Ruban V P, Podolsky D I, Rasmussen J J
L.D. Landau Institute for Theoretical Physics, 2 Kosygin Street, 117334 Moscow, Russia.
Phys Rev E Stat Nonlin Soft Matter Phys. 2001 May;63(5 Pt 2):056306. doi: 10.1103/PhysRevE.63.056306. Epub 2001 Apr 16.
Models of inviscid incompressible fluid are considered, with the kinetic energy (i.e., the Lagrangian functional) taking the form L approximately integral k(alpha)/vk/2dk in 3D Fourier representation, where alpha is a constant, 0<alpha<1. Unlike the case alpha=0 (the usual Eulerian hydrodynamics), a finite value of alpha results in a finite energy for a singular, frozen-in vortex filament. This property allows us to study the dynamics of such filaments without the necessity of a regularization procedure for short length scales. The linear analysis of small symmetrical deviations from a stationary solution is performed for a pair of antiparallel vortex filaments and an analog of the Crow instability is found at small wave numbers. A local approximate Hamiltonian is obtained for the nonlinear long-scale dynamics of this system. Self-similar solutions of the corresponding equations are found analytically. They describe the formation of a finite time singularity, with all length scales decreasing like (t*-t)(1/(2-alpha)), where t* is the singularity time.
考虑无粘性不可压缩流体模型，在三维傅里叶表示中，动能（即拉格朗日泛函）具有形式(L\approx\int\frac{k^{\alpha}}{v^{k/2}}dk)，其中(\alpha)为常数，(0\lt\alpha\lt1)。与(\alpha = 0)的情况（通常的欧拉流体动力学）不同，(\alpha)的有限值会导致奇异的、冻结的涡旋丝具有有限能量。这一特性使我们能够研究此类丝的动力学，而无需对短长度尺度进行正则化处理。对一对反平行涡旋丝相对于静止解的小对称偏差进行了线性分析，并在小波数处发现了类似克劳不稳定的现象。得到了该系统非线性长尺度动力学的局部近似哈密顿量。通过解析方法找到了相应方程的自相似解。它们描述了有限时间奇点的形成，所有长度尺度都像((t^ * - t)^{1/(2 - \alpha)})那样减小，其中(t^*)是奇点时间。
