Experimental observations and numerical simulations of the large force inhomogeneities present in stationary bead packs are presented. Forces much larger than the mean occurred but were exponentially rare. An exactly soluble model reproduced many aspects of the experiments and simulations. In this model, the fluctuations in the force distribution arise because of variations in the contact angles and the constraints imposed by the force balance on each bead in the pile.

The inviscid damping of an asymmetric perturbation on a two-dimensional circular vortex is examined theoretically, and with an electron plasma experiment. In the experiment, an elliptical perturbation is created by an external impulse. After the impulse, the ellipticity ͑quadrupole moment͒ of the vortex exhibits an early stage of exponential decay. The measured decay rate is in good agreement with theory, in which the perturbation is governed by the linearized Euler equations. Often, the exponential decay of ellipticity is slow compared to a vortex rotation period, due to the excitation of a quasimode. A quasimode is a vorticity perturbation that behaves like a single azimuthally propagating wave, which is weakly damped by a resonant interaction with corotating fluid. Analytically, the quasimode appears as a wave packet of undamped continuum modes, with a sharply peaked frequency spectrum, and it decays through interference as the modes disperse. When the exponential decay rate of ellipticity is comparable to the vortex rotation frequency, the vorticity perturbation does not resemble a quasimode; rather, it is rapidly dominated by spiral filaments. Over longer times, linear theory predicts algebraic decay of ellipticity; however, nonlinear oscillations of ellipticity emerge in the experiment before a transition to algebraic decay would occur.

Vortex-in-cell simulations that numerically integrate the 2D Euler equations are compared directly to experiments on magnetized electron columns ͓K. S. Fine, A. C. Cass, W. G. Flynn, and C. F. Driscoll, ''Relaxation of 2D turbulence to vortex crystals,'' Phys. Rev. Lett. 75, 3277 ͑1995͔͒, where turbulent flows relax to metastable vortex crystals. A vortex crystal is a lattice of intense small diameter vortices that rotates rigidly in a lower vorticity background. The simulations and experiments relax at the same rates to vortex crystals with similar vorticity distributions. The relaxation is caused by mixing of the background by the intense vortices: the relaxation rate is peaked when the background circulation is 0.2-0.4 times the total circulation. Close quantitative agreement between experiment and simulation provides strong evidence that vortex crystals can be explained without incorporating physics beyond 2D Euler theory, despite small differences between a magnetized electron column and an ideal 2D fluid.

This paper further examines the rate at which potential vorticity in the core of a monotonic cyclone becomes vertically aligned and horizontally axisymmetric. We consider the case in which symmetrization occurs by the damping of a discrete vortex Rossby ͑VR͒ wave. The damping of the VR wave is caused by its stirring of potential vorticity at a critical radius r * , outside the core of the cyclone. The decay rate generally increases with the radial gradient of potential vorticity at r * . Previous theories for the decay rate were based on ''balance models'' of the vortex dynamics. Such models filter out inertia-buoyancy ͑IB͒ oscillations, i.e., gravity waves. However, if the Rossby number is greater than unity, the core VR wave can excite a frequency-matched outward propagating IB wave, which has positive feedback. To accurately account for this radiation, we here develop a theory for the decay rate that is based on the hydrostatic primitive equations. Starting from conservation of wave activity ͑angular pseudomomentum͒, an expression for the decay rate is derived. This expression explicitly demonstrates a competition between the destabilizing influence of IB wave emission, and the stabilizing influence of potential vorticity stirring at r * . Moreover, it shows that if the radial gradient of potential vorticity at r * exceeds a small threshold, the VR wave will decay, and the vortex will symmetrize, even at large Rossby numbers.