Scaling in Anti-Plane Elasticity on Random Shear Modulus Fields with Fractal and Hurst Effects

The scale dependence of the effective anti-plane shear modulus response in microstructures with statistical ergodicity and spatial wide-sense stationarity is investigated. In particular, Cauchy and Dagum autocorrelation functions which can decouple the fractal and the Hurst effects are used to describe the random shear modulus fields. The resulting stochastic boundary value problems (BVPs) are set up in line with the Hill–Mandel condition of elastostatics for different sizes of statistical volume elements (SVEs). These BVPs are solved using a physics-based cellular automaton (CA) method that is applicable for anti-plane elasticity to study the scaling of SVEs towards a representative volume element (RVE). This progression from SVE to RVE is described through a scaling function, which is best approximated by the same form as the Cauchy and Dagum autocorrelation functions. The scaling function is obtained by fitting the scaling data from simulations conducted over a large number of random field realizations. The numerical simulation results show that the scaling function is strongly dependent on the fractal dimension D, the Hurst parameter H, and the mesoscale δ, and is weakly dependent on the autocorrelation function. Specifically, it is found that a larger D and a smaller H results in a higher rate of convergence towards an RVE with respect to δ.

A Physically-Based Stochastic Boundary-Layer Perturbation Scheme. Part I: Formulation and evaluation in a convection-permitting model.

We present a simple, physically consistent stochastic boundary layer scheme implemented in the Met Office’s Unified Model. It is expressed as temporally correlated multiplicative Poisson noise with a distribution that depends on physical scales. The distribution can be highly skewed at convection-permitting scales (horizontal grid lengths around 1 km) when temporal correlation is far more important than spatial. The scheme is evaluated using small ensemble forecasts of two case studies of severe convective storms over the UK. Perturbations are temporally correlated over an eddy-turnover timescale, and may be similar in magnitude to or larger than the mean boundary-layer forcing. However, their mean is zero and hence they, in practice, they have very little impact on the energetics of the forecast, so overall domain-averaged precipitation, for example, is essentially unchanged. Differences between ensemble members grow; after around 12 h they appear to be roughly saturated; this represents the time scale to achieve a balance between addition of new perturbations, perturbation growth and dissipation, not just saturation of initial perturbations. The scheme takes into account the area chosen to average over, and results are insensitive to this area at least where this remains within an order of magnitude of the grid scale.

A Physically-Based Stochastic Boundary-Layer Perturbation Scheme. Part II: Perturbation Growth within a Super Ensemble Framework

Convection-permitting forecasts have improved the forecasts of flooding from intense rainfall. However, probabilistic forecasts, generally based upon ensemble methods, are essential to quantify forecast uncertainty. This leads to a need to understand how different aspects of the model system affect forecast behaviour. We compare the uncertainty due to initial and boundary condition (IBC) perturbations and boundary-layer turbulence using a super ensemble (SE) created to determine the influence of 12 IBC perturbations vs. 12 stochastic boundary-layer (SBL) perturbations constructed using a physically-based SBL scheme. We consider two mesoscale extreme precipitation events. For each we run a 144–member SE. The SEs are analysed to consider the growth of differences between the simulations, and the spatial structure and scales of those differences. The SBL perturbations rapidly spin-up, typically within 12 h of precipitation commencing. The SBL perturbations eventually produce spread that is not statistically different from the spread produced by the IBC perturbations, though in one case there is initially increased spread from the IBC perturbations. Spatially, the growth from IBC occurs on larger scales than that produced by the SBL perturbations (typically by an order of magnitude). However, analysis across multiple scales shows that the SBL scheme produces a random relocation of precipitation up to the scale at which the ensemble members agree with each other. This implies that statistical post-processing can be used instead of running larger ensembles. Use of these statistical post-processing techniques could lead to more reliable probabilistic forecasts of convective events and their associated hazards.

Variability and Clustering of Midlatitude Summertime Convection: Testing the Craig and Cohen Theory in a Convection-Permitting Ensemble with Stochastic Boundary Layer Perturbations

The statistical theory of convective variability developed by Craig and Cohen in 2006 has provided a promising foundation for the design of stochastic parameterizations. The simplifying assumptions of this theory, however, were made with tropical equilibrium convection in mind. This study investigates the predictions of the statistical theory in real-weather case studies of nonequilibrium summertime convection over land. For this purpose, a convection-permitting ensemble is used in which all members share the same large-scale weather conditions but the convection is displaced using stochastic boundary layer perturbations. The results show that the standard deviation of the domain-integrated mass flux is proportional to the square root of its mean over a wide range of scales. This confirms the general applicability and scale adaptivity of the Craig and Cohen theory for complex weather. However, clouds tend to cluster on scales of around 100 km, particularly in the morning and evening. This strongly impacts the theoretical predictions of the variability, which does not include clustering. Furthermore, the mass flux per cloud closely follows an exponential distribution if all clouds are considered together and if overlapping cloud objects are separated. The nonseparated cloud mass flux distribution resembles a power law. These findings support the use of the theory for stochastic parameterizations but also highlight areas for improvement.