Analytic solution of an oscillatory migratory alpha^2 stellar dynamo
aa r X i v : . [ a s t r o - ph . S R ] F e b Astronomy&Astrophysicsmanuscript no. paper c (cid:13)
ESO 2018Astron. Astrophys. 598, A117 (2017)
Analytic solution of an oscillatory migratory α stellar dynamo A. Brandenburg , , , Nordita, KTH Royal Institute of Technology and Stockholm University, 10691 Stockholm, Sweden JILA and Department of Astrophysical and Planetary Sciences, University of Colorado, Boulder, CO 80303, USA Laboratory for Atmospheric and Space Physics, University of Colorado, Boulder, CO 80303, USA Department of Astronomy, Stockholm University, 10691 Stockholm, SwedenReceived November 8, 2016, accepted December 1, 2016, Revision: 1.85
ABSTRACT
Context.
Analytic solutions of the mean-field induction equation predict a nonoscillatory dynamo for homogeneous helical turbulenceor constant α e ff ect in unbounded or periodic domains. Oscillatory dynamos are generally thought impossible for constant α . Aims.
We present an analytic solution for a one-dimensional bounded domain resulting in oscillatory solutions for constant α , butdi ff erent (Dirichlet and von Neumann or perfect conductor and vacuum) boundary conditions on the two boundaries. Methods.
We solve a second order complex equation and superimpose two independent solutions to obey both boundary conditions.
Results.
The solution has time-independent energy density. On one end where the function value vanishes, the second derivative isfinite, which would not be correctly reproduced with sine-like expansion functions where a node coincides with an inflection point.The field always migrates away from the perfect conductor boundary toward the vacuum boundary, independently of the sign of α . Conclusions.
The obtained solution may serve as a benchmark for numerical dynamo experiments and as a pedagogical illustrationthat oscillatory migratory dynamos are possible with constant α . Key words. dynamo – magnetohydrodynamics – magnetic fields – Sun: magnetic fields – stars: magnetic field
1. Introduction
The magnetic fields in stars and galaxies are believed to begenerated and maintained by large-scale dynamos that convertkinetic energy into magnetic energy through an inverse cas-cade (Pouquet et al., 1976). With the development of mean-fieldtheory (Parker, 1955; Steenbeck et al., 1966), this complicatedthree-dimensional process became amenable to simpler analyticand numerical treatments in one and two dimensions.The best known mean-field e ff ect is the α e ff ect, whichemerges from the parameterization of the turbulent electromo-tive force in terms of the mean field in the form u × b = α B − η t ∇ × B , (1)where u and b are the fluctuating velocity and magnetic fields,overbars denote averaging, and B is the mean magnetic field.Here, α quantifies the α e ff ect and η t is the turbulent magneticdi ff usivity. Both are in principle functions of position, but in thepresent paper we will treat them as constants.The earliest model of a dynamo for the Sun goes back toParker (1955), who considered the additional presence of di ff er-ential rotation, which is referred to as the Ω a ff ect. In the pres-ence of both α and Ω e ff ects, there are self-excited oscillatoryplain wave solutions in unbounded domains. They take the formof traveling waves (Parker, 1955). Specifically, if α is positive inthe north and negative in the south, and the di ff erential rotationhas a negative radial gradient, waves are traveling equatorward,providing thus an explanation for the shape of Maunder’s but-terfly diagram (Maunder, 1904). The first global axisymmetrictwo-dimensional models of such dynamos go back to the sem-inal work of Steenbeck & Krause (1969a). These dynamos arereferred to as α Ω dynamos. In the absence of di ff erential rotation, a plain wave solutionansatz leads to non-oscillatory dynamos if α exceeds a certainthreshold ( α > η t k , where k is the wavenumber). Such dynamosare referred to as α dynamos. The dynamo of the Earth is be-lieved to be an example of an α dynamo, because shear is ex-pected to be weak. Axisymmetric models of dynamos of thistype where presented by Steenbeck & Krause (1969b). The non-oscillatory property of such dynamos is consistent with the non-cyclic nature of the Earth’s magnetic field. In galaxies, on theother hand, shear is important, so they are examples of α Ω dy-namos. However, asymptotic solutions have shown that such dy-namos are non-oscillatory owing to the flat geometry in whichsuch dynamos are embedded (Vainshtein & Ruzmaikin, 1971).Numerical investigations of α dynamos revealed onlynonoscillatory solutions (R¨adler, 1980), until Shukurov et al.(1985) found that, under certain conditions, oscillatory solu-tions are here possible, too. They associated this with the non-selfadjointness of the problem. In fact, the possibility of oscil-latory solutions to an α dynamo was already mentioned earlierby Ruzmaikin et al. (1980) in a study of disk dynamos with astrongly localized α e ff ect. In 1987, there appeared two back-to-back papers that demonstrated conclusively that α dynamos canin principle be oscillatory provided the α e ff ect is non-constant(Baryshnikova & Shukurov, 1987; R¨adler & Br¨auer, 1987). Thispossibility remained mainly an academic curiosity without realastrophysical interest at the time.In subsequent years, attention was drawn to the possibil-ity that global dynamos with radially dependent α can exhibitoscillatory solutions (Stefani & Gerbeth, 2003). Meanwhile, di-rect numerical simulations of helically forced turbulence haveshown a strong similarity between α e ff ect dynamos and tur-bulent three-dimensional dynamos with fluctuating magneticfields and nonvanishing mean fields. These dynamos turned
1. Brandenburg: Analytic solution of an oscillatory migratory α stellar dynamo out to be equivalent to those predicted from α -e ff ect dynamos(Brandenburg, 2001). Mitra et al. (2009) applied such dynamosto spherical wedges with helically forced turbulence. Whenthe helicity of the forcing was assumed such that it changessign about the equator, Mitra et al. (2010) found oscillatory so-lutions with equatorward migration similar to what occurs inthe Sun. K¨apyl¨a et al. (2013) argued that such an e ff ect canexplain the equatorward migration in their spherical wedge-geometry dynamos, even though shear was still present and, asit turned out later, responsible for an α Ω -type dynamo in thiscase (Warnecke et al., 2014). In other simulations, however, theargument in favor of an α dynamo could still be supported(Masada & Sano, 2014).Corresponding mean-field solutions were presented byBrandenburg et al. (2009) for dynamos in Cartesian geometrywith α profiles proportional to z . Cole et al. (2016) showed thatsuch dynamos are not necessarily expected to operate in spheri-cal shells that extend all the way to the poles, unless the turbulentmagnetic di ff usivity becomes small at high latitudes. The trueapplicability of such α dynamos to stars remains therefore ques-tionable. Nevertheless, such dynamos are gaining in importancein view of the many numerical studies of turbulent dynamos,in which the helicity profile is non-uniform (Mitra et al., 2014;Jabbari et al., 2016) and / or the boundary conditions on the twosides of the domain are di ff erent (Jabbari et al., 2017). This hasled to the possibility that oscillatory α dynamos might actuallybe possible for constant α , provided the boundary conditions areindeed di ff erent and the two sides. If this is the case, it should bepossible to construct exact analytical solutions of such an oscil-latory migratory α dynamos. The purpose of the present paperis therefore to present such a solution. The fact that such a so-lution can be obtained analytically is significant not only as abenchmark for numerical studies, but also as a clear textbook-style demonstration of oscillatory α dynamos.
2. Statement of the problem
The equation for an α dynamo with total (sum of microphysicaland turbulent) magnetic di ff usivity, η T = η + η t , is given by ∂ A ∂ t = α ∇ × A − η T ∇ × ∇ × A , (2)where A is the mean magnetic vector potential in the Weylgauge, and the mean magnetic field is B = ∇ × A . We nondi-mensionalize by measuring lengths in units of k − , where k isthe wavenumber of the most slowly decaying mode, and time ismeasured in units of the turbulent–di ff usive time, τ td = ( η T k ) − .Velocities are measured in units of η T k , so in the following wedenote by α the nondimensional α e ff ect, α/η T k . We now con-sider a one-dimensional domain, so the governing equations are, ∂ A x ∂ t = − α ∂ A y ∂ z + ∂ A x ∂ z , (3) ∂ A y ∂ t = + α ∂ A x ∂ z + ∂ A y ∂ z , (4)and A z =
0. In the following, all quantities are dimensionless.We consider perfect conductor boundary condition on one sideof the domain ( z = xy plane vanishes on the boundary. Owing to the use of the Weylgauge, the electrostatic potential gradient is absent in Eq. (2), sothe perfect conductor condition implies that A x = A y =
0. On the other side of the domain, we assume a vacuum bound-ary condition. For our one-dimensional domain, this means that B x = B y = ∂ z A x = ∂ z A y =
0. The most slowly decaying mode is a quar-ter sine wave, that is, A x or A y are proportional to sin z in0 ≤ z ≤ π/
3. Complex notation and integral constraints
The basic approach used here is similar to that in other problemswith constant coe ffi cients and in finite domains with boundaryconditions, such as the no-slip condition in Rayleigh–B´enardconvection (Chandrasekhar, 1961) or the pole-equator boundaryconditions in α Ω dynamos (Parker, 1971). Unlike convection,which is non-oscillatory at onset, we allow here for oscillatorysolutions. Furthermore, we combine Eqs. (3) and (4) into a sin-gle equation for the complex variable A ≡ A x + i A y . (5)Thus, Eqs. (3) and (4) can be written as ∂ A ∂ t = i α ∂ A ∂ z + ∂ A ∂ z . (6)We now assume the solution to be of the form A ( z , t ) = ˆ A ( z ) e − i ω t , (7)where ˆ A ( z ) obeys the ordinary di ff erential equationˆ A ′′ + i α ˆ A ′ + i ω ˆ A = , (8)where primes denote z derivatives. The boundary conditions areˆ A = z = , (9)ˆ A ′ = z = π/ . (10)In general, ω can be complex, but since we are here interested inmarginally excited dynamos, we restrict ourselves in the follow-ing to ω being real.We now also assume that α is constant. In that case,oscillatory solutions were previously thought impossible(R¨adler & Br¨auer, 1987). Analogously to their approach, wemultiply Eq. (8) by ˆ A ∗ , where the asterisk denotes complex con-jugation, and integrate by parts. Using Eqs. (9) and (10), we ob-tain Z π/ ˆ A ′′ ˆ A ∗ d z = − Z π/ (cid:12)(cid:12)(cid:12) ˆ A ′ (cid:12)(cid:12)(cid:12) d z . (11)Furthermore, ( ˆ A ˆ A ∗ ) ′ = ˆ A ′ ˆ A ∗ + ˆ A ˆ A ′∗ = A ′ ˆ A ∗ ), soˆ A ′ ˆ A ∗ = (cid:18) (cid:12)(cid:12)(cid:12) ˆ A (cid:12)(cid:12)(cid:12) (cid:19) ′ + i Im( ˆ A ′ ˆ A ∗ ) . (12)Equation (8) yields altogether four terms, two of which are realand the other two imaginary. We obtain two integral constraints α = − Z π/ (cid:12)(cid:12)(cid:12) ˆ A ′ (cid:12)(cid:12)(cid:12) d z , Z π/ Im( ˆ A ′ ˆ A ∗ ) d z , (13) ω = − α (cid:12)(cid:12)(cid:12) ˆ A (cid:12)(cid:12)(cid:12) π/ ,Z π/ (cid:12)(cid:12)(cid:12) ˆ A (cid:12)(cid:12)(cid:12) d z , (14)
2. Brandenburg: Analytic solution of an oscillatory migratory α stellar dynamo where | ˆ A | π/ denotes the value of | ˆ A | on the second boundaryat z = π/
2. This implies that αω ≤ α ) and ω , | ˆ A | π/ > α , B ( z ) = i ˆ A ( z ). Unfortunately, the per-fect conductor boundary condition, i η T ˆ B ′ = α ˆ B , is more cum-bersome. Instead, one could formulate the problem for an artifi-cially modified boundary condition, ˆ B ′ = z =
0. Togetherwith the condition ˆ B = z = π/
2, the problem for ˆ B ( z )becomes equivalent to that for ˆ A ( z ). In either case, the integralconstraints are analogous to those of R¨adler & Br¨auer (1987);see Appendix A for details.
4. The solution
Given that Eq. (8) has constant coe ffi cients, it has solutions pro-portional toˆ A i ( z ) ∝ e i k i z , (15)where the index i denotes one of two independent solutions. The k i are in general complex and obey the characteristic equation k + α k − i ω = . (16)It has two solutions, k ± = − α/ ± p α / + i ω. (17)To satisfy the boundary conditions (9) and (10), we writethe solution as a superposition of e i k + z and e i k − z . Equation (9) isreadily satisfied by writingˆ A ( z ) = e i k + z − e i k − z , (18)where we have ignored the possibility of an arbitrary (complex)constant in front of ˆ A . To satisfy Eq. (10), we now require that D ( α, ω ) = k + e i k + π/ − k − e i k − π/ (19)vanishes. The existence of solutions to D ( α, ω ) = | D | ; see Fig. 1, where wealso plot separately the real and imaginary parts of D . We seetwo zeroes in D ( α, ω ), which is confirmed by the crossing ofthe lines where Re D and Im D vanish. [At α = ω =
0, there isno such crossing, so D (0) is not a solution.] The transcendentalequation relating α to ω can be written in more explicit form as e i π √ α / + i ω + (cid:16) α/ + p α / + i ω (cid:17) (cid:30) (i ω ) = . (20)To find solutions to D ( α, ω ) =
0, it is convenient to introduce thecomplex variable Z ≡ α + i ω. (21)We seek solutions to D ( Z ) = Z = Z − D ( Z − Z − ) / ( D − D − ) , (22)where subscripts 0 and − Z ∗ = α + i ω ≈ . − . , (23)with the corresponding complex wavenumbers k + ≈ . − . , (24) Fig. 1.
Plots of (a) real, (b) imaginary, and (c) absolute parts of D ( α, ω ). In (a) and (b), the zero lines are marked in white, whilein (c) those of Re D are dotted blue and those of Im D are solidred. k − ≈ − . + . . (25)The wavenumbers k + and k − obey the relation k + + k − + α = , (26)which follows from Eqs. (9) and (18). The critical values of α and ω were first obtained by Jabbari et al. (2017) using explicittime integration.Additional solutions exist in the second and fourth quadrantof the αω plane; see Fig. 2. They are all oscillatory, in agree-ment with the integral constraints; see Eqs. (13) and (14) and
3. Brandenburg: Analytic solution of an oscillatory migratory α stellar dynamo Fig. 2.
Similar to Fig. 1(c), but for the next higher modes ( + signs).Table 1. However, those higher modes would generally be un-stable in a nonlinear calculation and therefore only of limitedinterest (Brandenburg et al., 1989).The solution is now completely described by the value of Z ∗ .It is convenient to write the solution in the formˆ A = r A ( z ) e i φ A ( z ) , (27)where r A ( z ) and φ A ( z ) are amplitude and phase of ˆ A . In view ofcomputing magnetic field and current density, we also defineˆ B ≡ i ˆ A ′ = r B ( z ) e i φ B ( z ) (28)andˆ J ≡ − ˆ A ′′ = r J ( z ) e i φ J ( z ) , (29)respectively. In Fig. 3 we plot the moduli and phases of ˆ A ( z ),ˆ B ( z ), and ˆ J ( z ). Note that r A (0) =
0, as required by Eq. (9), and r ′ A ( π/ = φ ′ A ( π/ =
0, as required by Eq. (10). In general,however, ˆ J (0) ≡ − ˆ A ′′ (0) ,
0. The derivative of the phase is an“e ff ective” wavenumber, k ( B )e ff = d φ B / d z , and determines the z -dependent phase speed c = ω/ k ( B )e ff , which is positive for positive α , so the wave moves in the positive z direction.In Fig. 3(c) we plot the magnetic and current helicity densi-ties, as well as the z component of the Lorentz force, A · B = Re ˆ A ∗ ˆ B , J · B = Re ˆ J ∗ ˆ B , ( J × B ) z = Im ˆ J ∗ ˆ B , (30) Table 1.
Critical values of α and ω for the higher modes. mode α ω − . − . − . − . − . − . − . − . Fig. 3. (a) Moduli and (b) phases of ˆ A ( z ), ˆ B ( z ), and ˆ J ( z ), aswell as (c) normalized magnetic and current helicity densitiestogether with the z component of the Lorentz force.normalized by R B d z ≡ R | ˆ B | d z and R J d z ≡ R | ˆ J | d z for the first, and second and third quantities, respectively. TheLorentz force has a maximum at z = . z =
0. The ratio between the integrals of the two helicitydensities is k ≡ Z Re ˆ J ∗ ˆ B d z , Z Re ˆ A ∗ ˆ B d z , (31)where k m denotes the wavenumber of the mean field; see Eq. (25)of Blackman & Brandenburg (2002). For α dynamos in pe-riodic domains, one finds k m / k =
1, but here we obtain k m / k ≈ .
4. Brandenburg: Analytic solution of an oscillatory migratory α stellar dynamo Fig. 4.
Butterfly diagrams for B x and B y , with z increasing down-ward.magnetic Taylor microscale wavenumber of the mean field, k T , defined through k = R | ˆ J | d z .R | ˆ B | d z , i.e., k T = k m .Finally, for the fractional current helicity of the mean field(Blackman & Brandenburg, 2002), ǫ m = Z Re ˆ J ∗ ˆ B d z , Z | ˆ J | d z Z | ˆ B | d z ! / , (32)we find ǫ m ≈ . ǫ m = α dynamos in periodic domains (Blackman & Brandenburg,2002).To plot butterfly diagrams of B x and B y , we can now writethe fully time-dependent magnetic field as B x ( z , t ) = r B ( z ) cos[ φ B ( z ) − ω t ] , B y ( z , t ) = r B ( z ) sin[ φ B ( z ) − ω t ] . (33)This also demonstrates that the magnetic energy density, E M = B = r B ( z ) = E M ( z ) , (34)is independent of time and only a function of z . In fact, the mag-netic and current helicity densities, as well as the z componentof the Lorentz force, all shown in Fig. 3(c), are also independentof time. The results for B x ( z , t ) and B y ( z , t ) are shown in Fig. 4,where z increases downward so as to facilitate comparison withFig. 2 of Brandenburg et al. (2009), who adopted a perfect con-ductor boundary condition at high latitudes and a vacuum con-dition at the equator. In their case, however, α was non-constantand vanishing on the equator.
5. Discussion
The graphs of the solutions obtained here look rather simple,but would have been impossible to guess based on previousexperience with one-dimensional dynamos with vacuum fieldconditions on both ends of the domain. The field componentsof those dynamos are proportional to cos z e i z . Such dynamoshave been studied extensively in connection with demonstrating Table 2.
Values of α and | ω | using one-sided ( ) fi-nite di ff erence formulae on the boundaries and symme-try / antisymmetry ( s ) conditions for di ff erent meshpoint numbers N mesh . Agreement with the analytic solution (“exact”) is indi-cated in bold face. N mesh α ( ) | ω | ( ) α ( s ) | ω | ( s )
32 2.55213 1.4350 2.55228 1.4289128 2.55071 1.4298 2.55074 1.4297512 exact 2.55065 1.4297 the asymptotically equal growth rates of even and odd dynamomodes (Brandenburg et al., 1989), the behavior of dynamos inthe highly nonlinear regime (Meinel & Brandenburg, 1990), andthe e ff ects of magnetic helicity fluxes (Brandenburg & Dobler,2001). Thus, one might have expected that the solution to thepresent problem could have been expanded in terms of sine func-tions proportional to sin (2 n + z with integers n ≥
0. Suchfunctions obey the boundary conditions of A x on z = π/
2. However, one sees immediately that such a solution for A x would imply that A y has terms proportional to cos (2 n + z ,which would then violate the boundary conditions on A y on bothboundaries; see Appendix B for details. This is indeed be a prob-lem for spectral codes that employ sine or cosine transforms;see Vasil et al. (2008a,b) for detailed studies and alternative ap-proaches. It can also be a problem for codes that use symmetryconditions to populate the ghost zones outside the computationalmesh, as is done by default in the P encil C ode . This highlightsonce more the significance of having an independent and ana-lytic solution of such a dynamo. To demonstrate this, we sum-marize in Table 2 the values of α and | ω | for a marginally exciteddynamo obtained by using either one-sided ( ) finite di ff er-ence formulae on the boundaries or symmetry / antisymmetry ( s )conditions (Brandenburg, 2003) for di ff erent meshpoint numbers N mesh . The scheme does not restrict the second derivative andis found to be slightly better than the s scheme.We have here also been able to find higher order modes. Theyall lie in the same two quadrants in the αω plane. Thus, for pos-itive α , we always have ω <
0. When determining ω empiricallyfrom the period of the oscillation, it would not have a definitesign, although the sign has implications for the phase speed. For α Ω dynamos with di ff erential rotation gradient Ω ′ in periodicdomains with real wavenumber k , self-excited solutions existonly when sgn [( k α Ω ′ ) ω ] >
0; see Appendix C and Table 3 ofBrandenburg & Subramanian (2005). However, unlike α Ω dy-namos, where both migration directions are possible, dependingjust on sgn ( α Ω ′ ), for oscillatory α dynamos, the migration di-rection is always away from the (perfect) conductor toward thevacuum. This agrees with earlier findings for oscillatory α dy-namos with nonuniform α profiles (Brandenburg et al., 2009).In the context of oscillatory α Ω dynamos, boundary condi-tions have long been known to introduce behaviors that are notobtained for infinite domains (Parker, 1971). The antisymmetrycondition at the equator was found to play the role of an absorb-ing boundary that led to localized wall modes (Worledge et al.,1997; Tobias et al., 1997). Subsequent work using complex am-plitude equations for the envelope of a wave train demonstratedthat boundary conditions can play a decisive role in determin- https://github.com/pencil-code
5. Brandenburg: Analytic solution of an oscillatory migratory α stellar dynamo ing the migration direction of traveling waves (Tobias et al.,1998). They emphasized that the traveling wave behavior islinked to the symmetry-breaking in the mean-field dynamo equa-tions. This rather general result could explain the migration di-rection of the α dynamo studied here. The symmetry break-ing, which occurs here through the boundary conditions, mightalso be responsible for the occurrence of an oscillatory moderather than the non-selfadjointness mentioned in the introduc-tion (Shukurov et al., 1985).
6. Conclusions
The present work has shown that α dynamos with constant α can have oscillatory solutions provided the boundary condi-tions on the two ends of the domain are di ff erent. It is possi-ble to construct a one-dimensional analytic solution character-ized by a complex function ˆ A ( z ), which obeys Dirichlet and vonNeumann boundary conditions on the two ends of the domain.The solution has been obtained as a superposition of two har-monic functions with complex wavenumbers. In principle, wecould have solved the problem directly for ˆ B ( z ) = i ˆ A ( z ), butthe boundary condition on z =
0, namely i η T ˆ B ′ = α ˆ B , wouldbe more complicated. Integral constraints on ˆ B would then beharder to impose, unless one changed the perfect conductorboundary condition to ˆ B ′ =
0. In that case, the problem becomesequivalent to the one considered here if we replace ˆ
A → ˆ B . Inthis connection, it should be noted that the very assumption ofa finite α e ff ect on a perfect conductor boundary, while mathe-matically sound, is physically not strictly realistic, because animpenetrable boundary would necessarily make α anisotropicsuch that its tangential components would vanish (R¨adler, 1982).Nevertheless, various DNS with helically forced turbulence ex-tending all the way to the walls confirm the presence of oscil-latory migratory solutions (Mitra et al., 2010; Warnecke et al.,2011; Jabbari et al., 2017).Owing to our restriction to Cartesian geometry, the main ap-plication of this model lies in the comparison with other nu-merical solutions in the same geometry (see, e.g., Jabbari et al.,2017). The present solution demonstrates clearly that a modelwith constant α is possible and has time-independent magneticenergy density. Thus, when looking only at the rms value of themagnetic field or the volume-integrated energy, one will not no-tice the presence of an oscillatory solution.When the α dynamo is applied to a star, α would have theopposite sign on the other side of the equator (here for z > π/ α profile suggests that the an-tisymmetric solution is more easily excited (Brandenburg et al.,2009; Cole et al., 2016). Such solutions would have a discon-tinuity in the derivative of the current density at the equator.More dramatic, however, would be the case of symmetric so-lutions when a vacuum or vertical field condition is assumed onthe outer boundary, because in that case the current density itselfwould be discontinuous at the equator. Interestingly, the criticalvalues of α are the same in both cases. While a step functionprofile of α is artificial, it does pose a simple benchmark for nu-merical schemes. The analytic solution presented here appliesalso to this case. This solution may also serve as a pedagogicalillustration that oscillatory migratory dynamos with constant α are possible. Acknowledgements.
I thank Ben Brown, Matthias Rheinhardt, and an anony-mous referee for useful remarks. This work was supported in part by the Swedish Research Council grant No. 2012-5797, and the Research Council of Norwayunder the FRINATEK grant 231444. This work utilized the Janus supercom-puter, which is supported by the National Science Foundation (award numberCNS-0821794), the University of Colorado Boulder, the University of ColoradoDenver, and the National Center for Atmospheric Research. The Janus super-computer is operated by the University of Colorado Boulder.
Appendix A: Integral constraint in multi-dimensions
The purpose of this appendix is to demonstrate the analogybetween Eqs. (13) and (14) and the corresponding one ofR¨adler & Br¨auer (1987). However, instead of assuming the dy-namo region to be surrounded by vacuum and extending someof the volume integrals over all space, we adopt here perfectconductor and vertical field boundary conditions. In a multi-dimensional domain, the latter is no longer a proper vacuumcondition, but it can be motivated as being a more realistic rep-resentation of stellar surface fields a ff ected by magnetic buoy-ancy e ff ects (Yoshimura, 1975). Multiplying by ˆ B ∗ , the dynamoeigenvalue problem takes the form − ˆ B ∗ · ( ∇ × ∇ × ˆ B ) + ˆ B ∗ · ∇ × ( α ˆ B ) + i ω | ˆ B | = . (A.1)Using2i α Im (cid:16) ˆ B ∗ · ∇ × ˆ B (cid:17) = ∇ · (cid:16) α ˆ B × ˆ B ∗ (cid:17) − ∇ α · (cid:16) ˆ B × ˆ B ∗ (cid:17) , (A.2)but assuming now constant α in a volume V , we obtain α = − Z V (cid:12)(cid:12)(cid:12) ∇ × ˆ B (cid:12)(cid:12)(cid:12) d V ,Z V Im (cid:16) ˆ B · ∇ × ˆ B ∗ (cid:17) d V (A.3)and, as in R¨adler & Br¨auer (1987), ω = − α I ∂ V Im (cid:16) ˆ B × ˆ B ∗ (cid:17) · d S ,Z V (cid:12)(cid:12)(cid:12) ˆ B (cid:12)(cid:12)(cid:12) d V . (A.4)These equations are analogous to Eqs. (13) and (14). By com-parison, R¨adler & Br¨auer (1987) assumed a potential field on allboundaries, so ˆ B = − ∇ Φ , where Φ is the magnetic scalar po-tential. Writing the integrand of the surface integral in Eq. (A.4)as ∇ × ( Φ ∇ Φ ∗ ) and turning the surface integral back into a vol-ume integral, one sees that the divergence of the curl vanishes,and therefore ω =
0. However, this does not apply to our casewhere we have di ff erent boundary conditions on the two ends.By comparison, in one-dimensional dynamos with vacuum con-ditions on both ends, | ˆ A| has, in a non-transient state and withthe gauge R ˆ A d z =
0, the same value on both boundaries, soEq. (14) does indeed predict ω = Appendix B: Quarter sine wave expansion
In this appendix we give the results for a quarter sine wave ex-pansion of ˆ A ,ˆ A ( z ) = ∞ X n = ˆ A n sin (2 n + z , (B.1)where each element of the expansion obeys Eqs. (9) and (10).The coe ffi cients are given by ˆ A n = R π/ ˆ A sin (2 n + z . We havestrictly ˆ A ′′ (0) =
0, although the analytic value is nonvanishing,ˆ A ′′ (0) ≈ . − . A ′ (0) we haveˆ A ′ (0) → S N ≡ N X n = (2 n +
1) ˆ A n , (B.2)which converges extremely slowly to the analytic value obtainedfrom Eq. (18), which is ˆ A ′ (0) ≈ . + . S n and ˆ A n .
6. Brandenburg: Analytic solution of an oscillatory migratory α stellar dynamo Table B.1.
Coe ffi cients ˆ A n and S n . n Re ˆ A n Im ˆ A n Re S n Im S n .
512 0 .
493 2 .
512 0 . − .
052 0 .
557 2 .
355 2 . − .
114 0 .
054 1 .
788 2 . − .
024 0 .
013 1 .
622 2 . − .
015 0 .
006 1 .
486 2 . − .
006 0 .
003 1 .
418 2 . − .
005 0 .
002 1 .
358 2 . − .
002 0 .
001 1 .
287 2 . − .
001 0 .
000 1 .
242 2 . − .
000 0 .
000 1 .
060 2 . .
000 0 .
000 1 .
044 2 . −→ Appendix C: Comparison with the α Ω dynamo The purpose of this appendix is to show that for α Ω dynamos, αω Ω ′ k > α c Ω ′ >
0, where c = ω/ k is the phase speed.We assume a linear shear flow velocity U = (0 , x Ω ′ , Ω ′ is the velocity gradient. Using the advective gauge, U · A = ∂ A x ∂ t = − Ω ′ A y + η T ∂ A x ∂ z , (C.1) ∂ A y ∂ t = + α ∂ A x ∂ z + η T ∂ A y ∂ z . (C.2)The dispersion relation is then − i ω ≡ − i kc = − η T k ± ( − i k α Ω ′ ) / . (C.3)Using (2 i) / = + i and ( − / = (1 + i)i = − + i, we have − i ω ≡ − i kc = − η T k ± (cid:2) i − sgn ( k α Ω ′ ) (cid:3) (cid:12)(cid:12)(cid:12) k α Ω ′ / (cid:12)(cid:12)(cid:12) / . (C.4)For positive (negative) values of k α Ω ′ , only the lower (upper)sign yields marginally excited dynamos, sosgn ω = sgn ( k α Ω ′ ) and sgn c = sgn ( α Ω ′ ) . (C.5)Thus, the migration direction depends just on the sign of α Ω ′ ,but the frequency depends also on the sign of k . References
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