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A.5 Meridional circulation maintenance

An equation for the maintenance of the circulation p can be obtained by applying a curl to the momentum Equation (40View Equation) and then taking the zonal average of the zonal component. The result may be expressed as:
( ) @p G G.^c @t--= - rsinh \~/ . r-sin-h = - \~/ .G + rsinh-, (74)
where ^c = sinhr^+ cos h^h is a unit vector directed perpendicular to and away from the rotation axis. The flux vector G is given by
G = GRS + GAD + GBF + GMT + GVD, (75)
where the components include the Reynolds stress
( ) RS --(< ' '> < ' '>) -- < ' '> < ' '> <(v')2> G =_ r vrwf - vfw r ^r + r v hw f - vfwh - ------- ^h, (76) 2Hr
the advection of vorticity by the meridional circulation and differential rotation
-- -- ( ) -<v>2 GAD =_ r <vM> <wf >- r<vf> <wr> ^r + <wh> ^h + 2_O_0 - r----^h, (77) 2Hr
buoyancy
BF G =_ - <r> g^h, (78)
magnetic tension
[ ] r^ <BrBh >- <B2 > coth GMT =_ - --- \~/ . <BhB > + ------------f------- 4p r [ 2 < 2>] + -^h- \ ~/ . <B B >- <B-h> +-B-f-- (79) 4p r r
and viscous diffusion
n -- GVD =_ - ------ \~/ (rsin hp) + rnSVD, (80) rsinh
where the radial and latitudinal components of SVD are given by
( ) SVD =_ (H -1 - H -1)-@-<v >- 5-H -1 + H -1 1-@--<v > r r n @r h 3 r n r @h r [ ( ) - 1 -1] + H -1 H -1 + 3- + dH-r--+ H-n-- <vh> (81) r r r dr r
and
2 ( ) @ [ ( 6) dH -1 ]<v > SVhD =_ -- H -r1+ 3H -n1 ---<vr> + H -r 1 2H -n1- H -r1- -- - ---r-- --r-, (82) 3 @r r dr 3
and where Hr and Hn are the scale heights for the density and viscosity:
-- - 1 1-dr- -1 1dn- H r = r-dr, H n = ndr . (83)

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