1. The Structure and Stability of an
Idealised Hurricane
Laura Merchant
110009528
School of Mathematics and Statistics
University of St Andrews
This dissertation is submitted for the degree of
Master of Physics: Mathematics and Theoretical Physics
April 2015
2.
3. Declaration
I certify that this project report has been written by me, is a record of work carried out by
me, and is essentially different from work undertaken for any other purpose or assessment.
Laura Merchant
110009528
April 2015
4.
5. Acknowledgements
I would like to thank Professor David Dritschel for all of the helpful discussions and
encouragement he provided throughout the project. I am also grateful to The Robertson
Scholarship Trust for the financial aid they have given me over the past four years.
6.
7. Abstract
The quasi-geostrophic shallow water equations are used to examine the stability and nonlinear
evolution of an idealised hurricane. The model consists of a simple, axisymmetric annular
vortex with a predefined potential vorticity distribution. We present an alternative analysis to
Flierl’s (1988) work for a multi-layered annular vortex. Barotropic and baroclinic instabilities
are found to exist for thin annuli. These instabilities are either enhanced or diminished
depending on the choice of potential vorticity within the core, p. We extend the analysis to
include the full nonlinear evolution. Through this, we discern the five break down patterns
that arise from the variation of p, the Rossby deformation length, LD and the choice of
inner radius, a. For two layers, wherein the density, ρ, is assumed to be constant in each
layer, 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 constructed
phase diagrams, which illustrate two competing regimes (baroclinic dominant and barotropic
dominant), the nonlinear evolution is then examined for select p, a, α, h1 and LD. Through
the numerical simulations, we see that the annular structure, which breaks down into an
asymmetric vortex or completely disintegrates, remains predominantly barotropic with some
baroclinic tendencies.
13. Chapter 1
Introduction
Throughout time, mankind has sought to understand physical processes that occur within
Nature. One field of considerable interest is the field of atmospheric sciences. This is a
field that encompasses the study of the atmosphere, the processes within it, and how other
systems interact with it. In particular, meteorology (the specialisation of weather events
and forecasting) has become one of the most well-studied subjects due to the vast economic
and 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 the
previous day (image courtesy: Birmingham Museum, taken from wikipedia) (b) A recent satellite image of
Europe (image courtesy: RMI). A comparison of the two images shows how far meteorology has come with
the advance of technology.
From antiquity to the modern age, the field of meteorology has grown from a collection
of hypotheses written by Aristotle in his ‘Meteorologica’ to something capable of predicting
weather on a daily basis. The beginnings of meteorology - largely accredited to the ancient
Greeks (Bowker, 2011) - involved observations of clouds, winds, rain, and other weather
events. It was understood that these were linked in some way. Within ‘Meteorologica’,
Aristotle discussed theories on cloud formation; the properties of tornadoes; hurricanes and
lightning and, more generally, the earth sciences. Although it was Aristotle that coined
the phrase “meteorology”, many other philosophers before him (Thales, Democritus and
Hippocrates) were classed as meteorologists due to their work on various atmospheric
phenomena such as the water cycle; weather predication, and ‘Airs, Waters and Places’.
14. 2 Introduction
At the time, these observations and hypotheses were considered to be the commanding
authority (Ahrens, 2006) within the sciences and were used consistently through to the
Renaissance era - until the advancement of observational instruments. The most notable
inventions were the Hygrometer (Cusa, 1450), Thermometer (Galilei, 1593) and the Barom-
eter (Torricelli, 1643). Although primitive in their early stages, the creation of these should
be classed as pivotal moments in meteorological history as, without them, our fundamental
understanding of the processes in the atmosphere would have been left to the speculation of
natural philosophers.
Before these developments, other avenues of research were conducted. This consisted of
the collaborative efforts of different observers across the world; the telegraph was used to
transmit 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 the
1920s, Lewis Fry Richardson created the first numerical simulation to predict the weather. It
was a long, arduous process: it took 6 weeks to complete a 6-hour forecast, which was based
on previous weather data and ended with an unrealistic result. The theory was ahead of the
available technology. However, in this case, there was a short turnover period and within 30
years 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 which
resulted in a 24 hour forecast (taking 24 hours to complete using data from the previous
day) (Charney et al., 1950). This was a remarkable advancement as it meant that, with
faster computers, the weather could be predicted more frequently, which is evident in the
satellite 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 Katrina
at 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).
15. 3
There is one enigma that has been documented throughout history that still baffles
many researchers today: the tropical cyclone. The difficulty in studying this phenomenon
boils down to the inability to recreate the required conditions that form the cyclone in a
laboratory setting. In addition, it is difficult to numerically model all physical processes
that are inherent in the cyclone (Emanuel, 1991). With all of these issues, why is there such
invested interest in the study of these deadly vortices? Every year, tropical cyclones (see
Fig. 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 advanced
methods of detecting, preventing and tracking cyclones, which would ultimately help reduce
the 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 core
with little chaotic motion (calm weather). Rainbands are formed around the eye by the inwards spiralling of
air. 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 dragged
upwards leaving a region of low pressure at the water’s surface. As the air rises, converging
winds 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 if
favourable 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 of
the Norwegian cyclone model (Bjerknes, 1919). This describes the structure and evolution
of extratropical cyclones over continental landmasses. This model is largely considered the
foundation of modern meteorological analysis (Schultz & Vaughan, 2011), and is included in
many introductory meteorology texts. However, there are some pitfalls within this model and
it fails to accurately describe oceanic midlatitude cyclones, meaning a different model was
required. In the 1990s, the Shapiro-Keyser model (Shapiro & Keyser, 1990) was formulated
and is based on oceanic cyclones. There are several differences between the models: the
16. 4 Introduction
overall evolution (see Fig. 1.4); what sort of front is formed in each model, and how the
cyclone 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 are
that either surface friction or the embedded large-scale flow determines the evolution and
structure of cyclones. There has been plenty of work on both avenues but there still appears
to be some disparity. (Schultz & Zhang, 2007)
Figure 1.4: Models of tropical cyclone evolution: a comparison of the different front structures. (a) Norwegian
Cyclone Model: (i) incipent frontal wave with cold (triangles) and hot (circles) fronts, (ii) and (iii) narrowing
warm sector due to faster spin of cold front, (iv) mature cyclone with occluded front. A Norwegian cyclone
is typically orientated north-south with a more intense cold front. (b) Shapiro-Keyser Model: (i) incipient
cyclone developes cold and warm fronts, (ii) cold front moves perpendicular to the hot front - the fronts
never meet resulting in frontal T-bone. (iii) frontal fracture leads to a back-bent front, (iv) warm seclusion
due to cold air encircling warmer air near the low centre. A Shapiro-Keyser cyclone will be elongated
east-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 weather
became part of everyday life and many researchers began applying the same numerical
methods 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 in
cyclone formation (Emanuel, 1991). As a result there has yet to be a complete model for
tropical cyclones. Serious work began on numerical modelling of cyclones towards the end
of 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 cyclone
structures 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 they
were 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 we
are now able to run high resolution simulations that include complex representations of the
physical processes. Despite this, simple models are still used today to develop our basic
understanding of cyclones (Mai et al., 2002) as can be readily shown by a comparison of
Ooyama’s (1969) and Emanuel’s (1989; 1995) model. The representation of moist convection
became 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 structure
and stability of an idealised hurricane (where hurricanes are tropical cyclones located in a
specific region, such as the North Atlantic ocean). In chapter 2, the underlying mathematics
of vortex dynamics will be explored, this being the simplification of the shallow water
equations under quasi-geostrophic theory and the layer approximation. Thereafter, two
models that sequentially build up from one another will be examined. Chapter 3 considers
17. 5
a single-layered rotating annulus (no forcing), and chapter 4 looks at the second model, a
two-layered rotating annulus (no forcing). The final chapter (chapter 5), will remark on
both models as well as possible extensions.
18.
19. 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 conservation
of momentum (2.1), mass (2.2) and energy (2.3) of the system:
Du
Dt
+ f × u = −
∇p
ρ
− ∇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 of
specific heats and L is the total energy loss function. Equation (2.1) describes the acting
forces: Coriolis force (f × u, f = 2Ω), pressure gradients (−∇p/ρ), gravitational force
including Newtonian gravity and centrifugal force (−∇Φ) and the final term (F), which
encompasses all other forces (viscosity, friction). In our problem, we restrict our discussion
to 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 look
at one of the simplest geophysical fluid models: the shallow water model. The rotating
shallow water equations (SWE) (Vallis, 2006) are derived by applying (i) the hydrostatic
approximation, 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 further
and further simplify our analysis. We apply another approximation to the SWE: the
quasi-geostrophic approximation.
20. 8 Mathematical Preamble
2.1.1 Quasi-Geostrophic Model
Quasi-geostrophic (QG) theory, which was first devised by Charney (1948), is one of the
simplest 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 to
retain the associated time evolution that would otherwise be omitted in a pure geostrophic
flow (Warneford & Dellar, 2013). Due to this, QG is particularly relevant when dealing with
numerical simulations because it reduces the dynamical degrees of freedom involved and
ultimately cuts the computational expense (Williams et al., 2010). As we want to look at
the overall stability of a hurricane, it would be advisable to use a model that is easy to solve
analytically, as well as one that can explore the full nonlinear evolution. The QG model
meets these criteria.
We apply several approximations to the primitive equations (in this case, the SWE) to
derive 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 and
length. A point of note is the typical value of the Rossby number for hurricanes. In some
instances, the Rossby number can become comparable to, or larger, than one. Yet QG
theory 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 ≈ 996km
thus meaning Ro ≈ 0.7. Although this is comparable to 1, QG theory is still applicable as
we 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 two
layered shallow water equations and what these equations become in the QG regime. The
derivations will be brief and the interested reader can find the full version, including the
scale analysis, non-dimensionalisation and algebra, in Vallis (2006).
21. 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 derived
by nondimensionalising the SWE (taking all quantities such that u = U ˜u and dividing by
the dominant scale) and expressing the result in terms of the small Rossby number,
Ro
D˜u
D˜t
− ˜v = −
∂˜η
∂˜x
(2.7)
Ro
D˜v
D˜t
+ ˜u = −
∂˜η
∂˜y
(2.8)
ϵ
D˜η
D˜t
+ (1 + ϵ˜η) ˜∇ · ˜u = 0 (2.9)
where ∼ denotes the nondimensional quantities and ϵ ≪ 1 is the typical scale of the free
surface variation η. Assuming that ϵ = O(Ro), we expand ˜u, ˜v and ˜η in powers of Rossby
number. Looking at the leading order, the above reduces to geostrophic balance with the
consequence of incompressibility. However, this means we have an insufficient number of
equations to solve our problem and are subsequently motivated to look at a higher order
Rossby number (namely the first order). At this order, equations (2.7)-(2.9) become
D0 ˜u0
D˜t
− ˜v1 = −
∂˜η1
∂˜x
(2.10)
D0˜v0
D˜t
+ ˜u1 = −
∂˜η1
∂˜y
(2.11)
Bu−1 D0 ˜η0
D˜t
+ ˜∇ · ˜u1 = 0 (2.12)
where Bu−1
= gH/(fL)2
is the dimensionless constant known as Burger’s number and D0/D˜t
is the 2D convective time derivative with ˜u0 = ( ˜u0, ˜v0). The final step in the derivation is
to 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 streamfunction
such 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 equations
into (2.12), the nondimensional form of the PV inversion equations are found
D0
˜Q0
Dt
= 0, ˜Q0 = ˜ζ0 − Bu−1
˜η0 (2.13)
22. 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
H
η (2.14)
= ∇2
ψ −
1
L2
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). With
the derivation complete, the SLQGSWE can be summarised as follows,
Single-Layered Quasi-Geostrophic Shallow Water Equations (SLQGSWE)
DQ
Dt
= 0
Q = ∇2
ψ −
1
L2
D
ψ
u = −
∂ψ
∂y
v =
∂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 as
reference to derive the equations for this model. The SWE are nearly identical to those used
above 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)
23. 2.1 Shallow Water Model 11
To derive the explicit equations needed to investigate this model, PV is defined in its full
dimensional 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 small
Rossby number approximation (i.e. applying the QG approximation), the PV reduces to
q1 =
f
H1
+
∇2
ψ1
H1
−
fη
H2
1
(2.19)
q2 =
f
H2
+
∇2
ψ2
H2
+
fη
H2
2
(2.20)
where ζj = ∇2
ψj and ∇2
ψj ≪ f. Continuing with the derivation, we need to calculate an
explicit form of the layer depth variation in terms of the streamfunction, ψj. Initially, we
have to obtain an expression for the layer-dependent pressure which can be achieved by
integrating 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 and
ψ2. 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/∂y
and vj = (fρj)−1
∂p/∂x. Applying the above definitions of p for each layer, we obtain the
following,
Lower layer : u1 = −
1
ρ1f
∂ps
∂y
v1 =
1
ρ1f
∂ps
∂x
(2.21)
Upper layer : u2 = −
1
ρ2f
∂ps
∂y
+
ρ1
ρ2
g
f
∂η
∂y
v2 =
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 that
uj = −∂ψj/∂y and vj = ∂ψj/∂x and so comparing this with Eqs. (2.21) and (2.22) we
arrive at the solutions for ψ1,2:
ψ1 =
ps
ρ1f
ψ2 =
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
24. 12 Mathematical Preamble
that Qj ≡ Hj(qj − f/Hj), linearise qj, and substitute in the explicit form of η and qj for
each layer:
Q1 = ∇2
ψ1 −
fη
H1
= ∇2
ψ1 −
f2
H1
ρ1ψ1 − ρ2ψ2
ρ1g
(2.25)
Q2 = ∇2
ψ2 +
fη
H2
= ∇2
ψ2 +
f2
H2
ρ1ψ1 − ρ2ψ2
ρ1g
(2.26)
If we were to consider how to solve this problem in the single-layered case, we would have
been able to invert the Poisson equation to obtain the streamfunction. However, here we
have a coupling between the two layers (as evident from the presence of both streamfunctions
in (2.25) and (2.26)), meaning we have to decouple the equations. The simplest method of
doing 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, we
obtain the following Poisson equation
∇2
ψt = 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 for
Q1 and Q2:
Qc = ∇2
ψc −
f2
gH2
ρ2
ρ1
+
H2
H1
ψc (2.28)
If we let 1/L2
D ≡ f2
/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 both
models with the same method. However, our set of equations for the two layer model is still
incomplete as two equations are needed for ψ1 and ψ2. These expressions are easily found
by solving simultaneously ψt and ψc. Before this, let us simplify the notation for these two
terms:
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 then
combine 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.
25. 2.2 Numerical Method 13
Two-Layered Quasi-Geostrophic Shallow Water Equations (2LQGSWE)
DQj
Dt
= 0 j = 1, 2
Qt = h1Q1 + h2Q2 ∇2
ψt = Qt
Qc = αQ2 − Q1 ∇2
ψc −
1
L2
D
ψc = Qc
ψ1 =
αψt − h2ψc
αh1 + h2
uj = −
∂ψj
∂y
ψ2 =
ψt + h1ψc
αh1 + h2
vj =
∂ψj
∂x
2.2 Numerical Method
In addition to the analytical work used to determine the linear stability of our idealised
hurricane, 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 in
the SLQGSWE and 2LQGSWE, given a PV distribution and then advect the solution to
the next time step. This is a near impossible task to do by hand and the challenge is made
easier by using a pre-existing numerical model to perform the calculations.
One method used to numerically solve the inversion and advection problem for vortex
patches (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 and
2LQGSWE) to calculate the velocity fields directly. PV is assumed to be piecewise constant
within the contours and the streamfunction is solved in terms of Green’s functions, which
then gets converted into an equation for the velocity field. However, there are several pitfalls
when 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 expense
problem. CS removes vorticity features that are smaller than a predefined scale (say δ) as
well 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 expensive
but play a negligible role.
The explicit details of both CD and CS as well as a review of the methods can be found
in papers by Zabusky et al. (1979), Dritschel (1988b, 1989) and Pullin (1992).
26.
27. 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 resembles
the 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 this
work so it includes the full nonlinear evolution of the rotating annulus through the use of
numerical simulations.
Figure 3.1: A schematic illustrating the structure of the single-layered axisymmetric vortex. A top down
view 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 rotating
annulus. Consider, initially, the basic state ¯Q in polar coordinates with the following
configuration (Fig. 3.1),
¯Q =
p r < a
1 a ≤ r ≤ 1
0 r > 1
(3.1)
where we vary −1 ≤ p ≤ 1 and 0 ≤ a ≤ 1. A small perturbation (primed quantities) is
applied to the basic state (bar quantities) such that:
Q = ¯Q + Q′
ψ = ¯ψ + ψ′
28. 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:
∇2 ¯ψ −
1
L2
D
¯ψ = ¯Q (3.2)
∇2
ψ′
−
1
L2
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 point
in our derivation of the eigenvalue problem for σ. Later in this chapter, the derivation of the
functional form of ψ′
for two cases of LD (finite and infinite) will be shown explicitly with
the latter used to reproduce results published by Dritschel (1986) and as an introduction to
the analytical method.
3.1 Infinite LD
3.1.1 Linear Stability
First, we must obtain the explicit form of ¯uθ from (3.2) using LD = ∞. In this limit, the
potential vorticity simplifies to the vertical vorticity, ζ, and so here we have to solve
¯ζ =
1
r
d
dr
r
d ¯ψ
dr
=
1
r
d
dr
(r¯uθ)
in each region and match the solution at the boundaries. The final result for the tangential
velocity is then given by
¯uθ =
1
2
pr r < a
1
2r
(p − 1)a2
+
1
2
r a ≤ r ≤ 1
1
2r
(p − 1)a2
+
1
2r
r > 1.
(3.4)
Equation (3.3) reduces to the typical Laplace equation (with LD = ∞) ∇2
ψ′
= 0, which in
2D polars is written as,
1
r
∂
∂r
r
∂ψ′
∂r
+
1
r2
∂2
ψ′
∂θ2
= 0. (3.5)
We then consider ‘plane-wave’ solutions of the following form, ψ′
= ℜ( ˆψ(r)ei(mθ−σt)
). The
(complex) amplitudes must be taken as a function of r due to the non-constant coefficients
29. 3.1 Infinite LD 17
in equation (3.5). The θ component of the above can be shown explicitly by considering
the separation of variables method (i.e. ψ′
= H(r)G(θ)) on equation (3.5) and enforcing
periodic 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 ˆψ,
r2 d2 ˆψ
dr2
+ r
d ˆψ
dr
− m2 ˆψ = 0. (3.6)
From inspection, the solutions are of the form ˆψ(r) ∝ rm
, r−m
but the Laplacian operator
is a linear operator meaning a complete solution would be a superposition of the two,
ˆψ(r) = Airm
+ Bir−m
i = 1, .., 3 (3.7)
for the three regions. Using the radial solution, (3.7), the equations are matched in each
region at the interface boundaries to obtain two equations relating the coefficients. To
simplify the matching procedure, we apply the condition that ˆψ is bounded: r → 0 (when
r < a) and r → ∞ (when r > 1). This means that B1 and A3 are both zero:
r < a ˆψ(r) = A1rm
a ≤ r ≤ 1 ˆψ(r) = A2rm
+ B2r−m
r > 1 ˆψ(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 small
perturbation 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) +
∂
∂r
ψ′
(r, θ, t) leads to,
uθ(¯r + ηk, θ, t) = ¯uθ(¯r) + ℜ
d ¯ψ
dr
(¯r)ei(mθ−σt)
+
d ¯uθ
dr
(a)ηk. (3.12)
Looking at (3.12) above (+) and below (-) the boundaries, we plug in (3.4), (3.8), ηk for
each region and equate the solutions:
¯r = a:
uθ(a−
+ η1, θ, t) =
1
2
pa + ℜ[mA1am−1
ei(mθ−σt)
] +
1
2
pℜ[ˆη1ei(mθ−σt)
]
uθ(a+
+ η1, θ, t) =
1
2
pa + ℜ[(mA2am−1
− mB2a−m−1
)ei(mθ−σt)
] + (1 −
1
2
p)ℜ[ˆη1ei(mθ−σt)
]
⇒ (1 − p)ˆη1 = m((A1 − A2)am−1
+ B2a−m−1
). (3.13)
30. 18 Single-Layered Rotating Annulus
¯r = 1:
uθ(1−
+ η2, θ, t) =
1
2
(1 + a2
(p − 1)) + ℜ[m(A2 − B2)ei(mθ−σt)
] +
1
2
(1 + a2
(1 − p))ℜ[ˆη2ei(mθ−σt)
]
uθ(1+
+ η2, θ, t) =
1
2
(1 + a2
(p − 1)) + ℜ[−mB3ei(mθ−σt)
] −
1
2
(1 − a2
(1 − p))ℜ[ˆη2ei(mθ−σt)
]
⇒ ˆη2 = m(B2 − A2 − B3). (3.14)
There are still two equations needed before the results can be combined together to obtain an
analytical solution for the frequency σ. Exploiting the material character of the boundaries
and 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),
−iσˆη1 +
1
2
ipmˆη1 = −imA1am−1
(3.16)
−iσˆη2 +
1
2
im(1 + a2
(p − 1))ˆη2 = −imB3. (3.17)
The final step is to combine equations (3.9),(3.10),(3.13),(3.14),(3.16) and (3.17). We start
by substituting (3.9) and (3.10) into (3.13) and (3.14) and rearrange to give equations for
A1 and A2. These are then substituted into (3.16) and (3.17) and after some algebra, we
arrive at an eigenvalue problem for σ,
A11 A12
A21 A22
ˆη1
ˆη2
= σ
ˆη1
ˆη2
(3.18)
where A11 =
1
2
(pm + (1 − p))
A12 = −
1
2
am−1
A21 =
1
2
(m(1 + a2
(p − 1)) − 1)
A22 =
1
2
(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 polynomial
must be solved numerically. Once we obtain an equation for σ, the linear stability of the
annulus 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 PV
core. σ is then given by
σ1,2(m, a) =
1
4
m(1 − a2
) ±
1
2
(1 −
1
2
m(1 − a2))2 − a2m (3.19)
31. 3.1 Infinite LD 19
0.0 0.2 0.4 0.6 0.8 1.0
a
0.0
0.5
1.0
1.5
2.0
2.5
σr
(a)
0.0 0.2 0.4 0.6 0.8 1.0
a
−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. The
plots show (a) the real frequencies, σr, and (b) the growth and decay rates, σi vs. the inner radius of the
annulus, 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 growth
rate σi = ℑ(σ) and the frequency σr = ℜ(σ) with changing a for azimuthal wave modes
m = 2, ..., 6. From this, we can see that m = 2 contains no instabilities (i.e. no growth
or decay) for p = 0 (which is true for all p > 0) and for all values of a. If we consider
what happens to the other modes, they all become linearly stable, for all a, when we take
p > 0.78. All modes will be linearly unstable for a range of a when we take p < 0 and as
p → −1, the instabilities grow in strength. To illustrate the effect changing p will have
on σi, we consider the m = 4 wave mode (see Fig. 3.3). As we move from negative to
positive values of p, the instabilities become weaker, eventually leading to complete lin-
ear stability. This is not only true for m = 4: the general trend can be seen in all wave modes.
With our initial analysis complete, we now compare our theory to the full nonlinear
evolution. In the next subsection, we will look at several numerical simulations and discuss
the results for select values of p and a.
32. 20 Single-Layered Rotating Annulus
0.0 0.2 0.4 0.6 0.8 1.0
a
−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 Evolution
First, consider an annulus with a zero PV core (p = 0) with a = 0.78. Figure 3.4a shows
the progression of this annular set-up. Initially, the ring of the annulus breaks up into five
separate vorticity patches, each rotating around their own centre of mass. However, together
they rotate in the same ring formation as the original annulus. After some time, this vortex
patch formation breaks down even further (see panel 3 of Fig. 3.4a) resulting in one large
vortex patch orbited by much smaller patches. As the simulation runs to late times, the new
formation 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 within
the core. These patches then merge together leaving behind 5 satellites. As we run into late
times, 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 (with
LD = 10, which is effectively infinite LD). The annular ring breaks into a thinner ring with
six satellites forming in the core (which is expected from the stability analysis graphs) and
as the simulation progresses, the ring remains intact. For later times, the satellites reform to
create an asymmetric core dotted with very small PV=0 vortex patches. We have obtained
a case where our original axisymmetric vortex becomes asymmetric. It would be worthwhile
to 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 Stability
The majority of the analytical method having been illustrated in the previous section, we
will now briefly discuss the differences in the approach for finite LD. This time we must
33. 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 indicate
different 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 differential
equations:
1
r
d
dr
r
d ¯ψ
dr
−
1
L2
D
¯ψ = ¯Q (3.20)
1
r
∂
∂r
r
∂ψ′
∂r
+
1
r2
∂2
ψ′
∂θ2
−
1
L2
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
34. 22 Single-Layered Rotating Annulus
obtain
¯ψ =
αI0(x) − L2
Dp r < a
βI0(x) + γK0(x) − L2
D 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 and
r → ∞) and where α, β, γ and δ are constants. Requiring ¯ψ and ¯uθ to be continuous across
the 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 2.0
r
−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 considered
LD = 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).
35. 3.2 Finite LD 23
To ensure that our results return to those in the infinite LD case (c.f Eq. (3.4)), we
compare the forms of the basic state tangential velocity (see Fig. 3.5). As expected when
taking 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′
= ˆψ(r)ei(mθ−σt)
and reduce the (3.21) to a differential equation solely in r
r2 d2 ˆψ
dr2
+ r
d ˆψ
dr
+
−
r2
L2
D
− m2
ˆψ = 0 (3.28)
The solution to the above is a superposition of modified Bessel functions of the first and
second kind
ˆψ(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 to
formulate the eigenvalue problem for σ are rederived:
B2 =
(A1 − A2)Im(a/LD)
Km(a/LD)
(3.30)
B3 = A2
Im(1/LD)
Km(1/LD)
+
(A1 − A2)Im(a/LD)
Km(a/LD)
(3.31)
κ1 ˆη1 =
1
LD
[(A1 − A2)I′
m(a/LD) − B2K′
m(a/LD)] (3.32)
κ2 ˆη2 =
1
LD
[B3K′
m(1/LD) − A2I′
m(1/LD) − B2K′
m(1/LD)] (3.33)
−iσˆη1 + i
α
LDa
mI1(a/LD)ˆη1 = −i
m
a
A1Im(a/LD) (3.34)
−iσˆη2 − i
δ
LD
mK1(a/LD)ˆη2 = −imB3Km(1/LD) (3.35)
where κ1 =
(β − α)
2L2
D
(I0(a/LD) + I2(a/LD)) +
γ
2L2
D
(K0(a/LD) + K2(a/LD))
and κ2 =
(γ − δ)
2L2
D
(K0(1/LD) + K2(1/LD)) +
β
2L2
D
(I0(1/LD) + I2(1/LD))
.
After substituting (3.30) and (3.31) into (3.32) and (3.33), the Wronskian identity
Im(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 A12
A21 A22
ˆη1
ˆη2
= σ
ˆη1
ˆη2
(3.36)
36. 24 Single-Layered Rotating Annulus
where A11 = m
α
aLD
I1(a/LD) + mκ1Im(a/LD)Km(a/LD)
A12 = −
m
a
κ2Im(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 previous
result is recovered (c.f. Eq. 3.18), the stability of the annulus is then investigated for various
values 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 1.0
a
−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 1.0
a
−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 for
p = 0 (Fig. 3.6), there are instabilities present in all wave modes (m = 2, .., 6) for thin
annuli (0.6 ≤ a ≤ 1). However, for p → 1, the instabilities gradually arise for thicker annuli
and 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 by
the fact that ¯uθ → 0 as LD approaches zero (see Fig. 3.5). Considering p < 0, we see that
all modes have instabilities that, again, increase in strength (σi increases) as p → −1 (when
we 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 instabilities
for a range of a, with m = 2 having an instability over the largest range (0.2 ≤ a ≤ 1). The
majority of the analysis discussed for LD = 0.1 is applicable to LD = 1: the instabilities
37. 3.2 Finite LD 25
vanish as p → 1 and amplify as p → −1. The differing factor is that for larger LD values
the instabilities are inherently stronger for all values of p.
0.0 0.2 0.4 0.6 0.8 1.0
a
−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 1.0
a
−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 1.0
a
−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 1.0
a
−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. the
inner 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 for
discussion.
For LD = 10, we arrive at a particularly note-worthy result. This value for the Rossby
deformation 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, albeit
weak, instability (c.f Fig. 3.2 and Fig. 3.8). We further our investigation of this anomaly
38. 26 Single-Layered Rotating Annulus
0.0 0.2 0.4 0.6 0.8 1.0
a
−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 1.0
a
−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 series
expansions 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) =
1
x
+
x
4
(2 ln(x) + 2λ − 1 − ln(4)) (3.41)
K2(x) =
2
x2
−
1
2
(3.42)
where x = 1/L or x = a/L and λ is the Euler-Mascheroni constant (≈ 0.577). Substituting
equations (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 for
the m = 2 wave mode. For LD = 10, such that the arguments of the Bessel functions are
considered small, the asymptotic expansion is almost identical to the exact solution (see Fig
3.9b).
Furthermore, we calculated the value of p that causes the m = 2 mode to become
completely void of instabilities - p must be greater than 0.0041. Therefore, it only takes a
small 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. This
occurs when disturbances of the same wave number, m, on two interfaces move together
and 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 is
presented for the general annular case. In brief, to perform the analysis, the total angular ve-
39. 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 = 2
wave mode corresponds to a long wave, so the phase-locking argument requires the coupling
between 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 increasing
values of LD, the instabilities exist for lower values of a, which eventually leads to the
infinite LD case. The variation of p again shows that for p → 1 the instabilities vanish and
for p → −1 intensify.
0.0 0.2 0.4 0.6 0.8 1.0
a
−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 1.0
a
−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 1.0
a
−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 Evolution
Having completed the stability analysis, numerical simulations are considered for select
values of p and a with LD = 0.1, 1, 10. For each run, the results are compared with the
theoretical stability analysis to check whether we achieve the wave mode corresponding to
the parameters we chose. The stability analysis for the finite LD case indicates that for thin
annuli, we will have instabilities for all wave modes except for select p values (c.f. previous
section) 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
40. 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 indicate
different 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 completely
into five separate vortex patches which corresponds to our earlier stability analysis (see
Fig. 3.7 at a = 0.78). This indicates that the annular set-up was unstable and that the
vortex moved into a more stable configuration. Here, (un)stable does not refer to linear
stability - the vortex patches arise due to instabilities. We merely describe the patterns
produced as stable or unstable with respect to remaining in the annular set-up or not.
41. 3.2 Finite LD 29
As time progresses, these patches recombine into four unequal patches and for the rest of
the simulation time the paw-print like pattern remains present. If we look at the stability
graph again at a = 0.78, two wave modes have nearly the same growth rate: m = 4 has
σi = 0.292516 and m = 5 has σi = 0.278189. Although wave mode five prevails initially, it
is 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 the
original core. These satellites appear to form their own deformed annular configuration
and 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 plots
cross one another, indicating that either mode could dominate at this radius. However, it is
clear from the simulation that m = 5 wins and we obtain 5 distinct satellites. To further
show that m = 5 is marginally stronger than m = 6, we evaluate and compare the stability
graphs for both wave modes. From this, we see that m = 5 has a growth rate, σi that is
0.0006 larger than the growth rate for m = 6.
For other runs, the annulus was given a value of p = −0.25 to simulate what happens
for an annulus with a negative PV core (see Fig. 3.11c). For this case, the annulus initially
breaks into five satellites (PV=1) that almost immediately leave the annular set-up, thus
causing a further three distinct satellites (PV=-0.25) to form. The five (PV=1) satellites
recombine into four patches, which are orbited by several patches PV=-0.25, and as time
progresses, the vortex patches spread out while moving as dipoles. This simulation is an
instance of complete destruction of the hurricane-like vortex; possible reasons for this are
the instabilities are stronger than in the p ≥ 0 case, and the negative PV value cancels out
other vortex patches.
As stated in the linear stability section, the phase locking mechanism can be seen in
the simulations. Figure 3.11d shows that selecting p = 0, a = 0.5 and LD = 1, gives us a
m = 2 wave mode instability. Panel 2 of 3.11d shows an elliptical distortion that arises from
phase locking. Both interfaces move together and this results in each interface acting upon
the other, causing further deformation. As time progresses, the ellipses become elongated
until the ellipses turn over, resulting in thin vacillations. The ellipse’s semi-major axis then
shortens. 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 the
annular 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 with
p = 0 where a weak instability (σi = 0.009) is present for finite LD. If we take any other
value of p, simulations for infinite and finite LD will be closely similar.
42.
43. 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 illustrating
the effects of both rotation and stratification (Polvani, 1991). These are properties inherent
in a real hurricane. Figure 4.1 shows the set up that we are adopting for this chapter: the
two layers have mean depths (H1,H2) and constant densities (ρ1,ρ2) and here the ratio
of these quantities are varied along with the inner radius of the annulus and the Rossby
deformation length. The PV for both layers is assumed to be identical and has the same
form as Eq. (3.1).
Figure 4.1: A schematic illustrating the structure of the two-layered axisymmetric vortex. A top down view
is also shown to indicate the exact distribution of the potential vorticity in both layers.
4.1 Linear Stability
The equations (2LQGSWE) that are be used within this chapter describe the same analytical
problem as in the previous chapter: the finite and infinite LD equations must be solved
together to obtain an analytical solution. However, this means that the stability analysis
need 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 over
the 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 fixed
Rossby deformation length (LD = 0.1, 1, 10) and azimuthal wave modes (m = 2, ..., 6). Phase
diagrams are constructed to illustrate the values of p and a that cause baroclinic instabilities
to 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
44. 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 dominate
over barotropic instabilities for LD = 1.
but at the cost of the range of a (the range decreases for the associated, increasing, p
values). The baroclinic dominant regime also exists for negative p values but exclusively
for the higher wave modes and for a very short range of a, which means there is only a
small set of parameters that cause multiple wave modes to be excited for negative p. The
phase diagrams for LD = 0.1 and LD = 10 show similar variations. However, for decreasing
LD, the baroclinic dominant regime exists only for very thin annuli (which corresponds to
the stability analysis in chapter 3). For large LD, the maximum p values, that cause the
baroclinic 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 parameters
that can be examined has increased from three to five. This means that performing a
complete analysis involving a variation of each of the parameters would take a considerably
long 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 the
two layers (i.e. the baroclinic dominant regime).
Figure 4.3a illustrates what happens to the annular vortex for p = 0.25, a = 0.8 and
α = 0.5. In both layers, the annulus breaks into seven distinct satellites (PV=1) that rotate
within the core in the original ring structure. Panel 3 of Fig. 4.3a shows the variation
between the upper and lower layers. In the lower layer, the satellites have developed thin
45. 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 indicate
different values of PV: 0, 0.25, 0.5 and 1.
vacillations which have curled over to connect to the satellites that are neighbouring. This
creates the appearance of seven more satellites that have PV=0. This is inherently different
to the structure in the upper layer, where no vacillation ‘tails’ are produced. As time
progresses, the PV=1 satellites perturb from the ring and recombine into three patches.
During this recombination, the core becomes chaotic and eventually disintegrates. At late
times, 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) hangs
together.
46. 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 vacillations
that cause six satellites (with PV=1) to form. However, these satellites almost immediately
dissolve into the annular ring and create an asymmetric annular vortex. Similarly, the upper
layer also produces vacillations which result in six equal PV=1 satellites that are held within
the core and separated from the annular ring by a thin PV=0.5 border. The vacillation
process happens again and broadens the gap between the satellites and the ring. These
satellites 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, an
asymmetric vortex forms.
47. Chapter 5
Concluding Remarks
Through the use of two sequential models, we have presented an extension of previous
works 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 analysing
the 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 for
thin annuli, will either be enhanced (p < 0) or diminished (p > 0). We also investigated
the finite LD parameter. For LD → 0, instabilities only arise for exceptionally thin annuli
until they disappear entirely, while large LD causes the stability analysis to revert to the
infinite case. Based on our stability analysis, numerical simulations were run for select p, a
and LD. These simulations showed that the initial state of the annulus could develop into
five distinct states: vortex patches rotating around a central patch; vortex patches rotating
within the original core; an asymmetric annulus; dipole translation and miniature annuli
rotating around the original core. We intend to carry out further simulations to ascertain
the parameter ranges that lead to these different states.
We extended the first model to include two layers with the same PV profile in each
layer. Due to this, we found that the linear stability analysis was identical to the first
model. Using the analysis, we investigated the possible parameters that cause the baroclinic
mode to dominant the barotropic mode. We found that the baroclinic dominant regime
exists 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 a
that facilitate this is very small. We chose to restrict our numerical treatment by keeping h1
and LD fixed while varying a, p and α. This restriction is due to the computational expense
that would occur if we investigated all possible variations of the parameters in addition to
time constraints of the project. Based on the constructed phase diagrams, we selected values
of a and p that were most likely to cause the baroclinic dominant regime and ran simulations
for these values. The simulations showed that the annulus structure remained inherently
barotropic with some slight baroclinic variations. In this model, we saw the initial annulus
develop into two distinct states: an asymmetric annulus and complete destruction of the
annulus into a swirling, chaotic vortex. There are several avenues that could be investigated
further. 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 simulate
a more realistic hurricane. Previously (see chapter 1), we discussed the processes involved in
hurricane formation, such as convection and air-sea interactions. The next logical step would
be to include these processes in our analytical model and investigate the flow modification
induced by these changes. However, to include all physical processes in one step would
prove overly complex. Therefore, it is advisable to introduce the processes in several steps.
48. 36 Concluding Remarks
Initially, we could modify the second model to include the compressible form of the
QGSWE such that a more realistic atmosphere is simulated. The next step would be to
include diabatic forcing through heating in the lower layer and cooling in the upper layer.
This would crudely simulate convection processes. Convection is particularly crucial in
the formation of hurricanes due to its role in the initial intensification. Considering a
model that includes convection may give rise to interesting dynamical behaviour for our
idealised hurricane. Two further modifications that could be made are introducing the
air-sea boundary and exploring the effect of large scale environmental shear as both of these
also 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 to
investigate the structure and stability of a more realistic hurricane.
49. References
Ahrens, C.D. 2006 Meteorology Today: An Introduction to Weather, Climate, and the
Environment, 8th edn. Cengage Learning.
Anthes, R.A., Rosenthal, S.L & Trout, J.W. 1975a Preliminary results from an
asymmetric model of the tropical cyclone. Mon. Wea. Rev. 99, 744–758.
Anthes, R.A., Rosenthal, S.L & Trout, J.W. 1975b Comparisons of tropical cyclone
simulations 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 the
barotropic vorticity equation. Tellus 2, 237–254.
Dritschel, D.G. 1986 The nonlinear evolution of rotating configurations of uniform
vorticity. J. Fluid. Mech. 172, 157–182.
Dritschel, D.G. 1988b Contour surgery: A topological reconnection scheme for extended
integrations using contour dynamics. J. Comput. Phys. 77, 240–266.
Dritschel, D.G. 1989 Contour dynamics and contour surgery: Numerical algorithms
for 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 of
a 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 scheme
based 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.
50. 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 hurricane
model. 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 balanced
low-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-layer
v-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 on
low-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 fresh
look 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 Extratropical
Cyclones: 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-scale
circulation. Cambridge University Press.
Warneford, E.S. & Dellar, P.J. 2013 The quasi-geostrophic theory of the thermal
shallow 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-geostrophic
regime. J. Fluid. Mech. 649, 187–203.
Yamasaki, M. 1968a Numerical simulation of tropical-cyclone development with the use of
primitive equations. J. Meteorol. Soc. Jpn. 46, 178–213.
Yamasaki, M. 1968b A tropical-cyclone model with parameterized vertical partition of
released latent heat. J. Meteorol. Soc. Jpn. 46, 202–214.
51. References 39
Yamasaki, M. 1968c Detailed analysis of a tropical-cyclone simulated with a 13-layer
model. Papers in Meteorolgy and Geophysics 20, 559–585.
Zabusky, N.J., Hughes, M.H. & Roberts, K. 1979 Contour dynamics for the euler
equations in two dimensions. J. Comput. Phys. 30, 96–106.
Zhu, H., Smith, R.K. & Ulrich, W. 2001 A minimal three-dimensional tropical cyclone
model. J. Atmos. Sci. 58, 1924–1944.
