4h magnetohydrodynamics

4H MAGNETOHYDRODYNAMICS

1. INTRODUCTION

What is magnetohydrodynamics?

Magnetohydrodynamics (or MHD for short) is the fluid mechanics of electrically con-ducting fluids. These include liquid metals (such as mercury, gallium, sodium ormolten iron) and ionised gases (sometimes called plasmas) such as the Solar atmo-sphere.

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 densityplasmas. 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 muchmore 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 herethat it is believed that the Earths magnetic field is generated. Studying and solving theequations of MHD should permit us to explain such phenomena as the gradual change ofthe field with time and the infrequent and irregular reversals of the field. This is an areaof 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 ar-eas of interest. First there is the convection zone. In this, or just below it, the Solar mag-netic field is generated. The basic mechanism (interaction of a moving electrically con-ducting fluid with a magnetic field) is similar to that operating in the Earths core butresults in a rather different magnetic field. The Solar field reverses regularly on a 22 yearcycle. Second, the Solar atmosphere (chromosphere and corona) is much less dense thanthe convection zone. Here, features such as flares and prominences can be observed andstudied. One of the major problems to be explained is the heating of the corona whichreaches temperatures of up to 106K while the photosphere (the narrow region separatingthe convection zone from the chromosphere) is only at a few thousand degrees K.

Industry Here there are many applications. For example electromagnetic forces can beused to pump liquid metals (eg. in cooling systems of nuclear power stations) without theneed for any moving parts. They can shape the flow of a molten metal and so aid control-ling its shape once solidified, and can even levitate and heat a sample of metal to preventany contact with (and consequent contamination from) a container.

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4H MAGNETOHYDRODYNAMICS - Introduction

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

Equations Governing MHD

Maxwells EquationsTo describe an electromagnetic field, we use a number of variables:

B formally the magnetic flux density but we shall refer to it as the mag-netic field

H the magnetic field strengthE the electric fieldD the electric displacementj the electric current densityc the electric charge density

These are related through Maxwells equations, the equations governing the evolutionof electric and magnetic fields:

D = c , (1.1)

E = Bt

, (1.2)

B = 0 , (1.3)

H = j + Dt

, (1.4)

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

D = E , B = H , (1.5, 6)

where is the permittivity (or dielectric constant) and is the magnetic permeabilityof the medium. For our purposes and can be approximated by their values in avacuum:

0 = 8.854 1012Fm1, 0 = 4 107Hm1 .

The speed of light c = (00)1/2 = 2.998 108ms1.

The Navier-Stokes equation

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The equation governing the flow of a fluid is the Navier-Stokes (or momentum) equa-tion

(ut

+ (u )u)

= p + 2u + other forces,

where is the fluid density, u its velocity, p the pressure, and the kinematic viscos-ity.

When the fluid contains electrical charge c per unit volume, then there is a force perunit volume of

cE .

When an electric current density j flows through the fluid, there is a force per unitvolume of

j B .

Then, the Navier-Stokes equation becomes

(ut

+ (u )u)

= p + 2u + cE + jB + other forces. (1.7)

Ohms LawOhms Law asserts that the total electric current flowing in a conductor is propor-tional 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 anadditional electric field uB. Ohms law then gives

j = (E + uB) , (1.8)

where the constant of proportionality is called the electrical conductivity.

Simplifications - The Magnetohydrodynamic ApproximationThe equations we have introduced above are capable of describing a great range ofphenomena. One example of course is electromagnetic waves. The wave equation thatdescribes these is derived from Maxwells equations.

In the applications that we are interested in here, speeds are very small comparedwith the speed of light c. We can make use of this to introduce substantial simplifica-tions to the governing equations.

To demonstrate this, let U be a typical fluid speed, T a typical time scale, L a typi-cal length scale (with U = L/T ), B a typical magnetic field strength and E a typicalstrength of the electric field. Equation (1.2) then requires that

E

L B

T E

B L

T= U (1.9)

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Let us then consider equation (1.4).

|D/t||H|

D/TH/L

=0E/T

B/0L= 00

E

B

L

T 1

c2L2

T 2=

U2

c2.

For speeds small compared with that of light (i.e. for U c), the displacement cur-rent 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)

|c| D

L, (1.11)

and from (1.10)

|j| HL. (1.12)

Then, making use of (1.9), (1.11) and (1.12)

|cE||jB|

(D/L)E(H/L)B

=0E

2

B2/0= 00

E2

B2 U

2

c2.

Hence the electric force cE can be neglected compared with the magnetic force (usu-ally called the Lorentz force) jB in (1.7), giving

(ut

+ (u )u)

= p + 2u + jB + other forces. (1.13)

The Magnetic Induction Equation

Having made the simplifications above, we can eliminate E completely from the prob-lem by combining (1.2), (1.3), (1.8) and (1.10).

Using B = 0H and Ohms law (1.8), (1.10) becomes

10B = (E + uB) ,

orB = (E + uB) , (1.14)

where =

10

, (1.15)

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Taking (1.14):(B) = (E + uB) .

Assuming to be constant, and using a standard vector identity, this becomes

((B) 2B

)= E +(uB) .

Using (1.2) and (1.3), this becomes

2B = Bt

+(uB) ,

or

Bt

= (uB) + 2B . (1.16)

where is called the magnetic diffusivity.

Equation (1.16) is known as the magnetic induction equation. It describes the evolu-tion of the magnetic field B. The first term on the right-hand side is the inductionterm that describes the interaction of the field with the flow u. It is the only termthat 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 decayof the field.

Mass ContinuityMass can neither be created or destroyed, so the density can only be changed bythe redistribution of matter.

Consider a volume V (not moving with the fluid) enclosed by a surface S with unitnormal vector n.

The rate of decrease of mass in V must equal the rate of flow of mass out of V , i.e.

t

V

dV =S

u n dS .

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Using the divergence theorem, this becomes

t

V

dV =V

(u) dV .

Since the volume V is fixed, this implies thatV

(

t+u

)dV = 0 . (1.17)

Further, since the volume V is arbitrary, (1.17) can only be true if

t+u = 0 . (1.18)

holds everywhere.

ClosureEquations (1.13), (1.16) and (1.18) form the basis for describing the flow of an elec-trically conducting fluid. Together they provide 7 scalar equations in the 8 unknownsB, u (3 scalar components each), p and . We therefore require an additional equa-tion to close the system. (More than one will be required if they introduce furtherunknowns.)

We shall consider two cases:

Incompressible FlowHere, the extra equation is simply

= const.

Then, (1.18) reduces tou = 0 . (1.19)

Compressible FlowHere, in general, p = p(, T ) where T is the temperature, so more than one extraequation is usually required because temperature is introduced as a new variable. Insome circumstances, however, p = p() is a good approximation. Examples of this are:

p = const. the isothermal gas law, (1.20)

andp = const. the adiabatic gas law, (1.21)

where is the ratio of specific heats. Much more detail can be found in Chapter 2 ofPriest.

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Summary

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

(ut

+ (u )u)

= p + 2u + 10

(B)B + other forces. (1.22a)

Bt

= (uB) + 2B . (1.22b)

and eitheru = 0 . (1.22c)

for an incompressible fluid, or

t+u = 0 , and some gas law , (1.22d)

for a compressible fluid.

Magnetic Field Visualisation

Magnetic Field LinesA magnetic field line is a line drawn such that the tangent to the line at any pointon the line is in the direction of the magnetic field B. In Cartesian coordinates, fieldlines are given by

dx

Bx=

dy

By=

dz

Bz. (1.23)

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 FluxThe magnetic flux F crossing a surface S is defined to be

F =S

B n dS , (1.24)

where n is the unit normal to S.

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Magnetic Flux TubesA magnetic flux tube is the volume enclosed by a set of field lines which intersect a simpleclosed curve.

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

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4h magnetohydrodynamics

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