238 22. Kinematic Dynamo Theory; Mean Field Theory Dynamo Solutions We seek solutions to the Induction (dynamo) equation ∂B/∂t = λ∇2B + ∇ x (u x B) (22.1) that do not decay with time and have no external exciting field. These are called a dynamo. Obviously the induction term must offset diffusion. To order of magnitude, we must have ∇ × ( u × B ) ~πuBL~λ∇2B ~π2λBL2⇒uLλ~π (22.2) where L is some characteristic size of the region in which the field is generated. The dimensionless number uL/λ is called the magnetic Reynolds number and it must be sufficiently large (about 10 or more) in order that a dynamo exist. However, the existence of a dynamo turns out to be a matter of some subtlety because it depends on the form of the flow as well as the magnitude. Most simple flows do not produce dynamos irrespective of their magnitude. In this chapter we consider kinematic dynamos. These are solutions where we specify the velocity field (without asking where it came from). However, we will focus on physically plausible motions, especially those relevant to mean field models. Mean field is standard physics jargon for any situation where small scale fluctuations are averaged, yielding a large scale outcome. (In this case, it means there are small scale motions exciting a large scale field). A fully dynamical dynamo is one where the velocity field is determined from solution of the equation of motion (which includes the Lorentz force arising from the field). In the next chapter, we will talk about the form that convection takes in the presence of a magnetic field and discuss a little the fully dynamical dynamos (for which only numerical solutions exist).239 A “Simple” Example of Dynamo Action (The α effect) Actually, there are no really simple examples of dynamo action since they all involve 3D velocity fields and there is no “closure” of the dynamo equation when the scale length of the flow is similar to the scale length of the convection. (By lack of closure, we mean that the action of the flow on a specified field might reproduce the original field but in addition produces a field that is usually more spatially complex than the one we are trying to make.) But let’s take the simplest case known, which turns out to be a case where we assume a small scale flow and attribute to it the property of helicity (defined later). Specifically, consider a flow field and magnetic field of the forms:  u = u ( q ∑ q )ei q . r ;  B = B 0ei k . r +σt+ b ( r )eσt (22.3) where q>> k is assumed (i.e. the flow is small scale but part of the field is large scale, i.e. small wavevector). The idea is that the flow u acts on the large scale field B to produce a small scale field b (which we will compute). The flow then acts on the small scale field to reproduce the large scale field. Let’s see how this works:  u × B = eσt[ u ( q ) × B 0] q ∑ei ( q + k ). r ∴ b ( r ) = b ( q  q ∑)ei( q + k ). r ; σ b ( q ) ≈ −λq2 b ( q ) + i q × [ u ( q ) × B 0]∴ u × b =1(σ+λq2) q , ′ q ∑.i u ( ′ q ) × { q × [u( q ) × B 0]}ei( q + ′ q + k ) (22.4) (The assumption q>>k is used again in the second line.) In the spirit of mean field theory, we focus on those contributions that can affect the large scale field. So we choose q′ = -q, and we see that u x b can be written in the form α.B where α is a tensor :  α =1(σ+λq2) q ∑.i[ u (− q ) × u ( q )] q (22.5) (making use of the fact that q.u = 0 for incompressible flow). Now u(q).[q x u(-q)] = q.[u(-q) x u(q)] so this tensor clearly involves a measure of the helicity, defined as u.(∇xu), the dot product of vorticity and flow. The name240 given to this scalar quantity is self-evident if one thinks about the properties of a fluid element that follows a helical path. The crucial idea is that this can have a non-zero mean (as well as fluctuating parts) but the mean field part is most important since it can lead to a generation of large scale fields. In general, this alpha model (as it is so called) yields an equation of the form: ∂ B ∂t=λ∇2 B + ∇ × ( α . B ) (22.6) In the particular case where we treat alpha as a scalar (i.e., only diagonal elements, all of the same size), we can visualize the alpha effect as in the following cartoon: A current is created that is parallel (or antiparallel) to the existing field.241 Mathematically, this alpha effect can by itself sustain a dynamo: (σ+λk2) B 0= iα k × B 0⇒ (σ+λk2) k × B 0= iα k × ( k × B 0) = −iαk2 B 0= −iαk2{iα k × B 0(σ+λk2)}⇒ (σ+λk2)2=α2k2∴σ> 0 ⇒αλk> 1 (22.7) This last requirement for dynamo growth is equivalent to exceeding a critical magnetic Reynold’s number (since alpha has dimensions of a velocity and k is an inverse length). Note however that alpha is not the fluid velocity, in fact it is roughly fluid velocity times a small scale magnetic Reynold’s number (α/λq), which may well be a small number. So in this model, at least, the criterion for a dynamo is something like (small scale Magnetic Reynold’s number) x (large scale magnetic Reynold’s number) > 10. This alpha effect is popular in mathematical models. There is some doubt whether it is the dominant process in actual dynamos, at least in planets. (It is popular in stellar dynamo models). The dominant process may not be simply characterized; it is complex. But one other effect is likely to be important: Differential rotation (shear). The ω -effect (Omega Effect).242 In the context of the Cartesian model discussed above, this is a large scale flow that converts one large scale component into another. Specifically, consider the flow in the x-direction in the form ωz, and suppose the initial field (magnitude B0) is purely in the z-direction. The induction effect is ∂ B ∂t= ∇ × (ωz ˆ x × B0 ˆ z ) =ωB0 ˆ x