Magnetohydrodynamics

(MHD)

&

Plasma Kinetic Theory Dr. D. N. Gupta

Faculty in ScienceDepartment of Physics and

AstrophysicsUniversity of Delhi

Delhi-7, India

There are three basic theories to understand the plasma dynamics:

Fluid TheoryKinetic TheoryMagnetohydrodynamics Theory

Fluid TheoryFluid Theory

Fluid models describe plasmas in terms of smoothed quantities like density and averaged velocity around each position.A more general description is the two-fluid picture, where the electrons and ions are described separately. Fluid models are often accurate when collisionality is sufficient high to keep the plasma velocity distribution close to the Boltzmann distributionBecause fluid models usually describe the plasma in terms of a single flow at a certain temperature at each spatial location, they can neither capture velocity space structure like beams.Easy to understand and implementationApplication to understand the nonlinear characteristic of plasmas such as plasma oscillations, self-focusing, harmonic generation, and high-energy physics

Kinetic TheoryKinetic Theory

Kinetic models describe the particle velocity distribution function at each point in the plasma, and therefore do not need to assume a Boltzmann distribution

A kinetic description is often necessary for collisional plasmas

PIC (particle-in-cell) technique is known to describe the kinetic theory, which includes the kinetic information by following the trajectories of a large number of individual particles.

Kinetic models are generally more computationally intensive than fluid models. The Vlasov equations may be used to describe the dynamics of a system of charged particles interacting with an electromagnetic field.

Magnetohydrodynamics TheoryMagnetohydrodynamics TheoryIntroduction

This theory has been used to explain the spontaneous generation and subsequent evolution of magnetic field within stellar and planetary interiors, to accounting for the gross stability of magnetically confined thermonuclear plasma.

Often, it has been said that the scale length of many instabilities and waves which are able to grow or propagate in a system are comparable with the plasma size. This shows that MHD is capable of providing a good description of such large scale disturbance, indicating that the MHD account of plasma behavior is necessarily a macroscopic one.

In essence, the theory is a connection between fluid mechanics and electromagnetism.Compare to fluid and kinetic theory, MHD is a relatively simple but not accurate.

MHD-Significance in fusion plasma

Lawson Parameter

The role of macroscopic and microscopic theory

Hence, the particle density tends to be influenced by the macroscopic properties of the plasma

The general problems of particular importance to which answers may be sought using MHD are:

(a) Finding magnetic field configurations capable of confining a

plasma in equilibrium (b) The linear stability properties of such equilibrium (c) The nonlinear development of instabilities and their consequence

Why MHD?

Of three levels of plasma description-Vlasov, two-fluid, and MHD- Vlasov is the most accurate and MHD is the least accurate. So why use MHD?

The answer is that because MHD is a more macroscopic point of view, it is more efficient to use MHD in situation where the greater detail and accuracy of the Vlasov or two-fluid theory are unnecessary.

MHD is particular suitable for situation having complex geometry because it is very difficult to model such situation using the microscopically oriented Vlasov or two-fluid approaches and because geometrically complexities are often most important at the MHD level of description.

The equilibrium and gross stability of three-dimension, finite extent plasma configurations are typically analyzed using MHD.

Fluid and kinetic theories are more accurate and reliable but these more suitable questions can be addressed after an approximation understanding has first been achieved using MHD.

MHD is especially relevant to situations where magnetic forces are used to confine or accelerate the plasma.

For example magnetic fusion confinement plasma, solar and astrophysical plasma, planetary and stellar dynamics and arc.

In fact, the MHD description is actually more appropriate and more accurate for macroscopic plasma than it is for ordinary plasma.

What is MHD?

MHD provides a macroscopic description of an electrically conducting fluid in the presence of magnetic fields.

The separate identities of the positively and negatively charged species do not feature in the formulation, so the conducting medium is a continuum or fluid through which the magnetic field lines are exist. This field may be externally applied, produced by current flowing in the fluid or a combination of both

MHD-Properties

MHD approximation for Maxwell’s equations (a) Assumption of charge neutrality in MHD

(b) Displacement current for MHD (high frequency term will be neglected)

MHD is restricted to phenomena having characteristic velocity slow compared to the speed of light in vacuum.

Such a single-fluid description will only be valid provided the plasma is collision dominated. The MHD time scale must be sufficient long for there to be adequately many collisions between particles. The MHD time scale will certainly be much larger than the time required for light to traverse the plasma. This allows the displacement current to be negligible from Maxwell’s equations.

MHD process has the conservation properties following the fluid mechanics and electromagnetism, namely:

(a) Conservation of mass

(b) Conservation of momentum

(c) Conservation of energy

(d) Conservation of magnetic flux

There are three waves which are capable of propagation in plasma via MHD:

(a) Sound wave (caused by vibration of fluid)

(b) Alfven wave (due to the vibration of magnetic field in the presence of fluid) (c) Magnetosonic wave (a coupling of above)

These are three routes via which energy may be transferred in MHD.

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In fluid theory, the relevant dependent variables, such as density, fluid velocity and pressure, are function of distance and time only. Hence, the average velocity distribution is assumed everywhere.The collisions between the plasma particle are usually sufficient frequent to maintain the average particle distribution.In high-temperature plasmas, however, deviations from local thermodynamics equilibrium can be maintained for long time.Hence, fluid theory can be applied by considering the average plasma parameters. The fluid theory is valid only if collisions are frequent enough (specially, the mean-free path is much shorter than some characteristic distance along the magnetic field). In the case where there are no collisions (very less collisions), the individual particles making up the plasma will freely stream for large distances along the field. To treat such problems, we need a kinetic theory in which individual particles velocities are taken into account.

The need for a kinetic theoryThe need for a kinetic theory

Such a theory will be needed to treat problems involving flow across a magnetic field in the case where the magnetic field is very weak, in the sense that the gyration period and gyration radius are not small compared with the characteristic time-scale and length-scale of the flow.In summary, kinetic theory is needed to treat (a) problem involving flow along a magnetic field (or in the absence of a magnetic field),

(b) problem of high-frequency ( ), flow across a

magnetic field (c) wave damping is due to wave-particles resonance, to treat

this we need to keep track of the particle distribution in velocity space.

c

The particle distribution function

The basic element in the kinetic description of a plasma is the distribution function that describes how particles are distributed in both physical space and velocity space.

The particle distribution function represents the number density of particles found near a point in the six dimensional space