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The vertical momentum equation has the hydrostatic or
quasihydrostatic balance on the right hand side. This discretization
guarantees that the conversion of potential to kinetic energy as
derived from the buoyancy equation exactly matches the form derived
from the pressure gradient terms when forming the kinetic energy
equation.
In the ocean, using zcoordinates, the hydrostatic balance terms are
discretized:

(2.92) 
In the atmosphere, using pcoordinates, hydrostatic balance is
discretized:

(2.93) 
where
is the difference in Exner function between the
pressure points. The nonhydrostatic equations are not available in
the atmosphere.
The difference in approach between ocean and atmosphere occurs because
of the direct use of the ideal gas equation in forming the potential
energy conversion term
. The form of these conversion
terms is discussed at length in Adcroft [2002].
Because of the different representation of hydrostatic balance between
ocean and atmosphere there is no elegant way to represent both systems
using an arbitrary coordinate.
The integration for hydrostatic pressure is made in the positive
direction (increasing kindex). For the ocean, this is from the
freesurface down and for the atmosphere this is from the ground up.
The calculations are made in the subroutine CALC_PHI_HYD. Inside this routine, one of other of the
atmospheric/oceanic form is selected based on the string variable buoyancyRelation.
Next: 2.14 Fluxform momentum equations
Up: 2. Discretization and Algorithm
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