We just finished talking about the classical, spherically symmetric, (quasi) time-steady solar interior. In reality, it s not any of those things: Helioseismology: the Sun pulsates & jiggles like a big ball of jello, and we use those pulsations to learn about the interior. The solar dynamo: the Sun s complex internal motions drive internal currents that generate its magnetic field. Helioseismology: When perturbed, stars oscillate sinusoidally around a mean hydrostatic (u = 0) steady state. For the Sun, the oscillations have small amplitudes ( linear perturbations ). Some other kinds of stars pulsate with large amplitudes (brightness variations of several visible magnitudes). What do the pulsations look like? At any fixed location (r,θ,φ), one sees sinusoidal oscillations in ρ, P, T (oscillating around a mean value), and u (oscillating around zero). For example, ρ(r,θ,φ,t) = ρ 0 (r) + ρ 1 X(r)Y(θ,φ)e iωt At the surface, we observe variations in brightness & Doppler shift (i.e., line-of-sight projected velocity) in spectral lines. The oscillations have discrete frequencies because they re in a bound cavity. (Think of nodes of vibration in a string when held fixed at both ends.) Solar oscillations also come in several different types, depending on the dominant force(s) inside the star: In inner regions, gravity is the main restoring force. In convectively stable regions, blobs bob up & down due to slight variations in buoyancy: internal gravity waves, or low-frequency g-modes. In outer regions, the pressure gradient is the major restoring force. The resulting acoustic waves, or high-frequency p-modes, are what we tend to observe in the Sun. 7.1
Cravens Chapter 5.1.4 deals with gravity waves, but those still haven t been directly observed in the Sun. (What perturbs the interior in the first place? For the Sun, it s the chaotic rising & falling of blobs in the convection zone!) If the p-modes are like acoustic waves (see Cravens Chapter 4.8.1), they propagate at a phase speed given by the sound speed, γp c s = ρ = γk B T µm H where the quantities ρ, P, and T are the time-averaged background ( zero-order ) quantities, not the fluctuating ones. In a bound object like the Sun, the waves aren t sinusoids that go on forever. In r, they ve got standing wave nodes at the surface and center. In θ,φ they behave like spherical harmonics Y lm (θ,φ). You ve seen the Y lm s if you ve seen solutions to Poisson s equation ( 2 f = 0) or the Schrödinger equation in 3D spherical coordinates. The l = m = 0 solution is spherically symmetric (Y lm = constant). l 0 is the degree. m (varies from l to +l) is the azimuthal order. l measures the overall horizontal wavelength on the surface, but doesn t specify whether the angular fluctuations are along longitude or latitude lines. m tells us how many periods circle the equator, since Y lm e imφ. i.e., l > 0, m = 0 l > 0, m < l l > 0, m = l 7.2
If there s no rotation nor other symmetry-breaking effects to define the equator, then all m solutions for a given l tend to be excited equally. The sinusoidally oscillating quantities thus vary like ρ(r,θ,φ,t) = ρ 0 (r) + ρ 1 X n (r)y lm (θ,φ) exp( iω nl t) and the frequencies ω depend on l (and details of the radial part of the solution, X n (r)), but not on m.... What about those X n (r) functions? In general, they depend on a solution of a 4th order differential equation. With proper boundary conditions applied, there s a discrete spectrum of radial eigenfunctions, each corresponding to its own eigenfrequency ω. They re counted by radial order n = {# of nodes from core to surface} The observed p-modes on the Sun mostly have n 1. In this limit, the modes start to behave more like actual waves, propagating through a slowly varying background. In fact, for n 1 p-modes, the frequency is given by a wave-ish dispersion relation, ω = c s k is also expressible as f = ω 2π = c s λ 7.3
and for n nodes from core to surface, λ R n Really, though, it s more like where q is an order-unity constant. i.e., f so f = nc s R. f ( n+ l 2 +q )[ R # of nodes from core to surface wave transit time from core to surface 0 ] 1 dr c s (r) If one measures multiple frequencies of neighboring high-order modes, one can use the so-called large frequency separation ω = ω n+1,l ω n,l to probe the radial dependence of c s (r) i.e., we measure T(r). Also, scaling arguments from our stellar interiors work tell us that dr R R R 3 t D! c s P0 /ρ 0 T0 GM But if we know a star s R (from its L & T eff ), this helps us solve for the asteroseismic mass.... Measuring pulsation frequencies also lets us measure details about the rotation of a star. Rotation is one effect that definitely sets up a clear-cut equatorial plane of symmetry. Recall that Y lm e imφ, so a perturbed fluid variable Φ will have a combined longitude & time dependence of Φ exp[ i(mφ+ωt)] i.e., m 0 modes march around in longitude. The classical solutions (at least for slowly rotating stars) are all applied in the star s own rotating frame of reference. 7.4
For an external observer, we see a given stellar longitude φ march around due to rotation itself: φ inertial = φ + Ωt. Thus, the fluid variation that we see is Φ exp[ im(φ inertial Ωt) iωt] and since we re looking into the system at a fixed φ inertial, the fluctuations we see go as Φ exp[ iω obs t] where ω obs = ω mω This rotational splitting helps us measure Ω(r, θ). From helioseismology, we ve seen that Ω isn t constant... the Sun undergoes differential rotation: Contours show constant Ω. Radiative interior is rigid. Convection zone: equatorial Π 25 days, polar Π 35 days. Ω/Ω 0.3 Why is this how it is? Still a topic of vigorous research & debate! In just a few rotations, the equator laps the higher latitudes! The interior gets twisted up... we ll see soon how this affects the magnetic field. Note the tachocline: strong shear layer at the base of the convection zone; this is where dynamo activity appears to be concentrated (more later). 7.5
There s also meridional circulation; i.e., nonzero values of u r and u θ. The magnitudes are even smaller than convective blob rise/fall times (u merid 10 2 km/s), so it takes decades to transport plasma over distances of order R. Recent work suggests there may be multiple meridional circulation cells in the convective zone, not just one. The solar dynamo In the convection zone, the plasma is in motion... in the r direction (rising/falling blobs), in both r and θ (meridional circulation), & in the φ direction (differential rotation). If any of those motions give rise to currents, then they can influence B via Ampere s law. Recall the MHD induction equation: B = (u B) + D B 2 B t where 1st term shows how the field is frozen-in to the flow, and the 2nd term describes magnetic diffusion (i.e., gradual spreading and dissipation of the field, due to resistivity & particle collisions). Given a tiny seed field, one can imagine the right set of motions (u) and fields (B) that would make the first term contribute to a net amplification ( B/ t > 0). 7.6
Over long times, the 1st term can generate B, and the 2nd term destroys it... dynamic equilibrium = dynamo! When we see magnetic fields in the universe, we can compare the object s age to the time it would take for D B to destroy any fossil field from the object s birth; B t D B 2 B i.e., B t D B B l 2 t diffuse l2 D B where l system size, and D B is a function of plasma ρ, T, & composition. If the system s age t diffuse then a dynamo must be there to maintain the field we see. This seems to be the case for the Sun, but what s the u? It s still a topic of active research, but I ll take you through the overall process that most believe is going on: the αω dynamo.... Start with observations! But what s going on with the magnetic field over the 11 (or 22) year cycle? 7.7
George Ellery Hale noticed several patterns... Hale s law(s) The B-field reorganizes itself on a roughly 11-year cycle... but not exact clockwork repetition; it s 11 ±{1 or 2}. Sunspot minimum: dipole field, aligned with rotation axis. Sunspot maximum: most of the field is concentrated into sunspot pairs (one +, one ) with the leading/following polarities of the opposite sense in the N/S hemispheres. Polarities flip every 11 years, so one full cycle is really 22 years. Note the trailing polarity in each hemisphere becomes the next cycle s polar cap polarity. Joy s law: Soon after Hale s original work, Alfred Joy noticed that most sunspot pairs tilt towards the equator. The leading spot tends to be at lower latitude than the following spot. [no matter their polarities!]... Next, can the so-called αω dynamo theory explain all these trends? 7.8
(1) The Ω effect: Start with a poloidal field (i.e., B r and B θ components only; field lines stay in planes of constant φ. (Example: the solar minimum dipole field.) Because the solar interior has β = P gas /P mag 1, interior motions of the gas (u 0) push around the magnetic field lines. Differential rotation stretches the field lines by carrying plasma at equator to higher φ faster than the plasma nearer the poles. Poloidal field gets turned into toroidal field, & intensified: all B φ.... (2) The α effect: To maintain a cyclic regeneration of the Sun s field, the toroidal field should be turned back into poloidal (preferably with stronger field strength) so the process can keep going indefinitely. This is the step with the most controversy. One thing we know, however, is that there are many ways for small blobs to rise up from the base of the convection zone, and carry magnetic fields with them. (high β) Are the blobs just the turbulent convective blobs themselves (i.e., they rise because they re hot and in pressure equilibrium)? Or are they knots of unusually strong B, which in total P equilibrium have lower density (high P mag, low P gas ) and thus rise buoyantly, and stretch the field into loops? ( Kinked into an α shape?) 7.9
When the field is mostly toroidal, the magnetic field gets pushed up by a rising blob, and forms a sunspot pair. If B φ > 0, the leading spot has negative polarity. If B φ < 0, the leading spot has positive polarity. The observed Hale polarity law is consistent with this, e.g., In 1996, north polar cap is + Ω effect makes B φ < 0 in north hemisphere At next solar maximum, the leading spots in north are + But we haven t examined how the toroidal field is converted back to poloidal! No matter what the rising blobs are, their motions are affected by the Coriolis force, a cor = 2Ω u. The rising speeds are slow... the dominant motion is due to differential rotation (i.e., u φ > 0 near equator in a rotating frame). If u φ was the only component, this pretty much explains Joy s law (tilts). a cor,θ = 2Ωu φ cosθ. 7.10
Babcock & Leighton noticed that tilted sunspot pairs eventually DECAY (i.e., the fields diffuse away, D B 0). The leading spots are nearest the equator, and they cancel out with their trans-equatorial counterparts. However, the trailing spots are nearer to the poles, and diffusion may be enough to ooze their polarity up to the poles i.e., making new poloidal field. It agrees with Hale s polarity laws, and we sort of see it: That s one idea. Parker had an alternate idea. He noted that there are other components of u that are affected by the Coriolis force. Deep down (near base of convection zone), the blobs probably have strong fields; i.e., high P mag & thus low P gas, if in total pressure equilibrium. Low-pressure regions are sites of converging flow, and the Coriolis force turns those into cyclonic flow (see right side of above cartoon). 7.11
Sense of twist is similar to hurricanes on Earth (counter-clockwise in north, clockwise in south). Again, the B-field gets dragged along: Because the sign of B φ is different in north & south, the new poloidal fields line up in the opposite direction as the original poloidal field. Eventually, the fragmented bits of poloidal field merge together & migrate to the poles, to produce a new dipolar poloidal field. This process occurs with roughly 11-year periodicity, so the full solar cycle (to get back to the same poloidal polarity) is really 22 years in duration. Both ideas are summarized in this cartoon: (Upper branch: Parker s cyclonic blobs; lower branch: Babcock/Leighton diffusing fields) 7.12
Note: There is still no complete theory of the dynamo! These basic steps must be going on, but we still don t know why the period is 10 years, and not 1 or 100.... Still, this process does enhance the field magnitude, and thus is a true dynamo. Schematically, the field is morphed as follows: The field is stretched by the bulk flows (i.e., kinetic energy is transferred to magnetic energy). New magnetic energy is redirected in space by twisting and folding the field, which further amplifies it. Magnetic diffusion smears out the localized twists and folds, thus leading to merging of the field on large spatial scales. To maintain a steady solar cycle, all that buildup must be balanced by losses. That s accomplished by some combination of Diffusivity, which drains magnetic field from the system in addition to facilitating the merging above. The Sun also sheds magnetic fields up through its surface in the form of highly twisted plasmoids (e.g., prominences, coronal mass ejections). More about those later. 7.13