# Equations of Motion

Equations of Motion

## Contents

The primitive equations in Cartesian coordinates are shown here. The momentum balance in the - and -directions are:

 $StartFraction normal partial-differential u Over normal partial-differential t EndFraction plus ModifyingAbove v With right-arrow dot normal nabla u minus f v equals minus StartFraction normal partial-differential phi Over normal partial-differential x EndFraction minus StartFraction normal partial-differential Over normal partial-differential z EndFraction left-parenthesis ModifyingAbove u prime w prime With bar minus nu StartFraction normal partial-differential u Over normal partial-differential z EndFraction right-parenthesis plus script upper F Subscript u Baseline plus script upper D Subscript u$ (1)
 $StartFraction normal partial-differential v Over normal partial-differential t EndFraction plus ModifyingAbove v With right-arrow dot normal nabla v plus f u equals minus StartFraction normal partial-differential phi Over normal partial-differential y EndFraction minus StartFraction normal partial-differential Over normal partial-differential z EndFraction left-parenthesis ModifyingAbove v prime w prime With bar minus nu StartFraction normal partial-differential v Over normal partial-differential z EndFraction right-parenthesis plus script upper F Subscript v Baseline plus script upper D Subscript v$ (2)

The time evolution of a scalar concentration field, (e.g. salinity, temperature, or nutrients), is governed by the advective-diffusive equation:

 $StartFraction normal partial-differential upper C Over normal partial-differential t EndFraction plus ModifyingAbove v With right-arrow dot normal nabla upper C equals minus StartFraction normal partial-differential Over normal partial-differential z EndFraction left-parenthesis ModifyingAbove upper C prime w prime With bar minus nu Subscript theta Baseline StartFraction normal partial-differential upper C Over normal partial-differential z EndFraction right-parenthesis plus script upper F Subscript upper C Baseline plus script upper D Subscript upper C$ (3)

The equation of state is given by:

 $rho equals rho left-parenthesis upper T comma upper S comma upper P right-parenthesis$ (4)

In the Boussinesq approximation, density variations are neglected in the momentum equations except in their contribution to the buoyancy force in the vertical momentum equation. Under the hydrostatic approximation, it is further assumed that the vertical pressure gradient balances the buoyancy force:

 $StartFraction normal partial-differential phi Over normal partial-differential z EndFraction equals minus StartFraction rho g Over rho Subscript o Baseline EndFraction$ (5)

The final equation expresses the continuity equation for an incompressible fluid:

 $StartFraction normal partial-differential u Over normal partial-differential x EndFraction plus StartFraction normal partial-differential v Over normal partial-differential y EndFraction plus StartFraction normal partial-differential w Over normal partial-differential z EndFraction equals 0$ (6)

For the moment, the effects of forcing and horizontal dissipation will be represented by the schematic terms and , respectively. The horizontal and vertical mixing will be described more fully in Horizontal Mixing and Vertical Mixing Parameterizations. The variables used are shown here:

 $script upper D Subscript u Baseline comma script upper D Subscript v Baseline comma script upper D Subscript upper C Baseline$ diffusive terms $script upper F Subscript u Baseline comma script upper F Subscript v Baseline comma script upper F Subscript upper C Baseline$ forcing terms $f left-parenthesis x comma y right-parenthesis$ Coriolis parameter $g$ acceleration of gravity $h left-parenthesis x comma y right-parenthesis$ bottom depth $nu comma nu Subscript theta Baseline$ molecular viscosity and diffusivity $upper K Subscript m Baseline comma upper K Subscript upper C Baseline$ vertical eddy viscosity and diffusivity $upper P$ total pressure $upper P almost-equals minus rho Subscript o Baseline g z$ $phi left-parenthesis x comma y comma z comma t right-parenthesis$ dynamic pressure $phi equals left-parenthesis upper P slash rho Subscript o Baseline right-parenthesis$ $rho Subscript o Baseline plus rho left-parenthesis x comma y comma z comma t right-parenthesis$ total in situ density $upper S left-parenthesis x comma y comma z comma t right-parenthesis$ salinity $t$ time $upper T left-parenthesis x comma y comma z comma t right-parenthesis$ potential temperature $u comma v comma w$ the ($x comma y comma z$) components of vector velocity $ModifyingAbove v With right-arrow$ $x comma y$ horizontal coordinates $z$ vertical coordinate $zeta left-parenthesis x comma y comma t right-parenthesis$ the surface elevation

These equations are closed by parameterizing the Reynolds stresses and turbulent tracer fluxes as:

 $ModifyingAbove u prime w Superscript prime Baseline With bar equals minus upper K Subscript upper M Baseline StartFraction normal partial-differential u Over normal partial-differential z EndFraction semicolon ModifyingAbove v prime w Superscript prime Baseline With bar equals minus upper K Subscript upper M Baseline StartFraction normal partial-differential v Over normal partial-differential z EndFraction semicolon ModifyingAbove upper C prime w Superscript prime Baseline With bar equals minus upper K Subscript upper C Baseline StartFraction normal partial-differential upper C Over normal partial-differential z EndFraction period$ (7)

An overbar represents a time average and a prime represents a fluctuation about the mean.

## Vertical Boundary Conditions

The vertical boundary conditions can be prescribed as follows:

top (:
 $StartLayout 1st Row 1st Column Blank 2nd Column upper K Subscript m Baseline StartFraction normal partial-differential u Over normal partial-differential z EndFraction equals tau Subscript s Superscript x Baseline left-parenthesis x comma y comma t right-parenthesis 2nd Row 1st Column Blank 2nd Column upper K Subscript m Baseline StartFraction normal partial-differential v Over normal partial-differential z EndFraction equals tau Subscript s Superscript y Baseline left-parenthesis x comma y comma t right-parenthesis 3rd Row 1st Column Blank 2nd Column upper K Subscript upper C Baseline StartFraction normal partial-differential upper C Over normal partial-differential z EndFraction equals StartFraction upper Q Subscript upper C Baseline Over rho Subscript o Baseline c Subscript upper P Baseline EndFraction 4th Row 1st Column Blank 2nd Column w equals StartFraction normal partial-differential zeta Over normal partial-differential t EndFraction EndLayout$ (8)
and bottom ():
 $StartLayout 1st Row 1st Column Blank 2nd Column upper K Subscript m Baseline StartFraction normal partial-differential u Over normal partial-differential z EndFraction equals tau Subscript b Superscript x Baseline left-parenthesis x comma y comma t right-parenthesis 2nd Row 1st Column Blank 2nd Column upper K Subscript m Baseline StartFraction normal partial-differential v Over normal partial-differential z EndFraction equals tau Subscript b Superscript y Baseline left-parenthesis x comma y comma t right-parenthesis 3rd Row 1st Column Blank 2nd Column upper K Subscript upper C Baseline StartFraction normal partial-differential upper C Over normal partial-differential z EndFraction equals 0 4th Row 1st Column Blank 2nd Column negative w plus ModifyingAbove v With right-arrow dot normal nabla h equals 0 EndLayout$ (9)

The new variables above are:

Variable Description
surface concentration flux
surface wind stress
bottom stress

The surface boundary condition variables are defined in the table above. Since is a strong function of the surface temperature, we usually choose to compute using the surface temperature and the atmospheric fields in an atmospheric bulk flux parameterization. This bulk flux routine also computes the wind stress from the winds.

On the variable bottom, , the horizontal velocity components are constrained to accommodate a prescribed bottom stress which is a choice of linear, quadratic, or a log layer, depending on the Options. The vertical scalar concentration fluxes may also be prescribed at the bottom, although they are usually set to zero.

## Horizontal Boundary Conditions

As distributed, the model can easily be configured for a periodic channel, a doubly periodic domain, or a closed basin. Code is also included for open boundaries which may or may not work for your particular application. Appropriate boundary conditions are provided for , and , as described in Boundary Conditions.

The model domain is logically rectangular, but it is possible to mask out land areas on the boundary and in the interior. Boundary conditions on these masked regions are straightforward, with a choice of no-slip or free-slip walls.

If biharmonic friction is used, a higher order boundary condition must also be provided. The model currently has this built into the code where the biharmonic terms are calculated. The high order boundary conditions used for are on the eastern and western boundaries and on the northern and southern boundaries. The boundary conditions for and are similar. These boundary conditions were chosen because they preserve the property of no gain or loss of volume-integrated momentum or scalar concentration.