中性流体和等离子体的守恒正则化
Conservative regularization of neutral fluids and plasmas
摘要：
理想的方程组（例如欧拉方程和磁流体力学）可能会产生奇异结构，例如冲击、涡流/流片。其中，涡奇异性的出现是由于涡的拉伸，这可能会导致无限增长的拟能。粘性和电阻率提供了这些奇异性的耗散正则化。与一维无粘Burgers方程的色散KdV正则化类似，我们提出了理想三维可压缩流、MHD和2流体等离子体的局部守恒正则化（具有低耗散高涡度流的潜在应用）。正则化涉及到在速度方程中引入一个涡旋“旋度”项ω 2 w x旋度w。截止长度λ必须与密度的平方根成反比，以确保“漩涡”能量的守恒。后者包括积极的动力，压缩，磁和涡的贡献，从而导致先验的界限拟能。扩展到2流体等离子体涉及额外的磁'旋转'条款的离子和电子速度方程和螺线管除了电流安培定律。以涡旋能为哈密顿量，建立了一个哈密顿-泊松括号公式。我们还建立了旋转正则化的极小性质。一个旋流速度场被示出为传输涡流和磁通量管以及W/ρ和B/ρ，从而推广了Kelvin-Helmholtz和Alfven定理。用定常正则化方程模拟了旋转涡、磁流体夹点和涡面。我们的正则化可以方便的数值模拟和统计处理的涡流和电流丝在3D。最后，我们简要地描述了一个保守的正则化的冲击类奇点在可压缩流推广KdV和非线性薛定谔方程的绝热动力学的气体在3D中。
英文摘要：
Ideal systems of equations such as Euler and MHD may develop singular structures like shocks, vortex/current sheets. Among these, vortical singularities arise due to vortex stretching which can lead to unbounded growth of enstrophy. Viscosity and resistivity provide dissipative regularizations of these singularities. In analogy with the dispersive KdV regularization of the 1D inviscid Burgers' equation, we propose a local conservative regularization of ideal 3D compressible flows, MHD and 2-fluid plasmas (with potential applications to high vorticity flows with low dissipation). The regularization involves introducing a vortical `twirl' term lambda^2 w x curl w in the velocity equation. The cut-off length lambda must be inversely proportional to square root of density to ensure the conservation of a `swirl' energy. The latter includes positive kinetic, compressional, magnetic and vortical contributions, thus leading to a priori bounds on enstrophy. The extension to 2-fluid plasmas involves additionally magnetic `twirl' terms in the ion and electron velocity equations and a solenoidal addition to the current in Ampere's law. A Hamiltonian-Poisson bracket formulation is developed using the swirl energy as Hamiltonian. We also establish a minimality property of the twirl regularization. A swirl velocity field is shown to transport vortex and magnetic flux tubes as well as w/rho and B/rho, thus generalizing the Kelvin-Helmholtz and Alfven theorems. The steady regularized equations are used to model a rotating vortex, MHD pinch and vortex sheet. Our regularization could facilitate numerical simulations and a statistical treatment of vortex and current filaments in 3D. Finally, we briefly describe a conservative regularization of shock-like singularities in compressible flow generalizing both the KdV and nonlinear Schrodinger equations to the adiabatic dynamics of a gas in 3D.
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