9 
Courant-Friedrichs-Lewy type, i.e. an inequality constraint involving a non-dimensional 
ratio of the relevant physical quantity (velocity, viscosity, diffusivity) to the pertinent figure 
of resolvability - a product of powers of mesh size and time step.) 
To put it simply, explicit treatment will work as long as significant response is guaranteed 
to take much longer than the beat of time resolution. An explicit numerical scheme must 
work with sufficient time resolution while an implicit procedure in principle needs not. 
Therefore, in principle, all processes that may possibly cause such difficulties are treated 
implicitly. Especially the coefficient (viscosity, diffusivity) of vertical coupling may vary on 
a scale covering several magnitudes. That is why vertical coupling, both shear and 
diffusion, is treated fully implicitly, i.e. over the complete water column at a time. 
An explicit treatment is used for horizontal coupling. In this context, a stringent stability 
constraint comes from the propagation speed of gravity waves, which constitute the 
fastest process and hence dictate the basic time step. As the time stepping scheme for 
the basic shallow water system is symmetric in time (forward-backward), central 
differences are employed for most spatial derivatives. A (non-conservative) explicit 
vector upwind scheme is used merely for transport of momentum. 
3.2 Specialities 
The transport algorithm for salinity and temperature is based on the budget equations for 
salt and heat, respectively. The numerical scheme is conservative, fully explicit, truly 
multi-dimensional (no fractional steps as to flow directions), shape-preserving and of low 
numerical diffusion. Its essential elements are, firstly, the upwind calculation of fluxes 
(van Leer’s 2D-formula (1984) generalised to 3D), secondly, the local gradient 
approximation by the zero average phase error method of Fromm (1968) and, thirdly, the 
Zalesak limiter (1979). The upwind bias of spatial discretisation is associated with the 
asymmetry (explicit) of time discretisation. As in any other explicit method, the time step 
is restricted by a stability condition. However, relying on an estimation of practical flow 
velocities, its regime can be kept well away from the stability bound. Of course, flux- 
corrected transport is computationally burdensome. However, in our implementation 
the computation load is eased by integrating several basic time steps into one larger 
time step, which allows more effective use of the stability restriction, well away from the 
constraint. 
A special problem is posed by the ice dynamics equations. In our view, an explicit 
method is out of the question because any disturbance propagates extraordinarily quickly 
and, moreover, the propagation speed depends notably on the ice properties. The 
unwieldy nature of the model is due to the constitutive plasticity of ice, which makes the 
problem highly non-linear. Therefore, iteration is required and an implicit method is used 
which is sufficiently robust and economical. The BSH model employs a fully implicit 
scheme working stably with any time step. The method used is successive approximation 
with the non-linearity lagged. The chosen time step of 15 minutes is much larger than any 
conceivable explicit value, but short enough to follow all variations of the atmosphere 
above the ice and of the sea below.