Three-dimensional Quasi-geostrophic Staggered Vortex Arrays

Abstract

We determine and characterise relative equilibria for arrays of point vortices in a three-dimensional quasi-geostrophic flow. The vortices are equally spaced along two horizontal rings whose centre lies on the same vertical axis. An additional vortex may be placed along this vertical axis. Depending on the parameters defining the array, the vortices on the two rings are of equal or opposite sign. We address the linear stability of the point vortex arrays. We find both stable equilibria and unstable equilibria, depending on the geometry of the array. For unstable arrays, the instability may lead to the quasi-regular or to the chaotic motion of the point vortices. The linear stability of the vortex arrays depends on the number of vortices in the array, on the radius ratio between the two rings, on the vertical offset between the rings and on the vertical offset between the rings and the central vortex, when the latter is present. In this case the linear stability also depends on the strength of the central vortex. The non-linear evolution of a selection of unstable cases is presented exhibiting examples of quasi-regular motion and of chaotic motion.

APPENDIX. INVARIANTS AND ACCURACY

We briefly discuss the accuracy of the calculations. To this end, we verify that the flow invariants are conserved within acceptable limits for the practical relevance of the discussion. Similarly to the classical planar, two-dimensional vortex dynamics, the QG vortex dynamics conserves some fundamental invariants, namely, the linear impulse $\boldsymbol{I}$, defined for a set of discrete point vortices as

$$\boldsymbol{I}=(I_{x},I_{y})=\sum_{i=1}^{N}\kappa_{i}(x_{i},y_{i}),$$

(A.1)

the angular impulse

$$J=\frac{1}{2}\sum_{i=1}^{N}\kappa_{i}(x_{i}^{2}+y_{i}^{2}),$$

(A.2)

and the interaction energy (which is also the Hamiltonian of the system)

$${H}=\sum_{i=1}^{N}\sum_{j=i+1}^{N}\frac{\kappa_{i}\kappa_{j}}{|\boldsymbol{x}_{i}-\boldsymbol{x}_{j}|}$$

(A.3)

Figure 28 shows the evolution of the linear impulses $I_{x},I_{y}$ and of the scaled angular impulse $J/J_{0}$ and interaction energy $H/H_{0}$ for the case presented in Fig. 21 with $n=2$, $\Delta=0$, $r_{i}/r_{e}=0.2$ and no central vortex (top panels) and for $n=9$, $\Delta=0$, $r_{i}/r_{e}=0.8$, and a central vortex $(\kappa_{0},z_{0})=(1,0)$ presented in Fig. 26. Here $J_{0}$ and $H_{0}$ are the initial values of the angular impulse and the interaction energy, respectively. Results show that the linear impulse (initially zero by symmetry) is conserved within machine precision. Angular impulse $J$ is conserved within $0.012\%$ for the case with $n=2$, while the energy is conserved with $0.2\%$ for a time step $\Delta t=0.075$ which is enough for the purpose of the discussion. In particular, for the initial evolution $t<50$ (which includes the first phases of the non-trivial unstable evolution), accuracy is much higher. For $n=9$ accuracy is higher as the time step used is smaller $\Delta t=0.0025$ (the vortex velocities are larger in this case). Small errors are associated with the finite accuracy of the time integration, especially when some vortices get close together and their velocity increases. For long integration periods, these errors could be mitigated by using an adaptive time step. We are, however, only interested in the early evolution of the arrays.