Salhi A, Baklouti F S, Godeferd F, Lehner T, Cambon C
Département de Physique, Faculté des sciences de Tunis, 1060 Tunis, Tunisia.
Université de Lyon, Laboratoire de Mécanique des Fluides et d'Acoustique, UMR 5509, Ecole Centrale de Lyon, CNRS, UCBL, INSA F-69134 Ecully Cedex, France.
Phys Rev E. 2017 Feb;95(2-1):023112. doi: 10.1103/PhysRevE.95.023112. Epub 2017 Feb 22.
Magnetic Archimedes Coriolis (MAC) waves are omnipresent in several geophysical and astrophysical flows such as the solar tachocline. In the present study, we use linear spectral theory (LST) and investigate the energy partition, scale by scale, in MAC weak wave turbulence for a Boussinesq fluid. At the scale k^{-1}, the maximal frequencies of magnetic (Alfvén) waves, gravity (Archimedes) waves, and inertial (Coriolis) waves are, respectively, V_{A}k,N, and f. By using the induction potential scalar, which is a Lagrangian invariant for a diffusionless Boussinesq fluid [Salhi et al., Phys. Rev. E 85, 026301 (2012)PLEEE81539-375510.1103/PhysRevE.85.026301], we derive a dispersion relation for the three-dimensional MAC waves, generalizing previous ones including that of f-plane MHD "shallow water" waves [Schecter et al., Astrophys. J. 551, L185 (2001)AJLEEY0004-637X10.1086/320027]. A solution for the Fourier amplitude of perturbation fields (velocity, magnetic field, and density) is derived analytically considering a diffusive fluid for which both the magnetic and thermal Prandtl numbers are one. The radial spectrum of kinetic, S_{κ}(k,t), magnetic, S_{m}(k,t), and potential, S_{p}(k,t), energies is determined considering initial isotropic conditions. For magnetic Coriolis (MC) weak wave turbulence, it is shown that, at large scales such that V_{A}k/f≪1, the Alfvén ratio S_{κ}(k,t)/S_{m}(k,t) behaves like k^{-2} if the rotation axis is aligned with the magnetic field, in agreement with previous direct numerical simulations [Favier et al., Geophys. Astrophys. Fluid Dyn. (2012)] and like k^{-1} if the rotation axis is perpendicular to the magnetic field. At small scales, such that V_{A}k/f≫1, there is an equipartition of energy between magnetic and kinetic components. For magnetic Archimedes weak wave turbulence, it is demonstrated that, at large scales, such that (V_{A}k/N≪1), there is an equipartition of energy between magnetic and potential components, while at small scales (V_{A}k/N≫1), the ratio S_{p}(k,t)/S_{κ}(k,t) behaves like k^{-1} and S_{κ}(k,t)/S_{m}(k,t)=1. Also, for MAC weak wave turbulence, it is shown that, at small scales (V_{A}k/sqrt[N^{2}+f^{2}]≫1), the ratio S_{p}(k,t)/S_{κ}(t) behaves like k^{-1} and S_{κ}(k,t)/S_{m}(k,t)=1.
磁阿基米德科里奥利（MAC）波在诸如太阳较差自转层等多种地球物理和天体物理流中普遍存在。在本研究中，我们运用线性谱理论（LST），研究了布辛涅斯克流体在MAC弱波湍流中逐尺度的能量分配。在尺度(k^{-1})下，磁（阿尔文）波、重力（阿基米德）波和惯性（科里奥利）波的最大频率分别为(V_{A}k)、(N)和(f)。通过使用感应势标量（对于无扩散布辛涅斯克流体而言，它是一个拉格朗日不变量[Salhi等人，《物理评论E》85，026301（2012年）PLEEE81539 - 375510.1103/PhysRevE.85.026301]），我们推导了三维MAC波的色散关系，推广了包括f平面磁流体动力学“浅水”波[Schecter等人，《天体物理学杂志》551，L185（2001年）AJLEEY0004 - 637X10.1086/320027]在内的先前关系。考虑磁普朗特数和热普朗特数均为1的扩散流体，解析地推导了扰动场（速度、磁场和密度）的傅里叶振幅解。考虑初始各向同性条件，确定了动能谱(S_{κ}(k,t))、磁能谱(S_{m}(k,t))和势能谱(S_{p}(k,t))的径向谱。对于磁科里奥利（MC）弱波湍流，结果表明，在大尺度下，即(V_{A}k/f≪1)，如果旋转轴与磁场对齐，阿尔文比(S_{κ}(k,t)/S_{m}(k,t))的行为类似于(k^{-2})，这与先前的直接数值模拟[Favier等人，《地球物理和天体物理流体动力学》（2012年）]一致；如果旋转轴垂直于磁场，则类似于(k^{-1})。在小尺度下，即(V_{A}k/f≫1)，磁能和动能分量之间存在能量均分。对于磁阿基米德弱波湍流，结果表明，在大尺度下，即((V_{A}k/N≪1))，磁能和势能分量之间存在能量均分，而在小尺度下((V_{A}k/N≫1))，(S_{p}(k,t)/S_{κ}(k,t))的行为类似于(k^{-1})且(S_{κ}(k,t)/S_{m}(k,t)=1)。此外，对于MAC弱波湍流，结果表明，在小尺度下((V_{A}k/\sqrt{N^{2}+f^{^{2}}}≫1))，(S_{p}(k,t)/S_{κ}(t))的行为类似于(k^{-1})且(S_{κ}(k,t)/S_{m}(k,t)=1)。
