@��a@t

+r ���u���a

�= �r � ��uP

�+ k � r � �X (17)

while r�(16) gives the divergence eq

@D

@t� k � r �

��u���a

�+r2 ~B = k � r � ��uP

��r2

�1

2u0�u0

�+r � �X : (18)

It may be�depending on the situation� that the eddy terms in the mean diver-gence 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 downgra-dient part, and only the latter appears related to processes such as theenstrophy cascade and dissipation;

2. the advected quantity ��a is the �raw�mean absolute vorticity, while theadvecting velocity is transformed, and so further eddy terms (those involvein 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 + k�r � �u� 6= ���a . This introduces extra eddyterms,

@~��a@t

+r ���u�~��a

�= �r � ��uP

���@� 00

@t+r � (�u�� 00)

�+ k � r � �X (19)

where

� 00 = ���a � ~��a

= ���1k � �r� u� k:r� �u

��

= k � u0 �r (�0=��) ;

which is an eddy term (in the sense that it is O (eddy amplitude)2 ). Whilethe time derivative will vanish for statistically steady eddies, the advective

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term remains in addition to the PV �ux. While we are left with a sinlgeeddy forcing term (the sum of �rst 3 terms on the RHS of (19)) and aclosed set of equations for the mean state, provided one also makes thetransformation ��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 benegligible. Note that to keep the mean thickness eq. (13) consistent withthis 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).

8.4 3D transport in (PV, �) coordinates

We will discussKushner & Held, J. Atmos. Sci., 56, 948-958 (1999).

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