K. \underline{Values of Physical Constants}

Conservation of momentum, heat and freshwater between CSM component models requires that certain physical quantities be kept constant in time and space throughout all the component models and the Flux Coupler. To illustrate, suppose a model is integrated by a first order time scheme over two time steps from n − 1 to n + 1, with an incoming flux F between n − 1 and n and an equal but opposite outgoing flux −F between n and n + 1. In the absence of all other transports, a prognostic variable, X, will change according to

X_{n+1} = X_{n−1} + \frac{F Δt}{Δz}(C_{n−1} − C_{n})

where C is the coefficient that transforms the flux into a kinematic flux of property X. Conservation, i.e. X_{n+1} = X_{n−1}, is satisfied if and only if C_{n} = C_{n−1}. For X being potential temperature, C = (ϱC_{p})^{−1} is the reciprocal of density times heat capacity. Although these are not in general constant, the conserved quantity is ϱ C_{p} θ, and when θ is made the prognostic variable, ϱ C_{p} is implicitly assumed constant, taken outside both time and space derivatives and divided out of all equation terms except the flux term shown above. For X being salinity in the ocean and F the flux of water of salinity S_{b}, C becomes (S_{o} − S_{b})ϱ_{f}^{−1}, where conservation is satisfied for a constant ocean salinity, S_{o}, freshwater density ϱ_{f} and S_{b}. Thus, water fluxes of different salinities, S_{b}, must first be converted into an equivalent freshwater flux by multiplying by a factor (S_{o} − S_{b})S_{o}^{−1}, before being combined into a single freshwater flux.

Similar arguments concerning heat and water fluxes back and forth across an interface with intervening lateral transport, show that the constants should also be spatially invariant. The latent heat of vaporization is another physical parameter that must remain spatially and temporally invariant. Then all heat lost from the ocean via evaporation appears at the point of condensation in the atmosphere.

The important constants in the CSM model are listed below in Table 1 along with their numerical values, which are set in the Flux Coupler. The last coefficient, C_{w}, is a special case because it is also used in the ice model, and must have the same value there for consistency. This constant occurs in both the Flux Coupler and sea–ice models because the determination of the ice–ocean stress is a special case in the present CSM setup, see Section G.

K-1 Table 1.

Constants Set in the CSM Flux Coupler and Their Values

ϱ_{a} - air density  1.2 kg/m^{3}
C_{pa} - heat capacity of air  1005 J/kg/K
(ϱ C_{p})_{a} - air density × heat capacity  1206 J/m^{3}/K
C_{pv} - heat capacity of water vapor  1810 J/kg/K
C_{vir} - (C_{pv}/C_{pa} − 1)  0.80
Z_{vir} - R_{v}/R_{a} − 1  0.606
Λ_{V} - latent heat of vaporization  2500000 J/kg
Λ_{F} - latent heat of fusion  334000 J/kg
Λ_{S} - latent heat of sublimation  2834000 J/kg
ϱ_{f} - density of freshwater  1000 kg/m^{3}
ϱ_{o} - density of ocean water  1026 kg/m^{3}
C_{po} - heat capacity of ocean water  3996 J/kg/K
(ϱ C_{p})_{o} - water density × heat capacity  4.1 × 10^{6} J/m^{3}/K
T_{f} - freezing point of ocean water  -1.8 C
S_{o} - salinity of ocean water  34.7 ppt
ϱ_{i} - density of sea-ice  905 kg/m^{3}
C_{pi} - heat capacity of sea-ice  2100 J/kg/K
(ϱ C_{p})_{i} - ice density × heat capacity  1.9 × 10^{6} J/m^{3}/K
T_{m} - melting point of sea-ice  0 C = 273.16 K
S_{i} - salinity of sea-ice  4 ppt
ϱ_{s} - density of snow  330 kg/m^{3}
C_{ps} - heat capacity of snow  2090 J/kg/K
(ϱ C_{p})_{s} - snow density × heat capacity  690000 J/m^{3}/K
g - gravitational acceleration  9.80616 m/s^{2}
κ - von Karman constant  0.40
σ - Stefan-Boltzmann constant  5.67 × 10^{−8} W/m^{2}/K^{4}
Ω - rotation rate of the earth  2π/86164 /s
C_{w} - coefficient in ice-ocean stress  0.6524 kg/m^{2}/s

K-2