2.2 The coupling between a charged fluid and the magnetic field

Magnetic fields are ubiquitous in the Universe and are dynamically important. At high frequencies, kinetic effects are dominant, but at frequencies lower than the ion cyclotron frequency, the evolution of plasma can be modeled using the MHD approximation. Furthermore, dissipative phenomena can be neglected at large scales although their effects will be felt because of non-locality of non-linear interactions. In the presence of a magnetic field, the Lorentz force $j × B$ , where $j$ is the electric current density, must be added to the fluid equations, namely

$[ ] @u 2 ( j ) 1 r ---+ (u . \~/ ) u = - \~/ p + j \~/ u + q + -- \~/ ( \~/ .u) ---B × ( \~/ × B), (6) @t 3 4p$

and the Joule heat must be added to the equation for energy

$[ ] 2 rT @s-+ (u .\ ~/ )s = sik @ui-+ x\ ~/ 2T + --c---( \~/ × B)2, (7) @t @xk 16p2s$

where $s$ is the conductivity of the medium, and we introduced the viscous stress tensor

$( ) sik = j @ui-+ @uk- - 2-dik \~/ .u + qdik \~/ .u. (8) @xk @xi 3$

An equation for the magnetic field stems from the Maxwell equations in which the displacement current is neglected under the assumption that the velocity of the fluid under consideration is much smaller than the speed of light. Then, using

\~/ × B = m0j

and the Ohm’s law for a conductor in motion with a speed $u$ in a magnetic field

j = s (E + u × B) ,

we obtain the induction equation which describes the time evolution of the magnetic field

$@B-- 2 @t = \~/ × (u × B) + (1/sm0)\ ~/ B, (9)$

together with the constraint $\~/ .B = 0$ (no magnetic monopoles in the classical case).

In the incompressible case, where $\~/ .u = 0$ , MHD equations can be reduced to

$@u --- + (u . \~/ ) u = - \~/ Ptot + n\ ~/ 2u + (b .\ ~/ ) b (10) @t$

and

$@b ---+ (u . \~/ ) b = - (b . \~/ ) u + j\ ~/ 2b. (11) @t$

Here $Ptot$ is the total kinetic $Pk = nkT$ plus magnetic pressure $Pm = B2/8p$ , divided by the constant mass density $r$ . Moreover, we introduced the velocity variables $V ~ ---- b = B/ 4pr$ and the magnetic diffusivity $j$ .

Similar to the usual Reynolds number, a magnetic Reynolds number $Rm$ can be defined, namely

c L Rm = -A--0, j

where $V ~ ---- cA = B0/ 4pr$ is the Alfvén speed related to the large-scale $L0$ magnetic field $B0$ . This number in most circumstances in astrophysics is very large, but the ratio of the two Reynolds numbers or, in other words, the magnetic Prandtl number $Pm = n/j$ can differ widely.

The change of variable due to Elsässer (1950 ), say $z± = u ± b'$ , where we explicitly use the background uniform magnetic field $b'= b + c A$ (at variance with the bulk velocity, the largest scale magnetic field cannot be eliminated through a Galilean transformation), leads to the more symmetrical form of the MHD equations in the incompressible case

$± ( ) @z--± (cA .\ ~/ ) z± + z± . \~/ z± = - \~/ Ptot* + n± \~/ 2z ± + n± \~/ 2z ± + F ± , (12) @t$

where $Ptot*$ is the total pressure, $2n ± = n ± j$ are the dissipative coefficients, and $F ±$ are eventual external forcing terms. The relations $\~/ .z ± = 0$ complete the set of equations. On linearizing Equation (12

) and neglecting both the viscous and the external forcing terms, we have

@z± ----± (cA . \~/ ) z± -~ 0, @t

which shows that $- z (x - cAt)$ describes Alfvénic fluctuations propagating in the direction of $B0$ , and $+ z (x + cAt)$ describes Alfvénic fluctuations propagating opposite to $B0$ . Note that MHD Equations (12 ) have the same structure as the Navier-Stokes equation, the main difference stems from the fact that non-linear coupling happens only between fluctuations propagating in opposite directions. As we will see, this has a deep influence on turbulence described by MHD equations.

It is worthwhile to remark that in the classical hydrodynamics, dissipative processes are defined through three coefficients, namely two viscosities and one thermoconduction coefficient. In the hydromagnetic case the coefficients considerably increase. Apart from few additional electrical coeffcients, we have a large-scale (background) magnetic field $B 0$ . This makes the MHD equations intrinsically anisotropic. Furthermore, the stress tensor (8 ) is deeply modified by the presence of a magnetic field $B0$ , in that kinetic viscous coefficients must depend on the magnitude and direction of the magnetic field (Braginskii, 1965 ). This has a strong influence on the determination of the Reynolds number.