Linearized Physics for Data Assimilation at ECMWF - Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications - page 250

In each of these situations, an estimation of the derivative close to the discontinuity
point will be different between the non-linear model (in terms of finite differences)
and the TL model. All of this makes the tangent linear approximation less valid
when the linearized model includes physical parameterizations compared to the
adiabatic version only. To treat the described problems, it is important to regularize,
i.e. to smooth the parameterized discontinuities in order to make the scheme
as much differentiable as possible. One should recognize that it is often quite
difficult to achieve a tradeoff between a physically sound description of atmospheric
processes and a well-behaved linear physical parametrization. However, without
a proper treatment of the most significant thresholds, the TL model can quickly
become too inaccurate to be useful. Therefore a lot of effort was devoted by
a number of investigators to deal with discontinuities present in parameterized
physical processes (e.g. Zou et al. 1993 ; Zupanski and Mesinger 1995 ; Tsuyuki
1996 ; Errico and Reader 1999 ; Janiskova et al. 1999 ; Mahfouf 1999 ; Laroche et al.
2002 ; Tompkins and Janiskova 2004 ; Lopez and Moreau 2005 ).
To illustrate a potential source of problem in the linearized model, the rain pro-
duction function, describing which portion of the cloud water is converted into pre-
cipitation, is shown in Fig. 11.1 . An increase of cloud water mixing ratio by a small
amount

dx (Fig. 11.1 a) leads to a small change in the precipitation amount dy NL in

the case of the non-linear (NL) model, but to a much larger change ( dy TL )inthe

case of the TL model. As a possible solution, one can modify the function to make it
less steep (dotted line on Fig. 11.1 b ). In this case, the resulting TL increment will be
significantly smaller ( dy TL 2 ). However, the required modification can be substantial

and it can deteriorate the overall quality of the physical parametrization itself.
Therefore one must always be careful to keep the right balance between linearity
and realism of the parametrization schemes. In the future, the better the non-linear
forecast model will become, the smaller 4D-Var analysis increments and hence the
hope to have less difficulties with using linearized physical processes will be.

11.4 Methodology for the Development of Linearized

Simplified Parameterizations

There are several problems with including physics in adjoint models. The devel-
opment requires substantial resources and it is technically very demanding. The
validation must be very thorough and it must be done for the non-linear, tangent-
linear and adjoint versions of the physical parametrization schemes. The compu-
tational cost of the model with physical processes can be very high despite some
possible simplifications applied. One must be also very careful with the non-linear
and threshold nature of physical processes which can affect the range of validity of
the tangent-linear approximation as mentioned above.
The development of a new linearized physical parametrization can be divided
into four main stages: