8 
The grid cells serve as control volumes. They play the role of finite volumes for which 
integral forms of the budget equations and fluxes through cell faces are formed. Spatial 
derivatives are approximated by finite differences. 
For the spatial discretisation of differential operators (divergence, gradient), staggered 
gridding is natural. The gridded quantities in the BSH model are situated on the Arakawa 
C lattice. 
3.1.2 Temporal scheme 
The model is composed of a number of inter-related elements. It comprises several 
quantities and processes, each of which has its typical time scale. Physically, time scales 
are characterized by response to forcing and the effects of interaction. Time scales can 
also be related to spatial scale(s). 
An appropriate numerical treatment should be based on an expedient implementation of 
all actions, reactions and interactions involved. To organize the time stepping procedure, 
the model is structured as a network of elementary units. These individual components 
are treated successively, one by one, during a complete time step cycle. For reasons of 
convenience and flexibility, it is desirable to work with small units, i.e. modular design. 
Different equations of the system are, therefore, treated individually neglecting the 
coupling for a basic time step. Furthermore, advection and diffusion are separated. This 
is done because, firstly, they are physically different, secondly, there is no need to treat 
them jointly and, thirdly, there are expedient schemes for each. 
However, some couplings must be retained at the most basic level of coding so that 
there is a lower limit to decomposition. Consider, e.g., the heat budget of sea ice where 
incoming and outgoing heat fluxes typically are almost balanced so that the net 
input/output of heat which is responsible for ablation/growth is delicate. Therefore, it is 
treated as a whole. 
The modular decomposition should not introduce instability or any computational mode. 
Apart from that, one is guided by convenience. Fractional steps should only cause minor 
de-coupling of the system, if any. Any change makes itself felt in all other components in 
every full time step cycle. 
In the BSH model, the numerical time step is kept fixed, regardless of the intrinsic time 
scales of the system, some of which are variable. Especially flow velocities and 
exchange coefficients dictate time scales that are subject to the general evolution. For an 
explicit method, i.e. without taking into account changes of tendency, there is a maximum 
admissible time step which is related to the time scale of the modelled process. 
Conversely, a problem emerges when this time scale is variable, controlled by complex 
interactions and potentially close to the limit that is tolerated by the scheme. Whenever 
such a time scale comes close to the numerically given constraint the scheme is prone to 
lose stability. (The time step of an explicit method is restricted by a condition of the