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Fig. 13.10
Effects of small errors in the initial conditions (dashed line) and imperfect knowledge of
the governing equation (dash-dotted line) on the prediction of a time series generated by
the difference equation (13.69). (Courtesy of E. Kallen.)
shows a solution in which the original initial data are used but the coefficient a is
perturbed by 0.001.
For the first several steps the two “predictions” are very close to the observed
values. However, after about 15 steps they begin to diverge. Thus, errors in either
the initial conditions or the governing equation can produce comparable results—
the departure of a predicted solution from the observed sequence increases rapidly.
For the case shown here only about 20 steps are required for the three sequences to
lose all resemblance to each other. Such behavior is characteristic of a wide variety
of systems governed by deterministic equations, including atmospheric flows.
Estimates of how this sort of error growth limits the inherent predictability of
the atmosphere can be made using primitive equation forecast models. In these
predictability experiments a “control” run is made using initial data corresponding
to the observed flow at a given time. The initial data are then perturbed by introduc-
ing small random errors, and the model is run again. The growth of inherent error
can then be estimated by comparing the second “forecast” run with the control.
Results from a number of such studies indicate that the doubling time for the root
mean square geopotential height error is about 2 to 3 days for small errors and
somewhat greater for large errors. Thus, the theoretical limit for predictability on
the synoptic scale is probably about 2 weeks.
Actual predictive skill with present deterministic models is, however, less than
the theoretical limit imposed by inherent error growth. A number of factors are
probably responsible for the failure of current models to reach the theoretical
forecast skill limits. These include observational and analysis errors in initial
data, inadequate model resolution, and unsatisfactory representation of physical
processes involving radiation, clouds, precipitation, and boundary layer fluxes.
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