4.3.3 Dynamic boundary conditions

The purpose of this section is to discuss the dynamic boundary conditions, which are conditions that prescribe the momentum flux through the model's side, bottom, and top boundaries.

As discussed in Section 7.4.1, bottom stress arises from both resolved topography, as well as unresolved or sub-grid scale (SGS) topography and bottom boundary layer effects. In MOM, it is possible to parameterize the SGS bottom stress either as a free-slip bottom drag,

$${\bf\tau}_{bottom-sgs}$$ = 0, (4.30)

or in terms of the flow near the bottom

 

$${\bf\tau}_{bottom-sgs} \rho_{o}C_D \vert{\bf u}_h\vert{\bf u}_h.$$ = (4.31)

Issues related to the stress arising from resolved topography with the full and partial cells are discussed in Chapter 26, and the bottom boundary layer issues are discussed in Chapter 36.

The momentum flux through the sea surface ${\bf\tau}_{surf}$( $\mbox{dyn}\,\mbox{cm}^{-2}$) comes from two sources:

 

$${\bf\tau}_{surf} {\bf\tau}_{winds} + {\bf\tau}_{fresh},$$ = (4.32)

which are the wind stress ${\bf\tau}_{winds}$ and momentum transfer in connection with a fresh water flux, ${\bf\tau}_{fresh}$. The dominating mechanism is the wind stress which comes from the interaction of the wind field with the ocean surface waves. Since the atmosphere-ocean boundary layer is not resolved by the model, it is parametrized, e.g., as function of the wind speed in some reference height in the boundary layer. A simple example is

 

$${\bf\tau}_{winds} \rho_{a} \, C_D^{wind} \, \vert{\bf u}^{wind}\vert{\bf u}^{wind},$$ = (4.33)

where $\rho_{a}$ is the density of the air, ${\bf u}^{wind}$ is the wind speed and C_D^wind a drag coefficient which depends on the wind speed, but also on the stability of the atmospheric boundary layer and the wave height. Generally, the physically correct calculation of the wind stress is not well known. Such uncertainty has prompted some climate modelers to consider coupling their ocean model to a surface wave model. The wave model then directly feels the winds from the atmosphere and is able to more accurately compute the surface stress field for use in the ocean model. The other mechanism for the vertical momentum transfer is fresh water flux. The fresh water volume flux through the air-sea interface carries a momentum which is approximately

 

$${\bf\tau}_{fresh} q_w \, \rho_f \, {\bf u}^{wind}.$$ = (4.34)

As discussed in Section 7.4.2, MOM identifies this flux with $q_w \, \rho_o \, {\bf u}(z=\eta)$, where ${\bf u}(z=\eta)$ is the horizontal current at the ocean surface. With a resolved boundary layer model, such as a wave model, this identification would not necessarily be exact.

Momentum flux through lateral boundaries is given by no-normal flow as well as no-slip boundary conditions. Therefore, all velocity components next the side walls are set to zero. The means for doing so are through the model's land-sea mask. Although the model employs no-slip next to the side boundaries, all that is necessary for formulating the solution methods for the tracer and momentum equations is the no-normal flow condition. This is an important point since the distinction made in MOM between ``side'' and ``bottom'' is possible only through its use of artificial stepped topography. In the real ocean, there clearly is no distinction. In principle, therefore, the methods employed in MOM can be used for a free-slip model with a smooth representation of the bottom. The details on how the prescribed momentum flux through the model boundaries is linked with the model variables are described in Sections 6.4.1.