Advection of the disturbance fields in stochastic parameterisations

A relatively new method to estimate the uncertainties of the weather prediction is the use of ensembles. In this case, the numerical weather prediction (NWP) model is run multiple times with slightly varying settings. A popular choice is to vary the initial conditions (prognostic variables) of the model, within the range of their uncertainty. It seems like this method does not bring enough variability: the ensemble members are still relatively similar to each other and the observation is often still not in the ensemble.

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.

Online generation of temporal and spatial fractal red noise

It is relatively easy to generate fields with fractal noise using Fourier, wavelet or cascade algorithms (see also my page on cloud generators). However, these noise fields have to be calculated fully before their first use. This can be problematic in case a 2D or 3D spatial field is needed as input to a dynamical model that should also have a temporal fractal structure. Often the number of temporal time steps is so large, that the noise field become impractically large. Therefore, this essay introduces an online method to generate fractal red noise fields, where the new field is calculated from the previous field without the need to know all previous fields; only the current and the new field have to be stored in memory.
The algorithm involves two steps. The spatially correlated red noise is calculated from a white noise field using Fourier filtering. The white noise field evolves temporally correlated by addition of Gaussian noise. In other words, every pixel of the white noise field represents a fractal Brownian noise time series. Fortunately, the spatially correlated noise field retains the fractal nature of the temporal structure of the white noise field.