Three models of a partially ionised fluid are considered by examining together three sets of (M)HD equations for the neutral, ionised, and electron components of a fluid. The first assumes low ionisation and isothermality leading to the one-fluid, isothermal model where all three non-ideal terms–resistance, the Hall effect, ambipolar diffusion–appear in the induction equation. New quantities introduced include: the ambipolar force density; coupling, rate, and ambipolar coefficients; and resistivity, all helping to determine the relative role of each non-ideal term. For resistive MHD, the Sweet–Parker model for magnetic reconnection, and dynamo theory are discussed. For the Hall effect, a two-fluid, isothermal model is introduced that refines the Sweet–Parker model to give a reconnection time scale in better keeping with observations of solar flares. Finally, the section on ambipolar diffusion derives the full two-fluid, non-isothermal model applicable for a fluid with arbitrary ionisation. Here, exchange terms are introduced to account for mass, momentum, and energy transfers when neutrals ionise or ions recombine.

This chapter looks at four important fluid instabilities – Kelvin–Helmholtz, Rayleigh–Taylor, magneto-rotational, and Parker–where normal mode analysis of the lin-earised equations is taught using each instability as an exemplar. All are examined from the linear regime in which conditions for instability and rates of growth of the fastest mode are developed from first principles. For the KHI, RTI, and MRI, numerical simulations are presented which recover the results of linear analysis from the early stages of a non-linear calculation. For the KHI and RTI, numerical simulations well into the non-linear regime are presented where the onset of fluid turbulence is noted. For the MRI, a section describing how it solved the angular momentum transport problem for accretion discs is included. For the Parker instability, an account is given how this purely astrophysical phenomenon explains the clumpy structure of the interstellar medium.

After some historical perspective on the subject, the introduction attempts to define, distinguish, and link in the broadest terms the various areas of physics related to fluid dynamics. These include fluid mechanics, hydrodynamics, gas dynamics, magnetohydrodynamics, and plasma physics. In particular, the link between ordinary hydrodynamics and magnetohydro-dynamics is made, and the approach this text takes in teaching both, namely wave mechanics, is revealed.

This chapter returns to the zero-field limit of MHD replacing the isotropic pressure force density in ideal HD with force densities arising from the viscous stress tensor for viscid HD. As tensor analysis is not a prerequisite for this course, the stress tensor is developed purely from a vector analysis of all stresses applied at a single point in a viscid fluid. This leads to the introduction of bulk and kinetic viscosity in a Newtonian fluid and the identification of ordinary thermal pressure with the trace of the stress tensor. Various flavours of the Navier–Stokes equation are developed including compressible and incompressible forms. The Reynold’s number is introduced as a result of scaling the Navier–Stokes equation which leads to a qualitative discussion on turbulent and laminar flow. Numerous examples are given in which a simplified form of the Navier–Stokes equation can be solved analytically, including plane-parallel flow, open channel flow, Hagen–Poiseuille flow, and Couette flow.

This chapter begins with a formal definition of a fluid (what it means to be a continuum rather than an ensemble of particles) followed by a review of kinetic theory of gases where the connections between pressure and particle momentum and between specific energy (temperature) and average particle kinetic energy are made. A distinction is made between extensive and intensive variables, from which the Theorem of Hydrodynamics is postulated and proven. From this theorem, the basic equations of ideal hydrodynamics (zero-field limit of MHD) are derived including continuity, total energy equation, and the momentum equation. Alternate equations of HD such as the internal energy, pressure, and Euler’s equations are also introduced. The equations of HD are then assembled into two sets–conservative and primitive–with the distinction between the two explained.

In a steady-state, axisymmetric atmosphere surrounding a gravitating point mass, three constants of flow along lines of induction (equivalently, streamlines) are identified, collectively referred to as the Weber–Davis constants. The MHD Bernoulli function, the fourth constant along a line of induction, is derived from examining Euler’s equation in a rotating reference frame, and a link is made between the centrifugal terms and the magnetic terms found in an inertial reference frame. From the four constants, two types of magneto-rotational forces arise which, acting in tandem, can accelerate material from an accretion disc to escape velocities provided the line of induction emerges from the disc at an angle less than 60°. Two astrophysical examples are then described. The first is a quantitative account of Weber and Davis’ model for a stellar wind, including the derivation of specific fluid profiles along a poloidal line of induction. The second looks at how the four constants can arise naturally in an axisymmetric, non-steady-state simulation of an astrophysical jet.

This chapter starts by distinguishing between the primitive and conservative equations of MHD in 1-D, emphasising that the former deal only with continuous flow, whereas the latter admit flow discontinuities. The first application is to MHD waves including Alfvén, slow, fast, and magneto-acoustical waves. An intuitive analogy is given describing what one might experience in an MHD atmosphere when a “thunder clap” occurs. The MHD Rankine–Hugoniot jump conditions for MHD are introduced and solved (using difference theory) revealing tangential/contact/rotational discontinuities and, most importantly, shock waves including slow, intermediate, and fast shocks. In the context of the not strictly hyperbolic nature of the MHD equations, both the entropy and evolutionary conditions are used to determine the physicality and uniqueness of the shock solution. Finally, discussion of MHD shocks includes the special cases of switch-on/off shocks and Euler shocks.