The geodynamo for non-geophysicists

Figure 1 shows a cross-section of the Earth. The core, which has a radius of 3480 km, consists mainly of iron and is electrically conductive. (The conductivity is about 1/10 of that of iron under normal conditions.) The mantle consists of rock whose electrical conductivity is about 5 orders of magnitude smaller. We can consider it an insulator. The inner core with a radius of 1210 km is solid, the outer core is liquid. The viscosity of the iron in the outer core is not very different from that of liquid iron on the Earth's surface. The viscosity of the mantle, on the other hand, is so high that we can consider it solid. The temperature in the core and mantle decreases from the inside to the outside. In the center of the Earth, it is about 6000 K, at the surface of the core it is slightly more than 4000 K. Since a heat transport takes place from inside to outside, the Earth cools down slowly. This cooling process, however, has a duration comparable to the age of the Universe. We can, therefore, assume a temperature gradient from the inside to the outside that is constant in time on timescales relevant to the dynamo process.

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In the region we are interested in, namely the outer core the heat is transported by convection: the liquid metal moves at a speed of a few km a⁻¹. (Compared to other 'geological' velocities, such as those of the continental plates, this velocity is quite high.) However, this convection current is not simply up and down. Due to the rotation of the Earth, any translational motion is overlaid by a rotational motion, a phenomenon that we know from the air currents in the atmosphere. Altogether we have a screw-shaped motion.

Where the field can be easily observed, i.e. outside of the Earth, it is approximately a dipole field, similar to the field of a bar magnet or a cylindrical coil. But inside of the Earth, the spatial distribution of the field strength is far from being that of a dipole field. Figure 2 shows calculated field lines [2, 6]. It has an irregular and intertwined structure with variations on a length scale of about 100 km. For more recent calculations of the magnetic field see for example [7].

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If one examines the field in a time interval that is short according to human time scales, one gets the impression that the field is constant in time. However, when looking at it over several hundred thousand years, the time development of the field can be seen [6, 8].

The geodynamo basically works like a self-excited technical, i.e. artificial generator. Let us briefly recall how such a generator works. The stator is an electromagnet that creates the magnetic excitation field. A current is induced in the rotor coil, which rotates in the stator field. This same current is passed through the stator coil.

The self-excited dynamo has some characteristics that we also find in the geodynamo, and which we would like to highlight here.

• In order for the self-excited dynamo to start, the stator must first be supplied from another source, because as long as no current flows in the stator coils, no current is induced in the rotor coils.
• If the dynamo runs too slowly, it can 'extinguish'.
• If the rotor runs sufficiently fast, an arbitrarily small magnetic field is sufficient to start the dynamo effect. The faster the dynamo runs, the more unstable becomes the currentless state.

For an artificial dynamo, the path of the electrical current is well-defined, as is the movement of the electrical conductors. The Earth achieves the same result, although the conductors seem to move without any plan, and although the electric current has no predetermined paths. In order to understand how it performs this miracle, let us first discuss a situation in which there is no dynamo effect, and understand the reason for it.

In the following we will look at the fields at one position. We are not asking how these fields are related to those in the neighborhood. So we will not be able to make any statements about the distribution of the field in space. In addition, we are only looking at one point in time. This means that the processes that we present in the following as a causal chain take place simultaneously at the same place. One can also see it that way: we break down a complex process into additive components or contributions. Or more precisely: we break down the relevant vector quantities velocity, magnetic field strength and electric current density into components whose combined action is easier to understand than the process as a whole.

We call the corresponding field strengths B₀, B₁, and so on. Accordingly, we designate electric currents with j₀, j₁, ... (the symbol of the current density) and movements with v (the symbol of velocity).

To get a dynamo effect, we have to assume that there is a magnetic field at the beginning. Let us call it B₀. We will discuss in section 5 where this initial field may come from. For the moment it is just 'fallen from the sky'. We now hope to get a new magnetic field out of it by a suitable movement of an electrically conductive liquid, so that the induction process continues by itself even though we subsequently switch off the source of B₀.

We begin by moving our liquid within the initial field perpendicularly to the direction of the field vector B₀, figure 3. (Black arrows represent movements, the red tubular shapes represent the magnetic flux and blue arrows the electric current density vector.)

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The resulting Lorentz force on the charge carriers causes an electric current perpendicular to B₀ and to the velocity according to the right-hand rule. According to Ampère's law this current causes a magnetic field B₁. What is the effect of the movement of the liquid in B₁? Above j₁ the Lorentz force creates a current j_a, which flows in the same direction as j₁, and below there is a current j_b, which flows in the opposite direction. The result of all three currents j₁, j_a and j_b together is an upwards shift of the total current. In other words: the liquid tries to drag the current j₁ with it. The greater the electrical conductivity, the better it succeeds in doing so. In the ideal case of a perfect conductor, this effect would be complete: the electric current and the magnetic field would be 'frozen' in the liquid.

But what happens when we now switch off B₀? The induced current is displaced a little by the liquid and, at the same time, it decays. There is no dynamo effect. The reason: the movement was too simple.

In fact, the dynamo effect cannot arise if the velocity field is curl-free or in other words, if there is no shear in the field. We therefore add a second, independent movement to the translational movement: a rotation around the direction of the translational velocity. The result is a helical motion. In the following, we explain how such a movement produces a magnetic field that has the same direction as the initial magnetic field.

Figure 4 shows the process, broken down into 4 separate steps. In each of them we consider only part of the problem: first only the translational component of the movement, then only the rotation, then again only the translation and finally once more the rotation. We also decompose the magnetic field and the electric current density vectors into components and consider only one of them at a time. This is permitted because the different components of the vector quantities magnetic field strength and electrical current density add up linearly and therefore do not influence each other. Of course, we only get a small part of the total solution of the problem. The directions of the arrows in the figure that represent the B and j vectors follow from the well-known right-hand rules.

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Figure 4(a) shows the initial magnetic field B₀ and the translational motion of the liquid. This movement within the magnetic field results in an electric current j₁, figure 4(b). For the sake of clarity, B₀ and the translational motion v_trans are no longer shown in the next figure 4(c). In the following, we are only interested in the effects of the current j₁. This current generates a magnetic field, figure 4(c). Figure 4(d) does no longer show j₁. Instead, we consider the effects of the rotational motion of the liquid within the field B₁. Figure 4(e) sketches the resulting electric current j₂. In figure 4(f), B₁ and v_rot are no longer shown, but instead the magnetic field B₂ caused by j₂ is represented. In figure 4(g) j₂ is omitted and the translational movement is considered again. It leads to a new current j₃, figure 4(h). In figure 4(i) B₂ and v_trans are no longer shown but instead, the magnetic field B₃ generated by j₃. Next, the influence of the rotational movement is again taken into account, figures 4(j) and (k): a current j₄ is generated. The last picture shows how the magnetic field B₄ is generated by j₄. The sequence of the cause-effect steps is summarized in table 1.

We now have the desired result: B₄ has the same direction as B₀. So we no longer depend on the original field B₀. Our 'dynamo' continues to run, provided, of course, that the liquid moves fast enough.

The representation in figure 4 may have created a false impression. It looks as if we had 'constructed' a stationary dynamo. The figure seems to show how a spiral movement reproduces the initial field B₀. It is true that our model shows that such a field is generated. However, that does not mean that the field configuration is stationary. The reason is that we have ignored several other processes. In figure 4(d), for example, we only considered the rotation, but not the translation. The translational movement also generates currents by means of B₁, and these, in turn, have their field, etc in figure 4(g), we only considered the effect of the translation, but not that of the rotation. Again, we have disregarded other current components, etc.

Furthermore, we have not taken into account another effect that makes the field more complicated: we have assumed the flow to be given. We imposed it on the liquid. This is not true either. The magnetic fields have an effect on the liquid. The thermodynamic driving engine, i.e. convection, supplies the energy dissipated by the electric currents. It is thus subjected to a braking effect, like an ordinary technical dynamo to which a load is connected. This effect changes the flow of the liquid.

The actual process is therefore, more complex than figure 4 suggests. More intricate structures are created and it cannot be expected that there is a stationary state. What we have looked at is only a small part of the overall picture. However, this contribution is important for an understanding of the geodynamo because we now see that the initial field B₀ is no longer needed.

We leave aside the problem of the Earth's magnetic field for a moment and look at an RL circuit: the terminals of a coil (inductance L) are connected to the terminals of a resistor (resistance R). We ask for the behavior of this arrangement when it is enlarged geometrically: when all linear dimensions are multiplied by one and the same factor k. We ask in particular how the decay time will scale. The original decay time being

$$\begin{equation*}\tau =\frac{L}{R}\end{equation*}$$

we ask for the decay time

$$\begin{equation*}\tau \prime =\frac{L\prime }{R\prime }\end{equation*}$$

after the circuit has been enlarged by a factor k. For this purpose, we need to calculate how the resistance and the inductance are scaling.

The resistance R of a geometrically simple resistor can be calculated as

$$\begin{equation*}R=\rho \cdot \frac{l}{A},\end{equation*}$$

where ρ is the electrical resistivity, l the length and A the cross-sectional area.

Increasing the size of the resistor results in

$$\begin{equation*}R\prime =\rho \cdot \frac{l\prime }{A\prime },\end{equation*}$$

where l^'= k · l and A^' = k² · A.

Since the increased resistor consists of the same material as the original one, the specific resistance ρ is not scaled. So we get

$$\begin{equation*}R\prime =R/k.\end{equation*}$$

If the resistor is increased in this way by a factor of ten, the resistance R is reduced to 1/10.

Accordingly, we calculate how the inductance L scales. From the formula for the inductance

$$\begin{equation*}L={\mu }_{0}\cdot {N}^{2}\cdot \frac{A}{l},\end{equation*}$$

where μ₀ is the magnetic constant, N the number of turns of the coil, A its cross-sectional area and l its length, we get

$$\begin{equation*}L\prime =k\cdot L,\end{equation*}$$

i.e. if a coil is enlarged by a factor of ten, the inductance increases tenfold.

With these results, we get for the decay time

$$\begin{equation}\tau \prime =\frac{L\prime }{R\prime }=\frac{k\cdot L}{R/k}={k}^{2}\cdot \tau .\end{equation} \tag{ 1 }$$

The decay time of the RL circuit thus increases with the square of the scaling factor. Let us look at an example: we assume an RL circuit of laboratory size, with a linear dimension of about 0.1 m and a decay time of 1 millisecond. We do not even need a coil. Each closed circuit has an inductance and a resistance. We now imagine enlarging this circuit to 100 km, i.e. by a factor of 10⁶. Equation (1) tells us, that the decay time will increase to 10⁹ seconds or about 300 years. Closed currents on this scale thus remain practically constant for times of the order of 10 years. Significant changes are only observed in time intervals of the order of a hundred years.

We can transfer this result to the electric currents of the geodynamo. Due to the large size of the Earth, the decay time of an electric current within the outer core is of the order of years.

In section 2, we have seen how a dynamo effect can occur in a moving, electrically conductive liquid. However, there is one problem we have not yet addressed. The dynamo effect cannot occur if the newly created field is smaller than the initial one. This would mean that the dynamo would extinguish. To prevent this happening, some conditions must be met.

One of these conditions is that the individual currents do not die away too quickly. Their decay time must be large. As we have seen, the decay time of a circuit is greater the larger its geometric dimensions are. Moreover, τ is great when the liquid is a good electrical conductor.

Furthermore, the functioning of the dynamo depends on the velocity of the liquid. Just like a technical dynamo, a geodynamo can only work when the velocity of the moving electrical conductors is sufficiently high.

So we have identified three parameters on which the functionality of the dynamo depends (i) the geometric extension l, (ii) the electrical conductivity σ and (iii) the velocity v of the liquid.

These three conditions are related in the simplest way one can imagine: the product of v, l and σ must reach a certain minimum value. Multiplying this product by the magnetic field constant μ₀ results in a dimensionless quantity, called the magnetic Reynolds number:

$$\begin{equation}{R}_{\mathrm{m}}={\mu }_{0}\cdot \sigma \cdot \text{v}\cdot l.\end{equation} \tag{ 2 }$$

The theoretical treatment of the problem shows that the minimum value for the occurrence of the dynamo effect is of the order of 10. However, for a geodynamo to run safely, R_m must even be larger.

For the Earth's outer core there is σ = 10⁶ Ω⁻¹ · m⁻¹, v = 5 × 10⁻⁴ m s⁻¹, and l = 1000 km.

With these values we obtain from equation (2) R_m = 10¹⁰, i.e. a value much greater than ten.

We can also understand why the effect does not show up in the bucket with the saltwater. With σ = 1 Ω⁻¹ · m⁻¹, v = 0.5 m s⁻¹, and l = 0.2 m we get R_m = 10^–7, a value that is far too small for a dynamo effect.

Somewhat more realistic would be the expectation that the movement of the water of the seas leads to a dynamo effect. Here a typical length would be about 100 m. But the magnetic Reynolds number is still far below the minimum value. So we should not be surprised that we do not encounter this effect anywhere on Earth except in the real geodynamo.

This seems to show that it is hopeless to realize a geodynamo effect in a laboratory experiment. However, such experiments have proved successful [9]. The facilities are several meters in size and work with liquid sodium, which is pumped to a speed of up to 20 m s⁻¹. The path of the flowing sodium is predetermined. A magnetic Reynolds number of about 10 is achieved in these experiments.