Introduction to Strong Alfvénic MHD Turbulence

Abstract

Many astrophysical fluids are magnetized and turbulent. Such fluids can be often described by magnetohydrodynamics (MHD). In this review, we mainly consider MHD turbulence with a strong mean magnetic field whose energy density is greater than or equal to the local kinetic energy density. In these fluids, the MHD waves, especially Alfvén waves, play a dominant dynamical role. Alfvén waves travel along magnetic field lines and collisions of opposite-traveling Alfvén wave packets are essential for turbulence cascade. We focus on strong turbulence regime, where nonlinear interaction during the collision is sufficiently strong and thus one collision is enough to complete turbulence cascade. We will cover the following types of turbulence. First, we review strong Alfvénic MHD turbulence. If the mean magnetic field is very strong, wave packets move very fast and duration of collision is too short to complete turbulence cascade. Even in this case we will show that strong turbulence regime emerges on a small scale. Second, we will consider small-scale MHD turbulence, where interaction of small-scale variants of Alfvén waves (i.e., whistler waves) is important. Third, we review scaling relations in strong relativistic force-free MHD turbulence, where interaction of relativistic Alfvén waves is important. Finally, we briefly discuss scaling relations in compressible MHD turbulence, where interaction of Alfvén waves is still important.

Figures (13)

• Figure 1: Energy cascade in hydrodynamic turbulence.
• Figure 2: Alfvén wave packets. (Left) An Alfvén wave packet moves at the Alfvén speed $V_A=B_0/\sqrt{ 4\pi \rho}$. When we use a system of units in which $4 \pi$ does not appear and $\rho=1$, it follows that $4 \pi \rho =1$ and hence $V_A=B_0$. Since Alfvén waves are non-dispersive, an Alfvén wave packet does not change its shape during the propagation in an incompressible fluid. (Right) When two opposite-traveling Alfvén wave packets collide, non-linear interactions happen.
• Figure 3: Collision of wave packets. Shapes shown below the arrows are cross-sectional shapes. (Left) Before the collision, we assume perpendicular shapes of the eddies are isotropic. (Right) During the collision, an eddy is distorted by shearing motion of the other eddy.
• Figure 4: (Left) Snapshot of magnetic field strength. The black arrow represents the direction of the mean magnetic field. (Right) Kinetic and magnetic spectra. They are compatible with Kolmogorov spectrum and hence agree with the GS95 prediction. From CV00ani.
• Figure 5: (Left) Illustration of scale-dependent anisotropy. The black arrow represents the direction of the mean magnetic field. (Right) Measured anisotropy of eddy shapes. The parallel and perpendicular sizes of eddies, $l_\|$ and $l_\bot$ respectively, follow the relation $l_\| \propto l_\bot^{2/3}$, which agrees with the GS95 prediction. From CV00ani.