Role of weak Casimirs in constraining stratified thermal IL^{(0,1)}QG flow

Determine the role of the weak Casimir integrals of motion I_F[bar{ξ},ψ_σ,ψ_{σ^2}] = ∫_𝒟 bar{ξ} F(ψ_σ − (2/3) ψ_{σ^2}) d^2x in constraining the dynamics of the stratified thermal quasigeostrophic model IL^{(0,1)}QG, including whether and how these integrals can be used to derive formal or Lyapunov stability conditions or other invariant bounds for stratified flows.

Background

In the IL^{(0,1)}QG model, the authors identify an infinite family of conserved quantities I_F = ∫ bar{ξ} F(ψσ − (2/3) ψ{σ^2}) d^2x, which they term weak Casimirs because they do not lie in the kernel of the full Lie–Poisson bracket for (bar{ξ}, ψσ, ψ{σ^2}), although they do form the kernel for a reduced bracket involving (bar{ξ}, χ) with χ := ψσ − (2/3) ψ{σ^2}.

While standard energy–Casimir methods (Arnold’s method) successfully yield formal and Lyapunov stability for certain basic states, attempts to include the weak Casimirs did not produce positive-definite pseudoenergy–momentum functionals except in a restrictive case. The authors therefore highlight that the dynamical significance of these weak Casimirs for constraining stratified flows in IL^{(0,1)}QG remains unresolved.