## 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.

## The Structure and Properties of the Sun

In my most recent post, I discussed the characteristics of the sun’s 11 and 22 year cycles, the observed laws which describe the behavior of the sunspot cycle, how proxy data is used to reconstruct a record of solar cycles of the past, Grand Solar Maxima and Minima, the relationship between Total Solar Irradiance (TSI) and the sunspot cycle, and the relevance of these factors to earth’s climate system. Before elaborating on the sun’s role in climate change, I’d like to take a look at the mechanism in terms of which the magnetic cycles underlying these solar cycles actually arises, but in order to do that, it’s necessary to first go over some basics:

The Structure of the Sun

The Core: The core of the sun is where pressures and temperatures are high enough to facilitate the nuclear fusion reactions which power the sun (Eddington 1920). The sun is so hot that there are few (if any) actual atoms of hydrogen and helium gas (Bethe 1939). They exist in a plasma state; the gases are ionized and cohabit with free electrons. So protons are being collided and fused into helium nuclei in what’s known as the proton-proton chain (PPO Chain), which is the dominant fusion process in stars of masses comparable to (or less than) the sun. In the PPO chain, two protons fuse and release a neutrino. The resulting diproton either decays back into hydrogen via proton emission, or undergoes beta decay (emitting a positron), which turns one of the protons into a neutron, thus yielding deuterium. The deuterium then reacts with another proton, producing 3He and a gamma ray. Two 3He from two separate implementations of this process then fuse to produce 4He plus two protons (Salpeter 1952).

Image by UCAR: Randy Russell (Windows of the Universe project)

This region comprises about the first 0.2 of the solar radius, and exhibits temperatures on the order of 14 – 15 million Kelvin.

The Radiative Zone: From about 0.2 – 0.7 solar radii from the center is the radiative zone. The nuclear fusion reactions in the core produce radiation which gets reiteratively absorbed and reemitted by various particles in this zone in a random zig-zag pattern. It can take hundreds of thousands of years for photons in this region to reach the surface in this manner. This succession of absorptions and reemissions also results in the photons which escape into the convective zone being of longer wavelength and lower energy than the gamma ray photons that were initially emitted from the nuclear fusion reactions in the core. The temperatures here are still on the order of a few million Kelvin.

The Layers of the Sun. (Image by Kelvinsong).

The Tachocline: The interface layer at the boundary separating the radiative zone and convective zone is called the Tachocline. The radiative and convective zones obey different rotational laws, and the advection of angular momentum in the Tachocline (which acts as a transition layer between them) is controlled by horizontal turbulence (Spiegel 1992). Changes in fluid flow velocities across this layer can twist magnetic field lines.

The Convective Zone: This zone runs from about 0.7 solar radii up to the sun’s surface (the Photosphere). As the name implies, heat pockets convect through the ionized gas in this region towards the surface in a manner similar to a boiling pot of water. The convective zone is just cool enough that many of the heavier ions are able to retain at least some of their electrons, which means that the material in the convective zone is more opaque, and thus it’s harder for radiation to get through it. Consequently, a lot of heat gets trapped in this zone, which causes the material to “boil” (or convect). As we’ll see soon enough, this property is important in making possible the solar dynamo mechanism that underlies the solar cycle. The temperature gradient of the convection zone ranges from around 2 million Kelvin near the tachocline to roughly 6,000 Kelvin at the sun’s surface.

The Photosphere: This is the sun’s visible surface layer. The photosphere includes features such as the following:

Sunspots: these are dark regions representing high magnetic flux, and are associated with changes in polarity of the sun’s magnetic field (Hale 1908). I’ve already covered some of the properties of sunspots, the sunspot cycle, and how sunspot abundance can be used as a proxy for Total Solar Irradiance (TSI) in a recent post here, and will cover the mechanism underlying the sunspot cycle when I elucidate the Solar Dynamo  in a follow-up post.

Sunspot c/o NASA

Faculae: these are bright regions which are also highly magnetized, but whose magnetic fields are concentrated in considerably smaller bundles than in sunspots (Richardson 1933). During solar maxima, when abundant dark sunspots are blocking the emission of heat and light, these bright regions overpower the darkening effect of the sunspots, thus resulting in the net increase in luminosity that we observe during solar maxima (Spruit 1982).

Faculae c/o NASA

Granules: these are the tops of convection cells which cover nearly the entire photosphere in ever-changing grain-like patterns (Langley 1874). The bright center bulges of the granules are regions where the plasma is rising to the surface, whereas the darker boundaries around them are where the plasma is cooler and sinking back down.

Granules: Image by Goran Scharmer and Mats G. Löfdahl

Supergranules c/o NASA

And supergranules: these are huge polygonal convective cells which are larger and last longer than granules, and are outlined by the chromospheric network. They have an average diameter of about 13,000 – 32,000 km and last an average of about 20 hrs (Simon 1964, Hagenaar 1997).

Additionally, the photosphere is also where solar flares originate. For more on solar flares, check out this article from space.com.

The Chromosphere: This is an approximately 2,000 km layer of gas residing above the photosphere, in which temperatures run from around 6,000 – 25,000 K (Vernazza 1976, Carlsson 1994). As a consequence, hydrogen in this layer emits light of a reddish color in a process called H-alpha emission (Michard 1958). The electrons of a particular atom can only occupy certain specific allowed energy states. They are quantized. These states correspond to specific principle quantum numbers, n = 1, n = 2, n = 3 etc… When an electron drops from one allowed energy state to a lower level, it emits a photon whose wavelength (and thus color) corresponds to the difference in energy between those two allowed states. Contrastingly, only a photon corresponding to the exact energy difference between two states can be absorbed by the electron to push it up to the higher energy state. In the limited case of hydrogen-like atoms, the relationship is described by the Rydberg formula, which is as follows:

1/λ = RZ2(1/nf2 – 1/ni2),

where λ is the wavelength of the photon emitted or absorbed, Z is the atomic number of the hydrogen-like atom in question (1 in this case), R is the Rydberg constant (approximately 1.097*10^7 m-1 in S.I. units), ni is the quantum number of the electron’s initial state, and nf is the quantum number of its final state. H-alpha emission occurs when an electron of a hydrogen atom drops from its third lowest allowed energy level (ni = 3) to its second lowest (nf = 2). ). If you plug in these values, you get that λ = 656 nm, which is in the red light range.

This transition is part of what’s called the Balmer series, which consists of all the allowed transitions between ni ≥ 3 and nf = 2 (Bohr 1913).

Other chromospheric features include the chromospheric network, which is a web-like pattern outlining supergranule cells, which results from bundles of magnetic field lines concentrated in the supergranules (Hagenaar 1997). Spicules are jet like eruptions of hot gas which protrude from the chromospheric network thousands of kilometers above the chromosphere and into the corona. Filaments and prominences are huge plumes of gas suspended as loops above the sun by magnetic fields, which underlie many solar flares (Kiepenheuer 1951, Menzel 1960). Plage, which are also associated with concentrations of magnetic field lines, appear as bright spots surrounding sunspots (Leighton 1959).

The Transition Region: This thin region resides between the cooler chromosphere and the much hotter corona. For this reason, temperatures rapidly increase with radial distance outward, ranging from 25,000 K around the boundary of the chromosphere to about 10^6 K out near the corona (Peter 2001).

The Corona: This aura of plasma is the sun’s outer atmosphere. It reaches temperatures far greater than at the sun’s surface (on the order of 1 – 3.5 million degrees Celsius). The reasons for these extreme temperatures comprise a long standing puzzle in solar astrophysics known as the Coronal Heating Problem, which is beyond the scope of this brief outline. That said, these extreme temperatures result in jets of plasma at speeds of up to 400 km/s (Brueckner 1983). Consequently, some of this ionized gas overcomes the sun’s gravitational pull, escapes, and subsequently cools down (Hundhausen 1970). This is the solar wind (Brueckner 1983). Incidentally, there is evidence that the sun’s rotation rate was greater in the past, and that the solar wind is responsible for its subsequent loss of angular momentum (Durney 1977). The corona is also the region from which Coronal Mass Ejections (CME) emerge. As the name implies, these can involve the ejection of billions of tons of plasma as a result of the reconnection of opposite ends of complicated magnetic field loops in the corona, and often accompany strong solar flares and filament eruptions. Not all solar flares and filament eruptions result in a CME though. Solar flares typically involve the expulsion of long radio wave radiation all the way up the EM spectrum through visible light (or even gamma rays), as well as protons and electrons, the latter of which can result in x-ray emissions via bremsstrahlung radiation (Arnoldy 1968). Charged particles in flares are accelerated by a combination of electric fields and magnetohydrodynamic waves (Miller 1997). You can read more about CME events here, here and here and solar flares here and here.

Any one of these layers and properties I’ve described here could be elaborated upon in greater detail, but this should be sufficient for the purpose of seguing into an explanation of the solar dynamo: the physical mechanism in terms of which solar cycles arise, which will be the topic of my next installment of this series.

Related Articles:

References:

Arnoldy, R. L., Kane, S. R., & Winckler, J. R. (1968). Energetic solar flare X-rays observed by satellite and their correlation with solar radio and energetic particle emission. The Astrophysical Journal151, 711.

Bethe, H. A. (1939). Energy production in stars. Physical Review55(5), 434.

Bohr, N. (1913). The spectra of helium and hydrogen. Nature92, 231-232.

Brueckner, G. E., & Bartoe, J. D. (1983). Observations of high-energy jets in the corona above the quiet sun, the heating of the corona, and the acceleration of the solar wind. The Astrophysical Journal272, 329-348.

Carlsson, M., & Stein, R. F. (1995). DOES A NONMAGNETIC SOLAR CHROMOSPHERE EXIST?. The Astrophysical Journal440, L29-L32.

Durney, B. R., & Latour, J. (1977). On the angular momentum loss of late-type stars. Geophysical & Astrophysical Fluid Dynamics9(1), 241-255.

Eddington, A. S. (1920). The internal constitution of the stars. The Scientific Monthly, 297-303.

Hagenaar, H. J., & Schrijver, C. J. (1997). The distribution of cell sizes of the solar chromospheric network. The Astrophysical Journal481(2), 988.

Hale, G. E. (1908). On the probable existence of a magnetic field in sun-spots. The astrophysical journal28, 315.

Hundhausen, A. J. (1970). Composition and dynamics of the solar wind plasma. Reviews of Geophysics8(4), 729-811.

Kiepenheuer, K. O. (1951). The Nature of Solar Prominences. Publications of the Astronomical Society of the Pacific63, 161.

Langley, S. P. (1874). On the structure of the solar photosphere. Monthly Notices of the Royal Astronomical Society34, 255.

Leighton, R. B. (1959). Observations of Solar Magnetic Fields in Plage Regions. The Astrophysical Journal130, 366.

Menzel, D. H., & Wolbach, J. G. (1960). On the Fine Structure of Solar Prominences. The Astronomical Journal65, 54.

Michard, R. (1958). INTERPRETATION OF THE H* alpha/SPECTRUM OF THE CHROMOSPHERE. Compt. rend.247.

Miller, J. A., Cargill, P. J., Emslie, A. G., Holman, G. D., Dennis, B. R., LaRosa, T. N., … & Tsuneta, S. (1997). Critical issues for understanding particle acceleration in impulsive solar flares. Journal of Geophysical Research: Space Physics102(A7), 14631-14659.

Peter, H. (2001). On the nature of the transition region from the chromosphere to the corona of the Sun. Astronomy & Astrophysics374(3), 1108-1120.

Richardson, R. S. (1933). A Photometric Study of Sun-Spots and Faculae. Publications of the Astronomical Society of the Pacific45(266), 195-198.

Salpeter, E. E. (1952). Nuclear reactions in the stars. I. Proton-proton chain. Physical Review88(3), 547.

Simon, G. W., & Leighton, R. B. (1964). Velocity Fields in the Solar Atmosphere. III. Large-Scale Motions, the Chromospheric Network, and Magnetic Fields. The Astrophysical Journal140, 1120.

Spiegel, E. A., & Zahn, J. P. (1992). The solar tachocline. Astronomy and Astrophysics265, 106-114.

Spruit, H. C. (1982). The flow of heat near a starspot. Astronomy and Astrophysics108, 356-360.

Vernazza, J. E., Avrett, E. H., & Loeser, R. U. D. O. L. F. (1976). Structure of the solar chromosphere. II-The underlying photosphere and temperature-minimum region. The Astrophysical Journal Supplement Series30, 1-60.

Image Credits:

Layers of the Sun by Kelvinsong (Own work) [CC BY-SA 3.0 (http://creativecommons.org/licenses/by-sa/3.0)], via Wikimedia Commons

Granules by Goran Scharmer/Mats G. Löfdahl of the Institute for Solar Physics at the Royal Swedish Academy of Sciences.

Proton-Proton chain Image by UCAR: Randy Russell (Windows of the Universe project)

Sunspots, Faculae, and Supergranules by NASA