On the Arnold Stability Criterion

The Arnold stability criterion suggests that a stationary ow of an ideal incompressible uid is stable if a certain quadratic form is deenite. We show that, in three or more dimensions, this quadratic form is never deenite. Typically the form is indeenite, and the spectrum of the associated Hermitian operator ranges from ?1 to 1. The exceptional case is where the velocity eld is harmonic (solenoidal and irrotational) in which case the quadratic form is identically zero. 1. Background V.I. Arnold 1,2] was the rst to recognize and develop a new set of geometric ideas in the hydrodynamics of an ideal uid. In particular, his stability criterion 3] has been used in a great number of later publications on the subject of hydrodynamic stability. This criterion states that a velocity eld that is a steady-state solution to the hydrodynamic Euler equations is stable if a certain quadratic form, deened on the space of innnitesimal deformations to this velocity eld, is deenite. Arnold also gave important applications of his stability criterion for 2 dimensional ows. He essentially proved nonlinear stability under the conditions of the criterion in a certain metric, using a method that avoided problems related to the geometry of a foliated family of orbits in the vicinity of a stationary point 4,5]. We refer the reader to reference 6] for a discussion of the applications of the stability criterion to 2D and quasi-2D ows. H.K. Mooatt 7] gave an explicit computation of the quadratic form for ABC ows and found that it is indeenite for these important examples. In this paper we examine the applicability of the Arnold criterion in 3 or more dimensions. We nd that the quadratic form is never deenite. In a few special cases it is identically zero. In all other cases it is indeenite, and the spectrum of the associated Hermitian operator is not even bounded from above or below. It should be emphasized