K. 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
Xn+1 = Xn-1 + F__t____z(Cn-1 - Cn )
where C is the coefficient that transforms the flux into a kinematic flux of property X .
Conservation, i.e. Xn+1 = Xn-1 , is satisfied if and only if Cn = Cn-1 . For X being
potential temperature, C = (%Cp )-1 is the reciprocal of density times heat capacity.
Although these are not in general constant, the conserved quantity is % Cp , and when is
made the prognostic variable, % Cp 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 Sb ; C becomes
(So - Sb )%-1f , where conservation is satisfied for a constant ocean salinity, So , freshwater
density %f and Sb . Thus, water fluxes of different salinities, Sb , must first be converted
into an equivalent freshwater flux by multiplying by a factor (So - Sb )S-1o , 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, Cw , 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________________________________
%a - air density 1.2 kg=m3
Cpa - heat capacity of air 1005 J=kg=K
(% Cp )a - air density x heat capacity 1206 J=m3 =K
Cpv - heat capacity of water vapor 1810 J=kg=K
Cvir - (Cpv =Cpa - 1) 0.80
Zvir - Rv =Ra - 1 0.606
V - latent heat of vaporization 2500000 J=kg
F - latent heat of fusion 334000 J=kg
S - latent heat of sublimation 2834000 J=kg
%f - density of freshwater 1000 kg=m3
%o - density of ocean water 1026 kg=m3
Cpo - heat capacity of ocean water 3996 J=kg=K
(% Cp )o - water density x heat capacity 4:1 x 106 J=m3 =K
Tf - freezing point of ocean water -1.8 C
So - salinity of ocean water 34.7 ppt
%i - density of sea-ice 905 kg=m3
Cpi - heat capacity of sea-ice 2100 J=kg=K
(% Cp )i - ice density x heat capacity 1:9 x 106 J=m3 =K
Tm - melting point of sea-ice 0 C = 273.16 K
Si - salinity of sea-ice 4 ppt
%s - density of snow 330 kg=m3
Cps - heat capacity of snow 2090 J=kg=K
(% Cp )s - snow density x heat capacity 690000 J=m3 =K
g - gravitational acceleration 9.80616 m=s2
- von Karman constant 0.40
oe - Stefan-Boltzmann constant 5:67 x 10-8 W=m2 =K4
- rotation rate of the earth 2ss=86164 =s
Cw - coefficient in ice-ocean stress 0.6524 kg=m2 =s
K-2