Trying a metric on atmospheric flows

Abstract

It is not obvious how to introduce, in a natural and intrinsic way, a metric structure in the configuration space of a fluid, i.e., in a space with (ρ, T,$\vec u$) ≡ (density, temperature, velocity) as coordinates. Such a lack of geometric structures imposes many restrictions on the kind of computations that one can perform: it is impossible to talk of “the distance” between two states of the fluid, i.e., between two positions with coordinates (ρ ₁, T ₁,${\vec u_1}$) and (ρ ₂, T ₂, ${\vec u_2}$) respectively; neither is it possible to talk of “the norm” of the vector formed by the rate of change of the dynamical variables (dρ/dt, dT/dt, d $\vec u$/dt); it is not possible to consider the angle between two such vectors; there is no volume element defined in configuration space, therefore, it does not make sense to talk about “the density” of a distribution of points in that space; etc. These limitations are too restrictive since, to name but a few examples: 1) A norm is needed when studying the (in)stability of flows, especially when the linear growth of the perturbations is only transient [[6]]; 2) The angle between two directions is required in order to determine how strongly a given perturbation projects along an optimal perturbation [[7]]; 3) A volume element is needed in order to determine the density distribution of ensemble simulations of flows [[8]]. Therefore, it is often opted to circumvent these limitations by introducing an acceptable, if somewhat arbitrary, metric tensor [[9, Hanifi A., Schmid P. and Henningson D.S., Transient growth in compressible boundary layer; Phys. Fluids 8 (1996) 826–837.]]. In the case of incompressible, isentropic flows, the kinetic-energy metric is justifiably employed [[6, 9]].