Abstract:
The stability of steady-state solutions of the equations governing two-dimensional,
homogeneous, incompressible fluid flow are analyzed in the context
of shear-flow in a channel. Both the linear and nonlinear theories are
reviewed and compared. In proving nonlinear stability of an equilibrium,
emphasis is placed on using the stability algorithm developed in Holm et al.
(1985). It is shown that for certain types of equilibria the linear theory is
inconclusive, although nonlinear stability can be proven.
Establishing nonlinear stability is dependent on the definition of a norm
on the space of perturbations. McIntyre and Shepherd (1987) specifically
define five norms, two for corresponding to one flow state and three to a
different flow state, and suggest that still others are possible. Here, the
norms given by McIntyre and Shepherd (1987) are shown to induce the same
topology (for the corresponding flow states), establishing their equivalence as
norms, and hence their equivalence as measures of stability. Summaries of the
different types of stability and their mathematical definitions are presented.
Additionally, a summary of conditions on shear-flow equilibria under which
the various types of stability have been proven is presented.
The Hamiltonian structure of the two-dimensional Euler equations is
outlined following Olver (1986). A coordinate-free approach is adopted emphasizing
the role of the Poisson bracket structure. Direct calculations are
given to show that the Casimir invariants, or distinguished functionals, are
time-independent and therefore are conserved quantities in the usual sense.
