Data Assimilation ofWeather Radar and LIDAR for Convection Forecasting and Windshear Alerting in Aviation Applications - Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications - page 539

The requirement that the velocity field remains divergence free, as implied by
( 22.17 ), is enforced either using a pressure correction method or a Poisson pressure
equation. In the pressure correction method, the momentum equations are integrated
first giving an estimate of the new velocity field u i

. This velocity field will in general
not be divergence free. The divergence becomes the source term in the pressure
correction equation, which is written

u i
t

@ 2 p 0
@x i

D @
@x i

: (22.24)

This is an elliptic equation, which is solved using the BiCGstab matrix equation
solver ( Nocedal 1980 ; Liu and Nocedal 1989 ). The resulting pressure correction
fields are then used to correct the pressure and velocity fields.
The pressure Poisson equation is written as

u i
t

@ 2 p
@x i

D @
@x i

@.

u i u j /
@x j

C ı i3 g . h i /
‚ ref

" ijk f j u k @ ij
@x j

(22.25)

This equation is solved before the momentum equations, to give a pressure field,
which, when used to calculate the pressure gradient in the momentum equations
ensures that the velocity field at the end of the next time step remains divergence
free. The pressure Poisson equation is used in the 4DVAR wind retrieval for the
following selected cases in this paper.

22.4.3.2 Adjoint Model Equations

The 4DVAR procedure uses an adjoint method to minimize the cost function .The

adjoint equations are derived by requiring that the first variation of the Lagrangian
L

J

. For conciseness, we present
the adjoint equations for the first-order Adam-Bashforth time integration scheme
( Shampine and Gordon 1975 ) as an example. For the first-order in time scheme, the
Lagrangian is defined as

with respect to all variables vanishes for t>0

N X

X

u i tF i / C Q n C 1 . n C 1 n tG n /

Πu n C i . u n C 1

i

L D J C

r

n D 0

C t p n C 1 P n (22.26)

; Q

Here u and p are the adjoint variables corresponding to u ; and p respectively,

P n are essentially the right hand
sides of the forward model equations with all variables evaluated at the

F n , G n and

and t is the time step. The functions

n th time
step. For details of the adjoint formulations, reference may be made to the appendix
in Newsom and Banta ( 2004 ). The current 4DVAR uses the adjoint equations for the