## The Solar Dynamo: The Physical Basis of the Solar Cycle and the Sun’s Magnetic Field

In my previous article, I laid out some basics about the sun’s structure and physical characteristics in order to set up the groundwork upon which I could then explain the physical mechanism which underlies the solar cycles I talked about in the article prior to that one. I understand that this is a bit more technical than most readers may be accustomed to, which is why I’ve included a simplified “tl; dr” version before delving deeper.

Solar Dynamo Theory

The leading scientific explanation for the mechanism by which these solar cycles emerge is the solar dynamo theory. It arises from an area of physics called magnetohydrodynamics, which is the field which studies the magnetic properties of electrically conducting fluids, and is covered in most university textbooks on plasma physics. So how does it work?

The tl; dr version is as follows: the convective zone of the sun is a plasma (ionized gas), and it moves around via turbulent convection currents. The flow of these charged particles generates electric currents. Those electric currents generate magnetic fields (via Ampere’s law). In turn, when those magnetic fields change, they induce electric currents (Faraday’s law). In this manner, the dynamo is self-reinforcing, and permits the continual generation of magnetic dipole fields over time. An analogy that helps some people is to think of the magnetic field loops as being like rubber bands. And the convection currents stretch and twist the magnetic field lines. Just as how stretching and twisting rubber bands will increase their tension, the stretching and twisting of magnetic field lines can make the field stronger at certain points and/or change the field’s direction. If this twisting and stretching is done in a particular way (i.e. in the manner which occurs in our sun), it produces a cycle of changing magnetic fields which corresponds to the 11 and 22 year solar cycles.

However, this is an extremely over-simplified version of what the theory entails. There constraints on what sort of velocity fields will produce the observed effects. Namely, the flow must be turbulent like a pot of boiling water (rather than like a stream or faucet). The flow must be three dimensional. That means that the flow must have components in the radial direction, along the meridians (north and south), and along the latitudinal lines (also referred to as the azimuthal direction). And the flow must be roughly helical (Seehafer 1996).

Another critical requirement is differential rotation. In other words, the angular velocities at which the different parts of the sun rotate vary both with radius and with latitude (Schou 1998). The rotation rate at the solar equator, for example, is faster than the rotation at the poles. This is possible for the sun because it is composed primarily of plasma rather than a solid like the Earth. In the convective zone, differential rotation is primarily a function of latitude, and varies only weakly with depth, while the tachocline exhibits a strong radial shear (Howe 2009). The reason for these requirements is that the motions of the plasma must be capable of converting a meridional (poloidal) magnetic field into an azimuthal (toroidal) magnetic field, and vice versa.

The Omega Effect

Basically, if we begin with a meridional magnetic field, the differential rotation of the sun twists and coils this field around the sun, which results in an azimuthal magnetic field. This phenomenon of converting a meridional magnetic field into an azimuthal one is called the Omega effect. Its relevance to the observed solar cycle is that the twisting of the magnetic flux strands in the azimuthal (toroidal) direction in shallow depths and low latitudes create concentrated magnetic “ropes,” which are brought to the surface via magnetic buoyancy to produce the bipolar magnetic fields associated with sunspots and other related activity of the solar cycle (Parker 1955, Babcock 1961).

The Alpha Effect

Contrastingly, the Alpha effect converts an azimuthal (toroidal) magnetic field into a meridional (poloidal) field. The precise mechanism by which this occurs is still not fully understood as of this writing, but it has to do with the interaction between the velocity field of the plasma, the rotation of the sun, the toroidal magnetic field, and the Coriolis Effect acting on rising flux tubes.

From a qualitative standpoint, suppose we have a sphere of hot plasma rotating at an angular velocity ω. Suppose also that the fluid convects, and that certain localized pockets are hotter than the surrounding fluid, and thus move radially outward at velocity u. Additionally, suppose the presence of a toroidal magnetic field which gets partially dragged by the motion of the fluid. Since the sphere is rotating, each of those pockets of fluid is acted on by the Coriolis force ω x u, and therefore twists as it moves upwards and expands. Consequently, the magnetic field lines twist as well. Since the signs of both the Coriolis force and the toroidal magnetic field are reversed in the northern versus the southern hemisphere, this results in small scale magnetic field loops of the same polarity in both hemispheres (Coriolis 1835). The idea then is that these small scale loops of magnetic flux gradually coalesce as a result of magnetic diffusivity, which therefore generates a large scale poloidal magnetic field (Parker 1955).

The Omega Effect and the Alpha Effect. Image by E. F. Dajka

In this manner, a poloidal magnetic field generates a toroidal magnetic field, which in turn regenerates the poloidal magnetic field, and so on and so forth. The poloidal fields predominate during solar minima, while the toroidal fields generate the sunspots and other activity associated with solar maxima. The cycle repeats with an approximately 11 year period, and the associated magnetic fields alternate polarity from one cycle to the next, thus producing the observed 22 year solar cycle. I should reiterate that there are other hypotheses than what I’ve described here, and unlike the Omega effect, which is better understood, no clear scientific consensus has yet emerged on the precise mechanism of the alpha effect. In recent years, a lot of focus has been placed on variants of what’s known as the Babcock-Leighton (BL) mechanism, which is described here.

The Fundamental Equations of Magnetohydrodynamics and the Solar Dynamo

Warning!! Vector partial differential equations ahead!

The mathematically faint of heart may want to scroll past this section!

The physics involved in the dynamo are described by the equations of magnetohydrodynamics (MHD), which derive primarily from classical electromagnetism, but also from fluid mechanics to some extent, because hot plasmas share certain dynamical behaviors with liquids. The relevant equations include the following:

E = J/σ − u × B,

where E represents the electric field, J is the electric current density (charge per unit time per unit area), u is the velocity of a fluid element of the plasma, B represents the magnetic field, and σ is the conductivity of the plasma (J can also be expressed as J = nqvd, where q = the charge of a given particle, n = the number of said particles present, and vd is the average “drift” velocity of the particles).

This actually derives from Ohm’s law. You may be more familiar with Ohm’s law in its common form V = IR, where V is voltage (or electric potential difference), I is the electric current, and R is the resistance. But this is just veiled form of a more fundamental form of the Ohm’s law equation. The current I can also be expressed as I = J·A (the dot product of the current density with area element), and the resistance R can be expressed as a property called the resistivity ρ of the conductor (in this case the plasma) times the length element L of a charged particle’s path divided by the path’s cross sectional area element A, (R = ρL/A).

Thus V = IR becomes V = J·A(ρL/|A|) = JρL. But the resistivity term ρ of a given medium is also the reciprocal of a quantity called its conductivity (denoted as 1/ρ = σ), and the dot product of J·A is just the product of their magnitudes, thus giving us V = JL/σ, or alternatively, J = σV/L. But in many conductive mediums, this term scales linearly with the electric field, and can be expressed as J = σE. However, that’s in a reference frame co-moving with the fluid element. From a fixed reference frame (assuming non-relativistic velocities), and with an external magnetic field B, an additional term must be added to account for the Lorentz force on the moving charges, and the equation becomes J = σ(E + u x B), where u is the velocity of the fluid element, and x is not a multiplication sign, but rather what’s called a cross-product operator.

Dividing both sides by σ, and subtracting u x B from both sides yields the aforementioned E = J/σ − u × B equation.

Another important equation in the magnetohydrodynamics of the solar dynamo is the pre-Maxwellian form of Ampere’s Law:

× B = μ0J,

where μ0 is the magnetic permeability constant, and × B operator represents what’s called the curl of the magnetic field B.

Finally, there’s Faraday’s Law, one form of which is × E = – ∂B/∂t, which is basically saying that the curl of the electric field is equal to the negative of the rate of change of the magnetic field with time.

But we already have another expression for E = J/σ − u × B.

By dividing both sides of our × B = μ0J equation by μ0 to get J = × B/μ0, and then substituting that for J into our Ampere’s Law equation E = J/σ − u × B, we get E = ( × B)/(μ0σ) – u × B.

We can then substitute into our Faraday’s Law equation × E = – ∂B/∂t, in which case we get

× [( × B)/(μ0σ) – u × B] = – ∂B/∂t.

Rearranging this, we get the following:

∂B/∂t  = × (u × B − η × B),

where η = 1/(μ0σ) is the magnetic diffusivity term.

This is the MHD induction equation. The first term on the right side of the MHD induction equation represents the induction via the flow of electrically charged constituents across the magnetic field, while the second term expresses Ohmic dissipation of the current systems supporting that magnetic field. The relative importance of these two terms is measured by what’s called a magnetic Reynold’s number: Rm = u0L/η, where u0 and L are characteristic values for the flow velocity and length scale of the system respectively. For solar dynamo action, where L is on the order of the solar radius, Rm is invariably much greater than 1. Ergo, the Ohmic dissipation is highly inefficient on this scale, and therefore maintaining a solar magnetic field against diffusion is no problem.

And now for something unrelated…

In the next installment, I’ll briefly go over an approach called Mean Field Theory, which astrophysicists and other scientists sometimes use to simplify their mathematical models of large complex systems.

Related Articles:

References:

Babcock, H. W. (1961). The Topology of the Sun’s Magnetic Field and the 22-YEAR Cycle. The Astrophysical Journal133, 572.

Coriolis, G. G. (1835). Théorie mathématique des effets du jeu de billard. Carilian-Goeury.

Howe, R. (2009). Solar interior rotation and its variation. Living Reviews in Solar Physics6(1), 1-75.

Parker, E. N. (1955). Hydromagnetic dynamo models. The Astrophysical Journal122, 293.

Parker, E. N. (1955). The Formation of Sunspots from the Solar Toroidal Field. The astrophysical journal121, 491.

Schou, J., Antia, H. M., Basu, S., Bogart, R. S., Bush, R. I., Chitre, S. M., … & Gough, D. O. (1998). Helioseismic studies of differential rotation in the solar envelope by the solar oscillations investigation using the Michelson Doppler Imager. The Astrophysical Journal505(1), 390.

Seehafer, N. (1996). Nature of the α effect in magnetohydrodynamics. Physical Review E53(1), 1283.

## A practical Introduction to Vectors

The mathematical concept of a “vector” is ubiquitous in the realms of physics, engineering and applied mathematics. Typically, the concept is first introduced (usually in a first semester physics course or perhaps a trigonometry or calculus course) as a quantity with both a magnitude and a direction, and is usually represented as a line segment with an arrow at the end of it (like a ray).

Often, these introductory treatments will distinguish what a vector is by contrasting it to what a scalar is. A scalar is just a quantity (albeit possibly with some units of measurement), such as length, weight, mass, speed, an amount of money, or a frequency. On the other hand, examples of vectors would be things like velocity, acceleration, force, momentum, and position (relative to some set of coordinates). In other words, they have a direction as well as a magnitude.

Although vectors can in principle be adapted to any coordinate system, for the sake of simplicity, we will presuppose a Cartesian coordinate system in either 1, 2, or 3 dimensions. In this representation, we’ll have a horizontal x axis and a vertical y axis for 2D, and for 3D we’ll have the y axis horizontal, the z axis vertical, and the x axis coming out of the page at the reader. This is just a matter of historical convention. Occasionally one might see the z axis being used as the one coming out of the page, but most of my pictorial examples involved the former and not the latter. This is a type of coordinate system with what’s called an “orthonormal” basis. I’ll cover the concept of a basis in a later post, but in practical terms, orthonormal just means that the coordinate axes are perpendicular to one another. Any points or vectors in the space are constructed by adding linear combinations of  unit vectors (magnitude equal to 1) in the x, y and z directions.

A vector can be represented by a letter with a little arrow on top of it. but since it’s a PITA to write them that way in this format, I’ll just use capital letters to name vectors for now. Suppose we have a vector, A. There are a number of common notational conventions for representing it.

A = [a1, a2, a3],

or

,

where the i, j and k with the little hats each represent a unit vector (a vector of unit length) in one of the component directions (x, y, and z respectively in this case), and the a1, a2 and a3, (or A_x, A_y and A_z) symbols represent the “components” of the vector in the x, y and z directions respectively. I will use the A = [a1, a2, a3] format for convenience.

So, for example, supposing you were traveling at 10 m/s in the x-direction. Then your velocity vector v = [10, 0, 0] m/s, since you’re not moving in the y or z directions at all.

Vectors can be added and subtracted just like scalar quantities, but there is a procedure to it. The process is to add or subtract each component of the vector.

For example, if you had a vector, A = [a1, a2, a3], and another vector, B = [b1, b2, b3].

Then A + B = [a1 + b1, a2 + b2, a3 + b3].

Similarly, A – B = [a1 – b1, a2 – b2, a3 – b3].

Let’s try a more concrete example. Supposing your positive x component corresponded to East, your positive y component corresponded to North, and your positive z compodent corresponded to up in the air. Supposing you had two cars driving in an open field. Car A has a velocity v_a = [40, 30, 0] km/hr, (that’s diagonal motion of 40 km/hr east and 30 km north) relative to an observer on the ground, and car B is moving at a velocity v_b = [0, -40, 0] km/hr.

If you wanted to know A’s velocity from the perspective of B, then you’d subtract the velocity of B from the velocity of A.

v_a – v_b = [ 40 – 0, 30 – (-40), 0 – 0] km/hr = [40, 70, 0].

The following Khan Academy video explains some more examples of this:

Something that is typically not mentioned in such introduction (mostly for the sake of expediency), is the fact that this definition of a vector is really only a special case of a much broader-reaching concept rife with many other special cases, each comprising their own unique attributes and applications. Later on, it is customary for the student to learn a more generalized concept of vector “spaces,” whereby a vector space is defined by a list of rules by which the elements of a space must abide in order to qualify as a vector space. This is traditionally taught in a first course on linear algebra, along with concepts such as “rank,” “dimension,” “linear independence,” and a “basis,” and opens the door to a lot of vector spaces which wouldn’t necessarily fit the description of a vector that students are typically taught in introductory physics. These included things like vector “fields” and topological vector spaces, of which metric spaces are a subset, of which normed vector spaces are a subset, of which inner product spaces and banach spaces are subsets, of which Hilbert spaces are a subset (etc). The latter, (Hilbert Spaces), are ubiquitous in quantum mechanics and quantum field theory for example. There may also come a point at which a student comes to learn that some vectors are also a subset of a class of mathematical objects known as “tensors.” Not all vectors are tensors, (but all first order tensors are actually vectors), and not all scalars are tensors, (but all zeroth order tensors are scalars).

In actuality, the hierarchy is a bit more involved than depicted here, and understanding it would require covering some concepts such as “completeness” from a branch of math known as functional analysis, which I will not attempt to cover here.

Don’t worry about not knowing what distinguishes these types of spaces from one another right now. The point is to state up front that there multiple levels to this concept of vectors, instead of simply conveniently neglecting to mention that until it can’t be put off any longer, as is often done in the earlier college courses. So rather than getting lost down that rabbit hole, just know there are a few basic operations and notational conventions to understand for vectors in the sense of a “quantity possessing a magnitude and a direction,” but that those are but a special case of a broader and very useful mathematical concept that has applications in various scientific sub-disciplines.

In a subsequent lesson, I can go over vector multiplication (i.e. dot products/inner products and cross products), scalar projections, vector projections, vectors as functions of independent variables, vector calculus, the concept of a vector space, and various useful operations involving vectors and vector functions.