joanne mason and edgar knobloch- long dynamo waves

8/3/2019 Joanne Mason and Edgar Knobloch- Long dynamo waves

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Physica D 205 (2005) 100124

Long dynamo waves

Joanne Masona,b,, Edgar Knoblocha,c

a Department of Applied Mathematics, University of Leeds, Leeds LS2 9JT, UKb High Altitude Observatory, National Center for Atmospheric Research,

Boulder, CO 80307, USAc Department of Physics, University of California, Berkeley, CA 94720, USA

Available online 5 February 2005

Abstract

A simple mean-field model of magnetic field generation in the Sun is considered. The model is characterized by spatially

disjoint locationsof the and effects,believed to take place in thesolar convection zone andin thesolar tachocline,respectively,

and includes -quenching. The model admits a long wave dynamo instability, whose evolution is described by a perturbed mKdV

equation. Solutions of this equation, the so-called snoidal waves, describe nonlinear waves of magnetic activity migratingtowards

the equator, as observed in the Sun.

2005 Elsevier B.V. All rights reserved.

Keywords: Mean field dynamo theory; mKdV equation

1. Introduction

Surface observations of the Suns magnetic activity reveal an array of features varying on many spatial and

temporal scales. One of the most widely studied surface manifestations of the Suns large-scale magnetic field are

sunspots. Sunspots are regions of intense magnetic field, and exhibit a 22 year cycle that takes the form of a wave

of magnetic activity. They are born in pairs of opposite polarity, and at the beginning of a cycle appear at about

30 latitude. Their lifetimes are short, however, of the order of several days or weeks, and as they decay new spots

emerge at a slightly lower latitude. The resulting wave of activity reaches the equator in about 11 years, at whichpoint the process starts afresh, with spot pairs beginning to reappear near 30, but this time with reversed polarities.

Thus the whole cycle takes approximately 22 years, a period that fluctuates somewhat but is in fact remarkably

stable over long times [3].

Corresponding author. Tel.: +1 303 497 1503; fax: +1 303 497 1589.

E-mail address: [email protected] (J. Mason).

0167-2789/$ see front matter 2005 Elsevier B.V. All rights reserved.

doi:10.1016/j.physd.2005.01.006


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J. Mason, E. Knobloch / Physica D 205 (2005) 100124 101

Mean field dynamo theory [20,25] provides a metaphor for understanding the origin of the solar magnetic cycle.

The theory describes the evolution of the large-scale azimuthally-averaged magnetic field, and relies on two basic

mechanisms: the shearing of a pre-existing poloidal field by the Suns differential rotation to produce a toroidal

field (the

effect), and a process known as the

effect that is responsible for regenerating the poloidal field fromthe toroidal field. Helioseismology (see, for example, [24,27]) suggests that the most important shearing motion

is radial and occurs at (or just below) the base of the solar convection zone in a region called the tachocline.

In contrast, the classical effect of Parker [21] describes the net effect of convection in a stratified rotating

shell, and hence is distributed throughout the solar convection zone, although the sign of this effect is believed

to be correct for equatorward migration of active regions only towards the lower reaches of this region (see, for

example, [5,32]). Parker [22] has explored a simple model, known as the interface dynamo, that incorporates

the above ideas, and in particular the suggestion that the and effects are physically separated. Nonlinear

versions of Parkers model that include the nonlinear suppression of the effect by the mean magnetic field have

been explored by (among others) Charbonneau and MacGregor [9] and Tobias [28], showing that the interface

scenario is effective and may generate equipartition strength magnetic fields, and that solutions of the model

partial differential equations can resemble qualitatively a number of the observed properties of the solar magnetic

cycle [6].Our purpose in this article is not so much to model the solar cycle, but to draw attention to some intriguing

properties of the interface dynamo. In particular, following Mason et al. [19], we note that the equations admit both

long wave modes with small wavenumber, and short wave modes with wavenumber comparable to the depth of the

region responsible for the -effect. Traditionally it is believed that the latter modes are the relevant ones for the solar

dynamo; however, with the development of the interface dynamo it makes sense to consider the long wave modes

as well [29,30]. Somewhat unexpectedly (given the dissipative nature of the model) we find that the long wave

modes evolve according to the modified Kortewegde Vries equation, and trace this observation to the fact that the

mean-field dynamo equations are written in terms of the vector potential for the poloidal field. At leading order this

potential is independent of the depth and is therefore phase-like. Consequently the theory of the long wave mode

bears considerable similarity to phase dynamics, and in particular resembles the development of the celebrated

KuramotoSivashinsky equation as the phase equation for the evolution of the Eckhaus instability [18]. However,the fact that the sign of the magnetic field is arbitrary and that the waves have a preferred direction of propagation

(equatorward) changes the form of the phase equation that results, and leads to a modified Kortewegde Vries

equation at leading order, with weak damping and forcing at supercritical dynamo numbers entering only at higher

order.

The paper is organized as follows. In Section 2 we describe the basic problem we study, followed in Section 3 by

a summary of the basic properties of the linear dispersion relation. The bulk of the paper is contained in Section 4

where the leading order amplitude equation, the mKdV equation, is derived. Section 5 contains a derivation of

the perturbed mKdV equation and discusses its solutions; the relation of these solutions to the solar magnetic

activity cycle is summarized in Section 6. Certain aspects of the (somewhat lengthy) derivation are relegated to

Appendix A.

2. The model and governing equations

We consider an idealized nonlinear mean-field dynamo in which the and effects are spatially separated. For

simplicity we take both of these to be spatially localized, with the former located at z = 1 (representing the effect

of the convection zone) and the latter located at z = 0 (representing the solar tachocline). We write the magnetic

field B(x,z,t ) in the form

B = Aey + Bey, (2.1)


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102 J. Mason, E. Knobloch / Physica D 205 (2005) 100124

Fig. 1. The geometry of the model. Local Cartesian coordinates are defined on the interface of the convection zone and the tachocline at a point

in the northern hemisphere, with x increasing poleward and z with radius.

where A(x,z,t )ey is the vector potential of the poloidal magnetic field and B(x,z,t ) represents the toroidal field.

The dimensionless dynamo equations [15,20] then read

A

t = (z)B +2

A, (2.2)

B

t= DG(z)

A

x+2B, (2.3)

where (z) = (z 1)/(1+ B2), G(z) = (z) and D 0G0z30/

20 is the dynamo number. The form of represents

quenching of the effect as the field amplifies [13], and provides the sole nonlinearity in the problem.

The above equations are to be solved in the semi-infinite domain < x < , L z L for waves that

travelin the negativex direction, i.e., towardsthe equator (Fig.1). Here z = L > 1 represents thetop of theconvection

zone, while z = L lies in the radiative interior below the tachocline. In the following we adopt boundary conditions

obtained by matching the magnetic field inside the layer to an external potential field. In a thin layer geometry such

a procedure leads to the boundary conditions [23]

B(x, z = L, t) = 0,A

z(x, z = L, t) = 0. (2.4)

Thus the toroidal magnetic field is confined in L < z < L while the poloidal magnetic field is normal to the layer

at z = L at leading order in its aspect ratio. In the following we increase the dynamo number D to trigger the

onset of the dynamo instability.

It should be noted that, except at the locations of the and effects responsible for magnetic field generation,

the equations for A and B are diffusion equations. Solutions of these equations in the three regions L < z < 0,

0 < z < 1, 1 < z < L satisfying the boundary conditions at z = L are therefore simple to write down. These

then have to be matched across z = 0 and z = 1 subject to the requirement that A and B are continuous and their

derivatives satisfy the jump conditionsA

z

z=0

= 0,

B

z

z=0

+ DA

x

z=0

= 0, (2.5)

A

z

z=1

+B

1 + B2

z=1

= 0,

B

z

z=1

= 0, (2.6)

obtained by integrating the model equations across z = 0 and 1, respectively. We employ here the usual notation in

which square brackets denote the jump in a quantity across the specified surface.

Before continuing we draw the readers attention to an unusual feature of the above problem. Since the scalar

field A is a potential and subject to Neumann boundary conditions, it is only defined up to a constant. This fact


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implies that A behaves like a phase variable in phase dynamics, a fact that is responsible for a number of unexpected

properties of the above problem.

3. Linear theory

We begin by considering the linear problem, i.e., by replacing (z) by (z) = (z 1). In each of the three

regions we seek solutions of the form

A(x,z,t ) = a(z) exp(pt+ ikx), B(x,z,t ) = b(z) exp(pt+ ikx),

where p = + i is the complex growth rate. Applying the continuity conditions on A and B, and the matching

and boundary conditions, leads to the dispersion relation [19]

4q2 sinh 2qL ikD sinh[2q(L 1)] = 0, (3.1)

where q2 p + k2.

3.1. Threshold for instability

As shown by Mason et al. [19] this dispersion relation describes in general two types of modes, a long wave

mode with wavenumber k 1 and a short wave mode with k = O(1). We focus here on the former and take

k = 1.

To compute the marginal stability curve we set = 0 and compute D = Dc(k) in the form of a series in . Since

within the model the direction of the waves is arbitrary we anticipate that Dc will be even in kwhile will be odd.

We therefore write

Dc = D0 + 2D2 + , = 10 +

330 + ,

and expand the dispersion relation (3.1) in powers ofq2 i + 2 1:

4q2(2L+4

3L3q2 +

4

15L5q4 +

8

315L7q6 + ) ikD(2(L 1)+

4

3(L 1)3q2 +

4

15(L 1)5q4

+8

315(L 1)7q6 + ) = 0. (3.2)

At O() the resulting problem is purely imaginary and we obtain

D0 =4L10

L 1. (3.3)

At O(2) the problem is purely real and yields

10 =

3

2(2L 1).

Since L > 1 this quantity is real. In the following we suppose that D0 > 0 and hence choose the positive sign in

this expression. With this choice of sign all disturbances travel in the negative x direction, i.e., towards the equator.


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At O(3) we obtain

D2(L 1) 4L30 =4

1510L[4L

2 + 26L 13], (3.4)

while at O(4) we obtain

D2(L 1)3 4L30(L

2 + 2L 1) =4

10510L[12L

4 + 46L3 203L2 + 180L 45]. (3.5)

From these equations we readily deduce the values ofD2 and 30. In particular D2 > 0 for all L > 1.

3.2. Growth rate of supercritical dynamo waves

We next suppose that the dynamo number is supercritical, so that > 0. We again write k = , and suppose that

D = D0 + 2D2 + 4D4 + + 2,

where = O(), but is otherwise an independent small parameter. We anticipate that the growth rate is an even

function of while the frequency is odd in :

= 202 + 2222 +

404 +O(42; 24; 6),

= 10 + 330 +

212 +O(5; 32; 4).

The expansion for incorporates the fact that = 0 when = 0, i.e., on the neutral stability curve. Substituting

the above expansions into the dispersion relation and focusing on the coefficients of 2, 2, 22 and 4 we obtain

02 = 0, 12 =L 1

4L, 22 =

10(2L2 3L+ 1)

3L, 04 = 0, (3.6)

respectively. In summary,

= 10 + 330 +

212 +O(5; 32; 4), = 2222 +O(

42; 24; 6). (3.7)

It follows that when = d, d= O(1), the amplitude of the waves must depend on three distinct timescales: t=

O(1), t= O(3) and t= O(4). This fact complicates considerably the derivation of the amplitude equation.

4. The amplitude equation

We now turn to the nonlinear problem and write

(z) =(z 1)

1+ B2 (z 1)(1 B2).

The dynamo equations (2.2)(2.3) can be written in the matrix form

L = N(),


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where L is the matrix

L = t xx zz (z 1)D(z)x t xx zz

,Nis the vector of nonlinear terms

N=

(z 1)B3

0

,

and

=

A

B

.

The linear theory of the preceding section tells us that although waves set in as soon asD exceeds Dc these waves

have a O() wavenumber and a O() frequency. This fact suggests that the simplest way to obtain an amplitude

equation for the waves is to use multiple space and time scales. Indeed, as already mentioned, Eq. (3.7) suggests that

we introduce a large spatial scale X = x, together with the time scales T10 = t, T30 = 3t, T12 =

2t, T22 = 22t.

Here x is a short lengthscale and ta fast timescale, although neither scale will appear in the solutions that follow.

As in the linear problem is defined by k = , while 2 is the departure from criticality, i.e. D = D0 + 2D2 +

+ 2. Since as already mentioned the potential A is phase-like we seek a solution in the form

= A0

0 + A1

B1 + 2 A2

B2 + ,where A0 is O(1) but depends on x and t in the form (x + ct), as appropriate for a traveling wave. Despite

this unusual Ansatz it should be clear that whenever DDc = O(2) the physical magnetic field will in fact

be O() since it depends on the derivative of A0 with respect to x. To proceed we replace x by X and t by

T10 + 3T30 +

2T12 + 22T22 . The above matrix problem, with the boundary, jump and continuity conditions

specified in Section 2, now becomes a series of problems to be solved at each order in (where = d). Since these

computations are somewhat involved they are relegated to Appendix A. We summarize here the results.

At O(1) we conclude immediately that A0 = A0(x, t) only. At O() the linear problem for (A1, B1) is inhomo-

geneous and we must impose a solvability condition. This condition yields the relation

A0

T10 c0

A0

X= 0, (4.1)

where

c0 =D0(L 1)

4L. (4.2)

Note that, as expected, c0 = 10 /k.

This result suggests that the remainder of the calculation is best performed in a frame moving with a speed c,

where

c = c0 + 2c20 +

2c02 +


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and c0 is given by Eq. (4.2). When = d we can combine the two O(3) timescales in a timescale T3, relabel the

O(4) timescale T4, and write instead c = c0 + 2c2 + etc., where c2 c20 + d

2c02 = 30 + d212, etc. This

procedure permits us to look for solutions of the form A = A(, T3, T4), B = B(, T3, T4), where = X + cT10.

This is accomplished by replacing t by c + 3

T3 + and x by ; the remaining derivative with respect toT3 thus refers to any T3 dependence in addition to that arising through .

In the moving reference frame the solvability condition (4.1) is replaced by (4.2). The solution of the O()

problem is then (see Appendix A):

A1 = c0A0

z2

2 c0

A0

Lz + b4 in z > 1, (4.3)

A1 = c0A0

z2

2+ c0

A0

Lz + b5 in z < 1, (4.4)

B1 = D0

2

A0

z +

LD0

2

A0

in z > 0, (4.5)

B1 =D0

2

A0

z +

LD0

2

A0

in z < 0. (4.6)

Here

b4 = 2c0LA0

+ b5

but the function b5 remains undetermined at this order. Note that b5 is again phase-like, i.e., it is independent of z.

At O(2) the solvability condition for A2, B2 is automatically satisfied and A2, B2 are readily found. At O(3)

we once again require a nontrivial solvability condition:

A0

T3 a

A0

a

3A0

3+

6L2c0

2L 1

A0

3= 0. (4.7)

Here

a = c2 +L 1

4L(D2 + d

2) =c0

15(4L2 + 26L 13) > 0,

a relation that follows from Eqs. (3.4) and (3.6). In the following it will be useful to write this equation in the form

A0

T3 a

A0

a

3A0

3+ bA20

A0

= 0, (4.8)

where A0 A0 and b = 18L2c0/(2L 1) > 0. The variable A0 is analogous to the local wavenumber in phase

dynamics.

Note that the linearization of Eq. (4.7) with A0 exp(i3p3t+ i), = (x + ct), c = c0 +

2c2 + yields

p3 0, confirming that A0 exp[i(10 + 230 +

212 + )t+ ix] as assumed in Section 3.

4.1. The mKdV equation

Eq. (4.8) is the modified KdV equation and is completely integrable [1]. The equation describes a variety of other

systems as well, including ion acoustic solitons [26,31] and interfacial waves in two-layer liquids with gradually

varying depth [12]. However, in the present case a > 0, b > 0 and no solutions in the form of solitary or cnoidal


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waves are possible. Instead we find a class of uniformly traveling nonlinear solutions called snoidal waves. Since

these waves are nonlinear we expect that they travel at a speed that differs from the linear speed. To permit drift

with respect to the frame = const., we go into a reference frame traveling with speed v northwards, i.e., we let

= vT3, and look for steady solutions of a given period in

. These satisfy the equation

a3A0

3+ (a + v)

A0

bA20

A0

= 0, (4.9)

where A0 is now a function of alone. Integrating this equation twice we obtain

1

2A

20 +

1

2

1 +

v

a

A20

b

12aA40 = E, (4.10)

where the prime denotes derivative with respect to . This equation has the solution

A0 = N1/2 sn(, s), (4.11)

where

N=6(a + v)

b

s2

1 + s2

, 2 =

1 +

v

a

11+ s2

, E=1

2N2. (4.12)

Here sn is the elliptic function of the first kind and 0 s 1 is its modulus. Its period is 4K(s)/, where K(s) is

the complete elliptic integral of the first kind. It follows that solutions of period 2 travel with speed v given by

v = a

4(1+ s2)K(s)2

2 1

= a

3

2s2 +

27

32s4 +

. (4.13)

When s = 0 the solutions are infinitesimal in amplitude and sinusoidal; such solutions are stationary in the frame

traveling with the linear velocity c, and indeed v(0) = 0. The amplitude N and speed v of the solutions increase

monotonically with s. Thus nonlinear waves travel towards the equator more slowly than infinitesimal waves. In

contrast, on the real line we have a two parameter family of solutions, specified by s and v. It should be emphasized

that, to this order, both s and v are determined by initial conditions; these specify the initial energy E, as well as theinitial momentum. The resulting description is appropriate for the nonlinear evolution of dynamo waves for O(3)

times when D Dc = O(2). However, on longer times both forcing and dissipation enter in the description, and

produce a slow drift in the modulus s. The fixed point of this drift determines s in terms of the distance d2 above

the instability threshold, and hence the amplitude and speed of the waves, cf. [17]. This calculation is the focus of

the next section.

5. The perturbed mKdV equation

In order to determine the equilibrated state of the waves for supercritical values of the dynamo number we need

to include the behavior of the system on the time scale T4 = O(1). To perform this calculation we first solve forA3 and B3 (see Appendix A), and then proceed to O(

4). The solvability condition at this order yields an equation

involving both the undetermined function b5 and the amplitude A0 satisfying Eq. (4.7). It therefore determines the

unknown function b5:

b5

T3 a

3b5

3 a

b5

+ b

A0

2b5

+

2A0

T3+

A0

22A0

2+

4A0

4+

2A0

2+

A0

T4= 0,

(5.1)


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where

= c0

6(2L2 14L+ 7), =

9L2(2L2 + 2L 1)

2(2L 1)2, (5.2)

=1

420(2L 1)[120L4 + 740L3 742L2 + 372L 93],

= c2 +(2L3 3L+ 1)c0

24L(D2 + d

2).

Note that b5 = 0.

This result can also be compared with the linear theory discussed in Section 3. We suppose that A0, b5

exp(i3p3t+ i4p4t+ i), where = (x+ ct), c = c0 +

2c2 +O(4), and linearize Eq. (5.1). Using the results

(3.4)(3.5) relating 30 and D2 together with the result (3.6) for 12 now shows that p4 = id222, and hence that

A0, b5 exp[i(10 + 2

30 +

2

12 + )t+ ix+ 2

2

22t] as assumed in Section 3.To obtain an evolution equation that includes the behavior of the system on the T4 timescale we follow Aspe and

Depassier [2] and reconstitute the amplitude equation. In order to do this we first differentiate Eq. (5.1) with respect

to , and rewrite it in terms of A0 A0 , b5 b5 . We then return to Eqs. (4.3)(4.4) for A1 in z > 1 and z < 1

and observe that these equations contain a common phase-like quantity b45 (b4 + b5)/2 = c0L(A0/) + b5.

Thus we set C = A0 + b45 and construct an evolution equation in the moving reference frame for the redefined

phase-like variable C(, ):

C

a

C

a

3C

3+ bC2

C

+ f = O(2). (5.3)

Here /= /T3 + /T4 + , and

f = (a + ) 2C

2+ (a + )

4C4

+ ( b) 2

2

13

C3

. (5.4)

We refer to Eq. (5.3) as the perturbedmKdV equation.

5.1. The bifurcation diagram

In this Section we use the perturbed mKdV equation to construct the bifurcation diagram for long dynamo waves.

As already noted we expect that the energyEor equivalently the modulus s of the waves will evolve as a consequenceof the perturbation f, cf. [14]. To determine the effect of this perturbation we write Eq. (5.3) in the moving frame

a3C

3+ (a + v)

C

bC2

C

= f+O(2), (5.5)

multiply by it Cand integrate over a period ofC. This procedure yields the following exactcondition for the presence

of a periodic solution of the perturbed mKdV equation:Cf d = O(). (5.6)

This condition can be approximated using C = A0 +O() and 4K(s)/ for the period, yielding the condition4K(s)/0

A0f d = O(), (5.7)


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Fig. 2. The amplitude N as a function of d2 computed from Eq. (5.8) (solid line) compared with the perturbation result (5.17) (dashed line)

(L = 10).

where A0 and are given by Eqs. (4.11)(4.12), with v given by (4.13). The quantity (5.7) is therefore a functionofs, h(s) say, and we therefore seek the root s = s ofh(s) = 0 as a function of the supercriticality parameter d2,

i.e., of[7]

h(s) 1

4E

+

b

a

1+

v

a

(a + )

v

a

NI2

+1

b

3a

(a + )

v

a

3

1 +

v

a

+

b

a

N2I4 +

2b

3a

+

b

a

N3I6 = 0. (5.8)

Here In =

4K(s)

0 snn d. From s we reconstruct the bifurcation diagram N(d2) (Fig. 2), and compute the speed

v (Fig. 3) as a function of d2. Fig. 4 shows the corresponding snoidal wave for d2 = 100 (s 0.79). Note that

dh/ds at s = s determines the stability of the solution: the solution is stable in time if dh/ds > 0 and unstable ifdh/ds < 0. Fig. 5 illustrates h(s) for d2 = 25, with the positive gradient at s 0.5 indicating a stable solution.

5.2. Weakly nonlinear theory

We can check the results of the preceding section using weakly nonlinear theory. We start with the first integral

of Eq. (5.5):

Fig. 3. The speed v as a function ofd2 computed from Eq. (5.8) (solid line) compared with the perturbation results (5.13) and (5.15) (dashed

line) (L = 10).


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Fig. 4. The snoidal wave A0 for d2 = 100 (s 0.79, L = 10).

a2C

2+ (a + v)C

1

3bC3 =

(a + )

C

+ (a + )

3C

3+ ( b)C2

C

+O(2), (5.9)

and suppose that

C = 1/2C0 + 3/2C1 +

5/2C2 + , v = v1 + 2v2 + , = c + 1. (5.10)

Substituting these expansions into Eq. (5.9) we obtain at O(1/2) the result

C0 + C0 = 0 (5.11)

with the solution C0 = R sin up to an arbitrary phase. At O(3/2) we obtain

C1 + C1 =b

3aC30 +

1

a v1C0 + (a + c)C0 + (a + )C

0 . (5.12)

The solvability conditions for this problem are

v1 =1

4bR2, c = . (5.13)

When these conditions hold we can solve for C1 and obtain C1 = (b/96a)R3 sin3. Finally, at O(5/2) we obtain

C2 + C2 =1

a[bC20C1 v2C0 v1C1 + (a + c)C

1 + (a + )C

1 + 1C

0 + ( b)C

20C

0]. (5.14)

Fig. 5. The function h(s) defined by Eq. (5.8) for d2 = 25, L = 10.


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The solvability conditions are now

v2 = b2

384aR4, 1 +

1

4( b)R2 = 0. (5.15)

The latter equation determines the direction of branching of long dynamo waves. We obtain

R2 =2L 1

9L21. (5.16)

Since 1 = (L 1)(2L 1)(c0/3L)(d2/) > 0 the primary bifurcation is supercritical for all values ofL > 1. In

Fig. 3 the dashed line represents the speed v as a function ofd2 as computed from the above expansion, while the

corresponding line in Fig. 2 represents the resulting amplitude N:

N=(2L 1)2(L 1)

27L3c0d

2. (5.17)

It is of interest to compare these results with the exact result computed in Section 4.1. From the expansion

(4.13) for v and the corresponding expansion for Nwe conclude that

s2 =

bR2

6a

5

82

bR2

6a

2+ , (5.18)

and hence that

v =1

4bR2 2

b2R4

384a+ ,

as obtained via perturbation theory. To obtain expression (5.18) we used the fact that sn(p, s) = [1+ (s2/16)+

]sin q + (s2/16+ )sin3q + , where q = (/2K)p.

Similarly, expanding the result (5.8) in powers ofs2 we obtain

h(s) =N

+

3

8( )s2 +

3a

2b(b )s2 +

.

The results (5.13b) and (5.15b) now follow on writing = c + 1, and using (5.18).

5.3. Wavelength selection

In Section 4 we imposed the wavelength of the solution; without loss of generality we chose this wavelength

to be 2. Although this is normal procedure in problems of this kind we expect that in the physical situation the

wavelength, or equivalently the wave speed v, will be selected by physical processes. To see how this comes aboutwe multiply Eq. (5.9) by Cand integrate the result over a period. We obtain

v =

[a(C

2 C2) + (b/3)C4] d

C2 d+O(2). (5.19)

This is an exact expression. Note, however, that the perturbation f has dropped out. For this reason this equation isidentically satisfied to O(), a fact that can be readily checked using an expansion of the form C = A0 + C1 +

in both (5.19) and (5.9). Thus any selection of the speed comes about at O(2), a calculation that is beyond the scope

of this paper. However, in contrast to mechanisms responsible for wavelength selection in other pattern forming

systems, exemplified by the so-called Busse balloon, we expect the selection based on Eq. (5.19) to be sharp.


13/25


Fig. 6. Profile ofB1(, z) at = /2 corresponding to a maximum value of A0 when d2 = 100. At = 3/2, corresponding to a minimum

valueofA0, the signofB1(, z) is reversed. Note that = /2 also corresponds to a zero ofA0 (with the mean subtracted off), and at = 0, ,

corresponding to minimum and maximum values of A0, respectively, B1(, z) = 0 (s 0.79, L = 10).

5.4. Physical manifestation of the solution

The corresponding solutions for the fields A0, A1, B1 can be reconstructed from the solution (4.11) to give

A0 =N1/2

sln[dn(, s) s cn(, s)],

to within an arbitrary constant of integration, and

A1 = N1/2(

1

2c0z

2 c0Lz + 2c0L)sn(, s)+ b5 in z > 1,

A1 = N1/2(

1

2c0z

2 + c0Lz)sn(, s) + b5 in z < 1,

B1 =1

2N1/2D0(L z)sn(, s) in z > 0,

B1 =1

2N1/2D0(L+ z)sn(, s) in z < 0.

Fig. 7. The toroidal (B/, solid line) and poloidal fields (|BP|/ for and|BP|/ for , |BP|2 2[(A1/z)

2 + A20], dashed line)

at z = 0 when d2 = 100 (s 0.79, L = 10).


14/25


Fig. 6 shows the resulting vertical profile of B1(, z) at = /2, chosen to correspond to a maximum of A0.

The leading order contributions to both the toroidal and poloidal fields are shown together in Fig. 7; both are

of order , despite the fact that A0 = O(1), with the toroidal field approximately twice as strong as the poloidal

field.

6. Discussion

In this paper we have derived an amplitude equation for slightly supercritical long wavelength dynamo waves.

The derivation was based on a simplified model of the mean-field dynamo equations, and led to a leading order

description of the waves in terms of a modified Kortewegde Vries equation. This equation is exactly solvable,

and in our case describes strongly nonlinear waves called snoidal waves. For positive dynamo numbers the waves

propagate in the negative x direction, corresponding to propagation towards the equator, as observed in the Sun. In

order to describe the growth and equilibration of the magnetic field we considered dynamo numbers O(2) above

critical, where 1 measures the wavelength of the waves (in units of the depth of the layer where dynamo action

takes place). At such values of the dynamo number the amplitude of the waves grows on a yet slower timescale andnonlinear quenching of the effect saturates the magnetic fieldat O() amplitude. The amplitude and speed of the

resulting dynamo wave are related to the distance d2 from threshold for the instability, and are approximated well

by second order perturbation theory. Our results indicate that the bifurcation producing these waves is supercritical,

suggesting that the waves are stable.

Because of their long wavelength the waves are described at leading order by an integrable amplitude equation,

with forcing and dissipation entering only at higher order. Consequently we have had to employ multiple scale

methods involving three distinct timescales (in addition to a slow spatial scale), followed by a procedure that has

been termed reconstitution to describe the effects of the higher order terms on the leading order dynamics. This

procedure is frequently used in applied mathematics, e.g., [2], although from the point of view of asymptotics the

results cannot be justified. However, in many instances an alternative iterative procedure leads to identical results,

e.g., [17], and in some of these the results can be justified rigorously via normal form theory [10,11]. We have not

attempted here these extensions of the theory.

Although our results are based on a specific and highly idealized model of the dynamo process, we believe that

many aspects of our results have general applicability. We have already noted that the structure of the expansion

owes much to the fact that the poloidal magnetic field is described in terms of a (vector) potential. As a consequence

the potentialA is phase-like, and the resulting expansion procedure resembles that familiar from studies of fixed-flux

convection [8,16] and of phase dynamics [18]. As is well known, in phase dynamics the leading order description

generally leads to Burgers equation (e.g., [4]), or if one deals with low-frequency long waves, the Korteweg

de Vries equation. The reason we obtain here the modified Kortewegde Vries equation can be traced to the

symmetry of the mean field equations under B B. These considerations suggest that long wave, low frequency

dynamo waves will in general be described by the modified Kortewegde Vries equation, albeit it with different

coefficients, and raise the possibility that solitary dynamo waves may be a general property of interface dynamo

models.

Acknowledgements

The second author spent 5 months in 1988 in the group of Professor Kuramoto as a JSPS Fellow [16], and

is grateful to Professor Kuramoto and all members of his group at that time for their warm and memorable

hospitality in Kyoto, and friendships that continue to this day. This work was supported in part by EPSRC

under grant GR/R52879/01, an EPSRC studentship, and the High Altitude Observatory. We are grateful to Prof.

D.W. Hughes and Dr S.M. Tobias for suggesting this investigation, and to Dr M.C. Depassier for a helpful

discussion.


15/25


Appendix A. Derivation of the mKdV equation

In this appendix we summarize the results of the various steps required to derive the amplitude equation (4.8).

We begin with the matrix problem

L = N(),

where L is the matrix

L =

t xx zz (z 1)

D(z)x t xx zz

,

Nis the vector of nonlinear terms

N=

(z 1)B3

0

,

and

=

A

B

,

together with the boundary conditions (2.4) and jump relations (2.5), (2.6). As explained in Section 4, we write

D = D0 + 2D2 + + 2 (where = d), replace x by X and t by T10 + 3T30 + 2T12 + 22T22 , andseek a solution in the form

= 0 + 1 + 22 + =

A0

0

+

A1

B1

+ 2

A2

B2

+

The matrix problem then becomes a sequence of problems to be solved at each order in , i.e. we solve

Lii = qi for i = 0, 1, . . .

We shall find that the linear operator L0 has a nontrivial nullspace (spanned by vectors of the form (A0, 0), whereA0 is independent ofz), so that we need to impose a solvability condition at each order. For this purpose we require

the solution of the adjoint problem, as described next.

A.1. The adjoint problem

To find the adjoint problem to

L0i = 0


16/25


(including appropriate boundary conditions, jump relations and continuity conditions) we use integration by parts

to write the scalar product , L0i in the form

, L0i = i,L0 + surface terms.

The adjoint problem is then defined to be

L0 = 0,

subject to boundary conditions that eliminate the surface terms for any i. Here

, L0i =

z

(1, 2)

zzAi (z 1)Bi

zzBi

=

z

(Ai, Bi)

zz1

zz2 (z 1)1

1

Ai

zAi

1

z+ 2

Bi

z Bi

2

z

L

L

. (A.1)

Since Bi(x, z = L, t) = 0 and (Ai/z)(x, z = L, t) = 0, the surface terms are eliminated by choosing 2(z =

L) = 0, 1/z(z = L) = 0, leaving the adjoint problem

zz 0

(z 1) zz

1

2

= 0,

1

z(x, z = L, t) = 0, 2(x, z = L, t) = 0.

In addition 1 and 2 must be continuous throughout the domain, and satisfy appropriate jump conditions derived

by integrating the equations across z = 1. We obtain

1 = independent ofz

2 =1(1+ L)z

2L+

1(1+ L)

2in z > 1, (A.2)

2 =1(1 L)z

2L

1(1 L)

2in z < 1.

A.2. The amplitude equation

The leading order problem is considerably simpler than those that follow due to B0 = 0. We are required to solve

2A0

z2= 0,

A0

z(x, z = L, t) = 0.

Since A0 is continuous throughout the domain it must be independent ofz, although it will depend on X, T10, etc.

At O() we obtain the problem

L01 = q1, (A.3)


17/25


where

L0 = zz (z 1)

0 zz ,

q1 =

T10 0

D0(z)X T10

A0

0

,

with the boundary conditions

B1(x, z = L, t) = 0,A1

z(x, z = L, t) = 0.

In addition A1 and B1 must be continuous throughout the domain and satisfy the jump conditionsB1

z

z=0

+D0A0

X(x, z = 0, t) = 0, (A.4)

A1

z

z=1

+ B1(x, z = 1, t) = 0. (A.5)

The problem (A.3) has a solution if and only if

, q1 = 0, (A.6)

where is given by Eq. (A.2). The solvability condition is thus

A0

T10

D0(L 1)

4L

A0

X = 0.

Equivalently, if we look for solutions in the form A0(X, T10, T30, . . .) A0(, T30, . . .), where = X + cT10, and

write

c = c0 + 2c20 +

2c02 +O(4),

we obtain at O() the problem

L0 A1

B1 = L1 A0

0 ,where L0 is as above and

L1 =

c0 0

D0(z) c0

,

subject to the boundary conditions

B1(x, z = L, t) = 0,A1

z(x, z = L, t) = 0. (A.7)


18/25


Moreover, A1 and B1 are continuous throughout the domain, and the jump conditions read

B1

z z=0 +D0A0

(x, z = 0, t) = 0, (A.8)

A1

z

z=1

+B1(x, z = 1, t) = 0. (A.9)

The solvability condition , L10 = 0 for this problem yields

A0

1+

D0(1 L)

4Lc0

= 0, (A.10)

or, equivalently,

c0 =D0(L 1)

4L,

as obtained above.

With this choice ofc0 we can solve the O() problem. The equation for B1 reads

2B1

z2= D0(z)

A0

.

Hence,

B1 = a2z + b2 in z > 0,

B1 = a3z + b3 in z < 0,

where a2, a3, b2, b3 are independent ofz but may be functions of the large space and slow time scales. The boundary

conditions (A.7) give

a2L + b2 = 0, b3 a3L = 0,

while continuity ofB1 across z = 0 gives

b2 = b3,

and the jump relation (A.8) yields

a2 a3 +D0A0

= 0.

These equations can be solved for the unknowns a2, a3, b2, b3 and we obtain

B1 = D0

2

A0

z +

LD0

2

A0

in z > 0,


19/25


B1 =D0

2

A0

z +

LD0

2

A0

in z < 0.

For A1 we solve

2A1

z2+ (z 1)B1 = c0

A0

,

obtaining

A1 = c0A0

z2

2+ a4z + b4 in z > 1,

A1 = c0 A0

z

2

2+ a5z + b5 in z < 1,

where a4, a5, b4, b5 are independent ofz. The boundary conditions (A.7) give

a4 = c0LA0

, a5 = a4.

Applying continuity in A1 across z = 1 gives

b4 = 2c0LA0

+ b5, (A.11)

and applying the jump relation (A.9) yields

A0

1

2D0(L 1) 2c0L

= 0,

a condition that is automatically satisfied by our choice of c0. Thus

A1 = c0A0

z2

2 c0

A0

Lz + b4 in z > 1,

A1 = c0A0

z2

2+ c0

A0

Lz + b5 in z < 1,

where b4 is related to b5 through Eq. (A.11). Note that b5 remains undetermined at this order.

At O(2) we obtain the problem

L0

A2

B2

= L1

A1

B1

+ L2

A0

0

,


20/25


where L0 and L1 are as above, and

L2 = 00

,subject to

B2(x, z = L, t) = 0,A2

z(x, z = L, t) = 0. (A.12)

As before A2 and B2 are continuous throughout the domain, and satisfy the jump conditionsB2

z

z=0

+D0A1

(x, z = 0, t) = 0, (A.13)

A2

z

z=1

+ B2(x, z = 1, t) = 0. (A.14)

The solvability condition for this problem reads

, L11 + L20 = 0,

and yields

z

(1, 2)

A0 c0A1

D0(z) A1 c0B1

= 0,

and is automatically satisfied. The O(2) problem is therefore solvable and we obtain

B2 = c0D0

4

2A0

2

2L3

3 Lz2 +

z3

4

+

D0

2

b5

(L z) in z > 0, (A.15)

B2 = c0D0

4

2A0

2

2L3

3 Lz2

z3

4

+

D0

2

b5

(L + z) in z < 0. (A.16)

The equation for A2,

2A2

z2 + (z 1)B2 = c0A1

2A0

2 ,

yields

A2 =c20

24

2A0

2z4

c20L

6

2A0

2z3 +

c0

b4

2A0

2

z2

2+ a8z + b8 in z > 1,

A2 =c20

24

2A0

2z4 +

c20L

6

2A0

2z3 +

c0

b5

2A0

2

z2

2+ a9z + b9 in z < 1.


21/25


22/25


where a = c0(4L2 + 26L 13)/15 > 0, and b = 18L2c0/(2L 1) > 0. With the solvability condition (4.7) we

can now solve the O(3) problem:

B3 = c

2

0D0240

3

A03

z5 + c

2

0LD048

3

A03

z4 +

D02

3

A03

+ c0 a6

z

3

6+

c0 b6

LD02

3

A03

z

2

2

+a10z + b10 in z > 0, (A.22)

B3 =c20D0

240

3A0

3z5 +

c20LD0

48

3A0

3z4 +

c0

a7

D0

2

3A0

3

z3

6+

c0

b7

LD0

2

3A0

3

z2

2

+a11z + b11 in z < 0, (A.23)

where a10, a11, b10, b11 are to be determined by continuity and the relations (A.18) and (A.19). The boundary

conditions (A.18) give

c20D0L5

60

3A0

3

D0L3

6

3A0

3+

c0L3

6

a6

+

c0L2

2

b6

+ a10L+ b10 = 0, (A.24)

and

c20D0L5

60

3A0

3

D0L3

6

3A0

3

c0L3

6

a7

+

c0L2

2

b7

a11L + b11 = 0. (A.25)

The remaining conditions lead to

a10 = a11 = 1

2

D0b9

+ (D2 + d2)A0

,

and

b10 = b11 = c20D0L

5

60

3A0

3+

D0L3

6

3A0

3

c0L3

6

a6

c0L2

2

b6

+

LD0

2

b9

+

L(D2 + d2)

2

A0

.

(A.26)

Similarly,

A3 =c3

0720

3A0

3 z6

c3

0

L

120

3A0

3 z5 +

c0

2b4

2 23A0

3 c0z4

24 +c0z

3

6a8

+ L3A0

3

+

c0

b8

2b4

2+ c2

A0

+

A0

T3

z2

2+ a12z + b12 in z > 1, (A.27)

A3 =c30

720

3A0

3z6 +

c30L

120

3A0

3z5 +

c0

2b5

2 2

3A0

3

c0z

4

24+

c0z3

6

a9

L

3A0

3

+

c0

b9

2b5

2+ c2

A0

+

A0

T3

z2

2+ a13z + b13 in z < 1, (A.28)


23/25


where a12, a13, b12, b13 are independent ofz and are related through conditions (A.18) and (A.19). The boundary

conditions (A.18) imply

a12 =

c30L5

30

3A0

3

c0L3

63A0

3 + c0

2b4

2

c0L2

2

a8

+ L2b4

2 c0

b8

c2

A0

A0

T3

,

(A.29)

and

a13 = c30L

5

30

3A0

3+

c0L3

6

c0

2b5

2+

3A0

3

c0L2

2

a9

+ L

c0

b9

2b5

2+ c2

A0

+

A0

T3

,

(A.30)

while continuity ofA3 across z = 1 gives

b12 =

c30L

60

3A0

3 +

c20

24

2b5

2 +

c0

6a9

2L

3A0

3+

c0

2

b9

1

2

2b5

2 + a13 + b13

c20

24

2b4

2

c0

6

a8

c0

2

b8

+

1

2

2b4

2 a12. (A.31)

The solvability condition (4.7) is recovered from the jump condition (A.20). Note that we now have three arbitrary

phase-like functions, namely b5, b9 and b13.

Finally, at O(4), we obtain the problem

L0

A4

B4

= L1

A3

B3

+ L2

A2

B2

+ L3

A1

B1

+ L4

A0

0

+N4,

where L0,L1,L2 are as above

L3 =

c2 T3 0

(D2 + d2)(z) c2 T3

, L4 =

T4 0

0 0

,

with T4 = d2T22 , and

N4 =

3(z 1)B21B2

0

.

The boundary conditions are

B4(x, z = L, t) = 0,A4

z(x, z = L, t) = 0, (A.32)

with A4 and B4 continuous throughout the domain, and satisfying the jump conditionsB4

z

z=0

+D0A3

(x, z = 0, t) + (D2 + d

2)A1

(x, z = 0, t) = 0, (A.33)

A4

z

z=1

+ B4(x, z = 1, t) 3B21(x, z = 1, t)B2(x, z = 1, t) = 0. (A.34)


24/25


The solvability condition for this problem reads

, L13 + L22 + L31 + L40 +N4 = 0,

and yields Eq. (5.1) relating b5 and A0.

As explained in Section 5, we then seek the reconstituted amplitude equation for the phase-like variable

C = A0 + (b4 + b5)/2. Differentiating (5.1) with respect to , multiplying by , and adding the resulting equation

to (4.8) yields

C

a

C

a

3C

3+ bC2

C

+ f = O(2), (A.35)

where /= /T3 + /T4 + , and

f =

2C

+

2C

2 +

2

2C3

3+

4C

4 . (A.36)

The first term in f involving the time derivative can be eliminated using Eq. (A.35), resulting in Eqs. (5.3) and (5.4).

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joanne mason and edgar knobloch- long dynamo waves

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