Another exact solution for two-dimensional, inviscid sinh Poisson vortex arrays

Arrays of vortices are considered for two-dimensional, inviscid flows when the functional relationship between the stream function and the vorticity is a hyperbolic sine. An exact solution as a doubly periodic array of vortices is expressed in terms of the Jacobi elliptic functions. There is a threshold value of the period parameter such that a transition from globally smooth distributions of vorticity to singular distributions occurs.

[1]  Cardoso,et al.  Quantitative experimental study of the free decay of quasi-two-dimensional turbulence. , 1994, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[2]  K. Chow.  A class of doubly periodic waves for nonlinear evolution equations , 2002 .

[3]  S. Dalziel,et al.  Source-sink turbulence in a stratified fluid , 1994, Journal of Fluid Mechanics.

[4]  Thierry Dauxois,et al.  Stability of periodic arrays of vortices , 1995, patt-sol/9510003.

[5]  Sean Oughton,et al.  Relaxation in two dimensions and the "sinh-Poisson" equation , 1992 .

[6]  G. J. F. van Heijst,et al.  On Chaplygin's investigations of two-dimensional vortex structures in an inviscid fluid , 1994, Journal of Fluid Mechanics.

[7]  Irene A. Stegun,et al.  Handbook of Mathematical Functions. , 1966 .

[8]  J. Rasmussen,et al.  Formation and temporal evolution of the Lamb-dipole , 1997 .

[9]  van Gjf Gert-Jan Heijst,et al.  Spontaneous spin-up during the decay of 2D turbulence in a square container with rigid boundaries , 1998 .

[10]  Solitons in (2 + 0) dimensions and their applications in vortex dynamics , 1997 .

[11]  J. Hesthaven,et al.  Dynamics of a nonlinear dipole vortex , 1995 .

[12]  Cardoso,et al.  Experimental study of freely decaying two-dimensional turbulence. , 1991, Physical review letters.

[13]  E. Hopfinger,et al.  Vortices in Rotating Fluids , 1993 .

[14]  Mark J. Ablowitz,et al.  Solitons and the Inverse Scattering Transform , 1981 .

[15]  S. Maslowe,et al.  A row of counter-rotating vortices , 1993 .

[16]  R. Pasmanter.  On long‐lived vortices in 2‐D viscous flows, most probable states of inviscid 2‐D flows and a soliton equation , 1994 .

[17]  T. Dauxois.  Nonlinear stability of counter‐rotating vortices , 1994 .

[18]  Jean-Marc Chomaz,et al.  Three-dimensional stability of periodic arrays of counter-rotating vortices , 2002 .

[19]  K. Chow,et al.  Inviscid two dimensional vortex dynamics and a soliton expansion of the sinh-Poisson equation , 1998 .

[20]  B. Kuvshinov,et al.  Double-periodic arrays of vortices , 2000 .