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Fig. 15.1
Time evolution of
the magnitude of the
perturbation to the
temperature and salinity
fields normalized by the
magnitude of their respective
initial perturbations
TLM stability
10
3
Temperature
Salinity
10
2
10
1
10
0
2468 0 2 4 6 8 0 2
Days in August, 2003
(and hence (
15.3
)) a small positive number is usually added to the denominator to
prevent it from vanishing. As mentioned above the depth increments in the vertical
discretization in NCOM depend on the time varying elevation only in the sigma
layers. In the z-level portion of the vertical grid, (
15.2
)and(
15.3
) take the form
2t
u
n
C
1
D
u
n
1
C
z
u
G
n
(15.4)
x
u
y
u
and
2t
u
n
C
1
D
ı
u
n
1
C
z
u
ıG
n
:
ı
(15.5)
x
u
y
u
As for the vertical mixing coefficients from the Mellor-Yamada turbulence closure
scheme, they are provided by the nonlinear model trajectory around which the model
is linearized.
The stability of the linearized model is assessed by the time evolution of small
perturbations: the tangent linear model is initialized by random three dimensional
perturbations of the temperature and salinity fields and integrated over time. At each
time step the norms of the perturbed temperature and salinity fields are computed
and divided by the norms of their respective initial perturbations. Results plotted in
Fig.
15.1
show that the linear perturbations are stable and bounded for about 12-15
days before they start to grow exponentially. Initial perturbations here are generated
by the adjoint integration forced by Dirac impulses at randomly selected grid points.
This process produces three-dimensional initial fields with dynamically coherent
structures compared to purely random fields. However, the TLM test with purely
random fields did not yield different results (not shown).
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