WIAS Preprint No. 302, (1996)

Viscous Perturbations of Vorticity Conserving Flows and Separatrix Splitting


  • Balasuriya, Sanjeeva
  • Jones, Christopher K. R. T.
  • Sandstede, Björn

2010 Mathematics Subject Classification

  • 76B47 76D99 86A05 37J40


  • two-dimensional vortices, splitting of separatrices, perturbation methods, extended Melnikov theory, forcing, Lagrangian transport, Melnikov function, meandering ocean jet


We examine the effect of the breaking of vorticity conservation by viscous dissipation on transport in the underlying fluid flow. The transport of interest is between regimes of different characteristic motion and is afforded by the splitting of separatrices. A base flow that is vorticity conserving is assumed therefore to have a separatrix that is either a homoclinic or a heteroclinic orbit. The corresponding vorticity dissipating flow, with small time-dependent forcing and viscous parameter ε, maintains an O(ε) closeness to the inviscid flow in a weak sense. An appropriate Melnikov theory that allows for such weak perturbations is then developed. A surprisingly simple expression for the leading order distance between perturbed invariant (stable and unstable) manifolds is derived which depends only on the inviscid flow. Finally, the implications for transport in barotropic jets are discussed.

Appeared in

  • Nonlinearity, 11 (1998), pp. 47-77

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