Nonlinear Stability of a Latitudinal Ring of Point-Vortices on a Nonrotating Sphere
Nonlinear Stability of a Latitudinal Ring of Point-Vortices on a Nonrotating Sphere
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非旋转球体上点涡纬度环的非线性稳定性
DOI：
10.1137/s0036139902399965
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发表时间：
2003
期刊：
影响因子：
--
通讯作者：
S. Boatto
中科院分区：
文献类型：
--
作者：
H. Cabral;S. Boatto
研究了球面上相同点涡构型相对平衡点的非线性稳定性。特别地，我们研究了稳定性是如何随经度$\ θ $和涡数n的变化而变化的。通过运动积分，我们将系统看作是在适当的旋转坐标系中，其中多边形涡构型是静止的。然后，在Dirichlet的充分判据之后，稳定性范围是$\theta$-区间，其中哈密顿量的Hessian -在平衡构型下评估-是正或负确定的。我们发现稳定性区间与由Polvani和Dritschel确定的线性稳定性区间一致[J]。流体机械。， 255 (1993), pp. 35—64]。对于$N=3$，我们恢复了Pekarsky和Marsden先前建立的结果[J]。数学。理论物理。， 39 (1998), pp. 5894—5907]。
We study the nonlinear stability of relative equilibria of configurations of identical point-vortices on the surface of a sphere. In particular, we study how the stability changes as a function of the colatitude $\theta$ and of the number of vortices N. By using the integrals of motion, we view the system in asuitable corotating frame where the polygonal vortex configuration is at rest. Then after a sufficient criterion due to Dirichlet, the stability ranges are the $\theta$-intervals for which the Hessian of the Hamiltonian---evaluated at the equilibrium configuration---is positive or negative definite. We find that the stability intervals coincide with those for linear stability determined by Polvani and Dritschel [J. Fluid Mech., 255 (1993), pp. 35--64]. For $N=3$ we recover the result previously established by Pekarsky and Marsden [J. Math. Phys., 39 (1998), pp. 5894--5907].