Data Assimilation ofWeather Radar and LIDAR for Convection Forecasting and Windshear Alerting in Aviation Applications - Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications - page 537

main difference between the two approaches lies in the number of control variables
employed. In Newsom and Banta ( 2004 ), the subgrid-scale fluxes of momentum
and heat are modeled through theoretical assumptions for turbulent eddy viscosity
and thermal diffusivity estimations, rather than treating directly the viscosity and
diffusivity as control variables. The advantage of using theoretical subgrid-scale
model is that the reduced number of control variable may improve the efficiency of
4DVAR calculation. However, the use of theoretical sub-grid scale model may not
be sufficient for resolving the turbulent eddy structures. This is due to the drawback
of the inability of using the subgrid-scale model to represent the turbulent field
correctly with a single universal constant, especially in strong shear, rotating flow,
near topography or transitional regimes ( Germano et al. 1991 ). In order to create
a computationally efficient analysis for our purposes, we have followed similar
approach as developed by Newsom and Banta ( 2004 ). For ensuring the correctness
of the retrieved eddy structures, the subgrid-scale model coefficients need to be
properly preset before performing LES.
The fundamental idea of the 4DVAR is to fit the prognostic/forward model to the
observations. This would rely on the estimation of the cost function to tell whether
the “fitting” is good enough. In our case, the cost function is given as follows.

J D Jr C Jd C Js (22.14)

The first term in ( 22.14 ), Jr , is the difference between forward model predicted radial
velocity and the LIDAR observations within the specified time window ( 3 min in

our cases). Jd is the divergence penalty term used for suppressing the divergence in
the initial field. Js is the smoothing penalty term and it helps to smooth the output a
little for easily identifying any possible eddy structures in the retrieved wind field.
Jr and Jd have the forms as taken from Newsom and Banta ( 2004 ) whereas the
Js is given by

X

Js D 1
2

w u . r 2 u / C w v . r 2 v / C w w . r 2 w Π/ (22.15)

i;j;k

The weighting factors w u and w v are normally set to 0.001 and w w is set to 0.5.
These are guess values for the time being. Further tests are required for determining
these weightings empirically. The governing equations and adjoint derivations are
summarized in the following subsections (the formulation is similar to Newsom and
Banta ( 2004 )).

22.4.3.1 Governing Equations

The governing equations are the Boussineq equations for a shallow atmospheric
boundary layer:

@ u i

@t

C @.

u i u j /
@x j

D @p

@x i C ı i3 g . h i /

" ijk f j u k @ ij

@x j ;

(22.16)

‚ ref