Data Assimilation ofWeather Radar and LIDAR for Convection Forecasting and Windshear Alerting in Aviation Applications - Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications - page 538

@ 1
@t

C @. 1 u j /
@x j

C w @ 0
@ z D @ @x j ;

(22.17)

@ u j

@x j D 0;

(22.18)

Here x i 's are the Cartesian components of the position vector x D Œx;y; z , u i 's the

Cartesian components of the velocity vector u D Œ u ; v ; w ; ı ij the Kronecker delta,

" ijk the permutation tensor, the acceleration due to

gravity and angled brackets represent averaging on horizontal planes. The pressure
p

f j the Coriolis parameter, g

is the non-hydrostatic component of the pressure normalized by the reference
density ref . Virtual potential temperature is decomposed as

.x;y; z ;t/ D 0 . z / C 1 .x;y; z ;t/; (22.19)

where subscript 0 refers to the initial base state profile and subscript 1 the dynamic
perturbations about the base state. ‚ ref is a reference virtual potential temperature

and is set to be equal to the virtual temperature at the reference level, namely,
the ground.

ij and are the turbulent fluxes of momentum and temperature

respectively.
The anisotropic component of the turbulent momentum flux is modeled as

ij 1

3 ı ij kk D 2K m D ij ;

(22.20)

where D ij is the strain rate tensor,

@

@x j C @

D ij D 1
2

u i u j

@x i

1

3 ı ij @

u k
@x k :

(22.21)

The eddy viscosity K m can be calculated using a number of different models. There

are models based on Troen and Mahrt ( 1986 )and Smagorinsky ( 1963 ). The former
model is used in this paper. The isotropic component of the turbulent momentum
flux 3 ı ij kk is absorbed into the pressure term.
Similarly, the turbulent flux of virtual potential temperature is modeled as

D K h @

@x j :

(22.22)

K h is given by

Here the eddy diffusivity

K h D K m
P r t

(22.23)

where the turbulent Prantle number P r t is typically set to 0.4.