Equations of Motion

Equations of Motion

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

$${\frac {\partial u}{\partial t}}+{\vec {v}}\cdot \nabla u-fv=-{\frac {\partial \phi }{\partial x}}-{\partial \over \partial z}\left({\overline {u'w'}}-\nu {\partial u \over \partial z}\right)+{\cal {F}}_{u}+{\cal {D}}_{u}$$

(1)

$${\frac {\partial v}{\partial t}}+{\vec {v}}\cdot \nabla v+fu=-{\frac {\partial \phi }{\partial y}}-{\partial \over \partial z}\left({\overline {v'w'}}-\nu {\partial v \over \partial z}\right)+{\cal {F}}_{v}+{\cal {D}}_{v}$$

(2)

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

$${\frac {\partial C}{\partial t}}+{\vec {v}}\cdot \nabla C=-{\partial \over \partial z}\left({\overline {C'w'}}-\nu _{\theta }{\partial C \over \partial z}\right)+{\cal {F}}_{C}+{\cal {D}}_{C}$$

(3)

The equation of state is given by:

$$\rho =\rho (T,S,P)$$

(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:

$${\frac {\partial \phi }{\partial z}}=-{\frac {\rho g}{\rho _{o}}}$$

(5)

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

$${\frac {\partial u}{\partial x}}+{\frac {\partial v}{\partial y}}+{\frac {\partial w}{\partial z}}=0$$

(6)

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

${\cal {D}}_{u},{\cal {D}}_{v},{\cal {D}}_{C}$	horizontal diffusive terms
${\cal {F}}_{u},{\cal {F}}_{v},{\cal {F}}_{C}$	forcing terms
$f(x,y)$	Coriolis parameter
$g$	acceleration of gravity
$h(x,y)$	bottom depth
$\nu ,\nu _{\theta }$	molecular viscosity and diffusivity
$K_{m},K_{C}$	vertical eddy viscosity and diffusivity
$P$	total pressure $P\approx -\rho _{o}gz$
$\phi (x,y,z,t)$	dynamic pressure $\phi =\left(P/\rho _{o}\right)$
$\rho _{o}+\rho (x,y,z,t)$	total in situ density
$S(x,y,z,t)$	salinity
$t$	time
$T(x,y,z,t)$	potential temperature
$u,v,w$	the ($x,y,z$) components of vector velocity ${\vec {v}}$
$x,y$	horizontal coordinates
$z$	vertical coordinate
$\zeta (x,y,t)$	the surface elevation

In the ocean, vertical mixing due to molecular viscosity is extremely weak compared to the turbulent mixing, so the terms involving $\nu$ and $\nu _{\theta }$ can be neglected.

To close these equations, parameterizations of the Reynolds stresses and turbulent tracer fluxes are introduced as functions of the other variables and vertical turbulent eddy viscosity and eddy diffusivity coefficients, $K_{M}$ and $K_{C}$, respectively.

$${\overline {u'w'}}=-K_{M}{\partial u \over \partial z};\qquad {\overline {v'w'}}=-K_{M}{\partial v \over \partial z};\qquad {\overline {C'w'}}=-K_{C}{\partial C \over \partial z}.$$

(7)

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

Details on ROMS options for computing $K_{M}$ and $K_{C}$ are described in Vertical Mixing Parameterizations.

Vertical Boundary Conditions

The vertical boundary conditions can be prescribed as follows:

top ($z=\zeta (x,y,t))$:

$${\begin{aligned}&K_{m}\,{\frac {\partial u}{\partial z}}=\tau _{s}^{x}(x,y,t)\\&K_{m}\,{\frac {\partial v}{\partial z}}=\tau _{s}^{y}(x,y,t)\\&K_{C}\,{\frac {\partial C}{\partial z}}={Q_{C} \over \rho _{o}c_{P}}\\&w={\partial \zeta \over \partial t}\end{aligned}}$$

(8)

and bottom ($z=-h(x,y)$):

$${\begin{aligned}&K_{m}\,{\frac {\partial u}{\partial z}}=\tau _{b}^{x}(x,y,t)\\&K_{m}\,{\frac {\partial v}{\partial z}}=\tau _{b}^{y}(x,y,t)\\&K_{C}\,{\frac {\partial C}{\partial z}}=0\\&-w+{\vec {v}}\cdot \nabla h=0\end{aligned}}$$

(9)

The new variables above are:

Variable	Description
$Q_{C}$	surface concentration flux
$\tau _{s}^{x},\tau _{s}^{y}$	surface wind stress
$\tau _{b}^{x},\tau _{b}^{y}$	bottom stress

The surface boundary condition variables are defined in the table above. Since $Q_{T}$ is a strong function of the surface temperature, we usually choose to compute $Q_{T}$ 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, $z=-h(x,y)$, 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 $u,v,C$, and $\zeta$, 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 $u$ are ${\frac {\partial }{\partial x}}\left({\frac {h\nu }{mn}}{\frac {\partial ^{2}u}{\partial x^{2}}}\right)=0$ on the eastern and western boundaries and ${\frac {\partial }{\partial y}}\left({\frac {h\nu }{mn}}{\frac {\partial ^{2}u}{\partial y^{2}}}\right)=0$ on the northern and southern boundaries. The boundary conditions for $v$ and $C$ are similar. These boundary conditions were chosen because they preserve the property of no gain or loss of volume-integrated momentum or scalar concentration.