Physics

Due to the one-dimensional character of GOTM, the state-variables listed above are assumed to be horizontally homogeneous, depending only on the vertical $z$-coordinate. As a consequence, all horizontal gradients have to be taken from observations, or they have to be estimated, parameterised or neglected.

For example, the surface slopes $\partial_x\zeta$ and $\partial_y\zeta$ representing the barotropic pressure-gradients may be determined by means of local observations or results from three-dimensional numerical models. It is also possible to prescribe a time series of the near-bed velocity components for reconstructing the barotropic pressure gradient, see Burchard (1999). The implementation of these options for the external pressure gradient is carried out in extpressure.F90, described in section 3.2.8. The internal pressure-gradient, which results from horizontal density gradients, can be prescribed from observations of horizontal gradients of $\Theta$ and $S$ or from three-dimensional model results (see intpressure.F90 in section 3.2.9). These gradients may also be used for horizontally advecting $\Theta$ and $S$ (see section 3.2.11 and section 3.2.12).

Another option in GOTM for parameterising the advection of $\Theta$ and $S$ is to relax the model results to observations. Evidently, this raises questions about the physical consistency of the model, but it might help to provide a more realistic density field for studies of turbulence dynamics. Nudging is also possible for the horizontal velocity components. This makes sense in order to initialise inertial oscillations from observed velocity profiles, see section 3.2.6 and section 3.2.7. In the momentum equations, advection and horizontal diffusion terms are neglected.

In hydrostatic ocean models, the vertical velocity is calculated by means of the continuity equation, where the horizontal gradients of $U$ and $V$ are needed. Since these are not available or set to zero, the assumption of zero vertical velocity would be consistent. In many applications however, a non-zero vertical velocity is needed in order to reflect the vertical adiabatic motion of e.g. a thermocline. In GOTM, we have thus included the option of prescribing a vertical velocity time series at one height level which might be vertically moving. Vertical velocities at the surface and at the bottom are prescribed according to the kinematic boundary conditions ($w=0$ at the bottom and $w=\partial_t\zeta$ at the surface), and between these locations and the prescribed vertical velocity at a certain height, linear interpolation is applied, see updategrid.F90 in section 3.2.4. This vertical velocity is then used for the vertical advection of all prognostic quantities.

Standard relations according to the law of the wall are used for deriving bottom boundary conditions for the momentum equations (see friction.F90 in section 3.2.10). At the sea surface, they have to be prescribed or calculated from meteorological observations with the aid of bulk formulae using the simulated or observed sea surface temperature (see section 5.2). In stratification.F90 described in section 3.2.15, the buoyancy $b$ as defined in equation (33) is calculated by means of the UNESCO equation of state (Fofonoff and Millard (1983)) or its linearised version. In special cases, the buoyancy may also be calculated from a simple transport equation. stratification.F90 is also used for calculating the Brunt-Väisälä frequency, $N$.

The turbulent fluxes are calculated by means of various different turbulence closure models described in great detail in the turbulence module, see section 4.7. As a simplifying alternative, mixing can be computed according to the so-called `convective adjustment' algorithm, see section 3.2.16.

Furthermore, the vertical grid is also defined in the meanflow module (see updategrid.F90 in section 3.2.4). Choices for the numerical grid are so-called $\sigma$-coordinates with layers heights having a fixed portion of the water depth throughout the simulation. Equidistant and non-equidistant grids are possible.