In this paper we define the Painlevé property for partial differential equations and show how it determines, in a remarkably simple manner, the integrability, the Bäcklund transforms, the linearizing transforms, and the Lax pairs of three well-known partial differential equations (Burgers’ equation, KdV equation, and the modified KdV equation). This indicates that the Painlevé property may provide a unified description of integrable behavior in dynamical systems (ordinary and partial differential equations), while, at the same time, providing an efficient method for determining the integrability of particular systems.

Freely evolving two-dimensional turbulence is dominated by coherent vortices. The density of these vortices decays in time as p tt with (=0.75. A new scaling theory is proposed which expresses all statistical properties in terms of g. Thus the average circulation of the vortices increases as t~t' and their average radius as t~. The total energy is constant, the enstrophy decreases as t~, and the vorticity kurtosis increases as t~. These results are supported both by numerical simulations of the fluid equations and by solutions of a modified point-vortex model. PACS numbers: 47.25.cBecause of its geophysical and astrophysical importance, the emergence and evolution of coherent vortices in freely evolving two-dimensional turbulence has been a subject of intense study in the last ten years. ' In thisLetter, we formulate a new scaling theory and present evidence for scaling behavior in both two-dimensional turbulence and a simple, punctuated-Hamiltonian, dynamical model of coherent vortices. In anticipation of this work on vortex dynamics, we have previously performed studies of punctuated-Hamiltonian models of aggregation in one-dimensional systems. The fluid-dynamical equations are g, +J(tlt, g) =v~( -1)~+'V t'tll, g-: V ttt, where ttt is the stream function, g is the vorticity, J(a, b) a"bJb, a~-is the Jacobian, V -=a"'+a, ' is the Laplacian, and v~i s the hyperviscosity for p a positive integer (p =2 here). The domain is a square of side 2trL and the boundary conditions are periodic in both x and y. Numerical solutions of Eq. (1) show that well separated, almost axisymmetric, coherent vortices emerge from structureless initial conditions. Between the vortices there is a background sea of small-scale, incoherent vorticity. After the emergence of the vortices the dynamics appears to be dominated by two processes: (1) mutual advection of well separated vortices in which Hamiltonian point-vortex dynamics is a good approximation, and (2) merger of like-sign vortices during close encounters.As a result of the mergers, the vorticity is concentrated in increasingly larger, fewer, and more widely separated vortices as time increases. A "vortex census" shows that, for a broad class of initial conditions, the number of vortices per area p(t) decreases according to p -t~, with g approximately 0.75. The focus of this Letter is on the scaling properties of this "dilute vortex gas. " There is a simple dimensional argument that predicts (=2. The kinetic energy per area 8 is invariant as v~0, ' where ] fO 2 Vy. Vydx 4 212+8-pg, ", a (3) This is easily seen as an approximation to the second expression in (2) for both spatial arguments within the same vortex (the self-energy 4, ). ' It also characterizes the contribution to 8 from arguments in separate vortices (the configuration energy 8"); 8, scales with the number of vortices, rather than the number of pairs, due to cancellations from vortices of opposite sign. We assume the contributions in (2) from arguments outside the vortices are negligible.Inviscid dynamics [vz =0 in Eq. (...

The stability properties and stationary statistics of inviscid barotropic flow over topography are examined. Minimum enstrophy states have potential vorticity proportional to the streamfunction and are nonlinearly stable ; correspondingly, canonical equilibrium based on energy and enstrophy conservation predicts mean potential vorticity is proportional to the mean streamfunction. It is demonstrated that in the limit of infinite resolution the canonical mean state is statistically sharp, that is, without any eddy energy on any scale, and is identical to the nonlinearly stable minimum enstrophy state. Special attention is given to the interaction between small scales and a dynamically evolving large-scale flow. On the b-plane, these stable flows have a westward large-scale component. Possibilities for a general relation between inviscid statistical equilibrium and nonlinear stability theory are examined.