A Weak Constraint 4D-Var Assimilation System for the Navy Coastal Ocean Model Using the Representer Method - Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications - page 362

15.3 The 4D-Var System

15.3.1 Linearization

Nonlinear terms in the model consist of all the advection terms in the momentum and
tracer equations, the horizontal mixing with the Smagorinsky formula, the curvature
correction, the vertical mixing with coefficients computed using the Mellor-Yamada
2.5 turbulence closure. Additional nonlinearities stem from the discretization in flux
conservative form where vertical increments

z in the sigma layers depend on the
free surface elevation. As a consequence, even the time discretization is nonlinear.
Nonlinearities also appear in the free surface, or barotropic mode, with the multiplica-
tion by the depth variables

u and v in ( 15.23 )and( 15.24 ). However, the barotropic

D D

u v

transports v are computed explicitly first, then the barotropic velocities

( u and v ) are derived by dividing the barotropic transports by the depth variable,
which is a nonlinear operation. The baroclinic pressure gradient is computed from the
density field obtained from the state equation as a nonlinear function of temperature
and salinity. Other nonlinearities appear in the various radiative conditions at the open
boundaries of the model domain mentioned above.
With the exception of the Mellor-Yamada turbulence closure, all of these
nonlinear terms are linearized according to the first-order Taylor's approximation
for the derivation of the tangent linear model.
For the sake of clarity, let's rewrite the leap-frog time discretization of ( 15.14 ),
see the appendix, in the form

D u and D

/ n 1 u n 1

u u

x y

/ n C 1 u n C 1 . z u z u D G n ;

. (15.1)

2t

G n represents the terms in the right hand side of ( 15.14 ) evaluated at time

where

/ n C 1 is available from a previously computed
elevation. The numerical model is updated by

z u level n , and the depth increment .

u G n

1 2t

x

u n C 1 D z u

/ n 1 u n 1 C

.

(15.2)

/ n C 1

u

y

. z u

The linearization of ( 15.2 )is

u ıG n

1 2t

x

u n C 1 D z u / n 1 ı u n 1 C .ı z u / n 1 u n 1 C ı .

/ n C 1

u

y

. z u

u G n

z u / n C 1

.ı

2t
x

/ n 1 u n 1 C z u . (15.3)

h
. / n C 1 i 2

u

y

z u

where u is the background solution, i.e. the solution around which the model is
linearized, G and z are computed using the background solution, and ı u , ıG

and ı z are the linear perturbations of u, G and z respectively. In both ( 15.2 )