As a consequence, meteorologists are starting to look at uncertainties in the model as an additional source of variability for the ensemble. This can be accomplished by utilising multiple models to create the ensemble (multi-model ensemble), or multiple parameterisation schemes or by varying the parameterisations of one model (stochastic parameterisations). This latter case is discussed in this essay.

## Stochastic parameterisations

In parameterisations the effect of subscale processes are estimated based on the resolved prognostic fields. For example, the cloud fraction is estimated based on the relative humidity at the model resolution. As the humidity also varies at scales below the model resolution (sub grid scale variability), it is possible to have clouds even though the average relative humidity is well below saturation. Such functional relations can be estimated by a large set of representative measurements or by modelling with a more detailed model. In deterministic parameterisations the best estimate of, for example, the cloud fraction is used. However, there is normally a considerable spread around this mean cloud fraction for a certain relative humidity. It is thus physically reasonable to consider the parameterised quantity as a stochastic parameter, with a certain probability density function (PDF).Ideally, one would like to use stochastic parameterisations that were specially developed for this application. Such a parameterisation could also take into account the relation between the PDF and the prognostic model fields. Developing parameterisations is a major task and normally performed by specialists of a certain process. Thus, to get a first idea of the importance of stochastic parameterisations, NWP-modellers started with more ad-hoc approaches. One can, for example, disturb the tendencies calculated by the parameterisations by introducing noise.

## Generating noise

The right type of noise is an important part of a stochastic parameterisation scheme. It is known that white noise has only little influence on the spread of the ensemble. The reason for this is still unclear. One reason might be that the model needs stochastic dynamic equations that explicitly take subscale variability into account. Another direction meteorologists are looking is using spatially and temporally correlated noise. For example, Susanne Theis (2005) here in Bonn works with noise that is fixed for a certain time and area. Glenn Shutts (2004) uses a cellular automaton (CA) based on Conrad’s game of life, to create correlated noise. The additional spread the correlated noise is able to generate is, however, insufficient to increase the spread of the ensemble enough, at least as long as the amplitudes and the correlation scales are chosen in a physically realistic range (Theis et al, 2005). See the next section for the main idea of this post that may increase the spread of the ensemble further in a physical manner.Given that the variability of many atmospheric constituents is well approximated by a fractal, self-similar structure, the most elegant correlated noise would be fractal red noise. Especially in the area of clouds and rain, which is of special interest here, there exist many studies showing a near fractal structure. Thus, such a structure seems a good first guess, as long as we do not have more information about its real structure.

It is easy to generate such a self-similar noise field with Fourier or wavelet algorithms. However, in this a case one needs to generate all disturbances in advance for all time steps of the model run. This is therefore not a practicable method. In another post, I present an algorithm that can generate such fractal fields online, i.e. needing storage only for the current and the new field.

## Physical background and advection

Up to now, our considerations were mainly statistical. However, the choice of the noise generator should also depend on the physical reasons one suspects behind the stochastic nature of a parameterisation.For example, in the parameterisation of rain one implicitly assumes a certain mean number of Cloud Condensation Nuclei (CCN), or a certain number of aerosols with a certain chemical composition. More (or less) CCN leads to a lower (or higher) rain conversion factor as the droplets stay smaller. Another physical influence could be a higher (or lower) than average variability of the subscale humidity field. This would influence the radiative budget via the cloud cover or cloud variability, and it will influence the rain rate as well. An important consequence of this physical picture is that you would expect the disturbance to advect like a passive scalar in the model.

Another noise source is the randomness of the turbulence that may accidentally increase of decrease the updraft speed of a convective cloud and thus the number of CCN. The same process may also perturb cloud cover. These disturbances are probably best modelled by non-advective noise. However, this noise category can be expected to have little or no temporal or spatial correlations. They are more like white noise, and thus likely not as important for the spread of the ensemble as disturbances that can be expected to be advected.

Thus, while I would not want to deny the existence of white noise processes that perturb the processes we would like to parameterise, I would expect that parameters that are advected are more important. Therefore, I would propose to treat the model disturbances as another prognostic atmospheric field that is advected and diffused like a passive scalar.

The advantage of this treatment of the disturbance is that the disturbance stays longer with its air mass and thus has more time to increase the spread of the ensemble without making the amplitudes and the correlations of the disturbance unphysically large.

Whether one implements (sub-scale) diffusion that will reduce the noise amplitude or not, will depend on whether one has a noise generator as a source of variability in the entire field. In case noise is introduced at the lateral boundaries only advection should be implemented.

## Conclusions

This essay put forward the idea of treating the disturbances used in stochastic parameterisation as an advecting 'prognostic' model field. Of course, this field is different for every ensemble member. In this sense, it is not a prognostic variable, we do not want to predict this field.As I do not have the skills nor the time to implement this idea, I hope that an NWP modeller is inspired by this text, and willing to try it out. I am curious, whether it will work. See also the general disclaimer to the essays.

## Further reading

If you are interested in this topic, you may be interested in the work of Susanne Theis and Petra Friedrichs of the working group on Climate Dynamics here in Bonn. And who knows, my work on cloud structure and radiation may be of interest as well.Acknowledgement

I got this idea during a presentation of Susanne Theis on her work. So her work had a major influence on my thinking. Furthermore, I would like to thank her and Felix Ament for useful discussions.

## References

Shutts, Glenn. A stochastic kinetic energy backscatter algorithm for use in ensemble prediction systems, ECMWF technical memoranda, Aug. 2000.Theis, S. E., A. Hense and U. Damrath (2005): Probabilistic precipitation forecasts from a deterministic model: a pragmatic approach. Met. Applications, submitted.

*This post was first published on my homepage on 07 May 2005.*

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