Linearized Physics for Data Assimilation at ECMWF - Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications - page 258

11.5.1.2 Vertical Diffusion

Vertical diffusion applies to wind components, dry static energy and specific
humidity. The exchange coefficients in the planetary boundary layer and the drag
coefficients in the surface layer are expressed as functions of the local Richardson
number,

,( Louis et al. 1982 ). They differ from the formulation of the operational
forecast model (i.e. full non-linear scheme) where for the unstable regime (

Ri

)
the Monin-Obukhov (M-O) formulation is used, together with a K-profile approach
for convectively mixed layer in the case of unstable surface conditions. In the
linearized model, the exchange coefficients are computed according to Louis et al.
( 1982 ). For the stable regime (

Ri <0

), diffusion coefficients according to the Louis
scheme are used close to the surface and above 300 m, then they tend asymptotically
to the M-O formulation. A mixed layer parametrization is also included. This is
consistent with the full non-linear model.
Analytical expressions are generalized for the situation with different roughness
lengths for heat and momentum transfer. For any conservative variable

Ri >0

(wind

vector components, u and v ; dry static energy, s ; specific humidity, q ), the tendency

produced by vertical diffusion is

@
@t

D 1

@
@

K.Ri/ @
@

(11.16)

z z

K

where is the air density. The exchange coefficient for heat and momentum

transfer is given by

K D l 2 ˇ ˇ ˇ ˇ

ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ f.Ri/

@ U

@

(11.17)

z

where U is the wind vector and represents the coefficient accounting for

the dependence of vertical turbulent diffusion on the local Richardson number,
either computed according to Louis et al. ( 1982 ),

f.Ri/

f L .Ri/ , or to the Monin-Obukhov

formulation, is the mixing length profile based on the formulation of

Blackadar ( 1962 ) with a reduction in the free atmosphere.
A continuous transition between Louis coefficients near the surface to about
300 m and M-O coefficients above is computed as

f MO .Ri/ . l

1
l p f.Ri/

1 1

p f MO .Ri/

D

z p f L .Ri/

C (11.18)

where is the Von Karman's constant, z is the height and is the asymptotic mixing

length.
To parameterize turbulent fluxes at the surface, the drag coefficient, C sf , (i.e. the

exchange coefficient between the surface and the lowest model level) is computed
as

C sf D g sf .Ri/ C N (11.19)