Statistically relevant and irrelevant conserved quantities for the equilibrium statistical description of the truncated Burger-Hopf equation and the equations for barotropic flow

The purpose of the current work is to study the statistical relevance and irrelevance of the additional conserved quantities in the simple models of actual weather systems. Here we consider the two models with the key features of statistical weather behavior: the truncated Burgers-Hopf (TBH) equation, which is, in fact, the Galerkin projection of the actual Burgers-Hopf equation on the finite Fourier basis; and the two different truncations of the equations for barotropic flow with topography—the traditional spectral truncation and the sine-bracket truncation. In the case with TBH the recently discovered Hamiltonian structure proposes the cubic Hamiltonian to be considered as an additional conserved quantity, since the equilibrium statistical theory developed for TBH is based on the conservation of energy. Thus, the question arises of the statistical significance of the Hamiltonian, beyond that of the energy. First, an appropriate statistical theory is developed which includes both the energy and the Hamiltonian. Then a convergent Monte-Carlo algorithm is developed for computing equilibrium statistical distributions. The probability distribution of the Hamiltonian on a microcanonical energy surface is studied through the Monte-Carlo algorithm and leads to the concept of statistically relevant and irrelevant values for the Hamiltonian. Empirical numerical estimates and simple analysis are combined to demonstrate that the set of statistically relevant values of the Hamiltonian has vanishingly small measure as the number of degrees of freedom increases with fixed mean energy. The predictions of the theory for relevant and irrelevant values for the Hamiltonian are confirmed through systematic numerical simulations. For statistically relevant values of the Hamiltonian, these simulations show a surprising spectral tilt rather than equipartition of energy. This spectral tilt is predicted and confirmed independently by Monte-Carlo simulations based on equilibrium statistical mechanics together with a heuristic formula for the tilt. For the equations for barotropic flow with topography, the two different spectral truncations are considered—the traditional truncation and the sine-bracket truncation. The main difference between the two is that, apart from the energy and the enstrophy, which are the conserved quantities for the traditional truncation, there is a vast number of additional invariants for the sine-bracket truncation, which, in fact, are the Casimir invariants for the Poisson bracket of the sine-bracket truncation. (Abstract shortened by UMI.)...