45 
EQUATION OF STATE 
The equation of state describes the relation between density P, temperature T and 
salinity S in the form p = p(S,T, p). Details are provided in Gill (1982). 
BUDGET EQUATIONS FOR HEAT AND SALT 
For a complete description of shallow water dynamics, the budget equations for 
temperature and salinity are required. 
The budget equation for temperature T reads: 
8T 1 д(иТ) 1 9(vcos<pT) d(yvT) _ 
dt R cos cp дЛ R cos cp d(p dz 
I t( K ‘ эг )| l 8 J.cospg, dT )+ d (K m) 
R cos <p дЛ R cos <p 8Л R cos cp 8<p R dtp dz dz 
where K h ,K v denote the horizontal and vertical diffusion coefficients, respectively. We 
assume that eddy diffusivity is related to eddy viscosity as follows. 
v- _ „ _ A v 
h 2 ’ v Pr 
The evolution of the temperature distribution is controlled not only by advection and 
diffusion inside the water body but also by heat fluxes through the sea surface and 
bottom. The computation of heat fluxes requires meteorological boundary values for 
wind, air temperature, cloudiness, and specific humidity. The detailed formulation of heat 
fluxes in the model has been described by Muller-Navarra and Ladwig (1997). Stipulating 
such boundary fluxes of heat amounts to a non-trivial specification of the term k v -%■ at 
the upper and lower surfaces. At the bottom, a rough parameterisation is used to take 
into account (diffusive) conduction and heat storage within the sediment. This was 
necessary particularly in the Wadden Sea tidal flats with their alternating phases of 
flooding and falling dry. 
Analogously, the equation for salinity S reads: 
dS 1 d(uS) 1 9(v cos^aS) d(wS) 
— + + + 
dt R cos cp 8Л R cos cp d(p dz 
1 d K h dS )+ 1 9 r cos<pK h dS } , 9 (K dS_ } 
R cos <p 8Л Rcos<p 8Л R cos cp dcp R d(p dz v dz 
The equations have a flux form, which is reasonable with respect to conservation and 
hence appropriate for the numerical treatment.