Magnetohydrodynamic turbulence concerns the chaotic regimes of magnetofluid flow at high Reynolds number. Magnetohydrodynamics (MHD) deals with quasi-neutral fluids with very high conductivity, like plasmas. The fluid approximation implies that the focus is on macro length-and-time scales which are much larger than the collision length and collision time respectively. Understanding MHD turbulence is fundamental because most of the visible matter in the universe is in the plasma state and this plasma is mainly turbulent.
Incompressible MHD equations The incompressible MHD equations for constant mass density, ρ m = 1 {\displaystyle \rho _{\text{m}}=1} , are
∇ ⋅ u = 0 , ∂ u ∂ t + u ⋅ ∇ u = − ∇ p + B ⋅ ∇ B + ν ∇ 2 u , ∇ ⋅ B = 0 , ∂ B ∂ t + u ⋅ ∇ B = B ⋅ ∇ u + η ∇ 2 B . {\displaystyle {\begin{aligned}\nabla \cdot \mathbf {u} &=0,&{\frac {\partial \mathbf {u} }{\partial t}}+\mathbf {u} \cdot \nabla \mathbf {u} &=-\nabla p+\mathbf {B} \cdot \nabla \mathbf {B} +\nu \nabla ^{2}\mathbf {u} ,\\[8pt]\nabla \cdot \mathbf {B} &=0,&{\frac {\partial \mathbf {B} }{\partial t}}+\mathbf {u} \cdot \nabla \mathbf {B} &=\mathbf {B} \cdot \nabla \mathbf {u} +\eta \nabla ^{2}\mathbf {B} .\\[5pt]\end{aligned}}}
where
u represents the velocity, B represent the magnetic field, p represents the total pressure (thermal+magnetic) fields,
ν {\displaystyle \nu } is the kinematic viscosity and
η {\displaystyle \eta } represents magnetic diffusivity. The third equation is the incompressibility condition. In the above equation, the magnetic field is in Alfvén units (same as velocity units). That is, B {\displaystyle \mathbf {B} } is normalized as B / μ 0 ρ {\displaystyle \mathbf {B} /{\sqrt {\mu _{0}\rho }}} . The total magnetic field can be split into two parts: B = B 0 + b {\displaystyle \mathbf {B} =\mathbf {B_{0}} +\mathbf {b} } (mean + fluctuations). The above equations in terms of Elsässer variables ( z ± = u ± b {\displaystyle \mathbf {z} ^{\pm }=\mathbf {u} \pm \mathbf {b} } ) are
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