By James R Holton
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Additional info for The Dynamic Meteorology of the Stratosphere and Mesosphere
41) 1 aci>z ) I aci>z I _ ocl>z' ocl>z' Ro ( --+u--+v --+w - - ay ax ot az + Br w' = Ro (J1 -ou + -1- -a (v I cos ()) + -1 -a (p0 w I) = 0 . 42) Here we have introduced the nondimensional zonal mean absolute vorticity - Ro 8 coso ay z =sin o- - - - (u cos 0). 43) This system is nearly identical to the set referred to as the "exact" perturbation system by Dickinson (1968b). Actually, this system is not really exact unless the mean meridional circulation vanishes. 43) will accurately describe any hydrostatic eddy motions on the spherical earth in the presence of an arbitrary mean zonal wind.
35) above]. However, in the tropical stratosphere the forcing terms J' and G' tend themselves to be O(Ro) and for short vertical wavelength modes Br may be as small as O(Ro2). 41) with the exception of the vertical advection of ci>2 which is actually smaller than indicated in our scaling because the vertical scale of the mean ftow is O(H) ~ D. The resulting perturbation equations are the same as Dickinson's "exact" model. 52) :r ' - 8 ) v, ( -8 +u- at ax 8 + [ Z- + COS (}- ay ( -8 - 8 ) at + U -ax m , 'Pz ( -ucos 8 o Geopotential is scaled for middle latitudes using geostrophic balance. Where necessary it is a simple matter to rescale geopotential for tropical motions. To avoid confusion we temporarily will designate dimensional variables by using asterisks. We let U, D, L designate the horizontal velocity, depth and horizontal length scales, respectively. L) is a Ross by number for the eddy motion. L) 2 , (Burger number)}. 18) we obtain the corresponding nondimensional set: Ro-8u ) v sm . () =- F-x Ro -bit - ( 1 + Dt cos () Ro 8 u ) Ro 3 -bv + ( 1 + -Dt D
Geopotential is scaled for middle latitudes using geostrophic balance. Where necessary it is a simple matter to rescale geopotential for tropical motions. To avoid confusion we temporarily will designate dimensional variables by using asterisks. We let U, D, L designate the horizontal velocity, depth and horizontal length scales, respectively. L) is a Ross by number for the eddy motion. L) 2 , (Burger number)}. 18) we obtain the corresponding nondimensional set: Ro-8u ) v sm . () =- F-x Ro -bit - ( 1 + Dt cos () Ro 8 u ) Ro 3 -bv + ( 1 + -Dt D