Mode control of Taylor–Couette flow using an

applied magnetic field

Anthony J Youd and Carlo F Barenghi

School of Mathematics & Statistics, Newcastle University, Newcastle upon Tyne,NE1 7RU, England, UK

E-mail: A.J.Youd@ncl.ac.uk

Abstract. Recent experiments in hydromagnetic finite aspect ratio Taylor–Couette flow have investigated the possibility of using an applied magnetic fieldto control flow states, as opposed to the usual methods of sudden starts of thecylinders or discontinuous jumps in the aspect ratio. We perform numericalsimulations to determine whether it is possible to arbitrarily transform normal oranomalous flow states into any other. We find that, with our chosen parameters, itis indeed possible to control flow patterns, but that not all transitions consideredare possible.

Keywords :flow instabilities; vortex dynamics; magnetohydrodynamics.

Mode control of Taylor–Couette flow using an applied magnetic field 2

1. Introduction

Couette flow of an incompressible viscous fluid contained in the gap between twoconcentric, rotating cylinders, has been at the forefront of hydrodynamic stabilitytheory for many years. Couette flow under the influence of an applied magneticfield was first studied in the pioneering work of Chandrasekhar (1961) and Donnellyand Ozima (1962), but was then almost forgotten until recent interest has arisenfrom astrophysical applications such as the magnetorotational instability (Balbus andHawley, 1991; Willis and Barenghi, 2002a; Rudiger et al., 2006; Liu et al., 2006;Szklarski and Rudiger, 2006) and dynamo action (Willis and Barenghi, 2002b; Dobleret al., 2002). The hydromagnetic application which motivates this paper is, however,the pioneering work of Benjamin (1978a,b) and Benjamin and Mullin (1981). Theyhighlighted the non-uniqueness of the Navier–Stokes equations and the discovery ofthe so-called ‘anomalous’ modes in the Couette geometry when end-effects are takeninto account. These flow solutions cannot be found in the most-studied case wherethe cylinders are taken to be of an infinite length. In a recent paper (Youd andBarenghi, 2006) we have shown that these anomalous modes are robust enough tosurvive external body forces, such as an applied axial magnetic field, giving rise tointeresting time-dependent solutions.

Recently, Kikura et al. (2005) have performed experiments with the aim ofcontrolling flow states in finite aspect ratio Taylor–Couette flow by the application ofa magnetic field. Electromagnetic forcing has also been used to generate and controlturbulent or chaotic flows (see, for example, Rossi et al., 2006, and references therein).Kikura et al. found that they could use the applied magnetic field as a ‘switch’ totransform normal or anomalous flow states into other normal or anomalous states whilealways keeping the aspect ratio, radius ratio and Reynolds number constant. Thisprocedure should provide a convenient alternative to sudden starts of the cylindersor discontinuous jumps in the aspect ratio — the usual methods for controlling flowpatterns (see, for example, Bolstad and Keller, 1987, and references therein). The aimof this paper is to explore this magnetic switch procedure by numerical simulations.

Kikura et al. used a magnet perpendicular to the axis of rotation, whereas weapply an axial magnetic field, which simplifies the symmetry of the problem. In thiswork both the flow and the external magnetic field are axisymmetric. The magneticfield in Kikura et al.’s experiment is non-axisymmetric. Kikura et al. state that theyuse a water-based magnetic fluid, with a 23.35% weight concentration of magnetiteparticles, having a sound velocity of 1450 ms−1. We formulate the problem for realisticparameters in liquid sodium. Given the differences between the experimental setup andour numerical simulations, we do not attempt to make a direct comparison betweenthem.

2. Governing equations and method of solution

We consider the flow of an incompressible viscous electrically conducting fluid ofconstant kinematic viscosity, ν, magnetic diffusivity, λ, magnetic permeability, µ0,and density, ρ, contained in the gap between two coaxial concentric cylinders of radiiR1 and R2. The cylinders can be either of infinite or finite height, h. We assume thatthe outer cylinder is at rest while the inner cylinder rotates with prescribed angularfrequency, Ω.

A magnetic field, B0 = B0z is applied in the axial direction, assuming cylindrical

Mode control of Taylor–Couette flow using an applied magnetic field 3

coordinates (r, θ, z). We make our equations dimensionless in a standard way usingδ = R2 −R1 as the length scale, δ2/ν as the time-scale, ν/δ as the velocity scale andB0 as the magnetic scale. The dimensionless parameters of the problem are the radiusratio, η = R1/R2, the inner cylinder Reynolds number, Re = R1Ωδ/ν, the aspectratio, Γ = h/δ, the Chandrasekhar number, Q = B2

0σδ2/ (ρν) (where σ = 1/ (λµ0)),

and the magnetic Prandtl number, Pm = ν/λ.Laboratory liquid metals have very small magnetic Prandtl numbers, for example,

Pm ∼ O(10−5) and O(10−7) for liquid sodium and gallium respectively. In the smallPrandtl number limit we expect that the induced field B ∼ O(Pm) and u ∼ O(1)where u is the fluid velocity. Letting B = Pmb, we have the following hydromagneticequations:

∂tu + (u · ∇)u = −∇p+ ∇2u +Q (∇ ∧ b) ∧ z, (1a)

∇ · u = 0, (1b)

∇2b = −∇ ∧ (u ∧ z) , (1c)

∇ · b = 0. (1d)

where p is the pressure. All fields are assumed to be axisymmetric, in agreement withexperimental findings at the low and moderate Reynolds numbers we consider here.

Full details of the numerical method can be found in our previous paper (Youd andBarenghi, 2006). Here it suffices to say that these equations are solved by a centred,equispaced, finite-difference scheme (second order accurate in both time and space)using the stream function–vorticity formulation (the magnetic equivalent of whichcorresponds to azimuthal magnetic field and current). Typically we use Nr = 80radial grid-points and Nz = NrΓ axial grid-points with time steps ∆t of the orderof 5 × 10−5. At the maximum Reynolds number of 500 used in the simulations, theboundary layer is approximately

√

1/Re = 4.5% of the gap width, and so with this gridresolution, there are three points inside the boundary layer. The Poisson equationsfor the stream function and azimuthal magnetic field and current are solved usingLapack linear algebra routines (Anderson et al., 1999).

The boundary conditions at the cylinder walls are straightforward. The boundaryconditions for the stream function and vorticity can be derived from the usual no-slip conditions, so ur = uz = 0 at r = R1 and R2, uθ = Re at r = R1, anduθ = 0 at r = R2. The boundary conditions for the magnetic field depend on theconductivity of the cylinders, as discussed in Roberts (1964). Hereafter, we assumeperfectly conducting cylinders and set (1/r)Bθ + ∂rBθ = 0 and Jθ = 0 at r = R1 andR2, where J = ∇∧B is the current. The boundary conditions at the ends, z = 0 andh, require more explanation and are discussed in the next section.

3. Anomalous modes

There has been much work done on Taylor–Couette flow at finite aspect ratio,stemming from the pioneering work of Benjamin (1978a,b) and Benjamin and Mullin(1981). They discovered solutions to the Navier–Stokes equations which do not existunder the assumption of infinite height cylinders — the so-called ‘anomalous’ modes.There are generally two options to consider when working with finite cylinders. Thefirst is the case where the end-plates are attached to the inner cylinder and so rotatewith it (see, for example Tavener et al., 1991); here it is found that the cells closest

Mode control of Taylor–Couette flow using an applied magnetic field 4

to the ends rotate in such a way so as to give rise to a radial outflow at the endwalls.The second is the case where the end-plates are attached to the (stationary) outercylinder, and this is the case we shall be considering in this paper. The classicalthinking in this case is that the centrifugal force exerted by the inner cylinder pushesthe fluid outwards toward the outer cylinder at mid-height, due to the braking effectof the stationary ends, then, to conserve mass, the fluid must move inwards at theends. The anomalous modes, however, appear to have an outflow at the ends. Closerinspection reveals the presence of small vortices, very much weaker than the fully-formed vortices, in the corners between the cylinders and the end-plates, which ensurethat there is, in fact, an inflow at the ends. (The situation is essentially reversed inthe case where the end-plates rotate with the inner cylinder.)

Schaeffer’s (1980) homotopy parameter τ is a useful tool in the study of anomalousmodes. It provides a means of continuously (theoretically) deforming the boundaryconditions at the ends from infinite, stress-free (τ = 0) boundary conditions, to finite,no-slip (τ = 1) boundary conditions. The boundary conditions at the ends can bewritten as

uz = 0,

(1 − τ)∂ur

∂z± τur = 0,

(1 − τ)∂uθ

∂z± τ uθ − F (r) = 0

(2)

for z = 0 and h, R1 6 r 6 R2, where the minus sign corresponds to z = 0 and the plussign to z = h, and where F (r) is some smooth function introduced to deal with thediscontinuity encountered where the cylinders meet the end walls. It can be verifiedthat when τ = 0 or τ = 1 the boundary conditions collapse to the correct forms forthe infinite and finite cylinder cases respectively.

Since we are dealing with an applied magnetic field which must also satisfyboundary conditions at the ends, we introduce an analogous representation of thevelocity boundary conditions for the magnetic field B and current J

(1 − τ)Bθ ± τ∂Bθ

∂z= 0,

(1 − τ)∂Jθ

∂z± τJθ = 0

(3)

for z = 0 and h, R1 6 r 6 R2, where again the minus sign corresponds to z = 0and the plus sign to z = h. Defining the boundary conditions in this way allows for asmooth transition between the boundary conditions for the infinite and finite cylindercases just as for the boundary conditions for the velocity components. An importantpoint to consider is the discontinuity present between the rotating inner cylinder andthe stationary end-plates where the azimuthal velocity increases from zero to Re. Thisdiscontinuity is present in any experiment where a small gap must be left between thecylinder and end-plate. We have discussed how we and others deal with this in ourprevious papers (Youd and Barenghi, 2005, 2006).

We choose to use a fixed aspect ratio Γ = 3 (the same as Kikura et al., 2005)and radius ratio η = 0.615. We use this radius ratio because of previous results (Cliffeand Mullin, 1985; Bolstad and Keller, 1987) showing the transitions to the anomalousmodes at the same radius ratio. Kikura et al. use a radius ratio of η = 0.6, but weshould only expect a slight difference in critical Reynolds numbers were a comparisonrequired.

Mode control of Taylor–Couette flow using an applied magnetic field 5

(a)

(b)

(c)

(d)

(e)

Figure 1. Computed flow states at Γ = 3, η = 0.615 in the absence of an imposedmagnetic field Q = 0. (a) normal 2-cell mode (N2, Re = 380), (b) normal 4-cellmode (N4, Re = 380), (c) anomalous 2-cell mode (A2, Re = 350), (d) anomalous3-cell mode (A3, Re = 240), (e) anomalous 4-cell mode (A4, Re = 400). Solidlines represent vortices rotating clockwise; dashed lines represent vortices rotatingcounter-clockwise.

Figure 1 shows all the possible flow states we shall consider in this paper at thechosen aspect ratio and radius ratio. Plotted are contours of the stream function inthe (r, z)-plane with the inner cylinder on the left and the outer cylinder on the right.No-slip boundary conditions (τ = 1) are imposed at the top and bottom boundaries.Figure 1(a) shows a normal 2-cell mode (N2), where the term ‘normal’ differentiatesfrom ‘anomalous’, and the number of cells refers to the number of fully formed largevortices. Figure 1(b) shows a normal 4-cell mode (N4); figure 1(c) shows an anomalous2-cell mode (A2); figure 1(d) shows an anomalous 3-cell mode (A3), and figure 1(e)shows an anomalous 4-cell mode (A4). The anomalous 3-cell mode of figure 1(d)is the only flow we consider which is asymmetric about the mid-plane. A mirrorimage anomalous 3-cell mode, where the small cell is at the top, can also be realiseddepending on the initial conditions.

4. Existing procedure to control flow states

Generating the flow states in figure 1 is a non-trivial task because we have five flowstates at the same aspect ratio and radius ratio. The flow state we end up withdepends heavily on the route taken through parameter space and so in table 1 weoutline a possible procedure to obtain the given flow patterns.

To explain the entries in the table we use the anomalous 2-cell mode as anexample. To begin with we set Q = 0, Γ = 2, Re = 300, and τ = 0 (so weassume infinite cylinders, initially). In the infinite cylinder case the bifurcation fromazimuthal Couette flow to Taylor vortex flow is a perfect pitchfork, one branch of whichcorresponds to a flow with an inflow at the ends, the other branch corresponding toa flow with an outflow at the ends. With careful choice of the initial conditions theseparameters lead to a 2-cell flow with an outflow at the ends (state A). (When we refer

Mode control of Taylor–Couette flow using an applied magnetic field 6

Table 1. Example routes through parameter space required to achieve the finalflow state in the absence of an imposed magnetic field Q = 0. The final threecolumns for each flow show the parameters at the state given in column one. Seethe text for a description of the route taken for the anomalous 2-cell mode.

State Γ Re τ State Γ Re τ

Normal 2-cell mode Normal 4-cell modeA 3 100 1 A 4 100 1B 3 380 1 B 4 380 1

C 3.5 380 1D 3 380 1

Anomalous 2-cell mode Anomalous 3-cell modeA 2 300 0 A 3 85 0B 2 300 1 B 3 240 0C 2 350 1 C 3 240 1D 3 350 1

Anomalous 4-cell modeA 3 400 1

to the ends in the case τ = 0 we actually mean the ends of the computational domain,which is the same as that for τ = 1 except the boundary conditions are stress-free).

Once this flow has saturated (i.e. the growth rate of the radial velocity at themidplane is on the order of 10−8) τ is switched instantaneously from zero to one. Inour case this has the same effect as smoothly changing τ . The resulting flow is ananomalous 2-cell mode exactly as in figure 1(c) except that Γ = 2 (state B). The thirdstep is to increase the Reynolds number to 350 so that the 2-cell anomalous mode stillexists when we change the aspect ratio (state C). The final step is to smoothly changethe aspect ratio from Γ = 2 to 3. Provided the jumps in aspect ratio are not too largethen it is possible for the anomalous 2-cell state to remain at the final parametersgiven in state D.

The other normal and anomalous flows are found in a similar way, as in table 1.

5. New magnetic procedure to control flow states

Once we have the desired initial (normal or anomalous) flow state from figure 1 wethen instantaneously apply an axial magnetic field (i.e. we restart our numerical codesetting Q 6= 0). The Reynolds number, aspect ratio and radius ratio are all held fixedthroughout this procedure. In the majority of the cases we examined the resultingflow pattern is a weak Ekman flow induced by the top and bottom ends (but seesection 6 for a discussion of some interesting phenomena where the intermediate flowis not an Ekman circulation). Once this Ekman flow is established, the magnetic fieldis instantaneously turned off (i.e. we again restart setting Q = 0). This flow state isthen left to saturate and we would like to determine whether or not it is the same asthe initial state.

At these early stages of investigation, rather than attempt to map out the wholeparameter space to determine critical Reynolds numbers and Chandrasekhar numbersfor a given initial flow state (which would be a substantial undertaking), we are moreinterested in discovering whether our new ‘magnetic switch’ procedure actually works.Is it possible to transform flow states just by switching a magnetic field on and off?

Table 2 summarises the possible outcomes we have found. The first column

Mode control of Taylor–Couette flow using an applied magnetic field 7

Table 2. Initial and final flow states under the influence of an applied magneticfield (A ≡ anomalous mode; N ≡ normal mode).

Initial state Final state Re Q B0(Gauss)A2 N2 350 100 79.09A2 N4 500 5000 559.29A2 A4 350 6000 612.67A3 N2 240 50 55.93A3 N4 500 5000 559.29A3 A4 400 5000 559.29N4 N2 380 50 55.93N4 A4 380 5000 559.29A4 N2 400 125 88.43A4 N4 500 5000 559.29N2 N4 500 5000 559.29N2 A4 380 5000 559.29

shows the initial flow pattern as produced by the traditional procedure outlined intable 1. The second column shows the final flow state after the magnetic field hasbeen switched on and off, as outlined in the procedure above. The third columnis the (fixed) Reynolds number throughout the procedure. The fourth column isthe Chandrasekhar number Q which was sufficient for the state change to occur —this is not necessarily the critical Q required, but rather a good order of magnitudeestimate. To illustrate the intensity of the magnetic field required to switch flowpatterns, the fifth column shows the value of B0 corresponding to the value of Qin the fourth column, assuming liquid sodium parameters: σ ≈ 107 Ω−1 m−1,ρ ≈ 0.92 × 103 Kg m−3, ν ≈ 0.68 × 10−6 m2 s−1, and a gap width δ ≈ 10−2 m,typical of experiments.

Our main result is that it is indeed possible to take any of the initial flow statesfrom figure 1 and transform it into another one through the procedure of switching amagnetic field on and off. The final state, however, is always either a normal 2-cellmode, a normal 4-cell mode, or an anomalous 4-cell mode. It was not possible toarbitrarily transform one flow state into any other. The applied magnetic field needsto be at least one order of magnitude greater in order to transform the initial flowinto an anomalous or normal 4-cell state than that required to transform the flowinto a normal 2-cell state. We are not able to transform the initial flow pattern intoeither the anomalous 3-cell mode or the anomalous 2-cell mode. Apparently, the routethrough parameter space necessary to find these flows (see table 1) is too complicatedfor our procedure to work. To obtain the initial anomalous 2-cell mode it is necessaryto alter the aspect ratio, and to obtain the initial anomalous 3-cell mode it is necessaryto smoothly increase the Reynolds number.

Figure 2 shows a time-series plot of the radial velocity ur measured in the middleof the gap at z = h/2, for the N2–A4 transition, and figure 3 shows computed contourplots of the stream function ψ at various times over the evolution. The contours arespaced linearly between the maximum and minimum values of ψ, which occur whenRe = 380 and Q = 0. In this case ψmax = 16.499 (and ψmin = −ψmax). In plots3(a), 3(b), and 3(d) we have used 40 contour levels, but in plot 3(c) (where Re = 380and Q = 5000), it is necessary to use 4000 levels to see the structure of the flowon the same scale as the other plots; here, ψmax = 0.078. From t = 0 to t = 1corresponds to state A of a normal 2-cell mode (cf. table 1); the contour plot 3(a)

Mode control of Taylor–Couette flow using an applied magnetic field 8

ur

t0.0 0.5 1.0 1.5 2.0 2.5

0

20

40

60

80

100

120

140

Figure 2. Time-series plot of the radial velocity ur measured in the middle ofthe gap at z = h/2, for the N2–A4 transition.

is at t = 1 when the flow has saturated. From t = 1 to t = 1.5 corresponds to thesame normal 2-cell mode but now at the larger Reynolds number of Re = 380; thecontour plot 3(b) is at t = 1.5. From t = 1.5 to t = 2 the magnetic field has beenswitched on (Q = 5000) and the resulting flow is an Ekman flow as shown in figure 3(c)at t = 2. Figure 4 compares this Ekman circulation to an Ekman circulation foundwhen no magnetic field is present. In figure 4(a) Re = 380 and Q = 5000, and in4(b) Re = 21 and Q = 0. In both cases the radial velocity measured in the middleof the gap at the mid-plane is approximately 0.065. Again, 40 contour levels areused, equispaced between the maximum and minimum values of ψ, which occur forthe flow with the larger Reynolds number (ψmax = 0.078; for the flow with the lowerReynolds number ψmax = 0.065). Note how the cells are heavily compressed towardsthe inner cylinder in the flow under the influence of a magnetic field — a phenomenonobserved in our previous paper (Youd and Barenghi, 2006), and also by Goodman andJi (2002). For a genuine Ekman circulation (without an imposed magnetic field) asshown in figure 4(b), the cells’ cores are more centrally located in the gap. The finalstage of the evolution from t = 2 to t = 2.5 is where the magnetic field is switched off(Q = 0) and the resulting flow is an anomalous 4-cell mode, as shown in figure 3(d)at t = 2.5.

Figure 5 shows a time-series plot of the total (dimensionless) kinetic energy Ek

for the same N2–A4 transition, where we define

Ek =1

2

∫

V

u · udV . (4)

Mode control of Taylor–Couette flow using an applied magnetic field 9

(a)

(b)

(c)

(d)

Figure 3. Computed contour plots of various flow states for the N2–A4 transition.(a) Re = 100, Q = 0, t = 1, (b) Re = 380, Q = 0, t = 1.5, (c) Re = 380, Q = 5000,t = 2, (d) Re = 380, Q = 0, t = 2.5. See the text for a detailed explanation ofthe figure.

(a)

(b)

Figure 4. Comparison of Ekman flows with and without the presence of amagnetic field. (a) Re = 380, Q = 5000, (b) Re = 21, Q = 0. In both cases, ur

measured in the middle of the gap at z = h/2 is 0.065.

6. Additional flow states

During the course of the numerical simulations, a number of interesting phenomenawere observed including bifurcations to time-dependent solutions.

As an example, consider the A2–N4 transition, which, in order to be successful,requires an imposed magnetic field strength of Q = 5000. This ensures that theintermediate flow is a weak Ekman circulation (similar to that shown in figure 4(a)).If the imposed magnetic field is not strong enough, however, it is possible to induce

Mode control of Taylor–Couette flow using an applied magnetic field 10

103E

k

t0.0 0.5 1.0 1.5 2.0 2.50

50

100

150

200

250

300

350

400

450

Figure 5. Time-series plot of the total (dimensionless) kinetic energy Ek for theN2–A4 transition. (The apparent discontinuities in the plot at times 1.0, 1.5, and2.0 are simply an artifact of the plotting.)

axisymmetric time-dependent motion (instead of an Ekman circulation).Figure 6 shows contour plots of this time-dependence in the case where Q = 50 —

two orders of magnitude lower than that required for the A2–N4 transition. The basicunderlying flow state is that of an anomalous 4-cell mode, and the most significantpart of the oscillation on top of this basic state takes place close to the midplane. Here,a vortex cell appears from the inner cylinder alternately just above the midplane, andjust below.

If the magnetic field strength is set lower still — Q = 10 — then the oscillation issuppressed. In this case, the intermediate flow pattern is that of a steady anomalous4-cell mode, rather than an Ekman circulation or a time-dependent flow. Once thisflow is established, it is then possible to instantaneously switch off the magnetic fieldand recover the same anomalous 4-cell state. This provides another possible route forthe A2–A4 transition at a slightly higher Reynolds number, but significantly lowerimposed field strength than that shown in table 2.

There is also some evidence of quasi-periodic motion when Q = 100. Figure 7shows contour plots similar to those in figure 6, and figure 8 shows a Poincare section.In this case the basic flow is a normal 2-cell mode, and a similar oscillation to theperiodic time-dependent case is observed, whereby new vortices appear from the innercylinder.

There was also evidence of time-dependent flows in the other transitions, butbuilding up a clear picture of exactly which parameters — in terms of Γ , Re, and Q— are required to induce particular states is a significant task.

Mode control of Taylor–Couette flow using an applied magnetic field 11

(a)

(b)

(c)

(d)

(e)

(f)

Figure 6. Contour plots of the time-dependent flow in the A2–N4 transition,induced by setting an imposed magnetic field strength of Q = 50. The period ofoscillation is approximately T = 0.1, and the snapshots are taken at (a) t/T = 0,(b) t/T = 0.20, (c) t/T = 0.40, (d) t/T = 0.73, (e) t/T = 0.80, (f) t/T = 0.95.

7. Conclusions

Direct comparison between our calculation and the experiment of Kikura et al. (2005)is not possible because they used a non-uniform magnetic field applied perpendicularlyto the axis, and there are differences between the magnetic properties of the fluids usedin the studies. Their results show transitions to flow states that we could not obtain,and vice versa. For example, they report a transition to the anomalous 3-cell mode,whereas they do not mention any transitions from anomalous 2-cell to normal 2-cellmodes. Despite the differences between our setup and theirs, our main conclusion isthat it is possible to control flow patterns using an applied magnetic field. On thenegative side, we were not able to arbitrarily change any state into any other arbitrarystate. Our numerical results and the experiment of Kikura et al., however, suggest

Mode control of Taylor–Couette flow using an applied magnetic field 12

(a)

(b)

(c)

(d)

(e)

(f)

Figure 7. Contour plots of the quasi-periodic flow in the A2–N4 transition,induced by setting an imposed magnetic field strength of Q = 100. The snapshotsare taken over a time interval of Tqp = 0.17. (a) t/Tqp = 0, (b) t/Tqp = 0.28, (c)t/Tqp = 0.47, (d) t/Tqp = 0.63, (e) t/Tqp = 0.81, (f) t/Tqp = 1.

that it is likely that some combination of the orientation of the magnetic field, theproperties of the magnetic fluid, and the values of the Reynolds number, aspect ratio,and radius ratio, would allow any state to be formed from any other.

It is interesting that time-dependent flows can also be realised under certainparameter regimes, but it is not yet clear exactly which parameters are required.

With the possibility of varying Γ , Re, and Q, there is huge scope for movementthrough parameter space. Perhaps a worthwhile starting point would be to fix one ofthese parameters, and concentrate on a restricted region of parameter space.

Ultimately, we would like to map out parameter space finding critical appliedmagnetic field strengths as well as allowing other orientations of the magnetic field.One particular magnetic field orientation worth considering is that of the imposedhelical magnetic field (B0z = B0, B0θ = βB0/r, where β is a measure of the relative

REFERENCES 13replacemen

uz

ur

2 4 6 8 10 12 14 16−10

−8

−6

−4

−2

0

2

4

6

8

10

12

Figure 8. Poincare section for the quasiperiodic flow in the A2–N4 transition,when Q = 100. The radial and axial velocities are plotted at r = (R1 + R2) /2,z = h/2, when the azimuthal velocity is zero at the same position.

magnitudes of B0z and B0θ) which has been used in studies of the MRI (see, forexample Hollerbach and Rudiger, 2005; Stefani et al., 2006). Fields of this formhave the advantage of being axisymmetric, making them more amenable to numericalsimulations, and it would be more interesting to exhaust all the axisymmetricpossibilities before attempting to tackle the fully 3D solutions required to directlymodel the experiments of Kikura et al..

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