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A steady ideal fluid flow on a surface corresponds to a geodesic in the area-preserving diffeomorphism group. The sign of the curvature operator along this geodesic has been of interest since Arnold noticed its connection to Lagrangian stability of the flow: nonpositive curvature implies by the Rauch comparison theorem that Lagrangian perturbations grow at least linearly in time. We obtain a new necessary and sufficient criterion for a steady flow with analytic stream function and isolated zeroes to have nonpositive curvature operator: either the surface is a flat torus, and the fluid flow has constant pressure; or the surface is a sphere, disc, or annulus with a globally-defined polar coordinate system such that the metric is d s 2 = d r 2 + ϕ 2 ( r ) d θ 2 . In the latter case, the velocity field must be of the form X = u ( r ) ∂ θ . Furthermore, the function Q = ( u ϕ ′ ) ′ / u ′ must be defined for every r and satisfy the differential inequality ϕ Q ′ + Q 2 ≤ 1 . This criterion is proved by using a new formula for the curvature of the area-preserving diffeomorphism group in the rotationally symmetric case, involving only first integrals in one variable, rather than infinite sums or the solution of a PDE. Elementary consequences of the criterion are also discussed: for example, there are no flows with nonpositive curvature operator on the standard round sphere; and on a flat surface, every rotationally symmetric flow has nonpositive curvature operator. Finally we show that if a steady flow satisfies both this nonpositive curvature criterion and the well-known Eulerian stability criterion of Arnold, then all Lagrangian perturbations grow polynomially in time, in the L 2 norm. Thus this is the first time methods of Riemannian geometry have given rigorous information on stability. |
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