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2.9 Nonhydrostatic formulation
The nonhydrostatic formulation reintroduces the full vertical
momentum equation and requires the solution of a 3D elliptic
equations for nonhydrostatic pressure perturbation. We still
integrate vertically for the hydrostatic pressure and solve a 2D
elliptic equation for the surface pressure/elevation for this reduces
the amount of work needed to solve for the nonhydrostatic pressure.
The momentum equations are discretized in time as follows:
which must satisfy the discreteintime depth integrated continuity,
equation 2.16 and the local continuity equation

(2.56) 
As before, the explicit predictions for momentum are consolidated as:
but this time we introduce an intermediate step by splitting the
tendancy of the flow as follows:
Substituting into the depth integrated continuity
(equation 2.16) gives

(2.59) 
which is approximated by equation
2.20 on the basis that i)
is not yet known and ii)
. If 2.20 is
solved accurately then the implication is that
so that the nonhydrostatic pressure field does not drive
barotropic motion.
The flow must satisfy nondivergence
(equation 2.56) locally, as well as depth
integrated, and this constraint is used to form a 3D elliptic
equations for
:

(2.60) 
The entire algorithm can be summarized as the sequential solution of
the following equations:



(2.61) 



(2.62) 



(2.63) 



(2.64) 



(2.65) 



(2.66) 



(2.67) 



(2.68) 



(2.69) 



(2.70) 



(2.71) 
where the last equation is solved by vertically integrating for
.
Next: 2.10 Variants on the
Up: 2. Discretization and Algorithm
Previous: 2.8 Staggered baroclinic timestepping
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