Geoscience Reference
In-Depth Information
all resolvable processes and sub-gridscale parameterization, errors are attributed to
the initial conditions and external forcing for all the dynamical equations, and the
derivation of their estimates is given below. Note that there is no external forcing
applied to the continuity equation, and thus it is not assigned a model error either,
as in
Jacobs and Ngodock
(
2003
).
Consider the momentum equation (
15.14
) in its non-discretized form
@
u
@t
C
:::
D
:::
C
1
F
(15.13)
represents the wind stress atmospheric forcing (in Nm
2
), the volume
flux source and the tidal potential, and
where
F
is the water density. The model error
at the surface consists of errors in the wind stress. For the subsurface, errors are
assumed to arise from the volume flux and the tidal potential terms. We consider
errors to be high in magnitude at the surface and decreasing with depth. Although
the wind stress varies in space and time, its associated error is assumed uniform
in the horizontal directions. The error magnitude is considered to be 50 % of the
actual wind stress at the surface and decreasing with depth in order to mimic the
decreasing impact of wind stress with depth. Two terms contribute to the forcing
for the temperature equation: the net longwave, latent and sensible heat flux on one
hand, and the solar radiation on the other hand. Both are assumed to be 30 % in error
and the sum of their errors constitutes the forcing error in the temperature equation,
with a spatial distribution similar to the one used for the errors in the momentum
equation. A similar approach is taken for the errors in the salinity equation, where
the forcing consists of the river inflow and evaporation minus precipitation. Forcing
terms here are also considered to be 30 % in error. Finally the standard deviations
for the initial condition errors are 1 m for the surface elevation,
ms
1
for both
components of the velocity field, 2 K for temperature and 0.5PSU for salinity. These
rather high errors indicate the lack of confidence in the forcing fields and initial
conditions. Spatial and temporal correlation scales in (
15.12
)aresetto10kmand
30 h. The errors and scales above are obviously arguable, and it is not our intention
to defend their choice. Rather, they are selected in this preliminary assimilation
setup to demonstrate the functionality of the NCOM 4D-Var system. Smaller errors
will be adopted when the system is used with real observations.
0:5
15.3.6
The Minimization
The solution of the assimilation problem is found by solving the Euler-Lagrange
(EL) system of equations associated with the minimization of the cost function
(
15.11
). The EL system is a linear yet coupled system between the adjoint and
state variables. The representer methods uncouples the system by expanding the
solution as the sum of a first guess and a finite linear combination of representer
functions, with the representer coefficients computed by solving a linear system in
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