The Structure and Stability of anIdealised Hurricane

Laura Merchant110009528

School of Mathematics and StatisticsUniversity of St Andrews

This dissertation is submitted for the degree ofMaster of Physics: Mathematics and Theoretical Physics

April 2015

Declaration

I certify that this project report has been written by me, is a record of work carried out byme, and is essentially different from work undertaken for any other purpose or assessment.

Laura Merchant110009528April 2015

Acknowledgements

I would like to thank Professor David Dritschel for all of the helpful discussions andencouragement he provided throughout the project. I am also grateful to The RobertsonScholarship Trust for the financial aid they have given me over the past four years.

Abstract

The quasi-geostrophic shallow water equations are used to examine the stability and nonlinearevolution of an idealised hurricane. The model consists of a simple, axisymmetric annularvortex with a predefined potential vorticity distribution. We present an alternative analysis toFlierl’s (1988) work for a multi-layered annular vortex. Barotropic and baroclinic instabilitiesare found to exist for thin annuli. These instabilities are either enhanced or diminisheddepending on the choice of potential vorticity within the core, p. We extend the analysis toinclude the full nonlinear evolution. Through this, we discern the five break down patternsthat arise from the variation of p, the Rossby deformation length, LD and the choice ofinner radius, a. For two layers, wherein the density, ρ, is assumed to be constant in eachlayer, the linear stability analysis is found to be identical to that in the single-layered case.This analysis is used to investigate the parameters that cause baroclinic (vertically-varying)instabilities to dominate barotropic (height-independent) instabilities. Based on constructedphase diagrams, which illustrate two competing regimes (baroclinic dominant and barotropicdominant), the nonlinear evolution is then examined for select p, a, α, h1 and LD. Throughthe numerical simulations, we see that the annular structure, which breaks down into anasymmetric vortex or completely disintegrates, remains predominantly barotropic with somebaroclinic tendencies.

Table of contents

List of figures xi

1 Introduction 1

2 Mathematical Preamble 72.1 Shallow Water Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1.1 Quasi-Geostrophic Model . . . . . . . . . . . . . . . . . . . . . . . . . 82.2 Numerical Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3 Single-Layered Rotating Annulus 153.1 Infinite LD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.1.1 Linear Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.1.2 Nonlinear Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.2 Finite LD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.2.1 Linear Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.2.2 Nonlinear Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4 Two-Layered Rotating Annulus 314.1 Linear Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314.2 Nonlinear Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

5 Concluding Remarks 35

References 37

List of figures

1.1 Advancements in weather imaging . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Hurricane Katrina . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Hurricane cross section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.4 Models of tropical cyclone evolution . . . . . . . . . . . . . . . . . . . . . . . 4

2.1 Shallow water model: single layer . . . . . . . . . . . . . . . . . . . . . . . . 92.2 Shallow water model: two layers . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.1 Diagram of single-layered rotating annulus . . . . . . . . . . . . . . . . . . . 153.2 Infinite LD: stability graphs for p = 0 . . . . . . . . . . . . . . . . . . . . . . 193.3 Effect of varying p on σi for infinite LD and m = 4 . . . . . . . . . . . . . . 203.4 Numerical simulations: various values of p and infinite LD . . . . . . . . . . 213.5 Comparison of uθ for infinite and finite LD . . . . . . . . . . . . . . . . . . . 223.6 Finite LD: stability graphs for p = 0 and LD = 0.1 . . . . . . . . . . . . . . 243.7 Finite LD: stability graphs for p = 0 and LD = 1 . . . . . . . . . . . . . . . 253.8 Finite LD: stability graphs for p = 0 and LD = 10 . . . . . . . . . . . . . . . 253.9 Comparison of the full analytical and asymptotic solution for m = 2 . . . . . 263.10 Effect of varying p on σi for finite LD and m = 4 . . . . . . . . . . . . . . . . 273.11 Numerical simulations: various values of p and finite LD . . . . . . . . . . . 28

4.1 Diagram of two-layered rotating annulus . . . . . . . . . . . . . . . . . . . . 314.2 Phase diagram for LD = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324.3 Numerical simulations: various values of p, a, α, LD = 1 and h1 = 0.5 . . . . 33

Chapter 1

Introduction

Throughout time, mankind has sought to understand physical processes that occur withinNature. One field of considerable interest is the field of atmospheric sciences. This is afield that encompasses the study of the atmosphere, the processes within it, and how othersystems interact with it. In particular, meteorology (the specialisation of weather eventsand forecasting) has become one of the most well-studied subjects due to the vast economicand social benefits that arise from it (Freebairn & Zillman, 2002; Frei, 2010).

(a) (b)

Figure 1.1: (a) Sir Francis Galton’s first weather map published April 1st 1875 based on data from theprevious day (image courtesy: Birmingham Museum, taken from wikipedia) (b) A recent satellite image ofEurope (image courtesy: RMI). A comparison of the two images shows how far meteorology has come withthe advance of technology.

From antiquity to the modern age, the field of meteorology has grown from a collectionof hypotheses written by Aristotle in his ‘Meteorologica’ to something capable of predictingweather on a daily basis. The beginnings of meteorology - largely accredited to the ancientGreeks (Bowker, 2011) - involved observations of clouds, winds, rain, and other weatherevents. It was understood that these were linked in some way. Within ‘Meteorologica’,Aristotle discussed theories on cloud formation; the properties of tornadoes; hurricanes andlightning and, more generally, the earth sciences. Although it was Aristotle that coinedthe phrase “meteorology”, many other philosophers before him (Thales, Democritus andHippocrates) were classed as meteorologists due to their work on various atmosphericphenomena such as the water cycle; weather predication, and ‘Airs, Waters and Places’.

2 Introduction

At the time, these observations and hypotheses were considered to be the commandingauthority (Ahrens, 2006) within the sciences and were used consistently through to theRenaissance era - until the advancement of observational instruments. The most notableinventions were the Hygrometer (Cusa, 1450), Thermometer (Galilei, 1593) and the Barom-eter (Torricelli, 1643). Although primitive in their early stages, the creation of these shouldbe classed as pivotal moments in meteorological history as, without them, our fundamentalunderstanding of the processes in the atmosphere would have been left to the speculation ofnatural philosophers.

Before these developments, other avenues of research were conducted. This consisted ofthe collaborative efforts of different observers across the world; the telegraph was used totransmit data between them, and this allowed rudimentary weather maps to be constructed(see Fig. 1.1a). The next step from this point was the prediction of the weather. During the1920s, Lewis Fry Richardson created the first numerical simulation to predict the weather. Itwas a long, arduous process: it took 6 weeks to complete a 6-hour forecast, which was basedon previous weather data and ended with an unrealistic result. The theory was ahead of theavailable technology. However, in this case, there was a short turnover period and within 30years the first computers were built and able to carry out numerical simulations. In 1950,a group at Princeton University carried out the first successful weather prediction whichresulted in a 24 hour forecast (taking 24 hours to complete using data from the previousday) (Charney et al., 1950). This was a remarkable advancement as it meant that, withfaster computers, the weather could be predicted more frequently, which is evident in thesatellite and radar images we see today (see Fig. 1.1b).

(a) (b)

Figure 1.2: Hurricane Katrina, the third deadliest hurricane in US history. (a) A satellite image of Katrinaat peak strength (image courtesy: MODIS, NASA). (b) The track of Hurricane Katrina through its lifetime:progression from tropical depression to category 5 Hurricane (image courtesy: ESL, Coastal Studies Institute,Louisiana State University).

3

There is one enigma that has been documented throughout history that still bafflesmany researchers today: the tropical cyclone. The difficulty in studying this phenomenonboils down to the inability to recreate the required conditions that form the cyclone in alaboratory setting. In addition, it is difficult to numerically model all physical processesthat are inherent in the cyclone (Emanuel, 1991). With all of these issues, why is there suchinvested interest in the study of these deadly vortices? Every year, tropical cyclones (seeFig. 1.2a) cause billions of dollars worth of damage as well as significant loss of life. Hence,more insight into how tropical cyclones develop and dissipate could lead to more advancedmethods of detecting, preventing and tracking cyclones, which would ultimately help reducethe devastation that cyclones cause in tropical regions and beyond.

Figure 1.3: The cross section of a hurricane (a type of tropical cyclone). The eye is a near singular corewith little chaotic motion (calm weather). Rainbands are formed around the eye by the inwards spiralling ofair. Image courtesy: the COMET program.

How do tropical cyclones form? Within tropical regions, at the air-water interface,the air is particularly warm. So, as a tropical storm passes over, the hot air is draggedupwards leaving a region of low pressure at the water’s surface. As the air rises, convergingwinds move in to replace it and eventually this hot air cools and condenses, releasing heat.A cyclical process evolves from the rising and sinking of hot and cold air so effectively,the tropical cyclone acts like a heat engine (see Fig.1.3). It has been shown that even iffavourable conditions (warm ocean waters; atmospheric instability; low vertical wind shear,and a few others (Gray, 1968, 1998)) are present, the cyclone may not form: these condi-tions are not necessarily sufficient in explaining the formation of a cyclone (Majumdar, 2003).

One (of the many) notable achievements in meteorological history was the formation ofthe Norwegian cyclone model (Bjerknes, 1919). This describes the structure and evolutionof extratropical cyclones over continental landmasses. This model is largely considered thefoundation of modern meteorological analysis (Schultz & Vaughan, 2011), and is included inmany introductory meteorology texts. However, there are some pitfalls within this model andit fails to accurately describe oceanic midlatitude cyclones, meaning a different model wasrequired. In the 1990s, the Shapiro-Keyser model (Shapiro & Keyser, 1990) was formulatedand is based on oceanic cyclones. There are several differences between the models: the

4 Introduction

overall evolution (see Fig. 1.4); what sort of front is formed in each model, and how thecyclone is orientated. Both models have their merit but one question remains unanswered:what determines which model a given cyclone adheres to? The two competing theories arethat either surface friction or the embedded large-scale flow determines the evolution andstructure of cyclones. There has been plenty of work on both avenues but there still appearsto be some disparity. (Schultz & Zhang, 2007)

Figure 1.4: Models of tropical cyclone evolution: a comparison of the different front structures. (a) NorwegianCyclone Model: (i) incipent frontal wave with cold (triangles) and hot (circles) fronts, (ii) and (iii) narrowingwarm sector due to faster spin of cold front, (iv) mature cyclone with occluded front. A Norwegian cycloneis typically orientated north-south with a more intense cold front. (b) Shapiro-Keyser Model: (i) incipientcyclone developes cold and warm fronts, (ii) cold front moves perpendicular to the hot front - the frontsnever meet resulting in frontal T-bone. (iii) frontal fracture leads to a back-bent front, (iv) warm seclusiondue to cold air encircling warmer air near the low centre. A Shapiro-Keyser cyclone will be elongatedeast-west along the strong warm front. Adapted from Schultz et al. (1998).

With the advent of the computer in the 1950s, the ability to forecast the weatherbecame part of everyday life and many researchers began applying the same numericalmethods to tropical cyclones in attempts to predict their lifetime as well as to track them(Ooyama, 1969). However, this proved to be an arduous task due to the physical processes(convection, boundary layers, rotation, stratification and the air-sea interaction) involved incyclone formation (Emanuel, 1991). As a result there has yet to be a complete model fortropical cyclones. Serious work began on numerical modelling of cyclones towards the endof the 1960s with both axisymmetric (Ooyama, 1969; Yamasaki, 1968a,b,c) and asymmetric(Anthes et al., 1975a,b) models being developed. It could be argued that tropical cyclonestructures are not purely axisymmetric. However, to understand the fundamental dynamics,this simplification is a very good approximation of the problem (Holton, 2004). In fact,many of the initial models were axisymmetric with limited vertical resolution and yet theywere able to, albeit simply, portray moist convection involved in tropical cyclone formation(Zhu et al., 2001). As computing power increased, models improved to the point that weare now able to run high resolution simulations that include complex representations of thephysical processes. Despite this, simple models are still used today to develop our basicunderstanding of cyclones (Mai et al., 2002) as can be readily shown by a comparison ofOoyama’s (1969) and Emanuel’s (1989; 1995) model. The representation of moist convectionbecame more sophisticated as our understanding evolved (Zhu et al., 2001).

For the rest of this thesis, a simple 2D model will be used to investigate the structureand stability of an idealised hurricane (where hurricanes are tropical cyclones located in aspecific region, such as the North Atlantic ocean). In chapter 2, the underlying mathematicsof vortex dynamics will be explored, this being the simplification of the shallow waterequations under quasi-geostrophic theory and the layer approximation. Thereafter, twomodels that sequentially build up from one another will be examined. Chapter 3 considers

5

a single-layered rotating annulus (no forcing), and chapter 4 looks at the second model, atwo-layered rotating annulus (no forcing). The final chapter (chapter 5), will remark onboth models as well as possible extensions.

Chapter 2

Mathematical Preamble

2.1 Shallow Water Model

As with most fields in Applied Mathematics and Physics, there exists a set of governing equa-tions (Vallis, 2006) that aptly describe the dynamics of fluid flows through the conservationof momentum (2.1), mass (2.2) and energy (2.3) of the system:

DuDt

+ f × u = −∇pρ

− ∇Φ + F (2.1)

∂ρ

∂t+ ∇ · (ρu) = 0 (2.2)

D

Dt

p

ργ

= −L (2.3)

with Dq

Dt= ∂q

∂t+ (u · ∇)q

where u = (u, v, w) is the velocity, p is the pressure, ρ is the density, γ is the ratio ofspecific heats and L is the total energy loss function. Equation (2.1) describes the actingforces: Coriolis force (f × u, f = 2Ω), pressure gradients (−∇p/ρ), gravitational forceincluding Newtonian gravity and centrifugal force (−∇Φ) and the final term (F), whichencompasses all other forces (viscosity, friction). In our problem, we restrict our discussionto an incompressible, inviscid and frictionless fluid with constant Coriolis frequency, f .

Our aim is to investigate a simple axisymmetric hurricane model and, as such, we lookat one of the simplest geophysical fluid models: the shallow water model. The rotatingshallow water equations (SWE) (Vallis, 2006) are derived by applying (i) the hydrostaticapproximation, dp/dz+ ρg = 0 and (ii) the long wave approximation, h/L ≪ 1 to equations(2.1)-(2.3) and have the form:

Du

Dt− fv = −g∂h

∂x(2.4)

Dv

Dt+ fu = −g∂h

∂y(2.5)

∂h

∂t+ ∇ · (hu) = 0 (2.6)

where h, u and v depend only on x, y and t. As it stands, we want to go one step furtherand further simplify our analysis. We apply another approximation to the SWE: thequasi-geostrophic approximation.

8 Mathematical Preamble

2.1.1 Quasi-Geostrophic ModelQuasi-geostrophic (QG) theory, which was first devised by Charney (1948), is one of thesimplest methods used to look at the synoptic scale motion in meteorology (Holton, 2004).It exploits the fact that these motions are in near-geostrophic balance, thus allowing one toretain the associated time evolution that would otherwise be omitted in a pure geostrophicflow (Warneford & Dellar, 2013). Due to this, QG is particularly relevant when dealing withnumerical simulations because it reduces the dynamical degrees of freedom involved andultimately cuts the computational expense (Williams et al., 2010). As we want to look atthe overall stability of a hurricane, it would be advisable to use a model that is easy to solveanalytically, as well as one that can explore the full nonlinear evolution. The QG modelmeets these criteria.

We apply several approximations to the primitive equations (in this case, the SWE) toderive the QG equations:

i. The Rossby number (Ro = U/fL) is small which enforces near-geostrophic balance.

ii. In the shallow water regime, the variations in the layer depth are assumed to be small(O(Ro)) compared to the total depth.

iii. Variations in the Coriolis parameter are small.

iv. The time scale is given by the advection term in Eq. (2.1): T = L/U .

Here we have used the typical characteristic scales, U and L for horizontal velocity andlength. A point of note is the typical value of the Rossby number for hurricanes. In someinstances, the Rossby number can become comparable to, or larger, than one. Yet QGtheory can still be a valid approximation. For example, consider hurricane Katrina (Fig.1.2a), which had maximum wind speeds of U = 77ms−1 and a horizontal scale of L ≈ 996kmthus meaning Ro ≈ 0.7. Although this is comparable to 1, QG theory is still applicable aswe want to look at a qualitative view of a hurricane to ascertain its structure and stability(Tsang & Dritschel, 2014).

In the following sections, we will discuss the derivations for both the single and twolayered shallow water equations and what these equations become in the QG regime. Thederivations will be brief and the interested reader can find the full version, including thescale analysis, non-dimensionalisation and algebra, in Vallis (2006).

2.1 Shallow Water Model 9

SWE: Single Layer

Figure 2.1: An illustration of the single layer shallow water model.

The single-layered quasi-geostrophic shallow water equations (SLQGSWE) are derivedby nondimensionalising the SWE (taking all quantities such that u = Uu and dividing bythe dominant scale) and expressing the result in terms of the small Rossby number,

RoDu

Dt− v = −∂η

∂x(2.7)

RoDv

Dt+ u = −∂η

∂y(2.8)

ϵDη

Dt+ (1 + ϵη)∇ · u = 0 (2.9)

where ∼ denotes the nondimensional quantities and ϵ ≪ 1 is the typical scale of the freesurface variation η. Assuming that ϵ = O(Ro), we expand u, v and η in powers of Rossbynumber. Looking at the leading order, the above reduces to geostrophic balance with theconsequence of incompressibility. However, this means we have an insufficient number ofequations to solve our problem and are subsequently motivated to look at a higher orderRossby number (namely the first order). At this order, equations (2.7)-(2.9) become

D0u0

Dt− v1 = −∂η1

∂x(2.10)

D0v0

Dt+ u1 = −∂η1

∂y(2.11)

Bu−1D0η0

Dt+ ∇ · u1 = 0 (2.12)

where Bu−1 = gH/(fL)2 is the dimensionless constant known as Burger’s number and D0/Dtis the 2D convective time derivative with u0 = (u0, v0). The final step in the derivation isto combine (2.10)-(2.12) to obtain a set of equations that describe the potential vorticity(PV) inversion problem. This is a method of inverting the PV to define a streamfunctionsuch that we can calculate the velocity field and all other fields (pressure etc.). Hence,rearranging (2.10) and (2.11) for v1 and u1 respectively and substituting these equationsinto (2.12), the nondimensional form of the PV inversion equations are found

D0Q0

Dt= 0, Q0 = ζ0 −Bu−1η0 (2.13)

10 Mathematical Preamble

where ζ0 defines the leading order vertical vorticity. Restoring the dimensions of Eq. (2.13)and using geostrophic balance (ψ = gη/f), we find

DQ

Dt= 0, Q = ζ − f

Hη (2.14)

= ∇2ψ − 1L2

D

ψ

where LD =√gH/f is the Rossby deformation length and describes the scale at which

rotational effects become comparable to buoyancy or gravity wave effects (Gills, 1982). Withthe derivation complete, the SLQGSWE can be summarised as follows,

Single-Layered Quasi-Geostrophic Shallow Water Equations (SLQGSWE)

DQ

Dt= 0

Q = ∇2ψ − 1L2

D

ψ

u = −∂ψ

∂yv = ∂ψ

∂x

SWE: Two Layers

Figure 2.2: An illustration of the two-layered shallow water model.

Figure (2.2) describes the two-layered approach needed for the SWE and will be used asreference to derive the equations for this model. The SWE are nearly identical to those usedabove but now we have to consider the densities (ρj with j = 1, 2) in each layer,

Duj

Dt− fvj = − 1

ρj

∂pj

∂x(2.15)

Dvj

Dt+ fuj = − 1

ρj

∂pj

∂y(2.16)

2.1 Shallow Water Model 11

To derive the explicit equations needed to investigate this model, PV is defined in its fulldimensional form,

Dqj

Dt= 0, q1 = ζ1 + f

H1 + η(2.17)

q2 = ζ2 + f

H2 − η(2.18)

where the free surface variation, η, is small compared with the layer depths Hj (i.e. η/Hj ≪1). It can then be shown by expanding the denominator of the above and using the smallRossby number approximation (i.e. applying the QG approximation), the PV reduces to

q1 = f

H1+ ∇2ψ1

H1− fη

H21

(2.19)

q2 = f

H2+ ∇2ψ2

H2+ fη

H22

(2.20)

where ζj = ∇2ψj and ∇2ψj ≪ f . Continuing with the derivation, we need to calculate anexplicit form of the layer depth variation in terms of the streamfunction, ψj. Initially, wehave to obtain an expression for the layer-dependent pressure which can be achieved byintegrating hydrostatic balance up through the layers.

z ≤ H1 + η : p = ps − ρ1gz

z ≥ H1 + η : p = ps − ρ1g(H1 + η) − ρ2gz

From this, we use the velocity field in each layer to define two streamfunctions, ψ1 andψ2. Considering geostrophic balance (equivalent to dropping the advection term in (2.15)and (2.16)), we obtain a general set of equations for the velocity field: uj = −(fρj)−1∂p/∂yand vj = (fρj)−1∂p/∂x. Applying the above definitions of p for each layer, we obtain thefollowing,

Lower layer : u1 = − 1ρ1f

∂ps

∂yv1 = 1

ρ1f

∂ps

∂x(2.21)

Upper layer : u2 = − 1ρ2f

∂ps

∂y+ ρ1

ρ2

g

f

∂η

∂yv2 = 1

ρ2f

∂ps

∂x− ρ1

ρ2

g

f

∂η

∂x(2.22)

In general, the velocity field can be expressed in terms of streamfunctions, ψj such thatuj = −∂ψj/∂y and vj = ∂ψj/∂x and so comparing this with Eqs. (2.21) and (2.22) wearrive at the solutions for ψ1,2:

ψ1 = ps

ρ1fψ2 = ps

ρ2f− ρ1

ρ2

gη

f(2.23)

Finally, the expression for η is obtained by eliminating ps in the streamfunction equations,

η = f

ρ1ψ1 − ρ2ψ2

ρ1g

(2.24)

Having obtained all the necessary equations, the derivation for the two-layered quasi-geostrophic shallow water equations (2LQGSWE) can be completed. We redefine PV such

12 Mathematical Preamble

that Qj ≡ Hj(qj − f/Hj), linearise qj, and substitute in the explicit form of η and qj foreach layer:

Q1 = ∇2ψ1 − fη

H1= ∇2ψ1 − f 2

H1

ρ1ψ1 − ρ2ψ2

ρ1g

(2.25)

Q2 = ∇2ψ2 + fη

H2= ∇2ψ2 + f 2

H2

ρ1ψ1 − ρ2ψ2

ρ1g

(2.26)

If we were to consider how to solve this problem in the single-layered case, we would havebeen able to invert the Poisson equation to obtain the streamfunction. However, here wehave a coupling between the two layers (as evident from the presence of both streamfunctionsin (2.25) and (2.26)), meaning we have to decouple the equations. The simplest method ofdoing this is to define ψ and Q for the barotropic and baroclinic modes.

For the barotropic (vertically-averaged) mode, we choose ψt = (H1ψ1 +H2ψ2)/(H1 +H2)and Qt = (H1Q1 +H2Q2)/(H1 +H2) such that, after substituting Q1 and Q2 into Qt, weobtain the following Poisson equation

∇2ψt = Qt (2.27)

Similarly, we define a new ψ and Q for the baroclinic (orthogonal, vertically-varying) case:ψc = (ρ2H2 − ρ1H1)/ρ1 and Qc = (ρ2Q2 − ρ1Q1)/ρ1 and substitute in our expressions forQ1 and Q2:

Qc = ∇2ψc − f 2

gH2

ρ2

ρ1+ H2

H1

ψc (2.28)

If we let 1/L2D ≡ f 2/gH2(ρ2/ρ1 +H2/H1), our two layer equations become comparable to

that of the single layer model (c.f. (2.14)), which means we can solve the equations in bothmodels with the same method. However, our set of equations for the two layer model is stillincomplete as two equations are needed for ψ1 and ψ2. These expressions are easily foundby solving simultaneously ψt and ψc. Before this, let us simplify the notation for these twoterms:

ψt = H1ψ1 +H2ψ2

H1 +H2= h1ψ1 + h2ψ2 (2.29)

ψc = ρ2ψ2 − ρ1ψ1

ρ1= αψ2 − ψ1 (2.30)

where we have defined hj = Hj/(H1 +H2) such that h1 + h2 = 1 and α = ρ2/ρ1. We thencombine equations (2.29) and (2.30) to obtain,

ψ1 = αψt − h2ψc

αh1 + h2(2.31)

ψ2 = ψt + h1ψc

αh1 + h2(2.32)

The final model is summarised below.

2.2 Numerical Method 13

Two-Layered Quasi-Geostrophic Shallow Water Equations (2LQGSWE)

DQj

Dt= 0 j = 1, 2

Qt = h1Q1 + h2Q2 ∇2ψt = Qt

Qc = αQ2 −Q1 ∇2ψc − 1L2

D

ψc = Qc

ψ1 = αψt − h2ψc

αh1 + h2uj = −∂ψj

∂y

ψ2 = ψt + h1ψc

αh1 + h2vj = ∂ψj

∂x

2.2 Numerical MethodIn addition to the analytical work used to determine the linear stability of our idealisedhurricane, we aim to look at the full nonlinear evolution of our rotating annulus configura-tions (see Figs. 3.1 and 4.1). Initially, we must solve the inversion problem, illustrated inthe SLQGSWE and 2LQGSWE, given a PV distribution and then advect the solution tothe next time step. This is a near impossible task to do by hand and the challenge is madeeasier by using a pre-existing numerical model to perform the calculations.

One method used to numerically solve the inversion and advection problem for vortexpatches (much like our idealised hurricane) is contour dynamics (CD) (Zabusky et al., 1979).The algorithm uses the inviscid, incompressible 2D PV equations (c.f. SLGQSWE and2LQGSWE) to calculate the velocity fields directly. PV is assumed to be piecewise constantwithin the contours and the streamfunction is solved in terms of Green’s functions, whichthen gets converted into an equation for the velocity field. However, there are several pitfallswhen using CD and one main cause for concern is the cost of computing. Dritschel (1988b)formulated an extension of CD known as contour surgery (CS) which solves the expenseproblem. CS removes vorticity features that are smaller than a predefined scale (say δ) aswell as allowing contours to merge (divide) depending on whether we approach (go below) δ.This effectively filters out the small scale motions that are very computationally expensivebut play a negligible role.

The explicit details of both CD and CS as well as a review of the methods can be foundin papers by Zabusky et al. (1979), Dritschel (1988b, 1989) and Pullin (1992).

Chapter 3

Single-Layered Rotating Annulus

In the 1980’s, several studies were conducted on the linear stability of (i) axisymmetric,piecewise constant potential vorticity patches (Dritschel, 1986; Flierl, 1988; Helfrich & Send,1988) and (ii) continuous vorticity distributions (Gent & McWilliams, 1986; Ikeda, 1981;McWilliams et al., 1986). Figure 3.1 illustrates the hurricane structure (single-layered,axisymmetric rotating annulus) that will be discussed in this chapter. It closely resemblesthe structure in Flierl’s (1988) paper. However, there are subtle differences between them.In Flierl’s work, the inner radius (a in Fig. 3.1) is held fixed (a = 1) and the outer radius,b, is varied 0 ≤ b ≤ 5 whereas we consider b = 1 and 0 ≤ a ≤ 1. We aim to extend thiswork so it includes the full nonlinear evolution of the rotating annulus through the use ofnumerical simulations.

Figure 3.1: A schematic illustrating the structure of the single-layered axisymmetric vortex. A top downview is also shown to indicate the exact distribution of the potential vorticity.

Firstly, let us discuss the general method for analysing the linear stability of our rotatingannulus. Consider, initially, the basic state Q in polar coordinates with the followingconfiguration (Fig. 3.1),

Q =

p r < a1 a ≤ r ≤ 10 r > 1

(3.1)

where we vary −1 ≤ p ≤ 1 and 0 ≤ a ≤ 1. A small perturbation (primed quantities) isapplied to the basic state (bar quantities) such that:

Q = Q+Q′

ψ = ψ + ψ′

16 Single-Layered Rotating Annulus

where we take Q′ = 0 everywhere, and instead perturb the vortex boundaries at r = a and 1.After linearising, the SLQGSWE reduce to the following:

∇2ψ − 1L2

D

ψ = Q (3.2)

∇2ψ′ − 1L2

D

ψ′ = 0. (3.3)

Equation (3.2) describes the basic state of the system and is used to calculate uθ (= ∂ψ/∂r)through integration and matching the solutions at the boundaries (r = a, r = 1) (see below).Since Q and ψ are independent of θ, the basic state radial velocity, ur (= r−1∂ψ/∂θ), is zero.Equation (3.3) is the second key equation for the stability analysis as it is the starting pointin our derivation of the eigenvalue problem for σ. Later in this chapter, the derivation of thefunctional form of ψ′ for two cases of LD (finite and infinite) will be shown explicitly withthe latter used to reproduce results published by Dritschel (1986) and as an introduction tothe analytical method.

3.1 Infinite LD

3.1.1 Linear StabilityFirst, we must obtain the explicit form of uθ from (3.2) using LD = ∞. In this limit, thepotential vorticity simplifies to the vertical vorticity, ζ, and so here we have to solve

ζ = 1r

d

dr

rdψdr

= 1r

d

dr(ruθ)

in each region and match the solution at the boundaries. The final result for the tangentialvelocity is then given by

uθ =

12pr r < a

12r (p− 1)a2 + 1

2r a ≤ r ≤ 1

12r (p− 1)a2 + 1

2r r > 1.

(3.4)

Equation (3.3) reduces to the typical Laplace equation (with LD = ∞) ∇2ψ′ = 0, which in2D polars is written as,

1r

∂

∂r

r∂ψ′

∂r

+ 1r2∂2ψ′

∂θ2 = 0. (3.5)

We then consider ‘plane-wave’ solutions of the following form, ψ′ = ℜ(ψ(r)ei(mθ−σt)). The(complex) amplitudes must be taken as a function of r due to the non-constant coefficients

3.1 Infinite LD 17

in equation (3.5). The θ component of the above can be shown explicitly by consideringthe separation of variables method (i.e. ψ′ = H(r)G(θ)) on equation (3.5) and enforcingperiodic solutions (θ corresponds to an angle and therefore G(θ) = G(θ + 2π). Plugging the‘plane-wave’ solution into Eq. (3.5), we arrive at a purely radial equation for ψ,

r2d2ψ

dr2 + rdψ

dr−m2ψ = 0. (3.6)

From inspection, the solutions are of the form ψ(r) ∝ rm, r−m but the Laplacian operatoris a linear operator meaning a complete solution would be a superposition of the two,

ψ(r) = Airm +Bir

−m i = 1, .., 3 (3.7)

for the three regions. Using the radial solution, (3.7), the equations are matched in eachregion at the interface boundaries to obtain two equations relating the coefficients. Tosimplify the matching procedure, we apply the condition that ψ is bounded: r → 0 (whenr < a) and r → ∞ (when r > 1). This means that B1 and A3 are both zero:

r < a ψ(r) = A1rm

a ≤ r ≤ 1 ψ(r) = A2rm +B2r

−m

r > 1 ψ(r) = B3r−m.

(3.8)

Continuing with our analysis, the solutions are matched at r = a and r = 1 resulting in

B2 = a2m(A1 − A2) (3.9)B3 = A2 + a2m(A1 − A2). (3.10)

The tangential velocity is then analysed at the interface boundaries by applying a smallperturbation to the radii of the annulus,

r = r + ηk(θ, t) k = 1, 2 (3.11)

where we take ηk = ℜ(ηkei(mθ−σt)) and r = 1, a. Taylor expanding uθ(r, θ, t) at the above

radii and substituting in uθ(r, θ, t) = uθ(r) + ∂

∂rψ′(r, θ, t) leads to,

uθ(r + ηk, θ, t) = uθ(r) + ℜ

dψdr

(r)ei(mθ−σt)

+ duθ

dr(a)ηk. (3.12)

Looking at (3.12) above (+) and below (-) the boundaries, we plug in (3.4), (3.8), ηk foreach region and equate the solutions:

r = a:

uθ(a− + η1, θ, t) = 12pa+ ℜ[mA1a

m−1ei(mθ−σt)] + 12pℜ[η1e

i(mθ−σt)]

uθ(a+ + η1, θ, t) = 12pa+ ℜ[(mA2a

m−1 −mB2a−m−1)ei(mθ−σt)] + (1 − 1

2p)ℜ[η1ei(mθ−σt)]

⇒ (1 − p)η1 = m((A1 − A2)am−1 +B2a−m−1). (3.13)

18 Single-Layered Rotating Annulus

r = 1:

uθ(1− + η2, θ, t) = 12(1 + a2(p− 1)) + ℜ[m(A2 −B2)ei(mθ−σt)] + 1

2(1 + a2(1 − p))ℜ[η2ei(mθ−σt)]

uθ(1+ + η2, θ, t) = 12(1 + a2(p− 1)) + ℜ[−mB3e

i(mθ−σt)] − 12(1 − a2(1 − p))ℜ[η2e

i(mθ−σt)]

⇒ η2 = m(B2 − A2 −B3). (3.14)

There are still two equations needed before the results can be combined together to obtain ananalytical solution for the frequency σ. Exploiting the material character of the boundariesand Taylor expanding the RHS,

Dηk

Dt= ur(r + ηk, θ, t) .= ur(r, θ, t) (3.15)

we obtain the last two equations by substituting the radii: (3.11), and ηk into (3.15),

−iση1 + 12ipmη1 = −imA1a

m−1 (3.16)

−iση2 + 12im(1 + a2(p− 1))η2 = −imB3. (3.17)

The final step is to combine equations (3.9),(3.10),(3.13),(3.14),(3.16) and (3.17). We startby substituting (3.9) and (3.10) into (3.13) and (3.14) and rearrange to give equations forA1 and A2. These are then substituted into (3.16) and (3.17) and after some algebra, wearrive at an eigenvalue problem for σ,(

A11 A12A21 A22

)(η1η2

)= σ

(η1η2

)(3.18)

where A11 = 12(pm+ (1 − p))

A12 = −12a

m−1

A21 = 12(m(1 + a2(p− 1)) − 1)

A22 = 12(1 − p)am+1

Due to the form of (3.18), σ can be obtained by solving the characteristic polynomial|(A−σI)| = 0. However, due to the number of parameters involved in (3.18), the polynomialmust be solved numerically. Once we obtain an equation for σ, the linear stability of theannulus can be analysed.

Returning to the original aim of infinite LD case (reproduction of documented results),we simplify the parameters by taking p = 0 to investigate a rotating annulus with zero PVcore. σ is then given by

σ1,2(m, a) = 14m(1 − a2) ± 1

2

√(1 − 1

2m(1 − a2))2 − a2m (3.19)

3.1 Infinite LD 19

0.0 0.2 0.4 0.6 0.8 1.0a

0.0

0.5

1.0

1.5

2.0

2.5σr

(a)

0.0 0.2 0.4 0.6 0.8 1.0a

−0.20

−0.15

−0.10

−0.05

0.00

0.05

0.10

0.15

0.20

σi

(b)

Figure 3.2: The linear stability of an annulus with p = 0 for: m = 2, m = 3, m = 4, m = 5, m = 6. Theplots show (a) the real frequencies, σr, and (b) the growth and decay rates, σi vs. the inner radius of theannulus, a. In this instance, the instabilities arise for annuli with a > 0.5.

which is as expected (see Dritschel (1986)). Figure 3.2 shows the variation of the growthrate σi = ℑ(σ) and the frequency σr = ℜ(σ) with changing a for azimuthal wave modesm = 2, ..., 6. From this, we can see that m = 2 contains no instabilities (i.e. no growthor decay) for p = 0 (which is true for all p > 0) and for all values of a. If we considerwhat happens to the other modes, they all become linearly stable, for all a, when we takep > 0.78. All modes will be linearly unstable for a range of a when we take p < 0 and asp → −1, the instabilities grow in strength. To illustrate the effect changing p will haveon σi, we consider the m = 4 wave mode (see Fig. 3.3). As we move from negative topositive values of p, the instabilities become weaker, eventually leading to complete lin-ear stability. This is not only true form = 4: the general trend can be seen in all wave modes.

With our initial analysis complete, we now compare our theory to the full nonlinearevolution. In the next subsection, we will look at several numerical simulations and discussthe results for select values of p and a.

20 Single-Layered Rotating Annulus

0.0 0.2 0.4 0.6 0.8 1.0a

−0.3

−0.2

−0.1

0.0

0.1

0.2

0.3

σi

Figure 3.3: The form of the growth and decay rates, σi for m = 4 and a range of p values: p = −0.5,p = −0.25, p = 0, p = 0.25, p = 0.5.

3.1.2 Nonlinear EvolutionFirst, consider an annulus with a zero PV core (p = 0) with a = 0.78. Figure 3.4a showsthe progression of this annular set-up. Initially, the ring of the annulus breaks up into fiveseparate vorticity patches, each rotating around their own centre of mass. However, togetherthey rotate in the same ring formation as the original annulus. After some time, this vortexpatch formation breaks down even further (see panel 3 of Fig. 3.4a) resulting in one largevortex patch orbited by much smaller patches. As the simulation runs to late times, the newformation changes very little, thus indicating a ‘stable’ vortex patch pattern has been found.

Figure 3.4b illustrates another possible break down of the annular ring. In this instance,the ring is very thin (a = 0.9) and splits into 11, almost equal, satellites that rotate withinthe core. These patches then merge together leaving behind 5 satellites. As we run into latetimes, these patches continue to rotate within the original core.

One simulation run (Fig. 3.4c) that is particularly significant is p = 0.5 for a = 0.7 (withLD = 10, which is effectively infinite LD). The annular ring breaks into a thinner ring withsix satellites forming in the core (which is expected from the stability analysis graphs) andas the simulation progresses, the ring remains intact. For later times, the satellites reform tocreate an asymmetric core dotted with very small PV=0 vortex patches. We have obtaineda case where our original axisymmetric vortex becomes asymmetric. It would be worthwhileto consider runs with 0.5 < p < 1 to see if we can obtain more asymmetric annular vortices.

3.2 Finite LD

3.2.1 Linear StabilityThe majority of the analytical method having been illustrated in the previous section, wewill now briefly discuss the differences in the approach for finite LD. This time we must

3.2 Finite LD 21

(a) p = 0, LD = 100 and a = 0.78

(b) p = 0.25, LD = 10 and a = 0.9

(c) p = 0.5, LD = 10 and a = 0.7

Figure 3.4: Stills of numerical simulations at different times throughout the calculation: initial (far left),early intermediate (centre left), late intermediate (centre right) and final (far right). The colours indicatedifferent values of PV (vertical vorticity in this case): 0, 0.25, 0.5 and 1.

include the second term in equations (3.2) and (3.3) which results in the following differentialequations:

1r

d

dr

rdψdr

− 1L2

D

ψ = Q (3.20)

1r

∂

∂r

r∂ψ′

∂r

+ 1r2∂2ψ′

∂θ2 − 1L2

D

ψ′ = 0. (3.21)

As equation (3.20) describes the case with finite LD, we rederive the explicit form of uθ

by initially solving for ψ and then differentiating (w.r.t r). With x ≡ r/LD henceforth, we

22 Single-Layered Rotating Annulus

obtain

ψ =

αI0(x) − L2Dp r < a

βI0(x) + γK0(x) − L2D a ≤ r ≤ 1

δK0(x) r > 1

(3.22)

uθ =

α

LD

I1(x) r < a

β

LD

I1(x) − γ

LD

K1(x) a ≤ r ≤ 1

− δ

LD

K1(x) r > 1

(3.23)

where we have immediately enforced the finite solution criteria (i.e. ψ is finite as r → 0 andr → ∞) and where α, β, γ and δ are constants. Requiring ψ and uθ to be continuous acrossthe boundaries (r = 1, r = a), we obtain

α = LDK1(1/LD) − aLD(1 − p)K1(a/LD) (3.24)β = LDK1(1/LD) (3.25)γ = aLD(1 − p)I1(a/LD) (3.26)δ = aLD(1 − p)I1(a/LD) − LDI1(1/LD). (3.27)

0.0 0.5 1.0 1.5 2.0r

−0.1

0.0

0.1

0.2

0.3

0.4

0.5

uθ

Figure 3.5: Comparison between uθ for the infinite and finite LD cases. Four values of LD are consideredLD = 0.1, LD = 1, LD = 10 and LD = ∞ with p = 0 and a = 0.2 fixed. For increasing values of LD, uθ

converges towards the original case outlined in Eq. (3.4).

3.2 Finite LD 23

To ensure that our results return to those in the infinite LD case (c.f Eq. (3.4)), wecompare the forms of the basic state tangential velocity (see Fig. 3.5). As expected whentaking the limit (LD → ∞), regardless of which values of p ∈ [−1, 1] and a ∈ [0, 1] we take,uθ returns to the original form in Eq. (3.4).

As with the previous section, we assume a wavelike solution of the form ψ′ = ψ(r)ei(mθ−σt)

and reduce the (3.21) to a differential equation solely in r

r2d2ψ

dr2 + rdψ

dr+− r2

L2D

−m2

ψ = 0 (3.28)

The solution to the above is a superposition of modified Bessel functions of the first andsecond kind

ψ(r) = AiIm(x) +BiKm(x) i = 1, .., 3 (3.29)

where (again) x = r/LD and Ai and Bi are constants. Using equations (3.12), (3.15) and(3.29) as well as the method described in the infinite LD case, the six equations used toformulate the eigenvalue problem for σ are rederived:

B2 = (A1 − A2)Im(a/LD)Km(a/LD) (3.30)

B3 = A2Im(1/LD)Km(1/LD) + (A1 − A2)Im(a/LD)

Km(a/LD) (3.31)

κ1η1 = 1LD

[(A1 − A2)I ′m(a/LD) −B2K

′m(a/LD)] (3.32)

κ2η2 = 1LD

[B3K′m(1/LD) − A2I

′m(1/LD) −B2K

′m(1/LD)] (3.33)

−iση1 + iα

LDamI1(a/LD)η1 = −im

aA1Im(a/LD) (3.34)

−iση2 − iδ

LD

mK1(a/LD)η2 = −imB3Km(1/LD) (3.35)

where κ1 =(β − α)

2L2D

(I0(a/LD) + I2(a/LD)) + γ

2L2D

(K0(a/LD) +K2(a/LD))

and κ2 =(γ − δ)

2L2D

(K0(1/LD) +K2(1/LD)) + β

2L2D

(I0(1/LD) + I2(1/LD)).

After substituting (3.30) and (3.31) into (3.32) and (3.33), the Wronskian identityIm(x)K ′

m(x) − I ′m(x)Km(x) = −1/x is used to rearrange for A1 and A2. These equations

are then substituted into (3.34) and (3.35) and after some algebra we arrive at the following:(A11 A12A21 A22

)(η1η2

)= σ

(η1η2

)(3.36)

24 Single-Layered Rotating Annulus

where A11 = mα

aLD

I1(a/LD) +mκ1Im(a/LD)Km(a/LD)

A12 = −m

aκ2Im(a/LD)Km(1/LD)

A21 = amκ1Im(a/LD)Km(1/LD)

A22 = −m δ

LD

K1(1/LD) −mκ2Im(1/LD)Km(1/LD)

Again, σ is the eigenvalue and can be found by solving the characteristic polynomial,|(A − σI)| = 0, numerically. After confirming that, in the limit LD → ∞, our previousresult is recovered (c.f. Eq. 3.18), the stability of the annulus is then investigated for variousvalues of p and LD over our chosen range of a for wave modes m = 2, .., 6.

LD = 0.1

0.0 0.2 0.4 0.6 0.8 1.0a

−2.5

−2.0

−1.5

−1.0

−0.5

0.0

0.5

σr

(a)

0.0 0.2 0.4 0.6 0.8 1.0a

−0.06

−0.04

−0.02

0.00

0.02

0.04

0.06

σi

(b)

Figure 3.6: The linear stability of an annulus with p = 0 and LD = 0.1 for: m = 2, m = 3, m = 4, m = 5,m = 6. Instabilities for small LD are very weak and are forced to exist in thin annuli. The plots show (a)the real frequencies, σr and (b) the growth and decay rates, σi vs. the inner radius of the annulus, a.

The stability for this case was examined for multiple values of p. It was found that forp = 0 (Fig. 3.6), there are instabilities present in all wave modes (m = 2, .., 6) for thinannuli (0.6 ≤ a ≤ 1). However, for p → 1, the instabilities gradually arise for thicker annuliand complete linear stability (all wave modes contain no instabilities) is obtained at p = 1.Notably, the two real frequencies, σr, at p = 1 vanish for LD → 0 and this is reflected bythe fact that uθ → 0 as LD approaches zero (see Fig. 3.5). Considering p < 0, we see thatall modes have instabilities that, again, increase in strength (σi increases) as p → −1 (whenwe take LD fixed at 0.1).

LD = 1

Figure 3.7 shows the case of p = 0 and LD = 1. For each wave mode there are instabilitiesfor a range of a, with m = 2 having an instability over the largest range (0.2 ≤ a ≤ 1). Themajority of the analysis discussed for LD = 0.1 is applicable to LD = 1: the instabilities

3.2 Finite LD 25

vanish as p → 1 and amplify as p → −1. The differing factor is that for larger LD valuesthe instabilities are inherently stronger for all values of p.

0.0 0.2 0.4 0.6 0.8 1.0a

−1.0

−0.5

0.0

0.5

1.0

1.5

2.0

σr

(a)

0.0 0.2 0.4 0.6 0.8 1.0a

−0.20

−0.15

−0.10

−0.05

0.00

0.05

0.10

0.15

0.20

σi

(b)

Figure 3.7: The linear stability of an annulus with p = 0 and LD = 1 for: m = 2, m = 3, m = 4, m = 5,m = 6. Instabilities in this instance are far stronger (5x larger) than those in Fig. 3.6. The plots show (a)the real frequencies, σr, and (b) the growth and decay rates, σi, vs. the inner radius of the annulus, a.

LD = 10

0.0 0.2 0.4 0.6 0.8 1.0a

−0.5

0.0

0.5

1.0

1.5

2.0

2.5

σr

(a)

0.0 0.2 0.4 0.6 0.8 1.0a

−0.20

−0.15

−0.10

−0.05

0.00

0.05

0.10

0.15

0.20

σi

(b)

Figure 3.8: The linear stability of an annulus with p = 0 and LD = 10 for: m = 2, m = 3, m = 4, m = 5,m = 6. The plots, again, show (a) the real frequencies, σr, and (b) the growth and decay rates, σi, vs. theinner radius of the annulus, a. As LD has increased, the plots have begun to look more like the infinite LD

case which is what we would expect. However, there exists an anomaly for the m = 2 mode, see below fordiscussion.

For LD = 10, we arrive at a particularly note-worthy result. This value for the Rossbydeformation length should be large enough that we revert back to the infinite LD case.For the most part, this is true. However, for the m = 2 wave mode, we have an, albeitweak, instability (c.f Fig. 3.2 and Fig. 3.8). We further our investigation of this anomaly

26 Single-Layered Rotating Annulus

0.0 0.2 0.4 0.6 0.8 1.0a

−0.08

−0.06

−0.04

−0.02

0.00

0.02

0.04

0.06

0.08

σi

(a) LD = 1

0.0 0.2 0.4 0.6 0.8 1.0a

−0.010

−0.005

0.000

0.005

0.010

σi

(b) LD = 10

Figure 3.9: A comparison between the full analytical and asymptotic solutions for the p = 0, m = 2.

by carrying out asymptotics for the m = 2 mode using the first two terms in the seriesexpansions of the Bessel functions I0,1,2 and K0,1,2:

I0(x) = 1 + x2

4 (3.37)

I1(x) = x

2 + x3

16 (3.38)

I2(x) = x2

8 + x4

96 (3.39)

K0(x) = − ln(x) − λ+ ln(2) + x2

4 (− ln(x) − λ+ 1 + ln(2)) (3.40)

K1(x) = 1x

+ x

4 (2 ln(x) + 2λ− 1 − ln(4)) (3.41)

K2(x) = 2x2 − 1

2 (3.42)

where x = 1/L or x = a/L and λ is the Euler-Mascheroni constant (≈ 0.577). Substitutingequations (3.37)-(3.42) into the eigenvalue problem (3.36), we numerically solve for σr,i.Figure 3.9 shows a comparison between the full analytical and asymptotic solutions forthe m = 2 wave mode. For LD = 10, such that the arguments of the Bessel functions areconsidered small, the asymptotic expansion is almost identical to the exact solution (see Fig3.9b).

Furthermore, we calculated the value of p that causes the m = 2 mode to becomecompletely void of instabilities - p must be greater than 0.0041. Therefore, it only takes asmall deviation from a zero PV core for the mode to become linearly stable.

One possible reason that this mode has an instability for finite LD is phase-locking. Thisoccurs when disturbances of the same wave number, m, on two interfaces move togetherand amplify the instability. This simple mechanism was described in depth by Dritschel& Polvani (1992), wherein the calculation for the wave mode that causes phase-locking ispresented for the general annular case. In brief, to perform the analysis, the total angular ve-

3.2 Finite LD 27

locity (background: Ωθ = uθ/r, wave: Ωw = σ/2m and any other angular velocities present)is found at each interface edge and then equated to obtain an equation for m. The m = 2wave mode corresponds to a long wave, so the phase-locking argument requires the couplingbetween wave modes and cannot be solved by merely equating the total angular veloc-ity on each interface. However, phase-locking can still be seen from the numerical simulations.

In general, Fig. 3.10 summarises the effects varying LD and p has on σi. For increasingvalues of LD, the instabilities exist for lower values of a, which eventually leads to theinfinite LD case. The variation of p again shows that for p → 1 the instabilities vanish andfor p → −1 intensify.

0.0 0.2 0.4 0.6 0.8 1.0a

−0.04

−0.03

−0.02

−0.01

0.00

0.01

0.02

0.03

0.04

σi

(a) LD = 0.1

0.0 0.2 0.4 0.6 0.8 1.0a

−0.3

−0.2

−0.1

0.0

0.1

0.2

0.3

σi

(b) LD = 1

0.0 0.2 0.4 0.6 0.8 1.0a

−0.3

−0.2

−0.1

0.0

0.1

0.2

0.3

σi

(c) LD = 10

Figure 3.10: The form of the growth and decay rates, σi for m = 4 and a range of p values: p = −0.5,p = −0.25, p = 0, p = 0.25, p = 0.5.

3.2.2 Nonlinear EvolutionHaving completed the stability analysis, numerical simulations are considered for selectvalues of p and a with LD = 0.1, 1, 10. For each run, the results are compared with thetheoretical stability analysis to check whether we achieve the wave mode corresponding tothe parameters we chose. The stability analysis for the finite LD case indicates that for thinannuli, we will have instabilities for all wave modes except for select p values (c.f. previoussection) hence the simulations of note are for annuli with a > 0.5.

To illustrate the comparison between the linear stability graphs and the nonlinear simu-lations we will first discuss the case of an annulus with a zero PV core. Figure 3.11a shows

28 Single-Layered Rotating Annulus

(a) p = 0, LD = 1 and a = 0.78

(b) p = 0.25, LD = 0.1 and a = 0.8

(c) p = −0.25, LD = 1 and a = 0.86

(d) p = 0, LD = 1, and a = 0.5

Figure 3.11: Stills of numerical simulations at different times throughout the calculation: initial (far left),early intermediate (centre left), late intermediate (centre right) and final (far right). The colours indicatedifferent values of PV: −0.25, 0, 0.25, 0.5 and 1.

how this annulus evolves with time. Initially, we see the annulus break down completelyinto five separate vortex patches which corresponds to our earlier stability analysis (seeFig. 3.7 at a = 0.78). This indicates that the annular set-up was unstable and that thevortex moved into a more stable configuration. Here, (un)stable does not refer to linearstability - the vortex patches arise due to instabilities. We merely describe the patternsproduced as stable or unstable with respect to remaining in the annular set-up or not.

3.2 Finite LD 29

As time progresses, these patches recombine into four unequal patches and for the rest ofthe simulation time the paw-print like pattern remains present. If we look at the stabilitygraph again at a = 0.78, two wave modes have nearly the same growth rate: m = 4 hasσi = 0.292516 and m = 5 has σi = 0.278189. Although wave mode five prevails initially, itis short-lived and is subsequently replaced by a more ‘stable’ configuration.

Figure 3.11b shows the progression of the annulus for p > 0 and small values of defor-mation length. As time progresses, the annulus forms five satellites that rotate around theoriginal core. These satellites appear to form their own deformed annular configurationand survive for long periods of time. Comparing the simulation to our stability analysis,at a = 0.8, two wave modes are competing with one another: the m = 5 and m = 6 plotscross one another, indicating that either mode could dominate at this radius. However, it isclear from the simulation that m = 5 wins and we obtain 5 distinct satellites. To furthershow that m = 5 is marginally stronger than m = 6, we evaluate and compare the stabilitygraphs for both wave modes. From this, we see that m = 5 has a growth rate, σi that is0.0006 larger than the growth rate for m = 6.

For other runs, the annulus was given a value of p = −0.25 to simulate what happensfor an annulus with a negative PV core (see Fig. 3.11c). For this case, the annulus initiallybreaks into five satellites (PV=1) that almost immediately leave the annular set-up, thuscausing a further three distinct satellites (PV=-0.25) to form. The five (PV=1) satellitesrecombine into four patches, which are orbited by several patches PV=-0.25, and as timeprogresses, the vortex patches spread out while moving as dipoles. This simulation is aninstance of complete destruction of the hurricane-like vortex; possible reasons for this arethe instabilities are stronger than in the p ≥ 0 case, and the negative PV value cancels outother vortex patches.

As stated in the linear stability section, the phase locking mechanism can be seen inthe simulations. Figure 3.11d shows that selecting p = 0, a = 0.5 and LD = 1, gives us am = 2 wave mode instability. Panel 2 of 3.11d shows an elliptical distortion that arises fromphase locking. Both interfaces move together and this results in each interface acting uponthe other, causing further deformation. As time progresses, the ellipses become elongateduntil the ellipses turn over, resulting in thin vacillations. The ellipse’s semi-major axis thenshortens. This causes the core to become rugby ball shaped. This process of elongating,vacillations and shortening repeats three times before the core becomes chaotic and theannular ring completely distorts.

In this section, we have omitted discussion of LD = 10 due to the fact this value of LD

is almost identical to the infinite LD case and only differs for the m = 2 wave mode withp = 0 where a weak instability (σi = 0.009) is present for finite LD. If we take any othervalue of p, simulations for infinite and finite LD will be closely similar.

Chapter 4

Two-Layered Rotating Annulus

Our investigation of the idealised hurricane is continued by looking at an incompressible two-layered rotating annulus. The two-layered model is one of the simplest models illustratingthe effects of both rotation and stratification (Polvani, 1991). These are properties inherentin a real hurricane. Figure 4.1 shows the set up that we are adopting for this chapter: thetwo layers have mean depths (H1,H2) and constant densities (ρ1,ρ2) and here the ratioof these quantities are varied along with the inner radius of the annulus and the Rossbydeformation length. The PV for both layers is assumed to be identical and has the sameform as Eq. (3.1).

Figure 4.1: A schematic illustrating the structure of the two-layered axisymmetric vortex. A top down viewis also shown to indicate the exact distribution of the potential vorticity in both layers.

4.1 Linear StabilityThe equations (2LQGSWE) that are be used within this chapter describe the same analyticalproblem as in the previous chapter: the finite and infinite LD equations must be solvedtogether to obtain an analytical solution. However, this means that the stability analysisneed not be tackled again because the results have already been obtained. This section will,instead, discuss the parameters that cause the baroclinic mode (finite LD) to dominate overthe barotropic mode (infinite LD).

Using the analytical solutions from earlier, the characteristic polynomials given by (3.18)and (3.36) are solved numerically and the complex roots from both are compared for fixedRossby deformation length (LD = 0.1, 1, 10) and azimuthal wave modes (m = 2, ..., 6). Phasediagrams are constructed to illustrate the values of p and a that cause baroclinic instabilitiesto dominate over the barotropic instabilities (see Fig 4.2). We see that, for increasing m,the values of positive p that adhere to the baroclinic dominant regime increase in magnitude

32 Two-Layered Rotating Annulus

(a) m = 2 (b) m = 3 (c) m = 4

(d) m = 5 (e) m = 6

Figure 4.2: Phase diagram illustrating the values of p and a that cause baroclinic instabilities to dominateover barotropic instabilities for LD = 1.

but at the cost of the range of a (the range decreases for the associated, increasing, pvalues). The baroclinic dominant regime also exists for negative p values but exclusivelyfor the higher wave modes and for a very short range of a, which means there is only asmall set of parameters that cause multiple wave modes to be excited for negative p. Thephase diagrams for LD = 0.1 and LD = 10 show similar variations. However, for decreasingLD, the baroclinic dominant regime exists only for very thin annuli (which corresponds tothe stability analysis in chapter 3). For large LD, the maximum p values, that cause thebaroclinic dominant regime, are smaller than those in the LD = 1 case.

4.2 Nonlinear Evolution

In comparison to the nonlinear evolution of the single-layered, the number of parametersthat can be examined has increased from three to five. This means that performing acomplete analysis involving a variation of each of the parameters would take a considerablylong time. Hence, we restrict our numerical simulations. We keep the layer depths equal(h1 = 0.5), consider LD = 1 and only vary a, p and ρ2/ρ1 (= α). We use the phase diagram(Fig 4.2) to choose values of a and p that cause a visible variation in patterns between thetwo layers (i.e. the baroclinic dominant regime).

Figure 4.3a illustrates what happens to the annular vortex for p = 0.25, a = 0.8 andα = 0.5. In both layers, the annulus breaks into seven distinct satellites (PV=1) that rotatewithin the core in the original ring structure. Panel 3 of Fig. 4.3a shows the variationbetween the upper and lower layers. In the lower layer, the satellites have developed thin

4.2 Nonlinear Evolution 33

(a) p = 0.25, a = 0.8, LD = 1, α = 0.5 and h1 = 0.5

(b) p = 0.5, a = 0.72, LD = 1, α = 0.8 and h1 = 0.5

Figure 4.3: Stills of numerical simulations at different times throughout the calculation: initial (far left),early intermediate (centre left), late intermediate (centre right) and final (far right). The colours indicatedifferent values of PV: 0, 0.25, 0.5 and 1.

vacillations which have curled over to connect to the satellites that are neighbouring. Thiscreates the appearance of seven more satellites that have PV=0. This is inherently differentto the structure in the upper layer, where no vacillation ‘tails’ are produced. As timeprogresses, the PV=1 satellites perturb from the ring and recombine into three patches.During this recombination, the core becomes chaotic and eventually disintegrates. At latetimes, the annulus has been completely destroyed. Due to the complexity of the pattern(see final panel) at this stage, it can be assumed that this pattern (or one similar) hangstogether.

34 Two-Layered Rotating Annulus

Another simulation was run for p = 0.5, a = 0.72 and α = 0.8 (Fig. 4.3b). Initially,the annular ring distorts differently in each layer. The lower layer produces vacillationsthat cause six satellites (with PV=1) to form. However, these satellites almost immediatelydissolve into the annular ring and create an asymmetric annular vortex. Similarly, the upperlayer also produces vacillations which result in six equal PV=1 satellites that are held withinthe core and separated from the annular ring by a thin PV=0.5 border. The vacillationprocess happens again and broadens the gap between the satellites and the ring. Thesesatellites are perturbed from the initial configuration and begin to recombine. At late times,the patches in the upper layer dissipate leaving an asymmetric core. In both layers, anasymmetric vortex forms.

Chapter 5

Concluding Remarks

Through the use of two sequential models, we have presented an extension of previousworks on the structure and stability of an idealised hurricane. We used the first model(the single-layered case) to discuss an alternative method to Flierl’s (1988) for analysingthe stability of a rotating annulus for the barotropic and baroclinic cases. In each case,we found that, depending on the sign of p, the instabilities, which exist predominantly forthin annuli, will either be enhanced (p < 0) or diminished (p > 0). We also investigatedthe finite LD parameter. For LD → 0, instabilities only arise for exceptionally thin annuliuntil they disappear entirely, while large LD causes the stability analysis to revert to theinfinite case. Based on our stability analysis, numerical simulations were run for select p, aand LD. These simulations showed that the initial state of the annulus could develop intofive distinct states: vortex patches rotating around a central patch; vortex patches rotatingwithin the original core; an asymmetric annulus; dipole translation and miniature annulirotating around the original core. We intend to carry out further simulations to ascertainthe parameter ranges that lead to these different states.

We extended the first model to include two layers with the same PV profile in eachlayer. Due to this, we found that the linear stability analysis was identical to the firstmodel. Using the analysis, we investigated the possible parameters that cause the baroclinicmode to dominant the barotropic mode. We found that the baroclinic dominant regimeexists predominantly for thin annuli and for positive PV cores (p > 0). For larger m values,negative cores can also cause the baroclinic dominant regime but the range of values of athat facilitate this is very small. We chose to restrict our numerical treatment by keeping h1and LD fixed while varying a, p and α. This restriction is due to the computational expensethat would occur if we investigated all possible variations of the parameters in addition totime constraints of the project. Based on the constructed phase diagrams, we selected valuesof a and p that were most likely to cause the baroclinic dominant regime and ran simulationsfor these values. The simulations showed that the annulus structure remained inherentlybarotropic with some slight baroclinic variations. In this model, we saw the initial annulusdevelop into two distinct states: an asymmetric annulus and complete destruction of theannulus into a swirling, chaotic vortex. There are several avenues that could be investigatedfurther. For example, we could examine the effect that unequal layer depths has on the over-all structure of the hurricane in addition to running more simulations for the equal depth case.

There are several possible extensions that can be applied to our second model to simulatea more realistic hurricane. Previously (see chapter 1), we discussed the processes involved inhurricane formation, such as convection and air-sea interactions. The next logical step wouldbe to include these processes in our analytical model and investigate the flow modificationinduced by these changes. However, to include all physical processes in one step wouldprove overly complex. Therefore, it is advisable to introduce the processes in several steps.

36 Concluding Remarks

Initially, we could modify the second model to include the compressible form of theQGSWE such that a more realistic atmosphere is simulated. The next step would be toinclude diabatic forcing through heating in the lower layer and cooling in the upper layer.This would crudely simulate convection processes. Convection is particularly crucial inthe formation of hurricanes due to its role in the initial intensification. Considering amodel that includes convection may give rise to interesting dynamical behaviour for ouridealised hurricane. Two further modifications that could be made are introducing theair-sea boundary and exploring the effect of large scale environmental shear as both of thesealso play a role in the intensity (and, perhaps, structure) of a hurricane.

In future work, we aim to look at the compressible QGSWE with diabatic forcing toinvestigate the structure and stability of a more realistic hurricane.

References

Ahrens, C.D. 2006 Meteorology Today: An Introduction to Weather, Climate, and theEnvironment, 8th edn. Cengage Learning.

Anthes, R.A., Rosenthal, S.L & Trout, J.W. 1975a Preliminary results from anasymmetric model of the tropical cyclone. Mon. Wea. Rev. 99, 744–758.

Anthes, R.A., Rosenthal, S.L & Trout, J.W. 1975b Comparisons of tropical cyclonesimulations with and without the assumption of circular symmetry. Mon. Wea. Rev. 99,759–766.

Bjerknes, J. 1919 On the structure of moving cyclones. Geofys. Publ. 1, 1–8.

Bowker, D. 2011 Meteorology and the ancient greeks. Weather 66 (9), 249–251.

Charney, J.G. 1948 On the scale of atmospheric motions. Geof. Publ. 17, 3–17.

Charney, J.G., Fjörtoft, R. & von Neumman, J. 1950 Numerical integration of thebarotropic vorticity equation. Tellus 2, 237–254.

Dritschel, D.G. 1986 The nonlinear evolution of rotating configurations of uniformvorticity. J. Fluid. Mech. 172, 157–182.

Dritschel, D.G. 1988b Contour surgery: A topological reconnection scheme for extendedintegrations using contour dynamics. J. Comput. Phys. 77, 240–266.

Dritschel, D.G. 1989 Contour dynamics and contour surgery: Numerical algorithmsfor extended high-resolution modelling of vortex dynamics in two-dimensional, inviscid,incompressible flows. Comput. Phys. Rep. 10, 79–146.

Dritschel, D.G. & Polvani, L.M. 1992 The roll-up of vorticity strips on the surface ofa sphere. J. Fluid. Mech. 234, 47–69.

Emanuel, K.A 1989 The finite-amplitude nature of tropical cyclogenesis. J. Atmos. Sci.46, 3431–3456.

Emanuel, K.A 1991 The theory of hurricanes. Annu. Rev. Fluid. Mech. 23, 179–196.

Emanuel, K.A 1995 The behaviour of a simple hurricane model using a convective schemebased on subcloud-layer entropy equilibrium. J. Atmos. Sci. 52, 3960–3968.

Flierl, G.R 1988 On the instability of geostrophic vortices. J. Fluid. Mech. 197, 331–348.

Freebairn, J. W. & Zillman, J. W. 2002 Economic benefits of meteorological services.Meteorological Applications 9 (1), 33–44.

Frei, T. 2010 Economic and social benefits of meteorology and climatology in switzerland.Meteorological Applications 17 (1), 39–44.

Gent, P.R. & McWilliams, J.C. 1986 The instability of barotropic circular vortices.Geophys. Astrophys. Fluid. Dyn. 35, 209–233.

Gills, A.E. 1982 Atmosphere-Ocean Dynamics. Academic Press.

Gray, W.M. 1968 Global view of the origin of tropical disturbances and storms. Mon.Wea. Rev. 96, 669–700.

38 References

Gray, W.M. 1998 The formation of tropical cyclones. Meteorol. Atmos. Phys. 67, 37–69.

Helfrich, K.R & Send, U. 1988 Finite-amplitude evolution of geostrophic vortices. J.Fluid. Mech. 197, 349–388.

Holton, J.R. 2004 An introduction to Dynamic Meteorology,Volume 1 . Academic Press.

Ikeda, M. 1981 Instability and splitting of mesoscale rings using a two-layer quasi-geostrophic model on an f-plane. J. Phys. Oceanogr. 11, 987–998.

Mai, N., Smith, R.K., Zhu, H. & Ulrich, W. 2002 A minimal axisymmetric hurricanemodel. Q. J. R. Meteorol. Soc. 128, 2641–2661.

Majumdar, K. K. 2003 A mathematical model of the nascent cyclone. IEEE Trans.Geo-Science and Remote Sensing 41, 1118–1122.

McWilliams, J.C., Gent, P.R. & Norton, N.J. 1986 The evolution of balancedlow-mode vortices on the beta plane. J. Phys. Oceanogr. 16, 838–855.

Ooyama, K.V. 1969 Numerical simulation of the life cycle of tropical cyclones. J. Atmos.Sci. 26, 3–40.

Polvani, L.M. 1991 Two-layer geostrophic vortex dynamics. part 2. alignment and two-layerv-states. J. Fluid. Mech. 225, 241–270.

Pullin, D.I. 1992 Contour dynamics methods. Annu. Rev. Fluid Mech. 24, 89–115.

Schultz, D.M., Keyser, D. & Bosart, L.F 1998 The effect of large-scale flow onlow-level frontal structure and evolution in midlatitude cyclones. Mon. Wea. Rev. 126,1767–1791.

Schultz, D.M. & Vaughan, G. 2011 Occluded fronts and the occlusion process:a freshlook at conventional wisdom. Bull. Amer. Meteor. Soc. 92, 443–466.

Schultz, D.M. & Zhang, F. 2007 Baroclinic development within zonally-varying flows.Q. J. R. Meteorol. Soc. 133, 1101–1112.

Shapiro, L.J & Keyser, D. 1990 Fronts, jet streams, and the tropopause. In ExtratropicalCyclones: the Eric Palmen Memorial Volume (ed. C. W. Newton & E.O. Holopainen),pp. 167–191. American Meteorological Soceity.

Tsang, Y.K. & Dritschel, D.G. 2014 Ellipsoidal vortices in rotating stratified fluids:beyond the quasi-geostrophic approximation. J. Fluid. Mech. 762, 196–231.

Vallis, G.K. 2006 Atmospheric and oceanic fluid dynamics: fundamentals and large-scalecirculation. Cambridge University Press.

Warneford, E.S. & Dellar, P.J. 2013 The quasi-geostrophic theory of the thermalshallow water equations. J. Fluid. Mech. 723, 374–403.

Williams, P.D., Read, P.L. & Haine, T.W.N. 2010 Testing the limits of quasi-geostrophic theory: application fo observed laboratory flows outside the quasi-geostrophicregime. J. Fluid. Mech. 649, 187–203.

Yamasaki, M. 1968a Numerical simulation of tropical-cyclone development with the use ofprimitive equations. J. Meteorol. Soc. Jpn. 46, 178–213.

Yamasaki, M. 1968b A tropical-cyclone model with parameterized vertical partition ofreleased latent heat. J. Meteorol. Soc. Jpn. 46, 202–214.

References 39

Yamasaki, M. 1968c Detailed analysis of a tropical-cyclone simulated with a 13-layermodel. Papers in Meteorolgy and Geophysics 20, 559–585.

Zabusky, N.J., Hughes, M.H. & Roberts, K. 1979 Contour dynamics for the eulerequations in two dimensions. J. Comput. Phys. 30, 96–106.

Zhu, H., Smith, R.K. & Ulrich, W. 2001 A minimal three-dimensional tropical cyclonemodel. J. Atmos. Sci. 58, 1924–1944.