High resolution numerical dynamos in the limit of a thin disk galaxy

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High resolution numerical dynamos inthe limit of a thin disk galaxyMatthew R. Walker a & Carlo F. Barenghi aa Division of Applied Mathematics, Department of Mathematicsand Statistics , The University of Newcastle upon Tyne , NE17RU, EnglandPublished online: 19 Aug 2006.

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HIGH RESOLUTION NUMERICAL DYNAMOS IN THE LIMIT OF A THIN DISK GALAXY

MATTHEW R. WALKER and CARL0 F. BARENGHI

Division of Applied Mathematics, Department of Mathematics and Statistics, The University of Newcastle upon Tyne,

NEl 7RU England

(Received 19 November 1993; injnal form 2 May 2994)

We have developed a high resolution numerical method based on spectral approximations to study kinematic dynamo models of the galaxy in the geometry ofan oblate spheroid. The method has enabled us to investigate the onset of dynamo action in the limit ofa thin disk. In this limit we have solved the discrepancy between the asymptotic analysis of Soward and the previous numerical works of Stix and White.

KEY WORDS: Kinematic dynamics, galaxies, thin discs.

1. INTRODUCTION AND MOTIVATIONS

Our concern is the galactic dynamo. The very large scale of the galaxy seems to indicate that the magnetic diffusion time z is very long, and that no dynamo action is needed to explain the existence of the present magnetic field. It is often argued, however, that gas turbulence in the galactic disk should reduce z to a value shorter than the galaxy’s lifetime. A dynamo mechanism is then required to explain the observations. The simplest dynamo models are based on a mean field approach with alpha and omega effects. The strong differential rotation of the galaxy, which winds up poloidal field lines into azimuthal ones, suggests that the dynamo is of the CIW kind, in which the magnetic field is mainly in the azimuthal direction. Supernovae events and stellar wind from massive stars are the sources of the alpha effect, which, turning azimuthal field into poloidal field, completes the dynamo cycle. Observations show that there are some galaxies in which the magnetic field is bisymmetric, but here we concentrate our attention on the basic case of axisymmetric CIW models only. The simplest kinematic galactic models are based on plane layers or disks (Parker, 1971; Ruzmaikin et al., 1988), a torus geometry (Schmitt, 1990) and a sphere with the galactic disk embedded in it (Elstner et al., 1990; Moss and Tuominen, 1990). The perturbations which are most easily excited have long wavelength, i.e. small wavenumber k (Moffatt, 1978). These unstable modes can be either steady or oscillatory. Soward (1992 a,b) pointed out that particular care must be taken in considering the limit k + 0 for steady modes. Using asymptotic methods, he calculated the onset of dynamo action in the limit of a thin disk galaxy. Unfortunately the results did not agree with the pioneering numerical calculation of Stix (1975, 1978), who studied oblate spheroidal dynamos at values of aspect ratio as small as 1/30. The results of Stix were also confirmed by White (1977).

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266 M. R. WALKER AND C. F. BARENGHI

The aim of this paper is to solve the mathematical problem raised by Soward and to clarify the apparent disagreement between his asymptotic analysis on the one hand, and the numerics of Stix and White on the other. To reach our aim we have developed a high resolution numerical method to solve the magnetic induction equation in the oblate spheroidal geometry which is relevant to the galaxy. The new method has enabled us to attain convergence at very small aspect ratios, beyond the values available to Stix.

A brief summary of the paper is the following. The model’s equations are introduced in Section 2 and the method of solution is described in Section 3. The results and the comparison with the works of Stix and Soward are presented in Section 4. This section also contains tests of the method of solution against the spherical limit of Roberts (1972), results about the transition from spherical to spheroidal geometry, and a discussion of the numerical convergence, which is a crucial issue in the thin disk limit. Section 5 is devoted to conclusions and suggestions for further work.

2. THEMODEL

Following Stix (1975), we describe the galaxy using the oblate spheroidal coordinate system (5, q, rp), details of which are to be found in Flammer (1957). The ratio E of the semi-minor axis b to the semi-major axis R is given by

where 5 = to is the galactic surface. The large scale magnetic field B does not depend on cp and statisfies the magnetic induction equation

dB/dt = pV2B + V x (v x B + aB), together with

V-B = 0,

where t is time and p is the turbulent magnetic diffusivity. We are interested in velocity fields which are predominantly in the azimuthal direction and assume

where Q is the unit vector along cp and s is the distance from the rotation axis of the galaxy. The arbitrary functions a and w define the alpha effect and the differential rotation; a choice of these functions characterizes a particular model. The alpha effect must be antisymmetric with respect to the galactic plane z = O (Parker, 1971; Ruz- maikin et al., 1988). The simplest choice, widely used in the literature, is the simple scalar form a = aoz, which corresponds to a = aoxg in oblate spheroidal coordinates. We consider two forms for the omega effect. The first is the form used by Stix:

o = o,[constant + + s2 + O(s3)]. (2.5a)

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DYNAMOS OF A THIN DISK GALAXY 267

Stix assumed solid body rotation near the rotation axis and used an expansion in powers of s with no first order term; in this way the shear is equal to R times the amplitude wo at s = R. The expansion is truncated after the quadratic term. We use (2.5a) to make comparisons with the work of Stix.

The second form

w = 0, [ 1 + (s/s0)2] - ’/’ (2.5b)

is a more realistic rotation curve of the galaxy. Here so is the radius at which rigid body rotation close to the galactic centre becomes differential rotation (Brandenburg et al., 1990; Donner and Brandenburg, 1990). The shear is negative and tends to zero ass-’ in the outer region.

Neglecting the spherical halo, we assume that the region external to the spheroid is a pure vacuum. The internal and external field components must match continuously at the galactic surface and the field strength is zero on the axis of rotation.

It is convenient to introduce the decomposition

B = B$ + V x (A$), (2.6)

for which (2.3) is automatically satisfied and (2.2) reduces to two scalar equations for the poloidal and toroidal components A and B. We use the length scale R and time scale R2/p to make (2.2) dimensionless and introduce the Reynolds numbers R, = aoR/p and R, = woRZ/p. We write the alpha and omega effects in the form a = aoa*({, q ) and w = wow*(t , q) to distinguish between the amplitudes a, and o, and the functional shapes LY* and w*. The LYW limit is then obtained by assuming RJR, << 1. The Reynolds numbers R, and R, combine into a single driving parameter, the magnetic Reynolds number, R, = (I R, R,I)’/’ by use of the scalings A -, AR,1/4 R; ‘I4 and B -, BRA’4 R; 1/4

(Jepps, 1975). The magnetic Reynolds number is related to the dynamo number D by D = Ri. Finally, the scaling A -, R A ensures that A and B are dimensionally correct. The resulting am dynamo equations are

I a 2

all + (1 - l l ’ ) ” ’ ~ [ ( l + llZ)l/ZB] ,

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268 M. R. WALKER A N D C. F. BARENGHI

where x = (/to. Due to the symmetry in q of the linear equations (2.7) and (2.8), the magnetic field divides into two independent parts: the dipole type, where B(x,q) = - B(x, - q), A(x,q) = A(x, - q), and the quadrupole type, where B(x,q) = B(x, - q), A(x,v]) = - A ( x , - q). The limits of the model are the flat disk, ((,, = 0), and the sphere, (to = a), corresponding to aspect ratio E = 0 and E = 1 respectively.

3. THE NUMERICAL METHOD

The method of solution of Stix and White was based on finite differences in the radial direction. Following Barenghi and Jones (1991), we use a fully spectral method in both x and v] in order to achieve higher numerical resolution for the same computing power. We expand the unknown poloidal and toroidal fields as

m = M n = N

A(x,q,4 = c C amn(t)X%4 Y 3 q h (3.1) m = l n = 1

m = l n = l

where X t ( x ) , X:(x), Y;f(q) and Yf(q) are suitable spectral functions, M and N are the degree of the spectral truncations in the radial and meridional directions. We make the simplification that the electric current is zero in the disk exterior, which implies B""' = - Vt)""' for some scalar field $""'. This, together with (2.3), implies Vzt+hou' = 0; in oblate spheroidal geometry the solution of this equation is

n = l

where Pn and Qn are Legendre functions of the first and second kind respectively and A, are arbitrary coefficients. At x = 0 we impose that the functions X : and X t are zero for even m, and that d X 2 d x and dX:ldx are zero for odd m due to a coordinate discontinuity. For the quadrupole solution we use the expansions

m = M n = N

and for the dipole solution we use

(3.5a)

(3.5b)

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where T,,, are Chebyshev polynomials of degree m, P,' are Legendre functions of order one and degree n. All the boundary conditions are enforced by this choice, except the matching of the internal and external poloidal field at the galactic boundary. This condition is given by

where

for a quadrupole type solution, and

(3.7a)

(3.7b) m = l

for a dipole type solution.

Equations (2.7), (2.8) and (3.6) have the form

-L1A+L2B i n O < x < l , O < q < l , (3.8) dA -- at

- = L I B + L , A i n O < x < 1 , O < q < l , (3.9) at dB

L , A = O a t x = l , (3.10)

where L,, L,, L,, and L, are linear operators. These equations are integrated from time tk to time tk+ ' = tk + 6t using the second order, implicit, Crank Nicholson method. One finds

[I -$(Gt)L,]Ak" -$(Gt)L,Bk+' = [l +i(Gt)Ll]Ak+:(St)L2Bk, (3.1 1)

-$(bt)LgAk+' + [l -+(Gt)L1]Bk+l= :(6t)L,Ak + [l +i(Gt)L1]Bk, (3.12)

L,Ak+' =O. (3.13)

We substitute the spectral expansions (3.4a, b) and (3.5a, b) in (3.1 l), (3.12) and (3.7a, b) in (3.13) for the quadrupole and dipole type solutions respectively. The equations are then evaluated at the collocation points xi, uj (i = 1 to M ; j = 1 to N) , where xi are the positive zeros of the Chebyshev polynomial of the second kind of degree 2(M + l), and

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qi are the positive zeros of the Legendre polynomial of the first kind and degree 2N + 1. The problem is reduced to a linear eigenvalue equation of the form LVk+' = RVk, where L and R are matrices of size 2MN x 2MN and the vector V, of length 2 M N , contains the spectral coefficients urn, and bmn. The algorithm Vk + ' = (L- 'R)Vk defines a mapping from the solution at time t" to the solution at time t k + ' . The 2MN eigenvalues, A,,, of this mapping are found using a NAG routine. The growth rate Real (CT,,,,) and the frequency of oscillation Im(omn) of a mode are determined by A,,, = exp(om,6t).

A mode is unstable if Real (on,,,) > 0. The advantage of time stepping with the implicit Crank Nicholson method, rather than replacing terms like dA/& with aA, is that the implicit method protects against numerical instability of the diffusion operator; in this way the high order modes which are poorly resolved at a given truncation have the correct sign of Real (0). This greatly assists the search for the critical values of R,. The price to pay is that one to ensure convergence not only with M and N , but also with 6t, but this is rapidly achieved. Typically a time step of 6t = is sufficient.

4. THE RESULTS

All calculations are performed on Hewlett Packard model 730 and 735 workstations. We find the critical magnetic Reynolds number R,, at a given aspect ratio E. The solution of the kinematic dynamo problem splits into four types: steady quadrupole, steady dipole, oscillatory quadrupole and oscillatory dipole. For each type the solution is tested for convergence in M, N and 6t at all Reynolds numbers and aspect ratios considered. Our criterion of convergence at given M , N and 6t is that R,, is determined to three decimal places. Table 1 reports critical Reynolds numbers R,, and critical angular frequencies of oscillation Q, to illustrate convergence rates in M and N . It should be stressed that as the aspect ratio is decreased, a higher truncation is necessary

Table 1 Oscillatory quadrupole at E = 1.5/15. Convergence in N and M .

M = 8 10 11 12

a) Critical Reynolds number R , N = l O 1828.188 1827.166

20 2575.630 2577.598 21 2575.789 2577.849 22 2575.907 2577.950 23 2575.907 2577.956 25 2575.907 2577.956

b) Critical angular frequency of oscillation R, N = l O 3415.24 3410.25

20 5 149.00 5 152.09 21 5148.63 5 152.21 22 5148.77 5152.30 23 5149.33 5 152.32 25 5149.33 5 152.36

1827.176

2577.714 2577.843 2571.843 2571.843

3410.03

5151.84 5 152.42 5152.37 5152.37

1827.176

2577.843 2577.843 2577.843

3410.03

51 52.41 5152.36 5 152.36

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to achieve convergence. We test our numerical code in the limit of a sphere, E = 1, and compare the results with the calculations of Roberts (1972). We consider Roberts’ distribution a* = cos(0) and w* = r, where r is the spherical radius and 0 is the colatitude; this distribution was also used for testing purposes by Barenghi and Jones (1991) and by Hollerbach et al. (1992). Roberts’ model corresponds to a* = q, o* = x in oblate spheroidal coordinates. Table 2 confirms that we obtain the correct answer in the spherical limit.

It is known [see for example Barenghi and Jones (1991) and Barenghi (1993)] that ctw dynamos in spherical geometry, which is relevant to the Earth and the Sun, tend to have oscillatory solutions, while dynamos in flat astrophysical objects, galaxies for example, tend to be steady. We employ our code to study this change of behaviour. Using (2.5a) we find that oscillatory solutions are preferred for E > 0.23 and E > 0.44 for the dipole and quadrupole types respectively. The physical interpretation of this result is the following. In the spherical dynamo of Roberts, assuming that ct is locally constant, the gradient dw/dr drives dynamo waves which move in the perpendicular, meridional direction. In the case of the spheroid the relevant term is do/ds. If the aspect ratio is too small the dynamo waves are constrained in the perpendicular direction by the flatness of the spheroid and cannot propagate.

a) Stix’s model.

We make comparisons with the works of Stix and Soward and study galactic dynamos with w* resulting from (2.5a). Steady dynamos onset at a lower dynamo number than oscillatory ones, so we shall discuss them first. The steady quadrupole and dipole are obtained when aow0 c 0 and ctowo > 0 respectively. Table 3 compares our critical magnetic Reynolds numbers R , with the predictions of Soward at different values of the aspect ratio. The fact that the steady quadrupole is the most likely to occur is the

Table 2 The spherical limit at E = 0.999999. Critical Reynolds numbers R , and critical angular frequencies i2, for the various models.

Sign of uowo Rm 9

a) Quadrupole + 76.08 55.15

- 85.36 67.45 + 76.040 55.163

- 85.371 67.612

Roberts (1972)

Present work

b) Dipole + 87.13 68.75

- 74.39 54.07 + 87.091 68.810

- 74.399 54.166

Roberts (1972)

Present work

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Table 3 Steady solutions: R,, vs E and comparison with Soward (1992a).

Quadrupole (zowo i 0) Dipole (zow,, z 0)

E R m c Soward’s R , R, SowardS R,

2/15 = 0.133 251.906 238.219 - - 1.75115 = 0.1 I 7 298.582 278.142 931.66 1 706.410 1.5115 = 0.100 364.967 35 1.98 1 1058.428 870.397 1.25115 = 0.083 465.205 452.486 1286.710 1 1 16.73 1 1/15 = 0.067 630.008 617.463 1679.044 15 19.420 3/50 = 0.060 728.451 71 6.01 3 191 5.970 1759.044 1/20 = 0.050 938.900 926.483 2424.000 2269.987 1/25 = 0.040 1285.699 1273.224 326 1.43 5 3 107.950 1/30 = 0.033 1666.435 1653.667 4 179.309 4025.157 31100 = 0.030 1937.641 1924.761 4832.071 4677.11 1 1/40 = 0.025 2518.696 2488.007 6228.027 6071.057 1/50 = 0.020 3478.969 3465.531 8529.099 8368.796 31200 =0.015 5290.043 5276.077 12855.626 12689.855 ljl00 =0.010 9587.837 9572.946 23085.293 22909.906 11200 = 0.005 26699.529 26685.688 63628.493 63404.984

most physically important result and confirms previous work on this subject (Stix 1975, 1978; White, 1977; Soward l978,1992a, 1992b). At small E the agreement between our calculations and the asymptotic analysis is excellent. For example at E = 0.005 the residual difference between the numerics and the asymptotics for the steady quadrupole solution is only 0.052%. At higher values of E our critical magnetic Reynolds numbers are consistent with the work of Stix, who considered only aspect ratios as low as E = 1/30. Figure 1 and 2 show our results together with those of Soward and Stix in terms of the parameter Q = c3 Ric which tends to a finite limit as the aspect ratio tends to zero. The asymptotic curve bends at small values of E and it is apparent that one has to consider aspect ratios smaller than E = 1/30 investigated by Stix to be in the asymptotic regime. The spatial structure of the steady quadrupole and dipole are illustrated in Figures 3 and 4, in which we plot the curves B = constant on the right-hand side of the figure and the curves As = constant on the left. Our curves compare well with those of Stix. He states that for more oblate disks, the structure of the field is essentially the same, and our results agree with this. Another interesting feature of Figures 3 and 4 is that we can test Soward’s prediction that the magnetic field reaches maximum intensity at the point (R/$,bJ2/3). For example the dipole solution illustrated in Figure 4 has peak values of A and B at the points (5.87,O) and (5.68,0.59) respectively which is consistent with Soward’s prediction of (4.95,0.81) at this aspect ratio.

The oscillatory dipole and oscillatory quadrupole solutions are obtained by setting cxow0 < 0 and aow0 > 0 respectively. They onset at higher dynamo numbers than the steady solutions. Table 4 and Figures 5 and 6 show the results for the oscillatory dipole type. We can satisfy our strict convergence criterion only as far as E = 0.05 at which we need N = 40 and M = 14. In the limit of small E the agreement between our calculation and the asymptotic analysis is not as good as in the steady case because the slope of the

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0.0 0.05 0.1 0.1 5

aspect ratio Figure 1 line with solid circles: our results.

Q vs E for steady quadrupole; dotted line: asymptotics of Soward; stars: numerical results of Stk:

curve Q vs E is different (see Figure 5). At the lowest aspect 'ratio E = 0.05 we find R , = 6153 against Soward's R , = 6265 at the same E, which is a difference of 1.8%. Similary the frequency of os"dil1ation at E = 0.05 differs from Soward's by 1.9%.

Figure 5 shows that our values of Q at small E are much closer to Soward's Q = 4466 at E = 0 than to the value Q = 3676 obtained by using Parker's slab model, which ignores radial variation of the dynamo number and the effects of curvature (Parker, 1971; Soward 1992b). It is important to understand that Soward's asymptotic calculation of Q consists of two parts: the zeroth order term Q = 4466 and the first order correction dQ/dc. The former part is simple to determine, but the latter involves difficult analysis, which is likely to be responsible for the difference between the slopes. Our work has renewed Soward's interest in the calculation and stimulated further analysis (Soward, 1994). Finally we remark that Stix's values of Q disagree in a qualitative way with both our results and Soward's, because they increase rather than decrease as E is reduced.

Table 5 and Figures 7 and 8 refer to the oscillatory quadrupole solution. In this case we have reduced the aspect ratio as low as E = 1/15 for which we need N = 30 and M = 11. The calculation of Stix apparently gives critical magnetic Reynolds numbers closer in value to Soward's asymptotics than our results. We think that this is due to lack of convergence in Stix's calculation. At aspect ratio ~ = 2 / 1 5 Stix reports

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I I I I 0.0 0.05 0.1 0.1 5

aspect ratio Figure 2 Q vs E for steady dipole; dotted line: asymptotics of Soward; stars: numerical results of Stix; line with solid circles: our results.

Figure 3 Steady quadrupole solution at E = 1.75/15: curves of B = constant are plotted on the right, curves of As = constant are plotted on the left.

Figure 4 Steady dipole solution at E = 1.75/15: curves of B = constant are plotted on the right, curves of As =constant are plotted on the left.

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Table 4 Oscillatory dipole: Rm and R, vs E and comparison with Soward (1992b)

& Rm a, Soward's R _ Sowad's R,

2/15 =0.133 1457.882 1423.059 1542.705 15 19.875 1.75/15 =0.117 1767.248 1836.234 1860.118 1949.143 1.5/15 =O.lOo 2210.485 2470.266 2312.466 2604.000 1.25/15 = 0.083 2886.021 3518.493 2997.767 3679.200 1/15 =0.067 4010.43 1 5447. I54 4 129.906 5638.500 0.75/15 = 0.050 61 52.9 15 9641.050 6265.301 9828.000

Q

I I I I 0.0 0.05 0.1 0.1 5

aspect ratio Figure 5 Q VSE for oscillatory dipole; dotted line: asymptotics of Soward; stars: numerical results of Stix; line with solid circles: our results.

R,, = 1369. We obtain a similar result (R? = 1372)at M = 8 and N = 6, but at this level of truncation the solution has not yet converged. The problem is resolution in the meridional direction due the combination of two factors. First, as R, is increased, the field lines crowd more towards the boundaries (White, 1977); secondly, as the spheroid becomes flatter, the poloidal field lines must acquire tighter curvature. For example, at E = 2/15, at fixed truncation M = 8 in the radial direction we obtain R , = 1150, 1256, 1372 (close to Stix's 1369), 1471,1609,1681,1707, 1713,1714 and 1714 at N = 2,4,6,8,

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Figure 6 Oscillatory dipole at E = 1.75/15 at onset. Curves of B = constant (right)and As = constant (left) at times (a) t = 0; (b) t = 4 3 0 , ; (c) t = 2n/3Q; (d) 1 = n/Q; (e) t = 4n/3Q; (f) t = 571/3f& and (g) t = 2n/R,, where 24QC is the period of oscillation.

10,12,14,16,18, and 19 respectively. Our converged value at E = 2/15 is R,, = 1715 at M = 1 1 andN= 19.

The disagreement between our results and Soward’s is certainly due to the fact that the asymptotic analysis depends on the parameter p1 being small (Soward, 1992 a); p1 is linked to the radial variation of o. In the case of Stix’s model p1 = f i and the analysis

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Table 5 Oscillatory quadrupole: Rm and Q, vs E and comparison with Soward (1992a)

E Rm 9 Soward‘s R- Sowad’s Q‘

2/15 =0.133 17 15.425 2962.984 1185.651 1074.488 1.75/15 =0.117 2070.716 3827.672 1446.125 1402.116 15/15 =0.100 2577.843 5 152.360 1819.058 1906.584 1.25/15 =0.083 3347.507 7336.893 2386.663 2742.663 1/15 =0.067 462 1.042 11332.628 3328.643 4280.700

7000

Q 5000

t

........................................................... 4000! 3000 0.0 0.05 0.1 0.1 5

aspect ratio Figure 7 Q vs E for oscillatory quadrupole; dotted line: asymptotics of Soward; stars: numerical results of Stix; line with solid circles: our results.

is not really consistent (Soward, 1994). The asymptotic work is based on the assump- tion that the solution is an interior mode in the disk, and it is possible that the solution which we find depends crucially on the existence of the boundary.

b ) A more realistic rotation curve

We repeat the numerical calculations described above to study the effects of o* resulting from the more realistic rotation curve (2.5b). It is convenient to write o* as o* = (1 + E ~ S ~ / K ’ ) - 1’2 where s is the dimensionless radius and K, which can vary

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Figure 8 Oscillatory quadrupole as in Figure 6.

between 0 and (1 + (0 2)'i2, is the measure of where rigid body rotation changes to differential rotation. Since the shear dw*/ds is now negative-whereas w* resulting from (2.5a) implies a positive shear-we expect steady and oscillatory quadrupole solutions for aooo > 0 and aowo < 0 respectively. At aspect ratio E = 2/15, aooo > 0 and K = 1, 0.5, 0.2 we obtain steady quadrupole solutions at R,, = 312.484, 222.653,

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202.796 respectively. The dynamo can therefore onset at a magnetic Reynolds number which is lower than obtained using the rotation curve (2.5a). The field structure remains the same in all cases although the position of the maximum intensity varies: as K is decreased, the region of dynamo action moves towards the axis of rotation. The situation is similar for the steady dipole model (a,o, < 0). For oscillatory quadrupole and dipole type solutions (aooo < 0 and aooo > 0) the value of the critical magnetic Reynolds number at the onset of dynamo action is, as in the case of the Stix model, much higher than the required value for the steady type dynamo. The field structure is similar to that of the Stix model although the dynamo waves now move in the opposite direction due to the sign change in the rotation shear.

5. DISCUSSION AND CONCLUSION

Our primary aim was to solve the mathematical problem of the onset of dynamo action in a thin spheroidal disk raised by Soward (1992a, b) and to clarify the discrepancy between his asymptotic analysis and the calculations of Stix. For steady quadrupole and dipole dynamos our values of R,, vindicate Soward’s work. One has to decrease the aspect ratio to values smaller than those considered by Stix to be in the asymptotic regime. At these very small values of E the agreement with the asymptotic analysis is excellent, as is apparent from Figures 1 and 2. We have also found that the asymptotic analysis predicts correctly the location of the region of maximum magnetic field. At higher values of E the work of Stix, Soward and ourselves all give consistent critical magnetic Reynolds numbers and field structures. For oscillatory dynamos the agreement between our calculations and the asymptotic analysis is not so good. In the quadrupole case our values of Q are much larger than those predicted by Soward and found by Stix, but we believe that our solution is correct. This is because firstly it is apparent that the agreement between Stix and Soward is due to lack of numerical resolution in Stix’s work: we can recover Stix’s results at low truncation level. Secondly, Soward’s asymptotic analysis is not consistent when applied to Stix’s model. The fortuitous agreement between the results of Soward and Stix confirms the importance of testing a numerical approximation at all values of the parameters varied in the calculation. There are other examples in dynamo theory in which poor numerical resolution in an unexpected part of the computation gives quantitatively wrong results, as reported for example in the recent work of Jault (1994). In the case of the oscillating dipole the disagreement between our results and Soward’s is possibly due to the difficulty in computing the first order correction dQ/de to Q; further analytic work is clearly required. We have also studied a more realistic rotation curve and confirmed that the preferred mode to occur is the steady quadrupole; oscillatory modes onset at much higher magnetic Reynolds number. The field structures which we obtain are similar to those found using Stix’s rotation curve, the only difference being a change in position of the region of most intense activity.

The way is now open to study nonlinear dynamos in thin spheroids. The fact that the oscillatory modes are excited at higher dynamo numbers than the steady ones does not mean that the oscillatory modes are of no interest. Estimated galactic magnetic Reynolds numbers are rather high [ R , x 1600 in Moss and Tuominen (1990) and in

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Ruzmaikin, et al. (1988); R, z 4500 in Stix (1978)l and the solution is likely to be in a strong supercritical regime in which steady and oscillatory modes can interact. Using the more realistic rotation curve (2.5b) we find for example that at E = 2/15 the oscillatory quadrupole mode onsets at R,, = 1841 for K = 1, and R,, = 1134 for K = 0.5. The studies of Jennings and Weiss (1990) and Jennings (1991) in the context of the solar dynamo have showed that the most unstable modes of linear theory do not necessarily dominate the solution once nonlinear effects are taken into account. Moss and Tuominen (1990) have already considered a disk dynamo with nonlinear a- quenching. We plan to extend our work in this direction by determining the solutions’ bifurcation diagram and adding more physical ingredients to the model.

Finally, from a computational point of view, our work has established that high resolution spectral techniques, combined with the speed and memory of modern workstations, can address magnetohydrodynamic problems thought previously only to be asymptotically accessible. This is an important unifying idea for both the theorctician and the numericist.

Acknowledgements

We wish to thank Professor A. M. Soward for stimulating this work and for many discussions. One of us (CFB) acknowledges the support of SERC grant number GR/H 03278.

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Documents

High resolution numerical dynamos in the limit of a thin disk galaxy