4h magnetohydrodynamics 1. introduction

4H MAGNETOHYDRODYNAMICS 1. INTRODUCTION What is magnetohydrodynamics? Magnetohydrodynamics (or MHD for short) is the fluid mechanics of electrically conducting fluids. These include liquid metals (such as mercury, gallium, sodium or molten iron) and ionised gases (sometimes called plasmas) such as the Solar atmosphere. Note that not all phenomena observed in plasmas can be described by a fluid theory; it may be necessary to consider individual particles, especially in very low density plasmas. Time will not permit us to look at such effects in this course.

Applications MHD has applications in many areas. A few brief details are given below. For much more information, see the websites linked on http://www.maths.gla.ac.uk/∼drf/courses/mhd/websites.htm The Earth The outer core of the Earth is composed primarily of molten iron. It is here that it is believed that the Earth’s magnetic field is generated. Studying and solving the equations of MHD should permit us to explain such phenomena as the gradual change of the field with time and the infrequent and irregular reversals of the field. This is an area of very active current research. MHD can also be used to describe the ionosphere. The Sun Much of the Sun is composed of ionised hydrogen. For MHD there are two areas of interest. First there is the convection zone. In this, or just below it, the Solar magnetic field is generated. The basic mechanism (interaction of a moving electrically conducting fluid with a magnetic field) is similar to that operating in the Earth’s core but results in a rather different magnetic field. The Solar field reverses regularly on a 22 year cycle. Second, the Solar atmosphere (chromosphere and corona) is much less dense than the convection zone. Here, features such as flares and prominences can be observed and studied. One of the major problems to be explained is the heating of the corona which reaches temperatures of up to 106 K while the photosphere (the narrow region separating the convection zone from the chromosphere) is only at a few thousand degrees K. Industry Here there are many applications. For example electromagnetic forces can be used to pump liquid metals (eg. in cooling systems of nuclear power stations) without the need for any moving parts. They can shape the flow of a molten metal and so aid controlling its shape once solidified, and can even levitate and heat a sample of metal to prevent any contact with (and consequent contamination from) a container. 1

4H MAGNETOHYDRODYNAMICS - Introduction

Fusion The goal of copying the Sun; releasing huge quantities of energy from the fusing of hydrogen into helium, has so far eluded us. No material can withstand the huge temperatures required. One promising way around this problem is to contain the ionised hydrogen in a magnetic container, so that there is no contact between the hydrogen and any material container. Progress continues but so far the temperatures and containment times achieved have fallen short of break-even where the energy put in to the system equals the energy given out from fusion.

Equations Governing MHD Maxwell’s Equations To describe an electromagnetic field, we use a number of variables: B – formally the magnetic flux density but we shall refer to it as the magnetic field H – the magnetic field strength E – the electric field D – the electric displacement j – the electric current density ρc – the electric charge density These are related through Maxwell’s equations, the equations governing the evolution of electric and magnetic fields: ∇ · D = ρc , ∂B , ∇×E = − ∂t ∇·B = 0 , ∂D ∇×H = j + , ∂t

(1.1) (1.2) (1.3) (1.4)

where, in an isotropic medium (which we shall assume) D = E ,

B = µH ,

(1.5, 6)

where is the permittivity (or dielectric constant) and µ is the magnetic permeability of the medium. For our purposes and µ can be approximated by their values in a vacuum: 0 = 8.854 × 10−12 Fm−1 , µ0 = 4π × 10−7 Hm−1 . The speed of light c = (µ0 0 )−1/2 = 2.998 × 108 ms−1 .

The Navier-Stokes equation 2


The equation governing the flow of a fluid is the Navier-Stokes (or momentum) equation ∂u ρ + (u · ∇)u = −∇p + ρν∇2 u + other forces, ∂t where ρ is the fluid density, u its velocity, p the pressure, and ν the kinematic viscosity. When the fluid contains electrical charge ρc per unit volume, then there is a force per unit volume of ρc E . When an electric current density j flows through the fluid, there is a force per unit volume of j × B. Then, the Navier-Stokes equation becomes ∂u ρ + (u · ∇)u = −∇p + ρν∇2 u + ρc E + j × B + other forces. ∂t

(1.7)

Ohm’s Law Ohm’s Law asserts that the total electric current flowing in a conductor is proportional to the total electric field. In addition to the field E acting on a fluid at rest, a fluid moving with velocity u in the presence of a magnetic field B is subject to an additional electric field u × B. Ohm’s law then gives j = σ(E + u × B) ,

(1.8)

where the constant of proportionality σ is called the electrical conductivity.

Simplifications - The Magnetohydrodynamic Approximation The equations we have introduced above are capable of describing a great range of phenomena. One example of course is electromagnetic waves. The wave equation that describes these is derived from Maxwell’s equations. In the applications that we are interested in here, speeds are very small compared with the speed of light c. We can make use of this to introduce substantial simplifications to the governing equations. To demonstrate this, let U be a typical fluid speed, T a typical time scale, L a typical length scale (with U = L/T ), B a typical magnetic field strength and E a typical strength of the electric field. Equation (1.2) then requires that B E ∼ L T

E L ∼ = U B T

⇒ 3

(1.9)


Let us then consider equation (1.4). |∂D/∂t| D/T 0 E/T EL 1 L2 U2 ∼ = = 0 µ0 ∼ 2 2 = 2 . |∇×H| H/L B/µ0 L BT c T c For speeds small compared with that of light (i.e. for U c), the displacement current ∂D/∂t can therefore be neglected in (1.4), which becomes ∇×H = j.

(1.10)

Let us now consider the forces in (1.7). From (1.1) |ρc | ∼

D , L

(1.11)

|j| ∼

H . L

(1.12)

and from (1.10)

Then, making use of (1.9), (1.11) and (1.12) |ρc E| (D/L)E 0 E 2 E2 U2 ∼ = = µ ∼ . 0 0 2 |j × B| (H/L)B B 2 /µ0 B c2 Hence the electric force ρc E can be neglected compared with the magnetic force (usually called the Lorentz force) j × B in (1.7), giving ρ

∂u + (u · ∇)u ∂t

= −∇p + ρν∇2 u

+ j × B + other forces.

(1.13)

The Magnetic Induction Equation Having made the simplifications above, we can eliminate E completely from the problem by combining (1.2), (1.3), (1.8) and (1.10). Using B = µ0 H and Ohm’s law (1.8), (1.10) becomes 1 ∇×B = σ(E + u × B) , µ0 or η∇×B = (E + u × B) , where

1 , µ0 σ

η = 4

(1.14)

(1.15)


Taking ∇×(1.14):

∇×(η∇×B) = ∇×(E + u × B) .

Assuming η to be constant, and using a standard vector identity, this becomes η ∇(∇·B) − ∇2 B = ∇×E + ∇×(u × B) .

Using (1.2) and (1.3), this becomes

−η∇2 B = −

∂B + ∇×(u × B) , ∂t

or

∂B = ∇×(u × B) + η∇2 B . ∂t

(1.16)

where η is called the magnetic diffusivity. Equation (1.16) is known as the magnetic induction equation. It describes the evolution of the magnetic field B. The first term on the right-hand side is the induction term that describes the interaction of the field with the flow u. It is the only term that can generate field. The second term on the right-hand side is a diffusive term. In the absence of a flow u, we will show later that the diffusive term leads to a decay of the field.

Mass Continuity Mass can neither be created or destroyed, so the density ρ can only be changed by the redistribution of matter. Consider a volume V (not moving with the fluid) enclosed by a surface S with unit normal vector n.

The rate of decrease of mass in V must equal the rate of flow of mass out of V , i.e. Z Z ∂ − ρ dV = ρ u · n dS . ∂t V S 5


Using the divergence theorem, this becomes Z Z ∂ − ρ dV = (∇·ρu) dV . ∂t V V Since the volume V is fixed, this implies that Z ∂ρ + ∇·ρu dV = 0 . ∂t V

(1.17)

Further, since the volume V is arbitrary, (1.17) can only be true if ∂ρ + ∇·ρu = 0 . ∂t

(1.18)

holds everywhere.

Closure Equations (1.13), (1.16) and (1.18) form the basis for describing the flow of an electrically conducting fluid. Together they provide 7 scalar equations in the 8 unknowns B, u (3 scalar components each), p and ρ. We therefore require an additional equation to close the system. (More than one will be required if they introduce further unknowns.) We shall consider two cases:

Incompressible Flow Here, the extra equation is simply ρ = const. Then, (1.18) reduces to ∇·u = 0 .

(1.19)

Compressible Flow Here, in general, p = p(ρ, T ) where T is the temperature, so more than one extra equation is usually required because temperature is introduced as a new variable. In some circumstances, however, p = p(ρ) is a good approximation. Examples of this are: p = const. ρ

the isothermal gas law,

(1.20)

p = const. ργ

the adiabatic gas law,

(1.21)

and where γ is the ratio of specific heats. Much more detail can be found in Chapter 2 of Priest. 6


Summary Our system of an electrically conducting fluid moving with velocity u(x, t) through a magnetic field B(x, t) is described (in the limit |u| c) by ρ

∂u + (u · ∇)u ∂t

= −∇p + ρν∇2 u +

1 (∇×B) × B + other forces. (1.22a) µ0

∂B = ∇×(u × B) + η∇2 B . ∂t

(1.22b)

∇·u = 0 .

(1.22c)

and either for an incompressible fluid, or ∂ρ + ∇·ρu = 0 , ∂t

and some gas law ,

(1.22d)

for a compressible fluid.

Magnetic Field Visualisation Magnetic Field Lines A magnetic field line is a line drawn such that the tangent to the line at any point on the line is in the direction of the magnetic field B. In Cartesian coordinates, field lines are given by dx dy dz = = . (1.23) Bx By Bz Example 1.1 Sketch the field lines for the field B = B0 (y, x, 0) where B0 is a constant, taking care to indicate the direction of the field. ♦

Magnetic Flux The magnetic flux F crossing a surface S is defined to be Z F = B · n dS , S

where n is the unit normal to S.

7

(1.24)


Magnetic Flux Tubes A magnetic flux tube is the volume enclosed by a set of field lines which intersect a simple closed curve.

The flux through any cross-section S of a flux tube is independent of the choice of S.

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