Structure and stability of solutions of the Euler equations: a lagrangian approach

This paper reviews methods that are essentially lagrangian in character for determination of solutions of the Euler equations having prescribed topological characteristics. These methods depend in the first instance on the existence of lagrangian invariants for convected scalar and vector fields. Among these, the helicity invariant for a convected or ‘frozen-in’ vector field has particular significance. These invariants, and the associated topological interpretation are discussed in §§1 and 2. In §3 the method of magnetic relaxation to magnetostatic equilibria of prescribed topology is briefly described. This provides a powerful method for determining steady Euler flows through the well-known exact analogy between Euler flows and magnetostatic equilibria. Stability considerations relating to magnetostatic equilibria obtained in this way and to the analogous Euler flows are reviewed in §4. In §5 the related relaxation procedure is discussed; for two-dimensional and axisymmetric situations this technique provides stable solutions of the Euler equations for which the vorticity field has prescribed topology. The concept of flow signature is described in §6: this is the relevant topological characteristic for two-dimensional or axisymmetric situations, which is conserved during frozen-field relaxation processes. In §§7 and 8, the formation of tangential discontinuities as a normal part of the relaxation process when saddle points of the frozen-field are present is discussed. Section 9 considers briefly the application of these ideas to the theory of vortons, i.e. rotational disturbances that propagate without change of structure in an unbounded fluid. The paper concludes with a brief discussion, with comment on the possible development of the results in the context of turbulence.