Falkovich Gregory, Vladimirova Natalia
Weizmann Institute of Science, Rehovot 76100, Israel.
Institute for Information Transmission Problems, Moscow 127994, Russia.
Phys Rev E Stat Nonlin Soft Matter Phys. 2015 Apr;91(4):041201. doi: 10.1103/PhysRevE.91.041201. Epub 2015 Apr 29.
We consider developed turbulence in the two-dimensional Gross-Pitaevskii model, which describes wide classes of phenomena from atomic and optical physics to condensed matter, fluids, and plasma. The well-known difficulty of the problem is that the hypothetical local spectra of both inverse and direct cascades in the weak-turbulence approximation carry fluxes that are either zero or have the wrong sign; Such spectra cannot be realized. We analytically derive the exact flux constancy laws (analogs of Kolmogorov's 4/5 law for incompressible fluid turbulence), expressed via the fourth-order moment and valid for any nonlinearity. We confirm the flux laws in direct numerical simulations. We show that a constant flux is realized by a nonlocal wave interaction in both the direct and inverse cascades. Wave spectra (second-order moments) are close to slightly (logarithmically) distorted thermal equilibrium in both cascades.
我们考虑二维格罗斯 - 皮塔耶夫斯基模型中的充分发展湍流，该模型描述了从原子物理、光学物理到凝聚态物质、流体和等离子体等广泛的现象类别。该问题的一个众所周知的难点在于，在弱湍流近似下，逆级联和正级联的假设局部谱所携带的通量要么为零，要么符号错误；这样的谱无法实现。我们通过解析推导得出了精确的通量守恒定律（类似于不可压缩流体湍流的柯尔莫哥洛夫4/5定律），该定律通过四阶矩表示，并且对任何非线性情况都有效。我们在直接数值模拟中证实了这些通量定律。我们表明，在正级联和逆级联中，恒定通量都是通过非局部波相互作用实现的。在两个级联中，波谱（二阶矩）都接近略微（对数）扭曲的热平衡。
