TY - JOUR

T1 - Thresholds for the formation of satellites in two-dimensional vortices

AU - Turner, M.R.

AU - Gilbert, A.D.

N1 - © The Author(s) and Cambridge University Press, 2008

PY - 2008

Y1 - 2008

N2 - This paper examines the evolution of a two-dimensional vortex which initially consists of an axisymmetric monopole vortex with a perturbation of azimuthal wavenumber m = 2
added to it. If the perturbation is weak then the vortex returns to an axisymmetric state and the non-zero Fourier harmonics generated by the perturbation decay to zero.
However, if a finite perturbation threshold is exceeded, then a persistent nonlinear vortex structure is formed. This structure consists of a coherent vortex core with two satellites rotating around it.
The paper considers the formation of these satellites by taking an asymptotic limit in which a compact vortex is surrounded by a weak skirt of vorticity. The resulting equations match the behaviour of a normal mode riding on the vortex with the evolution of fine-scale vorticity in a critical layer inside the skirt. Three estimates of inviscid thresholds for the formation of satellites are computed and compared: two estimates use qualitative
diagnostics, the appearance of an infection point or neutral mode in the mean profile. The other is determined quantitatively by solving the normal mode/critical-layer equations numerically. These calculations are supported by simulations of the full Navier-Stokes equations using a family of proles based on the tanh function.

AB - This paper examines the evolution of a two-dimensional vortex which initially consists of an axisymmetric monopole vortex with a perturbation of azimuthal wavenumber m = 2
added to it. If the perturbation is weak then the vortex returns to an axisymmetric state and the non-zero Fourier harmonics generated by the perturbation decay to zero.
However, if a finite perturbation threshold is exceeded, then a persistent nonlinear vortex structure is formed. This structure consists of a coherent vortex core with two satellites rotating around it.
The paper considers the formation of these satellites by taking an asymptotic limit in which a compact vortex is surrounded by a weak skirt of vorticity. The resulting equations match the behaviour of a normal mode riding on the vortex with the evolution of fine-scale vorticity in a critical layer inside the skirt. Three estimates of inviscid thresholds for the formation of satellites are computed and compared: two estimates use qualitative
diagnostics, the appearance of an infection point or neutral mode in the mean profile. The other is determined quantitatively by solving the normal mode/critical-layer equations numerically. These calculations are supported by simulations of the full Navier-Stokes equations using a family of proles based on the tanh function.

U2 - 10.1017/S0022112008003558

DO - 10.1017/S0022112008003558

M3 - Article

SN - 0022-1120

VL - 614

SP - 381

EP - 405

JO - Journal of Fluid Mechanics

JF - Journal of Fluid Mechanics

ER -