source: (poseidon.hydro.f90) [ c_92_rp1 release ]

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HORIZONTAL HYDRODYNAMICS

This routine advances the state of the ocean due to horizontal hydrodynamics:

The state is advanced for one "timestep" by splitting the equations into an "internal" and "external" mode. The external mode is solved with much finer timesteps in order to resolve the fast external gravity waves.

The effects which are included here are:

- CORIOL
- coriolis effect ( plus momentum advection in vector invariant form )
- PRESS
- pressure gradient
- HDIFFUSE
- horizontal diffusion (explicit)
- *SFLUX*
- surface forcing
- PEN_RADIATION
- penetrating radiation
- *HOR_ADV*
- horizontal mass and heat advection

The effects which are not included:

- VDIFFUSE
- vertical diffusion and viscosity
- VVELOC and VERADV
- vertical remapping
- SHAPIRO and FILTER_H
- Shapiro filtering (except that shapiro filtering on h creates a bolus flux which is applied here)
- MIXDLYR
- Mixed layer

The momentum equations may be written in the form

{partial v}over{partial t} = - f k times v - g nabla %eta + A + P_1 + F + D

or

{partial v}over{partial t} = - f k times v - nabla p_B + g nabla D + A + P_2 + F + D

where A is advection of momentum, P_1 is pressure other than - g nabla eta P_2 is pressure other than - nabla p_B + g nabla D F is surface forcing and D is diffusion

We will solve this as

{partial v}over{partial t} = {partial v_E}over{partial t} - f k times v ' ' + {A}' + {P}' + {F}' + {D}'

where v_E is the external mode flow and X = overline{X} + X' is the separation in to vertical ' average and departure. Note that CORIOL routine computes both A and f k times v ' ' using the vector invariant form.

{partial v_E}over{partial t} is integrated in STEP_EXTERNAL :

{partial v_E}over{partial t} = - nabla p_B + g nabla D + f k times V_E overline{A} + overline{P_2} + overline{F} + overline{D}

or

{partial v_E}over{partial t} = - g nabla eta + f k times V_E overline{A} + overline{P_1} + overline{F} + overline{D}

The vertically integrated mass equation is

{partial H }over{partial t} = - nabla cdot v_E H

where H is the total mass in a water column. This equation is integrated in STEP_EXTERNAL, and later enforced in *HOR_ADV*.

During STEP_EXTERNAL, surface height %eta is diagnosed from

%eta = H ( 1 + overline{%alpha} ) - D

where overline{%alpha} is the mass-averaged value of %alpha - %alpha_o , and D is the depth of the ocean. In STEP_EXTERNAL, only v_E and H (and thereby %eta) are allowed to change.

To advance the equations from t^{n} to t^{n+1}, we use Bob Hallberg's (1997) ' split explicit scheme:

For the internal mode momentum equation: A Forward-Backward step is taken for the internal mode gravity wave equation A predictor-corrector step is taken for the momentum advection and Coriolis terms A forward step is taken for horizontal diffusion

For the internal mode mass (H) equation we use the forward-backward

The external mode is coupled to the internal mode

We advance the external mode overline{%alpha}, overline{A} , overline{P} , overline{F} , overline{D} and {A}' , {P}' , {F}' , {D}' based on the state at t^{n} (and t^{n-1} for diffusive terms)

The external mode equations for partial v_E / partial t and partial H / partial t are advanced with short steps from t^{n-1} to t^{n+1}.

partial v / partial t is evaluated using partial v_E / partial t and {A}' , {P}' , {F}' , {D}'

Horizontal advection of mass, heat and salt is treated, requiring that the vertically integrated mass fluxes from v^{n}h^{n} are equal to the integrated mass fluxes used in the external mode equations.

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