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The MHD Equations

1.1 Derivation of the MHD Equations

In this book, we will treat the description of equilibrium and stability properties of magnetically confined fusion plasmas in the framework of a fluid theory, the so-called Magnetohydrodynamic (MHD) theory. In this chapter, we are going to derive the MHD equations and discuss some of their basic properties and the limitations for application of MHD to the description of fusion plasmas. The derivation follows the treatment given in [1]. For a more in-depth discussion of the MHD equations, the reader is referred to [2]. Non-linear aspects of MHD are treated in [3]. A good overview of general tokamak physics can be found in [4].

1.1.1 Multispecies MHD Equations

As a magnetized plasma is a many-body system, its description cannot be done by solving individual equations of motion that would typically be a set of, say, equations1 that are all coupled through the electromagnetic interaction. Hence, some kind of mean field theory is needed.

Starting point of our derivation is the kinetic equation known from statistical physics. It describes the many-body system in terms of a distribution function in six-dimensional space , where

is the probability to find a particle of species at with velocity at time . Here, and are independent variables that, in the sense of classical mechanics, fully describe the system.

The basic assumption of kinetic theory is that fields and forces are macroscopic in the sense that they have already been averaged over a volume containing many particles (say, a Debye-sphere2) and the microscopic fields and forces at the exact particle location can be expressed through a collision term giving rise to a change of along the particle trajectories in six-dimensional space. We note that this has reduced the microscopic problem of the interactions to the proper choice of the collision term.

Evaluating the total change of along the trajectories and keeping in mind that along these, and , where is the force acting on the particle and its mass, the kinetic equation can be expressed as

where we have assumed that the only relevant force is the Lorentz force and hence explicitly neglected gravity (which is a good approximation for magnetically confined fusion plasmas, but generally not true in Astrophysical applications). According to the above-mentioned description of mean field theory, the fields and will have to be calculated from Maxwells equations using the charge density and current resulting from appropriate averaging over the distribution function in velocity space as will be described in the following.

The kinetic equation is used to describe phenomena that arise from not being a Maxwellian, which is the particle distribution in thermodynamic equilibrium to which the system will relax through the action of collisions. In fusion plasmas, this frequently occurs as the mean free path is often large compared to the system length as is for example the case for turbulence dynamics in a tokamak along field lines. Another important example is when the relevant timescales are short compared to the collision time, such as in RF (radio frequency) wave heating and current drive that can occur by Landau damping rather than collisional dissipation. Here, a description using the Vlasov or Fokker-Planck equation is needed.

However, in situations where is close to Maxwellian, one can average the kinetic equation over velocity space to obtain hydrodynamic equations in configuration space. When doing so, one encounters so-called moments of . The kth moment, which is related to the velocity average of , is given by

These moments are related to the hydrodynamic quantities used to describe the plasma in configuration space. For the zeroth moment, we obtain

which is the number density in real space. The first moment of is related to the fluid velocity in the centre of mass frame by

For the second moment, it is of advantage to separate the particle velocity into the fluid velocity and the random thermal motion according to

It is easy to show that , as expected for thermal motion, as

However, the quadratic average is non-zero, representing the thermal energy via

where is the Boltzmann constant and we have used the definition of the thermal energy density and its relation to the pressure for an ideal plasma. We note that this definition relies on the previous assumption that is close to Maxwellian. More generally, the second moment is defined as a tensor of rank 2, the pressure tensor

where denotes the dyadic product. The non-diagonal terms of this tensor are related to viscosity, whereas from Eq. 1.8, it is clear that the trace of is equal to , that is for an isotropic system, the diagonal elements of are just equal to the scalar pressure. Therefore, the pressure tensor is also often written as

where is the unit tensor and the anisotropic part of .

We now integrate the kinetic equation (Eq. 1.2) over velocity space3 to obtain

which is the equation of continuity for species . Here, we have assumed that the velocity space average of the collision term is zero, meaning that the total number of particles is conserved for each species. Should this not be the case (e.g. by ionization or fusion), the right-hand side would consist of a source term.

The next moment is obtained by multiplying the kinetic equation by and integrating over velocity space. This yields the momentum balance

where the friction force is the first moment of the collision term for collisions with species . We note that only collision with unlike particles lead to a net friction force while collisions within one species, which are important for thermalization, do not transfer net momentum to that species. This form is also called the conservative form as, like the equation of continuity, it relates the temporal derivative of a quantity (in this case, the momentum) to the divergence of a flux. However, this equation can be rearranged using the continuity equation into a form in which the dyadic product of the velocity can be absorbed in the derivative on the left-hand side:

which is usually called the force balance. Here, the operator

is called the substantial or convective derivative and measures the change along the trajectory of a fluid element in the laboratory frame. In ordinary hydrodynamics, Eq. 1.13 is called the Euler equation while equations in the co-moving frame are referred to as the Lagrangedescription.

The system of equations so far is not closed as a second moment appears in the first moment equation, just as the velocity as first-order moment occurs in the zeroth-order continuity equation. It is clear that this problem cannot be solved by adding the second moment of the kinetic equation as a third moment will appear. This is the closure problem of MHD, where at each step, an additional relation will be required to close the system. If we want to stop here, we obviously need a relation for the pressure, that is an equation of state. This could be the adiabatic equation

where is the adiabatic coefficient and we have assumed that we only deal with the scalar pressure in Eq. 1.13. Together with Maxwell's equations for the fields and , we now have indeed a closed system. However, we will still simplify this system for a two-component plasma in Section 1.1.2.

1.1.2 One-Fluid Model of Magnetohydrodynamics

For the case of a two-component plasma consisting of one ion species and electrons, the system of two-fluid equations can be combined to give a set of one-fluid equations. Here, owing to the large mass difference between the two species, the mass and momentum are more or less contained in the ions, whereas the electrons guarantee quasineutrality and lead to an electrical current if their velocity is different from that of the ions. In the following, we will assume a hydrogen plasma, that is charge number . Specifically, the one-fluid variables are the mass density

where we have used charge neutrality (, the centre of mass fluid velocity

and the electrical current density

The one-fluid equations are obtained by adding or subtracting the continuity and force balance equations for the individual species and expressing them in the one-fluid variables, neglecting terms of the order . In this process, addition will give a one-fluid equation for the velocity, whereas the subtraction will yield one for the current density.

Adding the continuity equations yields

that is a one-fluid continuity equation while subtracting them leads to

which is the continuity equation for the electrical current. As we assume the plasma to be quasi-neutral, the electrical charge density vanishes and the equation just reads .

Adding the force equations leads...