Geoscience Reference
In-Depth Information
the solution to (
7.58
)is
L
X.t/
D
X
0
c
0
L
:
(7.63)
L
In this case, the phase locations of the disturbances move away from their initial
locations faster than exponential with time. This can be seen by the fact that
disturbances in linear shear flow approach infinity, i.e.
X
!1,as
t
!1,but
Œt
1
D
L
2
=.c
0
X
0
/
disturbances in quadratic shear flow reach infinity in finite-time
.
t
L
2
=.c
0
X
0
/
Note that in the limit of small time, i.e.,
then equation (
7.63
)
behaves approximately as
X.t/
X
0
C
c
0
t
L
2
X
0
;
(7.64)
which is obviously non-Gaussian even when
X
0
is normally distributed and becomes
increasingly non-Gaussian as time goes on.
Hence, the structure of the shear flow the disturbances are being advected within
will determine whether the resulting phase distribution will be Gaussian or non-
Gaussian at some time later. Moreover, even though (
7.57
) is an equation linear
in the amplitude of the disturbance, its characteristics maybe be non-linear, which
could lead to non-Gaussian distributions. This implies that it is not sufficient to
simply note that the physical system (
7.57
) is linear in amplitude in order to assess
whether a Kalman filter will be optimal or not. The correct condition is that both
the model and its characteristics must be linear and the initial distribution one draws
from must also be Gaussian. Given the severity of these conditions it would appear
that it is unlikely that the phase distributions of actual flow features found in nature
would always maintain a normally distributed character. Rather, it would seem more
likely that the atmosphere would evolve through time and flow configurations in
which a Gaussian initial phase distribution would be altered to have non-Gaussian
characteristics.
References
Anderson JL, Anderson SL (1999) A monte carlo implementation of the nonlinear filtering problem
to produce ensemble assimilations and forecasts. Mon Weather Rev 127:2741-2758
Anderson J, Hoar T, Raeder K, Liu H, Collins N, Torn R, Avellano A (2009) The data assimilation
research testbed: a community facility. Bull Amer Meteor Soc 90:1283-1296
Bishop CH, Etherton BJ, Majumdar SJ (2001) Adaptive sampling with the ensemble transform
Kalman filter. Part I: Theoretical aspects. Mon Weather Rev 129:420-436
Burgers G, Van Leeuwen PJ, Evensen G (1998) Analysis scheme in the ensemble Kalman filter.
Mon Weather Rev 126:1719-1724
Chen Y, Snyder C (2007) Assimilating vortex position with an ensemble Kalman filter. Mon
Weather Rev 135:1828-1845
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