Anisotropic Continuous-in-Scale Universal Multifractal Cascades: Simulation,Analysis and Correction Methods

Continuous-in-scale multifractal cascades has long been an attractive choice for mathematically modeling turbulent and turbulent-like geophysical fields. These fields are usually anisotropic as they are subject to both stratification and rotation, thereby questioning the isotropy assumption often made to model them. The self-affine and generalized scale invariance approaches to scaling are used here to introduce anisotropy in such models. These anisotropic simulations have (1) unresolved large-scale features and (2) statistics that deviate from the desired power-law scaling mainly in the small scales. The former issue is solved via nesting, whereas the latter is attempted to be overcome using singularity correction methods. While earlier studies have proposed isotropic correction methods, here they have been generalized to correct anisotropic simulations. These singularity corrections seem to improve the small-scale statistical properties of mildly anisotropic simulations; nesting, on the other hand, appears to enhance statistics over almost all scales even for strongly anisotropic simulations. Both the correction and nesting techniques lead to a reduction in computational time and memory usage suggesting that nested singularity-corrected cascades offer a better framework for quantitatively modeling the atmosphere, ocean, solid earth, and associated fields.