Superfluid Turbulence in the Low Temperature Regime

低温下的超流湍流研究超流体湍流的两种最常用的方法是HVBK理论和经典涡丝理论。这些模式有许多缺点，这使它们不切实际和即兴。我们建议的主要目的是使用基于某些形式的非线性薛定谔方程(NLS)的第三种方法，用于两个密切相关的目的：为HVBK理论和经典涡丝理论提供微观上真实的参数，以及使用NLS的新变体来解释超流体湍流，这些新的变体更接近真实的氦II。更具体地说，我们计划：(1)使用NLS方程来足够详细地研究涡线重联的过程，以定量地描述伴随的声波和开尔文波的辐射。这些结果将使我们能够定义重联规则，当经典理论应用于超流体湍流时可以使用这些规则。(2)用NLS理论模拟低温下的超流体湍流，研究超流体中弱湍流态和强湍流态之间的转变。(3)分析了考虑超流体与正常流体相互摩擦的耗散NLS模型。(4)对强迫和耗散NLS方程的吸引子、分叉序列和通向混沌的途径进行了研究和分类。然而，我们的主要目标是阐明低温条件下的超流体湍流，在低温条件下，实验目前处于计划阶段。超流湍流经过多年的实验研究，现已成为低温物理学的一个重要分支。除了这门学科本身的智力挑战之外，还有几个原因。氦被用作超导磁体和红外探测器的冷却剂，仅举几个工程应用中的两个例子。在这些情况下，流动通常会变得湍急。这一主题还具有超出氦研究领域的意义，例如在高温超导、磁自旋系统、晶体的熔化转变以及中子星自转中毛刺的起源的研究中。超流体湍流还可以提供对经典流体湍流的洞察，特别是在高雷诺数时，涡量具有间歇的、分形的特征。

SUPERFLUID TURBULENCE IN THE LOW TEMPERATIRE REGIME The two most common approaches to study of superfluid turbulence are the HVBK theory and the classical theory of vortex filaments. There are many shortcomings in these models that make them unrealistic and ad hoc. The main objective of our proposal is to use a third approach based on certain forms of nonlinear Schroedinger equation (NLS) for two closely related purposes: to provide microscopically realistic parameters for HVBK theory and the classical theory of vortex filaments and to elucidate superfluid turbulence using new variants of the NLS that are more faithful to real helium II. More specifically we plan: (1) use NLS equations to study the process of vortex line reconnection in sufficient detail to characterize quantitatively the accompanying radiation of sound and Kelvin waves. The results will allow us to define reconnection rules that could be used when classical theory is applied to superfluid turbulence. (2) Use NLS theory to simulate superfluid turbulence in the low temperature regime to study the transition between weak and strong turbulence states in the superfluid. (3) Analyze dissipative NLS models that include damping from the mutual friction between superfluid and normal fluid. (4) Study and classify attractors, bifurcation sequences, and routes to chaos in solutions of forced and dissipative NLS equations. Our main goal is, however, to elucidate superfluid turbulence in the low temperature regime in which experiments are currently in the planning stage. Superfluid turbulence has been studied experimentally for many years and has by now become a major branch of cryogenic physics. In addition to the intrinsic intellectual challenges of the subject, there are several reasons for this. Helium is used as a coolant for superconducting magnets and for infrared detectors, to name just two of several engineering applications. The flow commonly becomes turbulent in these contexts. The subject also has implications beyond the field of helium research, such as in the study of high temperature superconductivity, systems of magnetic spins, melting transitions of crystals, and the origin of glitches in neutron star rotations. Superfluid turbulence may also provide insights into classical fluid turbulence, especially at high Reynolds numbers, where the vorticity has an intermittent, fractal character.