1  Introduction

1.1 What is MHD?

Magnetohydrodynamics (universally abbreviated as MHD) is the study of electrically conducting fluids including liquid metals and plasmas.

Liquid metal applications of MHD

These include geophysics (the Earth’s magnetic dynamo is driven by flow of molten iron) and engineering (pumping, stirring, casting and liquid metal cooling of nuclear reactors).

Plasma is the “fourth state of matter”. When gases are heated to very high temperatures, molecules come apart leading to free electrons and ions. These are charged particles so the plasma conducts electricity. Typically the energy required is high enough that light is emitted.

Sketch of the states of matter.

Figure 1.1: States of matter.

Examples of plasmas

Natural plasmas on Earth include lightning, flames, St Elmo’s fire, the aurora (Northern/Southern lights), and the ionosphere (outer atmosphere). Artificial examples include fluorescent lights or neon signs, not to mention controlled thermonuclear fusion devices.

In fact, 99% of (baryonic) matter in the universe is in the plasma state. This includes the Sun and other stars, the solar wind, interstellar and intergalactic plasmas.

Many of the illustrative examples in this course will be taken from the Sun’s atmosphere [my area of research].

In this course we will derive the fundamental equations of MHD and study their properties and solutions. We will see that the interplay between the fluid velocity \(\ub\) and the magnetic field \(\Bb\) leads to rich dynamical phenomena beyond those of regular fluids.

The name “magnetohydrodynamics” was first used by Alfvén in 1942, although there were some experiments much earlier – for example, Faraday in 1832.

See https://xkcd.com/1851/ for a physicist’s view of MHD…

1.2 What are we aiming to cover?

During the course we will derive and study the following MHD equations: \[ \ddy{\rho}{t} + \nabla\cdot\big(\rho\ub\big) = 0, \tag{1.1}\] \[ \rho\left(\ddy{\ub}{t} + (\ub\cdot\nabla)\ub\right) = -\nabla p + \textcolor{red}{\Jb\times\Bb} \tag{1.2}\] \[ \ddy{p}{t} + \ub\cdot\nabla p + \gamma p\nabla\cdot\ub = 0, \tag{1.3}\] \[ \textcolor{red}{\ddy{\Bb}{t} = \nabla\times\big(\ub \times\Bb\big) + \eta\Delta\Bb} \tag{1.4}\] \[ \textcolor{red}{\nabla\times\Bb = \mu_0\Jb} \tag{1.5}\] \[ \textcolor{red}{\nabla\cdot\Bb = 0}. \tag{1.6}\] From Fluid Mechanics III, you should recognise the density \(\rho(\xb,t)\), the velocity field \(\ub(\xb,t)\) and the pressure \(p(\xb, t)\). The new variables are magnetic field \(\textcolor{red}{\Bb(\xb,t)}\) and current density \(\textcolor{red}{\Jb(\xb,t)}\).

Consider these equations in turn:

  • Equation 1.1 is the familiar (mass) continuity equation, which is the local expression of mass conservation.
    • Recall that integrating Equation 1.1 over a fixed volume gives \(\displaystyle \ddt{}\int_V\rho\dV = -\oint_{\partial V}\rho\ub\cdot\dS\).
  • Equation 1.2 is the Euler equation for an ideal fluid, which is just Newton’s Second Law applied to a fluid parcel. The left-hand side is mass (per unit volume) times acceleration (moving with the fluid). The right hand side is the force per unit volume; a conducting fluid feels an additional Lorentz force, \(\fb=\Jb\times\Bb\).
    • Recall that the left-hand side of Equation 1.2 may be written as \(\displaystyle \rho\DDt{\ub}\), where \[ \DDt{f} \equiv \ddy{f}{t} + \big(\ub\cdot\nabla\big)f \] is the material derivative: the rate of change of \(f\) seen by an observer moving with the fluid.

For an ideal fluid as in Equation 1.2, the appropriate boundary condition on a solid boundary is \(\nb\cdot\ub=0\).

In this course we will neglect gravity and viscosity but in the presence of both we would replace Equation 1.2 by the Navier-Stokes equation \[ \rho\left(\ddy{\ub}{t} + (\ub\cdot\nabla)\ub\right) = -\nabla p + \Jb\times\Bb + \rho\gb + \frac{\mu}{3}\nabla\big(\nabla\cdot\ub\big) + \mu\Delta\ub. \] In the presence of viscosity, the appropriate condition on a solid boundary becomes the no-slip condition \(\ub=\bfzero\).

  • Equation 1.3 is the adiabatic energy equation, where \(\gamma\) is a constant adiabatic index. For compressible plasmas, this is typically a better approximation than the simpler barotropic fluids \(p=P(\rho)\) studied in Fluid Mechanics III.

    • Recall that an incompressible fluid (e.g. a liquid) would have \(\rho=\rho_0\) constant, so Equation 1.1 would imply \(\nabla\cdot\ub=0\). But in compressible fluids, \(\rho\) generally varies in space and time.

  • Equation 1.4 is the induction equation that describes the change in \(\Bb\) over time.

  • Equation 1.5 is the differential form of Ampère’s Law relating \(\Jb\) to \(\Bb\).

  • The solenoidal condition Equation 1.6 is a physical constraint on \(\Bb\) rather than an evolution equation per se. It will allow us to simplify the mathematics.

The plan of this term is roughly as follows:

  1. Introduction – this lecture.
  2. Magnetic fields – introduce Equation 1.6 and Equation 1.5 and the basic mathematics of magnetic fields.
  3. Kinematics – derive Equation 1.4 and study its behaviour when \(\ub\) is prescribed.
  4. Statics – introduce the Lorentz force and study static equilibria of Equation 1.2.
  5. Dynamics – study the full equations including Equation 1.2 and Equation 1.3; focus on energy conservation and linear MHD.