willis phd

The Hydromagnetic Stability of

Taylor–Couette Flow

Ashley Phillip Willis

Thesis submitted for the degree of

Doctor of Philosophy

*Department Of Mathematics

University Of Newcastle Upon TyneNewcastle Upon Tyne

United Kingdom

May 2002

Acknowledgements

First and foremost I would like to thank my supervisor Carlo Barenghi forguidance, support and enthusiasm throughout this research project. I wouldalso like to express my gratitude to Graeme Sarson, Anvar Shukurov, and thepostdocs at Newcastle for many helpful comments and suggestions, and alsoto David Samuels for first introducing me to research.

I am grateful to the Engineering and Physical Sciences Research Counciland the School of Mathematic and Statistics for funding during my time atNewcastle.

For support, encouragement and much besides I thank my family, to whomI will always be indebted.

Finally, for many enjoyable hours, thank you to all the postgrads and staff,past and present, of the Hotspur contingent. Cheers all!

Abstract

The stability of flow between rotating cylinders is a classical problem in hydrodynamics,and in his landmark 1961 book on stability theory, Chandrasekhar devoted equal attentionto the hydrodynamic and the hydromagnetic Couette problems. In the latter case the fluidis an electrical conductor. Despite an early interest in hydromagnetic Couette flow, whichincluded experiments with mercury between cylinders, the literature has primarily focusedon linear stability.

To extend these studies we develop a formulation suitable for solution of the full nonlinearhydromagnetic equations. Our study of nonlinear flows is motivated by the development ofnew flow acoustic visualisation techniques for opaque fluids. Lack of flow visualisation hasheld back progress in the hydromagnetic case relative to the hydrodynamic case.

Using our formulation we consider Taylor–Couette flow in the small magnetic Prandtlnumber limit, relevant to laboratory fluids. In the presence of an imposed axial field finiteamplitude solutions are computed, and we investigate the secondary instability to wavymodes.

At finite magnetic Prandtl numbers, we study the destabilisation of Rayleigh-stableTaylor–Couette flows by the presence of a magnetic field — the magneto-rotational in-stability.

We are also motivated by the renewed interest in hydromagnetic flows in confined ge-ometries which arises from current and planned experiments to produce dynamo action inthe laboratory. The ability of Taylor–Couette flow to drive dynamo action is investigatedvia kinematic dynamo simulations. Nonlinear saturated solutions are also computed. Likethe magneto-rotational instability we seek the parameters for which the dynamo is mostlikely to be realisable in the laboratory.

CONTENTS

1 Introduction 3

2 Equations and boundary conditions 9

2.1 Equations governing hydromagnetic flow . . . . . . . . . . . . . . . . . . . . 92.2 The small Prandtl number limit . . . . . . . . . . . . . . . . . . . . . . . . . 102.3 Conditions at insulating boundaries . . . . . . . . . . . . . . . . . . . . . . 10

3 Formulation and solution 12

3.1 The magnetic field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.2 Numerical method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.3 The velocity field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.4 Numerical tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.4.1 Linear stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.4.2 Nonlinear axisymmetric flow . . . . . . . . . . . . . . . . . . . . . . 173.4.3 Wavespeeds in wavy Taylor–vortex flow . . . . . . . . . . . . . . . . 18

3.5 Comparison with experiment . . . . . . . . . . . . . . . . . . . . . . . . . . 193.6 Summary and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4 The stability of laboratory fluids 22

4.1 An imposed axial field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.2 Stability of CCF in a weak magnetic field . . . . . . . . . . . . . . . . . . . 244.3 Nonlinear axisymmetric TVF . . . . . . . . . . . . . . . . . . . . . . . . . . 254.4 The stability of TVF to wavy perturbations . . . . . . . . . . . . . . . . . . 304.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

5 The magneto-rotational instability 34

5.1 Dissipation in accretion discs . . . . . . . . . . . . . . . . . . . . . . . . . . 345.2 The Rayleigh stability criterion . . . . . . . . . . . . . . . . . . . . . . . . . 355.3 The MRI in Taylor–Couette flow . . . . . . . . . . . . . . . . . . . . . . . . 35

5.3.1 Destabilisation and restabilisation by an imposed magnetic field . . 365.3.2 Dependence of the MRI on the magnetic Prandtl number . . . . . . 37

5.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

6 A Taylor–Couette dynamo 41

6.1 Kinematic dynamo instability in Taylor–Couette flow . . . . . . . . . . . . . 426.1.1 Dependence of the dynamo on the flow . . . . . . . . . . . . . . . . . 426.1.2 Dependence of the dynamo on the magnetic Prandtl number . . . . 45

6.2 Self-consistent dynamo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

CONTENTS 2

6.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

7 Conclusions and future work 52

A Differential operators in cylindrical–polar coordinates 54

B Evaluation of nonlinear terms 56

C Evaluation of Bessel functions 58

D Free decay of the magnetic field 60

E Integral evaluations 64

E.1 Integration over a Chebyshev expansion . . . . . . . . . . . . . . . . . . . . 64E.2 Parseval’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

1

INTRODUCTION

The motion of an incompressible viscous fluid between concentric rotating cylinders (figure1.1) is one of the most important problems of fluid dynamics and is much studied as abenchmark in issues of stability and nonlinear behaviour. Taylor (1923) found that if therotation of the inner cylinder is greater than some critical value, then the purely azimuthalcircular–Couette flow (CCF) becomes unstable to axisymmetric perturbations. A secondaryflow appears which has axial and radial motion in the form of pairs of toroidal vortices (figure1.2), now known as the Taylor–vortex flow (TVF). If the inner cylinder is driven faster thenthis flow becomes unstable to non-axisymmetric perturbations. Azimuthal waves appearin the Taylor–vortices and the whole pattern rotates at some wavespeed (wavy modes, seefigure 1.3). Taylor–Couette flow is one of the most important examples of fluid systemsthat exhibits the spontaneous formation of more and more complex dynamic flow structures,which take place as the drive is increased (Andereck, Liu & Swinney 1986, see figure 1.4).A great number of variations of the original pattern observed by Taylor (1923) have beeninvestigated (Egbers & Pfister 2000).

Studies on the linear stability of hydromagnetic CCF

In his landmark 1961 book on stability theory, Chandrasekhar devoted equal attention tothe hydrodynamic and the hydromagnetic Couette problems; the latter is the case in whichthe fluid is an electrically conducting liquid (e.g. mercury, liquid gallium, liquid sodium) anda magnetic field is applied externally. Chandrasekhar (1952) calculated the linear stabilityof circular–Couette flow to axisymmetric perturbations when the outer cylinder is fixed,and in Chandrasekhar (1961) extended this to co- and counter-rotating cylinders.

Donnelly & Ozima (1962) and Donnelly & Caldwell (1963) performed experiments withmercury between Perspex and stainless-steel cylinders. The results confirmed the theoreticalresults of Chandrasekhar (1961), although little difference was found in the results betweenthe two types of boundary conditions (figure 1.5).

Roberts (1964) derived boundary conditions for cylinders of arbitrary conductivity, andextended Chandrasekhar’s theory to non-axisymmetric perturbations. When the outercylinder is fixed the circular–Couette flow was found to be unstable to non-axisymmetricperturbations just above the critical value for axisymmetric perturbations. Chang & Sartory(1967) also considered the stability to non-axisymmetric perturbations with co- and counter-rotating cylinders. Just like the hydrodynamic case it was shown that with counter rotationnon-axisymmetric perturbations could go unstable first.

Chapter 1. Introduction 4

PSfrag repla ementsR1 R2

1 2Figure 1.1. Schematic of the Taylor–Couette domain. The axial dimension is usually large

relative to the width of the gap and is often modelled as infinite.

(a) (b)

PSfrag repla ements zR1 R2

2PSfrag repla ements

Re1Re

StableUnstable

Figure 1.2. (a) Sketch of Taylor–vortex flow. The pattern is axially periodic with associatedwavenumber α and is axisymmetric. (b) The driving required to destabilise circular–Couette flowis a function of α, where Re1 is a measure of the rotation of the inner cylinder (defined in §2.1). IfRe1 is increased slowly Taylor–vortex flow onsets at the critical value Rec with wavelength 2π/αc.


Figure 1.3. Projection of a wavy Taylor–vortex flow. The pattern repeated six times in theazimuthal direction (m = 6). The narrow dark bands are out-flows and the wider grey bands arein-flow regions. Shown over three axial periods.

PSfrag repla ementsCouette Flow

Couette FlowCouette Flow

UnexploredFeaturelessTurbulen eSpiral

Turbulen eTurbulen e

InterpenetratingIntermitten y

Wavelets

Re2

FlowVortex

RippleModulatedWavesWavy In ow+TwistsWavy In owWavy Vorti esCorks rew

SpiralsTurbulentTaylorVorti esModulated WavesSpiralsWavy

SpiralsSpiralsSpiralsTwists

WavyOut ow

01000

Re1

100020003000

Wavy Wavy

Wavy

Wavy2000

1000

1000Figure 1.4. Stability diagram of supercritical Taylor–vortex flow for different driving of the in-ner, Re1, and outer, Re2, cylinders. Experiments of Andereck et al. (1986) with aspect ratio 30,R1/R2 = 0.833, αav = 3.14.


(a)

PSfrag repla ements00 1 2 3 42:0

2:53:03:54:0

logRe

logQ

(i) (ii)

(b)

PSfrag repla ements 00001

12

2

3

3

4

4

logQ

(i)(ii)

Figure 1.5. Dependence of (a) the critical driving, Rec, and (b) the wavenumber αc, on the strengthQ of an imposed axial magnetic field. Solid lines are the theoretical results of Chandrasekhar (1961)in the limit R1/R2 → 1 (narrow gap) for (i) conducting and (ii) insulating boundaries. Experimentalresults of Donnelly & Ozima (1962) with mercury, where R1/R2 = 0.95; , stainless-steel cylinders;2, Perspex cylinders.


Nonlinear hydromagnetic Taylor–Couette flow

For the magnetic case Tabeling (1981), using a method similar to Davey’s (1962) ampli-tude expansion, calculated effective viscosity of nonlinear axisymmetric flow in the Taylorvortex flow regime; he compared against Donnelly’s (1962) experiments which indicate thatthe onset of wavy vortices is significantly inhibited by the magnetic field.

Despite this early interest in the hydromagnetic Couette problem, most of the activityof the following years was devoted to the hydrodynamic case. As magnetic fluids areopaque, lack of flow visualisation has clearly held back progress in the hydromagneticCouette problem, compared to the hydrodynamic case, where there is a wealth of literatureconcerning nonlinear flows.

More recently Nagata (1996) has investigated nonlinear solutions in the planar geometry,and Hollerbach (2000a) has shown that Taylor cells also exist in spherical Couette flow.

Thesis structure

As the Taylor–Couette flow can be used to model a range of problems, much of the restof the introduction has been left to the appropriate following chapters. In chapter 2 wepresent the equations which govern magnetohydrodynamic (MHD) flows. The equationsare reformulated for the parameters relevant to fluids available in the laboratory. Guidedby the results of Donnelly & Ozima (1962), we derive the conditions valid at insulatingboundaries.

In an effort to extend the linear studies of Chandrasekhar (1961) into the nonlinearregime, in chapter 3 we develop a formulation, based on spectral methods, suitable forsolution of the full three dimensional hydromagnetic equations in cylindrical geometry. Theformulation for the velocity component is motivated by the formulation for the magneticfield. It is suitable for study at finite Prandtl numbers and in the small Prandtl numberlimit, relevant to laboratory liquid metals. The method is tested determining the onset ofaxisymmetric Taylor vortices, and finite amplitude solutions.

Having found a convenient numerical formulation of the hydromagnetic Taylor–Couetteproblem, in chapter 4 we investigate the stability of laboratory fluids. The aim of chapter4 is to continue the investigations into the three-dimensional time-dependent flow regimeof the wavy modes. It is important to have precise results at small Reynolds number MHDflows in order to develop and test modern acoustic flow visualisation techniques for opaquefluids (Kikura, Takeda & Durst 1999) which offer the best chance to detect patterns inMHD flows.

Astrophysical interest in the magneto-rotational instability (Ji et al. 2001; Rudiger &Zhang 2001) adds further motivation for the study of magnetic Taylor–Couette flow whenthe magnetic Prandtl number is finite. Taylor–Couette flow can be interpreted as a model ofKeplerian flow, and turbulence in accretion discs is thought to arise from a linear instabilityassociated with the presence of a magnetic field. In chapter 5 we explore the instabilityas a function of the speed of the outer cylinder in the Rayleigh-stable region, which isthe parameter space of astrophysical interest. We determine the effect of changing themagnetic Prandtl number over a wide range. Whilst for laboratory fluids the magneticPrandtl number is very small, in galaxies it is believed to be large (Kulsrud & Anderson1992).

Motivated by the recent success of MHD dynamo experiments (Gailitis et al. 2001; Stei-glitz & Muller 2001), in chapter 6 we explore the possibility that Taylor–Couette flow cansustain generation of magnetic field. Firstly, by solving the kinematic dynamo problem, we


widen the study of Laure, Chossat & Daviaud (2000) and identify the region of parameterspace where the magnetic field’s growth rate is higher. Secondly, by solving simultaneouslythe coupled nonlinear equations which govern velocity field and magnetic field, we find aself-consistent, nonlinearly saturated dynamo. Like the magneto-rotational instability weseek the parameters for which the dynamo is most likely to be realisable in the laboratory.

Finally, in chapter 7 we point to further work in view of the findings reported in thisthesis.

2

EQUATIONS AND BOUNDARY CONDITIONS

It is remarkable that so many hydromagnetic flows can be modelled accurately by so fewequations, from magnetic flows in the laboratory to a wide range of astrophysical flows.In this chapter we introduce a non-dimensionalisation of the standard MHD equationsappropriate for our geometry. Together with the boundary conditions for the velocity, andthe more unwieldy representation of the boundary conditions for the magnetic field, weform a complete statement of our problem.

2.1 Equations governing hydromagnetic flow

For fluid velocity u, and magnetic field B, the equations governing incompressible hy-dromagnetic flow are

∂tu + (u · ∇)u = −1

ρ∇p+ ν∇2

u +1

ρµ0(∇ ∧ B) ∧ B, ∇ · u = 0, (2.1a, b)

∂tB = λ∇2B + ∇ ∧ (u ∧ B), ∇ · B = 0, (2.1c, d )

where p is the pressure. The density ρ, ν the kinematic viscosity, λ the magnetic diffusivityand µ0 the magnetic permeability are assumed to be constant. The fluid is containedbetween two concentric cylinders of inner radius R1 and outer radius R2. The inner andouter cylinders rotate at constant angular velocities Ω1 and Ω2 respectively. We make theusual simplifying assumption that the cylinders have infinite length and use cylindricalcoordinates (r, θ, z).

Throughout the rest of this work we will make the variables dimensionless using thefollowing scales:

δ = R2 −R1, length (gap width); δ2/ν, time (viscous diffusion);ν/δ, velocity; B0 magnetic field.

We introduce the following dimensionless parameters: radius ratio (η), Reynolds numbers(Re1 and Re2), Chandrasekhar number (Q) and magnetic Prandtl number (ξ) defined as

η = R1/R2; Rei =Ri Ωi δ

ν, i = 1, 2; Q =

B20 σδ

2

ρν; ξ =

ν

λ, (2.2)

where σ = 1/(λµ0) is the electrical conductivity. The dimensionless forms of (2.1a,c) arethen

∂tu + (u · ∇)u = −∇p+ ∇2u +

Q

ξ(∇ ∧ B) ∧ B, (2.3a)

Chapter 2. Equations and boundary conditions 10

∂tB =1

ξ∇2

B + ∇ ∧ (u ∧ B). (2.3b)

A steady-state solution of the governing equations is circular–Couette flow, u = (0, uθ, 0)where uθ(r) = ar + b/r. The constants a and b are determined by the no-slip boundaryconditions and the dimensionless form for the azimuthal circular–Couette flow is

uθ =1

1 + η

[

(Re2 − ηRe1)r +η

(1 − η)2(Re1 − ηRe2)

1

r

]

. (2.4)

We set u = u + u′ and p = p + p′. The deviation, u′, then satisfies the homogeneousDirichlet boundary condition

u′ = 0 at R1 =

η

1 − η, R2 =

1

1 − η. (2.5)

Subtracting the Navier–Stokes equation for u from (2.3a), the evolution of u′ is now de-scribed by

(∂t −∇2)u′ = N − ∇p′, ∇ · u′ = 0, (2.6a, b)

(∂t −1

ξ∇2)B = NB , ∇ · B = 0, (2.6c, d )

with nonlinear terms,

N =Q

ξ(∇ ∧ B) ∧ B − (u · ∇)u′ − (u′ · ∇)u, NB = ∇ ∧ (u ∧ B). (2.6e, f )

2.2 The small Prandtl number limit

The magnetic Prandtl number ξ is very small in liquid metals available in the laboratory,so we set B = B0 + ξb. In the limit ξ → 0 the governing equations become

(∂t −∇2)u′ = N − ∇p′, ∇ · u′ = 0, (2.7a, b)

∇2b = NB, ∇ · b = 0, (2.7c, d )

where,

N = Q(∇ ∧ b) ∧ B0 − (u · ∇)u′ − (u′ · ∇)u, NB = −∇ ∧ (u ∧ B0). (2.7e, f )

Note that these equations are descriptive rather than predictive for b.

2.3 Conditions at insulating boundaries

The governing equations (2.6) represent a tenth order system in r and we therefore requireten boundary conditions. The first six are simply the no-slip condition, ur = uθ = uz = 0applied at the boundaries r = R1 and r = R2. The boundary conditions for the magneticfield depend on the nature of the container, as discussed by Roberts (1964), who determinedconditions for arbitrary values of electrical conductivity.

Boundary conditions for the magnetic field can have a huge influence on the flow, as seenby Hollerbach & Skinner (2001) when studying the analogous problem in spherical geometry.Here the axially imposed magnetic field does not penetrate the boundaries but there are

Chapter 2. Equations and boundary conditions 11

still differences between the theoretical results of Chandrasekhar (1961) for insulating andperfectly conducting cylinders. However, in experiments by Donnelly & Ozima (1962) usingmercury, only a small difference was found between results with Perspex and stainless–steelcontainers (see figure 1.5a). Hereafter we consider only the case of insulating boundaries.

Ampere’s law says that, J = ξ−1∇ ∧ B = 0, when r < R1 or r > R2, as the current

within an insulator must be zero. It follows that the magnetic field is irrotational and canbe expressed in terms of a potential, ψ, in the following way:

B = −∇ψ, −∇ · B = ∇2ψ = 0. (2.8)

This equation can be solved for ψ by separation of variables, ψ(r, θ, z) = R(r)Θ(θ)Z(z).In our periodic coordinates we obtain

Θ′′(θ)Θ(θ)

= −m2,Z ′′(z)Z(z)

= −α2, (2.9)

where m is integer. The equation for R(r) satisfies the modified Bessel equation,

1

rR′(r) +R′′(r) −

(

α2 +m2

r2

)

R(r) = 0. (2.10)

The boundary conditions for R(r) then depend on the type of solution.If ψ is independent of θ and z (m = α = 0) then ψ must be constant and so B = 0.

But this means that we have three conditions at each boundary, and we only need two.However, the divergence-free condition implies that a solution which is independent of θand z must have no radial component. Is it therefore sufficient to take

Bθ = Bz = 0. (2.11)

If ψ is independent of z but depends on θ (α = 0,m 6= 0) then R(r) = r±m. Recallingthat B = ∇ψ, we have

∂θBr = ±mBθ, Bz = 0. (2.12)

If ψ is z dependent (α 6= 0) then R(r) = Bm(r) where Bm(r) denotes either of themodified Bessel functions Im(αr),Km(αr). We obtain

∂zBr =∂rBm

BmBz,

1

r∂θBz = ∂zBθ. (2.13)

In the outer region r > R2 the field tends to zero, B → 0 as r → ∞, and in the inner regionr < R1 B must remain finite, which implies that we take

R(r) =

Im(αr)Km(αr)

orr+m

r−m onr ≤ R1

r ≥ R2(2.14)

The two relations (2.11), (2.12), or (2.13) applied at the points r = R1 and r = R2,given the appropriate function from (2.14), are equivalent to Roberts’ insulating boundaryconditions. In this way we have the remaining four boundary conditions which are required.

3

FORMULATION AND SOLUTION

A re-occurring difficulty with primitive-variable formulations is in ensuring the fields aredivergence-free. A possible solution is to combine time splitting and pressure projection.The divergence of the momentum equation gives a Poisson equation for the pressure whichis used to project the velocity into the space of solenoidal functions. No pressure termoccurs naturally in the induction equation, and if not removed divergence can build up inthe solution for the magnetic field, especially at larger magnetic Prandtl numbers. However,an arbitrary projection function could be added in order for the divergence to scale withthe timestep. Marcus (1984) used an influence matrix method in order to implement thecorrect boundary conditions for the pressure (Rempfer, 2002). This technique leads to adivergence that is zero to machine accuracy. In both methods an adjustment to the fieldis made at each timestep. However, for the hydromagnetic case there is no timestep in thesmall Prandtl number limit; without potentials it is difficult to invert the Poisson equationfor the magnetic field whilst simultaneously ensuring it is divergence-free. We propose aformulation able to cope with both finite Prandtl numbers and the small Prandtl numberlimit without significant adjustments.

Popular in MHD is the toroidal-poloidal potentials form where variables are decomposedas A = ∇∧ (ψe)+∇∧∇∧ (φe) where e is a vector constant. To eliminate the pressure inthe momentum equation it is commonplace to take the e-components of the first and secondcurls as the governing equations for the velocity. For the magnetic field it is sufficient totake the e-components of the induction equation and its first curl.

In spherical geometry one assumes e = rs, the spherical radius. Taking rs-componentsof successive curls, the Navier–Stokes and induction equations separate into one equationfor each of the potentials, Hollerbach (2000b). Unfortunately complications can arise if thismethod is used in cylindrical geometry. The choice e = z leads to separate equations foreach of the potentials but raises the order of the equations in r. However, Marques (1990)has derived the extra boundary conditions required for the hydrodynamic problem.

Taking the second curl leads to an operator acting on one of the potentials in the form ofdouble Laplacians. The equation itself is a parabolo-elliptic equation. In spherical geometryTilgner & Busse (1997) successfully implemented a second order code using this formulationwith stress-free boundaries. Hollerbach (2000b) also used this formulation with no-slipboundary conditions but found it to be unstable, even for very small timesteps, unless animplicit first order time discretisation was used. In cylindrical geometry Rudiger & Feudel(2000) used the formulation of Marques (1990) but similarly used a first order method to

Chapter 3. Formulation and solution 13

avoid numerical difficulties. It is not at all clear that higher derivatives necessarily entailnumerical instability in hydrodynamical solvers. However, for reasons that become apparentwhen the velocity field is discussed (§3.3), the second curl is avoided.

Instead we are motivated by a parallelism with the magnetic field and take only thefirst curl. As the pressure has not been eliminated, we also take the divergence of themomentum equation. A single-curl formulation was proposed by Glatzmaier (1984) in thespherical geometry and has proved very successful.

We have made the choice e = r where r is the cylindrical-polar radius. This choice givesequations which couple the potentials, but this is no particular problem; with primitivevariables r and θ components are coupled. Fortunately, although we still raise the orderof the equations, the extra derivatives appear in the periodic coordinates and so no extraboundary conditions are required. To ensure capture of all possible solutions extra termsalong θ and z are added, in order to accommodate solutions that are independent of bothθ and z (see appendix A). The full expansion of variables has the form

A = ψ0 θ + φ0 z + ∇ ∧ (ψr) + ∇ ∧ ∇ ∧ (φr), (3.1)

where ψ(r, t, z), φ(r, t, z) and ψ0(r), φ0(r) contain the periodic and non-periodic partsrespectively. We discuss first the formulation for the magnetic field, as it motivates themethod for the velocity.

3.1 The magnetic field

The magnetic field is expanded as

B = T0 θ + P0 z + ∇ ∧ (T r) + ∇ ∧ ∇ ∧ (Pr), (3.2)

and substituted into the induction equation, (2.6c). For the non-periodic potentials T0,P0

the governing equations are obtained from the θ, z components

(∂t −1

ξ(∇2 − 1

r))T0 = θ · NB, (3.3a)

(∂t −1

ξ∇2)P0 = z · NB . (3.3b)

with boundary conditions at R1, R2,

T0 = 0, P0 = 0. (3.4)

The periodic potentials T ,P are assumed to be of form ei(αz+mθ). In order to match theboundary conditions a spectral expansion will be required. There is no pressure term toeliminate here, so we take the r-components of the induction equation and its first curl

2

ξr2∂θzT −∇2

c(∂t −1

ξ∇2)P − 2

ξr3∂rθθP =

1

r2r · NB , (3.5a)

−∇2c(∂t −

1

ξ∇2)T − 2

ξr3∂rθθT +

2

r2(∂t −

2

ξ∇2)∂θzP =

1

r2r · ∇ ∧ NB , (3.5b)

where

∇2 = ∇2 +2

r∂r, ∇2

c =1

r2∂θθ + ∂zz. (3.6)


with boundary conditions

α = 0 : ∂θT = 0,(

∇2c ±

m

r∂r

)

∂θP = 0; (3.7a)

α 6= 0 :∇2

cT − 2

r2∂θzP = 0,

1

r∂θT −

( Bm

∂rBm∇2

c +2

r+ ∂r

)

∂zP = 0,

(3.7b)

where for ± we take + at R1, − at R2. The function Bm(r) denotes the modified Besselfunctions Im(αr) and Km(αr), and the boundary condition (3.7b) is evaluated on R1 andR2 respectively. Methods for evaluating the Bessel functions are given in appendix C.

This formulation is suitable for the small Prandtl number limit (2.7), relevant to labora-tory liquid metals. We expand b by the same potentials and make the replacements,

∂t → 0, − 1

ξ→ 1,

throughout. The field b satisfies the same boundary conditions as B.Having settled on a formulation of the equations, we now discuss the numerical method.

3.2 Numerical method

It is customary to adjust the radial range into the unit interval, so a new radial variablex is defined as

r = R1 + x, R1 = η/(1 − η), x ∈ [0, 1]. (3.8)

If a field is expected to have m1-fold rotational symmetry, such as the case of wavy modes,then variables are expanded as

A(x, θ, z, t) =

N∑

n=0

∑

|k|<K

∑

|m|<M

Ankm(t)T ∗n(x) ei(αkz+m1mθ) (3.9)

on the domain [0, 1] × [0, 2π/m1] × [0, 2π/α] where T ∗n(x) is the nth shifted Chebyshev

polynomial. As the variable A is real the coefficients satisfy Akm = A∗−k,−m for all n and

are said to be conjugate-symmetric. Therefore it is only necessary to store and time-evolvehalf the coefficients, usually taken as m ≥ 0. Variables are collocated on the N +1 extremaof TN (x),

A(xl, θ, z, t) =∑

|k|<K

∑

|m|<M

Alkm(t) ei(αkz+m1mθ) (3.10)

This arrangement of points is well suited to our problem with the points concentrated nearthe boundaries.

As the velocity and magnetic field are coupled by the nonlinear terms it makes senseto treat them explicitly. Nonlinear terms are evaluated pseudospectrally, where necessary.This method is detailed in appendix B. Large terms in the zero-modes, like circular–Couette flow or an imposed axial magnetic field, can be extracted and calculated exactly.Let q indicate the time discretisation tq = q∆t with q = 0, 1, 2, .... We choose to use secondorder Adams–Bashforth to estimate NB at the intermediate time q + 1

2 .


.BC BC

BC

BC

BCBC

=

T T

P P

N1

N2

Y

q+1 q q+ 1

2

X↓

→PN

n=0

N+1...2N+10...N1

x=0

xl

↓xl

↓n

+

Figure 3.1. Matrix-vector form for each k,m mode. The element Xln is the nth coefficient in thesum at xl.

The linear terms are easier to evaluate and are timestepped using the implicit Crank–Nicolson method. Substituting the spectral expansion in the governing equations (3.5) andthe boundary conditions (3.7), after collocation the problem for each k,m mode becomes

X

[

TP

]q+1

= Y

[

TP

]q

+

[

N1

N2

]q+ 1

2

, (3.11)

(see figure 3.1) where X and Y are matrices. The vector [T ,P]q contains the spectralcoefficients of the potentials at time tq. The nonlinear terms have been evaluated on thecollocation points. Despite the Fourier expansions, the matrices X,Y are real and arecalculated by the same routine, as they differ only by a scalar constant, namely ±∆t.

With this formulation modifications for the small Prandtl number limit (2.7) are relativelyminor. The magnetic field is now completely defined by the velocity at some particular timetq, For each k,m mode the problem in matrix-vector form is now simply

X

[

TP

]q

=

[

N1

N2

]q

. (3.12)

The matrices X are calculated by the same routines above, as, again they only differ byscalar constants.

3.3 The velocity field

Unless there is an externally imposed velocity field, there is no non-periodic pressure tobe eliminated. The non-periodic part of the velocity is then treated in the same manner asthe magnetic field, i.e.

(∂t − (∇2 − 1

r))ψ0 = θ · N , (3.13a)

(∂t −∇2)φ0 = z · N , (3.13b)

with boundary conditionsψ0 = φ0 = 0, (3.14)

on R1, R2.


For the periodic part we follow the procedure applied to the magnetic field and take onlythe first curl. As the pressure has not been eliminated, we also take the divergence of themomentum equation. The equations obtained are

2

r2∂θzψ −∇2

c(∂t − ∇2)φ− 2

r3∂rθθφ =

1

r2r · (N − ∇p), (3.15a)

−∇2c(∂t − ∇2)ψ − 2

r3∂rθθψ +

2

r2(∂t − 2∇2)∂θzφ =

1

r2r · ∇ ∧ N , (3.15b)

∇2p = ∇ · N . (3.15c)

These equations may look compicated enough, but taking the second curl they are muchworse! However, they simplify considerably for the axisymmetric problem. Note also thesimplification that the linear differential operators on the left-hand sides of (3.5a,b) and(3.15a,b) are the same with ξ → 1. Fourth order derivatives have been avoided, otherwiseas |dp T ∗

n(x)/dxp| = O(n2p) matrices can become difficult to invert accurately with largertruncations.

Every governing equation is only second order in r, and therefore all equations have thesame number of associated boundary conditions. This permits us to take the same radialtruncationN for all variables, so all matrices are likewise of the same size. This simplifies theactual implementation enormously! In fact, we timestep the governing equations (3.15a,b),the same way as the magnetic field,

X

[

ψφ

]∗= Y

[

ψφ

]q

+

[

N1

N2

]q+ 1

2

, (3.16)

and the matrices X−1,X−1

Y may be precomputed by the routine for the magnetic field asthe equations are the same, but for scalar constants. Thus, linear terms for the velocityare also timestepped using Crank–Nicolson. Using this method, Marcus (1984) found thata numerical neutrally stable oscillation can occur at large wavenumbers. Fortunately thisdoes not present a problem here, as the presence of the magnetic field tends to reduce thenatural wavenumber.

Together, the evolution equations (3.15a,b) are only fourth order in r for the potentials,but there are six boundary conditions for the velocity. They are timestepped with boundaryconditions uθ = uz = 0, or,

r∂zψ + ∂rθφ = 0, −∂θψ + (2 + r∂r) ∂zφ = 0. (3.17a, b)

The pressure–Poisson equation (3.15c) is inverted with the boundary condition ur = 0, or,

−r∇2cφ = 0, (3.17c)

essentially the no-penetration condition.Inversion of (3.15c) requires a boundary condition, indirectly determined by (3.17c),

in terms of p. In spherical geometry the governing equations separate in the linear partand Glatzmaier (1984) worked with a separate equation for each of the potentials. Here,coupling of the potentials leads to a boundary condition that is not immediately obvious.The adjustment for pressure is

[

ψφ

]q+1

=

[

ψφ

]∗− X

−1

[

1r ∂rp

0

]

. (3.18)


∆t % error in σ % error in σB

0.01 5.74×10−2 –0.003 5.18×10−3 1.48×10−2

0.001 5.76×10−4 1.65×10−3

0.0003 5.2×10−5 1.5×10−4

0.0001 6×10−6 2×10−5

Table 3.1. Error in growth and decay rates. N → ∞. For η = 1/1.444, α = 3.13, Re1 = 80,Re2 = 0 the growth rate is σ = 0.430108693 (Barenghi 1991). For the magnetic field m = 1,σB ξ = −14.055585, ξ = 1. The error is proportional to ∆t2.

In order to separate φq+1 from ψq+1 we must work with X−1 rather than X. For each k,m

mode ∇2c is just a scalar. Imposing −r∇2

cφq+1 = 0 the boundary condition becomes

X−1

(

1

r∂rp

)

= φ∗, (3.19)

where X−1

is the lower left quadrant of X−1. Condition (3.19) is implemented in the

usual manner by multiplying on the left with the coefficients T ∗n(x) at the boundaries. The

boundary conditions (3.17a-c) are then simultaneously satisfied on applying the adjustment(3.18).

3.4 Numerical tests

3.4.1 Linear stability

The linear part of the code is shared between the velocity and magnetic fields. Appropri-ate tests are determining the critical Reynolds number, Rec, for the onset of Taylor–vortexflow in the presence/absence of a magnetic field, and determining the growth/decay ratesof the fields.

An eigenfunction of the linearised equations grows or decays exponentially at a rate σ.Barenghi (1991) examined convergence with ∆t for the velocity by comparing against aknown growth rate. A simple initial disturbance to the appropriate mode is φ ∝ x2(1 −x)2 sinαz, or equivalently φ0,±1,0 = 3∆, φ2,±1,0 = −4∆, φ4,±1,0 = ∆, which satisfies theboundary conditions and mimics TVF surprisingly well.

To ensure the boundary conditions for the magnetic field have been set up correctly wecheck our method against analytically derived decay rates (see appendix D). Table 3.1shows results of the test of growth rates and the comparison with Barenghi (1991). Notethat the error is proportional to ∆t2.

To check the interaction of the two fields and the case ξ → 0 we compare the onset ofTVF against Roberts (1964). Table 3.2 shows the number of modes required to reproducea few of Roberts’ results to five significant figures.

3.4.2 Nonlinear axisymmetric flow

The saturation to a steady flow, for some not too large Re1 > Rec, provides a testingground for the evaluation of nonlinear terms. Barenghi (1991) gives values for velocities


Q αc N Rec

30 2.69 8 280.9710 281.05

100 1.73 8 463.2010 463.52

300 0.928 8 796.5210 798.5712 798.55

Table 3.2. Critical Reynolds numbers for varying numbers of modes and magnetic field strengths.η = 0.95 with insulating walls. For the largest number of modes in each case the values are thesame to five significant figures as the results of Roberts’ (1964) calculations.

N K ur

Re1 = 72.4569 106.066 150.000- - 4.23363 17.9705 33.6805

10 6 4.236577 17.94932 33.488698 4.236615 17.97669 33.66495

12 4.236616 17.97902 33.7022216 6 4.233596 17.94086 33.45839

8 4.233635 17.96816 33.6413512 4.233635 17.97046 33.67982

Table 3.3. Radial velocity at the outflow. ∆t→ 0. η = 0.5, x = 0.5, α = 3.1631, and a fixedouter cylinder. In the first row are the values of Jones (1985a).

at the outflow which are in agreement with results obtained by Jones (1985a) using adifferent method. In table 3.3 we examine convergence with N ,K. Generally we findthat convergence is quicker in z than r, and, for a given truncation accuracy decreases withincreasing Re1. More energy is found in the higher modes as Re1 is increased.

3.4.3 Wavespeeds in wavy Taylor–vortex flow

A simple small wavy perturbation to axisymmetric TVF that satifies the boundary con-ditions is ψ ∝ x2(1 − x)2 sinm1θ for some wavy mode m1. The pertubation will eitherdecay or grow and saturate depending on whether or not the parameters are in the wavyTVF regime. King et al. (1984) compared wavespeeds found from physical experimentswith numerical calculations. They found that “the wavespeed is a sensitive indicator ofthe accuracy of a numerical code”. . . “any compromise in numerical resolution changes thewavespeed by several percent”. They also argue that the wavespeed can be measured moreprecisely, both in experiment and numerically, than torques which are dependent on axialwavelength (see the comparison with torque experiments in §3.5).

A few of the results used by Marcus (1984) as a test for his numerical method are givenin table 3.4. Marcus’ numerical results were well within the range of experimental error


Re1/Rec 2π/α Marcus Measured

3.98 2.40 0.3443± 0.0001 0.3440± 0.00083.98 3.00 0.3344± 0.0001 0.3347± 0.00075.97 2.20 0.3370± 0.0001 0.3370± 0.0002

Table 3.4. Wavespeeds expressed as a fraction of the angular velocity of the inner cyliner.η = 0.868, Rec = 115.1, m1 = 6.

of about 1%. Calculations with our code gave results all within 0.1% of Marcus’ values.

3.5 Comparison with experiment

In this section we make our first direct comparison with results with results from hydro-magnetic experiments.

The torque per unit axial length on the inner cylinder is defined as

G =αr

2π

∫ 2π

0r dθ

∫ 2π/α

0dz

(

1

r− ∂r

)

uθ

∣

∣

∣

∣

∣

r=R1

. (3.20)

There is no magnetic torque with insulating boundaries. For an axisymmetric flow andgiven the expansion for the velocity (3.1), this simplifies to

G = 2πr2(

1

r− ∂r

)

(ψ0 + uθ)

∣

∣

∣

∣

r=R1

(3.21)

The ratio of the effective viscosity of the flow to the kinematic viscosity of the fluid is equalto G/G where G is the component of the torque due only to the underlying circular–Couetteflow, uθ.

Figure 3.2 shows experimental results obtained by Donnelly & Ozima (1962) with mercuryand Perspex cylinders. Also shown are torques for axisymmetric calculations with theiraspect ratio, η = 0.95, and results of an amplitude expansion calculated by Tabeling (1981)in the narrow gap limit.

As the Reynolds number is increased there is good agreement between our numericalmethod, Tabeling’s amplitude expansion and Donnelly’s experiment, until Donnelly’s re-sults deviate from both ours and Tabeling’s calculations. The points plotted in figure 3.2are time averages as significant fluctuations were observed. Tabeling conjectured that thisis due to the appearance of wavy modes. With Q = 0 the onset of wavy modes is not farabove the onset of TVF and in simulations of these modes we find a reduced torque.

Note that if an axial magnetic field is imposed the onset of wavy modes is significantlyinhibited. An investigation of the stability to is discussed in §4.4. Here it suffices to notethe good agreement between our calculations and the experiment in the weakly nonlinearaxisymmetric regime.

3.6 Summary and discussion

We have developed a formulation of the governing MHD equations of the cylindricalCouette geometry, suitable for timestepping in the nonlinear regime. Results agree wellwith experiments.



600 1200 1800 2400Re1

G=~ G

Figure 3.2. Comparison of torques. Experimental results with η = 0.95. , Q = 0; +, Q = 180;2, Q = 652. Solid line, our numerical results. Dashed line, Tabeling’s expansion about the Reynoldsnumber in the narrow gap limit.


Although the equations do not decouple in the linear part, and we must treat mean-flows separately, the formulation is similar to that used by Glatzmaier (1984) in sphericalgeometry. We use potentials for the velocity yet do not eliminate the pressure. This hasseveral advantages. Our motivation for adopting such a formulation is that the magneticfield then shares the same formulation as the velocity, dramatically reducing the potentialfor error. Only a relatively small part of our code is dedicated entirely to the magnetic field;this feature is important for testing, as there are fewer results against which to compareour results. Furthermore, it can also accomodate the small Prandtl number limit with onlyminor adjustments. The choice of governing equations which are only second order in rmakes the method accurate and matrices easily invertible. This feature also enables us totake the same radial truncation for all variables, if we desire, simplifying implementation agreat deal.

We have opted to use potentials which ensure divergence-free fields. Primitive variableformulations for time integration of the Navier–Stokes equations, such as Marcus (1984)and Quartapelle & Verri (1995) in this geometry, do not in general extend naturally to themagnetic field. In particular, they are not well suited to the small Prandtl number limit,relevant to liquid metals available in the laboratory.

For the axisymmetric case the expansion by potentials is essentially the same as thatused by Barenghi (1991) and Jones (1985a) and results appear to be very similar in termsof accuracy. Although we have one extra equation, the actual form of our equations issimpler because we avoided taking the second curl.

Our method is second order in time and exhibits good temporal stability, We have notencountered the difficulties experienced by Hollerbach (2000b) and Rudiger & Feudel (2000)with three-dimensional potential formulations and no-slip boundaries. Using the implictEuler method on the linear terms reduces the method to O(∆t/Re, ∆t2). With finite mag-netic Prandtl numbers the magneto-rotational instability (see chapter 5) leads to Reynoldsnumbers which can be surprisingly low and the O(∆t/Re) error would dominate.

Results obtained using our method compare well with existing hydrodynamic literaturewith respect to the nonlinear equilibration of Taylor-vortex flow (Barenghi 1991), the on-set of wavy modes (Jones 1985b) and the wavespeed of wavy modes (Marcus 1984). Inthe presence of a magnetic field the results also compare well for the linear stability ofcircular–Couette flow (Roberts 1964), and in the nonlinear range the amplitude expansionof Tabeling (1981) and experiments of Donnelly & Ozima (1962).

Results obtained using our method compare well with existing hydrodynamic literaturewith respect to the onset of TVF, the nonlinear equilibration of TVF, the onset of wavymodes and the wavespeed of wavy modes.

4

THE STABILITY OF LABORATORY FLUIDS

After the experiments of Donnelly & Ozima (1962) and Donnelly & Caldwell (1963) therehave been few studies of the hydromagnetic Taylor–Couette flow in the laboratory, and thesehave mostly focused on the linear stability of circular–Couette flow. Moving past the linearinstability, in this chapter we first calculate nonlinear steady axisymmetric solutions anddetermine how their strength depends on an applied axial magnetic field. These solutionsare then perturbed to find critical Reynolds numbers for the secondary instability. Theonset of wavy modes, and the related wavespeeds at onset, are calculated as a function ofincreasing magnetic field strength.

The magnetic Prandtl number, ξ, is very small in liquid metals available in the laboratory(mercury, ξ = 0.145 × 10−6; liquid sodium at 120C, ξ = 0.89 × 10−5), and the flow isaccurately described by the MHD equations in the small magnetic Prandtl number limit(§2.2). Typical solutions for nonlinear steady fields in the small magnetic Prandtl numberlimit are shown in figure 4.1. The main difference between the hydrodynamic and thehydromagnetic cases when an axial magnetic field is imposed is the axial elongation of theTaylor cells (see figure 1.5b). This occurs when the field is quite strong. However, themagnetic field does not need to be strong to have a significant stabilising affect.

4.1 An imposed axial field

The equation defining the magnetic field (2.7c), with the non-dimensional imposed fieldB0 = z, may be re-written as

∇ ∧ ∇ ∧ b = ∇ ∧ (u ∧ z). (4.1)

Note that since ∇ ∧ (u ∧ z) = 0 we can write (u ∧ z) = ∇Ψ. Equation (4.1) can beintegrated immediately to obtain

∇ ∧ b = (u′ ∧ z) − ∇Ψ, (4.2)

where Ψ is absorbed into the arbitrary function of integration Ψ. By taking the divergenceΨ solves

∇2Ψ = ω′z, for R1 < r < R2,

∂rΨ = 0, on r = R1, R2,

(4.3)

and ω′z is the z-component of vorticity. The function Ψ satisfies the homogeneous Neumann

boundary condition that there be no current through the boundaries, r · ∇ ∧ b = 0. Thefollowing analysis may differ slightly for non-insulating boundaries.

Chapter 4. The stability of laboratory fluids 23

PSfrag repla ements

u b b+ 2max jbjz

01

2

R1 R2Figure 4.1. Hydromagnetic flow in the presence of an imposed axial field. Q = 180, Re1 = 1.5Rec,Rec = 619, η = 0.95, α = 1.24. The rightmost plot demonstrates that the field lines are dragged bythe in and outflows.


The effect of the magnetic field on the fluid is introduced via the Lorentz force, whichcan now be written as

L = Q (∇ ∧ b) ∧ z = −Q[

u′r, u′θ, 0

]T −Q[

r−1∂θΨ, −∂rΨ, 0]T. (4.4)

The first term on the right hand side is directly proportional and opposed to any flow acrossthe imposed magnetic field lines. Axial flow is unaffected by the Lorentz force. Therefore itis not surprising that the flow pattern is found to elongate in this direction as the strengthof the imposed field increases (Chandrasekhar 1961).

It is well known that components of vorticity perpendicular to an imposed field are prefer-entially damped. The vorticity of the underlying circular–Couette flow (CCF) points in thez direction, so CCF is unaffected by the imposed magnetic field, and the effective viscosityof the fluid is unchanged by its presence until the appearance of the Taylor instability.

Taking the dot-product of u′ with (2.7a) and integrating over the volume gives an energybalance equation for the disturbance. The nonlinear term

∫

u′ · (u′ · ∇)u′ dV = 1

2

∫

∇ · (u′ 2 u′) dV (4.5)

vanishes due to the boundary conditions. This term represents the transfer of energy withinthe disturbance to higher modes (smaller length scales) by advection. Using the property

∫

u′ · ∇2

u′ dV =

∫

∇ · (u′ ∧ ω′) dV −

∫

ω′ · ω′ dV, (4.6)

where similarly the divergence integral vanishes, the change in kinetic energy can be writtenas

12 ∂t

∫

u′ · u′ dV =

∫

u′ru′θ

(

1

r− ∂r

)

uθ dV

−∫

ω′ · ω′ dV − Q

∫

(

u′ 2r + u′ 2θ − u′θ ∂rΨ)

dV. (4.7)

As the instability that first appears is initially axisymmetric, azimuthal derivatives havebeen ignored in (4.7). For a steady-state solution the terms on the right hand side mustbalance, hence these three terms can be identified as

E1 − E2 − E3 = 0. (4.8)

The first term, E1, is the supply of energy to the disturbance from radial shear of thecircular–Couette flow. The second term, E2, is the viscous dissipation and the third, E3, isthe magnetic damping term.

4.2 Stability of CCF in a weak magnetic field

The linear stability of this flow at various radius ratios (Soundalgekar, Ali & Takhar1994), for co- and counter-rotating cylinders (Chen & Chang 1998), and at asymptoticallylarge Q (Chandrasekhar 1961), is a well studied topic, so we focus briefly on the behaviourat small Q. In this and the following sections we find the magnetic field does not need to belarge, Q = O(10), in order to have a significant affect. By experimental standards this Qis not very large; experiments with Q > 103 were performed by Donnelly & Ozima (1962).


The disturbance to the circular–Couette flow can be decomposed into Fourier modes,

u′(r, z) = 1√

2u′0(r) +

∞∑

k=1

u′k(r) cosαkz, (4.9)

where 2π/α is the critical wavelength at which the fundamental disturbance (k = 1) firstappears. Due to orthogonality of the cosine function we have

12

∫

u′ · u′ dV =

π2

α

∑

k

∫

u′k · u′

k rdr, (4.10)

which allows us to consider the contribution of each mode to the energy (see appendix E).For a disturbance to the fundamental mode of infinitesimal amplitude A1, the nonlinear

term (4.5) will be very small and modes k > 1 can be assumed negligible. At the criti-cal Reynolds number this disturbance neither grows nor decays and can be considered inisolation.

With a fixed outer cylinder the energy, E1, supplied to the fundamental disturbanceis proportional to Re1, the driving imposed by the inner cylinder. The inviscid Rayleighcriterion predicts that the flow is unstable for any rotation of the inner cylinder. The viscousdissipation, E2, prevents the immediate onset of TVF. It originates from shears within, andso derivatives of, the disturbance. From (4.7) and (4.9) for the first mode, it is expectedthat E2 = O(α2A2

1). For a weak magnetic field the magnetic damping, E3, is O(QA21),

linearly proportional to Q. Provided the magnetic field is not large enough that it affectsthe wavenumber of the flow, we expect the critical driving, Rec, at which the instabilityfirst appears, to be delayed linearly with Q. Further, since α ≈ 3 for hydrodynamic flows,if Q = O(10) then E2 and E3 are expected to be of comparable size.

Results of calculations at radius ratios η = 0.65, 0.72, 0.83 are given in figure 4.2. Thewavenumber decreases approximately linearly by 15% over the not-so-small range ofQ in thefigure. For each η the wavenumber α = 3.13, 2.93, 2.70 ± 0.01 at Q = 0, 15, 30 respectively.The delay of transition to TVF remains almost linear in Q. In the limit Q → 0 we findthat Rec(Q)/Rec(0) = 1 + γ Q where γ ≈ 0.0181 for all η in figure 4.2. As no energy ispassed to higher modes for very small A1, the structure of the disturbance does not changeappreciably with Q or η. Therefore the ratio of magnetic to viscous dissipation remains thesame with different η.

4.3 Nonlinear axisymmetric TVF

In this section we study the nonlinear flows that occur above the first transition. If theinner cylinder is driven past the critical rate, then energy is quickly transferred to highermodes by the nonlinear advection. Isolating one of these modes, the dissipative terms areE2 = O(α2k2A2

k) and E3 = O(γ QA2k). Viscous dissipation quickly becomes dominant as

energy moves to higher modes. This can be seen in figure 4.3. However, magnetic dampingremains significant on the fundamental mode, draining energy before it can be passed tohigher wavenumbers. There the fluid viscosity determines the amplitude of the disturbance.The net result can be seen in figure 4.4 where we plot the amplitude of the disturbance atdifferent values of Q. A relatively wide gap, η = 0.65, was chosen for the calculations, asthe axisymmetric flow is stable to non-axisymmetric perturbations well beyond the onset ofTVF. The maximum axial velocity, rather than the radial velocity, was used to characterisethe amplitude of nonlinear TVF, as the outflow becomes increasingly jet-like as Re1 is


PSfrag repla ements00 8 16 24 3260

100140180

Re

Q

= 0.830.720.65

Figure 4.2. Delay of onset of TVF by an imposed axial field. Dotted lines are linearextrapolations from Q small.


PSfrag repla ements1050

00

510

15

2 4 6 8

10

10 12k

ln [E1(k)ln [E2(k)ln [E3(k)

Figure 4.3. Typical spectrum of energy terms. For the first few modes magnetic (dashed) andviscous (solid) dissipations are comparable. Viscous dissipation dominates the behaviour of highermodes, and E2/E3 ∝ k2. (Q = 30, η = 0.65, α = 2.71, Re1 = 1.5Rec, Rec = 113.4.)


PSfrag repla ements60 80

100120 140 160 180

0

00100100

200300400500

Re1

A2Q = 9 16.6

Figure 4.4. Square of maximum axial velocity versus Re1 in the presence of imposed axialmagnetic fields. η = 0.65 .

increased. Tabeling (1981) used an amplitude expansion about the point of criticality in

the narrow gap limit to show that in the weakly nonlinear regime A ∝ β(Re1 −Rec)1

2 , andthat β does not strongly depend on Q. From figure 4.4 we see that this approximation isvalid well beyond the critical point, even though the nonlinearity affects the flow pattern.

Kikura, Takeda & Durst (1999) performed experiments to measure fluid velocities innonlinear hydromagnetic flow. Their imposed field was different from ours, so we cannotmake a direct quantitative comparison. However, the same order-of-magnitude argumentson the energy of the harmonics still hold, and the numerical results presented in this sectionare qualitatively the same as their experimental results.

Since the magnetic damping is effective only on larger length scales, apart from a changein wavelength of the flow, the flow pattern of nonlinear axisymmetric TVF in the presenceof an imposed axial field is very similar to that in the hydrodynamic (non-magnetic) case.This is apparent from figure 4.4 where the main difference is only that the onset of TVFis delayed. This can also be seen in figure 4.5 where (Re1 − Rec(Q)) is the same forboth plots (with η = 0.83, in (a) Q = 0, Re1 = 1.5Rec(0), α = 3.13 and (b) Q = 20,Re1 = Rec(Q) + 0.5Rec(0), α = 2.85). Although the axial wavelengths differ, both areplotted the same size for comparison. The azimuthal flow is faster than circular–Couetteflow (u′θ > 0) at the outflow regions (z = π/α) and slower (u′θ < 0) at the inflow regions(z = 0, 2π/α). The only clear observable difference is that the fast azimuthal jet-flow regionthat occurs at the outflow is slightly larger for the magnetic flow, in addition to the stretchedaxial wavelength. However, figure 4.5b is almost indistinguishable from hydrodynamic flowat the longer wavelength, α = 2.85, where Re1 = 1.5Rec (not plotted). The deviation from


(a) (b)

PSfrag repla ements 00 R1R1 R2R2

22

Figure 4.5. Contours of u′θ

at η = 0.83 for (a) Q = 0, Re1 = 1.5Rec(Q = 0), α = 3.13 and (b)Q = 20, Re1 = Rec(Q) + 0.5Rec(0), α = 2.85.


circular–Couette flow is large; the maximum of u′θ is approximately 35% of the maximumof uθ = Re1. The maximum values of |u′r,θ,z| for all three sets of parameters are within2%, except for |u′z| in the magnetic flow which is 8% larger. The Lorentz force has noz-component and so does not damp axial flow, and therefore the curves in figure 4.4 arenot quite parallel. Despite only small differences in the flow patterns, only the magneticflow is stable to non-axisymmetric disturbances, which is the topic of the next section.

4.4 The stability of TVF to wavy perturbations

Finding the preferred mode and transitions between fully developed nonlinear wavy-modes is an enormous challenge. For the sake of simplicity, we investigate which modes arepossible by monitoring the growth or decay of infinitesimal non-axisymmetric disturbancesto the axisymmetric TVF. The disturbance translates in the azimuthal direction at somefraction of the rotation rate of the inner cylinder — the wavespeed s.

In the hydrodynamic case, Jones (1985,a,b) calculated stability boundaries and corre-sponding wavespeeds by solving the eigenvalue problem. The stability of axisymmetric flowwas found to depend strongly on the radius ratio. Figure 4.6 shows the stability boundariesand corresponding wavespeeds at onset of the wavy modes. The stability boundaries areshown relative to Rec for the onset of TVF (m = 0). For increasing Reynolds numbers andη > 0.75, we see that TVF can be destabilised to the m = 1 wavy mode (lower part ofthe boundary in figure 4.6a) and then be restabilised by a further increase of the Reynoldsnumber (upper part of the boundary). Such transition sequences predicted by Jones’ resultswere verified experimentally by Park & Jeong (1984).

The critical Reynolds number for the disappearance of m = 1, 2 modes (upper boundary)increases rather rapidly as η is increased past about 0.85. Therefore the choice η = 0.83was taken for our magnetic calculations.

The magnetic field, imposed axially, has the effect of increasing the critical wavelengthfor onset of TVF. Jones (1985b) held α fixed at 3.13, but this becomes inappropriate for ourmagnetic calculations. Figure 4.7 shows the stability of the equilibrated TVF that appearsat the critical wavenumber for the given field strength, α(Q). The dotted lines are thestability boundaries for bifurcation from circular–Couette flow, as calculated by Chen &Chang (1998). The difference between the dotted lines and the solid lines confirm Jones’assertion that it is essential to perturb TVF (which must be computed numerically) tounderstand the onset of wavy modes. The stability boundaries for the onset of wavy modesare sensitive to small differences in the parameters, and the magnetic field does not needto be strong to produce interesting effects.

The magnetic damping term in (4.7) damps the radial and azimuthal components ofthe axisymmetric flow. The curvature of the cylinders also has an additional effect on theviscous dissipation of these two components of the velocity. There is a clear similaritybetween increasing the imposed magnetic field strength and decreasing the radius ratio.We do not pursue this relationship too far as the stability boundaries have not been fullyexplained, even for hydrodynamic flows. However, significant progress on the mechanisminvolved was made by Jones (1985b).

Jones (1985b) noted that dissipation due to the radial and axial shear within TVF appearto be important factors in the stability of the azimuthal jet that appears at the in- and out-flows. The elongation of Taylor cells over the range of Q in figure 4.7 (by around 10%) mightbe a factor in the suppression of wavy modes. Antonijoan & Sanchez (2002) investigatedthe hydrodynamic stability to wavy modes as a function of α. The resulting stability


(a)


0:70 0:75 0:80 0:85 0:90Re 1=Re

m = 1

m = 2m = 3

(b)

PSfrag repla ements 0:20:30:40:5

0:70 0:75 0:80 0:85 0:90s

m = 1 m = 2m = 3

Figure 4.6. (a) Stability of axisymmetric TVF to non-axisymmetric perturbations over a rangeof η, as determined by Jones (1985b), α = 3.13. (b) Corresponding wavespeeds, s, at onset as afraction of Ω1. The dotted line is the intersection with the magnetic results at η = 0.83 .


(a)

PSfrag repla ements Re 1

0 5 10 15 2050100150200250300

Qm = 0

m = 1 m = 2m = 3 m = 123

(b)

PSfrag repla ements0 5 10 15 20

50 0:340:380:420:460:50

Q

s m = 1m = 2 m = 3

m = 123

Figure 4.7. For η = 0.83, α = αc(Q) (a) stability of hydromagnetic TVF to non-axisymmetricperturbations for increasing magnetic field strength, and (b) corresponding wavespeeds at onset.Dotted lines are bifurcations from CCF.


boundaries are qualitatively similar to figure 4.7a for increasing wavelength. However, fromthe results of Antonijoan & Sanchez (2002), the change in α over the range for Q in figure4.7a is a factor of 2-3 smaller than that required to suppress only the m = 1 mode. Theenhanced stability must also be attributed to magnetic damping of the disturbance.

The general trend in figure 4.6b is for the wavespeed, s, to decrease with decreasing η.For wider gaps there is more fluid in the slower outer regions and the mean azimuthal flowspeed decreases with increasing gap width, as does s. Also, as the Reynolds number isincreased the outflow region narrows and the inflow region becomes larger than that of theoutflow. Therefore the bulk of the fluid is in the slower inflow region and again the meanazimuthal flow is reduced. This mean-flow interpretation of the wavespeed is consistentwith the findings in figure 4.7b. Here the wavespeed decreases with increasing Reynoldsnumber, as the outflow still narrows and the inflow broadens, but the gap width is fixedand there is no additional drop in the pattern when the magnetic field strength increases.Compare figure 4.6b for decreasing η with figure 4.7b for increasing Q.

4.5 Summary

We have investigated magnetic Taylor–Couette flow in the nonlinear regime. In thepresence of an imposed axial magnetic field the Lorentz force is found to have a significantdamping affect, but only at larger length scales. This is a consequence of the small magneticPrandtl number limit, relevant to experiments with liquid metals, for which the magneticfield is completely defined by the velocity field.

When the imposed field is not too large, the axial wavenumber of the flow is not greatlyaffected. In this weak-field regime the stability of circular–Couette flow is enhanced linearlywith Q, Rec(Q)/Rec(0) = 1 + γ Q where γ appears to be independent of the gap width.

In the nonlinear regime, magnetic damping affects disturbances at the fundamental axialwavenumber, but the remaining energy passed to higher modes is dissipated mainly by thefluid viscosity. This determines the amplitude of the disturbance, which behaves approx-imately like β(Re1 − Rec(Q))

1

2 well beyond the critical point, where β does not dependstrongly on Q. This is consistent with the findings of the amplitude expansion by Tabeling(1981).

Our main finding is that the magnetic field has its most striking effect on the stability ofTVF to wavy modes. A small field is capable of pushing the secondary instability from onlya few percent above the first instability to several times past the critical Reynolds numberfor the onset of TVF. This is similar to the relative stability of TVF to wavy perturbationsin wide gaps. As in hydrodynamic flows, the wavespeed decreases with increased Reynoldsnumber, but the dependence of the wavespeed on the imposed field strength does not appearas strong as the dependence on the gap width.

The significant enhanced stability observed in the calculations above occurs at only rel-atively small imposed field strengths, well within experimental range. As a secondarybifurcation, the transition to wavy modes is difficult to detect accurately via torque mea-surements. A visualisation technique, using ultrasound, is being developed for opaque fluidsby Kikura, Takeda & Durst (1999). This type of transition could serve as a good test forflow visualisation in magnetic fluids, where the transitions are accurately defined. The newtechnique is of particular interest to those working on dynamo experiments.

5

THE MAGNETO-ROTATIONAL INSTABILITY

It is thought that turbulence in accretion discs arises from a magneto-rotational instability(MRI), where a magnetic field destabilises a rotating velocity field in which angular mo-mentum increases outwards (Balbus & Hawley 1991). This instability was discovered byVelikhov (1959) and Chandrasekhar (1961) when studying the motion of a magnetic fluidin the Taylor–Couette configuration. It is only years later, now that it has been realisedthe rotation of cylinders can be used to mimic Keplerian flow, that the implications forastrophysics are beginning to be appreciated.

5.1 Dissipation in accretion discs

Material falling onto a central object often has some rotation associated with it. Thematerial then tends to form a disc in which, due to viscosity angular momentum is trans-ported outwards and material spirals inwards, releasing large quantities of gravitationalpotential energy as it does so. This is believed to be the mechanism responsible for thelarge quantities of energy emitted in X-ray binaries and galaxies with active galactic nuclei.Due to the large mass M and/or the small radius R of the central object, there is a largegravitational potential −GM/R available for dissipation, where G is the gravitation con-stant. This dissipation is proportional to the viscosity of the disc material, but estimatesof the kinematic viscosity ν are much too small to explain the energy release seen in ob-servations of accretion discs. The presence of turbulence in the disc is believed to accountfor a larger effective viscosity. However, balancing centrifugal and gravitational forces, thesteady Keplerian flow at cylindrical radius r is described by

Ω(r) =

(

GM

r3

)1

2

, (5.1)

and by the Rayleigh stability criterion (§5.2) the flow is hydrodynamically stable (hereq = 3

2). The instability from which the turbulence arises is believed to be associated withthe presence of a magnetic field within the disc.

McIvor (1977) estimates the dissipation parameters for various regions in the interstellarmedium. For a warm weakly ionised disc, collisions in the neutral component occur fre-quently so that the dominant damping process is the kinematic viscosity. It is shown byMcIvor (1977) to be much larger than the Ohmic diffusivity under such conditions. There-fore η ≪ ν i.e. the magnetic Prandtl number ξ = ν/η is large. This large ξ is an importantfactor in the destabilisation of the flow (§5.3.2).

Chapter 5. The magneto-rotational instability 35

5.2 The Rayleigh stability criterion

Linear stability of a rotating fluid requires that the angular momentum of the flow in-creases radially. The condition for stability can be written

d

dr(r2Ω)2 > 0 ∀ r ∈ [R1, R2], (5.2)

and the flow is linearly unstable if the inequality is reversed for some r. Supposing nowthat the azimuthal flow takes the form

Ω(r) ∝ r−q, (5.3)

then Rayleigh’s criterion is satisfied for q < 2.The circular–Couette flow has angular velocity

Ω(r) = a+ b/r2, (5.4)

In the previous chapters only the case of a fixed outer cylinder was considered. As theangular momentum at the outer cylinder is zero, any rotation of the inner cylinder ren-ders the flow unstable according to the (inviscid) Rayleigh criterion. To venture into theRayleigh-stable regime we must consider co-rotating cylinders and introduce the rotationparameter

µ =Ω2

Ω1. (5.5)

The criterion for stability, (5.2), can be re-expressed for circular–Couette flow as

µ > η2, (5.6)

at all r. Circular–Couette flow does not precisely take the form (5.3), except for solid-bodyrotation, Ω(r) = a (q = 0), and on the Rayleigh line when Ω(r) = b/r2 (q = 2), butapproximately

q = − ∆ ln Ω(r)

∆ ln r=

lnµ

ln η, (5.7)

and the Taylor–Couette flow can be used to mimic Kepler flows. In the following sectionwe look for linear instabilities that violate the Rayleigh criterion.

5.3 The MRI in Taylor–Couette flow

As early as 1961, Chandrasekhar had shown a sufficient condition for stability in thepresence of a magnetic field is that Ω(r) is a monotonic increasing function of r, andremarked that the Rayleigh criterion is not recovered in the case of a limiting small field!The possibility that a magnetic field could destabilise a flow had also been observed byVelikhov (1959) and later Kurzweg (1963) calculated linear stability at varying magneticPrandtl numbers.

There have been many studies of the MRI recently, from numerical simulations of ac-cretion discs (Brandenburg et al. 1995) to nonlinear calculations in spherical geometry(Drecker et al. 2000). In particular, in a recent paper Rudiger & Zhang (2001) analysedthe linear stability of hydromagnetic Couette flow and showed that, with magnetic Prandtlnumbers less than unity, azimuthal Couette flow is easily destabilised by a magnetic field.


PSfrag repla ements00 2 4 6 8 10 12 146062

6466687072

pQRe 1 = 0 = 2

Figure 5.1. η = 0.5, ξ = 1. Critical Reynolds number of the inner cylinder (Re1c) versus appliedmagnetic field (

√Q); two cases are presented: outer cylinder fixed (µ = 0) and on the Rayleigh line

(µ = η2).

They found that the instability extends into the region of parameter space which is Rayleigh-stable in the absence of a magnetic field. Rudiger & Zhang also studied the MRI instabilityat magnetic Prandtl numbers ξ as small as 0.001, towards the limit relevant to liquid sodiumand gallium (ξ ≈ 10−5) which are used in current MHD dynamo experiments (Tilgner 2000).

The following sections extend the investigation of Rudiger & Zhang (2001). We explorethe instability as a function of the speed of the outer cylinder in the Rayleigh-stable region,which is the parameter space of astrophysical interest (Rudiger & Zhang considered onlyone nonzero ratio of outer to inner cylinder’s rotation). We also determine the effect ofchanging the magnetic Prandtl number, extending the range studied by Rudiger & Zhang.

5.3.1 Destabilisation and restabilisation by an imposed magnetic field

Figure 5.1 shows the result of our calculations for the stability of dissipative Couetteflow to axisymmetric perturbations, for radius ratio η = 0.5 and magnetic Prandtl numberξ = 1, as a function of the applied magnetic field. We plot the result in terms of

√Q rather

than Q in order to make direct comparison with the work of Rudiger & Zhang (2001). Thefirst curve refers to the case in which the outer cylinder is fixed, µ = 0, which is the moststudied case in the fluid dynamics literature. It is apparent that the presence of a magneticfield makes the flow more unstable. The critical Reynolds number, which is Re1c = 68.2for Q = 0, decreases with increasing Q and has a minimum at Q = 39. The most unstablemode is m = 0 over the range for Q in figure 5.1. From here we consider axisymmetricdisturbances only. The critical axial wavenumber α decreases significantly from 3.1 to 1.7


PSfrag repla ements020 40 60 80

100120000050100

150200250300Re 1

Re2

Q = 0 41639100100100400

Figure 5.2. η = 0.5, ξ = 1. Critical Reynolds number of the inner cylinder (Re1c) versus Reynoldsnumber of the outer cylinder (Re2) at different values of applied magnetic field (Q). The upperdotted line is the Rayleigh criterion. The lower dotted line is solid-body rotation.

over the range, and varies like 1/√Q thereafter. This stiffening eventually restabilises the

flow and Re1c increases like√Q for strong fields.

The initial destabilisation is consistent with the finding of Rudiger & Zhang (2001); thesmall difference between their Re1c and ours is certainly due to the different boundaryconditions for B (they assumed pseudo-vacuum conditions and we assume insulating con-ditions). The second curve of figure 5.1 refers to the case µ = η2 (the Rayleigh line), whichseparates stable and unstable regions in the absence of a magnetic field. The curve wellillustrates the striking destabilising effect of the magnetic field.

Figure 5.2 shows stability boundaries in the Re1 vs Re2 plane for different values of theimposed magnetic field. The axial wavenumber α does not vary a great deal along theboundaries, its dependence being principally determined by the strength of the field Q.

It is apparent that even a small value of Q is enough to make the boundary cross theRayleigh line (the upper dotted curve in the figure). The destabilising effect of the magneticfield is so large that the stability boundary drops toward the region of solid body rotation(Ω1 = Ω2 or µ = 1), which is the lower dotted line. However, if the applied field is strongenough the flow can be restabilised, in accordance with the results of figure 5.1.

5.3.2 Dependence of the MRI on the magnetic Prandtl number

The destabilisation becomes more important the larger the magnetic Prandtl numberξ. Figure 5.3 shows results for a moderate value of applied magnetic field, Q = 10, atincreasing values of ξ. It is apparent that the stability boundary drops much below the


PSfrag repla ements020 40 60 80

100120000050100

150200250300Re 1

Re2 = 0:010:1 1 10 100100

100Figure 5.3. η = 0.5, Q = 10. Critical Reynolds number of the inner cylinder (Re1c) versusReynolds number of the outer cylinder (Re2) at different values of magnetic Prandtl number (ξ).The upper dotted line is the Rayleigh criterion. The lower dotted line is solid-body rotation.


PSfrag repla ements 4 210 0012

2

34

4logRe 1

log Figure 5.4. η = 0.5. Critical Reynolds number of the inner cylinder (Re1c) versus magnetic

Prandtl number (ξ) on the Rayleigh line, µ = η2.

Rayleigh line µ = η2 and, for large enough ξ, becomes asymptotic to the solid body rotationline µ = 1.

In figure 5.4 we find that on the Rayleigh line the stability follows a power law

Re1c ∝ ξ−β, β = −0.500 ± 0.002, (5.8)

over a surprising range of ξ. The error in β is based on a fit through all points calculated.For η = 0.5 the minimum Re1c occurs at Q = 86 and α = 2.1. To verify this result weuse the WKB analysis of Ji, Goodman and Kageyama (2001). The local dispersion relationthey find for the Taylor–Couette flow is identical to the one derived for accretion discs inthe incompressible limit (Sano & Miyama 1999). From the dispersion relation the necessaryand sufficient condition for stability to axisymmetric perturbations is

(ξ + S2)2(1 + ǫ2) + 2ζRe2m − 2(2 − ζ)Re2

mS2 ≥ 0, (5.9)

where S is the ratio of diffusive to Alfven timescales, Rem is the magnetic Reynolds number,and ζ = 1/(rΩ) ∂r(r

2Ω). On the Rayleigh line Ω(r) takes the simple form, Ω = b/r2, andso ζ = 0 at all r. We said earlier that in figure 5.2 the wavenumber changes little alongthe boundary, as its dependence is principally on the strength of the imposed field. Thewavenumber parameter ǫ of Ji et al. (2001) then depends only on Q, so ǫ = ǫ(Q). Ignoringfactors that depend only on ǫ in the substitutions

S2 = ξ Q, Rem = ξ Re, (5.10)


(5.9) leads to the condition for stability

Re1 ≤ A(Q)√ξ, (5.11)

which confirms our result (5.8). This relation (5.11) should hold for all ξ as we have notbeen required to take any limits. The function A(Q) has a minimum for some Q and α(Q).For η = 0.5 we found the minimum occurs at Q = 86, α = 2.1.

Whilst the linear result (5.11) should hold for very small ξ, from the work of Wendt (1933)and Taylor (1936) a turbulent (nonlinear) instability is observed at large Reynolds numbers(Re1 & 2.5×105) with this radius ratio. Extrapolating down one magnitude to ξ = 10−5 forlaboratory fluids gives Re1c ≈ 2 × 105, a similar value, for the magneto-rotational (linear)instability.

A radial truncation of 12 Chebyshev modes was found sufficient for all calculations. Inparticular, for the calculation at ξ = 10−4 in figure 5.4, convergence was tested by increasingthe truncation and also decreasing the timestep. The fractional numerical error in Re1c isestimated at approximately 10−6.

5.4 Summary

Our calculations show that many rotation laws of the form Ω(r) = a + b/r2 which arehydrodynamically stable (that is to say, they satisfy the Rayleigh criterion) become linearlyunstable when a magnetic field is applied. Our results confirm the finding of Rudiger &Zhang (2001) and extend them in the Rayleigh stable region.

We have determined the instability at magnetic Prandtl numbers ξ one order of magnitudesmaller than Rudiger & Zhang’s, towards the small magnetic Prandtl number limit, whichis relevant to possible MHD dynamo experiments with liquid sodium and gallium. Althoughthe power law Re1c ∝ ξ−0.5 that we find on the Rayleigh line (µ = η2) is slightly differentfrom theirs (Re1c ∝ ξ−0.65 on µ = 1

3), it confirms their conjecture that the nonlinearinstability found by Richards & Zahn (1999) and the MRI are likely to occur at Reynoldsnumbers of the same order of magnitude.

We also find that the flow becomes particularly unstable if the magnetic Prandtl numberis greater than unity. The instability boundary in the Re1 vs Re2 plane rapidly tendstowards the solid body rotation line. This enhanced instability for large ξ is consistentwith earlier results of Kurzweg (1963). His boundary conditions were selected such as toavoid mathematical difficulties but for small ξ agreed well with the results of Chandrasekhar(1961). The significance of the instability in this case is linked to the possibility (Kulsrud &Anderson 1992; Brandenburg 2001) that large values of ξ exist in central regions of galaxies.

6

A TAYLOR–COUETTE DYNAMO

Even though astrophysical objects possessing observable magnetic fields are extremely di-verse, it is widely accepted that the physical mechanisms supporting the fields are fairlyuniversal and rely on features common to virtually all astrophysical objects (e.g. differentialrotation, convective or turbulent motions, etc.). MHD dynamo theory quantifies this ideaand states that astrophysical magnetic fields are created by inductive currents driven bymotions of electrically conducting plasmas (Moffatt 1978). Until recently, dynamo actionwas the subject of theoretical or numerical investigations only. The recent demonstration ofdynamo action in controlled laboratory experiments (Gailitis et al. 2001; Stieglitz & Muller2001) has stimulated interest in the study of dynamo action in confined geometries of po-tential laboratory interest, such as spheres or cylinders. The configuration of an MHD fluidconfined between concentric cylinders (Taylor–Couette flow) is particularly relevant. Dy-namo action is associated with spiral and sheared flows, which suggests that Taylor–vortexflow is good candidate for dynamo experiments in a simple geometry.

In particular we note the works of Dobler, Shukurov & Brandenburg (2002) who stud-ied the dynamo mechanism in the presence of an imposed axial flow (resulting in a screwdynamo) and of Laure, Chossat & Daviaud (2000) who concentrated on the basic Taylor–Couette configuration. The latter performed a kinematic dynamo calculation in this geom-etry and demonstrated that an imposed Taylor–vortex flow is capable of dynamo action.The combination of shear with simple roll flows has also been modelled by Dudley & James(1989) in spherical geometry. There it was demonstrated that magnetic field generationis sensitive to the nature of the driving flow. Our work differs from the studies of Dudley& James (1989) and Laure, Chossat & Daviaud (2000) in two important respects: first(kinematic dynamo) we use the velocity fields that are actual solutions of the Navier-Stokesequations to generate a magnetic field, not arbitrary imposed flow fields; secondly (fullyself-consistent dynamos) we let these velocity fields evolve alongside the magnetic field, thussolving the full MHD equations.

In this chapter we aim to widen the (linear) kinematic theory study of Laure, Chossat &Daviaud (2000) in the parameter space. The growth of the magnetic field as a function ofan imposed flow pattern. Secondly, in the spirit of Glatzmaier & Roberts (1995), Sarson& Jones (1999) and Hollerbach (2000b) for convection driven dynamos, we go beyond thelimits of linear theory and investigate the dynamically self-consistent dynamo mechanism.The magnetic fields and flow pattern evolve together and saturate nonlinearly.

Chapter 6. A Taylor–Couette dynamo 42

6.1 Kinematic dynamo instability in Taylor–Couette flow

As no magnetic field is externally imposed in our dynamo simulations, the Q parameter(§2.1) becomes redundant. Using (µ0ρ)

1

2 ν/δ as a scale for the magnetic field the dimen-sionless MHD equations become

∂tu + (u · ∇)u = −∇p+ ∇2u + (∇ ∧ B) ∧ B, ∇ · u = 0, (6.1a, b)

∂tB =1

ξ∇2

B + ∇ ∧ (u ∧ B), ∇ · B = 0, (6.1c, d )

and u′ = u − u is the disturbance to circular–Couette flow u.If the magnetic disturbance is small the Lorentz force can be assumed to be negligible.

For prescribed Reynolds number Re1, wavenumber α, rotation ratio µ and radius ratio ηthe Navier–Stokes equation (6.1a) is solved in the absence of a magnetic field. The flowu thus obtained is used when solving the induction equation (6.1c) for B, starting from asmall magnetic seed field of wavelength αB .

Note that u being fixed, (6.1c) is linear and has eigenfunction solutions B which grow ordecay exponentially. If the real part of the growth rate σB is positive then the magnetic fieldgrows (kinematic dynamo action). The circular–Couette flow u is not capable of dynamoaction and if u′ = 0 the magnetic field decays on the Ohmic timescale (see appendix D).Therefore, we always assume Re1 > Re1c, where Re1c is the critical Reynolds number atwhich circular–Couette flow is unstable to the formation of nonlinear Taylor vortices. Wealso assume α = 3.14, which corresponds to almost square cells.

6.1.1 Dependence of the dynamo on the flow

Following Laure, Chossat & Daviaud (2000) we assume αB = 12α, the characteristic length

for the magnetic field is the length of two pairs of Taylor–vortices. In several calculationswith αB = α we found the magnetic seed field decayed quickly. The axisymmetric flowcannot generate an m = 0 magnetic field. In figure 6.1 we see that in narrower gaps thedynamo prefers larger m. The dynamo is local here in the sense that the characteristiclength scale for the magnetic field comply with the scale of the flow. As the Taylor-vortexflow is itself unstable to non-axisymmetric perturbations in narrower gaps (see Jones 1985b,§4.4), we usually consider the case η = 0.5 where m = 1 is preferred. The relative stabilityof Taylor-vortex flow in wider gaps also allows for a clearer interpretation of the affect ofthe magnetic field in the nonlinear self-consistent solutions presented in §6.2.

Figure 6.2 shows that, not surprisingly, σB falls off sharply when the magnetic Prandtlnumber ξ < 1, in which case much larger Reynolds numbers (relative to Re1c) are neededfor dynamo action. Generally it is easier to generate a magnetic field in media with largermagnetic Prandtl numbers as is believed to be the case for warm weakly ionised gas, McIvor(1977), Kulsrud & Anderson (1992).

Figure 6.3 shows growth rates as a function of Re1 with Re2 = 0, η = 0.5 for a few valuesof ξ. Driving the flow harder seems to move the flow into a regime which is less favourablefor magnetic field generation when the outer cylinder is fixed. Allowing co-rotation, thestability boundary for onset of Taylor–vortex flow tends to the Rayleigh line µ = η2 forlarge Reynolds numbers. As circular–Couette flow alone is not capable of dynamo action,we must consider flows which are Rayleigh unstable, µ < η2. Figure 6.4 shows growth ratesas a function of Re1 for different values of µ. Again, we normalise the intensity of the driveRe1 by the critical value Re1c. It is apparent that magnetic field generation is easier with



1 2 3 4 5 6 B

m = 0:5 0:7 0:9

Figure 6.1. Growth rates σB for various η and m, where Re1 = 2Re1c(η), ξ = 2.


00

1

1 2 3 B

= 0:30:5 0:70:9

Figure 6.2. Growth rates σB for various η as a function of ξ, where Re1 = 2Re1c(η). CriticalReynolds numbers at η = 0.3, 0.5, 0.7, 0.9 are respectively Re1c = 73.3, 68.2, 79.5, 131.6.


PSfrag repla ements 1:00:50:00:00:00:51:0

1:01:5

1:5

2:0

2:0 2:5 3:0Re1=Re1

B = 3:0

Figure 6.3. Growth rates σB as versus Re1 at η = 0.5.


001

1

2

2

3

3 4 5Re1=Re1

B = 342 782122

Figure 6.4. Growth rates σB with co-rotation. η = 0.5, ξ = 2. At µ = 0, 1

2η2, 3

4η2, 7

8η2, critical

Reynolds numbers are Re1c = 68.2, 84.3, 112.7, 155.3 respectively.



2

2

3

3 4 5Re1=Re1

= 0122 342 782maxju0 j=maxju zj

Figure 6.5. Amplitude of the azimuthal disturbance u′θ

= uθ − uθ drops as µ→ η2 in the absenceof a magnetic field. At each Re1/Re1c the amplitudes of uz are close for all µ.

co-rotating cylinders and µ = 34η

2 in an important case. At this particular µ the amplitudeof the azimuthal disturbance u′θ = uθ − uθ and uz are similar, seen in figure 6.5. At a givenRe1/Re1c(µ), the components ur, uz appear to be very similar in structure and amplitude.However, uθ changes significantly with µ.

6.1.2 Dependence of the dynamo on the magnetic Prandtl number

Of the rotation ratios considered in figure 6.4, µ = 34η

2 is most favourable for magneticfield generation. At this µ we define Re∗1 as the critical Reynolds number at which σB = 0(marginal state for the onset of dynamo action). Figure 6.6 shows how Re∗1 depends onξ. Fitting the last three points for small ξ we obtain the slope −1.1. Approximately,Re∗1/Re1c ∝ 1/ξ, for the small ξ of laboratory magnetic fluids. Defining the magneticReynolds number Rem as

Rem =R1 Ω1 δ

λ= Re1 ξ (6.2)

we conclude that the critical magnetic Reynolds number Re∗m = Re∗1 ξ is approximatelyconstant and O(102) for suitably chosen η, µ. Here µ = 3

4η2 but the most suitable µ is

likely to vary with η. The Re∗m above are consistent with the results of Laure, Chossat &Daviaud (2000) who found Re∗m = O(102) in their calculations with µ = 0.


PSfrag repla ements 32 2 10

0 1

2

3

4

ln ln(Re 1=Re 1 1)

Figure 6.6. Dependence on ξ for the driving required for field generation. µ = 3

4η2, η = 0.5.

6.2 Self-consistent dynamo

In this section the equations (6.1a) and (6.1c) are solved simultaneously using as initialconditions the previously prescribed Taylor–vortex flow u and the accompanying eigenfunc-tion B. Figure 6.7 shows that after an initial transient the magnetic energy EB saturatesto a constant value, indicating dynamo action. At the parameters in figure 6.7 the calcula-tions required 16 Chebyshev modes radially, 16 axial and 12 azimuthal Fourier modes witha timestep of 10−4τ , where τ is the rotation period of the inner cylinder. The magneticenergy is plotted as a fraction of the energy of the driving circular–Couette flow ECCF, theenergy source for the dynamo. The energy sink is an increased viscous dissipation in thevelocity disturbance in addition to Ohmic dissipation. In these calculations the dynamo isdynamically self consistent (u and B evolve together).

Figures 6.8 and 6.9 show the typical field structure of the initial conditions. A fixedouter cylinder was taken for visualisation purposes as the surfaces are less self-obscuring.The initial flow pattern u has m = 0 symmetry and as Re1 > Re1c vortex cores areslightly shifted towards the outflow regions. The eigenfunction B in figure 6.9 has m = 1symmetry. The flow pattern, initially axisymmetric is deformed by the action of the Lorentzforce (∇ ∧ B) ∧ B and acquires an m = 2 contribution visible in figure 6.10. Figure 6.11shows the kinetic energy of the velocity disturbance and the magnetic energy of the variousazimuthal modes, E(m) and EB(m) respectively, for the saturated dynamo of figure 6.7a(see appendix E). The velocity has contributions m = 0, 2, 4, . . . and the magnetic field hasm = 1, 3, 5, . . . etc. The perturbation to the magnetic field is difficult to appreciate visuallyon the dominant m = 1 structure. It remains rather similar to figure 6.9.


PSfrag repla ements00 4 8 12 160:000:000:020:040:060:080:10

tE B=E CCF (a)

(b)Figure 6.7. Magnetic energy versus time as the magnetic field saturates. η = 0.5, ξ = 2. (a)µ = 0, Re1 = 1.5Re1c, ECCF = 4.15 × 104. (b) µ = 3

4η2, Re1 = 2Re1c, ECCF = 3.63 × 105.

6.3 Discussion

By solving the kinematic dynamo problem, we have determined that a Taylor-vortex flowpattern can sustain a growing magnetic field.

Like in the models of Dudley & James (1989) we also find that the dynamo is sensitiveto the flow. Further, for flows that are capable dynamo action we see that the growth rateis not a monotonic increasing function of the Reynolds number. This is not seen in Dudley& James (1989), most likely due to the prescribed form for the driving flow patterns.

In the Taylor-vortex flow the best growth rates have been obtained with co-rotation. Therelative magnitude of the shear and roll in the flow plays an important part in the successof the dynamo mechanism.

Solving the full MHD equations we have demonstrated the existence of a fully self-consistent nonlinearly saturated dynamo. Hopefully these results will stimulate experi-mental work on the problem. Future theoretical work will investigate dynamo action inhydrodynamically stable flows and address the nature of the magnetic field structure whenthe dynamo is driven harder – our dynamo is laminar – and most of the present work isconcerned with wider gaps.


Figure 6.8. Isosurface of helicity |u · ∇ ∧ u| at η = 0.5, Re1 = 1.5Re1c, µ = 0. Shown over twoaxial periods.


Figure 6.9. Isosurface of |B|. The field is m = 1. Same parameters as figure 6.8 with ξ = 2 (as infigure 6.7a).


Figure 6.10. Isosurface of helicity at magnetic field saturation. The flow is m = 2, looking thesame if rotated by 180. Parameters as in figure 6.7a.



00 2 4 6 8 10 12m

log [EB(m) =ECCFlog [E(m) =ECCF

Figure 6.11. Kinetic energy of the velocity disturbance and the magnetic energy of the variousazimuthal modes. Parameters as in figure 6.7a.

7

CONCLUSIONS AND FUTURE WORK

After stating the problem in chapter 2, our first task was to develop a numerical formulationfor solving the full MHD equations in the Taylor–Couette geometry (chapter 3). From there,the stability of hydromagnetic Taylor–Couette flow has been investigated in a variety ofcontexts.

The formulation we employ is also suitable for studying the flow in the small magneticPrandtl number limit. In chapter 4, Taylor–vortex flow in laboratory magnetic fluids wasrevealed to be stabilised significantly to non-axisymmetric perturbations. The startlingdegree of stabilisation, relative to the delay of the first instability, certainly deserves moreattention towards finding a satisfactory explanation. Whilst two flows may appear similarin structure and amplitude, the manner of the dissipation mechanism plays an importantpart in its relative stability.

This can be seen in chapter 5, where it is shown that an imposed axial field can alsodestabilise the flow. There is a strong dependence on the magnetic Prandtl number ξ.In particular, this has implications for flows in accretion discs where the magnetic ξ isthought to be large. For the much smaller ξ of laboratory fluids, in the vein of realising thisinstability in experiments, it would be interesting to see how the critical Reynolds numbersvaries with other parameters, gap widths for instance, and if the critical values of rotationΩ1 can be reduced.

Also possible when the magnetic Prandtl number is finite is magnetic field generation.In chapter 6 we presented the first fully self-consistent dynamo calculations. The ease ofmagnetic field generation was found to be a sensitive function of many parameters. Growthof the field is not necessarily a monotonic increasing function of the driving, and it is crucialto find a regime in which the dynamo mechanism is most robust. Again, the gap width, oreven counter-rotation and spiral flows, may prove to be more suitable for dynamo action.One of the beautiful features of Taylor–Couette flow is the many flow regimes it exhibits,despite its simple geometry.

The MHD Couette problem with axial flow, driven either by a pressure gradient or bytranslation of the cylinders, is relevant to dynamo experiments (see Dobler, Shukurov &Brandenburg 2002). These different mechanisms may prove more or less efficient, but areless simple in terms of experimental design. Either way, for the small ξ of laboratory fluidsthese configurations must be compared at larger Reynolds numbers.

To achieve larger Reynolds numbers it is clear that a higher spatial resolution is requiredthan used previously. For large truncations timestepping the linear part of the code is

Chapter 7. Conclusions and future work 53

most expensive: O(N2KM). My suggestion would be to keep the Fourier expansion in θand z, as anyway the boundary conditions must be met for the magnetic field. But in theradial dimension r, where there is currently a power two in the truncation parameter N ,finite differences could be used. Computation times could be reduced to O(NKM(log2K+log2M)), the time for the nonlinear evaluations.

In pushing towards more realistic (smaller) magnetic Prandtl numbers, we are forced inthe direction of large Reynolds numbers. Turbulence modelling is also need to gain realinsight before experimental design. Turbulence is easily generated in counter-rotation. Itwould be interesting to see the effects of an applied field when there are also centrifugalforces present. In this and all the above there is also the question of end effects to consider.

It seems there is plenty of scope for extensions of this thesis, and in the work presentedso far we have taken only the first steps in a wide range of possibilities for MHD Taylor–Couette flow.

A

DIFFERENTIAL OPERATORS IN CYLINDRICAL–POLAR

COORDINATES

Gradient

(∇f)r = ∂rf, (∇f)θ =1

r∂θf, (∇f)z = ∂zf. (A.1)

Laplacian

∇2f = (1

r∂r + ∂rr)f +

1

r2∂θθf + ∂zzf. (A.2)

Laplacian of vector

(∇2A)r = ∇2Ar −

2

r2∂θAθ −

Ar

r2,

(∇2A)θ = ∇2Aθ +

2

r2∂θAr −

Aθ

r2, (A.3)

(∇2A)z = ∇2Az.

Divergence

∇ · A = (1

r+ ∂r)Ar +

1

r∂θAθ + ∂zAz. (A.4)

Curl

(∇ ∧ A)r =1

r∂θAz − ∂zAθ,

(∇ ∧ A)θ = ∂zAr − ∂rAz, (A.5)

(∇ ∧ A)z = (1

r+ ∂r)Aθ −

1

r∂θAr.

Advective derivative

(A · ∇ B)r = Ar ∂rBr +Aθ

r∂θBr +Az ∂zBr −

AθBθ

r,

(A · ∇B)θ = Ar ∂rBθ +Aθ

r∂θBθ +Az ∂zBθ +

AθBr

r, (A.6)

(A · ∇B)z = Ar ∂rBz +Aθ

r∂θBz +Az ∂zBz.

Appendix A. Differential operators in cylindrical–polar coordinates 55

Toroidal–poloidal decompositionsRadial

A = ψ0 θ + φ0 z + ∇ ∧ (rψ) + ∇ ∧ ∇ ∧ (rφ), (A.7)

ψ0 = ψ0(r), φ0 = φ0(r), ψ = ψ(r, θ, z), φ = φ(r, θ, z).

∇2cf =

1

r2∂θθf + ∂zzf. (A.8)

Ar = −r∇2cφ,

Aθ = ψ0 + r∂zψ + ∂rθφ, (A.9)

Az = φ0 − ∂θψ + (2 + r∂r)∂zφ.

Axial

A = ∇ ∧ (zψ) + ∇ ∧ ∇ ∧ (zφ), (A.10)

ψ = ψ(r, θ, z), φ = φ(r, θ, z).

∇2hf = (

1

r∂r + ∂rr)f +

1

r2∂θθf. (A.11)

Ar =1

r∂θψ + ∂rzφ,

Aθ = −∂rψ +1

r∂θzφ, (A.12)

Az = −∇2hφ.

Components of various curls

A = A(r, θ, z), A0 = A0(r).

1

r2r · ∇ ∧ A =

1

r2∂θAz −

1

r∂zAθ,

1

r2r · ∇ ∧ ∇ ∧ A = −1

r∇2

cAr +1

r3(1 + r∂r)∂θAθ +

1

r∂rzAz, (A.13)

θ · ∇ ∧ A0 = −∂rAz,

z · ∇ ∧ A0 = (1

r+ ∂r)Aθ.

B

EVALUATION OF NONLINEAR TERMS

When using spectral methods the evaluation of nonlinear terms is usually the source of mostdifficulty, or at least is potentially the most computationally expensive part of a numericalcode.

After having expanded out nonlinear operators, consider the multiplication

∑

nkm

ankm(x)ei(αkz+m1mθ) ×∑

nkm

bnkm(x)ei(αkz+m1mθ), (B.1)

where ankm(x) and bnkm(x) are some combination of spectral coefficients and (possiblyderivatives of) the Chebyshev polynomials. We remove this complexity in the x part of thesums by collocation. We then wish to express the multiplication as a single sum,

∑

km

alkmei(αkz+m1mθ) ×∑

km

blkmei(αkz+m1mθ) =∑

km

clkmei(αkz+m1mθ). (B.2)

For each l a typical term is,

aprei(αpz+m1rθ) bqse

i(αqz+m1sθ) = c ei[α(p+q)z+m1(r+s)θ],

which suggests that

ckm =∑

p + q = k|p| , |q| < K

∑

r + s = m|r| , |s| < M

apr bqs. (B.3)

Direct evaluation of this sum for each l is an O(K2M2) process, with the additional com-plication of unusual limits.

In cylindrical geometry we can use the Fourier expansion in two of the dimensions. Weuse of the Fast Fourier Transform (FFT) in these dimensions and the resulting methodis pseudo-spectral. The sums for a and b are evaluated in physical space via the FFT,multiplied, then the result is transformed using the FFT back into spectral space. Thismethod and adjustments for isotropically truncated convolutions are described by Orszag(1971). Below is the extension to two dimensions using the notation adopted so far.

Consider the convolution

ckm =∑

p + q = k‖p‖, ‖q‖ ≤ K

∑

r + s = m‖r‖, ‖s‖ ≤ M

apr bqs, (B.4)

Appendix B. Evaluation of nonlinear terms 57

where ‖k‖ ≤ K means −K ≤ k < K. The akm and bkm are extended onto ‖k‖ ≤ K,‖m‖ ≤ M by appending c−K,m = ck,−M = 0 ∀m,k, with no effect on the sum. The two-dimensional discrete Fourier transform of spectral coefficients ckm at the points in physicalspace

θn = 2πn/(2Mm1) n = 0, ..2M − 1,zj = 2πj/(2Kα) j = 0, ..2K − 1,

(B.5)

is defined by

cjn =∑

‖k‖≤K

∑

‖m‖≤M

ckmei(αkzj+m1mθn), (B.6)

while the inverse transform is defined by

ckm =1

4KM

2K−1∑

j=0

2M−1∑

n=0

cjne−i(αkzj+m1mθn). (B.7)

Note that by the periodicity of the exponential we have that∑

‖k‖≤K ≡ ∑2K−1k=0 where

c2K−k = c−k. Having performed transforms on apr and bqs, we multiply in physical space;call cjn = ajn bjn. The inverse transform is then used to get the coefficients ckm in spectralspace. Altogether the following has been calculated:

ckm =∑

‖p‖,‖q‖≤K

∑

‖r‖,‖s‖≤M

apr bqs1

4KM

2K−1∑

j=0

2M−1∑

n=0

ei[α(p+q−k)zj+m1(r+s−m)θn]. (B.8)

By the property

1

4KM

2K−1∑

j=0

2M−1∑

n=0

ei[α(p−k)zj+m1(r−m)θn] (B.9)

=

1 k ≡ p(mod 2K) and m ≡ n(mod 2N)0 otherwise

,

we haveckm = ckm + other combinations ck±0,2K, m±0,2M . (B.10)

Due to the evaluation on discrete grid points, harmonics become indistinguishable and arealiased onto ckm. The simplest way of eliminating this effect is to use Orszag’s ‘three-halves’rule. Assume K is even. Then akm and bkm are extended onto ‖k‖ ≤ 3K/2, by appendingzeros, before taking the transform on 3K points. After performing the inverse transform ofcjn = ajn bjn wavenumbers greater than K are disgarded.

The FFT method reduces evaluation of the convolution from an O(K2M2) process toO(KM(log2K + log2M)). Recalling that our data are always either real or conjugate-symmetric also permits a factor of 2 reduction in the calculation of each of the transforms.Clearly the FFT method is more complicated than directly calculating the sum (B.3), andit is not necessarily competitive for smaller truncations. Our code compares the FFT anddirect sum methods for a given truncation, then selects the fastest.

C

EVALUATION OF BESSEL FUNCTIONS

The methods used are as suggested by Abramowitz & Stegun (1965, §9.12) with minoradjustments. For x ≫ m the modified Bessel functions behave like e±x/

√x and therefore

lead to potential over or underflow. We observe that the Bessel functions are required onlyas ratios in the boundary conditions (2.13). Instead of the functions explicitly, we calculateIm(x) = e−xIm(x) and Km(x) = exKm(x), as the exponentials cancel in the ratio.

To calculate Im=0,..M(x), we first set IM = 0, IM−1 = 1 where M > M . Then therecurrence (A&S 9.6.26),

Im−1 = Im+1 +2m

xIm, (C.1)

gives the remaining Im. The true Im(x) satisfy (A&S 9.6.37),

1 = I0(x) + 2I1(x) + 2I2(x) + . . . (C.2)

so the Im are rescaled such that they satisfy (C.2). Accuracy is inferred from the size ofIM before rescaling.

For Km(x) there is no suitable property for rescaling. Instead a “correct” starting value,K0, is set using the polynomial approximation (A&S 9.8.5-6),

e−xK0(x) = − ln(x/2)ex I0(x) −.57721566+.42278420(x/2)2 +.23069756(x/2)4

+.03488590(x/2)6 +.00262698(x/2)8

+.00010750(x/2)10 +.00000740(x/2)12

+O(10−8) 0 < x ≤ 2

x1

2 K0(x) = 1.25331414 −.07832358(2/x)+.02189568(2/x)2 −.01062446(2/x)3+.00587872(2/x)4 −.00251540(2/x)4+.00053208(2/x)6 +O(10−7) 2 ≤ x <∞

(C.3)

The second starting value, K1, may be obtained from (A&S 9.6.15),

I0 K1 + I1 K0 =1

x, (C.4)

Appendix C. Evaluation of Bessel functions 59

and the remaining Km from the recurrence

Km+1 =2m

xKm + Km−1 . (C.5)

To calculate the derivatives, once Im(x) and Km(x) are obtained we use recursively (A&S9.6.26),

2I ′m(x) = Im+1(x) + Im−1(x), −2K ′m(x) = Km+1(x) + Km−1(x). (C.6)

D

FREE DECAY OF THE MAGNETIC FIELD

The expected decay rates of the magnetic field when u = 0 are used as a first test of ournumerical method (§3.4.1). The natural decay rates are also indicative of which modes aremore or less likely to be excitable in kinematic dynamo calculations (chapter 6).

To derive this decay rate we use a mixture of analytical and numerical methods which isdifferent from the numerical technique used to solve the MHD equations. We express themagnetic field in terms of two scalar potentials

B = ∇ ∧ (T z) + ∇ ∧ ∇ ∧ (Pz), (D.1)

which guarantees a divergence-free field. We rely on the following property of cylindricalcoordinates

∇ ∧ ∇ ∧ (f z) = ∇(∂zf) − z∇2f. (D.2)

The induction–diffusion equation for a stationary conductor is

∂t B =1

ξ∇2

B = − 1

ξ∇ ∧ ∇ ∧ B, (D.3)

then using the property (D.2) and ignoring gauges we obtain

∂t T =1

ξ∇2T , ∂t P =

1

ξ∇2P. (D.4)

We next expand T , P and seek eigensolutions for the magnetic field of the form

A(r, θ, z, t) = Am(r) eσB t+i(αz+mθ), (D.5)

where −σB is the decay rate. Substitution into (D.4) yields

1

r∂r Am(r) + ∂rr Am(r) +

(

σ2 − m2

r2

)

Am(r) = 0, σ2 = −σB ξ − α2, (D.6)

Suppose that −σB ξ < α2 (α 6= 0), i.e. that decay is relatively slow. Then solutions to(D.6) are the sums of modified Bessel functions I(σr), K(σr). These functions and theirfirst derivatives are monotonically increasing and decreasing for I,K respectively. Thenthe magnetic energy must have the largest value at the boundaries. Since the diffusivityλ = 1/(µ0σ), where σ is the conductivity, as we consider electrically insulating boundaries

Appendix D. Free decay of the magnetic field 61

m 0 1 2 3k −σB ξ

0 (T ) 10.634504 2.525434* 6.259403 11.170948(P) 9.613411 10.634504 13.640533 18.474045

1 (1st) 14.919861 14.760834 16.619549(2nd) 20.431404 22.010271 25.611050

2 (1st) 45.851458 45.868780(2nd) 49.822104 51.678920

Table D.1. Decay of the magnetic field, −σB ξ, for η = 0.35, α = 3.13×k. When k = 0 toroidal andpoloidal modes separate and their slowest decaying modes are given; α is a redundant parameter inthis case. Otherwise they couple and the first two modes are given. The dominant mode is marked*.

this is not possible for a free magnetic decay mode. We therefore assume that −σB ξ > α2.The resulting Bessel equation (D.6) has solution

Am(r) = AJm Jm(σr) +AY

m Ym(σr). (D.7)

The magnetic field within the conductor may be expressed

Bm(r) =

imr−1 Tm(r) + iα∂r Pm(r)−∂r Tm(r) − αmr−1 Pm(r)

σ2 Pm(r)

. (D.8)

Matching these components to the boundary conditions for the magnetic field (2.13) at thetwo boundaries defines the problem

M(σ) [T Jm , T Y

m , PJm, PY

m ]T = 0, (D.9)

for the four unknown coefficients. The quantity M is a 4 × 4 matrix and is real if σ is real(non-oscillatory decay modes). The slowest decaying eigensolution for B is determined bythe smallest σ such that det M(σ) = 0. Typical results are shown in table D.1 and figureD.1 for non-axisymmetric modes. Figure D.2 shows the typical structure of one of thesemodes.

The choice of z in the decomposition (D.1) was taken in order to decouple the potentialsT and P (except in the boundary conditions) and caused the appearance of convenientBessel functions. In spherical geometry this aim is achieved by taking instead r. Thisleads to a problem where the toroidal component is trapped within a spherical conductor.Boundary conditions then separate for the toroidal and poloidal components, see Gubbins& Roberts (1987, §3.2). Unfortunately this does not occur in cylindrical geometry exceptin special cases.

One such case is that of z independent (α = 0) modes. We have argued that −σB ξ > α2

which suggests the dominant modes are likely to have low α. Matching to the boundaryconditions (2.12) gives,

[

Jm+1(σT R1) Ym+1(σT R1)Jm−1(σT R2) Ym−1(σT R2)

] [

T Jm

T Ym

]

= 0, (D.10a)


PSfrag repla ements 00000 112

2 3 45 51015202530

m = 3

B

Figure D.1. Decay rates for the slowest non-axisymmetric modes, m = 1, 2, 3, as a function of αat η = 0.5.

[

Jm(σP R1) Ym(σP R1)Jm(σP R2) Ym(σP R2)

] [

PJm

PYm

]

= 0, (D.10b)

where we have used the recurrence relation Bν±1(z) − νzBν(z) = ∓B′

ν(z) and B denoteseither J or Y . This case is therefore much simpler to state and solve than the coupledproblem.

Another special case is the narrow gap limit, η → 1. As η/(1 − η) < r → ∞, forfixed α,m we ignore terms with factor 1/r. Solutions to the reduced version of (D.6) arejust sines and cosines, and components of the magnetic field (D.8) become either purelytoroidal or poloidal. Substitution into the boundary conditions (2.13) and use of the limitsI ′ν(z)/Iν(z) → 1, K ′

ν(z)/Kν(z) → −1 as z → ∞ reveals that the toroidal decay rates arethe roots of sin σT = 0 i.e. σT = π, 2π, . . .; similarly for poloidal modes we obtain theroots of (σ2

P − α2) sin σP − 2ασP cos σP = 0.


Figure D.2. Isosurface of |B|; decay mode at α = 2.1, η = 0.5, m = 1, shown over two axialperiods, z = 0 . . . 4π/α.

E

INTEGRAL EVALUATIONS

E.1 Integration over a Chebyshev expansion

E =

∫ 1

0

∑

n

An T∗n(x) dx. (E.1)

The shifted Chebyshev polynomials T ∗n(x) are odd about x = 1

2 for n odd, so

E =∑

n even

An Tn, Tn =

∫ 1

0T ∗

n(x) dx. (E.2)

The coefficients Tn are constant and the sum is simple to evaluate accurately. The coeffi-cients are given below for n up to 64.

n −1/Tn n −1/Tn n −1/Tn n −1/Tn n −1/Tn

0 −1 14 195 28 783 42 1763 56 31352 3 16 255 30 899 44 1935 58 33634 15 18 323 32 1023 46 2115 60 35996 35 20 399 34 1155 48 2303 62 38438 63 22 483 36 1295 50 2499 64 4095

10 99 24 575 38 1443 52 270312 143 26 675 40 1599 54 2915

E.2 Parseval’s theorem

Consider the real vector field A which has been expanded on [R1, R2]× [0, 2π]× [0, 2π/α]as

A(r, θ, z) =∑

km

Akm(r) ei(αkz+mθ), Akm = A∗−k,−m . (E.3)

The energy integral associated with the field A is

E = 12

∫

A · AdV. (E.4)

Appendix E. Integral evaluations 65

The exponentials have the property

∫ 2π

0ei(m+n) dθ = 2π δm,−n ,

∫ 2πα

0eiα(k+j) dz =

2π

αδk,−j , (E.5)

and so, using (E.3),

E =2π2

α

∑

km

∫

|Akm|2 r dr, (E.6)

where|Akm|2 = |Ar km|2 + |Aθ km|2 + |Az km|2. (E.7)

Using the method of §E.1 to integrate radially, (E.6) is easy to evaluate accurately.If A is independent of θ, (E.4) may be written

E =

∞∑

k=0

Ek, Ek =

2π

α

∫

A20 r dr, k = 0,

4π

α

∫

|Ak|2 r dr, k > 0.

(E.8)

The m = 0 index has been dropped. Each Ek is the energy associated with the axialwavenumber k and (E.8) is a form of Parseval’s theorem. In a similar manner, when thefield A depends on θ, the energy can be split over the azimuthal m-modes,

E =

∞∑

m=0

Em, Em =

2π

α

∫

A200 r dr +

4π

α

∑

k>0

∫

|A2k0| r dr, m = 0,

4π

α

∑

k

∫

|Akm|2 r dr, m > 0.

(E.9)

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