Fractal-to-Euclidean crossover of the isotropy restoration feature in a family of fractal resistor networks

Fractal with microscopic anisotropy shows a unique type of macroscopic isotropy restoration phenomenon that is absent in Euclidean space [M. T. Barlow et al., Phys. Rev. Lett. 75, 3042]. In this paper the isotropy restoration feature is considered for a family of two-dimensional Sierpinski gasket type fractal resistor networks. A parameter xi is introduced to describe this phenomenon. Our numerical results show that xi satisfies the scaling law xi similar to l(-alpha), where l is the system size and alpha is an exponent independent of the degree of microscopic anisotropy, characterizing the isotropy restoration feature of the fractal systems. By changing the underlying fractal structure towards the Euclidean triangular lattice through increasing the side length b of the gasket generators, the fractal-to-Euclidean crossover behavior of the isotropy restoration feature is discussed.

Ensemble prediction for nowcasting with a convection-permitting model - II: forecast error statistics

A 24-member ensemble of 1-h high-resolution forecasts over the Southern United Kingdom is used to study short-range forecast error statistics. The initial conditions are found from perturbations from an ensemble transform Kalman filter. Forecasts from this system are assumed to lie within the bounds of forecast error of an operational forecast system. Although noisy, this system is capable of producing physically reasonable statistics which are analysed and compared to statistics implied from a variational assimilation system. The variances for temperature errors for instance show structures that reflect convective activity. Some variables, notably potential temperature and specific humidity perturbations, have autocorrelation functions that deviate from 3-D isotropy at the convective-scale (horizontal scales less than 10 km). Other variables, notably the velocity potential for horizontal divergence perturbations, maintain 3-D isotropy at all scales. Geostrophic and hydrostatic balances are studied by examining correlations between terms in the divergence and vertical momentum equations respectively. Both balances are found to decay as the horizontal scale decreases. It is estimated that geostrophic balance becomes less important at scales smaller than 75 km, and hydrostatic balance becomes less important at scales smaller than 35 km, although more work is required to validate these findings. The implications of these results for high-resolution data assimilation are discussed.