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Chapter 1

Introduction and motivation

Although Newton’s laws of motion were immediately successful when applied to

planetary motions, they were much less so when employed in the field of fluid me-

chanics. A good example of this is the 1999 solar eclipse which was predicted

correctly to the minute over a hundred years before the event. Whether the sky

would be clear or cloudy on the day however was not known even a day in advance.

Although both of these systems are governed by Newton’s second law of motion

the planetary orbit solutions are stable whilst those governing fluid flows in the

atmosphere are generally unstable.

In fluid dynamics stability takes the form of laminar flows whose clean, regu-

lar velocity profiles represent the most basic solutions to the hydrodynamic equa-

tions. Instability meanwhile is characterised by irregular, unpredictable turbulent

flows that generally break down into ever more complex patterns and structures.

This complicated behaviour is epitimised by the experimental results of Osbourne

Reynolds [1] who studied the, now classic, problem of flow in a pipe.

Owing to the unpredictable nature of turbulence, knowing where and when insta-

bility will appear is often of great interest in ‘real-life’ problems. In some situations,

e.g. aviation, turbulence is best avoided, whilst in other situations it may be essential

for the system to function correctly, e.g. mixing problems.

The study of hydrodynamic stability therefore is concerned with when and how

13

CHAPTER 1. INTRODUCTION AND MOTIVATION 14

these laminar flows break down, their subsequent development and eventual transi-

tion to turbulence. The subject has a long history beginning in the latter part of

the nineteenth century and, with many applications in fields such as engineering,

meteorology, oceanography, astrophysics and geophysics, it is widely recognised as

one of the fundamental problems of fluid mechanics.

1.1 Spherical Couette flow

The roots of the spherical Couette flow problem date back as far as 1890 when Mau-

rice Couette experimentally investigated the dynamics of a viscous fluid contained

between two concentric cylinders. Keeping the inner cylinder fixed whilst rotating

the outer one, Couette studied the moment of the drag exerted by the fluid on the

inner cylinder — his goal being to measure the viscosity of the fluid. He noticed

that for velocities above some threshold value, the rate at which this drag increased

with velocity changed significantly. This change in behaviour was attributed to a

transition of the system from steady, laminar flow to chaotic, turbulent motion.

The theoretical investigation of fluid flow bounded by concentric, rotating cylin-

ders was first considered by Lord Rayleigh (John Strutt, 3rd Baron Rayleigh) who

studied the stability of an inviscid fluid to axisymmetric disturbances (see [2],[3]).

Rayleigh concluded that for differentially rotating cylinders with a negative angular

momentum ratio, the flow would always be unstable. However for cylinders rotating

in the same direction, the flow could be stable if the angular momentum of the outer

cylinder exceeded that of the inner one. Roughly speaking this means that, in the

absence of viscosity, a basic state flow is stable if, and only if, the velocity profile

of the fluid exhibits the property that the absolute value of the angular momentum

increases monotonically outwards with radius. This statement can be thought of as

a direct application of Rayleigh’s circulation criterion (see [2],[4],[5]).


Taylor–Couette flow

The above ideas were developed further by Sir Geoffrey Taylor who extended Rayleigh’s

inviscid analysis to include the effects of viscosity in his ground breaking paper of

1923 [6]. Driven by the experimental findings of Couette (and others), Taylor con-

structed a more advanced and accurate apparatus with which to visualise viscous

fluid flow between concentric rotating cylinders — now known as Taylor–Couette

flow.

Taylor’s findings showed that once the angular momentum ratio of the cylinders

reached a critical point, i.e. when the magnitude of the angular momentum on the

inner boundary was sufficiently larger than that of the outer one, the (purely zonal)

basic flow became unstable to a series of cellular, axisymmetric, toroidal vortices.

These vortices are now known as Taylor vortices and are illustrated by a schematic

drawing and an experimental photograph in Figure 1.1(a) and 1.1(b) respectively.

An important outcome of this was the realisation that stability in a rotating

fluid can be characterised by a dimensionless quantity — now known as the Taylor

number, denoted Ta — which is, generally, proportional to the square of the Reynolds

number (although the precise definition may vary). Roughly speaking the Taylor

number of a system can be thought of as a measure of the ratio of the angular

momentum to the kinetmatic viscosity (or viscosity per unit density).

The key feature of this Taylor number is that all flows with Ta less than some

critical value Tac (say) will remain stable and the effect of viscosity will be to damp

any small disturbances by dissipation. However flows for which Ta exceeds Tac will

become unstable to axisymetric Taylor vortices as described previously. The finite

strength of the vortex structures will be a function of the angular velocity, and hence

the Taylor number, as shown by Taylor’s torque measurement experiments [7].

The experimental observations outlined above suggested some important simpli-

fications that could be made in the theoretical treatment of the problem and led

Taylor to seek disturbances to the basic flow in the form of axisymmetric vortices.

He expanded the, radially dependent, vortex amplitudes as series of Bessel functions,


(a) (b)

Figure 1.1: Taylor vortex instabilities in the classical, cylindrical, Taylor–Couette

flow configuration (a) shows a schematic illustration of the vorticies whilst (b) depicts

Taylor vortices arising in a laboratory experiment.

each with its own wavenumber km, before solving numerically. (The critical Taylor

number Tac in this case would be the global minimum of Ta over all wavenumbers

km.)

The agreement between the values of Tac calculated using this theoretical ap-

proach and those seen in the experimental results was really rather good. In ad-

dition agreement between Taylor’s viscous results and the findings of Rayleigh for

the inviscid case were very good when both cylinders rotated in the same direction.

Namely Rayleigh’s circulation criterion held in this case and as the angular velocity

of the outer cylinder Tac approached the line of equal angular momenta for the inner

and outer boundaries (i.e. Rayleigh’s neutral stability point) from above. However

Rayleigh’s conclusion that inviscid fluid between two differentially rotating cylinders

would always be unstable did not hold as the presence of viscosity serves to stabilise

the system.


Despite the generally excellent agreement between Taylor’s theoretical [6] and

experimental [6], [7] analyses, one key difference was evident; namely regarding the

temporal evolution of the system. Being concerned solely with whether or not dis-

turbances persisted or died out, Taylor’s analysis was limited to a linear theory

whereby he linearised his equations of motion about small amplitude disturbances.

This meant that, for unstable flows, Taylor’s theory predicted unbounded vortex

growth with time, in contrast to the existence of a finite amplite equilibrium sug-

gested by his experimental findings.

Following the development of weakly nonlinear theory in the 1950s, Davey [8]

employed an expansion about the small disturbance amplitude of the linear the-

ory and, by truncating to second order, constructed a weakly nonlinear analysis.

He identified a nonlinear amplitude equation governing the amplitude growth of

small Taylor vortex disturbances, now generally referred to as the Stuart–Landau

equation, whose solutions evolved towards a steady equilibrium as time progressed.

Spherical Couette flow

With the linear stability of cylindrical Taylor–Couette flow well known and under-

stood, the challange became to tackle the more complicated Spherical Couette flow

problem, which is more relevant in geophysical and astrophysical contexts.

The Spherical Couette flow problem consists of a viscous fluid, held at constant

temperature and pressure, confined between two concentric spherical shells of radius

R1 and R2 (with R1 < R2). The spherical shells rotate about a common axis with

angular velocities Ω1 and Ω2 respectively as depicted in Figure 1.2. However a

stationary outer sphere is generally favoured by experimentalists and so the majority

of numerical studies in this area have focused on this case, i.e. Ω2 = 0.

The basic flow is strongly zonal although, unlike the classical Taylor–Couette

problem, there is also a (much weaker) meridional circulation between the equator

and the poles. When the inner sphere rotates faster than the outer one, the centrifu-

gal forces that drive the instability are greatest in the vicinity of the equator. Once


Ω1

Ω2

R1

R2

εR1

Figure 1.2: The spherical Couette flow configuration. Fluid is confined between two

spherical shells of radius R1 and R2 (R1 < R2) that rotate about a common axis

with angular velocities Ω1 and Ω2 respectively. The configuration is characterised

by the dimensionless parameter ε, the ratio of the gap width to the radius of the

inner sphere.

again stability can be characterised by the dimensionless Taylor number Ta which is,

as in the cylindrical case, a measure of the circulation of the flow (roughly speaking

proportional to the square of the angular momentum and inversely proportional to

the square of the kinematic viscosity).

The findings of Marcus and Tuckerman [9], [10] suggested that configurations

could be classified according to the size of the gap aspect ratio ε, defined to be the

ratio of the gap width to the radius of the inner cylinder (i.e. R2 − R1 : R1). They

identified narrow, medium and wide gap regimes. In the narrow and medium gap

cases, where the ratio was approximately less than a quarter, axisymmetric Taylor–


Equator

Figure 1.3: A sketch showing equatorially symmetric, toroidal, Taylor vortices in

the spherical Couette flow configuration.

vortex type instabilities were observed at the onset of instability in the vicinity of

the equator.

These findings were confirmed by the theoretical and experimental investigations

of Buhler [11] who identified equatorially symmetric instabilities in which vortices

appeared in pairs with one either side of the equator. He isolated modes invoving

zero, one and two vortex pairs. Figure 1.3 shows a schematic representation of the

equatorially symmetric, single pair, vortex pattern. Buhler also identified asymmet-

ric vortex patterns whereby a single pair of vortices arose in the region of the equator

with one noticably larger in diameter than the other. In this case the streamline

defining the boundary between the vortex cells was located away from the equator.

1.2 The narrow-gap problem

The study of the stability of spherical Couette flow in the narrow-gap limit (ε small)

has a long history beginning with Walton [12]. When the gap aspect ratio ε is


small, the expected mode of instability takes the form of Taylor vortices whose

angular, latitudinal, length scale is comparable to the gap width, as Wimmer’s

[13],[14] experiments confirm.

In this small ε limit the local equatorial geometry resembles concentric cylinders.

Walton [12] extended this infinite cylinder idea by employing a multiple scale analysis

based on the small parameter ε and expanding about the local critical value of the

Taylor number Tac — the critical value for the cylinder problem.

This rather simplistic analysis failed to correctly incorporate the true influence of

the effects of boundary curvature and the basic state meridional circulation. These

are both significant features and contribute towards a physical mechanism known as

phase mixing. Phase mixing is characterised in the WKB formulation by a non-zero

frequency gradient at the equator and has the effect of shortening the latitudinal

length scale and increasing viscous dissipation. The consequence of this phase mixing

therefore is that, as time progresses, vortices generated at the local cylinder critical

value of the Taylor number, Tac, ultimately decay to zero. In fact the stabilising

process is so strong that it causes the global critical Taylor number Tag to exceed

the local cylinder critical value Tac even in the limiting case as ε tends to zero!

Walton failed to fully appreciate the significance of phase mixing, the result being

that the value of the critical Taylor number predicted by [12] was too low.

The correct asymptotic solution was later developed by Soward and Jones [15]

who undertook their linear analysis in the vicinity of a complex latitude away from

the equator. They obtained the true global critical Taylor number, Tag, without

expanding about local values, and showed that it exceeded the local cylinder value

Tac. Unfortunately attempts to apply these techniques to a weakly nonlinear exten-

sion, with the inclusion of the Stuart–Landau term identified by Davey [8], proved

too complicated.

To get around this problem Harris, Bassom and Soward [16] allowed the outer

shell to move and considered the case of almost co-rotation. In doing so they intro-

duced another small, dimensionless parameter related to the departure from unity


of the angular momentum ratio of the two shells. As this parameter approaches zero

then so does the difference between the global and local Taylor numbers, Tag − Tac

as well as the complex latitude value used by Soward and Jones [15]. Using these

asumptions they were able to extend the linear analysis of [15] into the weakly

nonlinear regime.

The governing equation studied by Harris, Bassom and Soward is the complex

Ginzburg–Landau equation

∂a

∂t=

(

λ − Υ2

ε x2 + ix)

a +∂2a

∂x2− |a|2a , (1.1)

rescaled here in keeping with this work (the true equation studied can be found

in Appendix A). The relevance of this equation, and the motivation behind its

existence, is described in Chapter 2 of this thesis; at this stage however, only an

understanding of what the individual terms represent is needed.

The equation (1.1) governs the amplitude modulation a(x, t) of toroidal, Taylor

vortex instabilities in space x and time t. The spatial variable x is related to the

latitude, albeit locally in the vicinity of the equator. The dimensionless parameter

Υε is a measure of the degree of co-rotation in the system and, being inversely pro-

portional to the departure from unity of the angular momentum ratio, is essentially

zero unless the angular velocities of the two spheres are very close. Meanwhile the

parameter λ is related to the excess Taylor number (i.e. the difference between Ta

and Tac) and can be thought of as driving the instability in the system. The dif-

fusion and temporal evolution of the amplitude function are governed by the terms

∂2a/∂x2 and ∂a/∂t respectively, whilst phase mixing enters the system through the

term ixa. Finally the term −|a|2a, the Stuart–Landau term obtained from Davey’s

weakly nonlinear analysis, captures the effects of nonlinearity in the system.

Pulse-train solutions

Following a complicated bifurcation sequence Harris, Bassom and Soward [17] (hence-

forth referred to as HBS) identified a global bifurcation to travelling wave solutions

valid in this co-rotation regime and located close to the equator. Being periodic in


(a) (b)

Figure 1.4: (a) An example of a pulse train solution. (b) An isolated pulse extracted

from the pulse-train solution shown in (a). The pulse has an amplitude structure

which is centered about the origin and is effectively zero outside the range [-2,2]

time these solutions can be expanded in terms of a series of Fourier modes whose

amplitudes are purely dependent on latitude. Upon closer inspection it seems that

these modes are only non-zero over a small range of latitude values with each mode

centred a distinct distance from the equator and oscillating at a distinct frequency.

Isolated functions such as those found above, where the amplitude is only signifi-

cantly different from zero over a small range of the independent variable, are known

as pulses. Additionally, a sequence of these pulses located at regularly spaced inter-

vals are known as pulse trains. An example of a pulse-train solution can be seen in

Figure 1.4(a) whilst an isolated pulse, extracted from the pulse-train in the vicinity

of the origin, can be found in Figure 1.4(b).

The results of HBS therefore suggested the existence of pulse-train solutions in

the spherical Couette flow problem for Taylor numbers close to the local, cylin-

der critical value Tac. This idea had previously been suggested by Soward [18]

who, whilst studying the related problem of thermal convection in a rotating, self-

gravitating sphere, proposed that the effects of phase mixing could be counteracted

by those of nonlinearity. However, despite some interest in the area, no such be-

haviour was ever identified. Furthermore the investigations of Ewen and Soward


[19], [20], who included a form of the HBS complex Ginzburg–Landau equation as

a special case, comprehensively ruled out the existence of pulse-trains.

In light of the above therefore, the results of HBS proved both interesting and

surprising. They pointed the way for Bassom and Soward [21] (henceforth referred

to as BS) to bypass the weakly nonlinear theory and develop solutions in the form

of inter-connected trains of pulses. It is the study of these pulse-trains, within the

context of equation (1.1), that provides the motivation for the first part of this

thesis.

1.3 The problem of thermal convection in a rapidly

rotating, self-gravitating sphere

Related to the spherical Couette flow problem is the problem of thermal convection

in a rapidly rotating, self-gravitating sphere. The problem is defined by a fluid

filled sphere which is uniformly heated throughout and maintained at a constant

temperature on the bounding surface. The fluid is assumed to satisfy the Boussinesq

approximation, which states that variations in density, and other parameters such

as viscosity, owing to changes in temperature can safely be ignored save for where

they appear in the external forcing, gravity, term (see [4] for further explanation).

Gravity is assumed to act radially inwards, as appropriate for a self-gravitating,

uniform density fluid, and the whole system rotates rapidly at a constant angular

velocity.

The dynamics of the problem gives it relevance to the study of many large

celestial bodies such as the Earth’s liquid core, the Sun and other stellar objects

as well as large fluid planets like Jupiter. Thermal convection in rapidly rotating

fluid spheres plays a key role in these astronomical flows where convection is often

the main source of mixing and heat transport. Therefore, knowledge of the general

behaviour of thermal convection in rapidly rotating systems is central to gaining an

understanding of these important geophysical and astrophysical problems.


Figure 1.5: A sketch showing non-axisymmetric convective motions in an internally

heated, rotating sphere arising in the form of tall, thin columns (after Busse [23]).

Early studies of buoyancy driven convective motions in rapidly rotating systems

showed that non-axisymmetric critical disturbances are generally preferred to ax-

isymmetric ones, see Roberts [22] and Busse [23]. Busse predicted that instability

first appears close to thin vertical cylindrical surfaces aligned parallel to the axis of

rotation, as shown in Figure 1.5 (see also [23] Figure 1). These columns intersect the

sphere at a latitude of roughly 63and propagate slowly in the direction of rotation.

This prediction was subsequently verified by the experimental findings of Busse and

Carrigan [24].

As a simplification to the spherical convection problem, Busse [23] approximated

the geometry by a cylindrical annulus about the axis of rotation with sloping upper

and lower boundaries. Taking gravity to act perpendicular to the axis of rotation and

by considering a small boundary slope (low curvature effects) the vertical velocity

will be small when compared with the horizontal velocities. This means that the

z-dependence can be integrated out of the problem and thus Busse’s analysis was


greatly simplified.

Yano [25] extended the Busse annulus model to include fully spherical boundaries

and showed that this modified Busse annulus model incorporated the effects of phase

mixing. He applied the z-averaging technique used in the original Busse annulus

model even though, with the small slope assumption no longer being valid, there

was little physical justification for it. Inspite of this Jones et al. [26] showed that

the approximation gives qualitatively reasonable results when compared with the

full three-dimensional linear analysis. This modified Busse annulus approximation

is now known as the quasi-geostrophic approximation (after Aubert et al. [27]).

Relaxation oscillations in rapidly rotating spherical systems

Three-dimensional time-dependent simulations of rapidly rotating spherical convec-

tion are limited by the availability of computational resources to somewhat mod-

erate values of the Ekman number E. In order to attain lower Ekman numbers

(or equivalently, higher Taylor numbers), closer to those believed to be appropri-

ate for astrophysical bodies, Cole [28] utilised the computationally less expensive

quasi-geostrophic model (see Appendix C).

He expanded his dependent variables in terms of Fourier functions in the az-

imuthal direction and, as a further simplification, considered only the behaviour at

onset by retaining only one azimuthal mode (the critical mode). In doing so he de-

veloped a sixteenth order PDE system of amplitude equations (see (C.3) and (C.4))

dependent on four non-dimensional parameters; namely the Rayleigh, Ekman, and

Prandtl numbers as well as the critical azimuthal wavenumber.

Cole sought travelling wave solutions to his system of equations which he used

as a starting point for the derivation of fully time-dependent solutions. Plotting the

kinetic energy of these solutions in time revealed a periodic time-series for which,

over the course of a period, the velocity component was only clearly distinguishable

from zero less than half of the time. This behaviour, where the amplitude of a

periodic (or quasi-periodic) system relaxes to a comparitively small magnitude over


Figure 1.6: Relaxation oscillation behaviour clearly visible in a plot of kinetic energy

versus time taken from Cole’s results [28]. The thick (lower amplitude) lines repre-

sent the mean part of the kinetic energy and thin (higher amplitude) lines represent

the fluctuating part (see Appendix C for more details).

the course of a period (or quasi-period), is called a relaxation oscillation and is

exemplified by the plot in Figure 1.6.

It is relaxation oscillations like the ones found by Cole that provide the moti-

vation for the second half of this thesis where we extend the HBS model (1.1) and

attempt to replicate this relaxation behaviour.

Although the Cole and HBS problems are not the same (at the very least the

Cole problem is non-axisymmetric) the amplitude equations studied by Cole, (C.3)

and (C.4), have a lot in common with those studied by HBS (1.1); in so much as

many of the features present in Cole’s model also play a part in HBS’s (e.g. phase

mixing, nonlinear forcing and diffusion).

We therefore extend the HBS model (1.1) as follows

∂a

∂t=

[

λ − Υ2

εx2 + ix

]

a +∂Θ

∂xa +

∂2a

∂x2, (1.2a)

1

κT

∂Θ

∂t=

∂

∂x|a|2 +

∂2Θ

∂x2. (1.2b)


The terms here are the same as in (1.1) save for the addition of Θ and κT which can be

thought of as the mean temperature profile and the thermal diffusivity respectively.

These equations will be motivated further in Chapter 8 but at this point we only

highlight the fact that we can easily recover the HBS equations by taking the infinite

κT limit.

It is the study of this extended HBS model (1.2) that comprises Part II of this

work. However it is important to be clear here that we do not solve the thermal

convection problem in this thesis, neither do we attempt to improve upon Cole’s

analysis. Instead we extend the HBS equations and attempt to capture some of the

features of Cole’s amplitude equations with the hope of reproducing this interesting

relaxation oscillation behaviour in our, relatively, much simpler system.

1.4 Outline of this work

In Part I of this thesis we extend the narrow-gap problem, outlined above in Section

§1.2, with a view to building upon the ideas introduced by BS [21] and HBS [17].

In Chapter 2 we expand upon the details given in Section §1.2 by reviewing

the ideas and developments made by other researchers working in this area. In

doing so we highlight the steps taken in [16], and prior publications, to derive the

complex Ginzburg-Landau equation governing the amplitude modulation of Taylor-

vortex type instabilities (1.1). We then set up the HBS co-rotation model and the

BS pulse-train model in non-dimensional units and briefly describe their findings

before ending the chapter with a summary of previous work on narrow-gap spherical

Couette flow.

Chapter 3 is concerned with drawing comparisons between the results of HBS

and BS and extending their ideas. New solutions of the HBS equations are sought

to provide a link between the two models and the methodology used to identify

these solutions is documented. We end the chapter by exploring the possibility of

extending the work of BS.

In Chapter 4 we explore the symmetries of the BS–model equations and develop a


general framework for identifying and classifying pulse-train solutions. New periodic,

pulse-train solutions found within this framework are reported in Chapter 6 whilst

the methodology used to obtain them is detailed in Chapter 5 along with a brief

outline of our tests for stability and robustness. We end Part I with some summary

remarks in Chapter 7.

In Part II of this thesis we study the extended HBS–model (1.2) and try to

reproduce some of the interesting relaxation oscillation behaviour identified by Cole

[28] in this much simpler, model system.

In Chapter 8 we further motivate the extended HBS equations (1.2) and recreate

some of the HBS solutions found in [17] by considering the infinite κT limit. We

investigate the behaviour of (1.2) when the parameter κT is small and identify chaotic

relaxation oscillations in this regime.

In Chapter 9 we study the behaviour of the system as the parameter κT is reduced

from infinity down into the relaxation regime for one particular set of parameter

values taken from the work of HBS. We document the bifurcation sequence taken

by the solution as it transitions from the regular HBS solution with large κT , to the

chaotic relaxation oscillation solution, at small κT . In doing so we identify some

interesting multi-frequency quasi-periodic behaviour which we investigate further in

Chapter 10 with a frequency power spectra analysis.

Finally, we summarise the findings of Part II with some concluding comments in

Chapter 11.

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