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Contents : If you have not already done so you are strongly encouraged to read the companion le on the non-divergent barotropic vorticity equation before proceeding to this shallow water case. We do not repeat the discussion that can be found there concerning spherical coordinates and the spherical harmonic spectral transform method. The Shallow Water Equations The shallow water equations describe the evolution of a hydrostatic homogeneous (constant density) incompressible ow on the surface of the sphere. The hydrostatic equation is accurate when the aspect ratio of the ow the ratio of the vertical scale to the horizontal scale is small. The shallow water equations are only relevant when the horizontal scale of the ow is much smaller than the depth of the uid. Hydrostatic balance is the statement that gravity balances the pressure gradient in the vertical equation of motion implying that vertical accelerations are negligible: p g (1) z If the density is constant then this equation implies that the horizontal pressure gradient is independent of z . One can therefore look for solutions in which the horizontal ow itself is independent of height. This is the key simpli cation that underlies the shallow water system. Since the horizontal ow is independent of height incompressibility implies that the vertical velocity is linear in z . The shallow water equations are utilized in di erent contexts in meteorology with di erent upper and lower boundary conditions. In the most familiar case of a free upper surface with imposed constant pressure ps at the upper surface integrating the hydrostatic equation down from the top we nd that the pressure at height z within the uid is ps + g (h z ) where h is the height of the interface. Therefore p (2) where gh. If the lower boundary is at then h H where H is the thickness of the uid layer. More generally H h hM where hM is the height of the lower boundary. In the default version of the code hM 0. 1 In the one and one-half layer con guration one assumes that there are two layers of uid with densities 1 and 2 with 2 1 . The top of the upper layer is assumed to be at a xed height zT . One assumes that the horizontal pressure gradient in the lower denser layer vanishes (as a shorthand one sometimes just states that the lower layer is at rest it s geostrophic ow is zero at least). Integrating up from a pressure p2 at some xed height z2 in the lower layer with the interface at z then at some height z1 in the upper layer p(z1 ) p2 ( z2 ) 2 g (z1 ) 1 g . The horizontal pressure gradient in the upper layer is then 2 1 1 p1 g g 1 1 (3) where g is referred to as the reduced gravity. We use the notation g (zT )g H where H is the thickness once again. In a rotating frame on the sphere the horizontal equations of motion are u u v u uv tan( ) 1 u fv + t a cos( ) a a a cos( ) and (4) v u v v v u2 tan( ) 1 f u t a cos( ) a a a (5) A constant radius a appears where the radial coordinate r would appear in full generality consistent with the assumption of a thin layer of uid. Vertical advection does not appear because u and v are assumed to be independent of height. The Coriolis and metric terms involving vertical motion are not included so as to maintain energy and angular momentum conservation laws. In a more convenient form 1 (E + ) u (f + )v t a cos( ) (6) v 1 (E + ) (f + )u t a (7) 1 v 1 u cos( ) a cos( ) a cos( ) (8) and where 2 and u2 + v 2 E 2 (9) The vorticity equation is then t (v(f + )) (10) 2 (11) while the divergence equation is D t where D A (E + ) 1 u 1 + v cos( ) a cos( ) a cos( ) (12) and A (f + )k v (f + )(v u) (13) Throughout these notes boldface is used for the component of a vector along the surface of the sphere the radial component if relevant is always written separately. The vorticity equation can be rewritten as (f + ) v t (f + ) (f + ) v (14) or D(f + ) (f + )D (15) Dt where D/Dt is the material derivative. On the sphere the vorticity and divergence completely de ne the ow. To obtain (u v ) from and D one can rst solve the Poisson equations for the streamfunction and velocity potential: 2 and 2 D. One then has 1 1 + (16) u a a cos( ) v 1 1 + a cos( ) a 3 (17) To complete the set of shallow water equations the momentum (or vorticitydivergence) equations must be supplemented with a st

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