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Klaus Fraedrich

Abstract

A parameterization scheme for cumulus convection is derived by averaging the conservative quantities, total energy and Ertel’s Wirbelinvariante, separately over the cloud and cloud-free areas and the lifetime of the convective elements. This leads 1) to one-dimensional models of a cloud ensemble where the mass fluxes within the convective elements and their life cycle (storage) are considered; and 2) to synoptic-scale equations including the related sources of heat, moisture and vorticity which are determined by the mechanisms of compensating subsidence and lateral mixing due to the convective activity. The simplified version of the interaction model is tested by mesoscale data of a cumulonimbus element and applied to the cloud cluster of a synoptic disturbance. Realistic results are obtained for the temperature, moisture and voracity distributions, the condensation rate and the convergence and storage of moisture (temperature) within the cloud ensemble, and for the large-scale sources of heat, moisture and vorticity.

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Klaus Fraedrich

Abstract

The dimensions of attractors are estimated from phase space trajectories of observed weather and climate variables (local surface pressure and relative sunshine duration, zonal wave amplitude; a δ18O-record). They provide primary information for descriptions of properties of the attractors of dynamical systems and give a lower limit to the number of the essential variables necessary to model the dynamics. These estimates are based on distance distributions of pairs of points on the single variable trajectory evolving in phase spaces which embed the attractor. One observes a low fractal dimensionality between three and four for the weather attractor, if interannual variability and seasonal changes are eliminated. The physical interpretation is based on the three dominating scales of cyclones, cyclone families and index-cycle; the irregularity of the flow and strong dependence on initial conditions amount for the fractal value. The climate variable also reveals a low dimensionality (between four and five) of the climate attractor. This is supported by an independent estimate based on eigenfunction expansion of the embedded phase space trajectory. These types of analyses suggest how to extend the standard data evaluation and model verification techniques to an analysis of the phase space behavior of observed and simulated dynamical systems.

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Klaus Fraedrich

Abstract

A parametric model of a transient cumulus cloud with the related cloud population is introduced. A closure condition is presented which couples the statistical, physical and geometrical parameters of the cloud ensemble with the large-scale circulation. The large-scale sources of latent and sensible heat, as well as the vorticity due to the convective activity of the transient cloud ensemble, can be deduced either with the aid of direct cloud observations or by using the closure condition. The latter approach is applied to undisturbed ATEX 1969 data.

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Klaus Fraedrich

Abstract

A parsimonious model is presented, leading to Schreiber’s aridity–runoff relation as equilibrium solution of the rainfall–runoff chain. The chain commences with a fast stochastic water reservoir of small capacity, representing interception and wetted ground in short time intervals. It feeds a slow (almost stationary) soil moisture reservoir of large capacity, balancing its runoff after long-term averaging. Parameterizing the fast reservoir’s capacity by the water equivalent of net radiation available for evaporation leads to a biased coin-flip surrogate for its “full” or “empty” states when rainfall is larger or smaller than the capacity. Rainfall surplus from the fast reservoir’s full state feeds the slow (almost stationary) soil moisture reservoir; with the residual evaporating the fast reservoir starts anew as empty. Rainfall below capacity evaporates completely and, leaving the energy surplus for sensible heat, the fast reservoir also starts anew as empty. Employing coin-flip occurrence probabilities from exponentially distributed precipitation yields Schreiber’s formula.

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Klaus Fraedrich

Abstract

A parametric model of an individual transient cumulus cloud is derived which allows the incorporation of direct cloud observations. An ensemble of these transient cumulus clouds is constructed using empirical cloud population statistics. It is shown that the mass; budget (vertical mass flux, lateral detrainment and final detrainment) of this cloud ensemble can be quantitatively determined by direct cloud observations (number of clouds, cloud cover) without additional aerological information.

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Klaus Fraedrich

Abstract

Predictability is deduced from phase space trajectories of weather and climate variables which evolve on attractors (local surface pressure and a δ 18O-record). Predictability can be defined by the divergence of initially close pieces of trajectories and estimated by the cumulative distance distributions of expanding pairs of points on the single variable trajectory. The e-folding expansion rates characterize predictability tune scales. As a first estimate one obtains a predictability time scale of about two weeks for the weather variable and 10–15 thousand years for the climate variable.

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Klaus Fraedrich

Abstract

Predictability time scales are estimated from annual time series of the El Niño/Southern Oscillation (ENSO). They we defined by the rate of divergence of initially close independent pieces of trajectories in phase space. Fitted stochastic processes and the nonlinear deterministic analysis of the empirical time series lead to e-folding predictability time scales up to 1.5 years (or one year of error doubling time) indicating that at least a skillful nowcasting of ENSO may be possible. Due to sparse data these estimates provide only weak bounds.

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Klaus Fraedrich and Martin Lutz

Abstract

Unfiltered, band- and low-pass filtered geopotential height values of the 500 mb level along 50°S are analyzed to evaluate teleconnections and zonal wave-propagation properties in the Southern Hemisphere (SH) midlatitudes. Correlation matrices of low-pass filtered data illustrate the dominant patterns of hemispheric and regional anomalies: the transpolar teleconnection, the South American anomaly and a stationary wavenumber 3 pattern (with contributions from wavenumbers 2 and 4). Longitude-time lag correlations of the low-pass filtered data document an eastward energy dispersion along a quasi-stationary zonal wave train; it emerges from the area of South America, has a wavelength of 90 to 100° longitude, and a group velocity of ∼10° per day. The bandpass filtered correlation matrices for this latitude circle reveal the sectors of the major storm tracks. The related longitude-time lag correlations show a wavelength of 70° to 80° longitude, an eastward propagation of ∼10° per day and a group velocity of ∼50° per day.

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Klaus Fraedrich and Horst Böttger

Abstract

A space-time spectral analysis is applied to the daily 500 mb geopotential field (gridded along 50° N) of five winter seasons (1972–77). The results are displayed in the wavenumber-frequency domain as two-sided frequency spectra of meridional geostrophic wind and one-sided spectra of geopotential. They show three isolated spectral peaks: stationary ultralong waves (k = 1–4, p ≈ 25 days), and eastward propagating long (k = 5–6, p ≈ 10 days, c ≈ 6 m s−1) and short waves (k = 7–8, p = 4–6 days, c ≈ 10 m s−1). In the wave-number-frequency domain only the magnitudes of the peaks vary from one season to the other.

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Klaus Fraedrich and Neville R. Smith

Abstract

The linear combination of two statistical forecast schemes of a single observable provides, in the average, a more accurate prediction than the individual forecasts alone. This method is applied to long-range forecasting of the monthly mean tropical Pacific sea surface temperatures.

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