DECAY OF CONFINED, 2-DIMENSIONAL, SPATIALLY PERIODIC ARRAYS OF VORTICES - A NUMERICAL INVESTIGATION

The disarrangement of a perturbed lattice of vortices was studied nume rically. The basic state is an exponentially decaying, exact solution of the Navier-Stokes equations. Square arrays of vortices with even nu mbers of vortex cells along each side were perturbed and their evoluti on was investigated. Whether the energy in the perturbation grows some what before it decays or decays monotonically depends on the initial s trength of the vortices of the basic state, the extent of lateral conf inement and the structure of the perturbation. The critical condition for temporally local instability, i.e. the critical amplitude of the b asic state that must be exceeded to allow energy transfer from the bas ic state to the perturbation, is discussed. In the strongly confined c ase of a square lattice of four vortices the appearance of enchancemen t of global rotation is the result of energy transfer from the basic s tate to a temporally local unstable mode. Energy is transferred from t he basic state to larger-scaled structures (inverse cascade) only if t he scales of the larger structures are inherently contained in the ini tial structure of the perturbation. The initial structure of the doubl e array of vortices is not maintained except for a very special form o f perturbation. The facts that large scales decay more slowly than sma ll scales and that, when non-linearities are sufficiently strong, ener gy is transferred from one scale to another explain the differences in the disarrangement process for different initial strengths of the vor tices of the basic state. The stronger vortices, i.e. the vortices per turbed in a manner that increases their strength, tend to dominate the weaker vortices. The pairing and subsequent merging (or capture) of v ortices of like sense into larger-scale vortices are described in term s of peaks in the evolution of the square root of the palinstrophy div ided by the enstrophy.