Fundamental Fluid Mechanics and Magnetohydrodynamics

Vector and dyadic (second-order tensor) fields are basic entities in Fluid Mechanics and MHD. The term “field” in this book means that the quantity is a function of position in three-dimensional space, in addition to any time dependence. This spatial dependence is conveniently represented by the associated position vector \(\mathbf{r}\) that may itself be a function of the time t (in a dynamical system), so any field is an invariant function of the form \({\varvec{\mathsf f}}(\mathbf{r}(t), t)\). On the other hand, vector and tensor representations depend upon the reference coordinate system chosen, and we discuss the representation of vectors as a useful prelude to the following sections on dyadics and their representation. The vector differential operator then introduced is used more extensively, from determining basis sets for coordinate systems to its role in so many mathematical expressions throughout the book. Although there is a brief section on the familiar special case of orthogonal curvilinear coordinates, it is notable that the previous and following sections on the integral theorems apply to any three-dimensional coordinate system (non-orthogonal curvilinear coordinate systems are particularly important in MHD). The remaining three sections on Green identities and Heaviside, Dirac and Green functions complete this chapter on relevant mathematical topics. The associated bibliography provides recommended sources on vectors and tensors, on distribution theory and mathematical methods, and on partial differential equations for further reading.

Conservation equations are the foundation for Fluid Mechanics and MHD, but others are needed to close the mathematical models. Although fluid pressure was at first assumed to be isotropic, when viscous stress was considered early in the eighteenth century it was evident that the assumption of incompressibility or the inclusion of a simple equation of state was no longer sufficient. The classical macroscopic equations (for mass, momentum and energy) follow from underlying microscopic theory, which also provides the relevant pressure tensor to incorporate viscosity. Except near magnetic null points, the pressure tensor for a plasma in a magnetic field is found to differ significantly from the classical shear viscosity form for a neutral fluid. There is also a brief introduction to the additional equation of magnetic induction required in MHD. The bibliography includes some references that provide further background to our presentation in this chapter, a worthy source on thermodynamics, and two books by Lamb and Prager particularly recommended for supplementary reading (more books on Fluid Mechanics are listed for Chap. 3).

Although the classical ideal fluid model entirely neglects fluid viscosity, it nevertheless describes some features in certain realistic flows or flow regions, and it is often applicable to wave motion as discussed in the next chapter. The inherent nonlinearity of this ideal model was addressed in remarkable ways by many famous mathematicians, who developed various concepts and results that still remain important. First integrals of the inviscid equation of motion (known as Bernoulli equations), aspects of vorticity, and potential theory for incompressible irrotational flow are landmarks of the classical ideal theory. An ordering procedure establishes that the incompressibility assumption applies in any subsonic flow, and confirms the relevant Bernoulli equation for the pressure variation in the ideal model. We then observe that the shear viscosity (whether large or small) must be included to account for the drag and enhanced vorticity in flow past an obstacle, and that perturbation or numerical methods are usually required since exact viscous solutions are rare. The chapter concludes with an optional (starred) section as an introduction to some simplified equations of motion in dynamical meteorology and oceanography, with some references for further reading. Some other notable fluid mechanics textbooks and several related sources (on asymptotic and perturbation methods, potential theory and hydrodynamic stability) are listed in the bibliography for this chapter.

Sound and water waves are familiar longitudinal and transverse disturbances relative to the direction of propagation in a fluid, respectively. Sound waves arise in a compressible fluid, but water (gravity) waves are well described in the subsonic incompressible approximation. Our main emphasis in this chapter is on linear wave theory, where the disturbances from an equilibrium or steady state are assumed small and solutions may be obtained by superposition as Fourier series or integral forms. Our analysis is extended to superposed fluids, where hydrodynamic instability may occur—and the other topics chosen either consolidate earlier concepts or are relevant to developments in the following two chapters. The additional bibliographic entries at the end of this chapter provide relevant supplementary reading.

We mentioned in the Preface that Fluid Mechanics and MHD often draw upon much the same mathematics and yield many closely related results. The mathematical kinship of the fundamental mathematical models was quite evident in Chap. 2, where novelties for MHD nevertheless emerged—viz. additional terms arising in the macroscopic equations (notably the Lorentz force in the equation of motion), the distinctive anisotropic plasma pressure tensor due to a magnetic field and the necessity to invoke suitable electromagnetic equations. This chapter explores the origin of the ideal and non-ideal MHD models briefly mentioned there, and then various important topics in MHD that are often prerequisite for our subsequent discussion of MHD stability theory. The additional bibliography for this chapter provides further background reading.

As in the earlier discussion on waves, disturbances from an initial equilibrium are usually assumed small enough to justify linearisation of the perturbed equations in the mathematical model, as an important first step to define the evolution of a complex system in this chapter. Normal mode analysis involving Fourier forms for the disturbance is therefore often invoked when the initial state is one-dimensional (dependent on only one spatial variable), except that the time exponent is assumed to carry a real part, often with the additional imaginary part (when there is an accompanying oscillation). The sign of the real part then determines whether or not the disturbance grows or decays exponentially—i.e. whether or not the system is linearly unstable or stable, respectively. Identification of the stabilising and destabilising forces is particularly important in MHD stability analysis, in both laboratory and astrophysical applications. In the case of the ideal MHD model, a variational formulation (an energy principle) permits the analysis of more complicated geometries, such as in modern experiments in controlled thermonuclear fusion research. Important stabilising and destabilising forces are identified under this formulation, which is then applied to investigate the stability of cylindrical and toroidal configurations. More direct analysis is usually followed to investigate instability in non-ideal MHD models, but we demonstrate an extension of the variational formulation that includes viscosity. Although the main application we consider is magnetic confinement in thermonuclear research, the final section on Hall instability includes an aspect relevant to laser-driven fusion, and some topics of interest in astrophysics and solar physics are discussed there and elsewhere. The additional bibliography for this chapter once again provides suggested further reading.