Nonlinear stability analysis of inviscid flows in three dimensions: Incompressible fluids and barotropic fluids

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Abstract
Using the energy‐conserved quantity method developed by Arnol’d [Dokl. Mat. Nauk 1 6 2, 773 (1965); Am. Math. Soc. Trans. 1 9, 267 (1969)] a study was made of the nonlinear stability of two inviscid fluid flows in three dimensions: (1) flow of a homogeneous fluid and (2) flow of a fluid whose energy density depends on the mass density alone (a so‐called barotropic fluid). In order to implement the Arnol’d technique one must identify the quantities conserved by the flow in addition to the total energy. In the case of the two flows considered, the conserved quantities cannot be expressed in terms of the usual Eulerian variables—fluid velocity and mass density—alone. Instead the introduction of the Lagrangian labels of the fluid elements is required. A complete description of these conserved quantities, in both Eulerian and Lagrangian specifications of the fluid, is provided. The phase space of the flow is the entire Hamiltonian phase space expressed in either canonical or noncanonical variables. The nature of the flows to which the Arnol’d method is applicable is discussed in some depth. It was discovered that only time independent Eulerian flows can be discussed by the method; this result is given a general Hamiltonian context. The allowed Eulerian equilibria are displayed in detail. Finally, having the formal structure of these flows well in hand, it is shown that they are not, in three dimensions, formally stable. This results from a particle vortex stretching mechanism which is identified. The nature of the indicated instability is not revealed by this work, but may well be a slowly evolving Arnol’d diffusion kind of breakdown of the equilibrium.

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