To avoid making this document seem too formidable, the algebra has been separated from the main text. You can find the details by clicking on words highlighted in blue, as they appear. I recommend reading the sections involving the mathematics, regardless of how long it has been since you took Calculus. After all, you will not be asked to derive anything, and there will be no quiz at the end, so you might as well wade through the entire thing. As a reward for doing so, you will find every algebraic step explained in English, and every so often you will find some associations between this material and other areas of geology (such as hydrogeology and seismology). But if you decide to skip the sections that deal with the details, you will still understand the material at a level higher than that provided in most Physical Geology textbooks. At the end I have included some comments intended to stimulate discussion of the material with students. Some of them can also be converted into test questions.

In the 17^th Century, Gilbert collected data on compass measurements from seamen and posited the existence of a magnetic field around the earth. He envisaged the earth as a giant lodestone with a permanent dipole field oriented parallel to the planet's axis of rotation. But in the 20^th Century, the decay time of the field around a magnet with the dimensions of the earth and a conductivity of the outer core was estimated to be of the order 10,000 to 15,000 years, which suggests that a planet whose age is a few billion years should have lost any original magnetic field long ago.

The reason for the decay is partly the fact that the temperature of the earth's core is well above the Curie temperature, and partly the existence of electric currents in the core, which generate heat because of electrical resistance, and therefore dissipate magnetic energy. For these reasons, the earth's magnetic field should not be sustainable. Yet paleomagnetic data show a continuous record of magnetic activity for at least 100 million years. Since the field has not dissipated, there must be a regeneration mechanism to sustain the source of magnetic energy. The dynamo mechanism is the most plausible one. In this tutorial, "dynamo" refers to a process, rather than to a particular model. The general principles that underlie the regeneration of a field are discussed. The discussion involves a descriptive approach based on some of the work of Michael Faraday and a more complicated approach based on Maxwell's equations. References are included at the end, from which more details can be obtained.

Description of Electromagnetic Phenomena

The standard description of electromagnetic phenomena is given by Maxwell's equations. In the mid-19^th Century, Maxwell synthesized some earlier work of Ampere, Faraday, and Gauss, and showed that a system of equations representing all of their individual studies provides a complete description of the interaction between electrical and magnetic fields. This "model" shows that electricity and magnetism are not separate entities, just different manifestations of a single phenomenon called electromagnetism.

Easily Done, Great Demonstration

Before getting to the system of equations, it is useful to think about some demonstrations that can be done with simple laboratory equipment. A number of years ago I spent several years teaching Physics in a high school. It was a tiring experience, but a fruitful one. I used to begin the unit on electromagnetism with a demonstration using a coil of wire that was wrapped around a wooden cylinder which was mounted to a circular wooden base. It looked like a large candle with a few thousand turns of wire wrapped around it. The base had a terminal on either side of the cylinder, and each end of the wire on the "candle" was attached to one of them. Connecting another piece of wire from one terminal to the other made a closed circuit. For the demonstration, I would insert a galvanometer into the part of the circuit formed by the second piece of wire. A galvanometer can be set up either as an ammeter or voltmeter (micro-amps and micro-volts, that is). For conceptual convenience, I set it up as an ammeter. From the zero position, the galvanometer's needle can move to the left or right, depending on the direction of the current in the circuit.

Because there was no power source, there was no current, so the needle on the meter was at zero. But when I took a bar magnet and inserted it into the hollow cylinder, the needle would move. This always got the students' attention because they could see that the circuit was not connected to a power source. Then I would remove the magnet, and the needle would move in the other direction. Clearly, a current was being generated in the coil of wire. A third thing I did was to vary the speed of the magnet. A slowly moving magnet produced a small movement of the needle (presumably, a small current); a rapidly moving magnet resulted in a large amount of movement (a large current).

I always began with this demonstration because I wanted the students to get over the "hump" involved with dealing with something that is fairly abstract - an electromagnetic field.
The demonstration made clear that something strange was happening; a current was "induced" in the coil of wire by the magnet. There was no physical contact between the magnet and the wire, so somehow, a force was applied to something in the wire, over a distance, and in a manner the students could not see.

When they saw for themselves that something puzzling was happening (after all, everyone knows you need a battery or a wall socket to produce a current), the students were ready for an explanation. The explanation involves three things: an electrical conductor, a magnet, and motion. The third component is just as important as the other two, because without motion, the electric and magnetic fields are independent. When they are in motion, they interact to produce a variety of effects. The demonstration provides the foundation needed to understand the dynamo process.

I should add that I do this demonstration in my introductory college Geology courses and it never fails to get the students' attention. In fact, a student once told me that a friend who recommended the course specifically mentioned the demonstration. I hope the friend remembered more about the course, but I take what I can get.

Standard Approach to Electromagnetism

During the first half of the 19^th Century, the experimental work of Faraday, Ampere and Gauss was "translated" into the language we call Partial Differential Equations. About the middle of the 19^th Century, Maxwell synthesized and modified slightly the earlier work, and came up with the system of equations that bear his name. Not many Geologists are comfortable dealing with partial differential equations, especially when they are expressed in the exotic notation that is used below, that of vector analysis, but if you keep the demonstration I described in mind, you will recognize the different parts of it in these equations.

Maxwell's Description of the Electromagnetic Field

A slightly restricted form of Maxwell's equations, which is suitable for studying the motions of an electrically conducting fluid in the earth's core is:

The difference between this version of Maxwell's equations and the complete system involves a term Maxwell added to Ampere's equation. It represents a current whose value is negligible when the motions are small relative to the speed of light, so it is customary to omit the term in studies of the earth's magnetic field.

A Walk Through the Equations

First look at Ampere's Law. It relates changes in the magnetic field strength B, to an electric current, J. We will get to what the word "changes" means shortly. In a laboratory experiment, the field strength determines how large an object the magnet can pick up.

The second equation, Faraday's Law, relates changes in the electric field strength E (think of it as the voltage) to the rate at which the magnetic field B changes.

Moving on, Ohm's Law states that the electric current J is proportional to the electric field strength (the "sigma E" term) and to a current induced by the effect of motion (with velocity u) on the magnetic field B (the sigma times [u x B] term has the units of current).

Finally, the two Gauss' Laws represent conservation conditions that B and E must satisfy. They will be used to simplify things later.

The Coefficients

The symbols and represent the magnetic permeability, the electrical conductivity, and the dielectric constant, respectively. The symbol q represents the charge density, a scalar function (which, like heat, can vary from place to place, but has no directional properties).

The magnetic permeability is a proportionality constant relating a magnetic field to an electric current. The conductivity (the reciprocal of the resistance) governs the strength of the electric current associated with a particular electric field. And the dielectric constant determines the maximum electric field strength.

The Vector Notation

The term (u x B) is the "cross," or vector product one learns about in introductory Physics courses. This term gives the voltage induced by the interaction between the motion of the fluid and the magnetic field. Multiplying by the conductivity gives units of current (recall the current "induced" in the coil of wire by the motion of the magnet).

The terms ( ) and () represent the vector operations called curl and divergence, respectively. They may look like the cross product and "dot" product used in Physics courses, but they are more involved. They are shorthand representations of ways to differentiate vector quantities. They usually are covered in third semester Calculus courses. Basically, we use the curl to represent shears or rotations, and the divergence to represent volume changes. Click here for more details on these operations.

What Do the Equations Say?

Looking back at Maxwell's equations now, it should be clear that they are coupled to each other. Faraday's law "says" that a rotation of the electric field lines (the curl of E) causes the value of the magnetic field strength to change with time (the dB/dt term). And Ampere's law "says" that a rotation of the magnetic field lines (the curl of B) produces (induces) an electric current, J. Finally, Ohm's law "says" that this current has two components, one caused by the electric field (the voltage), and one caused by the interaction between the motion of the fluid and the magnetic field.

Recall the Demonstration?

These statements are equivalent to what I said about the demonstration in the Physics classes. The key is motion. It couples things together. Look at Maxwell's equations again. If the magnetic field lines of B, in Ampere's equation, are not moving in space (that is, if ), there is no current J. If I had not inserted the magnet into the coil of wire (the motion), no current would have been generated. Or, we can say that if B in Faraday's Law is not changing with time, the electric field E is stationary (in which case (). If B does change with time, so does E. The faster I moved the magnet, the stronger the current induced in the coil. Motion is required for the interactions that characterize electromagnetic phenomena. That statement will be an important part of what follows.

Using Maxwell's Equations to Make a Dynamo

Now it is time to do some algebra. To use Maxwell's equations to describe the dynamo effect that produces the earth's magnetic field, we take advantage of the coupling between the equations. We eliminate some variables by substituting some of the equations into the others. Simplifying the results requires the use of some algebra involving the vector operations, curl and divergence. This is done elsewhere in this document. Click here to see the details. The end result of the algebra is an equation called the Induction Equation, described below.

Results of the Derivation

The algebraic operations result in a Partial Differential Equation called the Induction Equation. This equation "says" that the time history of the magnetic field has two components, one involving the dissipation of the field due to thermal effects and the other involving spatial variations in the flow.

The equation is similar in form to one used in Hydrogeologic studies. Click here to seen an association with that topic.

Solving the Induction Equation

The Induction Equation contains terms (the derivatives) which specify the manner in which the magnetic field varies with time and in space. A solution to this equation is a function B(x,y,z,t), which when graphed, shows a picture of the field in space as it changes with time. But solving the equation is complicated by the fact that although it is an equation for B, it also contains u. So we have one equation with two unknowns. There are two approaches to dealing with this problem: a complicated one (called the dynamic approach) that gives quantitative results, and a simpler one (called the kinematic approach) that gives more restricted results. Click here for an explanation of the two approaches.

Analyzing the Induction Equation

To extract information about the solution of the Induction Equation, we adopt here a version of the kinematic approach that I will call an "asymptotic" approach because we ask what would happen if some of the terms were extremely (asymptotically) large or small. The simplest way to find out what kinds of things the Induction Equation describes is to make some assumptions about the magnetic diffusivity , whose value depends on the electrical conductivity, or the resistance. By asking what happens when that coefficients is very large or very small, we can examine the contributions of each term in the equation and see what they indicate separately.

First assumption: infinite conductivity

When the conductivity of the fluid is very large - approaching an infinite value - the value of approaches zero. In that case the term in the equation can be neglected. This situation corresponds to a fluid with no electric resistance. In such a case there is no energy loss from thermal effects, currents persist for long periods of time, and the field does not "spread out" and dissipate (as it would if energy were lost to heat). In this case, the Induction Equation reduces to

This equation "says" that changes in B (the left hand side) are due only to the motion of the fluid (the meaning of the "gradient of u" term is basically, the "intensity" of the motion). Then the magnetic field lines must move with the fluid. They are "frozen" in the fluid. Recall that we assumed that the fluid is incompressible (), so the shape of the field is determined solely by the motion of the fluid (the term advection is usually used to describe this motion). Click here for a discussion of another consequence of dropping the Laplacian from the equation.

Alternate assumption: zero conductivity

The other assumption we can make is that the conductivity is very low, in which case the resistance is very large, and for all practical purposes, no induced electric currents will exist. The Induction Equation becomes

This is a diffusion equation. Its solution is a function whose shape "spreads out" in all directions with time. The magnetic energy will be dissipated by thermal effects in a time specified by the inverse of the coefficient "eta". That time is about 1 second for a copper sphere whose radius is 1 cm and about 10,000 years for the earth.

General Conditions

In reality, the conductivity of the earth's core is neither zero nor infinite, so both advection and diffusion occur. The movement of the fluid in the core is the ultimate source of the energy needed to regenerate the magnetic field. Recall that I said earlier that motion would be important. The three things said earlier to be necessary for a dynamo were an electrical conductor, a magnet and motion. In the core, we have a metallic fluid, which is in motion. The existing magnetic field is the third requirement.

What are the Sources of the Three Necessary Components?

1. The cause of the earth's large metallic core is felt to be chemical separation early in the planet's history. Although it was once thought that this process required a molten phase, presumably caused by impacts of large meteorites as the earth grew by accretion, it appears that a molten planet may not be necessary. In a planet with the composition of carbonaceous chondrites, geochemistry predicts a separation into three "layers," dominated by Mg-Fe silicates, FeS, and Fe metal. Provided the temperature is high enough, the separation might occur even in the solid state. The density of the core is consistent with a composition largely of iron, with some nickel and sulfur alloyed in. So any moderately large planet consisting of refractory materials should have a metallic core. A straightforward discussion of this point can be found in Mussett and Brown (1981).

2. The assumption that a magnetic field existed early in the earth's history seems reasonable considering that magnetic fields are associated with essentially every swirling dust cloud that can be observed with radiotelescopes.

3. Although the rotation of the earth may be important in keeping the axis of the magnetic field nearly parallel to the earth's rotation axis, the rotation cannot be the source of the motion in the fluid outer core that causes the induction; it is too symmetric to produce a magnetic field. Motion in the core itself due to thermal convection or to the release of gravitational energy as the outer core cools and solidifies is probably the source of the magnetic energy.

Estimates of the ability of thermal convection to stir the core sufficiently to produce induction require that a large amount of the earth's (radioactively decaying) potassium be present in the core. The amount required may be unrealistic, so it is not clear what contribution to regenerating the magnetic field thermal convection makes.

Gravitational energy seems more likely as a source of the needed energy. The process envisaged is based on the fact that the core must have some nickel alloyed into the iron to produce the correct density. This mechanism relies on the preferential deposition of a solid layer of nickel-rich material at the edge of the outer core that is slightly denser than the material remaining in the outer core (nickel is denser than iron). This deposition at the boundary will result in a layer of fluid just above the boundary that is depleted in nickel and therefore, has a lower density than the rest of the outer core. Being lighter, it will rise and produce convective circulation.

This gravitational convection will be aided by the release of latent heat as the fluid solidifies, and by the heat generated by the induced electric currents.

General Comments from which Test Questions Can be Devised

1. If the magnetic field exists because of fluid motion in the outer core, the field should have a finite lifetime. When the earth cools to the point that the entire core is solid, the field should decay in about 10,000 years. Whether this will ever happen is not clear because the Sun may go through a Nova stage before that occurs, making the subject academic.

2. The fluid zone in the earth exists because of the size of the earth. It takes a long time for a large object to cool down. A smaller planet, say Mercury, should have cooled completely, long ago. It should not have a fluid core. But it does have a (weak) magnetic field. And the Galileo spacecraft has determined that Jupiter's moon Ganymede also has a weak magnetic field. Perhaps cooling does not proceed in these bodies in the way we envisage (if so, they may contain fluid zones), or perhaps there is another way to maintain a magnetic field. In effect, the earth is one "data point." It would be useful to do some other experiments to get additional data points. Putting enough seismic instruments on Mercury to work out the internal structure would be an interesting experiment.

3. Nothing was said in this tutorial about reversals of the earth's magnetic field. They are a very interesting subject, and may contain some information about the regeneration mechanism. Because the fluid in the outer core is probably undergoing turbulent motion (the rate of motion does not have to be large in an object the size of the core for it to be turbulent), the reversals may be caused by changes in the spatial patterns. Think of water boiling on a stove. The spatial patterns develop, persist for a while, and then change. A significant change in the direction of motion in the core might change the polarity. The fact that it seems to be constrained to being sub-parallel to the planet's rotation axis is probably caused by the daily rotation.

4. Reversals probably occur because the spatial pattern of motion in the outer core changes enough that the regeneration mechanism is not sufficient to prevent decay of the field. So it begins to decay, but grows again as a new pattern of motion develops. We do not know if it always reverses when the motion changes - it could grow again in the same orientation. We would have no evidence of that except for a weak field strength. Not knowing how weak the field would have to be to indicate this was happening, we cannot know if this happens. But if reversals are associated with the decay process, they should occur in approximately the decay time - about 10,000 years. Is that long enough for the higher levels of cosmic and solar radiation that would strike the planet to affect evolutionary processes? As far as I know, no one has been able to correlate an extinction or speciation event with a reversal. But that may be because the correlation would depend on accurate dates for each event, something we do not have. Even if they are assigned the same dates, did they really occur simultaneously?

5. How often does Geology or Geophysics make the "front page?" The July 19, 1996 edition of the New York Times contained an article on the Inner Core (IC) of the earth on page one! It seems that some seismologists at Lamont-Doherty Observatory have determined that the IC rotates independently of the rest of the planet. The rate is such that it should "lap" the rest of the planet in a few hundred years - a very rapid rate.

The determination was made using a newly discovered lag time in seismic waves passing through the IC from north to south. It was known that the IC is anisotropic - seismic waves travel in the north-south direction more rapidly than in the east-west direction. But recently, they found that the travel time of north-south waves differed from that measured a few years ago. The explanation must involve rotation.

The reason for the change is that the rotation axis of the IC is tilted a bit relative to the north-south rotation axis of the planet. So the IC precesses a bit as it rotates. As it precesses, the "fast" and "slow" directions rotate in space, so after a while, waves traveling in the north-south direction sample material from a slightly different direction - and their travel time changes.

The cause of the rotation must be related to the Magnetic field. The field induced by motions in the outer core must be producing a torque that tweaks the IC and changes its angular momentum. In effect, the IC is behaving like the armature of an electric motor.

References

Most intermediate level books on Electrodynamics have a chapter on Magnetohydrodynamics, which deals with dynamo processes. They all should derive the Induction Equation (or at least, display it prefaced by the words "it can be shown"). But most of the treatments are concerned with plasma physics rather than fields generated in true fluids. An exception is the following book which contains an advanced discussion of topics related to the dynamo mechanism. And it has sections dealing with the earth's magnetic field.

Roberts, P.H., 1967, An Introduction to Magnetohydrodynamics: American Elsevier Publishing Co., New York.


For a recent, advanced discussion of current research on the magnetic field, the following book is recommended. In addition to the standard topics, the authors cover topics from Nonlinear Dynamics, such as chaotic motion in an electrically conducting fluid. They emphasize the importance to the induction process of complex motions in the core. The first chapter is fairly straightforward but the rest is very heavy going.

Childress, S and A. Gilbert, 1995, Stretch, Twist, Fold: The Fast Dynamo: Spring Verlag, Berlin.



There are not many good, largely non-mathematical introductions to Geophysics available. One such book with a good chapter on the thermal processes occurring in the Core is the following. Unfortunately, this edition of the book is dated. I understand a new edition is out but I have not seen it. For some reason, as each new printing of the original edition came out, few of the errors in the original were corrected. But chapter 6, on the earth's core is still reliable and worth reading. The section on chemical differentiation of the earth in chapter 5 is also worth reading. Anyone with introductory courses in Physics and Chemistry should be able to follow the material.

Brown, G. and A. Musset, 1981, The Inaccessible Earth: George Allen & Unwin, London.



The vector operations used to derive the Induction Equation are usually covered in a third semester Calculus course. A good discussion of them can be found in chapter 3 of the text I used in school. I believe it is still in print.

Kaplan, w, 1952, Advanced Calculus: Addison-Wesley Publishing Co., Inc., Reading, MA.

Return to the beginning.

Return to the Teaching Laboratory

Physical Significance of the Vector Operators Curl and Divergence

The curl consists of several terms of the form

The rates of change of the x component of B in the y direction and the y component of B in the x direction, describe a shearing motion. So the curl represents a rotation. The complete expression contains terms of the same form involving the x and z components, which are added to this one. So the vector notation, () saves a lot of space.

The divergence also represents rates of change but in this case, we have sums instead of differences.

The change in the x component of B in the x direction is determined and added to equivalent quantities in the y and z directions. Physically, this operation describes the compression or extension of a material. Note that
( something) = 0 is a conservation condition, as it denotes zero volume change.

So the curl and divergence allow us to represent rotations and volume changes in fluids. How do we go about using them? Read on.

Using the Curl and Divergence

The vector operators, curl and divergence provide more than a shorthand notation. They allow us to separate out different effects.

For example, if you want to determine whether or by how much a quantity is changing, you differentiate it. The derivative gives you an expression or a value for the rate of change. When you are dealing with vector quantities, you have more options. If a variable is incompressible (e.g. water), the divergence of the quantity will be zero. If the quantity is not rotating, taking the curl will give zero.

When you have an expression that may involve both volume change and rotation, you can study them separately by taking the divergence and curl separately. The substitutions that are used (below) to derive the equation that represents the earth's magnetic field would have resulted in something much too complicated to interpret. Taking the curl of both sides (if you do it to one side, you have to do it to the other) simplifies things. It allowed us to separate the volume effects from the rotation effects.

Return to the text. Or read on for another application.

Other Applications of the Curl and the Divergence

Curl and divergence are not limited to fluids. In the early part of the 19^th Century, Cauchy used these operations (though not the vector notation, which had not yet been developed) to study the properties of elastic solids. He started with an equation of motion (Newton's Second Law), invoked Hooke's Law to relate stress to deformation, and then applied the curl and divergence (separately) to the result. He ended up with three wave equations, one for compression and two for shear (two equations, because shear waves can be polarized, so vertically and horizontally vibrating shear motions are possible). In this way he was able to show that the only ways to deform an elastic solid are to compress it or shear it, and therefore only two kinds of wave motion can propagate through an elastic such as the earth - compressional and shear waves.

Return to the text.

Derivation of an Equation that Describes the Dynamo Mechanism

This section may seem too formidable to bother with. But you should wade though it anyway. In particular, you should not be concerned if you see no physical significance in some of the steps because in some cases, there is none. Sometimes we have to use mathematical identities to simplify complex terms. Like the curl and the divergence, these identities are introduced in third semester Calculus courses.

To begin, we try to eliminate as many of the variables as possible. We can get rid of the current J by taking Ohm's Law and substituting it into the right hand side of Ampere's Law. That gives

This does not appear to be simpler than the original version (instead of two variables, we now have four), but we'll get rid of the extra ones shortly. Now we take the curl of both sides (Click here to learn why).

The left hand side is simplified using the vector identity (one of those things that has logical but no physical significance)

where alone (on the right hand side) represents the gradient, not the divergence. It is the shorthand notation for the rate of change of a component in each of the three directions. It is not important here because Gauss' law tells us that that term is zero. Then we are left with the second term, called the Laplacian operator, which, when written out, has the form

This term's physical significance is related to the curvature of the magnetic field.

So the left hand side of the equation obtained by combining Ampere's and Ohm's Laws simplifies to the Laplacian. Looking at the right hand side, the curls of the two terms can be simplified somewhat. Faraday's Law lets us substitute - for the term. Then, bringing everything left over to the left hand side, we get

where = 1/ has dimensions L²/T, and is called the magnetic diffusivity.

This equation may not look simple, but it is down to two variables, u and B. It "says" that the rate at which the magnetic field B changes with time (the term) is affected by the motion of the conducting fluid (through its velocity u) in which the field is established, as well as by the dissipation of B (represented by the Laplacian) due to thermal effects.

The equation is called the Induction Equation because it shows that the value of the magnetic field strength B, is affected by the interaction between B and the motion of the fluid in which B has developed.

Interpretation of the Induction Equation is simplified if we revise the cross product term. Another useful vector identity is

This beast is more complicated than is necessary. On the right hand side, the term can be eliminated (by invoking Gauss' Law) and, if we assume that the conducting fluid is incompressible, we can set (recall the comment about divergence being a conservation condition). That leaves for the middle term in the Induction Equation, which takes the form

This is still not very simple, so now we adopt a Lagrangian viewpoint by imagining that we are moving along with a point in the fluid as it moves. To do this, we extend the concept of the derivative to include spatial as well as temporal changes. The operator

is called the "material" or convective derivative. The first term on the right hand side gives the time rate of change of a quantity such as B and the second term gives the change due to movement of the point as the fluid flows. The sum of the two describes the total change in the quantity. After incorporation of the term in the derivative, the Induction Equation takes the form

Now we see clearly that the time history of the magnetic field has two components, one involving the dissipation of the field due to thermal effects (the Laplacian) and the other involving spatial variations in the flow (the dot product).

Return to the text.

Relevance of the Induction Equation

The equation has the form of an advection-diffusion equation. Without the curl of (u x B) it would be a standard Parabolic partial differential equation, such as those describing diffusion of materials and the flow of heat. The term involving the curl represents advective or convective transport.

Equations of this form are commonly used in Hydrogeologic studies to represent the movement of a plume of a contaminant through groundwater. As the plume moves downgradient (the advective effect), it also spreads out (the effect of diffusion). The relative importance of the two phenomena on the shape of the plume depends on different properties of the medium. The relative magnitudes of the diffusion coefficient (which governs changes in shape) and the permeability (which governs horizontal and vertical transport of the entire mass of material) determine the concentration of the pollutant at some distance from the source, at some time after introduction to the groundwater. The Induction Equation does not have a permeability coefficient; the ratio of the inverse of the velocity u and the diffusion time might be the equivalent comparison.


Return to the text.

Solving the induction Equation

The dynamic approach utilizes a separate equation for the velocity u, an equation that must be solved simultaneously with the one for B. The "equation of motion" is a complicated form of Newton's second law, called the Navier Stokes equation (using vector notation it is one equation; it takes three equations if you write out expressions in all three dimensions separately). This equation for u is coupled to the one for B because the magnetic field affects the motion of a conducting fluid.

This equation of motion rarely can be solved because it is non-linear; some of the terms are squared. In addition, the boundary conditions (basically the geometry appropriate for the problem) often involves non-linear terms, which also complicates things. For these reasons, it is usually necessary to solve the problem (the coupled equations for u and B) numerically, on a computer This gives quantitative results, but is fraught with problems associated with the approximations needed to do the numerical integration.

The kinematic approach proceeds by making an assumption about the form of u instead of trying to solve for it. That way, you only need to deal with the equation for B. If the expression used for u is not too complicated, the kinematic analysis of the Induction Equation lets us determine the balance between the dissipation due to thermal losses and the regenerative effects of the motion.

Return to the text.

Consequences of Reducing the Order of a Differential Equation

Neglecting the Laplacian in the Induction Equation has serious consequences for anyone wishing to carry out a complete analysis of the Induction Equation. Derivatives tell us that a variable is changing smoothly, so ignoring a derivative is equivalent to introducing a discontinuity into the problem, normally at a boundary. "Boundary conditions" relate the mathematics problem to the real world. If we neglect the Laplacian and thereby reduce the "order" or the equation by two derivatives, we have to compensate by changing something at the boundaries also. In fluid mechanics problems, the correction usually consists of positing the existence of a "boundary layer," a narrow zone in which the properties of things (in this case, the magnetic field) change abruptly. Entire books have been written about the very subtle techniques called "singular perturbation methods" that have been devised to deal with boundary layers.

Return to the text.