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In our shallow water context, the Reynolds’ stresses are viewed as (horizontal and 
vertical) turbulent transfer of horizontal momentum. 
1, Horizontal turbulent exchange of momentum 
In choosing a parameterisation for the terms u'u',u'v',VV we follow Smagorinsky (1963) 
and introduce a traceless tensor, i.e. we put u'u' = -vV. Obviously this is a conceptual 
contradiction because it always makes the kinetic energy of fluctuations vanish. In our 
context, however, its purpose is to provide a sink of energy for the mean flow. Thus a 
calculation of the budget of the total kinetic energy JJÿ(u 2 + v 2 )cos<p • dA- dtp suggests 
the following specification in terms of stretching and shearing deformation rate of the 
mean flow 
u u = - v v 
-AÏ0 
1 du 
R cos (p dA 
1 dv 
R d<p 
tan tp 
R 
v) 
u'v' = - A s h h ( 
1 dv 1 du tan© . 
• + + — u) 
R cosç dA R dtp R 
The eddy viscosities A s h ',A s h for horizontal stretch and shear are set equal to 
some A h which, for material objectivity, should depend on a tensorial invariant of the strain 
rate. Again following Smagorinsky, we take its deviatoric invariant and put 
A h = (k h Lj 
du 
R cos tp dA 
1 dv tan© , 2 . 
-vY +( 
Rdtp R 
1 dv 1 du tan© 2 
— + h — u) 
R cos cp dA R dtp R 
,1/2 
Such an expression for the eddy viscosity amounts to a mixing length formulation. As in 
large eddy simulation, the pertinent length scale L h is related to the grid mesh length 
(R cos tp/S.A, R&tp) as filter scale. We opted for 
(,k h L h ) 2 = (3/cKcos<pA/l)(3/cRA<p) 
where the non-dimensional constant tc is 0.4 and is termed the Karman constant. The 
factor 3 was estimated by numerical experimentation. 
2. Vertical turbulent exchange of momentum 
To ensure that the vertical exchange of horizontal momentum is dissipative, an eddy 
viscosity formula is used for the stresses r i =-u'W, =-vV inside the water 
column. It reads 
du dv 
As for horizontal exchange, we refer to Prandtl’s mixing length hypothesis to specify the