Nonlinear stability or triad-wave interaction (Q919915)

Since V. I. Arnold used a variational principle to study the nonlinear stability of ideal inviscid fluid, there have been many papers published in this field. \textit{D. D. Holm} [with J. E. Marsden, T. Ratiu and A. Weinstein, Phys. Rep. 123, 1-116 (1985)] reviewed the recent results. For an ideal fluid system or wave system, the Hamiltonian is a conserved quantity and in an ideal fluid, the potential vorticity q, and the mass density p are conserved quantities. While using these conserved quantities, some new conserved quantities can be constructed. We can construct a new conserved quantity \(H_ D=H_ C+{\bar \psi}(g)\), where \(H_ C\) is Hamiltonian, \({\bar \psi}\)(g) is an arbitrary function of the conserved quantity g. By analysing \(\delta^ 2H_ D\) we can get some information about the nonlinear stability of the system. If \(\delta^ 2H_ D\) has a definite sign, we can judge that the system has nonlinear stability. If \(\delta^ 2H_ D\) has not a definite sign, we can judge that the system may have nonlinear instability (but the variational principle cannot prove this). As for systems with infinite number of degrees of freedom, formal stability does not mean the nonlinear stability and convexity estimate is needed, but for systems with finite number of degrees of freedom, formal stability means nonlinear stability and the convexity estimate is not needed. Nonlinear stability of wave interaction in triad is studied in this note. Lorenz has been first person who got the simplest equations of barotropic potential vorticity. He used some special form of solution and details to explain the ``index cycle'' which is important in atmospheric problems. What we want to do is just to use the variational principle to study the nonlinear stability of triad-wave interaction.