Statistical Theory of Anisotropic Magnetohydrodynamic Turbulence: An Approach to Strong Shear Alfvén Turbulence by Direct-Interaction Approximation
Statistical Theory of Anisotropic Magnetohydrodynamic Turbulence: An Approach to Strong Shear Alfvén Turbulence by Direct-Interaction Approximation
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各向异性磁流体动力湍流的统计理论：强剪切阿尔文湍流的直接相互作用近似方法
DOI：
10.1086/307702
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发表时间：
1999
期刊：
影响因子：
--
通讯作者：
Kunji Nakayama
中科院分区：
文献类型：
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作者：
Kunji Nakayama
我们在欧拉直接相互作用近似（DIA）的框架内发展了强、均匀和各向异性MHD湍流的统计理论。本文着重分析了平均磁场均匀的稳态剪切alfvsamn湍流。我们假设alfvsamn波和能量级联在含能范围（ECR）的时标远大于在惯性范围（InR）的时标。由此，我们可以得到控制传播子时间演化的DIA方程和解析可解的相关函数。DIA方程的解包括一个依赖于kz的alfv<s:1>振荡因子和一个依赖于k⊥的松弛因子。这里kz和k⊥分别是平行于和垂直于平均磁场的波数。将结果应用于光谱能量传递DIA方程，可以得到能量级联的高各向异性；也就是说，向更高kz模式的能量级联受到抑制，因此只会发生k⊥级联。因此，InR扩展到比ECR大得多的k⊥，而kz波段的展宽受到抑制。基于此，我们假设InR的能谱的函数形式为E(k⊥,kz)∝k-μ⊥δ(kz)，并发现μ = 5/2；因此我们得到相应的一维谱，k⊥∫∞-∞dkzE(k⊥,kz)∝k-3/2⊥。此外，我们还证明了三波共振对MHD湍流中能量级联的重要性。我们的理论可能受到虚假对流效应的影响，因此所得的能谱也是如此。尽管如此，它可以成为一个很好的起点，用于适用于实际天体物理MHD湍流的精细理论。
We develop a statistical theory of strong, homogeneous, and anisotropic MHD turbulence within a framework of the Eulerian direct-interaction approximation (DIA). Analysis is concentrated on stationary shear Alfvén turbulence of which the mean magnetic field is uniform. We assume that timescales of the Alfvén wave and the energy cascade in the energy-containing range (ECR) are much larger than the cascade timescale in the inertial range (InR). Thereby, we can obtain DIA equations governing the time evolution of the propagator and the correlation functions in analytically solvable form. The solutions of the DIA equations include an Alfvén oscillation factor depending on kz and a relaxation factor depending on k⊥. Here kz and k⊥ are the wavenumbers parallel to and perpendicular to the mean magnetic field, respectively. Applying the result to the DIA equation of spectral energy transfer, we can show high anisotropy of energy cascades; that is, energy cascades to higher kz modes are inhibited, hence only k⊥ cascades occur. Thus, InR extends to much larger k⊥ than ECR while the kz-band broadening is suppressed. Motivated by this, we assume the functional form of the energy spectrum of InR to be E(k⊥,kz) ∝ k-μ⊥δ(kz) and find that μ = 5/2; hence we obtain the corresponding one-dimensional spectrum, k⊥ ∫∞-∞ dkzE(k⊥,kz) ∝ k-3/2⊥. Furthermore, we show the importance of the three-wave resonances for energy cascades in MHD turbulence. Our theory probably suffers spurious convection effects, and therefore so does the resultant energy spectrum. Nevertheless, it can be a good starting point toward refined theories applicable to real astrophysical MHD turbulence.