Dimension and entropy in the soil-covered landscape

Both the Hausdorff dimension and the K-entropy supply a measure of the irregularity of the landspace surface. The relationship between the two measures is investigated over a variety of terrains in Britain and a method of calculating the entropy is checked against an independent estimate of the dimension with reasonable agreement. The calculation of the K-entropy requires that the landscape surface be represented by an homogenous ergodic random field. This condition is satisfied by the tendency of soil-covered terrains to progressively approximate to a form well represented by a Gaussian field. Gaussian random fields can either be very smooth, possessing derivatives of all orders at every point or they are highly irregular and non-differentiable everywhere. Within the regular conceptualization the Rice-Kac theory is used to predict the numbers of crossing points and the extent of excursion sets. These predictions are tested against an example terrain from the High Weald of East Sussex with very good agreement, apart from predictions of local maxima. A worked example of the calculation of the K-entropy is given as an appendix. The potential role of information theory in geomorphology extends beyond the use made of entropy in this investigation. In particular ergodic theory has important practical and theoretical implications.     

MODULATION OF ATMOSPHERIC THERMAL AND MECHANICAL FORCINGS AND NUMERICAL MODELING OF MEAN MERIDIONAL CIRCULATION

Upon investigating the relative locations of internal and external forcing and the resultant mean meridional circulation,it was found that thermal forcing and mechanical forcing for the formation of atmospheric mean meridional circulation are modulated by a certain ratio.This ratio is determined by the inherent baroclinity,static stability and absolute vorticity of the atmosphere.By employing a parameterization scheme for radiative heating and condensation heating,together with the analysisdata of the European Center for Medium-range Weather Forecasts,the mean meridional circulation for January wassimulated numerically.It was found that latent heat release in the tropics may result in the formation of double-layeredHadley circulation,so do the eddy momentum transfer processes.On the other hand,mean meridional circulations in extra-tropics are mainly determined by external momentum forcing and atmospheric properties of eddy momentum andheat transfer.     

Flow through anisotropic networks

This paper presents an analysis of Hagen-Poiseulle flow through plane random anisotropic networks of interconnected channels. Macroscopic permeability tensor of the network is expressed in terms of statistico-geometrical characteristics like the degree of anisotropy in channel orientations, average co-ordination number of the network and first two moments of channel length distribution. Analytical results are illustrated and verified using numerical analysis of flow in a simulated random network. The emphasis of the paper is on the effects of anisotropy on distributions of flow rates in channels. It is shown that, due to anisotropy the maximum flow rate generally occurs in channels that are not aligned along the direction of the macroscopic pressure gradient.     

NUMERICAL EXPERIMENT OF THE INFLUENCES OF HEAT AND WIND STRESS ON THE EQUATORIAL OCEAN CIRCULATION*

A time-dependent linear model on the equatorial beta plane is developed in this paper.Its vertical structure consists of two active layers with equal density and temperature above the thermocline and a quasi-stationary layer with constant density below the thermocline.The results of nurnericat experiments show that the direct influence of heat on the equatorial ocean is much smaller than that of wind stress.Nevertheless,through the tropical ocean-atmosphere interaction,the change of wind stress resulting from the thermal forcing may set a determinant effect on the equilibrium and anomalous development of the ocean-almosphere circulation.     

Entropy estimation of hydrological extremes

The method of Relative Entropy with Fractile constraints (REF method) is explained and applied to model extreme compound hydrological phenomena, such as extreme sea levels under storm conditions. Also presented is a simple method of Tail Entropy Approximation (TEA), which amounts to a correction of traditional statistical estimates for extreme observations.Distribution assumptions are necessary but downplayed in the REF method, relegating the prior distribution to the role of an extrapolation function. The estimates are objective in an information-theoretical sense. They also satisfy a strict requirement of self-consistency that is generally not satisfied by standard statistical methods: invariance under monotonic transformations of the random variable.Historical records of storm surge levels in the Netherlands and annual maximum tidal heights for Sheerness, UK, are used as examples. Comparison is made with distributions obtained using other methods.It is concluded that the tail entropy approximation provides simple, objective estimates of extremes in the tail beyond the range of observations.     

NUMERICAL EXPERIMENT OF THE INFLUENCES OF HEAT AND WIND STRESS ON THE EQUATORIAL OCEAN CIRCULATION

A time-dependent linear model on the equatorial beta plane is developed in this paper.Its verticalstructure consists of two active layers with equal density and temperature above the thermocline and aquasi-stationary layer with constant density below the thermocline.The results of nurnericat experiments show that the direct influence of heat on the equatorial ocean ismuch smaller than that of wind stress.Nevertheless,through the tropical ocean-atmosphere interaction,the change of wind stress resulting from the thermal forcing may set a determinant effect on the equilibriumand anomalous development of the ocean-almosphere circulation.     

Estimation of a multidimensional covariance function in case of anisotropy

A model of a multivariate covariance function with an ellipsoidal directional correlation scale has been developed. The axes of the ellipsoidal scale are related to the eigenvalues and eigenvectors of a matrix B which characterizes the ellipsoid of the range of influence. The matrix B is found to be related to a matrix T which can be estimated directly from sparse sampling data and can be used to determine estimates of the matrix B. The method has been applied to both two-dimensional and three-dimensional cases. The numerical results show that the satisfactory accuracy is obtained with sparse sampling data from an anisotropic random function.