Hydrodynamic Modelling in Nuclear Fusion

1 - The Fluid Picture Of Plasmas

 

Like ordinary gases, which are also large collections of particles, plasma can be treated like a fluid, although it has significant electrical properties. The equations that govern the fluid-like properties are obtained by taking velocity integrals or moments of the Boltzmann equation, which describes the statistical evolution of a group of particles. The equation is essentially an equation of motion in the phase space. In plasma physics, the relevant version of Boltzmann’s equation is often called the Vlasov equation with Landau collisional term. It takes into account the effects of electromagnetic fields and Coulomb collisions.

Eq. 1

 

Where f is the density of particles in the six dimensional phase space and the term in rhs is a collision operator that describes the effects of Coulomb interactions. q, m, and v are the charge, mass and velocity of the particles under consideration. In many cases in plasma physics, collisions are ignored, though the fluid picture is only valid formally, when collisions are frequent enough to keep the mean free path much smaller than the system size. The zeroth moment of the Vlasov equation corresponds to mass conservation or continuity, the first to momentum conservation or force balance, the second to energy conservation. In simplified form, the first two moments can be written as

Eq. 2

 

Where nm, the fluid mass density, and v is the fluid velocity. Note that the equation for each moment
refers to the next higher moment, the fluid velocity is needed to complete the continuity equation for
example. To be useful, the set of equations must be closed, typically by making simplifying assumptions, like an equation of state. The moment equations combined with Maxwell’s equations for electromagnetics form the basis for the fluid picture of plasmas called magnetohydrodynamics or MHD.

2 - Magnetohydrodynamics (MHD)

Magnetohydrodynamics is the macroscopic fluid theory of plasmas. The equations that govern MHD are Maxwell’s equations and moments of the Vlasov equation. These moments yield separate equations for ions and electrons, which are coupled through the fields generated and through collisions. It is common to use a single-fluid set of equations that can be derived by ignoring electron inertia, assuming that electron motion is fast compared to scales of interest. This is equivalent to restricting our interest to times long compared to the electron cyclotron frequency and plasma frequency. A further simplification, appropriate in many cases, is made by ignoring resistive effects, that is by assuming the plasma is a perfect conductor. The MHD equations are

- Continuity

- Momentum balance

- Equation of state

- Ampere's law

- Faraday's law

- Ohm's law

3 - Equilibrium

 

In the fluid picture, magnetic confinement is achieved through the balance between plasma pressure and the magnetic force. At equilibrium, the time derivatives vanish leaving:

From this equation, it is clear that both J and B must lie on the surfaces of constant p. These surfaces are usually called the flux surfaces and labeled with the enclosed magnetic flux. For a confined plasma, p will be a maximum near the axis and close to zero at the boundary. The current and field are related by Ampere’s law. For the case of a straight cylinder with a radial pressure gradient and no current parallel to B

This is called the diamagnetic current. It arises from the imbalance in gyro-orbiting
particles that is created by the pressure gradient. The pressure profile and parallel current can be considered free parameters in this equilibrium. Note that the magnetic field can be aligned in the z (axial) direction, with theta, or in some combination of the two. The essential pressure balance between the confining field and the plasma can be seen by substituting J from Amperes law, which after some vector algebra yields for the case of straight field lines

 

This relation suggests a definition for the normalized plasma pressure,

 

The importance of plasma pressure in the dynamics of a system are determined by beta.

Practical considerations require that a magnetic confinement device be toroidal. The toroidal geometry adds two complications to the simple equilibrium just considered. First, the magnetic field must have both poloidal and toroidal components to cancel the single particle drifts. Secondly, in addition to radial force balance (from plasma pressure that tries to expand in the r direction),
toroidal force balance must be considered as well. Two forces tend to expand the plasma in the R direction, one current driven and one driven by the plasma pressure. Toroidal current exerts a hoop stress, as a result of the self force between different current elements. The pressure imbalance arises because there is more surface area on the outside (large R) of the torus than the inside. Toroidal balance is achieved by the addition of a vertical magnetic field, which when crossed with the toroidal current produces a compensating force. In toroidal geometry, MHD equilibrium is calculated by the Grad-Shafranov equation.

4 - Stability

 

From the earliest studies into magnetically confined plasmas, researchers recognized that consideration of plasma equilibrium was not sufficient. Experimental plasmas could exhibit violent behavior sometimes losing their stored energy in a few microseconds. Further analysis showed that these plasmas were MHD unstable - like a ball sitting at the top of a hill, they were in a state of unstable equilibrium. Free energy for the instabilities comes from the plasma pressure and current. Pressure-driven modes exhibit “interchange” behavior, parts of the fluid moving toward the high pressure region while other parts move away. This phenomenon is analogous to the Rayleigh-Taylor instability that occurs when a glass of water is inverted. Current-driven instabilities often take the form of a kink. The plasma tries to twist itself into a corkscrew shape. Fortunately there are stabilizing forces that can come into play as the plasma moves away from equilibrium. Since the plasma is tied to magnetic field lines, these must be bent or compressed if the plasma is to move. Both processes require energy input and are thus stabilizing. Magnetic field curvature can be either
stabilizing, when the pressure gradient is away from the center of curvature or destabilizing otherwise. MHD stability is calculated by analyzing the effect of an infinitesimal displacement of the plasma. Destabilizing and stabilizing forces are summed up and found to move the plasma toward or away from equilibrium. Ideal MHD instabilities propagate at the Alfven velocity, which is on the order of 107 m/sec for a fusion plasma, and therefore they must be avoided.