38. DYNAMOS; MAGNETIC FIELD GENERATION AND MAINTENANCE

Transcription

1 38. DYNAMOS; MAGNETIC FIELD GENERATION AND MAINTENANCE (Much of this Section derives from H. K. Moffatt, Magnetic Field Generation in Electrically Conducting Fluids, Cambridge, 978) The study of dynamos is motivated by one of the most fundamental problems of physics: How to explain the structure, dynamics, and maintenance of magnetic fields in the universe. Dynamo action will be defined precisely later in this Section. For now, we say that a dynamo is a process that can generate and amplify magnetic fields in the presence of resistive diffusion. Although it is generally accepted that magnetic fields were not produced during the Big Bang, magnetic fields on the order of 0-24 T were likely created during the very early stages of the universe, before the onset of galaxy formation. The best current estimates of the magnetic field strength in existing galaxies is 0-0 T; how did the primordial field get amplified by 4 orders of magnitude? Further, galactic magnetic fields exhibit coherent structure on length scales that are much larger than the observed dynamical fluctuations. If the field is tied to the plasma, it should be tangled on relatively short length scales. How did this large scale structure come about, and how is it maintained? The resistive diffusion time (based on Coulomb collisions) for galaxies is estimated to be longer than the age of the universe, so there is no need to explain the lifetime of galactic fields. However, this is not the case for smaller astronomical bodies, such as the earth. The age of the earth is estimated to be about 4.5! 0 9 years (the age of the solar system), but the resistive diffusion time is only.5! 0 4 years, a difference of over 5 orders of magnitude. Any primordial field trapped during the earth s formation would have decayed by now, and yet the earth clearly remains magnetized. The terrestrial magnetic field is also dynamical; the geological record indicates that the polarity of the dipole field has reversed many times over the past several million years, with a mean period of about 4! 0 5 years. This record is indicated in the figure below.

2 The age of the sun is also about 4.5! 0 9 years, and the resistive diffusion time is on the order of 0 9 years, so, again, there is no clear need to explain the lifetime of the solar magnetic field. However, the solar field is incredibly dynamic and regular. Observations of the number of sunspots versus time show an almost regular period of years, as shown in the figure below. The peaks of the curve are called solar maxima, and the valleys are called solar minima. Solar maxima are associated with dynamical activity such as solar flares and coronal mass ejections that can cause disruptions to terrestrial communications, and damage satellites. The oscillations in the sunspot number are strongly correlated with reversals of the dipole field with a 22 year period. Further, during the beginning of a solar cycle (the end of solar minimum) the sunspots first appear at high latitudes (near the poles of the sun), and then migrate toward the equator as solar maximum is approached. During the next cycle the polarity of the spots is reversed, correlating with the reversal of the large scale dipole field. This behavior is captured in the butterfly diagram, which indicates the observed latitude of sunspots as a function of time. This is shown over several solar cycles in the figure below. What is the origin of this dynamical behavior? The solar magnetic field also exhibits structure on scale lengths much longer than the observed velocity fluctuations. The figure below on the left shows the shows what is called the granulation of the photosphere, or visible surface of the sun. These structures are thought to be the tops of thermal plumes generated by convection below the photosphere. Their characteristic size is about 0 6 m. Presumably this is the scale on which the coronal magnetic field is driven. The figure below on the right shows bright 2

3 loops that are the characteristic structures in the solar corona. They are believed to highlight the structure of the magnetic field. Their characteristic scale is around m. As is the case in galaxies, how can the short wavelength driving mechanism result in long wavelength coherent magnetic field structure? A mechanism for field generation and maintenance is also needed to explain the behavior of laboratory plasmas, in particular the RFP. In the figure below we sketch the toroidal flux as a function of time for an RFP discharge. The solid line represents experimental results. The dashed line indicates the results of a transport simulation using the experimental parameters. This calculation indicates that flux should rapidly decay as a result of resistive diffusion. In the experiment the flux is maintained for a much longer time. After toroidal field reversal is achieved in the RFP, only negative toroidal flux can be supplied by the external circuit. Without some field generation mechanism, the lifetime of the positive toroidal flux (due to the toroidal field between the reversal surface and the axis) is limited by the resistive decay of the toroidal field. This is especially apparent in the case where the toroidal field is held fixed (and negative) at the outer boundary. The toroidal flux will become negative on the resistive diffusion time, as indicated in the figure below. 3

4 Instead, we have seen in Section 37 that toroidal flux is generated during quasi-periodic bursts of activity, called sawteeth. The figure is repeated below. What is the relationship, if any, between relaxation and dynamo? What is the relationship between the behavior of the RFP plasma and astrophysical plasmas? For all the cases discussed above, the magnetic Prandtl number is very large, so that resistivity is the dominant dissipation mechanism. It is usual to ignore viscosity in the theory. Consider a fluid that occupies a volume V with surface S. It is characterized by a resistive diffusivity! = " / µ 0. It is surrounded by a vacuum of volume ˆV ; we take the volume V! = V + ˆV to comprise the entire universe! All flows V and currents J flow within V. The flow satisfies! " V = 0 within V, and V! ˆn = 0 on S. The fluid has a characteristic length scale L ~ V /3. The magnetic field B, which occupies V!, is produced entirely by J. Then at large distance from the fluid B must be a dipole field, so B ~ / r 3 as r! ". The field B evolves according to and!b!t = " # V # B ( ) + $"2 B!!!!!!!!!!!!!!!in V!!!, (38.)! " B = 0!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!in ˆV!!!, (38.2) 4

5 ! B" = 0!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!on S!!!, (38.3) where!..." indicates the jump across S ; Equation (38.3) says that there are no surface currents on S. Equations (38.) (38.3) are to be solved subject to an initial condition B(r,0) = B 0 (r). The magnetic energy is M ( t) = B 2 dv 2µ 0 "!!!. (38.4) V! It is finite since B 2 dv ~ / r 3 is bounded at infinity. We assume that initially M (0) = M 0 > 0. Clearly, if V = 0 then M (t)! 0 as t! ". Some non-zero velocity field is necessary to counteract the effects of resistive diffusion. We can now define dynamo action: For a given V and!, we say that V acts as a dynamo if M (t)! 0 as t! ". This includes cases where M (t) tends to a constant, where it fluctuates, and where it goes to infinity. What about the flow V(r,t)? Where did it come from? In classical dynamo theory, which we will review here, V is taken to be any kinematically possible flow field. What does this mean? The density and velocity must satisfy the continuity equation with!" + # $ "V = 0!!!, (38.5)!t V! ˆn = 0 (38.6) [ ] is said to be kinematically possible if Equations on S. The joint field V(r,t),!!(r,t) (38.5) and (38.6) are satisfied. We remark that only a small subset of these fields are also dynamically possible, i.e., satisfy the equation of motion; the allowable flows in classical dynamo theory are not so constrained. For this reason, classical dynamo theory is also called kinematic dynamo theory; the back reaction of the magnetic field on the flow, through the J! B force, is not considered. This may be valid if the kinetic energy is much greater than the magnetic energy. We also note that Equation (38.5) can be rewritten as d! dt = "!# $ V!!!, (38.7) where d! / dt = "! / "t + V # $! is the Lagrangian derivative. If the fluid is incompressible, so that d! / dt = 0, then kinematically possible flows are defined by the conditions! " V = 0 and V! ˆn = 0 on S. This will be the case in what follows. Why treat the velocity field this way? There are two reasons. First, the kinetic energy is much larger than the magnetic energy in many astrophysical settings, so it may be a good physical approximation. Second, and perhaps more important, the resulting theory is linear, and therefore amenable to analytic treatment; if V(r,t) is a given function, then Equation (38.) is linear in B. The questions classical dynamo theory 5

6 asks are: a) What kind of flows V(r,t) (if any) can produce a dynamo?; and, b) How can we test whether a given V(r,t) produces a dynamo? One constraint on the parameters of the problem can be obtained by considering the time evolution of the magnetic energy, given by Equation (38.4). Recall that in a previous Section we showed that dm dt = # V! J " BdV $ %# J 2 dv!!!, (38.8) V V where the integrals are taken over V because both the velocity and the current density vanish in ˆV. The first term is the rate of production of magnetic energy, and the second term is the rate of Ohmic dissipation. Their magnitudes can be estimated as and # V! J " BdV ~ VJB V ( ) L 3! V #!" J 2 dv ~!J 2 L 3 ~! & # B 2 L 3 & $ % µ 0 ' ( $ % µ 0 ' ( V $ B ' % & µ 0 L ( ) BL3 = V $ B 2 L 3 ' L % & ( ) * V M!!!, (38.9) L µ 0 L ) * M!!!. (38.0) 2 2 L For dynamo action we require dm / dt > 0, which yields VL /! " R m >, where R m is the global magnetic Reynolds number. A more rigorous estimate is R m >! 2 ~ 0. This result is not very insightful, but true nonetheless; we could have anticipated that the dynamics should dominate dissipation for magnetic field generation to occur. In practice, we expect that R m >> will be required. Now consider the important case where the system has some preferred axis, O-A, say, as in the figure. This may be because the system is rotating, as is the case for planets, stars, and galaxies, or because of inherent geometry, as in a magnetic fusion device. Such a configuration 6

7 has two types of magnetic fields: a poloidal field, B P, which lies in the (R,Z) plane everywhere parallel to the axis O-A; and, a toroidal field B T in the ê! direction, which encircles the axis O-A. How could a dynamo work in this situation? In order to sustain both fields, we need to devise a steady loop, denoted as B P! B T, in which poloidal field is converted into toroidal field, and toroidal field is in turn converted into poloidal field; they sustain each other. The first part of the loop, B P! B T, can be easily accomplished with differential rotation about the axis O-A, as shown below. If the velocity V T (r) = V! ê! does not correspond to rigid body rotation, then some of the poloidal field that was originally parallel to the axis will be bent into the toroidal direction, thus producing toroidal field. Now we just need to get B P from B T. It turns out that this is not simple. Now let the system have axial symmetry, so that everything is independent of the toroidal angle!. With B =! " A, the induction equation is!a!t = "E = V # B " $J!!!. (38.) We know from our previous studies of toroidal equilibria that in an axisymmetric system we can write! = RA ", so that and RB R =! "#!!!, (38.2) "z RB z =!"!!!, (38.3)!R µ 0 J! = "# * ( $ % "& ' ) * R &$ +, -!!!. (38.4) If we also decompose the velocity as V = V P + V T, then V! B = "V P # $%, and Equation (38.) becomes!"!t = #V P $ %" + &' * "!!!. (38.5) 7

8 We now assume that! " V =! " V P = 0!!!, (38.6) (so that V is kinematically possible) and V P! "# = 0!!!, (38.7) (so that! is advected with the fluid). We multiply Equation (38.5) by! / " and integrate over all space to obtain d dt $ 2! " 2 dv = % V #! " V P $ &'" dv + $ "( * " dv!!!. (38.8) V # V # Using Equations (38.6) and (38.7), the integrand in the first term on the right hand side can be written as (! / ")V P #$! = $ # V P! 2 / 2" ( ). Then, integrating the divergence, d dt 2! " $ 2 dv = % $ V # 2! " 2 V P & ˆndS + $ "' * " dv!!!. (38.9) S # V # As r! ",! ~ / r, so the surface integral scales like V P! ˆn / ", so the surface integral vanishes if V P! ˆn / " # 0 as r! ". This is trivially satisfied if V P! ˆn = 0 on S!, or V P = 0 in ˆV. Thus d dt 2! " 2 dv V # $ = "% * " dv V # $!!!. (38.20) From Equation (38.4), we can write! * " = # $ f, where f =!" / R. Then using the identity!" # f = " #! f ( ) $ f # "! and integrating the divergence, we find d dt $ 2! " 2 dv = V # The surface integral scales as / r 2 d dt S # $ R " ˆn %&" ds ' $ R &" 2 dv!!!. (38.2) V # as r! ", so, finally, 2! " 2 dv V # $ = % R &" 2 dv V # $!!!. (38.22) The integral on the right hand side is positive definite, so that the left hand side must continually decrease with time. Therefore! 2 (and hence B P and M (t) ) must vanish as t! ". Thus, dynamo action is impossible in axisymmetric systems. This famous result is known as Cowling s Theorem. Cowling s Theorem is one of many anti-dynamo theorems. Others state that B T cannot be maintained in axisymmetry, that dynamo action is impossible from purely toroidal flows, and that dynamo action is impossible from plane 2-D motions. Together, they imply that dynamos require some ingredient that is symmetry breaking, like threedimensionality. The step B T! B P must come from complex motions. So, dynamos may be possible, but they cannot be simple! 8

9 A relatively simple three-dimensional flow that can amplify magnetic field is called the stretch, twist, fold flow. It is illustrated in the figure below. The initial condition consists of a circular flux tube. It is then stretched to larger diameter. Since both the volume of, and the flux in, the tube must be conserved (the flow is incompressible), the stretching increases the magnitude of the magnetic field in the tube. The twist and fold stages return the tube to its original diameter, but with a larger magnetic field. The field will increase each time the process is repeated. This leads to field amplification, albeit on a small length scale. We remark that this flow is quite contrived, and unlikely to occur in nature exactly as diagramed above. But it is kinematically possible, and that s all that matters in classical dynamo theory. One could conceive of flows like this occurring irregularly as a result of turbulence. Then, on the average, field amplification might occur. The example of stretch, twist, fold flow implies that the axisymmetry required by Cowling s Theorem might be broken at short length scales if there is a small random (and therefore non-axisymmetric) component of the velocity field. As we did with turbulence, we write the flow and magnetic field as and V(r,t) = V 0 (r,t) +! V(r,t)!!!, (38.23) B(r,t) = B 0 (r,t) +! B(r,t)!!!, (38.24) where f 0 = f is an axisymmetric mean component, and! f is a small, threedimensional random component. Of course,! f = 0. The random component may arise from either turbulence, of a superposition of waves. For example, in the case of a thermally stratified fluid in the presence of gravity, if g! "T > 0 (strongly heated from below) the fluid is unstable (because of thermal expansion and buoyancy), and the random component will be turbulent; if g! "T > 0 (so that the hot fluid is above the cold fluid), the configuration is stable and the random component will be the result of waves. If L is the scale of spatial variation for the mean quantities, and l 0 is the scale of variation for the random component, then, for the case of turbulence, l 0 is the size of the energy containing eddies, and for the case of waves, l 0 is the wavelength. From our turbulence lectures (Section 36), we know that the mean and random components of the magnetic field evolve according to 9

10 and where!b 0!t = " # ( V 0 # B 0 ) + " # E + $" 2 B 0!!!, (38.24)!!B!t = " # ( V #! 0 B) + " # ( V! # B 0 ) + " # G + $" 2 B!!!,! (38.25) E =! V!! B!!!, (38.26) and G =! V!! B "! V!! B!!!. (38.27) The goal of the theory will be to express E in terms of V 0 and B 0, i.e., we will look for a closure relation. Suppose that! B = 0 at t = 0. Then, at t = 0,! B!!t = " # ( V! # B 0 )!!!, (38.28) so that! B is linear in B 0. It follows that E =! V!! B is also linear in B 0 (since! B is linear in B 0. We therefore anticipate a closure expression of the form E i =! ij B 0 j + " ijk #B 0 j #x k + $ ijkl # 2 B 0 j #x k #x l +...!!!. (38.29) It will be important to note that! ij,! ijk,! ijkl, etc., are pseudo-tensors, since E is a vector and B 0 is a pseudo-vector. Since! B depends on! V, V 0, and! (in addition to the linear dependence on B 0 ), we expect that the! ij,! ijk,! ijkl, etc., will be completely determined by V 0,!, and the statistical properties of! V. Now consider the case V 0 = constant. If we transform to a frame of reference moving with V 0, V 0! B 0 " 0, and B 0 evolves according to!b 0i!t! %!B = " ijk # kl B 0l + $ 0l ( jlm +...!x j & '!x m ) * + +,2 B 0i!!!. (38.30) If B 0 is weakly non-uniform, then!b 0l /!x j >>! 2 B 0l /!x j!x m and the first term dominates. (However,! jlm will still contribute to an eddy resistivity that will enhance!, i.e.,! e =! + ". We will ignore this effect from now on, and concentrate on the effect of!.) The we can write E i =! ij B 0 j!!!. (38.3) We can decompose! ij into symmetric and anti-symmetric parts as 0

11 ! (s) ij = ( 2! +! ij ji )!!!, (38.32) and! (A) ij = ( 2! "! ij ji )!!!. (38.33) The anti-symmetric part, Equation (38.33), can be written as where! (A) ij = "# ijk a k!!!, (38.34) a k =! 2 " lmk# lm!!!, (38.35) (you can work it out!) so that!! ij =! (s) ij " # ijk a k. Then Equation (38.3) becomes E i = (! (s) ij " # ijk a k ) B 0 j!!!,!!!=! (s) ij B 0 j " # ijk B 0 j a k!!!,!!!=! (s) ij B 0 j + ( a " B 0 ) i!!!, (38.36) so that the anti-symmetric part of! ij merely contributes an addition to the mean velocity, i.e., V 0! V 0 + a. Only the symmetric part contributes to the dynamo. We will therefore concentrate on the symmetric part of! ij ; from now on the notation! ij will refer to its symmetric part. When! V is statistically isotropic and homogeneous, the statistical properties of! V are invariant under rotation. In this case! ij must also be isotropic, so that! ij =!" ij!!!, (38.37) and a = 0. Now,! must be a pseudo-scalar. That means that it must change sign under coordinate inversion r! = "r. Since! depends on the statistical properties of V!, this implies that V! cannot be statistically invariant under inversions, i.e.,! can be non-zero only if V! lacks reflectional symmetry. This imposes another constraint on flows that can produce dynamo action. If! " 0, then E =!B 0 and J =!E =!"B 0!!!, (38.38) where! = / " is the electrical conductivity. The field E =!V!!B therefore produces a mean current that is parallel to B 0. Without fluctuations J =!V 0 " B 0, which is perpendicular to B 0. The appearance of this parallel mean current is called the! -effect. Since mean toroidal current due to the fluctuations is J T =!"B T, and since! " B P = µ 0 J T = µ 0 #$B T, the mean toroidal field acts as a source for the poloidal field,

12 and we have a way of closing the loop B P! B T and producing a dynamo. Now, all we need to do is to calculate!! We have seen that V! must lack reflectional symmetry in order to produce dynamo action. It turns out that this means that V!!!" = 0, where! = " #!V is the random vorticity. The quantity V! " is called the kinetic helicity (as opposed to the magnetic helicity). The physical picture is illustrated in the figure below. The flow on the left has vorticity but no kinetic helicity. The flow on the right has finite kinetic helicity. Therefore, the random flow field must have some mean twist, or handedness, in order to produce dynamo action. We now proceed to calculate!. In order to compute E =! V!! B, we must express!b in terms of B 0,!, and the properties of! V. The fluctuating field satisfied Equation (38.25). For the case V 0 = 0, which is all we will consider here, this is! B!!t = " # ( V! # B 0 ) + " # G + $" 2 B!!!.! (38.38) The term containing G =! V!! B "! V!! B will cause difficulty in solving this equation, and we look for circumstances under which it can be ignored. The magnitudes of the terms in this equation are, approximately,! B! = " # ( V! # B 0 ) "!t # % $ %%& + " # G +! $"2 " B!!!, (38.39) 2 $!B/l 0!B/t 0 B 0!V /l 0!V!B/l 0 where! B =! B 2 /2,! V =! V 2 /2, and l 0 and t 0 are the length and time scales associated with the fluctuations. There are two cases of interest: ) random waves, for which!vt 0 / l 0!! Vk 0 / " 0 << (or! V <<! 0 / k 0, the phase velocity of the waves); and, 2) turbulence, for which! Vt0 / l 0!. For the case of random waves, we have 2

13 ! " G #!B / #t $! V! B l 0 t 0!B =! Vt 0 l 0 <<!!!, (38.40) and the term! " G can be ignored, so that!!b!t = " # ( V! # B 0 ) + $" 2 B!!!.! (38.4) For the case of turbulence,! Vt0 / l 0! so that formally the term! " G must be retained. However, both! " G and!! B /!t can be neglected in comparison with!" 2! B if! B! /!t $! 2 B l 0 "# 2 B! t 0 " B! = Vl! 0 " <<!!!, (38.42) where! V = l0 / t 0. This is equivalent to R m0 <<, where R m0 is the magnetic Reynolds number at the l 0 length scale. In this case, 0 =! "! V " B 0 ( ) + #! 2! B!!!. (38.43) In both cases,! B is generated from! V. For the case of turbulence [Equation (38.43)] the process is instantaneous because of the dominance of diffusion. However, the solutions of Equation (38.4) for random waves should approach the solutions of Equation (38.43) in the limit R m0 <<. Therefore, we can study the case of random waves, and obtain the turbulent results in the limit! " #. These approximations are collectively called first order smoothing. An Aside on Some Statistical Definitions for Random Fields The purpose here is to introduce the kinetic helicity spectrum, which will appear in what follows. For many more details, see G. K. Batchelor, Theory of Homogeneous Turbulence, Cambridge, 953. and The Fourier transform pair for the velocity! V is!v ( k,! ) =!V r,t #i k$r#!t!v ( r,t)e ( 2" ) %% ( ) drdt!!!, (38.44) 4 ( ) =!V k,! Since V! r,t The correlation tensor is defined as $$ ( )e i( k"r#!t) dkd!!!!. (38.45) ( ) is real, V! (!k,!" ) = V! * ( k," ), and since! " V! ( r,t) = 0, k! V! ( k," ) = 0. R ij (!," ) = V i ( r,t)v j ( r +!,t + " )!!!, (38.46) where F =! F P(u,u 2,...)du du 2... is the probability average of F, and P is the joint probability distribution function. Now consider the mean quantity 3

14 Since!V i ( k,! ) i!v * j ( k,! ) =!!!!!!!!!! ( 2" ) 8 Vi ( r,t)! * #i k$r# % && && Vj i ( r,t) e ( k $ r %#!t #! % t %) drd r% dtd t%!!!. %% e i( k! k" )#x e i ( $! $ ")t dxdt = 2& we can write Equation (38.47) as!v i ( k,! )! * Vj i ( k,! ) = " ij k,! ( ) 4 ' k! k" (38.47) ( )' ( $! $ ")!!!, (38.48) ( )# ( k $ k% )# (! $!%)!!!, (38.49) ( ), which is called the spectrum tensor of V! ( r,t), is the Fourier transform where! ij k," of the correlation tensor, Equation (38.46). The spectrum tensor has the Hermitian property! ij (k,") =! ij (#k,#") =! * ji (k,"), and since! " V! ( r,t) = 0, k i! ij (k,") = k j! ij (k,") = 0. The energy spectrum is defined as ( ) = 2 " ii k,! E k,! # ( )ds!!!, (38.50) S k where S k is a sphere of radius k in k -space. Also, 2!V 2 = 2 R ii ( 0,0)!!!,!!!!!!!!!!!= 2 ##! ii ( k," )dkd"!!!,!!!!!!!!!!!= "" E( k,! )dkd!!!!. (38.5) Therefore,!E( k," )dkd" is to be interpreted as the kinetic energy in the range between k and k + dk, and! and! + d! ; E k,! ( ) > 0 for all k and!. ( ) = ik " V! ( k,! ), and ( k," ) = # imn # jpq k m k p $ nq ( k," ). Using the properties of! ij, one ( k," ) = k 2 # ii ( k," ), so that The vorticity is defined as! = " #!V, its transform is! k,! its spectrum tensor is! ij finds! ii 2! 2 = "" k 2 E( k,! )dkd!!!!. (38.52) The kinetic helicity spectrum is defined as ( ) = i " ikl k k # il k,! F k,! $ ( )ds!!!. (38.53) S k Then!V!!" = i# ikl %% k k $ il ( k," )dkd"!!!, (38.54) 4

15 !!!!!!!!!!!!= "" F( k,! )dkd!!!!. (38.55) From the above, it can be shown that! V!!" = 0 if! V is reflectionally symmetric. We ( ) " 0 have seen that lack of reflectional symmetry is required for the! -effect, so F k,! is an important property of flow fields that can produce dynamo action. Now let B 0 be uniform and steady, and! "! V = 0. Then, with first order smoothing, the induction equation is!!b!t " #$2! B = B0 % $ ( )! V!!!, (38.56) which is an inhomogeneous equation for! B. Consider the case of a single wave ( ) = V! 0 # sin kz! "t!v r,t $ ( )ê x + cos( kz! "t)ê y % &!!!, (38.57) with k = kê z and k > 0,! > 0, and assume a solution of the form!b ( r,t) = Re!B 0 exp $% i( k! r " #t)&'. Substituting into Equation (38.56), we find so that or!b 0 = ik! B 0 "i# + $k 2! V 0!!!, (38.58)!B ( r,t) = Re ik! B 0 "i# + $k 2 V! ( r,t)!!!, (38.59) where!b ( r,t) =!u r,t k! B 0 ( $"!V + #k!u )!!!, (38.60) " 2 + # 2 k 4 ( ) = V! 0 # cos kz! "t $ ( )ê x! sin( kz! "t)ê y % &!!!. (38.6) Notice that, in Equation (38.60), the diffusivity! introduces a phase shift between! B and! V. The computation of E =! ij =! (3) " i 3 " j 3, where!v!!b is now straightforward. The result for! ij is! (3) = " # V! 2 0 k 3!!!. (38.62) $ 2 + # 2 4 k Notice the following: ) The tensor! ij is anisotropic, i.e, the only non-zero component is! 33 ; it know about the direction of the wave. 2)! ij " 0 as! " 0, so that some diffusion is necessary for the! -effect. This is related to the phase shift noted above. Without this phase shift, we would have! V!! B = 0, and no possibility of a dynamo. 3) 5

16 E =! V!! B is uniform is space, so that G =! V!! B "! V!! B = 0, i.e, first order smoothing is exact in this case. This will not be true in general. The case of a superposition of random wave is similar, but requires more calculation. One must now take the Fourier transform of Equation (38.56), solve, and then invert the transform and compute E = V!! B!. The result, after a long calculation, is E i =!B 0i with where F k,! ( )! = " 3 # k 2 F k,$ %% $ 2 + # 2 k dkd$!!!, (38.63) 4 ( ) is the kinetic helicity spectrum defined in Equation (38.55). As anticipated, the! -effect requires a flow field with! V!!" # 0. We now consider the evolution of the axisymmetric mean fields in cylindrical coordinates. We write V = R!(R,z)ê " + V P and B = B(R, z)ê! + B P, where B P =! " $ % A( R,z)ê # & '. With E =!B, Equation (38.24) becomes the pair of equations and!b!t + RV "# $ B ' P % & R( ) = R B P " #* ( ) + ê + "#, (-B P ) $!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!+. # 2 / ' % & R 2 ( ) B!!!,!A!t + R V P "# RA ' ( ( ) = $B + % ) #2 & R 2 (38.64) * +, A!!!. (38.65) The toroidal field B has two sources, given by the first two terms on the right hand side of Equation (38.64). The first term, R B P!"# ( ), is due to differential rotation in the ( ), is directly from the poloidal field as a mean flow. The second term, ê! " # $ %B P result of the! -effect. The poloidal field, Equation (38.65), has a single source term,!b, directly from the toroidal field and the! -effect. This reconfirms Cowling s theorem that, without the symmetry breaking due to three dimensional motions (as encapsulated in! ), dynamo action is impossible. The ratio of the two sources of the toroidal field is ( ) ( ) ' LB "# P R B P! "# ê $! " % &B P & 0 B P / L = L2 "#!!!, (38.66) & 0 where! 0 is some characteristic value of!. If! 0 >> L 2 "#, then the effect of differential rotation is negligible, and the only source of toroidal field is!. Since! is also the source of poloidal field, these dynamos are called! 2 -dynamos. Conversely, if! 0 << L 2 "#, then differential rotation acts as a source of the toroidal field, and the 6

17 poloidal field is generated by!. These dynamos are called! " # dynamos. They are important in astrophysical systems that are dominated by rotation. So, what really happens at small scales? Can small scale turbulence really generate and sustain large scale fields? In recent years computers have become just barely powerful enough to begin to answer this question through direct numerical simulation or DNS) of the MHD equations. One starts with a small, random V! and B! fields with!v!!" # 0, integrates the resistive MHD equations with forward in time R m >> ( ~ 0 3 ), including the back-reaction, and looks for field amplification. The result is that there is a dynamo due to the! -effect, i.e., the initial magnetic energy is amplified. However, this energy is found to inevitably saturate, or stop growing, at a low amplitude. Further, it seems that the process generates more B 2 than it does B, i.e., it can generate energy but not mean flux. It appears that the way to generate and sustain mean flux is to have some around in the initial state, which begs the question of how it got there in the first place. At the present time it does not appear that small scale turbulence alone is sufficient to explain the sustainment and dynamics solar and other magnetic fields. Nonetheless, DNS does seem to do a pretty good job of reproducing the global properties of the geo-dynamo. The large scale field is sustained against resistive diffusion, and demonstrates reversals of polarity. People have also applied DNS to the global sun (as opposed to a small slice of it). They get a dynamical field, but not detailed agreement with observations. For example, they cannot reproduce the butterfly diagram. Worse, there is a profound disagreement with the results of helioseismology, from which one can deduce the internal structure of the sun. There is a result from the hydrodynamics of rotating systems called the Taylor- Proudman theorem (unproven here, but it is not difficult) that says that the angular velocity should be constant on cylindrical surfaces concentric with the axis of rotation. The global MHD simulations of the sun find the same result: the angular velocity in the interior of the sun is constant on cylinders. This seems good, because it agrees with the Taylor-Proudman result. However, helioseismology finds that the angular velocity in the solar interior is constant on cones, in complete disagreement with the MHD calculations! The measurements are very accurate, and everybody believes them, but the disagreement is not understood at all! What about laboratory plasmas? Historically, some sort of dynamo mechanism has been invoked to explain the generation and sustainment of the toroidal flux in the RFP. But what really happens? The RFP is a driven system. There is an applied toroidal voltage that drives the current. This voltage (or toroidal electric field) constantly supplies poloidal flux to the system (i.e., it drives current). As we saw in Section 37, this tends to drive the system away from its preferred, relaxed condition, resulting in the destabilization of long wavelength, low frequency (i.e., m = ) MHD modes. The nonlinear interaction of these modes produces a mean parallel electric field E =!V!!B that tends to suppress parallel current in the core and drive parallel current in the edge. This is precisely what is needed to produce toroidal flux. This is an! -effect that accounts the first part of the dynamo loop, B P! B T. Results from numerical simulation 7

18 of this process are shown in the figure below ( E f is the electric field due to the! - effect). But, as far as I know, there is no evidence of the second half of the loop, B T! B P. Instead, B T is simply dissipated resistively. Magnetic energy comes from the external circuitry into the discharge as poloidal flux. It is converted into toroidal flux by the! -effect produced by the non-linear interaction of long wavelength MHD instabilities. Small scale turbulence seems to play no role in this process. The toroidal flux is then dissipated by resistivity. If one turns off the driving force (the toroidal voltage), the discharge terminates immediately. In the end, the RFP seems to exhibit flux transport and conversion as an aspect of the relaxation process, but it does not appear to fit the definition of a classical dynamo. Of course, both the geo-dynamo and the solar dynamo also occur in driven systems, in both cases the drive is due to thermal convection as a result of strong heating from below. Perhaps similar processes as are observed in the RFP also occur throughout the universe. 8