The stochastic multiplicative cascade structure of deterministic numerical models of the atmosphere

Abstract

Abstract.By direct statistical analysis we show that over almost all their range of scales and to within typically better than ±1%, atmospheric fields obtained from analyses and numerical integrations of atmospheric models have the multifractal structure predicted by multiplicative cascade models. We quantify this for the horizontal wind, temperature, and humidity fields at 5 different pressure levels for the ERA40 reanalysis, the Canadian Meteorological Centre Global Environmental Multiscale (CMC, GEM) model, as well as the National Oceanographic and Atmospheric Administration Global Forecasting System (NOAA, GFS). We investigate the additional prediction that the cascade belongs to a universal multifractal basin of attraction. By demonstrating a "Levy collapse" of the statistical moments to within ±2 to ±5% over most of the range of scales, we conclude that there is good evidence for this. Finally, we discuss how this stochastic multiplicative cascade structure can be exploited in improving ensemble forecasts.

From molecules to meteorology via turbulent scale invariance

“…There have been attempts to examine the scaling of model fields, both on a case-study basis as described above and more generally. Lovejoy et al (2009c) and Stolle et al (2009) concluded that weather could be modelled as a cascade process, pointing out that whereas horizontal scaling could be investigated for models with regularly spaced grids, the general absence of such regularity with altitude in atmospheric models used for assimilation and forecasting hampered analysis of their vertical scaling. In this case, the cascades refer to the meaning whereby the same processes are at work scale by scale across the whole scaling range…”

“…The comparison suggests a new way of testing models against observations, in a way that incorporates the energy and mass transfers on all scales, which in this case seem to be inadequate quantitatively. Stolle et al (2009) have recently performed extensive studies of the horizontal scaling in ECMWF reanalyses.…”

The temporal cascade structure of reanalyses and global circulation models

“…In this paper we therefore often use C 1 and α to reduce the characterization of the scaling (via the exponents K(q)) to manageable proportions. In any event, we show that over significant ranges of scale, the data display 'Levy collapses' when the moments are normalized by the theoretical universal K(q), and this even for scales somewhat outside the scaling range (see Stolle et al (2009) and section 5 below). In other words, the universality relation Eq.…”

“…Recently the deterministic and stochastic approaches have been shown to be surprisingly compatible. Stolle et al (2009) analysed both meteorological reanalyses (European Centre for Medium range Weather Forecasting's (ECMWF) reanalysis (ERA40)) and meteorological forecasting models ((Global Forecast System (GFS) and Global Environmental Multiscale (GEM)) and showed that the spatial statistics of the turbulent fluxes associated with the temperature, zonal wind and humidity were indeed very close to those predicted by cascade models with deviations typically < ± 1% over the range of 5000 km down to the model/reanalysis (hyper) viscous dissipation scales. Lovejoy and Schertzer (2011) found similar results for the ECMWF interim reanalysis products, extending these results to the geopotential height, to the vertical wind, and to the meridional wind fields as well as the corresponding turbulent fluxes.…”

“…In section 2 we present the numerical outputs, which we analyse, and we explain the technique for estimating the fluxes in both space and in time, touching on other relevant cascade studies. In sections 3 and 4 we recapitulate pertinent spatial cascade results from Stolle et al (2009) and we then systematically examine the temporal cascade structure for the different fields (horizontal wind, temperature, humidity), different models and as functions of altitude. In section 5 we compare the statistics of the fluxes estimated in the spatial domain at the dissipation scale with those estimated in the temporal domain in the scaling regime.…”