SUBROUTINE NCALFA

The model may be run with the users choice of conditions at the open boundaries. Instabilities often occur at or close to these boundaries. In fjord studies we have for instance found it difficult to get fresh water fronts associated with river runoff smoothly through the open boundary at the mouth of the fjord. In such studies we have applied relaxation of the variables through FRS-zones, see Martinsen and Engedahl [25], which at least allows the production of reasonable model results in the model domain interior to the FRS-zone, see [5,7]. Each prognostic variable, $\phi$, in the zone is after each model time step updated according to

$$\begin{eqnarray*} \phi & = & (1 - \alpha)\phi_{M} + \alpha \phi_{F}, \end{eqnarray*}$$

where $\phi_{M}$ contains the unrelaxed values computed by the model and $\phi_{F}$ is a specified forced solution in the zone. The relaxation parameter $\alpha$ varies from 1 at the model boundary to 0 at the end of the zone facing the interior model domain. The quality of this flow relaxation scheme depends strongly upon the quality of the specified forced solutions, $\phi_{F}$, chosen for the prognostic variables.

The SUBROUTINE NCALFA computes the relaxation parameters ALPHA(LB) and ALPHAE(LB), where LB is the width of the FRS-zone, to be used for 3-D variables and 2-D variables respectively.

It may be shown, see [25], that to use this boundary conditions is equivalent to adding a Newtonian friction term of the form

$$\begin{eqnarray*} & & -K (\phi_{M} - \phi_{F}), \end{eqnarray*}$$

to the right hand side of the equations with

$$\begin{eqnarray*} K & = & \frac{\alpha}{\Delta t (1 - \alpha)}, \end{eqnarray*}$$

where $\Delta t$ is the time step used when propagating the variable $\phi$. Thus the strength of Newtonian forcing is dependent on the time step, and we are thus attempting to solve a different mathematical problem when we vary the time step. We have also seen in practice that by keeping the $\alpha$ arrays fixed and varying the time step, different solutions may be produced. This problem is not explicitly discussed in [25].

In NCALFA we therefore first compute the $\alpha$ arrays with one of the formulas from [25]. The $tanh$ formulation is normally used. We assume that this produces a reasonable Newtonian friction throughout the FRS-zone for the time step $\Delta t_0$. The values of $K$ throughout the zone are computed from the above formula. Then we recompute the $\alpha$ arrays, one for the 3-D variables that are propagated with time steps DT and one for the 2-D variables that are propagated with time steps DT/N2D, according to

$$\begin{eqnarray*} \alpha & = & \frac{\Delta t K}{(1 + \Delta t K)}. \end{eqnarray*}$$

The results are placed in the model variables ALPHA and ALPHAE respectively. By doing this the Newtonian friction is unaltered as the time steps are modified and we may at least hope to demonstrate convergence in time as the time step is reduced.

NCALFA may be unchanged from one application to another, but it may be advisable to test the sensitivity of the model results both to the representation of the $\alpha$ arrays and to the choice of $\Delta t_0$. As a rule of thumb choose $\Delta t_0$ close to the value DX / $\sqrt{2 \times \mbox{GRAV} \times \mbox{DEPMAX} }$ where DEPMAX is the maximum depth over the model region. If the user experiences problems at or close to the boundaries, the above options may be attempted. If the quality of the forcing field $\phi_{F}$ is poor, this may also reduce the quality of the model outputs and possibly cause instabilities. See [5] for a suggestion on how to produce $\phi_{F}$ in idealized fjord studies.