The coarse-grained hydrodynamics of turbulent superfluid fluids describes the fluid flow in terms of two equations for the averaged normal and superfluid velocities. These two equations are coupled via the mutual friction term, which contains the quantity $${\mathcal {L}}(r,t)$$ L ( r , t ) –the vortex line density (VLD) of the vortex tangle. The question of how to treat the quantity $${\mathcal {L}}(r,t)$$ L ( r , t ) –the so-called closure procedure is crucial for the correct description of flow of turbulent superfluids. The article provides a critical analysis of several approaches to the closure procedure. The first one, which is usually referred to as HVBK method suggests to express the quantity $${\mathcal {L}}(r,t)$$ L ( r , t ) via coarse-grained vorticity $$\nabla \times {\mathbf {v}}_{s}$$ ∇ × v s using the famous Feynman rule. This method and idea on the vortex bundle structure, which justifies the use of the HVBK approach, are analyzed and discussed in detail. Another approach that has been popular before, but is still used sometimes, is called the Gorter–Mellink relation. This method suggests that the VLD $${\mathcal {L}}(r,t)$$ L ( r , t ) is proportional to the squared relative velocity between normal and superfluid components $$\propto (v_{n}-v_{s})^{2}$$ ∝ ( v n - v s ) 2 . One more variant of the closure procedure, discussed in the paper, is based on the method in which the vortex line density $${\mathcal {L}}(r,t)$$ L ( r , t ) is not expressed directly via the velocity (and/or vorticity) field, but is an independent equipollent variable, controlled by a separate equation. The latter approach is called as hydrodynamics of superfluid turbulence (HST). The advantages and disadvantages of each method are discussed.

中文翻译：

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湍流超流体粗粒流体动力学的封闭问题

湍流超流体流体的粗粒流体动力学根据平均法向速度和超流体速度的两个方程描述流体流动。这两个方程通过相互摩擦项耦合，其中包含量 $${\mathcal {L}}(r,t)$$ L ( r , t ) – 涡旋缠结的涡线密度 (VLD)。如何处理量$${\mathcal {L}}(r,t)$$L ( r , t )——所谓的闭合过程对于正确描述湍流超流体的流动至关重要。本文对关闭程序的几种方法进行了批判性分析。第一种，通常称为 HVBK 方法，建议通过粗粒度涡度 $$\nabla \times 表示数量 $${\mathcal {L}}(r,t)$$ L ( r , t ) {\mathbf {v}}_{s}$$ ∇ × vs 使用著名的费曼规则。详细分析和讨论了这种关于涡束结构的方法和想法，这证明了使用 HVBK 方法是合理的。另一种以前流行但有时仍在使用的方法称为 Gorter-Mellink 关系。这种方法表明 VLD $${\mathcal {L}}(r,t)$$ L ( r , t ) 正比于正常和超流体分量 $$\propto (v_{n}- v_{s})^{2}$$ ∝ ( vn - vs ) 2 . 论文中讨论的另一种闭合程序变体基于涡线密度 $${\mathcal {L}}(r,t)$$L ( r , t ) 不直接表示的方法通过速度（和/或涡度）场，但它是一个独立的等波变量，由单独的方程控制。后一种方法称为超流体湍流 (HST) 的流体动力学。