Intermittent Vector Fields: A Challenge for Mathematical Geophysics?

Abstract:

Geophysical fields display strong intermittency over a wide range of scales. Multifractals has become a standard tool to analyze and simulate this key phenomenon for scalar fields. However, fields of interest, e.g. the velocity and the magnetic fields are vector fields. Some time ago, "Lie cascades" were introduced to deal with such fields by considering exponentiation from a stochastic element of a Lie algebra to its corresponding Lie group of transformations. The concerned transformation corresponds to the fine graining/downscaling of the field to higher and higher resolution. Unfortunately, developments were paused due to the possible large number of degrees of freedom of the latter, in particular with respect to the information that can be easily extracted from a d-dimensional vector field. In short, some physics was missing.

In this communication, we point out the interest of the Clifford algebra Cl_p,q to make concrete progress. Ironically, these algebra were mentioned at once as rather straightforward generalizations of the scalar complex cascades, but they were not investigated. On the contrary, the particular case of the "pseudo-quaternions" l(2,R)=Cl_2,0=Cl_1,1 has been often used to generate generalized scales to analyse and simulate anistropic scaling (scalar) fields. The latter is in fact illustrative of the basic property of the Clifford algebra $Cl_{p,q}$ to be generated by a quadratic form $Q$ whose signature ($p,q$) is fundamental.