The formulation of the vertical turbulent mixing is taken almost directly from the very simple scheme used in the LMD climate model. The effect of turbulent mixing on momentum and potential temperature is evaluated by the mean of a classical diffusion equation
where the turbulent mixing coefficient is computed as
in term of mixing length l and a diagnostic estimate of the turbulent energy
(the minimum value of the kinetic energy was set to
m
s
).
In the case where
,
the vertical mixing coefficient can be expressed as a function
of the Richardson number
as
For both momentum and potential temperature,
the turbulent surface flux is computed as the product between the vertical
gradient of the quantity
(estimated between the surface value and that in the first
atmospheric layer) and a drag coefficient given by:
where is the wind in the first atmospheric layer and
m s
.
The diffusion equation is integrated with an
implicit time scheme.
In the simulations presented below, l was fixed to 35 m and
to
2
, typical values used for terrestrial surfaces in the
climate GCM.
we use the formulation developed by [8] for the terrestrial planetary boundary layer, in the form which has been implemented at ECMWF.
The drag coefficient is given by
where is the roughness length, z is the height of the middle
of
the first atmospheric layer, k=0.4 is the von Karman constant and
is the Richardson number
where and ||V|| are the potential
temperature and wind
velocity in the first atmospheric layer, and
the surface
temperature.
Two different functions f are used for momentum (
) and
potential
temperature (
). In either case, a different formulation is
used
depending on whether the atmosphere is stable (
)
or unstable ( )
As in the original terrestrial version, we use C=b=d=5.