Abstract
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.
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Notes
- 1.
MHD is sometimes called hydromagnetics, less frequently but rather more comprehensively magneto-fluid-dynamics, and occasionally magnetogasdynamics in the context of dense partially ionised gases.
- 2.
Since \(\mathbf E\) and \(\mathbf B\) appear only as derivatives in (5.16)–(5.18), it might seem that we could add constants to \(\mathbf{E}'\) and \(\mathbf{B}'\), but this possibility is ruled out because the Lorentz force \(\mathbf{E} + \mathbf{v} \varvec{\,\times \,}\mathbf{B}\) on a particle must be invariant.
- 3.
The conducting liquid is often but not always treated as uniform, such that all essential coefficients (of viscosity, resistivity or the Hall term) are taken to be known and constant. A non-uniform liquid with position-dependent coefficients may be considered, to discuss MHD motion in the Earth’s core for example—given of course suitable modifications and additional definitive equations such as \(d \rho /dt=0\) to account for the variable density and the variable coefficients.
- 4.
However, not all plasmas are simple and fully ionised—e.g. in the MHD power generation context, the MHD fluid is a complex mix of partially ionised plasma and burning coal dust!
- 5.
R.L. Dewar (Nuclear Fusion 18, 1541–1553, 1978).
- 6.
This is discussed further in B.F. McMillan, R.L. Dewar and R.G. Storer (Plasma Physics and Controlled Fusion, 46, 1027–1038, 2004).
- 7.
A detailed discussion of case (c) is given in S. R. Hudson, R. L. Dewar, G. Dennis, M. J. Hole, M. McGann, G. von Nessi and S. Lazerson (Physics of Plasmas 19, 112502, 2012) and references therein; and we also note that the “stepped pressure model” discussed there was envisaged much earlier by David Potter (in Methods in Computational Physics 16 edited by John Killeen, pp. 43–83, Academic Press, 1976).
- 8.
This result is amusingly called the “Hairy Ball Theorem,” for it may be visualised as the impossibility of combing finite-length hairs on such a ball flat to its surface everywhere—it must have a bald spot or cowlick.
- 9.
This is essentially the “imagined experiment” described by M.D. Kruskal and R.M. Kulsrud, (Physics of Fluids 1, 265–274, 1958).
- 10.
This is often called “Taylor relaxation”. For a review see J.B. Taylor (Reviews of Modern Physics, 58, 741–763, 1986).
- 11.
The parameter s has been adopted in some important computer codes, and of course should not be confused with our previous usage as a “line length”.
- 12.
We have followed the convention illustrated in Fig. 1 of R. C. Grimm, R. L. Dewar and J. Manickam (Journal of Computational Physics 49, 94–117, 1983). The choice of direction of \({\varvec{\nabla }}\theta \) is not universal in the literature—cf. O. Sauter and S. Yu. Medvedev (Plasma Physics Communications 184, 293–302, 2013). However, it is usual for \(\{{\varvec{\nabla }}s,{\varvec{\nabla }}\theta ,{\varvec{\nabla }}\zeta \}\) to be a right-handed set, so if the angle \(\theta \) is reversed then so too is \(\zeta \)—i.e. to increase in the opposite sense to \(\phi \).
- 13.
Some authors reserve the terminology “magnetic coordinates” for straight-field-line coordinates, referring to the more general coordinates with arbitrary \(\theta \) and \(\zeta \) as “flux coordinates”.
- 14.
The Kruskal=Shafranov condition was earlier identified in stability analysis for simpler cylindrical geometry—cf. Sect. 6.6.
- 15.
By analogy with \(\hbar \) we use the notation \({\;\iota \!\!}\) to denote \(\iota /2\pi \), where \(\iota \) is the increment in poloidal angle in radians after one toroidal circuit.
- 16.
A more general mathematical perspective is provided by the theory of matched asymptotic expansions—cf. [9, pp. 321–342]. This theory gives the leading order “outer region” behaviour—and when more physics is included as in Sect. 5.16, further resolves the discontinuity through an “inner region” expansion on smaller length scales and longer timescales.
- 17.
Dirac delta function behaviour is ruled out for the fundamental fields \(\mathbf{v}\) and \(\mathbf{B}\) on physical grounds (the kinetic and magnetic energies must be finite), such that both of these fields are in \(L^2\)–space. While it is sometimes useful to imagine a delta function mass density for mathematical convenience, there is no physical motivation for this and for simplicity we do not allow for that here.
- 18.
The product of a Heaviside step function and a delta function can usually be interpreted using \(\delta (\cdot )H(\cdot ) = \frac{1}{2}\delta (\cdot )\) that follows from \(\delta (x)f(x)=f(0)\delta (x)\) with \(f(x)=H(x)\) defined in (1.68), although we could work with conservation forms such as (5.41) to avoid having to use this formula.
- 19.
- 20.
The implications of this condition for the existence or otherwise of equilibrium solutions in non-symmetric systems are discussed in M. McGann, S.R. Hudson, R.L. Dewar and G. von Nessi (Physics Letters A, 374, 3308–3314, 2010).
- 21.
Unless the coils are superconductors, this is only an acceptable approximation over times that are short compared with the resistive skin time \(\delta ^{2}/\eta \), where \(\delta \) is the thickness of the conductor and \(\eta \equiv 1/(\mu _{0}\sigma )\) is the resistivity introduced in Sect. 2.12.
- 22.
In passing, we note that this pressure continuity condition does not challenge the traditional notion of plasma confinement by a magnetic field, because any jump in the magnetic field envisaged under the ideal MHD model (when \(\mathbf{j}_*\ne 0\)) corresponds to a continuous but steep variation in the magnetic field under any resistive MHD model—i.e. the surface current in the ideal MHD boundary condition (5.99) occupies a thicker transition layer than that envisaged under condition (5.127), but which is nevertheless less than the macroscopic length scale of any field quantity outside it. The resistive diffusion time is also relatively long in most time-dependent scenarios—e.g. much longer than the time scale of the ideal and non-ideal MHD instabilities considered in the following chapter.
Bibliography
H. Alfvén, Cosmical Electrodynamics (Oxford University Press, Oxford, 1950). (Pioneering work, including a discussion of MHD waves and solar physics)
D. Biskamp, Nonlinear Magnetohydrodynamics (Cambridge University Press, Cambridge, 1997). (Textbook with introductory chapters that may be read in conjunction with our presentation, prior to the discussion of nonlinear processes)
T.G. Cowling, Magnetohydrodynamics (Adam Hilger, Bristol, 1976). (Revision of an early concise text on MHD, with an emphasis on geophysical and astrophysical applications)
P.A. Davidson, An Introduction to Magnetohydrodynamics (Cambridge University Press, Cambridge, 2001). (Introductory textbook on MHD mentioned in the Preface)
J.P. Goedbloed, R. Keppens, S. Poedts, Advanced Magnetohydrodynamics; with Applications to Laboratory and Astrophysical Plasmas (Cambridge University Press, Cambridge, 2010). (Extends “Principles of Magnetohydrodynamics” to deal with streaming and toroidal plasmas, and also nonlinear dynamics)
J.P. Goedbloed, S. Poedts, Principles of Magnetohydrodynamics (Cambridge University Press, Cambridge, 2004). (First referenced in Chapter 2)
R.M. Kulsrud, Plasma Physics for Astrophysicists (Princeton University Press, Princeton, 2004). (The graduate textbook mentioned in the Preface, emphasising physical intuition in developing the analysis with astrophysical applications in mind)
G.E. Marsh, Force-Free Magnetic Fields: Solutions, Topology and Applications (World Scientific, Singapore, 1996). (A brief history of force-free magnetic fields before a discussion of field topology, helicity and multiply connected domains)
P.D. Miller, Applied Asymptotic Analysis (American Mathematical Society, Providence, 2006). (First referenced in Chapter 3)
E.N. Parker, Spontaneous Current Sheets in Magnetic Fields with Applications to Stellar X-Rays (Oxford University Press, Oxford, 1994). (Spontaneous tangential discontinuities (current sheets) in a magnetic field embedded in highly conducting plasma may explain the activity of external magnetic fields in astrophysics and the laboratory)
J.A. Stratton, Electromagnetic Theory (Adams Press, Chicago, 2007). (Reprint of the classic text originally published by McGraw-Hill in 1940, which covers fundamental electromagnetic theory in considerable depth)
J. Wesson, D.J. Campbell, Tokamaks, 3rd edn. (Oxford University Press, Oxford, 2004). (Surveys the important features of tokamak experiments, including equilibrium and confinement, heating, instabilities, surface interactions and diagnostics)
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Hosking, R.J., Dewar, R.L. (2016). Magnetohydrodynamics (MHD). In: Fundamental Fluid Mechanics and Magnetohydrodynamics. Springer, Singapore. https://doi.org/10.1007/978-981-287-600-3_5
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