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.

In this chapter, everything is brought together to solve the MHD Riemann problem, the most general 1-D MHD problem one can solve semi-analytically. Non-linear waves are introduced in which the 1-D primitive equations are neither steady-state nor linearised. The fast and slow eigenkets are evaluated and their normalisation to account for the not-strictly hyperbolic nature of MHD is emphasised. A method to determine profiles of the primitive variables across slow and fast rarefaction fans is described, including Euler, switch-on, and switch-off fans. A strategy for solving the MHD Riemann problem follows, including use of a multi-variate secant root finder, sixth- order Runge–Kutta, and inverting a 5 × 5 Jacobian matrix with emphasis on characteristic degeneracy and matrix singularity. The chapter concludes with an explicit algorithm for an MHD Riemann solver including numerous examples using a solver developed by the author.

This chapter introduces the magnetic induction, B̅, to fluid dynamics. After a brief introduction establishing the ubiquity of magnetism in the universe, the ideal induction equation is derived from the idea of electromagnetic force balance and Faraday’s law. By proposing and proving the flux theorem, Alfvén’s theorem is proven to show that in an ideal MHD fluid, magnetic flux is conserved and frozen-in to the fluid. It is further shown how the introduction of Bti introduces the Lorentz force density to the momentum equation and the Poynting power density to the energy equation. Two variations of the equations of MHD are assembled, both involving the conservative variables. Finally, the vector potential, magnetic helicity, and magnetic topology are introduced in an optional section where the link to solar coronal flux loops is made.

By linearising the equations of HD developed in Chapter 1, the wave equation for the propagation of sound is derived. This is examined from two approaches: direct solution of the wave equation and examining normal modes to convert the problem to one of linear algebra. This introduces the very important concepts of eigenvalues (characteristic speeds) and eigenkets (right eigenvectors) along with the role they play in examining fluid dynamics in terms of waves. From the 1-D, non-linearised, steady-state equations, the Rankine–Hugoniot jump conditions are derived from which the conditions for tangential/contact discontinuities and shocks are developed. An optional section considers the phenomenon of bores and hydraulic jumps, while the last section introduces concepts such as streamlines and stream tubes culminating with Bernoulli’s theorem applied to an incompressible fluid, a subsonic compressible fluid, and a supersonic compressible fluid.

This chapter serves as the “practice chapter” for the main goal of Part I: solving the MHD Riemann problem. Lagrangian and Eulerian frames of reference are introduced from which the three Riemann invariants of HD are identified. Space-time diagrams are introduced as a useful visual and conceptual aid in understanding the role of characteristic paths through a continuum, which is in keeping with the text’s underlying approach of treating fluid dynamics as a form of wave mechanics. The Riemann problem for HD is defined and a method of characteristics is introduced whose main purpose is to understand qualitatively how the solution to the HD Riemann problem begins to unfold. In so doing, shocks and contact discontinuities are rediscovered and rarefaction fans are introduced. It is shown how examining the eigenkets leads to profiles of the primitive variables across a rarefaction fan which ultimately leads to a semi-analytic solution to the HD Riemann problem.