dynamics of vortex rossby waves in tropical cyclones
TRANSCRIPT
D ynam ics o f vo rtex R ossb y w aves in trop ica l cyclon es
by
Lidia N ik itin a
A thesis submitted to
the Faculty of Graduate and Postdoctoral Affairs
in partial fulfillment of
the requirements for the degree o f
Doctor of Philosophy
in
Mathematics
Carleton University
Ottawa, Ontario
© 2013
Lidia Nikitina
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A b stract
This thesis describes an analytical study of vortex Rossby waves in tropical cy
clones. Observational analyses of hurricanes in the tropical atm osphere indicate the
existence of spiral rainbands which propagate outwards from th e eye and affect the
structure and intensity of the hurricane. These disturbances may be described as
vortex Rossby waves.
The aim of this research is to study the propagation of vortex Rossby waves in
tropical cyclones and wave-mean-flow interactions near the critical radius where the
mean flow angular velocity matches the phase speed of the waves. Depending on
the wave magnitude, the problem can be linear or nonlinear. Analytical techniques
including Laplace transforms, multiple scaling and asym ptotic expansions are used to
obtain approximate solutions of the governing linear and nonlinear equations. In this
study we carry out asymptotic analyses to examine the evolution of the interactions
near the critical radius in some two-dimensional configurations on an /-p lane and a
/?-plane.
The results are used to explain some features of the tropical cyclone’s develop
ment, namely, the change of angular wind in the critical layer, the secondary eyewall
formation and the eyewall dynamics.
A ck n ow led gem en ts
I would like to express my deepest gratitude to my supervisor, Prof. Lucy Camp
bell, and thank her for her perm anent support and patience, for her wisdom and
kindness, for her high-level of professionalism both in research and teaching, for the
great time I spent at the School of M athematics and S tatistics during the course of
my thesis work.
I appreciate the time and effort the external examiner Prof. Simon Clarke and the
committee members Prof. Dave Amundsen, Prof. Arian Novruzi, and Prof. Abhijit
Sarkar put into reading and reviewing my thesis in spite of the ir busy schedules.
I would like to thank Dr. G ilbert Brunet for bringing this interesting problem to
our attention. I also would like to thank him for helpful discussions.
This thesis project was supported by N atural Sciences and Engineering Council
of Canada (NSERC) for three years with Alexander G raham Bell Canada Graduate
Scholarship.
C ontents
T itle page i
A bstract ii
A cknow ledgm ents iii
List o f Tables v i
List o f F igures v ii
List o f A ppendices v iii
1 In troduction 1
1.1 Motivation for the study and overview of the t h e s i s .................................. 1
1.2 Study of tropical cyclones .................................................................................. 4
2 B asic con cep ts o f geophysica l fluid dyn am ics 13
2.1 Basic equations of fluid m echan ics........................................................................ 14
2.2 Effects of ro ta t io n ................................................................................................... 16
2.3 The Coriolis effect for a thin layer on a ro tating s p h e r e ................................ 21
2.4 Waves in the ocean and the a tm osphere ..............................................................23
iv
V
3 V ortex dynam ics in th e a tm osph ere 26
3.1 Barotropic vorticity eq u a tio n ................................................................................... 26
3.2 Barotropic Rossby w a v e s .......................................................................................... 34
3.3 Vortex Rossby w a v e s .................................................................................................36
3.4 Previous theoretical studies of tropical cyclone d y n a m ic s ............................ 39
4 Steady linear problem 46
4.1 Steady linear solution for some special c a s e s .....................................................46
5 T he linear tim e-d ep en d en t prob lem 64
5.1 Laplace transform of the linear time-dependent e q u a t i o n ............................ 65
5.2 Evaluation of the inverse Laplace transform .....................................................75
5.3 Time-dependent solution close to the critical la y e r ...........................................88
5.4 The inner layer solution .......................................................................................... 91
5.5 Discussion of the linear time-dependent s o lu t io n ..............................................94
6 Effect o f non linearity and /3-effect 97
6.1 Weakly-nonlinear a n a l y s i s .......................................................................................97
6.2 Effect of the variation of the Coriolis f o r c e ...................................................... 101
6.3 Combined effects of nonlinearity and the /3-effect............................................ 109
6.4 Evolution of the mean flow in the inner la y e r ................................................... 114
7 C onclusions 118
7.1 S u m m a r y ....................................................................................................................118
A H ypergeom etric fu n ction s 122
A .l Generalized hypergeometric fu n c tio n s .................................................................122
A.2 The solutions of the hypergeometric equation ................................................123
vi
B N onlinear evo lu tion o f th e m ean flow 128
C T he phase sh ift for th e stead y so lu tion 131
B ibliography 134
List o f Tables
6.1 Wavenumbers corresponding to the term s in equation (6.49)
List o f F igures
1.1 Tropical cyclone s t r u c t u r e ................................................................................. 7
1.2 Hurricane Ivan 2004, secondary eyewall fo rm a tio n .........................................11
2.1 Vector in a rotating f r a m e .....................................................................................17
2.2 Centripetal acceleration ........................................................................................ 19
2.3 Local Cartesian c o o rd in a te s ................................................................................. 21
4.1 The domain of the problem ................................................................................. 48
4.2 The angular velocity profile ................................................................................. 50
4.3 Intervals for the steady s o lu t io n ...........................................................................57
4.4 The am plitude of the steady solution....................................................................62
4.5 Contour plots for the steady s o lu t io n .................................................................63
5.1 The contour of integration in the complex s-plane. Case 1 and 2. . . . 69
5.2 The contour of integration in the complex s-plane. Case 3 ..........................70
6.1 Influence of /3-effect on the vortex Rossby wave propagation ..................106
6.2 Schematic diagrams for w avenum bers...............................................................I l l
6.3 Dynamics of the critical radius location............................................................. 117
A .l The hypergeometric functions............................................................................... 127
viii
ix
List o f A ppendices
A Hypergeometric functions.........................................................................................122
A 1 Generalized hypergeometric functions .........................................................122
A 2 The solutions of the hypergeometric equation ..........................................123
B Nonlinear evolution of the mean flow .................................................................. 128
C The phase shift for the steady solution .............................................................. 131
C hapter 1
Introduction
1.1 M otivation for th e s tu d y and overv iew o f th e
th esis
The effect of the Coriolis force gives rise to large scale oscillations in the atmosphere
and ocean which are known as Rossby waves. These waves affect weather and cli
mate and play a role in various observed phenomena including the development of
hurricanes or tropical cyclones. The dynamics of tropical cyclones have been studied
extensively during the past few decades using observational data, theoretical mod
eling and numerical simulations. The motivation for understanding and simulating
hurricanes is based on the profound effects th a t hurricanes can have on human life as
one of the most devastating disasters. But a full theory of hurricane dynamics is far
from complete.
Vortex Rossby waves propagating within tropical cyclones are known to play an
im portant part in cyclone development and evolution. The waves affect the strength,
structure and dynamics of a cyclone. In particular, vortex Rossby waves are believed
1
C H APTER 1. INTRO D U CTIO N 2
to play a role in the formation of the secondary eyewall, an outer ring of intense
thunderstorms th a t develops outside the initial eyewall of the hurricane and eventually
replaces the original eyewall.
In this thesis we present a simple two dimensional configuration th a t represents
Rossby wave propagation in a cyclonic vortex and describes a possible mechanism for
the secondary eyewall formation. Approximate analytic solutions are derived using
asymptotic methods and weakly-nonlinear analyses. Previous analytic investigations
of this mechanism have been based on linearized equations th a t did not include the
effects of the gradient of the Coriolis force. The addition of nonlinearity and Coriolis
effects are im portant features of this thesis project.
An overview of the thesis is as follows. The rest of this chapter gives background
information about tropical cyclones. It includes a description of cyclones and the
physical processes involved in the development of the tropical cyclones, as well some
basic information about the structure of cyclones, their life cycle, intensity. In partic
ular, the secondary eyewall replacement cycle is described.
Chapter 2 gives a mathem atical description of the basic features of geophysical
fluid dynamics, including the effect of the E arth ’s rotation, and a description of Rossby
waves.
Chapter 3 deals with analysis of vortex dynamics in the atmosphere. This chapter
includes the derivation of the basic equations th a t are solved in the thesis. These
equations describe tropical cyclones as well as the propagation of vortex Rossby waves
in tropical cyclones. Chapter 3 also gives information about some basic approaches
to the study of the dynamics of tropical cyclones.
Chapters 1-3 present introductory m aterial on hurricanes, geophysical fluid dy
namics, and Rossby waves based on standard texts and papers on these topics. The
derivation of the basic equations follows K undu and Cohen (2004), Holton (1992),
C H APTER 1. INTRO DU CTION 3
Landau and Lifshitz (1953), and Gill (1982). The equations for the vortex waves are
based on the studies of Montgomery and Kallenbach (1997) an d Brunet and Mont
gomery (2002).
The description of my original work sta rts in C hapter 4. I represent a tropical
cyclone as a vortex in a two-dimensional configuration. The vortex Rossby waves
are considered as small perturbations to the basic rotation of the cyclone, which
are periodic in the azimuthal direction and have am plitude th a t varies in the radial
direction. I derive a solution of the governing equation for a configuration in which
the wave am plitude is steady. The waves propagate on an /-p lane , a horizontal plane
in which the Coriolis force is taken to be a constant. The waves are assumed to be
of small amplitude and the governing equation is linearized. T he problem is defined
in term s of polar coordinates. The waves are generated by a boundary condition,
periodic both in time and in the azimuthal direction, a t some fixed distance from the
centre of the vortex.
The steady linear problem is governed by an ordinary differential equation which
describes the wave amplitude in term s of the radial variable. T he equation is singular
at the radius where the cyclone angular velocity is equal to the phase speed of the
waves. This is called the critical radius and the region close to this radius is called
the critical layer. The solution of the ordinary differential equation is expressed in
terms of hypergeometric functions.
In Chapter 5 a Laplace transform is used to derive an approxim ate time-dependent
solution for the linear problem. I derive an approxim ate solution valid in the outer
region, far from the critical layer, as well as in the inner region inside the critical layer.
In Chapter 6 I include the effects of nonlinearity and the “/3-effect” which arises
from the latitudinal variation of the Coriolis force. In section 6.1 I carry out a weakly-
nonlinear analysis of the nonlinear equation considering th e nondimensional wave
C H APTER 1. INTRO D U CTIO N 4
amplitude e as a small param eter w ithout taking into account th e /3-effect. In section
6.2 the /3-effect is included in the linear problem and in section 6.3 I discuss the effect
of including both nonlinearity and the /3-effect. The nonlinear interaction between
vortex waves and the mean flow changes the angular m om entum and changes the
mean flow speed in the inner layer. This is discussed in section 6.4. I also give some
explanation for the secondary eyewall formation and the eyewall replacement cycle.
Appendix A describes the properties of the hypergeometric equation, its singular
points, and its solutions th a t I use in my thesis. Appendix B gives the derivation of
the angular momentum equation th a t I use in Chapter 6 to examine the nonlinear
evolution of the angular velocity of the mean flow. Appendix C explains the phase
shift of a singular term in the steady solution.
1.2 S tu d y o f trop ica l cyclon es
A cyclone is a system of ro tating fluid flow w ith sustained w inds in the form of an
axisymmetric vortex. In this section a brief description of tropical cyclones, their
development and their structure is given following Holton (1992), Ogawa (1992) and
the review article of Emanuel (2003). The eyewall replacem ent cycle is described.
Tropical cyclones
A tropical cyclone is a cyclone th a t originates over tropical oceans and is driven by
heat transfer from the ocean. Tropical cyclones are categorized according to their
maximum wind speed, defined as the maximum speed of th e wind a t an altitude of
10 m. Tropical cyclones w ith maximum wind speeds in the range 1 5 -1 7 ms-1 , are
known as tropical depressions; when their speeds are in the range of 18 to 32 ms-1 ,
they are called tropical storms. Tropical cyclones with m aximum winds of 33 ms-1 or
C H APTER 1. INTRO D U CTIO N 5
greater are called hurricanes in the western North A tlantic and eastern North Pacific
regions; in the South Pacific and Indian Oceans they are usually called cyclones; in
East Asia, e.g. Japan and China, they are usually called typhoons. O ther names used
include “cordonazo” in Mexico, “bagyo” in the Philippines, and “tainos” in Haiti.
The kinetic energy of these large scale vortex motions is abou t 1018 Joules. Trop
ical cyclones are among the most spectacular and deadly geophysical phenomena.
Tropical cyclones th a t cause extreme destruction are rare, although when they occur,
they can cause great amounts of damage or thousands of fatalities. The 1970 Bhola
cyclone is the deadliest tropical cyclone on record, killing more than 300,000 people
and potentially as many as 1 million after striking the densely populated Ganges
Delta region of Bangladesh on 13 November 1970. The N orth Indian cyclone basin
has historically been the deadliest. The G reat Hurricane of 1780 is the deadliest A t
lantic hurricane on record, killing about 22,000 people in the Lesser Antilles in the
Caribbean. A tropical cyclone does not need to be extremely strong to cause memo
rable damage; deaths can be caused by rainfall, flood, or mudslides. The Galveston
Hurricane of 1900 is the deadliest natural disaster in the U nited States, killing an
estimated 6,000 to 12,000 people in Galveston, Texas. Hurricane K atrina is estimated
as the costliest tropical cyclone worldwide, causing $ 81.2 billion in property damage
with overall damage estimates exceeding $ 100 billion. K atrina killed a t least 1,836
people after striking Louisiana and Mississippi as a major hurricane in August 2005.
Damage from Hurricane Sandy in 2012 is estim ated as $ 50 billion. Hurricane An
drew (1992) is also one of the most destructive tropical cyclones in U.S history, with
damages totaling $ 40.7 billion.
Up until World War II, detection of tropical cyclones relied on reports from coastal
stations, islands, and ships a t sea. At this time, it is likely th a t many storms escaped
detection entirely, and some of them were observed only once or a few times during
C H APTER 1. INTRO DU CTION 6
the course of their life cycle. During World War II, m ilitary aircraft were tasked with
finding tropical cyclones tha t could pose a danger to naval operations. An aircraft
was able to penetrate the interior of strong storm s for the first tim e in July 27, 1943,
flying into the eye of a Gulf of Mexico hurricane. At about th e same time, the first
radar images revealed the structure of tropical cyclones including the eye and spiral
rainbands. A great step forward came in 1960, when the first image of a tropical
cyclone was transm itted from a polar orbiting satellite. Now all tropical cyclones are
recorded by satellite based measurements.
T h e s t ru c tu re o f a tro p ic a l cyc lone
The radius of a tropical cyclone is generally between 100 and 2000 km. A cyclone of
high intensity tends to develop an eye, an area of relative calm (and lowest atmospheric
pressure) a t the center of circulation. The eye is often visible in satellite images as a
small, circular, cloud-free spot. Surrounding the eye is the eyewall, an area about 16
km to 80 km wide in which the strongest thunderstorm s and winds circulate around the
storm ’s center. The maximum sustained winds in the strongest tropical cyclones have
been estimated at about 85 ms-1 . Maximum upflow occurs in the eyewall. In intense
storms, clouds over the eyewall may form a nearly concentric ring. There may be two
or even three eyewalls present a t the same time, evolving through a characteristic cycle
in which the outer eyewalls contract inward and replace dissipating inner eyewalls.
The earliest radar images of tropical cyclones showed th e outward-propagating
spiral rainbands tha t accompany them. They consist of bands of cumulus clouds,
some tens of kilometers wide with radial propagation speeds of about 4 ms-1 , some
times connecting with the eyewall. Outside the eyewalls the w ind speed and intensity
decrease gradually at first and then fall off faster a t large radius.
Tropical cyclones are formed in four m ajor phases. The first stage occurs when a
CHAPTER 1. INTRO D U CTIO N 7
D*m* Cirrus Ovsrcsst
Figure 1.1: Tropical cyclon e stru ctu re. The main parts of a tropical cyclone are the eye, the eyewall, and the rainbands. This figure is taken from http://serc.carleton.edu/research education/katrina/understanding.htm l.
collection of small thunderstorm s forms over the tropical ocean. The hurricanes th a t
threaten the East and Gulf coasts of the United States, for example, are the product
of storms th a t first form over the coast of western Africa and head west across the
Atlantic Ocean. The cyclones th a t strike East Asia and A ustralia form over the central
Pacific. Australia is also affected by cyclones which form in th e North-W estern Indian
Ocean and Torres Strait. Tropical cyclones are formed between 5° and 20° latitude,
because of the high humidity, light winds, and warm sea surface tem peratures present
in these locations. Cyclones and hurricanes can only form w hen the ocean water is
around 26.5°C or higher, and these conditions are present in these latitudes during
certain months of the year.
Weather conditions in the central Atlantic and Pacific Oceans are most favorable
for hurricane formation during the summer and early fall m onths, which is why the
period of June to October is often referred to as “Hurricane Season” , in the north
C H APTER 1. INTRO D U CTIO N 8
ern hemisphere. The opposite situation occurs in the southern hemisphere since the
tropical waters of Australia are warmest from December to April.
A tropical disturbance officially enters the second stage of development called a
tropical depression once the sustained wind speed w ithin the storm reaches 15 ms-1 .
The term tropical depression refers to the falling surface pressures measured in the
region surrounding the storm. The pressure drop occurs as w ater vapor within the
storm condenses into water droplets and releases latent heat into the atmosphere.
Latent heat is defined as the energy released when a substance changes phases, such
as gas to liquid in this case.
This process becomes a chain reaction th a t pulls hot, humid air from the surface of
the ocean up to high altitude where the air becomes cold and w ater vapor condenses
into thick clouds. This growing air mass becomes increasingly dense causing the
atmospheric pressure to grow. The increasing pressure pushes the growing mass of
clouds outward away from the center to create the spiraling bands of clouds th a t
hurricanes are known for. As the air mass spirals outward, its pressure decreases and
the dense air plunges back towards the ocean surface where i t started. It now picks
up vapor again from the warm waters below and is sucked back into the center of the
depression to begin its journey anew. As the cycle continues, the surface pressure at
the center drops lower, causing the circulation of the air to strengthen and the winds
to grow increasingly stronger.
Once the sustained wind speed increases to 18 ms-1 , the tropical depression enters
the third stage of development called a tropical storm. Until now, the storm has not
had any formal designation. Tropical depressions are usually numbered in a sequential
order as they appear, but it is not until they become tropical storm s tha t they receive
an official name. The tradition of naming storms dates back to the arrival of the
Europeans in the Americas, bu t the current m ethod of system atically naming tropical
CH APTER 1. INTRO DU CTION 9
storms was not formalized until 1953. Female names were used exclusively until the
late 1970s when the use of both male and female names in an alternating format was
adopted.
It is also during the tropical storm phase th a t the storm begins to take on its
characteristic appearance as a ro tating spiral w ith bands of clouds circulating around
the point of lowest surface pressure a t its center.
When the wind speed reaches 33 m s-1 , the tropical storm is officially classified as
a tropical cyclone or hurricane. This is the fourth stage of the hurricane development.
The storm now develops an eye a t the center of the circulating spirals where there
is very little wind, few clouds, and extremely low atm ospheric pressure. The eye
may extend from 8 to 65 km in diam eter. Surrounding the eye is an area called the
eyewall tha t circulates around the eye and contains the highest clouds. The eyewall
is the region containing the strongest winds, heaviest precipitation, and most intense
thunderstorms of any part of the hurricane. The most destructive region of the cyclone
is the portion of the eyewall ro tating in the direction of the forward motion of the
storm since the force of the circulating winds is combined w ith the motion of the
storm. Radiating outward from the center are the spiraling bands of clouds tha t
produce heavy rain and wind but are still considerably weaker than the eyewall.
Cyclones typically move westward a t a speed of about 4 m s-1 during the early
stages of formation. The storms are steered in this direction by the trade winds tha t
occur near the equator and blow towards the west. As a sto rm moves further from
the equator, its course is turned towards the poles and the cyclone curves towards the
north in the northern hemisphere or towards the south in th e southern hemisphere.
Tropical storms tha t form in the northern hemisphere always ro tate in a counter
clockwise direction due to the Coriolis force induced by the E arth ’s rotation, while
tropical storms of the southern hemisphere ro tate clockwise.
C H APTER 1. INTRO D U CTIO N 10
E yew all rep la cem en t c y c le
Eyewall replacement cycles occur naturally in intense tropical cyclones. Fortner (1958)
was the first to document the concentric eyewall and eyewall replacement cycle in
Typhoon Sarah in 1956. W hen a tropical cyclone reaches its highest intensity, the
asymmetric ring of outer rainbands contracts and strengths and organizes into a ring
of thunderstorm s called an outer eyewall, or secondary eyewall, around the existing
inner eyewall (Willoughby et al., 1982). As a result, the inner eyewall weakens and is
eventually replaced by the more intense outer eyewall. This process takes place on the
order of a day or two (Sitkowski et al., 2011). This process of weakening followed by
reintensification is called the eyewall replacement cycle. Recent measurements have
shown tha t nearly half of all tropical cyclones attaining a t least 60 ms-1 wind speeds
undergo the concentric eyewall replacement cycle (Hawkins and Helveston, 2008).
In these hurricanes the asymmetric outer rainbands form the ir own convective ring
(secondary eyewall) in coincidence w ith a local tangential wind maximum around the
inner eyewall (Willoughby et al., 1982). Figure 1.2 shows a rad ar image of Hurricane
Ivan (2004) with a developed secondary eyewall.
The motivation to understand and forecast the eyewall replacement cycle is high,
because the process can have very serious consequences arising from dram atic intensity
and structure changes. The formation of an outer eyewall can cause a decrease in the
maximum wind speed at the location of the initial eyewall, while the outer wind
maximum tends to broaden the hurricane wind field which can lead to an increase of
integrated kinetic energy (Maclay et al., 2008). This has profound impacts both near
and far from coastal regions. A sudden expansion of hurricane force winds near landfall
can impact a larger coastal area while reducing preparation tim e. W hen a hurricane
is farther from shore, an increased hurricane wind field is likely to lead to a greater
C H APTER 1. INTRO D U CTIO N 11
04Q9MI Ivan• 4 • t - s t ’i j I T ' *c '3JS 38* '*>»>, /WO•1- 1; -.m r t f ’~& ■-. > ’ PM *•!«. i - - h r * * ‘ e t j r r t * ^*«wnr»
Secondary eyewall
Primary eyewall
Figure 1.2: H u rr ic a n e Ivan . Radar image of Hurricane Ivan (2004) showing the secondary eyewall formation around the prim ary eyewall.This image is taken from the website of the NOAA H urricane Research Departm ent http://www.aoml.noaa.gov/hrd/Storm_pages/ivan2004/i040909il640_1700c.jpg
C H APTER 1. INTRO D U CTIO N 12
storm (Irish et al., 2008). In addition, the contraction of an outer eyewall near the end
of an eye replacement cycle can sometimes lead to rapid intensification resulting in a
more intense hurricane than when the process of eyewall form ation began. This was
the case for Hurricane Andrew (1992), which intensified to a category 5 hurricane as
it approached the southeast coast of Florida during the end of a n eyewall replacement
cycle (Willoughby and Black, 1996). Observations of the hurricane season of 2005
(Houze et al., 2007) showed th a t the winds in the secondary (or outer) eyewall were
initially weaker than those in the original prim ary (or inner) eyewall, bu t as the
secondary eyewall contracted, the storm reintensified. The processes occurring in the
moat region between the old and new eyewalls became dynam ically similar to those
in the eye.
C hapter 2
Basic concepts o f geop h ysica l fluid
dynam ics
In this chapter I give a basic description of geophysical fluid dynam ics and its appli
cation to wave and vortex processes in the atmosphere.
The term geophysical fluid dynamics refers to the dynam ics of fluid flows, partic
ularly air and water, in the E arth ’s environment. The discipline includes the study of
both fluid phases: liquids (waters in the ocean) and gases (in th e E arth ’s atmosphere
and in the atmospheres of other planets). Usually geophysical fluid dynamics deals
with large-scale flows, and the rotation of the E arth or density differences (warm and
cold air, fresh and saline waters) are im portant for these processes. Typical problems
in geophysical fluid dynamics concern the variability of the atm osphere (weather and
climate dynamics), of the ocean (waves, vorticity and currents),the motions in the
E arth ’s interior responsible for the magnetic dynamo effect, vortices on other planets
(such as Jupiter’s G reat Red Spot), and convection in stars (the Sun, in particular).
We live within the atmosphere and are ever affected by th e weather and its rather
chaotic behaviour. The motion of the atmosphere is connected with th a t of the ocean,
13
C H APTER 2. BA SIC CO NCEPTS OF G EO PH YSICAL FLUID D YN AM IC S 14
with which it exchanges fluxes of momentum, heat and m oisture, and it makes the
dynamics of the ocean as im portant as the atmosphere.
The two features th a t distinguish geophysical fluid dynam ics from other areas of
fluid dynamics are the rotation of the E arth and the vertical density stratification of
a medium because of the vertical gradient of tem perature or salinity.
The effect of the rotation of the E arth takes the form of th e Coriolis force and
gives rise to large scale oscillations in the atmosphere and ocean which are known
as Rossby waves. The effects of stratification and the gravitational force give rise to
internal gravity waves and surface gravity waves. These waves have profound effects
on weather and climate and play a role in various observed phenomena including
the development of tropical cyclones. This thesis deals with vortex Rossby waves
which propagate in a tropical cyclone and have an influence on the cyclone strength,
structure and dynamics.
The motion of fluid in the atmosphere and the ocean is naturally studied in a
coordinate frame rotating with the Earth. In this chapter we discuss the basic theory
of Rossby waves in a geophysical fluid th a t is affected by th e Coriolis force. We
describe these effects in the governing equations following Landau and Lifshitz (1953),
Kundu and Cohen (2004), Holton (1992), and Gill (1982).
2.1 B asic equations o f fluid m ech an ics
The study of fluid mechanics is based on the laws of conservation of mass, momentum,
and energy.
C H APTER 2. BA SIC CO NCEPTS OF GEO PH YSICAL FLUID D YN AM IC S 15
C on serva tion o f m ass
The conservation of mass equation is
^ + V • (pu) — 0, (2.1)
where p is the density and u is the fluid velocity. This equation is called the continuity
equation and expresses the differential form of the principle of conservation of mass.
C on serva tion o f m o m e n tu m
The momentum equation
Dui dp d~ + P 9 i +Dt dxj dxj
. dui duj . _ _+ 3 ' , v ' u '5”
(2 .2 )
is known as the Navier-Stokes equation. The notation refers to the full derivative,
D d d dxi .D t = dt + d x i~ d t ' ^
Here p is the pressure, g is the gravity acceleration, p is the coefficient of dynamic
viscosity, i and j correspond to coordinate axes, the repeated indices mean summation
over i — 1 ,2 ,3 or j = 1,2,3. For an incompressible fluid the momentum equation is
written in the form:DUi d p cy
p ^ t = ~ a i i + l,9- + 'i V ( 2 ' 4 )
or in vector form asD u 0
p— = - V p + pg + p V u. (2.5)
For atmospheric dynamics the viscous term in the m om entum equation is small
CH APTER 2. BA SIC CO NCEPTS OF G EO PH YSICAL FLUID D YN AM IC S 16
relative to other terms. For hurricane dynamics the length scale is L ~ 105m and
the velocity scale is U ~ 10 m s-1 . The kinematic viscosity of the atmosphere is
u = n /p ~ 10~5m2s_1. Hence, the Reynolds number which is a measure of the ratio
of inertial forces to viscous forces Re = U L /u ~ 1011 is very large for hurricane-scale
flows, and we can neglect the viscous term s in the momentum equation
D u _ , ,p— = - Vp + pg. (2.6)
C o n serv a tio n o f e n e rg y
The conservation of energy equation is
p|5 = - „ ( V . u ) + 4 > - | | , ( 2 .7 )
where e is the internal energy of the system, vector q is the h ea t flux through a unit
area, (j> is the rate of viscous dissipation. This is called the therm al energy equation.
The set of equations (2.1), (2.2) and (2.7) is used as the s ta rting point for studies of
fluid dynamics.
2.2 E ffects o f ro ta tion
In studying the motion of the fluid in the atm osphere we often have to take into
account the rotation of the Earth. The E arth ’s rotation is im portan t when the scales
of the processes under consideration are large and comparable w ith the E arth ’s radius.
Following the description given in Kundu and Cohen (2004) and Holton (1992), we
consider a frame of reference ( a q , :^ ,^ ) ro tating w ith a uniform angular velocity f I
CH APTER 2. BA SIC CO NCEPTS OF G EO PH YSICAL FLUID D YN AM IC S 17
Figure 2.1: V ector in a ro ta tin g fram e. The vector i ro ta tes with the reference frame. O is the angular velocity of the frame rotation, a is th e angle between the vector i and the axis of rotation. The change in the angle of ro tation d9 causes the change in the vector di.
with respect to a fixed frame. Any vector P is represented in th e rotating frame by
P — P iii + P 2I2 + ^313- (2 .8 )
To a fixed observer the directions of the rotating unit vectors i i , i2 and i3 change with
time. To this observer the time derivative of P is
dPi dP2 dPi dt 12 dt 13 dt
where we have used the notation dij /dt , j = 1,2,3, to represent the rates of change
of the unit vectors. To an observer in the ro tating frame, th e rate of change of P is
CH APTER 2. BA SIC CO NCEPTS OF G EO PH YSICAL FLUID D YN AM IC S 18
the sum of the first three terms in (2.9), so th a t
(f~)F = ( ^ ) c Pl + P2 + Ps ’ ( 2 ' 1 0 )
where the subscript of F refers to the fixed frame and the subscript of R to the
rotating frame.
Each unit vector i traces a cone with a radius of sin a, where a is a constant angle
between the vector and axis of rotation (Figure 2.1). The m agnitude of the change of i
in time dt is |di| = sin adO, which is the length traveled by the t ip of i. The magnitude
of the rate of change is therefore |d i/d i| = sin a{d6fdt) = |Q| s in a , and the direction
of the rate of change is perpendicular to the (fi, i)-plane. Thus, (d ijd t ) = SI x i for
any rotating unit vector i. The sum of the last three term s in (2.10) can thus be
w ritten as P\Pt x i t + P2Pt x i2 + P^Yl x i3 = f t x P . E quation (2.10) then becomes
( S ) , - (? )„ * » -We can use this argument to derive an expression for the Coriolis force. Consider
the point in the atmosphere with position vector r. The application of the rule (2.11)
to the vector r gives
u f = u jj + 11 x r (2-12)
where u F = ( ^ ) F and u F = ( ^ ) R- Applying (2.12) to the velocity vector u F gives
du .p \ { d u p \ . .+ £2 x u F. (2.13)
dt ) F y dt j ft
C H APTER 2. BA SIC CO NCEPTS OF G EO PH YSICAL FLUID D YN AM IC S 19
Q
R -Q2R
Figure 2.2: C e n tr ip e ta l a c c e le ra tio n . The centripetal acceleration — fl2R is perpendicular to the axis of rotation. f2 is the angular velocity of the rotating frame, r is the position vector of a point in the frame, a is the angle between the vector f and the axis of rotation, R is the projection of r onto the plane perpendicular to the axis of rotation.
We then substitute the expression (2.12) into the right-hand side of (2.13) to get
So the acceleration in the fixed frame is given in term s of th a t in the rotating frame
— ) + n x u f l + f l x Q x r .d t J R
f i x r ) + f2 x (u r + SI x r)R
(2.14)
by
= a# + 2 fI x u r + f i x (f i x r). (2.15)
The last term in (2.15) can be w ritten in terms of a vector R drawn perpendicular
C H APTER 2. BASIC CO NCEPTS OF G EO PH YSICAL FLUID D YN AM IC S 20
to the axis of rotation (Figure 2.2). Clearly from Figure 2.2, f t x r = O x R because
|0 x r | = |0 | | r | s i n a = |0 | |R |. (2.16)
Using the vector identity A x (B x C) = B (A C ) — C (A B ) we can rewrite the last
term in (2.15) as
n x (ft x R ) = f t( f tR ) - |0 |2R . (2.17)
Since O and R are perpendicular, f t ■ R = 0, and equation (2.15) becomes
ap = + 2O x u/j — |0 |2R . (2.18)
So the inertial acceleration in the fixed frame ap equals the acceleration measured
in the rotating system, plus the Coriolis acceleration 2f2 x u and the centripetal
acceleration — |0 |2R.
In the momentum equation (2.2) we replace the term p ^ f - by p™ + 2 0 x u —
|0 |2R . The centripetal acceleration —|0 |2R is included in gravity for the geopotential
surface of geoid. So in the ro tating frame of reference, the m om entum equation (2.6)
can be written as
= ~ p V p ~~ 2Q X U + g ’ 2 ' 19^
where the term 2 0 x u is defined as the Coriolis acceleration.
C H APTER 2. BA SIC CO NCEPTS OF G EO PH YSICAL FLU ID D YN AM IC S 21
noflhpole
Figure 2.3: L ocal C a r te s ia n c o o rd in a te s . The local coordinate system rotates with the Earth. Q is the angular velocity of the E arth ro tation. 0 is the latitude of the origin of the local coordinate system, x is the west-east horizontal variable, y is the south-north horizontal variable, z is in the vertical direction. This figure is taken from Gill (1982).
2.3 T he C oriolis effect for a th in layer on a ro ta tin g
sphere
The atmosphere and the ocean are very th in layers in which the depth scale H of
the flow is a few kilometers, but the horizontal scale L is of the order of hundreds
or thousands of kilometers. We consider a coordinate frame ro ta ting with the Earth.
The Earth rotates with an angular speed |f2| — Q. — 0.73 • 10-4 s-1 . In the local
coordinate system (x , y , z ), the angular velocity is Cl = (0, f lc o s# , fls in # ), where 6 is
latitude (see Figure 2.3). The Coriolis force F is
F — 2pfl x u = 2p(wQcos9 — u fls in# , u fls in # , — uQcosO). (2.20)
C H APTER 2. BA SIC CO NCEPTS OF G EO PH YSICAL FLUID D YN AM IC S 22
Since the vertical component of the velocity is small com pared with the horizontal
components w v, w u, the Coriolis force can be approxim ated as
2 pfl x u « p(—2vfl sin 0, 2 ufi sin 0, —2uflcos 0).
The quantity f = 2Q sin 0 is called the planetary vorticity, or th e Coriolis parameter.
B a ro tro p ic flow
A barotropic flow is a fluid flow in which the pressure is a function of the density only,
i.e. p — p(p). It is a flow in which isobaric constant pressure surfaces are isopycnic
surfaces (surfaces of constant density). A two-dimensional flow on a horizontal plane
where density variations are neglected is an example of a barotropic flow.
T he /-p la n e approxim ation
The Coriolis param eter / = 2 fis in# varies w ith latitude 0. The variation is im portant
only for processes having very long time scales or very long length scales. For small
scale processes the Coriolis param eter / can considered to b e constant: f = fo =
2flsin0o where 0O is a fixed latitude under consideration. T his is called the /-p lane
approximation.
T he /3-plane approxim ation
A better approximation for the Coriolis param eter / which takes into account the
variation of / w ith latitude can be obtained by linearizing / ab o u t some fixed latitude
do. We consider a plane tangent to the surface of the E arth a t the latitude On and
define Cartesian coordinates x going from west to east and y going from south to
C H APTER 2. BASIC CO NCEPTS OF G EO PH YSICAL FLUID D YN A M IC S 23
north on this plane. We write
/ = 20 sin d — 20. sin[(0 — 0O) + 0O] = 2 0 [sin(0 — 60) cos do +• eos(0 — do) sin 0O] •
(2 .21 )
For latitudes d close to d0, d — d0 is a small angle, so cos(0 — do) ~ 1 and sin(0 — do) ~ 0.
We use the arclength formula to express the south-north coordinate on the plane in
terms of the latitude by y = Rsin(d — do) where R is the radius of the Earth. The
Coriolis param eter / can then be approxim ated as
/ = 2f2(sin 0O + ^ cos d0) = fo + @y, (2.22)
where we define the constants fo and ,6 as
fo = 20sindo, (2.23)
l i
This is called the ^-approximation. I t approximates the curved surface of the Earth
by a horizontal plane.
2.4 W aves in th e ocean and th e a tm osp h ere
Barotropic vortex Rossby waves are examples of waves th a t occur in the atmosphere.
M athematically they are represented as sinusoidal perturbations with specified wave
lengths and amplitudes. We write a sinusoidal one-dimensional wave in the form
CH APTER 2. BASIC CO NCEPTS OF G EO PH YSICAL FLUID D YN AM IC S 24
where the argument 2 n (x —ct)/X is called the phase of the wave, and a is the amplitude.
The param eter A is called the wavelength. The wavenumber is defined as
k = y , (2.26)
which is the number of complete waves in a length of 2tt. The period T of a wave is
the tim e required for the wave to travel one wavelength:
T = (2.27)
The number of oscillations per unit time at a fixed point in space is the frequency,
given by
v = (2.28)
Clearly, c — Xv. The quantity
oo — 2it v — kc (2.29)
is called the circular frequency. The speed of propagation of th e wave is given by
c = y - (2.30)k
This is called the phase speed, as it is the ra te a t which the ‘phase’ of the wave (i.e.
the crests and troughs) propagates. In general the circular frequency is a function
depending on the wavenumber oo = oo(k). The relation between oo and k is called
a dispersion relation oo = oo(k). If the phase speed oo/k is not constant then the
wave is called dispersive. If we consider a wavepacket, which comprises a spectrum of
waves of different frequencies and wavenumbers, dispersive waves from a wavepacket
propagate with different speed depending on the frequency, and the observed pulse
C H APTER 2. BASIC CO NCEPTS OF G EO PH YSICAL FLUID D YN AM IC S 25
changes shape. If ui/k is constant, the wave is nondispersive, and all waves in a
wave packet propagate with the same speed. For the wavepacket we define the group
velocity as
c» = | (2.31)
For non-dispersive waves the group velocity cg is equal to the phase velocity c. For
dispersive waves cg ^ c.
C hapter 3
V ortex dynam ics in th e atm osphere
3.1 B arotropic v o rtic ity eq u ation
The goal of this thesis is to investigate the dynamics of hurricanes and vortex waves
using a mathematical framework based on the equations of m otion for a barotropic
fluid flow on a horizontal plane tangent to the surface of the E arth . The model com
prises a vortex with Rossby waves which are generated near th e centre and propagate
outwards. It includes time dependence, the effect of the Coriolis force and nonlinear
interactions between the vortex and the waves. However, it is simple enough to allow
us to derive approximate analytical solutions which give us som e insight into possible
mechanisms tha t can lead to the secondary eyewall formation and replacement cycle.
In this chapter we derive the governing equations for our model, i.e. the equa
tions for barotropic vortex Rossby waves, in both rectangular and polar coordinates.
We also discuss previous work on the subject of vortex Rossby waves and hurricane
dynamics.
Analyses of tropical dynamics (Holton, 1992, section 11.2) show th a t in the absence
of condensation heating, the synoptic-scale dynamics of the tropical atmosphere is
26
C H APTER 3. VO RTEX D YN A M IC S IN T H E A T M O SP H E R E 27
approximately barotropic when the vertical scale is comparable with the scale height
of the atmosphere. So in this thesis we shall describe the hurricane dynamics by the
equations of motion for a barotropic fluid flow. We derive the equations in terms of
Cartesian coordinates, x and y in a horizontal plane tangent to the surface of the
Earth, and z in the vertical direction. The corresponding com ponents of the velocity
u are u, v, and w.
We derive the potential vorticity equation following H olton (1992) and Kundu
and Cohen (2004). Large-scale atmospheric flow can be considered as barotropic and
incompressible (see, e.g., Holton, 1992, section 4.5). For such a flow the continuity
equation (2.1) simplifies todu dv dw „ ,Tx + Ty + l T z = 0 - ^
We consider a shallow layer of fluid of depth h, where h is a function of x, y, and t. Let
y (x ,y , t ) be the vertical displacement of a fluid particle in th e flow, and h ( x , y , t ) =
7y2(x,y, t) — rji(x ,y ,t) , where rp is the vertical displacement o f the liquid particle at
the bottom of the layer, and % is the vertical displacement a t its surface. In the
shallow wave approximation it is assumed th a t if there are waves in the flow, their
wavelength A is much larger than h, the depth of the layer (h/X <C 1). In this
case the vertical velocities are much smaller than the horizontal velocities. Then the
acceleration d w /d t is negligible in the vertical component of th e momentum equation
(2.6). The pressure distribution is assumed to be hydrostatic,
P = Po + P9V, (3.2)
where po is the pressure a t some reference height. The horizontal pressure gradients
CH APTER 3. V O RTEX D YN A M IC S IN T H E A T M O SP H E R E 28
are therefore
1 dp dp p d x 9 d x ’ 1 dp dp p d y 9 d y '
(3-3)
Holton showed (Holton, 1992, section 11.2) th a t in the tropical troposphere a
typical scale of the vertical velocity is 0.003 ms-1 . For the hurricane dynamics a
typical horizontal velocity scale U is 10 ms-1 . The vertical length scale H is 104m,
equation is W /H which is 3 x 10-7s-1 . Scale of the horizontal derivatives and
is U /L which is 10-4s-1 .
So we see tha t W /H -C U /L and can consider to be close to zero, and trea t
the vertical velocity as being independent of 2 . Hence, from th e continuity equation
(3.1) the sum ^ is independent on 2 as well. Integrating (3.1) with respect to 2
across the layer of fluid from z — p 1 a t the bottom to 2 = % a t the surface, we obtain
a t the bottom of the layer. The difference in velocity between the surface and at the
bottom is given by
and the horizontal length scale L is 105 m. So the scale of the te rm “ in the continuity
k d i + n&y + ~ = ° ’(3-4)
where w(p2 ) is the vertical velocity a t the surface and w(pi) is the vertical velocity
w(p2) - w(pi) =P ( V 2 ( x , y , t ) - p i(x ,y , t ) ) = D h (x ,y , t ) = dh
D t D t d td h dh.
+ U — b V-
CH APTER 3. V O R T E X D YN A M IC S IN TH E A T M O SP H E R E 29
and the sum of equations (3.4) and (3.5) can be rew ritten as
^ + - k iKh] + d i lvh] = 0- <3'6)
Writing the pressure in term s of rj (3.4) for a fluid shallow flow in a rotating frame,
we obtain the horizontal components of the momentum equation (2.19) in the form
du du du „ dn ,a i + u a i + v B y ^ v = - s 8 i ' (3'7)dv dv dv „ duM + u a x + v a i + f u = ~ % - (38)
A vorticity equation can be derived by differentiating (3.7) w ith respect to y , (3.8)
with respect to .x, and subtracting. We denote the vertical com ponent of the relative
vorticity V x u bydv du
i = Tx -ay- ( 3 ' 9 )
In the /3-plane approximation / depends on y only, / = /o + Py- Then from (3.7),
(3.8) and (3.9) we obtain
<3 >°>
We can write equation (3.6) as
D h , / du d v \D i + h \ d i + d ^ J = ° ( 3 U )
and then use it to eliminate the horizontal divergence d u /d x + d v /d y from (3.10) to
C H APTER 3. V O RTEX D YN A M IC S IN T H E A T M O SP H E R E 30
obtain
We make use of
to write (3.12) as
D t h D t
This equation can be w ritten in the compact form
DC (C + f ) D h _ B fm = — T - D i ~ v T v <312>
~Dt = ~dy (3' 13)
D(C + / ) _ (C + /)D fc
D t \ h
This is the potential vorticity conservation law in the shallow-water approximation.
If the flow is horizontal or at least approxim ately horizontal then we can consider
w = 0 and from (3.4) obtain tha t d u /d x + d v /d y = 0. In this case (3.10) is simplified
to
I + ”| = °- <3-16>
and the barotropic potential vorticity equation is
^ ( C + / ) = 0 . (3.17)
This equation could also be derived by using the baroclinic potential vorticity equation
(Holton, 1992, section 4.6):
§ t (Q + f ) + (C» + / ) v , - u = - k • V„ x , (3.18)
where k is the unit vector in 2-direction, 9 is the potential tem perature, and the
CH APTER 3. VO RTEX D YN A M IC S IN TH E A T M O SP H E R E 31
subscript 9 means tha t the derivatives are taken on a surface w ith constant 9. The
potential tem perature is by definition the tem perature tha t a parcel of dry air at
pressure p and temperature T would acquire if it were expanded or compressed adia-
batically to the reference pressure ps = 1000 mb:
0 = t ( ^ J (3.19)
where k = R /c p, R is the gas constant for the dry air, and cv is the specific heat at
constant pressure. The notation ^
D & „ ,_ _ _ _ + u . Ve (3.20)
is the to tal derivative on a surface where 9 = constant.
If we make the assumption th a t the flow is barotropic, equation (3.18) simplifies
to the equation (3.15) which we derived using the shallow w ater approximation. To
show this, we note tha t
£^((0 + /) + (Ce + f ) ^ e ■ u = + / ) • (3.21)
We consider a tropical cyclone to be a two-dimensional vortex in a horizontal plane.
For an ideal gas, 9 is constant on a barotropic surface. For th is case the operator V#
is simplified to the horizontal component of the operator V, and the vertical vorticity
Q — £. We consider adiabatic flow, so D 9 /D t — 0. In th is case, equation (3.18)
reduces to equation (3.15)
J^ (C + / ) = 0 . (3.22)
We will use this equation to obtain an equation for the propagation of barotropic
C H APTER 3. VO RTEX D YN A M IC S IN TH E A T M O SP H E R E 32
Rossby waves.
We will consider a tropical cyclone as a vortex where the vertical scale H is smaller
than the horizontal scale L, H /L <C 1. As discussed above in this section, tropical
dynamics can be considered to be a barotropic flow and we can neglect the vertical
variation of the flow and consider the problem as two-dimensional. So the continuity
equation can be approximated by
where the subscripts denote partial differentiation. This allows us to define a stream-
function 'S>(x,y,t) by
ux + vy = 0, (3.23)
u = - tyy , v = \&x. (3.24)
The vorticity can then be expressed in terms of the stream function as
C = VX ~ U y = V 2<£ . (3.25)
We can rewrite equation (3.22) in term s of the stream function as
# ( V 2* + / ) QDt
(3.26)
On a /3-plane the Coriolis param eter f — fo + fly, so we can write
(3.27)
or in the form
(3.28)
CH APTER 3. VO RTEX D YN A M IC S IN TH E A T M O SP H E R E 33
This equation is called the barotropic vorticity equation. We will use it to describe
barotropic Rossby waves.
To simplify further consideration of the problem let us consider the nondimensional
form of the barotropic vorticity equation. Let us denote all dimensional variables by
an asterisk * and write the dimensional barotropic vorticity equation as:
v (■W ) " ar A ) + s ^ v U f ) = - / r ( 3 ' 2 9 )
Suppose the characteristic length scale of our problem is L and the characteristic
velocity is U . Then the characteristic time scale is T = L /U and the characteristic
streamfunction is ^ = UL. We define the nondimensional variables in terms of the
corresponding dimensional ones as follows:
This means th a t the derivatives in term s of the non-dimensional variables will be:
d I d dx* L dx ’A - J Ady* L dy ’ d_ _ \^d_
dt* ~ T d t
(3.31)
(3.32)
(3.33)
We write equation (3.29) in term s of the non-dimensional variables as
After simplification, we obtain the nondimensional barotropic vorticity equation in
CH APTER 3. VO RTEX D Y N A M IC S IN TH E A T M O SP H E R E 34
the form
V 2* , - tfvV 2# x + 'M V 2'^ ) - (3.35)
The parameter f3 in (3.35) is the nondimensional gradient of th e Coriolis param eter.
To determine the magnitude of /3 we note th a t the dimensional /?*, according to (2.24),
is
3 ’ = ^ 3 3 c o Se0, (3.36)f i 'E a r th
where 6q is the latitude of the centre of the vortex. Since the vortex is located in the
tropics, Oq is close to zero. So, with REarth ~ 6.3 x 106km an d the angular velocity
of the E arth ’s rotation P lE a r th ~ 7 x 10"5 s” 1, the value of (3* is close to 2.2 x 1CT11
m _1s_1. The dimension of /3* is m -1s-1 . So, the characteristic value of (3 is B — U /L 2
where U is the characteristic velocity of a hurricane, and L is its radius.
A typical radius of the vortex L is about 2 — 3 x 105 m, and a typical speed of the
tangential wind U is about 30 — 100 m s-1 . So the nondimensional j3 is
/3 2QE a r th L 2 2 f r , o ^ n ,
v j i ? = 77 ' (3-37)
We can therefore consider j3 to be a small param eter in our problem , i.e. j3 <C 1.
3.2 B arotropic R ossb y w aves
The barotropic vorticity equation (3.28) can be used to describe the propagation of
waves th a t are known as barotropic Rossby waves. The stream function T and vorticity
V 2T axe considered to be the to tal stream function and to ta l vorticity, respectively,
each comprising a mean quantity corresponding to the background fluid flow, and a
perturbation quantity representing the waves. If the wave am plitude is considered
C H APTER 3. V O RTEX D YN A M IC S IN TH E A TM O SP H E R E 35
small relative to the magnitude of the mean flow, then the equation can be linearized.
If the background flow is taken to be the zonal-mean of th e zonal wind and the
mean speed assumed to be a constant f/, then the waves can be represented in the
form
$ = A e i { k x + l y - * t ) (3 .38)
where x and y are rectangular coordinates, the wavenumbers k and I determine the
direction of phase propagation in the x and y directions, A is th e wave amplitude, and
u is the wave frequency. Substituting this form of the wave into the vorticity equation
(3.28) and linearizing it we obtain the dispersion relation for barotropic Rossby waves
“ = V k - J h - <M9>
The group velocity is the gradient of to in the wavenumber space
. dui . du>Cs = l l d k + i2 ~dl• (3'40)
The phase speed isuikij. + Ii2)
c = e + p (3-41)
and the phase speed in the x-direction can be w ritten in the form
kM U k2 (3k2Cx~ k 2 + P ~ k 2 + P (fc2 + /2)2'
(3.42)
The negative sign of the /3-term shows th a t the phase propagation is westward relative
to the mean flow. If the eastward current cancels the westward phase speed, giving
cx = 0, stationary waves are formed.
C H APTER 3. VO RTEX D YN A M IC S IN TH E A T M O SP H E R E 36
3.3 V ortex R ossb y w aves
For studying vortex dynamics it is convenient to use polar coordinates, r and A, where
r is the radial distance from the centre of the vortex, and A is the angle measured
from a reference direction, usually to the east. The polar coordinates are defined
as r = y /x 2 + y2, A = arc tan(x /y ), or x — r cos A and y = rsinA . The velocity
components are vradiai and vaziTnuthai in the directions of r and A respectively. In polar
coordinates the continuity equation
V u = 0 (3.43)
is w ritten as1 9 1 Q~ ~a~(j”Vradial) 4 ~^rVazimuthal 0- (3.44)r a r r o A
The relations between the stream function T (r, A, t) and the velocity components are
1 TVradiai * A 5r
Vazimuthal ^ r- (3.45)
The x and y derivatives of the stream function can be w ritten in polar coordinates as
4/^ — v — Vradiai cos A “I- sin A 4/a cos A “I- sin A,r
t y y = - u = vazimuthai sin A - Vradiai cos A = 4>r sin A + - ' F a c o s A. (3.46)
Substituting (3.46) into (3.28), we can write the barotropic vorticity equation in polar
coordinates as
V 2^ t - - ^ AV 2^ r + - * rV 2tfA - —4/a sin A + /M r cos A = 0. (3.47)r r r
CH APTER 3. V O RTEX D YN A M IC S IN TH E A T M O SP H E R E 37
The basic flow in a cyclonic vortex is generally taken to be a uniform rotation of
the vortex depending on the radial variable r and independent of the azimuthal angle
A and time t, and it can thus be w ritten as (0, v(r)). The stream function ip(r) for the
basic flow is given by
v(r) = (3.48)
where the prime denotes differentiation w ith respect to r. We also define the angular
velocity D(r) of the basic flow by v = f2(r)r.
The total streamfunction ^ ( r , A, t) and angular and radial velocity components are
each decomposed into a contribution from the steady basic flow and a time-dependent
perturbation representing the waves as follows,
\&(r, A, t) = ^ ( r ) + e^ (r, \ , t ) , (3.49)
where
V a z i m u t h a l ( r , A, t) = v{r) + £v(r, A, t ) (3.50)
and
Vradial{r, A, t ) = eu(r, A, t ) . (3.51)
It is assumed tha t u, v and ip are 0 (1 ), so th a t the param eter e gives a measure
of the magnitude of the waves relative to th a t of the basic flow. Using (3.49) - (3.51),
the equation (3.47) can be w ritten in terms of the stream function perturbation as
d v d \ 2 , 1 / ® /- l - \ P , ■ i a i \ P - x— + V ip ip \Tr(vr + - v ) W\ sin A -I- +p-ipr cos A—v cos A =at r oX J r or r r e
- ip \V 2ipr)- (3.52)
The f term in (3.52) is present because the basic flow velocity v(r) does not satisfy
C H APTER 3. VO RTEX D YN A M IC S IN TH E A T M O SP H E R E 38
equation (3.47) with the /3 term s included.
We s tart our analyses of equation (3.47) by considering th e case of waves on an
/-p lane, where the Coriolis param eter / is approxim ated by / 0, a nonzero constant.
In this configuration we set /3 = 0 in equation (3.52) and obtain
d d d A 1 d . 1 , s . r* o , . .— -I- ) V ip ii)xw-{vr + - v ) = — (VvV ipr). (3.53)at r oX J r dr r r
It assumed also th a t the wave am plitude is small relative to the magnitude of the
background flow. So the param eter e <C 1. This allows us to linearize equation (3.53)
and obtain
( l + IOv2 ^ ({v+ )=o' <3'54)
We derive approximate analytical solutions to this linear equation first for the case of
waves with an amplitude th a t does not depend on time (chapter 4), and then for the
case where the wave am plitude evolves with tim e (chapter 5). In the time-dependent
configuration we derive asymptotic solutions th a t are valid for late time. In the limit
of infinite time these time-dependent solutions approach th e corresponding steady
solutions derived in chapter 4. Section 6.1 gives a weakly-nonlinear analysis of the
nonlinear equation (3.53) in which e is considered to be a sm all param eter but the
/3-effect is neglected. This corresponds to the situation where /3 -C e -C 1.
The /3-effect is included in sections 6.2 and 6.3. The goal is to include both
nonlinearity and the /3-effect; however as a first step, we investigate the effect of
adding /3 terms to the linear problem (section 6.2). This corresponds to a situation
where e /3. In tha t case the term proportional to | is larger than the 0 (1 ) terms
in the solution and can not satisfy the specified 0 (1 ) boundary conditions. In order
to derive a solution in this case we need to neglect this term. We can do so by adding
an extra term B(r, A) = —-ncos(A ) to (3.47). The main focus of this thesis is on the
C H APTER 3. VO RTEX D Y N A M IC S IN TH E A T M O SP H E R E 39
configuration where /3 <§; e which is the more likely situation to occur in our problem
where /? ~ 1CT2 for hurricanes with a length scale L ~ 105m. In th a t case the 0(/3/e)
term in equation (3.52) gives rise to an extra term in the solution but does not change
the qualitative behaviour of the solution. We will discuss the influence of 0 (/? /e) term
on the solution in section 6.3.
3.4 P revious th eo retica l stu d ies o f tro p ica l cyclon e
dynam ics
Several theories have been advanced to describe hurricane dynam ics, the physics of
outward propagating hurricane bands, secondary eyewall form ation and other features
of the hurricane structure. Some theories describe the spiral bands as inertia-gravity
waves (Abdullah, 1966, K urihara and Tuleya, 1974, K urihara, 1976). Some more
recent papers describe them as potential-vorticity disturbances (Guinn and Shubert,
1993), while another theory hypothesizes th a t symmetric instability plays an impor
tan t role in the spiral band formation (Willoughby at ah, 1984).
MacDonald (1968) was the first to propose and qualitatively describe the spiral
bands as vortex Rossby waves. Montgomery and Kallenbach (1997) extended the the
ory of vortex wave propagation by unifying the physics of barotropic vortex axisym-
metrization and vortex wave propagation in rapidly ro tating vortices. Montgomery
and Kallenbach (1997) developed an inviscid model where the waves propagate radially
outward from the original eyewall region as spiral bands and participate in wave-mean
flow interactions. They integrated the linearized vorticity equation on an /-p lane for
the case of a symmetric vortex with a vorticity profile decreasing monotonically with
radius. They derived an exact solution for the wave com ponent corresponding to
C H APTER 3. V O RTEX D YN A M IC S IN TH E A T M O SP H E R E 40
wavenumber one and an approximate solution for the higher wave numbers.
This solution was extended by Schecter and Montgomery (2004). They studied the
conditions under which the potential vorticity in the core of a tropical cyclone becomes
vertically aligned and horizontally axisymmetric. They showed th a t the amplitude of
vortex Rossby waves decays exponentially outside the core of the cyclone, and the
decay rate depends on the gradient of potential vorticity a t th e critical radius.
Brunet and Montgomery (2002) studied the linear initial-value problem for waves
propagating on a smooth circular vortex on an /-p lane. They developed a theory for
a barotropic configuration for a vortex of finite depth w ith a nonzero Rossby radius
where p is the density, D is the depth of the flow, and / is the Coriolis parameter. It
was shown tha t the non-dimensional evolution equation for the potential vorticity per
turbation depends on one param eter, involving the azimuthal wavenumber, the basic
state potential vorticity gradient and the Rossby radius of deformation. The time-
dependent initial-value problem for this configuration was solved in Hankel transform
space.
Brunet and Montgomery (2002) and Schecter and M ontgomery (2004) discussed
the influence of inertia-buoyancy oscillations, i.e. gravity waves, on the vortex dynam
ics. They concluded tha t these oscillations are im portant for cyclones in the middle
latitudes. Schecter and Montgomery (2004) showed th a t the influence of vortex waves
exceeds the effect of gravity wave propagation on the vortex dynamics. Hendricks et al.
(2010) included the inertia-gravity waves in their numerical sim ulations of the vortical
motion in a hurricane. They concluded th a t inertia-gravity waves are insignificant in
the process of intensification or decay of the vortex.
Martinez et al. (2010a), M artinez et al. (2010b), M artinez et al. (2011) used
the empirical normal mode technique to isolate vortex waves from other datasets.
of deformation. The Rossby radius of deformation is proportional to lr
C H APTER 3. VO RTEX D YN A M IC S IN TH E A TM O SP H E R E 41
They studied the impact of vortex waves on the hurricane dynam ics and intensity,
concentric eyewall genesis, and other main features of a m ature tropical cyclone. The
results of the analyses of their simulations led them to conclude th a t the main features
of tropical cyclones can be explained by vortex Rossby waves.
In summary, the conclusions of the above mentioned studies in the hurricane dy
namics suggest th a t vortex Rossby waves play a more significant role than inertia-
gravity waves. This justifies (at least as a first approxim ation) the representation of
the hurricane dynamics using a barotropic model th a t includes propagating Rossby
waves on a horizontal plane bu t no inertia-gravity waves or o ther three-dimensional
effects.
The subject of secondary eyewalls and their formation is one of the most im portant
research topics in the dynamics of tropical cyclones. However, there is as yet no unified
theory to explain secondary eyewall formation. Kossin and Sitkowski (2009), Sitkowski
et al. (2011) developed empirical models based on observational datasets to predict the
hurricane dynamics and the appearance of the secondary wind maximum. Willoughby
and Shapiro (1982) proposed a model where the eyewall form ation is the response of
a tropical cyclone to circularly symmetric, convective heat sources. Willoughby et
al. (1982) presented numerical simulations which showed th a t the tangential wind
increases rapidly inside the radius of maximum wind and decreases inside the eye near
the central axis of the vortex. According to their numerical simulations and aircraft
observations, the secondary eyewall often contracts as it intensifies. A secondary
eyewall is frequently observed to become narrower and replace a pre-existing eyewall.
Montgomery and Kallenbach (1997) hypothesized th a t th e secondary eyewall for
m ation may take place because of vortex wave propagation in tropical cyclones. The
waves transfer angular momentum to the mean flow at the critical radius where their
phase velocity matches the mean flow angular velocity.
C H APTER 3. VO RTEX D YN A M IC S IN T H E A T M O SP H E R E 42
Another theory is based on the work of Nong and Em anuel (2003) in which it is
proposed tha t the instability mechanism leading to the secondary eyewall formation
results from wind-induced surface heat exchange.
Kuo et al. (2008) and M artinez et al. (2011) used numerical simulations to
investigate the mechanisms by which vorticity perturbations around a strong tropical
cyclone may form a ring of enhanced vorticity. Martinez e t al. (2011) applied a space
time empirical normal mode statistical technique to study the genesis of a secondary
eyewall in a simulated hurricane.
High-resolution, full-physics numerical simulations have also been used to study
secondary eyewall formation by Terwey and Montgomery (2008) who hypothesized
tha t the secondary eyewall is the result of an anisotropic upscale energy cascade of
convectively generated vorticity anomalies and vortex Rossby wave propagation.
Abarca and Corbosiero (2011) carried out numerical sim ulations of hurricanes R ita
and K atrina and investigated the secondary eyewall formation. The results of their
numerical simulations support the idea th a t vortex Rossby waves play an im portant
role in the process. Secondary eyewall formation in their model occurs a t a radius of 65
(80) km in K atrina (Rita) close to the hypothesized critical radius of the vortex waves.
The secondary eyewall formation in their simulations is characterized by maxima in
convective activity and potential vorticity m axim a in the lower troposphere. Their
results support the notion th a t secondary eyewall formation is intimately related to
bands of high precipitable water and potential vorticity variance emanating from the
primary eyewall.
The debate about the mechanisms for the secondary eyewall formation is still very
active, as seen for example in the recent polemic paper of Terwey et al. (2012) where,
based on the numerical simulations in Terwey and M ontgomery (2008), the authors
argue against some conclusions of Ju d t and Chen (2010) an d present arguments to
C H APTER 3. V O RTEX D YN A M IC S IN TH E A T M O SP H E R E 43
show the importance of vortex Rossby waves in the secondary eyewall formation.
One of the key points to explain the secondary eyewall form ation mechanism is
the interaction between vortex Rossby waves and the mean flow of the cyclone. Wave-
mean flow interactions generally take place when waves reach a location in the atmo
sphere where the background wind speed equals the wave phase speed. This location
is called a critical latitude or critical level, or in the case of vortex waves, a critical
radius. Montgomery and Kallenbach (1997) developed a balanced numerical model
based on vortex wave propagation and their experiments dem onstrated th a t the in
teraction between the vortex waves and the mean flow at the critical radius may be
the primary mechanism for the increase in the angular wind in the vortex. They also
suggested th a t vortex wave dynamics may be a key factor to explain the secondary eye
wall formation. Since th a t work, the concept of the critical layer interaction has been
used to explain some features of tropical cyclone dynamics. B runet and Montgomery
(2002) discussed critical layer interaction in their solution an d showed its influence
on the cyclone dynamics. Martinez et al. (2011) used the em pirical normal-mode ap
proach to demonstrate th a t the maximum cyclonic angular m om entum is transported
to the location where the secondary eyewall forms. The fact th a t the critical radius
for some modes of vortex waves is contained inside the region where the secondary
eyewall forms led researchers to suggest th a t a wave-mean flow interaction mechanism
may be suitable to explain dynamical aspects of concentric eyewall genesis.
In this thesis we will use the concept of the critical layer interaction to investigate
the secondary eyewall formation. Interest in critical layer interactions has been stimu
lated by the suggestion th a t this mechanism can explain certain phenomena observed
in the atmosphere and ocean. For example, Holton and Lindzen (1972) argued tha t
critical layer absorption of gravity waves might be the cause of the changes in mean
flow momentum tha t produce the quasi-biennial oscillation o f easterly and westerly
C H APTER 3. VO RTEX D YN A M IC S IN TH E A T M O SP H E R E 44
winds tha t are observed in the middle atmosphere.
From a mathematical point of view, the critical layer occurs a t the point where the
steady linear inviscid equation for the wave am plitude has a singularity. The singu
larity can be avoided by introducing to the critical layer one or more the effects th a t
have been neglected, namely time-dependence, nonlinearity, an d viscosity. The intro
duction of time-dependence was first considered by Booker and Bretherton (1967) in
their study of the vertical propagation of gravity waves in a stratified shear flow. They
considered a linear, inviscid model and showed th a t near the critical layer the waves
are absorbed and are unable to penetrate above the layer. Analyses by Stewartson
(1978) and W arn and W arn (1976, 1978) used asym ptotic m ethods to investigate a
nonlinear Rossby waves critical layer regime defined by a balance between tempo
ral evolution and nonlinearity. There have since been a num ber of analytical and
numerical studies on the development of Rossby waves critical layers. For example,
Campbell and Maslowe (1998) and Campbell (2004) which exam ined a configuration
with a spatially-localized Rossby wave packet.
These investigations all involved Rossby waves in rectangular geometry. A recent
study by Caillol (2012) deals w ith vortex waves in a rapidly rotating vortex. His
simplified analytical model is based on the propagation of a vortex Rossby wave in
a barotropic and axisymmetric vortex on an /-p lane. The wave-flow interaction is
described in the critical layer where the viscous term is included in the governing
equation. This study describes the change of the mom entum flux and predicts the
existence of multiple vortices in the critical layer.
My thesis investigates the role of vortex Rossby waves in tropical cyclones dynam
ics and includes:
• a mathematical description for vortex Rossby wave propagation in a cyclonic
two-dimensional vortex,
C H APTER 3. VO RTEX D YN A M IC S IN TH E A TM O SP H E R E 45
• solutions for the linear steady problem for waves on an /-p lane in term s of
hypergeometric functions,
• solutions for the linear time-dependent problem for waves on an /-p lane far
away from the critical layer (the outer region) and in th e critical layer (the
inner region),
• a weakly-nonlinear analysis of the wave-mean flow interaction for waves on an
/-plane,
• an analysis of the linear equation for waves on a /3-plane, i.e. including the effect
of the variation of the Coriolis param eter with latitude,
• an analysis of the combined effects of nonlinearity an d the variation of the
Coriolis param eter (the /3-effect) on the vortex dynamics,
• an investigation of the change of the mean angular velocity in the critical layer
which can explain the formation of the secondary eyewall and its dynamics.
C hapter 4
Steady linear problem
4.1 S tead y linear so lu tio n for som e sp ec ia l cases
In this chapter we derive solutions for the linear problem involving vortex waves on
an /-plane. The governing equation for this configuration is (3.54):
We consider waves with the stream function rp(r, A, t) generated a t some fixed distance
r = ri from the centre of the vortex, corresponding to th e prim ary eyewall, and
propagating outwards in the direction of increasing r. The dom ain for the problem is
defined in terms of polar coordinates r and A by r i < r < oo, 0 < A < 2-7T with r = 0
corresponding to the centre of the vortex (see Figure 4.1). T h e waves are generated
by a steady oscillatory boundary condition for the pertu rbed streamfunction of the
(4.1)
which can be written in term s of the angular velocity fl(r) = v ( r ) / r as
(4.2)
46
C H APTER 4. S T E A D Y LIN E A R P R O B L E M 47
form
iP(n, A, t) = A (ei(fcA- wt) + e - i{kX~u t ) ) , (4.3)
where A is a constant, k is the azim uthal wave number, k > 0, and cu is the circular
frequency of the wave. We require t h a t T is finite as r —>■ oo. T he to tal streamfunction
ip(r, \ , t ) is, as defined in (3.49),
V ( r , \ , t ) = tp(r) + e ip (r , \ , t ) , (4.4)
where e « l and ip(r, A, t) is 0 (1 ). Thus A is taken to be 0 (1 ), and we can set A = 1.
W ith the boundary condition (4.3) we can derive a normal m ode solution to equa
tion (4.2) in the form
ip(r, A, t) — ^ (r)e ^ fcA_w + c.c., (4.5)
where c.c. denotes the complex conjugate of the preceding term and <p(r) is the
complex wave amplitude. By adding the complex conjugate we ensure th a t is real
although <f> generally has a nonzero im aginary part. The form (4.5) assumes tha t the
wave amplitude is a function of the radial variable only and does not evolve with time.
In chapter 5 we consider the case where the am plitude evolves with time.
After substituting this normal mode form into the linear equation (4.2) we obtain
an ordinary differential equation for the wave am plitude <i>(r) ,
S' e *(fi”(r) + 3S2'/r)0 + 7 ~ ^ + — z r r m — * = 0 ’ <4-6>
with boundary conditions d>(r\) = 1 and <p(r) finite as r —» oo.
We see th a t the equation is singular a t r = r c, where f2(rc) = This radius is
the radius where the angular velocity of the vortex Ll(r) equals the phase speed of the
waves and is called the critical radius, and the region surrounding the critical radius
CHAPTER 4. S T E A D Y LIN E A R PR O BLE M 48
/* ~ » r = r ,r?f
1 s /» » * * * *
Figure 4.1: T h e d o m a in o f th e p ro b le m . The domain is r x < r < oo, 0 < A < 2-7T with r = 0 corresponding to the centre of the vortex, r and A are polar coordinates. Waves are generated a t the prim ary eyewall r = r x.
is called the critical layer.
We only consider cases where the mean flow angular velocity is an analytic function
of r. This means tha t a solution of equation (4.6) in the vicinity of the critical radius
could be derived as a series in powers of (r — r c) using the m ethod of Frobenius. A
second solution would include a logarithmic term log(r — rc).
We examine some specific profiles of angular velocity Ll(r) — v / r which allow
an exact solution of (4.6) to be obtained. These profiles are approximations of the
observed profile of the angular velocity in a tropical cyclone and have been studied
by previous researchers by analytical methods and numerical simulations.
We first consider the case where fl(r) is constant and the re is no critical radius.
We then consider the case where Ll(r) is a quadratic function of r. This is the profile
tha t will be used throughout the thesis.
C H APTER 4. S T E A D Y LIN EAR P R O B L E M 49
T h e case w h ere f2 = f2o, a c o n sta n t
If the angular velocity Q is constant, then the equation for the wave am plitude d>(r)
(4.6) becomes
r r2(4.7)
which is a Cauchy-Euler equation. Its solution is
4> = C xr k + C2r~k (4.8)
where C\ and C2 are constants. In order for (f> to be bounded a t large r, C\ = 0. The
boundary condition 4>{ri) = 1 tells us th a t C2 = T and so th e solution is wave with
amplitude decreasing with distance from the hurricane center
T h e ca se w h ere Q is a q u a d ra tic fu n c tio n o f r
Martinez et al. (2010a), M artinez et al. (2010b) used an exponential profile of the
basic flow angular velocity in their numerical simulations of hurricane dynamics. This
profile is based on typical flows observed in tropical cyclones. Brunet and Mont
gomery (2002) argued th a t the angular velocity of a cyclone can be approximated by
a quadratic function of r. Based on th a t, we will use
as the angular velocity profile. This gives an approxim ation for the exponential profile
Ll0e~ar2 for r < 1 / -J a (see Figure 4.2).
(4.9)
Q(r) — Q0( l — £*r2), a > 0, (4.10)
C H APTER 4. S T E A D Y LIN E A R P R O BLE M 50
co/k
1/Va r
Figure 4.2: T h e a n g u la r v e lo c ity p ro file . The angular velocity Ct(r) = fio(l — otr2) used in our investigation is an approxim ation for the exponential profile Q(r) = Q0e~ar , r i is the location of the prim ary eyewall, rc is the critical radius and corresponds to the point where f2(rc) = oo/k.
W ith the angular velocity profile (4.10) the equation (4.6) becomes
8kaD0b + — — —<p
oj - D o k (I ~ otr2)0 = 0. (4.11)
This equation has regular singular points a t the center of the vortex where r = 0 and
at r —> oo. The regular singular point a t r = 0 is not in the domain of our problem
and does not need to be considered. There may also be a regular singular point at
the critical radius,
where the denominator of the last term in (4.11) becomes zero. The critical radius is
located where the angular velocity of the vortex equals the phase speed of the waves
u> = flofc(l — otr2). (4.13)
CH APTER 4. S T E A D Y L IN E A R PR O BLE M 51
The form of the solution of (4.11) depends on the choice o f u>, fi0, and k. There
are three possibilities: u = Qok, u> > Ll0 k, and u < Ll0 k.
T h e ca ses w h ere u = Qok a n d to > Qok
If uj = Llok, then equation (4.11) becomes
aj k 2 + 8
4>" 4--------------^— 4> = 0, 0 < ri < r < oo, (4-14)
which is an Euler equation. There are no singular points in the dom ain of the problem.
The general solution is
<f>{r) = C lr'/WT* + (4.15)
In order for the solution to be bounded for large r, Ci — 0 and the solution satisfying
the boundary condition <j){ri) = 1 is
cj) = ( r / r 1) - ' /P T I. (4.16)
If u) > Q0 k, then again there are no singular points in the dom ain of the problem. If
rq is small then we can examine the behaviour of the solution for small r. In tha t
case we can write equation (4.11) as
+ £ _ ( V + ~ ~ b T r2) ^ + ° ( r2) = ° ‘ (4 ' 17)r \ u — ilok J
So a t leading-order, the solution can be expressed in term s of modified Bessel functions
of the first and second kind
\ t ( I SkaLlo \ —T ts ( I SkaLt0 \ .m = C J t (V z r ^ k T) + c *K k n < r «*• (418)
C H APTER 4. S T E A D Y L IN E A R P R O B L E M 52
T h e case w h ere u < Qok
The most interesting case for our problem is th a t in which ui < Q 0 k. This is the focus
of our investigation for the rest of the thesis. In this case equation (4.11) has a regular
singular point atf k n 0 - u \ 1 / 2
r° V k a ^o ) '
We will show tha t equation (4.11) can be transformed into a hypergeometric equation.
In order to do so we rewrite it as
(uj — Llok + Qqkar2) ^ <j>" + —— ~2 ^ j ~ = 0, (4.20)
A A „ + *; _ 8(i = 0> m i }or
( - -\ kaO 0 J \ r r
and then make the change of variables
f = T/ r ‘ = (4-22)
where r c is the critical radius given by (4.19). We also define a new variable <p by
4>{r) = 4>{t).
The derivatives of 4> in term s of the new variables are
0 '(r) = —0 '(r), (f>"(r) = \<j>”{r). (4.23)rc r*
In terms of <j> and r, equation (4.21) can be w ritten as
( - l + r2) ( V + f - ^ < A - 8 0 = 0 (4.24)
C H APTER 4. S T E A D Y LIN E A R P R O B L E M 53
or
+ J - ^ = ° ' (4'25)
This equation has regular singular points at r = 1 where the denom inator in the last
term of equation (4.11) is zero, corresponding to the critical radius r = rc.
Following Erdelyi (1953), we seek a solution for (4.25) in th e form
(f>(r) = rpf ( f 2), (4.26)
in order to reduce equation (4.20) to a hypergeometric equation. The first and the
second derivatives of (f)(f) with respect to r are
4>'(r) — prv~ 1 f { f 2) + 2 rp+lf ' ( r 2) (4-27)
and
4>"(r) = p(p — l ) r p~2/ ( r 2) + 2(2p + l ) r p/ ' ( r 2) + 4 r p+2 f " ( r 2), (4.28)
where the primes on the right-hand side represent derivatives with respect to f 2.
Substituting (4.26)-(4.28) into (4.24) we obtain
r 2 (r 2 — l)[p(p — 1 ) f p ~ 2 f { f 2) + 2(2 p + l ) r p/ ' ( r 2) + 4rp+2/ " ( r 2)]
+ f 2( r2 - l)[pfp~2f (r2) + 2rp/ ' ( r 2)] - k 2 rp(r 2 - 1 ) / ( r 2) - 8r p+2/ ( r 2) = 0. (4.29)
After some simplifications we get
4f4(f2 - l ) / " ( r 2) + 4 r2(r2 - 1 ) (p + 1 ) / ' ( r 2) + r 2[p2 - k 2 - 8] / ( r 2)
+ [ k 2 - p 2] f ( f 2) - 0. (4.30)
C H APTER 4. S T E A D Y LIN E A R PR O B LE M 54
This equation can be transformed into a hypergeometric equation if the last term is
set to be zero. This means
p2 = k 2 (4.31)
and so p is either k or — k.
We also introduce the new variable 2 = f 2, so the equation for / can be w ritten as
z(z - 1 ) f ' { z ) + (z - l)(p + 1 ) f ( z ) - 2f ( z ) = 0, (4.32)
where the primes denote differentiation with respect to 2 . T his is a hypergeometric
equation. A hypergeometric equation is generally w ritten in the form (see Abramowitz
and Stegun, 1964)
2(1 — z ) f " + (c — (a + b + 1 ) z ) f — (ab)f = 0. (4.33)
Writing equation (4.32) in the form (4.33)
2(1 - z ) f " + ((p + 1) - (p + 1)2) / ' - ( - 2 ) / = 0, (4.34)
we see that
a + b + 1 = p + 1 and ab = —2. (4.35)
k + y/k 2 + 8 , k - V k 2 + 8 , ,If p = k, a = , b — --------- , an d c = k + 1. (4.36)
tc » - k + y/k 2 + 8 —k - \Jk 2 + 8If p — —k, a = --------- , b — , and c = —k + 1. (4.37)
Equation (4.34) has three singular points, a t 2 = 0, 2 = 1 and 2 —» 00. As already
noted, 2 = 0 corresponds to r = 0 and it is not in the dom ain of the problem. The
solution of the hypergeometric equation close to point 2 = 1 can be written in the
C H APTER 4. S T E A D Y L IN E A R P R O B LE M 55
form of hypergeometric functions (see the detailed description of the hypergeometric
equation and its solutions in Appendix A). Two linearly independent solutions of the
hypergeometric equation (4.34) for z close to 1, 0 < \z — 1| < 1, are
f m (z) = (1 - z )F(a + 1, 6 + 1; 2; (1 - z)) (4.38)
and
/ 2(i)(z) = (1 - z ) F { a + 1, 6 + 1; 2 ; (1 - z ))lo g (l - z) +
( 1 _ z ) p ^ ± ^ _ p u ( 1 _ z r
(x(a + 1 + n) - x(a + 1) + x (6 + 1 + n) - x(b + 1) -
X(2 + n) + x(2) - x ( n + 1) + x ( l ) ) + (4.39)
where (a)n = II”r01(a + i) is the Pochham mer symbol, and F is the hypergeometric
function (see Appendix A) given by
o o ,
F ( a , b , c , z ) = J 2 ^ f z n. (4.40)^ cW n!
The function x(z) is
x W = M M i . (4.41)
The solutions fi(i)(z) and f 2 (\){z) are valid for 0 < \z — 1| < 1. Following Abramowitz
and Stegun (1964) we use the subscripts 1(1) and 2(1) to indicate th a t the solutions
(4.38) and (4.39) are valid close to the singular point z — 1. So with the constants a,
b, c given by (4.37) and (4.36), the two linearly independent solutions of (4.32) can
C H APTER 4. S T E A D Y L IN E A R PR O B LE M 56
be written in terms of r 2/ r 2 as
/l(l) 1 21 -
/±fc+\/fc2+ 8
£n=0+ l)(n) r 2\ "
0 < < 1 (4.42)
and
/ 2(i) ( ^2
+ 1r* \ soo (, g + J n ) (
± k - V k * + 82-'n=0 (2)(n)n!
+ !)(») ^
'±fc-v/fc +8
(2)(n)n!
/ ±fc + V F + 8 , \ ( ± k + \ / F + 8 ,X o + 1 + n - x --------- =---------+ 1
( ± k - \ / F T 8 , ^+X ---- ----o-----------h 1 + n
[ ± k - V k * T 8 \ .x I --------- + 1 I — x (2 + n)
+ X ( 2 ) - x ( « + l) + X(l)].
0 < 1 - — < 1(4.43)
These solutions are valid in the intervals r\ < r < rc and rc < r < \/2 rc (see Figure
4.3).
Recall th a t we searched for a solution of the form of <f>{r) = <p(r) = fp/ ( r 2)
(4.26). Thus, the solution of equation (4.20) close to r = rc is proportional to a linear
combination of / 1(1) and / 2(q
<p(r) =±k
am f m + a 2( i) /2(i) (4.44)
where aqj) and a2(i) are constants. Applying the boundary condition th a t 4>{r\) = 1
C H APTER 4. S T E A D Y L IN E A R PR O B LE M 57
/fl(=c)5 f-2 (x)
2 2Figure 4.3: Intervals for th e s tea d y so lu tion . The solutions / i (i)(^2) and / 2( i)(^ ), given by (4.43) and (4.42), are valid for ry < r < \ /2 rc, r ^ rc. T he solutions /poo) (( 5) and / 2(oo)(^i), given by (4.51) and (4.52), are valid for r > r c. In the black interval, rc < r < \ / 2r c, the solutions overlap.
we can write the solution in the form
f r \ ± k f m ( g ) + * 1/ 2(1) ( s )0( r ) = r 7 - 7 ^ — ’ ( 445)
' x' f m ( j i ) + a xf 2(i) y i j
where a\ = The solutions /p i) and /p p are valid for 0 < |1 — ^ | < 1, i.e. in the
intervals r x < r < rc and r c < r < y/2rc. The leading-order term s of the solution are
C H APTER 4. S T E A D Y LIN E A R P R O B L E M 58
or, taking into account that
tog 1 - -o = loe (r c - r ) + log(rc + r) - log(r^), (4.47)
we can write the leading-order term s of (4.44) in the form
4 >(r) ~ r±k [0 (1) + 0 ((rc - r) log(rc - r) + 0 (rc — r ) ] ,
0 <1 c
< 1. (4.48)
The logarithmic term in (4.45) is (1 — 4 )k> g(l — ^ ) where
log 1 = log 1 + % arg ( 1
For r < rc, arg (l — 4 ) = 0; for r > r c, a rg (l - T-^) = ± 7r. T he sign of the argumentr c r c
is determined by considering the tim e-dependent problem (see Appendix C of this
thesis). It is found tha t arg(l — rj) = —7t. So there is a phase shift of —7r across ther c
critical layer from r < rc to r > rc and the am plitude of the solution is discontinuous
across the critical layer. This is analogous to the conclusions of previous researchers
for the problem of forced Rossby waves in a rectangular dom ain (e.g. W arn and Warn,
1978).
For z > 1 the solution of (4.32) is proportional to a linear combination of the
functions defined in (A. 17) and (A. 18):
/i(oc)(z) = 2 aF (a ,a - c + I; a - b + 1; - ) , |z | > 1, (4.49)
h(oo){z) = ^ hF(b,b - c + 1 ;6 - a + 1; - ) , |z | > 1. (4.50)
C H APTER 4. S T E A D Y LIN E A R PR O B LE M 59
In term s of our problem these two solutions are
oo / -fc-Vfc^+ix /fe-vF+gx^ /"»V hi f r . ( 4 52)
' n ! ( - > / F T 8 + l)„ V r ? ' ’ ;
r*2 \ -«2
The solution of equation (4.20), valid in the interval |^-| > 1, or r > r c (see Figure
4.3), is proportional to a linear combination of /i(oo) and / 2(oo)
<Kr ) = r,.
±k«l(oo)/l(oo) ( r 2 j + a2(oo)/2(oo) f 2 , r > rc, (4.53)
where ai(oo) and a2(oc) are constants. The leading-order term s in the solution are
±k1 /l(oo) I ,
±k / 2r \ ( rrc
fe) 1 + 0
1 + 0- 2
- 2
r > r c, (4.54)
and
±fc
/2(oo) (0 ±fe x 2N i * * v p > *r \ ( r ' 2
1 + 0
1 + 0
+ P +8
r > r. (4.55)
The term with the positive power is divergent for r —¥ oo. So th e coefficient <22(00) = 0
CH APTER 4. S T E A D Y LIN E A R P R O B L E M 60
and the solution is
^(r ) ~ ( r l al(oo)/l(oo) ( 2 O ©-v/P+8'
r > rc,
as expected from (4.16), and satisfies the condition <f>(r) -> 0 as r —> oo.
We need to match the solution (4.56) with the solution (4.48). The solution (4.48)
is valid in the intervals ri < r < rc and rc < r < y/2rc (see Figure 4.3), while the
solution (4.56) is valid for r > rc . The two solutions should m atch in the overlapping
interval rc < r < y/2rc (the black interval on Figure 4.3). The solution (4.56) decreases
in this interval with increasing r , so the solution (4.48) should also be decreasing in
the interval. To satisfy this condition we have to choose the factor r~k in (4.48).
W ith the negative sign of k, the wave am plitude (4.45) in th e interval r\ < r < rc
and rc < r < y/2 rc is
4>{r)a x
0 < < 1, (4.56)
where Qj is the denominator of (4.45)
a i - /i(i) + a i / 2(i) ^ (4.57)
So the solution of (4.20) is
C H APTER 4. S T E A D Y L IN E A R PR O B LE M 61
where 0 i(r) is
rr, «i(i)/i(i) + 02(i)/2(i) , (4.59)
where and / 2(i) are given by (4.42) and (4.43) with the negative sign of k. Func
tion (f>2 (r) is given by
/ „ \ — Vfca+8 oo ( -fc+>/fc2+ 8 \ ( k + \ / k 2 + 8 \ / 2 \ —71, r > r c . (4,0)
n=0 n
The solution for the steady-state case is plotted in figures 4.4 and 4.5 for a normal
mode with wavenumber k = 2. From the graph it is evident th a t the amplitude of
the wave decreases across the critical layer from r < rc to r > r c where the waves are
partially absorbed.
C H APTER 4. S T E A D Y LINEAR PR O B LE M 62
<f>( r)1.4
1
0 r rl c
Figure 4.4: T h e a m p litu d e o f th e s te a d y so lu tio n . Profile of the wave amplitude 4>{r) (4.45) of vortex waves which is the solution of (4.20) w ith k — 2, ai = 1. The eyewall location r\ is taken r i = 0.1 r c. The dashed line shows th a t the solution is singular in the inner region close to the critical radius rc.
CH APTER 4. S T E A D Y LIN E A R PR O B LE M 63
rc
.
71
r l2710
Figure 4.5: C o n to u r p lo ts fo r th e s te a d y so lu tio n . Contour plots for the steady solution ip(r, A) — (j>{r)etkX+ c.c., where the am plitude of vortex waves <j>(r) (4.45) is the solution of (4.20) with k = 2, a\ — 1. The eyewall location ri is taken ri = 0.1 r c. The contour plots are presented in a rectangular domain (the upper panel) and a circular domain (the lower panel) for r\ < r < V 2 rc, 0 < A < 2 ir. T he plots dem onstrate the phase shift in the solution across the critical level a t r = rc an d resulting attenuation of the wave amplitude for r > rc.
C hapter 5
T he linear tim e-d ep en d en t problem
In this chapter and the rest of the thesis we consider the tim e-dependent problem in
which the wave amplitude is allowed to vary w ith time. In th is chapter we solve the
linear time-dependent problem on an /-p lane, i.e. we neglect nonlinearity and the
/^-effect. The problem is given by equation (3.54)
( I + + 18*= ° (5-i)
in the domain r i < r < o o , 0 < A < 27t, t > 0. As in chapter 4 the boundary
condition is
i ;(ru \ , t ) = Aei{kX- “t ) +c.c., (5.2)
where A is a constant th a t we set to 1. We also require th a t ^>(r, A, t) and its derivatives
be bounded as r —> oo. The initial condition is th a t the wave vorticity V 2ip = 0 at t =
0. As before, the mean flow is given by the angular velocity profile fi(r) = fl0(l ~ a r 2)
and the azimuthal velocity v ( r ) = Cl(r)r. Since the governing equation (5.1) is linear,
64
C H APTER 5. THE LIN E A R TIM E-D EPEN D EN T P R O B L E M 65
the boundary condition (5.2) means th a t the solution of (5.1) m ust take the form
V>(r, A, t) — (f){r, t )elkX + c.c. (5.3)
Substituting this into the governing equation (5.1) gives an equation for the wave
amplitude <p(r,t),
^ 4>rr + i <j>r — + &ikQo®4> = 0 , (5.4)
where the subscripts of r denote partial differentiation with respect to r. The bound
ary conditions are <j>(ri,t) = 4 >(r,t) —> 0 as r —» oo and the initial condition is
r(f>r + r 2 (f>rr — k 2 (f> = 0 at t = 0. We solve this problem by m aking use of a Laplace
transform.
5. 1 Laplace transform o f th e linear tim e-d ep en d en t
equ ation
Let (f>(r,s) be the Laplace transform of the function 0 (r, t), defined as
poo
4>(r,s) = I <j)(r,t)e~sidt, (5.5)Jo
where s is a complex variable. Taking the Laplace transform of (5.4) and applying
the zero initial condition on the vorticity we obtain
- ( ~ I ~ k 2 ~\(s -(- ikLl) ( 4>rr -I— <f>r (j> J + 8 ikLloa<t> = 0. (5.6)
C H APTER 5. THE LIN EAR TIM E -D E P E N D E N T P R O B L E M 6 6
The boundary condition for s) is
r ° ° 1I e ’" ‘e std t = —. (5.7)
Jo S + ZUJ
We can rewrite equation (5.6) in the form
( ~ 1 - k 2 ~\(s + ikLlo — ikPloar2) f cj)rr H— 4>r ------<f> j + 8 ikQ 0 acf> = 0. (5-8)
Using the fact th a t ui — kCl(rc) — A:ST0( 1 — a f 2), we can also w rite (5.8) in term s of uj
( _ I - k 2 ~\(s + iu> + ik f lo a(r2 — r 2)) ( <f>rr -\— 4>r ----- (f> J + 8 ikQ,0 a<P = 0. (5.9)
Following the same procedure we used for the steady-state case in chapter 4 we intro
duce a new variable r, which now depends on s,
f { s ) = { t t S c = (510)
where. . iPLoOtk
*•> = i r k k - <511>We also define <j>{r,s) = <p(r,s). In term s of the wave angular frequency u>, we can
write
= s + J , n: t k a r r ( 5 ' 1 2 )
Note th a t for s = —iui, we obtain
7 ( - i u ) = (5.13)r c
C H APTER 5. THE LIN EAR T IM E-D EPEN D EN T P R O B L E M 67
In terms of the new variables r and 4>, we can rewrite equation (5.8) as
(5.14)
where the subscripts denote partial derivatives of <fi w ith respect to f.
Equation (5.14) has the same form as the steady-state equation (4.25) and the
solution is of the same form, <f>{r) = r p/ ( f 2) with p = —k, as for the steady-state case.
We have to consider different regions in the complex s-plane to find the solution of
this equation and evaluate the inverse Laplace transform. E quation (5.14) has three
singularities.
• r = 0, or = 0. This case corresponds to r —» 0 an d we do not consider
it because it is not in the domain of our problem.
• r = 1. Near this singular point, the solution is proportional to a linear combi
nation of /i( i)(r2) and f 2 (i){f2), where /q q and f 2(\) a re given by (A .15) and
(A. 16) in Appendix A. So the solution is
where aqi) and a2(i) are functions of s th a t can be determ ined from the boundary
conditions. This solution is valid for 0 < |1 — r 2| < 1 (see Abramowitz and
Stegun, 1964) which corresponds to the region
4>(r,s) = (V 7 (s)r) k [a i( i)(s)/i(1)(7 (s )r2) + a2(i) (s ) /2(i)(7 (s )r2)] , (5.15)
s + iilok ' s + iilok ^(5.16)
or
s + > -Qo&kr2, r ^ rc. (5.17)
C H APTER 5. TH E LIN E A R TIM E-D EPEN D EN T P R O B L E M 6 8
In the complex s-plane this is the region in the complex s-plane outside the
circle of radius | Lloakr2 centred a t s = — as shown in Figures 5.1 and 5.2.
We shall refer to this as region I.
• f -> oo. For r » 1 the solution of equation (5.14) is proportional to a linear
combination of the two functions /q o o )^ 2) and / 2 (<x>)ir2), where /i(oo) and / 2(oo)
are given by two functions (A .17) and (A .18) in A ppendix A. So the solution is
4>{r, s) = (V 7 (s)r)~k [a1(oo)( s ) /1(oo)(7 (s )r2) + a2 (oc){s)f2 (oo)(l(s)r2)] , (5.18)
where oo) and a 2(oo) are functions of s th a t can be determ ined by matching
this solution to (5.15).
This solution is valid for |r2| = [7 (s )r2| > 2 (see Abramowitz and Stegun, 1964).
|s + zfl()fc| < ctkr2. (5.19)
This is the region in the complex s-plane inside the circle of radius ^Ll0 a k r 2
centred a t s — —iTlnk (the grey shaded region in Figures 5.1 and 5.2). We shall
refer to this as region II.
To obtain <p(r, t ) we need to compute the inverse Laplace transform of tp{r. s). In
doing so, we have to take into account the fact th a t there a re two different regions
in the complex s-plane th a t need to be considered separately. Region I is outside of
the circle |s + iQok\ > ^Doakr 2 where the solution has the form (5.15), and region II
is the region inside the circle |s + iQ0 k\ < | f 20cxkr2 where th e solution has the form
(5.18) (see Figures 5.1 and 5.2).
CH APTER 5. THE LIN E A R T IM E-D EPEN D EN T P R O B L E M 69
Im(s)Im(s)
Re(s)Re(s)
s— -iOok(l-ar2) s= -i<a = -iQ0k(l-arc2)
- s—iQokCl-ar2)- s=-k>= -ifiok(l-arc2)
■ s-ifio k Q -ari2)
Figure 5.1: T h e contour o f in tegration in th e com p lex s -p la n e for th e inverse Laplace transform for th e cases w here th e p o le is in reg ion I.The shaded region is region II where the solution (5.40) is valid. The region outside the circle is region I where the solution (5.39) is valid. The pole P, s — —ioj, is in region I. It gives the steady term of the tim e-dependent solution in the form (4.59), which is valid for r\ < r < V 2 rc, r ^ rc.(a) If r > \ f i r i, the solution (5.39) has two branch points, s = — iQ0 k ( l — a r 2) and s = — iLlok(l — a r 2), which are surrounded by the contours C 3 and C2.(b) If y/2ri < r < \ /2 rc, r ^ rc, the solution (5.39) has one branch point in region I, s = — iLlok(l — a r 2), surrounded by contour C3.
C H APTER 5. TH E LIN E A R TIM E-D EPEND ENT P R O B L E M 70
Im(s)
- s= -iQok(l-ar2)
s= -ioo = -iQok (l-arc2)
Figure 5.2: T h e c o n to u r o f in te g ra tio n in th e co m p lex s -p la n e fo r th e in v e rse L ap lace tra n s fo rm fo r th e case w h e re th e p o le is in r e g io n II .The shaded region is region II where the solution (5.40) is valid. The region outside this disc is region I where the solution (5.39) is valid.The pole P, s = —iu>, is in region II. It gives the steady term in the time-dependent solution in the form (4.60), which is valid for r > y/2rc. T here is one branch point, in region I, s = — Q0&(1 ~~ <*r2), surrounded by contour C3.
C H APTER 5. THE LIN EAR TIM E-D EPEN D EN T P R O B L E M 71
R eg io n I
In region I (outside the grey disc in Figures 5.1 and 5.2), where 0 < j 1 — 7 (s )r2| < 1,
the solution is proportional to a linear combination of the two linearly independent
solutions (see Appendix A, A. 15 and A. 16):
OO / - f c W f c j + 8 , - | \ , - k - V k i + 8 , I ' l
Aii) (7 ( s y ) = (1 - 7 ( V ) E 2------ ^ (1 - 7 M - Tn = 0 K * H n ) n .
(5.20)
and
h { i) (7 (s)r2) = ~
+ (1 _ 7 (s )r2) £ + +n=0
(1 - 7 (s )r2)n log ( l - 7 (s )r2)
+ (1 - 7(*)r2) E ( 2 g ; ( )n , 2------ ^ (1 - 7 ( s ) r T71— 1
( r H v /F + 8 1 , - k + \ Z W + %( x( j + ! + «)- x( g +
/ —& - \/A:2 + 8 — fc - \ / k 2 + 8+X ( j + ! + « ) “ x( g + X) “ X(2 + n)
+ X ( 2 ) -X ( « + 1 ) + X ( 1 ) ) - (5.21)
Both series (5.20) and (5.21) converge for all r in the interval 0 < 11 — 7 (s )r2| < 1
(see Abramowitz and Stegun, 1964). The solution is
4>i(r,s) = (V*7 (s)r) *[a1(i ) /1(i)(7 r 2) + a 2(i)/2(i)(7^2)], (5.22)
where ai(i) and a2(i) are functions of s which can be determ ined by boundary condi
tions.
C H APTER 5. THE L IN E A R T IM E-D EPEN D EN T P R O B L E M 72
The leading-order terms of / 1(1) and / 2(i) are
fn i ) (r , s ) ~ (1 - l ( s ) r 2) + 0 ( ( 1 ~ l ( s ) r 2)2) , |1 - 7 ( s ) r2| < 1, (5.23)
s ) ----- ^ + (1 - 7 (s )r2) log(l - 7 (s)r2) + 0(1 - 7 (s )r2),
|1 - 7 ( s ) r 2| < 1. (5.24)
For small values of j 1 — 7 (s)r2| where |1 — 7 (s )r2| < 1, the solution /^ j) <C / 2(i)- The
leading-order term s for the solution are
M r < s ) ~ ( V l ( s ) r )~k + (1 - 7 {s)r2) log (1 - 7 (s )r2) + 0 (1 - 7 (s )r2)^ ,
|1 - 7 (s )r2| < 1. (5.25)
After applying the boundary condition <j>(r\,s) = , we obtain
~ 1 f r \ ~ k - 5 + (1 - 7 ( s ) r 2) lo g (l - n / j s y 2) + 0(1 ~ i ( s ) r 2)1 r,S s + fc u V ri/ - | + (1 - 7 (s)rf) log(l - 7 (s)rf) + 0(1 - 7 ( s ) r f ) ’
|1 - 7 (s )r2| < 1. (5.26)
This solution has two singular points, a t s = —iQ0 k( 1 — a r 2) where 1 — 7 (s)r 2 = 0
and s = — a r 2) where 1 — 7 (s )r2 = 0 .
R eg io n II
Region II is the grey shaded disc shown in Figure 5.1, where |7 (s )r2| > 2, or js +
< \Ll0 a k r 2. The point s — —iQ0k is the centre of the disc. At the point
s = —iQ0 k , 7 (s) —> 00 and for s ^ —iQ0 k, 7 (s) — is finite.
C H APTER 5. THE LIN EAR TIM E-D EPEND ENT P R O B L E M 73
For s = —iQok equation (5.6) becomes
0rr + ~4>r ~ - - ^ - 0 = 0 (5.27)r r*
and the solution is
4>(r, - iQ0 k ) = a 1r ' /PT1 + a2r - ' /P T i (5.28)
where a\ and a2 are constants. The solution should be finite as r —» oo, so we must
set ai = 0. To find a2 we apply the boundary condition (5.7) a t r = rj:
4 >(ru - i f l 0 k)-iQ0k + iu iQ0 a k r %
and hence obtainI / r \ - V f c 2 + 8
M r . - i d o = - <5-29)
Equation (5.27) has no singularities in the domain of our problem, so s = —iQok is
not a singular point for the inverse Laplace transform.
For s 7 —ikQo, when y /y (s)r is large bu t finite, the solution of (5.6) has a form
similar to the steady-state solution for large r (4.53):
4>n(r,s) = ^-z ( v /;:Ks)^)“ *:[ai(oo)(s)/i(oo)(7 (s )r2) + a 2(oo)( s ) /2(oo)(7 (s )r2)],
|7 (s )r2| > 2, (5.30)
where a^oo) and a2(oo) are functions of s, and /i(oo) and / 2(oo) are the functions given
by (A. 17) and (A. 18) in Appendix A:
/To—— °° ( — 8 \ / fc+\/fc2+8
h U < ‘ P ) = h ( s y ) ~ - g „,2(^ ) : 8 + ^ " ( v T M r r .
|7 (s )r2| > 2 , (5.31)
CH APTER 5. THE LIN EAR T IM E-D EPEN D EN T P R O B L E M 74
. ^ /TJTZ 00 ( ~k-V&+&\ / fc -y ^ + 8 \A ( . ) h W = 2,," n!( —vA;2 + 8 + 1)„
|7 (s )r2| > 2. (5.32)
So in region II the solution 0 //(r , s) is a linear combination of th e functions
( y / r ( s j r ) kf\(oo)('y(s)r2) ~ (VtCs)*-) k{ l ( s ) r 2) k ~ ( y / r ( s ) r ) -v/F+i
|7 (s )r2| > 2 , (5.33)
and
{ V l ( s ) r ) - kf 2 (oo)(7 (s)r2) ~ ( \ / l < s ) r ) - k( j ( s ) r 2 ) h±:/f I i ~ (■V CK *)r)WE3+*,
|7 (s )r2| > 2. (5.34)
The function (5.34) is divergent for r —>• 00. So the coefficient <22(00) (s) = 0 and the
solution (5.30) is
4>n(r,s) ~ (vS < s)r) *ai(oc)/i(oo)(7 (s )r2)0 0 f - k - V k 2 + 8 \ / f c - y 4 ? + 8 \
~ « . M ( ^ W r ) +- , r a E - K - ^ + S + l) , ( 7 ( , ) r 2 r "n=0 n
/ f s + i k Q ( ^ m ) n )( ^ ± g ) u / a + t fcn 0 1 \ "Gl(°°) i t o f l o r y ^ n ! ( - \ A 2 + 8 + 1)„ \ r 2 /
|s + ikLlo| < i/ufloQ;?'2- (5.35)
CH APTER 5. THE LIN E A R TIM E-D EPEN D EN T P R O B L E M 75
After applying the boundary condition (5.7) we obtain
/ — k — \/fc2 +8 \ f k — v /k2 -4-8 \ / \ n. . -V P +8 r 00 ( " 1 ....)n’( ' 2 ( s+ ik Q p 1 \
7 , , 1 / V \ Z ^ n = 0 n ! ( - V f c 2 + 8 + l ) „ V i k a Q o r 2 J<Pn(r,s) — 1 1
s + i u ) \ r j ( -A t^ E tj)wi(t z ^ ± i )n / S+Ifcn„ x \ » !Z ^ n = 0 r i! ( -v 'fe i + 8 + l ) „ \ tfcaUo r f /
|s + iA;Oo| < - k f l o a r 2. (5.36)
The leading-order terms of the solution are
.) ~ ^ (iV U + o ( ^ i ) + O
fo r|s + zfcfi0| < -^kLl^cur2. (5.37)
s + i u ) \ r i J V V ikaClo r
We observe th a t this solution gives us (5.29) in the limit s -» —iLl0 k. So the solution
4>ii(r j s) is continuous in region II where it is defined. We have now obtained the
solution 4 >(r, s ) everywhere in the complex s-space and can find <fi(r, t) by making use
of an inverse Laplace transform.
5.2 E valuation o f th e inverse L aplace transform
In section 5.1 we derived the solution of equation (5.6) in different regions in complex
space,
4>[(r,s), for |s -I- ifcflol > ^kLtoar 2 (Region I),H e s) = •{ _ l (5.38)
i|i/;(r,s), for \s + ikLlo\ < -kL lo ^ f 2 (Region II),
C H APTER 5. TH E LIN EAR TIM E-D EPEN D EN T P R O B L E M 76
where
~ 1 / r \ * - § + ( ! - 7(g)?~2) log(l ~ l { s ) r 2) + Q (( 1 - 7 (s )r2))
/r,S s + i u Vr , / - 5 + (l-7(s)»'i)log(l-7(s)r?) + 0(( l -7(s)r?))
To find <j>(r,t), the solution of equation (5.1), we evaluate the inverse Laplace
integration will lie to the right of all the singularities of the integrand. The integral is
evaluated along the closed Bromwich contour in the complex s-plane shown in Figures
5.1 and 5.2. The contour of integration is closed around the pole of the integrand and
deformed to go around any branch points of the integrand. Following the standard
procedure, the radius of the large semicircle is allowed to become infinite while the
radii of the circles around the branch points go to zero. M aking use of the residue
theorem the integral is found to be equal to the sum of the contributions from the
singularities.
We evaluate the inverse Laplace transform in the complex s-plane for each value
of r , rq < r < oo. Depending on the value of r , we have three different configurations
in the complex s-plane. Case 1 for rq < r < i /2 r i, Case 2 for y/r\ < r < y/2rc, r ^ rc,
1 - 2(1 - g g g .r3 ) log(l - ^ r 2) + Q (( l - g g f e H ) )ikQ q q 2
s+ ik Q o 1ik f t q q 2 s-Mfcf2o 1
tfcOpq 2 s+ikClo 1
and
Vf?+8
+ o (s -b ikfyo 1
ika^lo) ) - ( 5 .4 0 )
transform of 4>(r, s) using the Laplace inversion integral
(5.41)a —ioc
where cj>(r, s ) is given by (5.38). Here a is a real number chosen so th a t the contour of
CHAPTER 5. THE LIN EAR T IM E-D EPEN D EN T P R O B L E M 77
and Case 3 for r > rc. The union of the intervals in Case 1 and Case 2 is the interval
ri < r < V2rc, r ^ rc, which is where the steady solution (4.59) is valid. And we
shall see tha t in the limit as t -* oo the time-dependent solution th a t is valid in this
interval converges to the steady solution (4.59). The interval in Case 3 is r > \ /2 rc
which is where the steady solution (4.60) is valid. And we shall see th a t in the limit
as t -> oo, the time-dependent solution th a t is valid in this interval converges to the
steady solution (4.60). The three cases are summarized below and are represented in
Figures 5.1 and 5.2.
C ase 1 . ri < r < y/2r\
In this case the integrand in (5.41) has
• a pole P at s — —iu in region I, outside the grey disc,
• a branch point a t s — —iQok(l — a r 2), o r s = — i u + ikQoa(r 2 — r 2),
• a branch point at s = — iQok(l — a r 2), or s = —iuj -I- ik f lo ^ i^ i ~ r2).
All the points lie outside the grey disc, so they are in region I. This is shown in Figure
5.1a.
C ase 2 . y/2r\ < r < y/2rc, r ^ rc
In this case the integrand in (5.41) has
• a pole P a t s — — iu in region I, outside the grey disc,
• a branch point a t s = —rf20A:(l — a r 2).
The point (s — —iflok(l — a r 2)), in this case, lies inside the grey disc, i.e. in region
II. Since it is not in region I, it does not contribute to the solution. This is shown in
Figure 5.1b.
CH APTER 5. TH E LIN EAR TIM E-D EPEND ENT P R O B L E M 78
C ase 3. r > s /2 rc
In this case the integrand in (5.41) has
• a pole P at s — — iu which is now in the region II,
• a branch point of the solution (5.39) a t s = —iQok(l — a r 2).
As in case 2, the point (s — —ifl 0 k( 1 — a r 2)) lies inside the grey disc, i.e. in region
II. Since it is not in region I, it does not contribute to the solution. This is shown in
Figure 5.2.
We now describe the evaluation of the contributions from the pole and branch
points.
T h e resid u e a t th e p o le P , s = —iu
In all 3 cases the integrand in (5.41) has a pole at s = — iu = — iQ0 k ( l — a r 2). The
residue at the pole is
For cases 1 and 2, where < r < V2rc, the pole P is in region I so we use
R e s ^ ^ ) = lim^ (j>(r, s)est j = 0 (r, - i u ) e lwt (5.42)
<f>(r,s) = cf)j(r,s) in (5.42). Recalling tha t 'y(-iuj) = (5.13), we obtainr c
Res {s=- iu)( ^ ( r , s ) ) = Ress=_iW(<^/(r, s))
(5.43)
For cases 1 and 2 the residue corresponds to the steady-state solution (4.59) which is
valid for r\ < r < s/2rc, r / rc. For case 3, where r > s/2.rc, th e pole —iu is in region
CH APTER 5. THE LIN EAR TIM E-D EPEND ENT P R O B L E M 79
II (Figure 5.2) so we use <j>(r,s) = <fin(r,s) in (5.42) and therefore
Res(*=_*,)(<A(r,s)) = Ress=-tw(4>n(r, s))t — k — \/fc2 + 8 \ / k — \/fc +8 \ / -> \ 1
-VF+8 r ° ° (----2------)"■(—*1---- )"r \ /s n = 0 n ! ( - \ / f c 2+ 8 + l ) „ \ r 2)
n ) 7 r ? \ W2^n=0 n^-v^S+g+i^ ^
This residue corresponds to the steady-state solution which is valid for r > y/2rc.
T h e c o n tr ib u tio n s from th e b ran ch p o in ts
To evaluate the inverse Laplace transform we integrate (5.39) along the contours
shown in Figures 5.1 and 5.2. Let us consider the integral for the case 1 where the
integrand has a pole and two branch points. For case 1 the contour consists of the 7
contours C 1 -C 7 which are shown in Figure 5.1a.
So the solution is
4>(r, t ) = Res(s=_iw) ( j : <ft(r, s) \ - ~ V ] f s)estds. (5.45)\ s T iuj J 2m J c . s + iuj
T h e co n to u rs C \, C2 , an d C3
The contour C\ can be described in polar coordinates as
s ^ a + R e 16, | < 9 < Y ■ (5-46)
So
3 tt/2
— —~ 4 >{r, s)estds =iCx
(5.44)
O / l / £,
L ^ {r’S)e“ d S = I a + R l - + ^ ' W ' ]‘^ ' a + R ‘ ', ) m < t d S ‘ (5-47)7r/2
C H APTER 5. TH E LIN EAR T IM E-D EPEN D EN T P R O B L E M 80
As R —> oo, the integrand in (5.47) is proportional to eRe'a or eRcos8, and cos#
is negative since | < 0 < So the integral along the semicircular contour C\
converges to zero as the radius of the semicircle becomes infinite.
The contour C2 is the circle surrounding the branch point s = —iQ0 k( l — a r 2). It
can be described as
and the integral converges to zero when e —> 0 .
The contour C3 is the circle surrounding the branch point s = —ifl0^ ( l — a r 2).
The integral along C3 takes a similar form to (5.49) and converges to zero as the
radius of the circle converges to zero.
So the inverse Laplace transform for case 1 is equal to th e contribution from the
residue at the pole (5.43) plus the contributions from the contours C4 and C5 near the
branch point .s = —iQ0 k( 1 — a r 2) and from the contours C6 and C 7 near the branch
point s = — iQok(l — a r 2).
s — —iLlok(l — a r 2) + ee10, 7r < 6 < 37r. (5.48)
So
/ 4 >(r, —iQ0 k( 1 — a r 2) + ee10) —iQok(l — a ( r 2 — r^)) + ee*0&
-in 0fc(l-a(r2- r 2))+ee‘s eei e ^
7T
T h e co n to u rs C4 and C5
The contours C4 and C5 correspond to cuts to the branch point s = —i i \ k ( l —a r 2). or
s — —iui + ikLl0 a (r 2 — r 2). Along the contour C4, s = s + = pein — iu> + ikLloa(r2 — r2),
C H APTER 5. THE LIN EAR TIM E-D EPEN D EN T P R O B L E M 81
and p changes from oo to 0. Along the contour C5, s = = pe %1X—iuj+ikQoa(r2—r 2),
and p changes from 0 to oo. Hence
{La Sc) -iut^ikQ.QCt(r2 — r 2)t (- k
+ I I (0 (r > s)est)ds ~ e/c4 Jo*,/ Vr l
f°° 1 1 — 2(1 — 7 (s )r2) log(l — 7 (s )r2) + 0 (1 — 7 ( s ) r2)
Jo/o (s + i u ) l - 2(1 - 7 (s)r?) log(l - 7 (s)rf) + 0 (1 - 7 (s)r?)e ptdp. (5.50)
For large t, the dominant contribution to this integral comes from small p because of
the factor of e~~pt, so we can obtain an approxim ation valid for large t. Let us estimate
the integrand for small values of p. We can write
1 1 1
s + iu ik f t0a(r + rc)(r — rc) — p ikDoa(r + r c)(r — rc) 1 ----- e-
1 A 1
ikfloa(r + rc)(r — rc) \ ikLl0a ( r + rc) r — rc
(5.51)
and expand it as a geometric series
1 1
s + iuj ikLl0a(r + rc)(r — rc)
O O / \ 7
(5.52)
This expansion is valid for j -j <C 1. The other term s in the integrand (5.50) can also
be approximated for small p. Along the contours C4 and CV,, 7 (5) can be expressed
in terms of p as
7 (5 )if loak
ikCl0a r 2 — p (5.53)
C H APTER 5. THE LIN EAR TIM E-D EPEN D EN T P R O B L E M 82
In the numerator of the integrand in (5.50) we have
1 — 7 (s)r2 = -------- —------ ~ ------- —— (1 H-------—-------1-n ' ikQ0a r 2 - p ikD0a r 2 V ikD0a r 2 ’) ’
p « 1. (5.54)
In the denominator of the integrand in (5.50) we have
2 f r 2 - r? pem \ f pen 2,+ + , « 1 . (5.55)
Thus, the denominator of the integrand in (5.50) is
1 - 2(1 - 7 (s)rJ) log(l - 7 (s)r?)
~ 1 - 2 ^ log (d ^ i) _ 2£_d (log (d_d) + t) + 0 ( p 2 ) ,
p < 1. (5.56)
The sum of the integrals along C4 and C5 is proportional to th e phase shift between
the two branches of the logarithm,
log(l — 7 (s+)r2) — log(l — 7 ( s - ) r 2) = —27ri, (5.57)
and the sum of the integrals is
CH APTER 5. THE LIN E A R TIM E-D EPEN D EN T P R O B L E M 83
where
= 2 hi(r)1 (ikLl0a ) 2(r + rc)r2 ’
__ h2(r)2 (ikLloa)3(r + rc)r4 ’
r2 _ „2 /„2 _ „2\ \ - 1r i i / r r iM r ) = ( 1 - 2— log J J , (5.60)
h2(r) = 2hi(r) - 2r 2 + 2 /i i ( r ) ^ |( r 2 - r 2) ^log ~ ^ + 1 ^ . (5.61)
The expansion (5.58) is convergent for \T0.Tc\ < 1 and so we can interchange the order
of the integration and summation in (5.58). For each n, we can express the integral
in terms of the gamma function T(n) as
fJopn~1e~ptdp — (5.62)
and we evaluate the integral (5.50) along the contours C4 and C5 as
( f + I ) r’ s)eStds\ J c 4 J C i /
/ \ —fc 4 OO 4~ 2 ( ^ j - ^ a j ( n + l ) ( i ( r _ ^ ) )n , fo r |r - rc\t > l.(5.63)
T h e co n to u rs Cq and C7
The contours C§ and Cj correspond to cuts to the branch poin t s = —iQ0k ( l — a r \ ),
or s = —iu — i£lQOtk{r . — rf). Along the contour C$, s = s+ = pet7r — iLl0k( 1 — a rf ) ,
and p changes from 00 to 0. Along the contour C7, s = s - = pe~t7r — iQok(l — a r 2),
C H APTER 5. TH E LIN E A R TIM E-D EPEN D EN T P R O B L E M 84
and p changes from 0 to oo. Hence
+ J ^ 0 (r, s)estds ~ e- ^ t eikQoa(rj-r^t
f ° ° 1 1 ~ 2(1 - 7 (s )r2) log(l - 7 (s )r2) + 0 (1 - 7 ( s ) r 2) _ p t ( .Jo (s + iu) 1 - 2(1 - 7 (s)r^)log (l - j( s ) r? ) + 0 (1 - y (s )rf)
The dominant contribution to the integrand comes from small p because of the factor
of e~pt. So for t 1 we can approximate the integrand by considering small p. The
term ^4— in the integral is approxim ated by
1 1 , 2 ' + O 0 > 2) V (5.65)s + itu ikLl0a(r j — r%) y ikQ.Qa{r^ — r\)
Along the contours C6 and C7, 7 (5) is w ritten in term s of p as
iD0a k7 = I t ? w r 7 (5 66)
and for small p it can be approxim ated as
7(s) = ~ (7 ( 1 " i , / i k n °ar2 ) + 0 { f ? ) ■ (5 67)
We also make the approximation
(1 - 7 (»)r2) ~ + ( l + + 0 (p 2) ) , (5.68)
So the num erator of the integrand (5.64) becomes
1 — 2(1 — 7 (s )r2) log(l — 7 (s )r2) ~
1 - o- ( ^ h +
C H APTER 5. THE LIN E A R TIM E-D EPEN D EN T P R O B L E M 85
The sum of the integrals along the contours and C 7 depends on the phase shift
between two branches of the logarithm. So again we have
log(l - 7 (s+)^i) - log(l - 7 (« - ) r i) = (5.70)
So for small p we can write the leading-order terms of the sum of the integrals along
C 6 and C 7 in the form
(.L+D4>(r, s)estds
g - i u t g i k Q o a ( r j - r % ) t
where
( y j ( - 2in) J ptdp, p<^i 1, (5.71)
6l(r) ( i k n 0a )2(rj - r l ) r \ (5'?2)
^3(r ) = 2 ( l — 2^-i—2— log ( 9— ^ \ (5.73)
with
So the sum of integrals along C 6 and C 7 is
( / + / ) 4>{r,s)estds ~ - 2i7re - iu;V fcnoa(r2- r")t ( y j , t » (5.74)
So we have obtained approxim ations for the tim e-dependent solution, valid for
t » 1, by evaluating the inverse Laplace transform in each of the cases shown in
Figures 5.1 and 5.2.
C H APTER 5. THE LIN E A R TIM E-D EPEN D EN T P R O B L E M 86
C ase 1, 7~i < r < \f2r\
For ri < r < \ /2 r i , the configuration for the inversion of the Laplace transform in the
complex s-plane is shown in Figure 5.1a. The pole is in region I and there are two
branch points. In this case the solution is
(j>(r,t) ~ $ oc(r)e_lwt
+e- ^ t eiknoa(r>~r*c)t ( l Y " ( 2h2(r)\ r i / \ t 2(ikQaa)2(r2 — r2)r2 t3{ikLloa)3{r2 — r2)2rA
-iuit ik S lo a (r \—rj?)t { ^ \ ( ^ 3 (0 , h 2 (r^)+e~ e 0 Kl~ c> — ^ ----------------;rr—« +r i ) \ t 2(ikf l0a)2(r2 — r2)r\ t3(ikQ0a ) 3(r2 — r2)2r \ /
t » 1,(5.75)
where <Lco(r) is the steady-state solution (4.59).
C ase 2, \ / 2 ri < r < \ /2 rc
For \/2 r i < r < y/2r c, r ^ rc, the configuration for the inversion of the Laplace
transform in the complex s-plane is shown in Figure 5.1b. T he pole is in region I.
There is only one branch point and the solution takes the form
<j>(r,t) ~ 4>oo(r)e lut
+ e -i*teifenoa(ra-r2)t ( L V V + 2M 0r i / \ t 2{ikQ,oa)2{r2 — r2)r2 t3(ikf loa)3(r2 — r2)2r4 /
1,(5.76)
where <Foc(r) is the steady-state solution (4.59).
CH APTER 5. TH E LINEAR TIM E-D EPEND ENT P R O B L E M 87
C ase 3, r > y/2rc
For r > V2rc, the configuration for the inversion of the Laplace transform in the
complex s-plane is shown in Figure 5.2. The pole is in region II. There is only one
branch point and the solution is
4>(r,t) ~ $oo(r)e ''twt
2fti(r)________ 2 h2(r) \t2{ikLlaOt)2{r2 — r 2)r2 t 3 (ikQoa)3 (r2 — r 2)2r4 )
t > 1(5.77)
where <F0c(r) is the steady-state solution of the form (4.60).
We observe th a t each of these solutions (5.75)-(5.77) comprises a term 4>00(r)e~'*u,t,
where $ 00(r) is the corresponding steady-am plitude solution from chapter 4, and one
or two time-dependent terms with frequencies coi = u — kl laa ( r 2 — r2) — k d 0a r 2 and
u>2 = u) — kLloa{r\ —r2) — kLl0a r 2.
The solutions (5.75), (5.76) and (5.77) give a complete description of the solution
of the linear time-dependent problem (5.1) for r\ < r < 1 / y / a , r ^ rc, the interval
shown in Figure 4.2. The solution (5.76) is valid around th e critical radius r = r c,
but not a t r — rc. In order to find a solution a t r = r c, we need to consider an inner
region, defined by a critical layer around the point r = rc.
In deriving the solution (5.76) we considered < 1 in order to make the
approximation (5.52). Integrating the geometric series (5.52) term by term gave us
series of term s proportional to [(r — rc)t]~n which is convergent for \r — rc\t > 1. Thus,
the solution (5.76) is valid for t » 1 and |r — rc\t > 1, outside of a layer of thickness
I centred a t r = rc. On the edge of this layer where (r — rc)t ~ 0 (1 ) and inside the
layer where |r — rc\t < 1, the solution (5.76) is not valid. T he region \r — rc\ > | is
_|_g-iwteifeno«(r2—r1)t
C H APTER 5. THE LINEAR T IM E-D EPEN D EN T P R O B L E M 88
considered to be the outer region.
In the inner layer where |r — rc\t ~ 0 (1 ) we can not use th is solution because of
the singularity The solution in the inner layer will be considered in the next
section.
5.3 T im e-d ep en d en t so lu tio n close t o th e critical
layer
To find the solution a t the edge of the critical layer where |r — r c\t ~ 0 (1 ), we return
to the evaluation of the integral (5.58). In the outer region we considered 1
and approximated by
-— E f V ) " e~ptdP> I— r < 1- (5-78)s + iuj r ~ r c ./o \ r ~ rcJ |r - rc|
Close to the critical layer |r — rc\t ~ 0 (1 ) and ~ 0 (1 ), so we need to retain all
the terms in the expansion (5.78).
We also expand f2(r) = fl0(1 — o r 2) in powers of r — rc. Since Ll(r) is analytic at
r c, we can use its Taylor series expansion
Q(r) ~ D(rc) + Q'(rc)(r - rc) + O ( (r - r c)2) ~ ^ - 2aTt0rc{r - rc) + O ((r - rc)2) .
(5.79)
Recall, tha t u / k — f i(rc).
C H APTER 5. THE LIN E A R TIM E-D EPEN D EN T P R O B L E M 89
Now we can write (5.58) in the form
/ + 1 H r ' s )eStdsJcA JCsJ/ \ ^ /»0O / z t x 7T
iuit e ikU 0a ( r 2- r 2 )t ( £_ \ (2z7r) J E°° 0e er j / Jo n \ i k Q o a 2 r c( r — r c)
( 2 p h \ ( r ) + p 2h j ( r ) \ g\(zA:fioCi)2r 22rc(r — r c) (ifcf2oo03r 42rc(r — r c) / ’
p < 1. (5.80)
where
/i4(r) = hx(r) - 4hf(r)(zfi0a ) 2r?2rc(r - r c) ^log - - - - - - + 1^ . (5.81)
So
( [ + [ } H C s ) e Std S ~ e - i “ t e - 2 i k n 0a r c (rc - r ) t (£ _ j\Vc'4 Jcs /
k
Jr ( 2ki v
Jo ^(^A:flott)2r 2(rc - r) ^(p )n+1
(zA:floQ:)2r2(r c — 0 (ikD0a2rc{rc — r))n
2hn (p)”"*~2 \( ikf l0a )3rA2rc(rc — r ) ■*—' (ifcQ0o:2rc(rc — r ) )n J 6 P> P ^ (5 82)
Again making use of (5.62), we can write (5.82) in term s of th e gamma function as
( f + f ) H r , s ) e stds ~ ( - ) — - - (2mt)\ J c 4 J c 5/ \ r i / ( i k n 0a r 2)
f 2 hi _______ r(n + 2)____________1 ikLlootr2 “ J (2z/d2oCM"c)n+1(rc — r )n+1fn+2
2/i4 (r) r (n + 3) t ^ > 1 (5 83){ikDoOi)2rA (2ifcf20a r c)n+1(rc — r )n+1t n+3 J
C H APTER 5. THE LINEAR TIM E-D EPEN D EN T P R O B L E M 90
Close to the critical layer where |r — rc\t ~ 0 (1 ), the solution is
( ^ J + J s)estds ~ e- ^ e- 2tfen°^c(rc-r)t k 2 mr\ J (i k Q 0a r 2)
2h i(r) y - r ( n + 2)( 9 . H c O n f y r A n + 1 ( r ^ — « O n - t - l / n + likQo(xr2t (2ikDoarc)n+1(rc — r )n+1tT1
2 hi{r) ^ T(n + 3)V 1 P L 1 - 1 (A 84'!(ikLl0ct)2r4t2 ^ (2iA:f2oQ!rc)n+1(rc — r ) n+1£n+1y
Note th a t all the terms in both series are of the same order in th e critical layer where
|r — rc\t ~ 0 (1 ). Following Campbell and Maslowe (1998), the series in (5.84) can be
expressed in term s of exponential integrals which are defined as
ro o - z u
E£(z) = I —k~du, Re(z) > 0 (5.85)J i u
andro o zu
E k ( z ) = / — du, Re(2:) > 0 . J i u
After integrating by parts we get
(5.86)
< - >
So close to the critical layer where (r — rc)t ~ 0 (1 ), the solution is
0 (r , f) ~ - e - “ ‘ ( ^ ( ^ ± - E y ^ ‘ \ 2ikr?Q |rc _ r |()
2 h j{r) j?egn(r-rc) (0i{ikLl0a r 2)2t2
E sgn(r '•c)(2tfcr*a|rc _ r |^ + q , (5.88)
where hi and are defined by (5.60) and (5.81) and 4>oc O'") is th e steady-state solution
which was derived in chapter 4. Close to the critical layer, the solution ^ ^ ( r ) is in the
C H APTER 5. TH E LIN EAR TIM E-D EPEN D EN T P R O B L E M 91
form (4.59). The solution (5.88) consists of the steady part and term s proportional to
negative powers of t. As t —> oc the solution converges to the steady state solution.
So, we have found the solution (5.76) valid for |r — rc\t » 1, t » 1, and the
solution (5.88) valid for |r — rc\t ~ 0 (1 ), 1.
5.4 T he inner layer so lu tio n
The time-dependent solution (5.76) and its derivatives with respect to r are singular
a t r — rc. The steady term includes a term proportional to (1 — 4 ) lo g ( l _ r |)r c r c
which is singular a t r = rc and the time-dependent term s of solution (5.76) include
terms proportional to ~ r - In order to find a solution th a t is nonsingular a t r — rc
we have to consider a so-called inner layer around r = rc.
We can write the solution (5.88) for r close to r c in terms of (r — rc) as
« r , t) ~ * _ ( * - « - ( T ) " ( j M L B r ^ \ 2 , k r ^ K - r\t)
2hi(r):E.'lgn{r Tc){2ikr2ca\rc - r\t) + O ( , (5.89)
(ikLl0a r 2)2t2 c \ t 3
where $oo(r ) is defined by (4.59). The leading-order term s of $ 00(r') also can be
w ritten in terms of (r — rc) as
< M r ) ~ e - ia,% ( r c) ( y )-k r 4
~ r ) lo§ ( ~ ) + ° ( E ~ r )2 ri rc
1 - ^ ( r c - r ) lo g (rc - r)
, |r - r c| < 1, (5.90)
wherer 2
M r c) = ( ^ l - 2 ^ 1 —^ J l o g ^ l - ^ J J . (5.91)
We wish to find a nonsingular solution valid in the inner region, where |r — rc| <
C H APTER 5. THE LIN E A R TIM E-D EPEN D EN T P R O B L E M 92
for t 1. On a time scale of t ~ 0{j i 1), where / i d , the thickness of the critical
layer a t time t is \r — rc\ ~ O(fi). So we define a new radial variable
Z = - ( r - r c) (5.92)
and a slow-time variable
T - nt. (5.93)
At the edge of the critical layer \Z \T ~ 0 (1 ), and inside the critical layer \Z\T < 0 (1 ).
We rewrite (5.88) in terms of Z, T and /r as
<f>(Z, T , t) = e~lujth\ (r) ( l + n \ o g i ^ ( —\ Z + Z + — Z lo g (-Z )\ \^*c/ \ \ ^ c TCJ rc
- R V 2 ( t k D 02 a r t:jZ\T) + 0 ( ^ 2). (5.94)
Making use of the following property of the exponential integral E<i (Abramowitz and
Stegun, 1964, equation 5.1.12) tha t
OO / \nE f { z ) ~ ± ( z \ o g ( ± z ) + (~fE - l ) z ) ~ ...............7 ".> (5-95)
where 7e is Euler’s constant, we can rewrite (5.94) as
r \ k /4 /r i 2<f>(Z,T,t) ~ J e l“l ^ /q ( rc) + /ulog/r J Z + /j, lo g — j Z
+ R ^ Z log( - Z ) - , 1 (ik2Q0a r cZ 'I ') lo g ( -z fc 2 fW cZT)
+“ l ^ F ^ ~ 1)l-ik2^ r‘Z T ) +
2/ii(rc) ^ (—2ikQ0a r cZ T ) 2 \ Ofifik S l^ lT Z . ' ( „ - ! ) „ ! + ° ^ ) | - (5-96>
w c ~ _ n * > - ^ 1 v / /
C H APTER 5. TH E LIN EAR TIM E-D EPEN D EN T P R O B L E M 93
After some simplifications, (5.96) becomes
4>{Z, T, t ) ~ e - l“l hi{rc) + filogfj, ( Z + n log — 1 Z
+/z— Z lo g ( -Z ) - n - — ^ - Z log(2ikQ0a r cT) - fJ -h--~^- Z \ o g ( ~ Z ) rr. r, r.
4hi(rc) 2hi(rc) E72—0, 711
( - 2 ik n 0a rcZ T ) T (n — l)n! + o (m2)
\ZT\ < 1. (5.97)
We see th a t the terms with Z log( - Z ) cancel each other and so we obtain a nonsingular
inner solution
fanner(Z,T,t) ~ e \ e lwth x (rc) ( l + n log / i — Z + ( — log — ) Z\ r xJ \ tc \ r c rc
4 4—fi—Z lo g (—2ikLloarcT) — n —(7 j5 — 1 )Z
Tr Tr
+/iT
2 + 8 ik n 0a Z 2T 2 + Ak2Q2° al rcZ3T 3 + o ( z 4r 4)ikLloar^
,(5.98)
In term s of the variables (r — r c) and t, (5.98) can be w ritten as
-kf a n n e r ( r , t) ~ e lut J e luJth x{rc) ^1 + logH— {rc - r)
4 2 \ 4 4— log — ) (rc - r) 4 (rc - r) log( - 2 ikfiQ0a r ct) + — (7^ - l) ( rc - r)+
+ - ikQ0ar%+ 8iA;floc*(rc ~~ r) t +
Ak2PtQa2rc{rc — r)3t3+ 0 ( ( r c - r) t ) ( 5.99)
This solution is valid for \Z \T < 1, or |( r — r c) |t < 1. In contrast with the outer
region where the solution tends to a steady state as t —> 00, th e inner solution grows
like log(f). From this we note th a t the qualitative behaviour of the solution for this
linear configuration for vortex waves with /3 — 0 is the sam e as for Rossby waves
CH APTER 5. THE LINEAR T IM E-D EPEN D EN T P R O B L E M 94
(with /3 ^ 0) in a configuration w ith rectangular geometry (Dickinson, 1970, Warn
and Warn, 1976).
In the next chapter we introduce nonlinearity and the /3-effect to the vortex wave
problem.
5.5 D iscussion o f th e linear tim e-d ep en d en t so lu
tion
We discuss the linear time-dependent solution we have obtained here and compare
with the conclusions made by other researchers who have exam ined similar configura
tion. The problem of wave-mean flow interactions a t a critical layer has been studied
by many previous researchers in the context of barotropic Rossby waves in a rectangu
lar domain (Dickinson, 1970, W arn and W arn, 1976, W arn and W arn, 1978, Campbell
and Maslowe, 1998, Campbell, 2004). We see some similarities between the solutions
we have derived in this chapter and those obtained in these previous studies. In the
rectangular configuration as well, the tim e-dependent solution comprises the steady
solution and time-dependent term s which are arises from th e branch points of the
integrand in the inverse Laplace transform . One branch point arises from the effect of
the mean flow, the other from the effect of the boundary condition. In both problems
in the outer layer time-dependent term s decrease w ith time as 1/t, and the inner layer
solution is valid for |r — rc\t < 1. We note th a t in the rectangular configuration these
effects arise because of the variation of Coriolis force, in our problem they arise even
in the /-p lane configuration.
The problem of deriving analytical solutions for vortex wave propagation in cy
clones was considered by other researchers (see, e.g., M ontgomery and Kallenbach,
CH APTER 5. TH E LIN E A R TIM E-D EPEN D EN T P R O B L E M 95
1997 and Brunet and Montgomery, 2002). We can compare ou r results with the re
sults obtained by Brunet and Montgomery (2002) whose configuration is the most
closely related to ours. They solved the initial-value problem on an /-p lane and used
an angular velocity profile in the form of a quadratic function, sim ilar to the profile we
used. They solved the initial-value problem using a Hankel transform . Their solution
for large r in the notation of this thesis is w ritten as
4>{r,t) ~ (1 + i f )-/i/4_1 (—r 2( l + i t ) )k/2 M ( ^ ^ + 8 , k + 1, - r 2(l + it)), (5.100)z z
where M(a, b, z) is Rum m er’s function.
Let us compare this result with the solution we derived in th is thesis. For negative
values of the real part of z, as \z\ —> oo, Rum m er’s function has the approximation
(Abramowitz and Stegun, 1964, equation 13.1.5)
M ( a , M ~ W ^ o ) {~ Zra[1 + 0 ( |Z |" 1)] (5’101)
for \z\ —> oo, where Re(z) < 0. So the asymptotic behaviour o f the solution for large
values of |r 2(l + i t ) | is
<f)(r, t) ~ O ^ [ —r 2(l + it)]k^2 [—r 2( l -I- it)] 2 2 ^ ~ O ^ r ” '/fc2+8[l + i t ^ ■
(5.102)
For large t this solution is proportional to 0 (r~ v'fc2+8) and it has the same qualita
tive behaviour as the steady-state solution (4.56) we obtained for r > rc which is
proportional to r -v,fe2+8.
To figure out the reason for the constant of 8 in this expression, we have to go
CH APTER 5. THE LIN EAR TIM E-D EPEN D EN T P R O B L E M 96
back to the beginning of chapter 4, to the equation we solved for the steady case
6' k2 - 4 - { v ' + v / r )
The numerator in the last term of equation (5.103) is + v/ r) . This expression
can be written in terms of the stream function of the mean flow ip(r) as
- S - ( v ' + v / r ) = -^-{iprr + ~i>r) = (5.104)r or r or r r or
which is proportional to the radial gradient of the vorticity of the mean flow. W ith
our choice of the angular velocity profile Cl = fl0( l — a r 2), we have v(r) = fi0(r — a r 3)
and so the gradient of the mean flow vorticity (5.103) is —8f2oar. This is the reason
for the 8 th a t appears in the power of r in the solution. We conclude th a t the rate
of attenuation of the vortex wave am plitude depends on the wave number k and the
gradient of the vorticity of the mean flow.
C hapter 6
Effect o f nonlinearity and /3-effect
6.1 W eakly-nonlinear an alysis
The linear time-dependent solutions derived in chapter 5 can b e considered as a first
approximation to the solution of the nonlinear equation (3.53)
J ; + = - - ( ^ r V 2^A - ^ AV V r)- (6.1)ot r oX J r or r r
To obtain a better approximation we can consider (3.53) an d carry out a weakly-
nonlinear analysis for £ < 1 . We can express the solution of (6.1) in powers of e
4>(r,\,t) ~ ip(0\ r , X , t ) + e ^ (1)(r, A,t) + 0 ( £ 2), (6.2)
where X,t) ~ 0 (1 ). The leading-order term satisfies th e boundary condition
^ ° ) ( r i , A,t ) = elkX + c.c. and can be thus w ritten as
tp ^ ( r , X, t ) = t)elkx + c.c. (6.3)
97
C H APTER 6. EFFECT OF N O N L IN E A R IT Y A N D 0 -E F F E C T 98
When the expansion (6.2) is substituted into the nonlinear equation (6.1), at 0 (1 ) we
obtain the linear equation (3.54) which tells us th a t ^*0)(r, X,t) is the linear solution
(5.76) tha t was derived in chapter 5.
At 0(e) we obtain
d v d \ 2 m 1 m d , 1 .Wt + r T \ ) V 'P d - C ' + C
= - - ( ^ w f - (6.4)
with the boundary conditions ip^(r-i, A, t) = 0. Substituting (6.3) into (6.4) gives
d v d \ 2 m 1 (i) d , 1 .7J7 + V i)( ] - - 1p x w - ( v r + - v ) ot r o X J r or r
( + W tT + V>«<°> - 0,<o|« + «(0>4? + V(0VS>ik» . T t t t t • t r t r r _ ~ t t t t t - t t t t t t - t t * __r \ \ r r
„ 2 i k A ( ( 0 ( O ) ) 2 + ^ ( O ) 0 ( O ) _ I 0 ( O ) 0 ( O ) _ V > 0 £ >
r r
_ e - 2 i k X ( ( ^ ( 0 } ) 2 + 0 * ( O ) ^ ( O ) _ ^ * ( 0 ) ^ ( 0 ) _ I ^ * ( O ) 0 J 4 O )
( 21m - e2lfe* ( ( 0 W)2 + - V > < ^ > - V 0> ^ ) '
_ e ~ 2 i k \ f { € ( 0 ) ) 2 + ^ ( O ) 0 * ( O ) _ I ^ * ( O ) 0 * ( O ) _ I ^ ( O ) 0 ( ; o A \ ( g 5 )
This implies th a t takes the form
rp^\r , A, t) = </>^(r, t) + <p^(r, t)e2lkX + c.c., (6 -6)
where 4>^(r, t) and <f> (r, t) are complex functions. The first te rm in (6.6) is the zero-
wavenumber component which arises from the wave-mean flow interactions. This term
satisfies
Jl V V > = ?i*im( 2 .W (»>^"))). (6.7)
CH APTER 6. EFFEC T OF N O N L IN E A R IT Y A N D (3-EFFECT 99
The evolution of this component is discussed in more detail in section 6.4. The
second term in (6.6) and its complex conjugate are the leading-order contributions to
the second harmonic. The function (f> (r, t) satisfies
( d v d \ _2 ,m 1 ,(i) d , 1 .{ a t + r d - \ ) V * A
= - ( § ( ( 4°>)2 - ). (6 .8 )
In the outer region where \r — rc\t 1, for t 1, the linear solution t) is (5.76)
-iu>t
, iknoarH ( r \ k ( 2^1_(r)_______ 2^2 (r)_______v i / \ t 2(ifcf7oQ:)2)(^2 — r 2)r2 t3(ikDoa)3(r2 — r 2)2r 4
+ c«h w ?. ( l Y N _____Mr)______ + j ( 6 . 9 )\ r i / \ f 2(?A;f2oa:)2) ( r2 — r 2) r 2 t3 ( ikf loa)3 (r2 — r 2)2r\ )
where 4>oo(r) is the steady-state solution. In the inner region where |r — rc\t <C 1, the
linear solution is
4>{0}(r,t) = cf>inner(r,t) ~ e ^ h i ( r c) ^1 + ^ (r _ rc)
4 4 (r - rc) log( - 2 ikQ0arct ) {^B - l ) ( r - r c)
+ S i/M M r - ^ ) 2i2 - ~ rc)3^ + 0 ((r - rc)V )ikLioar2 (.6 .10)
The nonlinear terms on the right-hand side of (6.5) contain derivatives with respect
to r. In the outer region where 0 ^ ( r , t) is given by (6.9), we observe tha t each time
th a t t) is differentiated with respect to r, the term s w ith the factor of elkn°ar2t
are multiplied by a factor of t. This means th a t for < > 1, th e leading-order terms on
C H APTER 6. EF FEC T OF N O N L IN E A R IT Y A N D (3-EFFECT 1 0 0
the right-hand side of (6.5) are proportional to
„jfc00Q(r2— r*) 0 (^2)
In the inner layer where <f>(°\r,t) is given by (6.10), we observe th a t (p^ (r , t ) is
0 ( l ) + 0 ( r —r c) + 0 ( lo g t) (r — rc) (5.99) and so the first derivative of (p(0) w ith respect
to r is 0 (1 ) 4- O (logt). We can write it in the form
4/ij(rc) / . 2 1(j>i0)(r, t ) -------1— E- ( lo g ------ 1 + t -ei k Q o 2 a rc( r ~ r c)t
rc V rc ikQ,Q2arc{r — rc)t
+ log( - ikQ 02arct) + (1 - 7b ) t 1
n = 1
ikLlo2arc{r — rc)t
rc)t)n (n + 1)!
(—ikQo2arc(r — rc)t)n \ . . . .
In the first derivative (6.11) there is the term proportional to eika-°a2rc{T-rc)t_
(Pr is differentiated with respect to r, it is multiplied by a factor of t. So, the second
derivative with respect to r is 0( t) , and the th ird derivative is 0 ( t 2).
Thus, for large t, the term s on the right-hand side of equation (6.1) are 0 ( t 2).
For large t the nonlinear term s increase as 0 ( t 2) and become 0 (1 ) on a time-scale
of t ~ 0 ( e ” ,//2). We can conclude tha t the nonlinear expansion (6.3) is valid for
t < 0{e~1/2). To examine the late-time nonlinear solution we would have to define a
slow-time variable r = el/2t.
All the solutions derived up to this point are for vortex wave propagation on an
/-plane where the Coriolis param eter is assumed to be a nonzero constant. In the
next section we reintroduce the /5-effect into the governing equations and examine
first a linear and then, in section 6.3, the nonlinear configuration.
C H APTER 6. EFFECT OF N O N L IN E A R IT Y AN D /3-EFFECT 101
6.2 Effect o f th e variation o f th e C oriolis force
The vortex Rossby wave equation with the /3-effect included is (3.52)
( d v d \ o , I , d 1 P , ■ i 0 - x a , x) V ' lP V\w-{Vr + - v ) sin A + -u c o s A + /3^y cos A =\ o t r oX J r or r r e- - ( t f v V 2^ A- ^ A V V r ) . ( 6 .12)
r
The problem is governed by two nondimensional param eters (3 and e. In section
3.1 we demonstrated th a t (3 can be considered as a small param eter given the typical
length scale for our problem, which gives /3 ~ 10~2. In general, wave amplitudes
are small relative to the magnitude of the mean flow so the param eter e can also be
considered to be small. T hat justifies the use of weakly-nonlinear analyses such as
the analysis th a t was carried out in section 6.1. The first step in our investigation
of the /3-effect shall be to examine the linear problem with th e /3-effect added. This
corresponds to a configuration where /3 e and so the nonlinear O(e) term s in (6.12)
appear at higher order than the 0(/3) terms. However this scaling means th a t the
0 ( /3e~1) term in (6.12) becomes the dominant term in the equation. Since this term
cannot satisfy the specified 0 (1) boundary conditions, the linear problem is not well-
defined in th a t case. In order to overcome this issue we would have to neglect the
0(/3e_1) term. We can do this by adding an additional term B(r, A) = —|u co sA to
the governing equation as discussed in section 3.1. In section 6.3 we consider the full
nonlinear equation (6.12) with both param eters e and /3 included. We shall assume
there tha t /3 <gC e which is the more likely physical configuration.
The linear equation with the addition of the term B becomes
C H APTER 6. EFFECT OF N O N L IN E A R IT Y A N D f3-EFFECT 1 0 2
To solve (6.13) we look for a solution of the form
ip(r, A, t) = ^ (0)(r, A, t) + /30(1)(r, A, t) + 0(/32), (6.14)
where tp^ (r , A, t) is the linear solution of (3.54) obtained in chapter 5 for the /-p lane
configuration. As in section 6.1, we write 0(°)(r, A, t) as
A, t) = < ^ ( r , t)etkX + c.c. (6.15)
After substitution of (6.14) into the governing equation (6.13) we obtain at 0(0):
+ I V V (1) ~ = ~ COS \ (6.16)d v d dt + r d A
with the boundary conditions -i//b(r, A,t) — 0. W riting cos A = (ezX + e zX)/2 and
sin A = (elX — e~zX)/2i and substituting (6.15) into (6.16), we obtain
+ ) V V (1) - ^ v )d v d dt + r d A _ 1
~ 2V ° > - ^ 0)) e*(fe+1)A _ °) + 0 (0)(r ) ) + C.C. (6.17)
This tells us tha t A, t) can be w ritten as
0 (1)(r, A, t) = ^ (la)(r, A, t) + 0 (16)(r, A, t ), (6.18)
where ^ la> satisfies
1 ( k2 \ r
0 ( ° ) - 0 (o ) )eW ) A+ c.c . (6.19)
C H APTER 6. EFFECT OF N O N L IN E A R IT Y A N D /3-EFFECT 103
and satisfies
^ V > + 4 0)( r ) ) e ^ - ^ + c.c., (6 .20)
with boundary conditions tp^la\ r i , A, t) — 0 and ip(lb\ r i , A , t ) = 0.
Each of the equations (6.19) and (6.20) is a nonhomogeneous differential equation,
so we can write the general solution in each case as the sum of the general solution
of the corresponding homogeneous equation and a particular solution of the nonho
mogeneous equation, which is proportional to e^fc±1 A -f c.c. Recall th a t the function
is the sum of (f>00(r)e lujt and two time-dependent term s th a t go to zero as
t oo. Therefore, a particular solution of each of (6.19) and (6.20) is proportional
to e*((fc±1)A- wt) for t 1. The zero boundary condition at r = rq tells us tha t the
solution of the homogeneous equation is also proportional to eI((fc±1)A- wt) for f > i.
Thus the solutions of the homogeneous equations corresponding to each of (6.19) and
(6.20) are of the same form as the solution bu t w ith the wavenumber k replaced
by k ± 1. For f > 1 the solution of the homogeneous equations a t leading order are
(6 .21 )
and
(6 .22 )
where <^la) and 4>(lb> satisfy equations
CH APTER 6. EF FEC T OF N O N L IN E A R IT Y A N D (3-EFFECT 104
and
*<>»" + _ A l l > > » _ ? 5 £ 2 _ ---------- = 0 . (6.24)r H (H0(l — a r 2)(k — 1) — uj)
Equation (6.23) has a singular point a t the value of r where uj = (k + l)flo (l — a r 2),
i.e. the mode with wavenumber k + 1 has a critical radius at
1 ^ - n 1 ' • (6.25)c*+1 a 1/2 \ (A; + l)fl0/
Similarly, the mode w ith wavenumber (k — 1) from (6.24) has a critical radius at
1 / \ 1/21 / . uj \ '
rCk = — I 1
The wavenumber k is the number of wavelengths in a full circle 0 < A < 2tt, s o k is a
positive integer. We can consider two cases k = 1 and A; > 1. If k = 1, then k + 1 = 2
and A: — 1 = 0. This means th a t a zerowavenumber (non-oscillatory) component is
generated by the /3-effect. This component represents a change in the mean flow and
will be discussed in section 6.4. If k > 1 then k + 1 and A: — 1 are both nonzero and
there is no effect on the mean flow.
The critical radii for the 3 modes are related by
rCk_, < r c < rCk+1 (6.27)
as shown in Figure 6.1. All three waves (with wavenumbers A;, A: + 1 and k — 1)
experience critical layer absorption a t their respective critical radii. This means tha t
as the waves propagate outwards from the eye of the vortex, the (A; — 1) mode is
absorbed first at rCJc_1 and its am plitude is greatly reduced while the other waves
C H APTER 6. EF FEC T OF N O N L IN E A R IT Y A N D (3-EFFECT 105
continue to propagate. Then the k mode is absorbed a t rc while the (k + 1) mode
continues to propagate and is absorbed a t rCk+l. This m eans th a t a t r c there are
only two components of the solution with non-negligible am plitude, corresponding to
wavenumbers k and (A; + 1). In the critical layer around r c, th e inner layer solution
for the mode k is valid, but to obtain information about the effect th a t the (k + 1)
mode has on the evolution of the critical layer around r c, we need an outer solution
for the (k + 1) mode.
In the critical layer near r c, the (k + 1) mode is described by its outer solution.
We write it as
^ la\ r , A, t) ~ [^(T O e-*** + Yparticular] ei(fc+1)\ t » 1, (6.28)
where 0 x ( r ) is the solution of (6.23), and can be w ritten as a linear combination of
the hypergeometric functions of the form
/ 2 ( 1 ) W J*
-{k+1)
- f c - i
Cfc+1+ 0 1 -
Cfc+1
(6.29)
. (6.30)
To find Y p a r t i c u l a r we return to the nonhomogeneous equation (6.19) and substi
tu te in the inner solution for 0 °* (r, t) since we are considering the critical layer near
r c. The inner solution for 0 (o (r, t) is (5.99). Substituting this into the right-hand side
CH APTER 6. EF FEC T OF N O N L IN E A R IT Y A N D (3-EFFECT 106
\ j ' c (m o d e 'k + 1 )
le k-1
Figure 6.1: S ch em a tic d ia g ra m o f th e k, k — 1, k + 1 m o d e s p ro p a g a tio n .Schematic diagram of the propagation of the waves w ith wavenumber k (arising from the boundary forcing a t r — r x), and wavenumber k ± 1 (arising from the /3-effect). The shaded regions show the inner layers for the critical radii o f each wave. The arrows show propagation of waves up to their critical radii. T he wave with wavenumber k + 1 is influenced by (3 effect at the critical layer for wave with wavenumber k.
C H APTER 6. EFFECT OF N O N L IN E A R IT Y A N D /3-EFFECT 107
of (6.19) gives
(f)(0) __ 0(0) ^ e - i u td ) ( r ^ 1^ ~ C'W /ll(rc)~ ( r ~ rc) + C (t)hi{rc) + 0 ( l / t )
+ Cn[(r - rc)(t)]n ) , t » 1, (6.31)n=l
where
C (t) = — ( \ o g ( - i k n 0ar^t) + (7^ - 1)) .T* n
(6.32)
Then we use the variation of param eters formula to obtain th e particular solution
Y p a r t i c u l a r ( r , t ) ~ ( ^ J j y ^ y / l ( l ) /2(1)
^ y / 2( i ) ( ^ (0) - / 1(1), t » 1, (6.33)
where W(r) is the Wronskian of the two functions / ip ) and f 2(i). So, the general
solution for the nonhomogeneous equation can be w ritten in th e form
^ y / i ( i ) ( V 0) - 4 0) )dr + A 2 ) / 2(1)
- ( / Mi) ( V 0) ~ (t>{r ] d r - f m , t > 1. (6.34)
Since we are evaluating the solution near r c, we write (6.29) and (6.30) in terms
of (r — rc) and get
/i(p ~ r {k+1) 1 - 21og 1 - 1 - + O I -T' f 1 4”' c fc+l / \ c* + 1 / \ cfc+l
~ r~ (fe+1)[a0 + a ^ r _ r c)], |r - r c| < 1 (6.35)
C H APTER 6. EFFECT OF N O N L IN E A R IT Y A N D (3-EFFECT 108
and
/ 2(i) ~ r {k+1) ( 1
where
c k + 1
~ r (*+1)[a2 + a3(r - r c)], |r - r c| < 1,
Cfc+ 1 ck+ ir-2
ai = ^ - ( 21og(lCk+1 Ck+1
0,2 — 1
03
c*+i2r c
r 2C f c + 1
The Wronskian for the functions /i(i) and / 2(2) is
W(r)/l(l) / 2(1)
/ l( l) /2(1)
~ r 2(fc+0 (aoa3 _ a ia 2), |r — r c| < 1.
is
— icot [/(ao + a ^ r - r c ) ) ^ 2^ 2) ( k u , s k------------------------- C2 h+i) ~ h n r ) - C { t ) h i - ( r - rcaoa3 — Oia2 r ^ k+1> \ r r
+C(t)hi + ^ Cn((r — rc)t)n J dr + A\
I71— 1
r fc [a2 + a3(r - r c)]
(a2 + a3(r — ^c)) r^2k+2'> ( k , . . . k~ ----- “ ------- ~(2 k+i) r - C < r - r c + C i /ma0a3 — a ia 2 r'.2'c+1) \ r r
+ Y 2 cn((r - rc)t)n \ d r + A 2n=l
r [a0 + a i ( r - r c) ] , |r — rc\ < 1, t » 1.
(6.36)
(6.37)
(6.38)
)
(6.39)
C H APTER 6. EFFECT OF N O N L IN E A R IT Y A N D (3-EFFECT 109
To find (^la*(r) we have to evaluate the integrals in (6.39). Note th a t
/ j T c n[ ( r - r c)t}ndr = [ cn[ ( r - r c)t}ndr = y ^ < [ ( r - r c)t]n ~ O f y") . (6.40)^ n—1 •* n= 1 n=2 ' '
After integrating (6.39) we obtain
<j>{la\ r , t ) ~ e-»wfr (_fe) [a2 + a3(r - r c)] - A2[a0 + Gi(r — rc)] + ^ r2C'(i )^ >
(6.41)
or,
<f la\ r , t ) ~ [0(1) 4 - 0 ( r — r c) + 0 (lo g t)]e_lwt. (6.42)
This solution represents the leading-order contribution to the solution near the crit
ical layer (near rc), arising from the variation of the Coriolis param eter. The wave
amplitude increases as log t because of the /3-effect, and it is th e result of interaction
between the k mode and the k + 1 mode in the critical layer for the k mode.
6.3 C om bined effects o f n o n lin ea r ity and th e /3-
effect
We now consider equation (3.52) which includes both nonlinearity and the /3-effect.
3 , » 9 ) „ 2 i 1 ^ /- . * 0 i -v , 0,Tr: + - TTT v ^ -i ’\ T r ( vr + ~v) V’a sin A + - v cos A -I- /3^y cos Aat r o A / r or r r e
= - - ( ^ V V a - ^ aV V t-)- (6.43) r
The param eters e and (3 are both small, the solution of (6.43) can be w ritten as
0 ~ + £ipH) 4. (3i(j(la) 4- 4- 0 ( e 2) -f 0 ( e ( 3) 4- 0(/32), (6.44)
CHAPTER 6. EFFECT OF N O N L IN E A R IT Y A N D 0 -E F F E C T 110
where i p ^ \ ip(la\ i p ^ are functions of r , A, and t.
As noted in chapter 3 and section 6.2, w ith the reference length scale for our
problem it is reasonable to consider the situation where « £ < 1. In th a t case
0e~l <C 1 and so the 0e~ l term th a t arises in the solution is not a leading order term.
We first investigate the 0(e) and 0 ( 0 ) term s in the solution an d then a t the end of
this section we will add the 0e~ l term to the analysis.
We can rewrite equation (6.43) in the form
L(ip) + 0B(xp) = — eN(i>, ip), (6.45)
where L, B, and N are the partial differential operators given by
. . ( d v d \ 2 1 d 1
m = \ m + r T \ r ^ - r ^ r ^ + r ^
N t y u l h ) = ~ ?/hAV2t M ,
B(ip) = ~ —ip\ sin A + "i/y cos A = -^r. tp\(elX - e~lX) + \ ipr (elX + e~lX) r 2i 2
Substituting (6.44) into equation (6.45), we obtain
L(ip{0)) + 0L(ip{la)) + 0L(ip{lb)) + eL(ipw ) + 0B(ip{o)) + e0B(iP{1))
= - e N ( ip i0), i p ^ ) - e 0 N ( ^ \ ^ ) - £0N('4>{o\'4>W)
- e 2N ( ^ ° \ 0<x>) + O ( e 20 ) + O(e02) + 0 ( e 3) + O ( 0 3). (6.49)
To analyze the order of the different term s in (6.49), their wavenumbers and their
possible interactions, we first note th a t a t leading order L ( i p ^ ) = 0 , where i p ^ is
the linear solution defined in chapter 5 w ithout the /1-effect and corresponds to the
modes ± k specified a t the forced boundary r = r j. We note also th a t the 0(e) term
(6.46)
(6.47)
(6.48)
C H APTER 6, EFFEC T OF N O N L IN E A R IT Y AN D /3-EFFECT 1 1 1
corresponds to wavenumbers 0 and ± 2 k, as shown in section 6.1. The 0(/3) term
corresponds to wavenumbers ± (k + 1) and ± ( k — 1), as shown in section 6.2, the
0 ( e 2) term corresponds to wavenumbers L k and ± 3 k, the 0 ( e 8 ) term corresponds
to wavenumbers ±1, ± (2 k + 1) and ± (2 k — 1), and the 0(/32) term corresponds to
wavenumbers ±fc, L (k + 2) and ± ( k — 2). This information is summarized in Table
6 . 1.
If f3 ~ e, e <C 1 and 8 -C 1, then e and j3 are the same order of magnitude and the
solution up to 0 (e ) and 0 (8 ) is simply the sum of the solutions obtained respectively
in sections 6.1 and 6.2.
Table 6.1: Wavenumbers corresponding to the term s in equation (6.49)
order terms wavenumber wavenumber special case k = 1
£ N ( ^ ° 8 i ’(0)y, L (ip^) 0 ;± 2 k 0, ±2
8 T ( ^ la>); L(-0<16>); B (^ (0)) ± (k — 1); ±(/c + 1) 0, ±2
82 B (^ (la)); 5 ( ^ (lb)) ± ( k - 2 ) ] ± k ; ± ( k + 2 ) ±1, ±3
e8£ ( ^ (1>) ±1; ± (2 k — 1); ±(2k + 1) ± 1 ,± 3
iV('0(o), ^ (la)); yv(^(0), ^ (lb)) ± l ;± ( 2 f c - l ) ;± (2 fc + 1) ± 1 ,± 3
£■2 ± 3 k ± 1 ,± 3
-k-2 -k-1 -k+1 -k+2 k-1 k+1 a
...........
Figure 6.2: S ch em atic d ia g ra m for w av en u m b e rs . Schematic diagram for the wavenumbers in equation (6.49) of the term s of 0 (e ) and 0(/3), the case where k > 0, k ^ 1. The 0 ( e ) terms include a component with zero wave number, but 0 (8 ) do not. There is no interaction between the e and 8 term s in th e mean flow.
C H APTER 6. EFFEC T OF N O N L IN E A R IT Y A N D /3-EFFECT 1 1 2
If k = 1, then the 0 (e ) and 0 (3 ) term s both correspond to the same wavenumbers,
0 and ±2.
If /? -C £ 1 then the solution can be w ritten in the form
rjj ~ ^(°) + + /3ip<'la'1 + + 0(ef3) + 0 (e 2) + 0(/?2). (6.50)
Substituting this equation into (6.49) we obtain
0 (1 ) ; L(^°>) = 0,
0 ( e ) : L ( ^ ) = - iV ( ^ (0), ^ (0)),
0(13): L (^ (la)) + L(x/j{lb)) = - £ ( ^ (0)),
0(e/3) : L(4>{2eO) = - iV ( ^ (0), ^ (la)) - Ar(^(o),-0(16)) - B(ip{1)),
0 ( e 2) : L ( ^ {2e)) = -7 V(^(o),-0(1)) - N ^ l ' i p W ) ,
0((32) : L ( ^ ) = -B ( i / i {la) + ^ u)). (6.51)
If A; > 1, then, up to the orders shown in the table, there are no interactions
between the terms of each order because they all correspond to different wavenumbers.
The larger the value of k is, the further apart the wavenumbers for the 0 (e ) and 0(/3)
terms are. This situation is illustrated in Figure 6.2. Continuing to higher orders of
e and (3 the solution can be represented as a discrete spectrum of contributions from
several wavenumbers. Since the 0 (e ) term includes a zero wavenumber component, it
means tha t nonlinearity affects the mean flow. W ith k > 1, however, the 0 (3 ) term
has no zero wavenumber component, so the /3-effect does not affect the mean flow.
The 0(/32) and 0 ( e 2) include term s corresponding to the fundam ental wavenumber
± k . All the 0(/32), 0 (e /3) and 0 ( e 2) term s contribute higher wavenumbers to the
solution. Figure 6.2 demonstrates th a t there are no further contributions to the zero
CHAPTER 6. EF FEC T OF N O N L IN E A R IT Y A N D /3-EFFECT 113
wavenumber component until higher orders. So we can conclude th a t if k > 1 the
mean flow is influenced only by nonlinearity.
The only situation where term s of different orders have the same wavenumbers is
the special case where k = 1. In th a t case the O(e) and 0((3) term s both correspond
to the wavenumbers k = 0 and k — 2, and so the solution u p to this order is the
sum of both effects. Since the 0{e) and 0(/3) term s both have zero wavenumber
components, we conclude th a t both nonlinearity and the /3-effect influence the mean
flow. The evaluation of the mean flow is discussed in section 6.4.
We now investigate the 0{/3e~l ) term in the solution
ip ~ _|_ e^(!) + /3t/;(la) + /3'ip(lb) + 0 { e2) + 0(s/3) + 0(/32). (6.52)e
The function A, t) satisfies the equation
0(J3/e) : L (^ > ) = - V- { e iX + e ^ ) . (6.53)
Thus ip(B\ r , A, t) corresponds to a wavenumber of 1 and m ust take the form
= <pB(r, t)e tX + c.c. (6.54)
with the boundary condition tfiB (rx,t) = 0. The solution o f the corresponding ho
mogeneous equation will take the same general form as tha t obtained for the leading
order solution and can be expressed in term s of hypergeometric functions as it was
done in chapter 4. The solution has the same qualitative form as the steady solution
4>oo. Since the nonhomogeneous term is independent of tim e, we can find a steady
particular solution to this equation. Because of the zero boundary condition, the
homogeneous solution should be steady as well. So the function <pB is steady and
CH APTER 6. EF FEC T OF N O N L IN E A R IT Y A N D (5-EFFECT 114
satisfies the equation
& + & - V + ^ r(Br = - f ■ (6.55)r z uj — i lk 2r
Its solution has only a quantitative effect on the solution for th e modes ±1. If k = 1,
<pB has the same wavenumber as the fundam ental mode. If k ^ 1, then the 0 (3 e ~ l )
term does not interact with the fundamental mode or any of th e 0(e) terms (which
correspond to wavenumbers 0, ±fc, ± 2 k).
6.4 E volu tion o f th e m ean flow in th e inner layer
From equation (6.7) we see th a t the mean flow changes w ith tim e due to the nonlinear
interaction. We also saw in section 6.3 th a t in the case where k = 1, the /3-effect
contributes to the mean flow change. In th a t case the change in the mean angular
velocity is the sum of the changes due to the nonlinear and /3 effects. Let us consider
first the change of the mean flow th a t results from the nonlinear wave-mean flow
interaction.
According to (3.49) the to tal stream function is
^ ( r , A, t ) = xj;(r) + etp(r, A, t ) . (6.56)
The perturbation eip(r, A, t) now includes a zero wave num ber component which we
will call ipo(r, t). Thus the mean stream function a t time t is th e sum of the initial mean
streamfunction 4>(r) and the time-dependent mean stream function rp0(r, t) induced by
the nonlinear interaction. We let /iptotai('i',t) be the mean stream function at time t.
Then we can write
Aotai (r , t) = i ) { r ) + -ip0{ r , t ) . (6.57)
CH APTER 6. EFFEC T OF N O N L IN E A R IT Y A N D [3-EFFECT 115
Similarly, the mean azimuthal velocity is
Vtotai(r, t) = v(r) + v0(r, t ) (6.58)
where v0(r, t) is the wave-induced mean velocity. We can ob ta in an equation for the
evolution of the mean flow either from equation (6.7) or d irectly from the azimuthal
momentum equation as done in Appendix B. In either case, we end up with (B.16):
^ r = e2h ^ {2kr ta^<0)« <0|)>' <6'59)
where Vtotai is the azimuthal velocity of the main flow.
To find the change in the azim uthal velocity in the inner layer where |r - rc\t < 1
we use the leading-order solution 0 ^ ( r , t) (5.99)
^(°) ^ e-«wt hi ^ + bi(rc - r ) - ib2^ ( r c - r)
+b2(rc - r) log(f) - i y + ib4t(r - rc)2 + 0 ( ( r - r c)3t2)^ (6.60)
where 6, for i = 1 ,2 ,3 ,4 are real numbers which are defined as
4
&i = —(log(kO0otrl) + - 1),
kQ0ar% ’64 = 8kCloa. (6.61)
CH APTER 6. EFFEC T OF N O N L IN E A R IT Y A N D 0 -E F F E C T 116
The derivative of 0 * ^ w ith respect to r in the inner layer is
# ( 0) ^ e- t ^ h i ^ _ 6i _ ib^ _ _ h lQg(t)
- 2 ib4(rc - r )t + 0 ( ( r - rc)2t2)) . (6.62)
Substituting this into (6.59) gives
* > - ■“ ( - t , : + + 0 « r - r j V ) I . (6.63)dt r% \ r i ) \ 2 t
So the time derivative of the mean azim uthal velocity satisfies
9 V t o t a l
dt~ e2 (o(l) + O + 0 ( t 2(r - r c)2) ) . (6.64)
Integrating with respect to tim e from the initial tim e to tim e t gives an equation for
the azimuthal velocity of the mean flow in the critical layer as a function of tim e t,
W , t ) ~ e(r) - ( j - l t - - ^ ( l o g ( ) 2) • (6.65)
So we see tha t the azimuthal velocity decreases w ith time.
We saw in section 6.3 th a t in the case where k = 1, nonlinearity and the /5-effect
both influence the mean flow. In this case we find th a t
Oi? f 4 7T \Vtotai(r,t) ~ v(r) - e2— h\ I — ~ t + O (logf)2 I + £0O{logt)elult + c.c. (6.66)
rc \ r c 4 J
So the mean flow change consists of two parts: a term decreasing with tim e and
some oscillations with am plitude proportional to log t. This change in the azimuthal
velocity means tha t the mean angular velocity Cltotai (r, t) also decreases with time in
the critical layer and hence the location of the critical radius changes with time.
C H APTER 6. EFFECT OF N O N L IN E A R IT Y A N D (3-EFFECT 117
For any initial Ototai(r) profile th a t decreases monotonically with r (such as the
profile Qtotai(r, 0) = Cl(r) = f2o(l _ £*r2) used in our study), a decrease in Ototai(r,t)
means tha t the critical radius moves inwards, towards the centre of the vortex. This
is seen in Figure 6.3. This result is in agreement w ith hurricane observations where
the secondary eyewall moves toward the centre of the hurricane (see, e.g., Sitkowski
et al., 2011).
» • • • *
r'r r, r
Figure 6.3: D ynam ics o f th e critica l radius location . T h e initial critical radius is given by £2 — uo/k, where £2(r) is the initial angular velocity. If the angular velocity decreases in the critical layer near r c, as shown by the dashed line, the critical radius moves towards the centre of the vortex, from location r c to r'c < rc.
C hapter 7
C onclusions
7.1 Sum m ary
In this thesis we presented approxim ate asym ptotic solutions for a problem represent
ing vortex Rossby waves propagation in a tropical cyclone. The cyclone was considered
on a horizontal plane defined by polar coordinates r and A. T he cyclone was repre
sented as a vortex with angular velocity 0 ( r ) in a domain given by t\ < r < oo and
0 < A < 2n. At r = r lt a sinusoidal wave was generated by a boundary condition
— e^kX wtK We considered waves propagating outw ards in the cyclone in
the form 0 (r, \ , t ) = 4>(r,t)etkX + (p*(r,t)e~lkX. We derived equations for the wave
amplitude starting from the governing equations for fluid dynam ics in a coordinate
frame rotating with the Earth. Several different configurations were examined.
First, we considered the linear problem where the wave am plitude 0 is independent
on t. In this case 0 is given by an ordinary differential equation in r, which is singular
at rc, where the phase speed of the wave mode is equal to the angular velocity of
the cyclone rotation. This equation was solved for a quadratic profile of the angular
velocity. The solution dem onstrates the attenuation of the waves in the cyclone at
118
CHAPTER 7. CONCLUSIONS 119
the critical radius.
Next we studied the linear time-dependent problem where th e wave amplitude <f>
depends on both r and t. We derived the solutions valid for la te time for the different
regions in the tropical cyclone: the region between the eyewall and the critical layer,
the region inside the critical layer, and the region outside the critical layer, far from
the centre of the vortex. These solutions were expressed in te rm s of hypergeometric
functions. Each solution comprises a term with steady am plitude and some time-
dependent terms th a t go to zero as t —> oo.
To find a nonsingular solution near the critical radius r = rc, we introduced an
inner layer and an inner variable Z — (r — rc)jfx w ith a corresponding slow time
variable T = fit, where p is the thickness of the inner layer. W e found th a t the inner
solution contains term s proportional to log(t). Then we reintroduced the nonlinear
terms to the problem and observed th a t these term s would grow like t2 in both the
outer and inner solutions. This means th a t the nonlinear solution is valid for t <
0 (e~ x/2). The nonlinear effects arise in the wave mode with wavenumber 2k and in
the mean flow with wavenumber zero.
We also added a /3-term into consideration to evaluate the effects of the variation of
the Coriolis force on the vortex Rossby wave dynamics. Our analyses showed tha t the
/3 term gives rise to wave modes with wavenumbers (fc+ l) and (/c —1). All three waves,
modes (k + 1), k and (k — 1), experience critical layer absorption a t their respective
critical radii. While the waves propagate outwards from the eye of the cyclone, the
(k — 1) mode is absorbed first at its critical layer rCk , and its am plitude is reduced
a t this radius. The k mode is absorbed a t rc while the (k + 1) mode continues to
propagate and is absorbed a t rCk+\. So there are two modes, with wavenumbers k
and (k + 1), in the critical layer r c and the wave am plitude o f the k + 1 mode grows
like log(f) in the vicinity of the critical layer r c. We also considered the more general
C H APTER 7. CONCLUSIONS 1 2 0
configuration th a t includes both nonlinearity and the /3-effect.
We concluded tha t if the fundamental mode has wavenumber k > 1, the /3-effect
does not influence the mean flow. If k — 1 then the mean flow is influenced by both
effects. The azimuthal velocity of the mean flow decreases w ith time because of the
nonlinear interaction, and the /3-effect adds oscillations to the mean flow.
The main features of our solutions, namely the critical layer absorption of the
waves, the development of concentric rings of high wave activity and the changes in
the location of these rings with time, can be used to explain some of the observed char
acteristics of tropical cyclones, such as the formation and evolution of the secondary
eyewall.
An im portant issue in the study of vortex Rossby wave dynamics in tropical cy
clones is the interaction between the waves and the mean flow in the vortex. The
problem of wave-mean flow interaction has been studied by m any previous researchers
in the context of barotropic Rossby waves in a rectangular domain (Dickinson, 1970,
Warn and Warn, 1976, Warn and Warn, 1978, Campbell and Maslowe, 1998, Camp
bell, 2004). We see some similarities in our conclusions com pared with these previous
studies. They found th a t wave-mean flow interactions occur in the critical layer, and
the thickness of the critical layer is 0 ( e -1/2) which is similar to our conclusions for
the vortex problem. In both problems there is also the phase shift in the logarithmic
term of the solution which defines the attenuation in the wave amplitude. The time-
dependent solution also has similar features. The mean flow changes with time due
to nonlinearity.
It is interesting to note th a t in rectangular coordinates these features of the solu
tion arise because of the /3-effect. In the context of vortex dynamics these effects are
obtained even without the inclusion of the /9-effect.
Our results give some insight into the secondary eyewall cycle. We have identified
CH APTER 7. CONCLUSIONS 1 2 1
at least two mechanisms tha t could result in changes in the in itia l eyewall configura
tion. One is the development of multiple concentric rings of critical layers a t different
orders, due to the inclusion of the /3-effect; these could be seen as concentric secondary
eyewalls. Another observation is the modification of the mean flow by the wave mo
mentum flux divergence which changes the location of the critical radius for the 0 (1 )
wave (the strongest of the concentric eyewalls) for tim e scales u p to t ~ 0 (e ~ 1/2). For
longer time scales when the nonlinear term s have become the sam e order as the linear
terms, a late-time solution may be possible through the introduction of a “slow” time
variable. This is one of the possibilities for further work on th is problem. Numerical
simulations can also be used to shed further light on the evolution of the waves and
their effects on the mean flow.
A possible further extension of this problem would be the introduction of a verti
cal coordinate, making the problem three-dimensional and allowing the introduction
of internal gravity waves in addition to vortex Rossby waves. There has been con
siderable debate in the past few decades as to which type of waves, vortex Rossby
waves or vortex gravity waves, play a greater role in hurricane dynamics. Brunet and
Montgomery (2002), and Schecter and Montgomery (2004) discussed the influence
of inertia-buoyancy oscillations, i.e. gravity waves, on the vortex dynamics. They
conclude tha t these oscillations are im portant for cyclones in the middle latitudes.
Schecter and Montgomery (2004) showed th a t the influence of vortex waves exceeds
the effect of gravity wave propagation on the vortex dynamics. Hendricks et al. (2010)
included the inertia-gravity waves in their numerical simulations of the vortical mo
tion in a hurricane. They concluded th a t inertia-gravity waves are insignificant in the
process of intensification or decay of the vortex. An investigation th a t includes both
types of waves could provide some insight to help address th is question.
A ppendix A
H ypergeom etric functions
A .l G eneralized hyper geom etr ic fu n ctio n s
The generalized hypergeometric function has the form (see Erdelyi, 1953)
pFq(au a2...,ap-b1,b2,...,bq;z) = (A>1)^__0 V°1 ) n - - \ P q ) n n .
where 2 is a complex variable, a*,, with k — 1 , . . . ,p, and bj w ith j = 1, ,q, are
complex constants. The notation (a)„ is the Pochhammer symbol,
(a)„ = n ”T01(a + f), (A.2)
and (a)0 = 1, (0)0 = 1. For the case where p — 2 and q — 1, the generalized
hypergeometric function is called simply the hypergeometric function F and is given
by
F 2 Fy{au a2-,by\z) = E (l )V i " / ‘ A -3)„=0
1 2 2
AP P E N D IX A. H YP E R G E O M E T R IC FU NCTIONS 123
A .2 T he so lu tion s o f th e h yp erg eo m etr ic equ ation
The hypergeometric function (A.3) is a solution of the hypergeometric differential
equation
where 2 is a complex variable, and a, b and c are complex constants. This equation
has three regular singular points, a t 2 = 0 ,1 ,00 (see Abramowitz and Stegun, 1964).
The solutions depend on the values of the constants a, b, an d c. In our problem
(chapter 4, p.55), c — a + b + l — p + 1 , ab = —2. We find th a t p = —k or p = k so
the values of a and b are
2(1 — z ) f " + (c — (a + b + 1 ) z ) f ' — ab f = 0, (A-4)
ak + V W + %
2
b =f c - N / F + 8
2
c — k + 1 (A-5)
for p = k. And
—k + \ / k 2 + 8a =
2- j f c - -n/ P T 8
c = —k + 1 (A-6)
for p = —k. Near the singular point 2 = 0 equation (A.4) has two linearly independent
solutions valid for \z\ < 1 (equations 15.5.18 and 15.5.19 from Abramowitz and Stegun,
APPEND IX A. H YP E R G E O M E T R IC FU NCTIONS 124
1964). These solutions are
t r \ 7“t/ I 1 . I \ jr fk + \ / k 2 + 8 k - y /k2 + 8/i(o)(2) = F (a , 6; 1 + fc, z) = F ( ------- , , 1 + k, z)
V - (A .7)^ ( l+ fc )„ n ! '
/ 2(o) = F{a, b-l + k, z) log 2 +
[X(a + n) - * (a) + x(& + n ) ~ x{b)~
X { k + l + n) + x (k + 1) - x (n + 1) + x ( l) ]
^ ( 1 - a U l - » C ■ (A 8 )71=0
F(a\ 6; c; z) in solutions (A.7) and (A.8) is a hypergeometric function which is defined
by the formula
F(a;b;c;z) = f ) (A.9)n=0 1 /"
where (a)„ is the Pochhammer symbol,
{a)n = n ”^ 1 (a + i ), (A. 10)
with (a)o = 1, (0)o = 1. In our problem the singular point 2 = 0 is not in the domain
of the problem. So we do not need solutions (A.7) and (A.8).
Near the other singular point 2 = 1 we introduce a new variable Zi = 1 — 2 and
write (A.4) as
21(1 - z x) f " + (0 - { - k + 1 ))21/ / - 2 / = 0. (A.11)
In this case the solution can be w ritten following equations (15.5.20) and (15.5.21) in
APPEND IX A. H YP ERG EO M ETRIC FU NCTIONS 125
Abramowitz and Stegun (1964) for the case where c = 0. For p — —k two linearly
independent solutions of the hypergeometric equation (4.34), valid for \z — 1| < 1, are
/i( i) ( l ~ z ) = = (1 - z )F (a + 1, b + 1; 2; (1 - z)) (A.12)
and
where
/ 2(i)(l ~ z ) = h (\)(z i) = (1 - z)F (a + 1, b + 1; 2; (1 - z)) log(l - z)
+ ( 1 ~ z ) h — s a — ( 1 - z)[x(a + 1 + n) - x(a + 1) + x ( b + 1 + n ) - x ( b + 1 )-
x(2 + n) + x(2) - x (n + 1) + x(l)] + (A.13)
x W _ (a .i4)
These solutions are valid for 0 < \z\ < 2, \z\ ^ 1. In our problem a and b are defined
by (A.5) and (A.6). For p = these solutions can be w ritten as
° ° / drfc+VA44-8 I 1 A ( i i k —V A 4+8 i -I A
- ^) = (1 - 0 2 3 -— 3------ o f 2------ ± - 4 M ( l - 2)“ (A.15)71=0
APPEND IX A. H YP E R G E O M E TR IC FU NCTIONS 126
and
/ 2 ( i ) ( l - z) = ~ 2
oo f ± k + V k 2+ 8 , 1\ / ± k - V k * + 8 , i \+(1 - z) E ( ~T^ ^ (1 - *>" iog(i - z ) +
t ' o (2)(")n!00 ( ± k + V W +8 | n /-±fc-y^+8 | i \(i _ 2) g 2 ± i M _ J ±1>M(1 _ 2)»
( » l ^ f ± g + 1 + n)- x (±t + f ^ + 1)
±A: — Vfe2 + 8 . ±fc — \/fc2 + 8 . . .+X ( j + ! + « ) - X( j + ^ ~ X( + n )+
x ( 2 ) - x ( r a + l ) + x ( l ) ) . ( A . 16)
These two functions /m ) and / 2(i) are plotted in Figure A .l for z = ^ . We see th a t
the function / ; is an order of m agnitude smaller than / 2(i)-
The third singularity of equation (A.4) is the singular po in t a t infinity. The so
lutions for z —> 00 are given by equations (15.5.7) and (15.5.8) in Abramowitz and
Stegun (1964):
fi(oo)(z) = z~aF(a, a - c + 1; a - b + 1; g ) , (A.17)
f 2 (ao){z) = z~hF(b,b - c + 1; 6 — a + 1; g ) . (A.18)
W ith the values of a, b, and c for our problem (A.5) and (A.6) and w ith p = ± k these
solutions are
j - t . V n f z A + V j i E ± I ( k + V k ^ + 8 \
/,,„,(*) = ( Z ) * ^ Y , M - (A.19)
APPEND IX A. H YP E R G E O M E TR IC FUNCTIONS 127
1.5
1
f 2 (1)
0.5
f l ( l )
0rl rc
Figure A .l: T h e h y p e rg e o m e tr ic fu n c tio n s . The hypergeometric functions/i( i)(r2/r;?) and h(\){r2/ r l ) in the interval r x < r < rc, k = 2, r x = 0.1 rc.
and
r - x — 00 ( - k - s / k 2+ 8 \ / k — \ / k ? + & \
/ „ - , ( * ) = ,A ,0 ,
These solutions are valid for \z\ > 1.
A ppendix B
N onlinear evolu tion o f th e m ean
flow
The angular velocity of the background flow changes with tim e due to the momentum
flux divergence at the critical layer (Montgomery and Kallenbach, 1997). To derive the
equation for the time evolution of the angular velocity let us w rite the A component
of the momentum equation (2.6) in term s of the stream function ^ ( r , A, t),
d ^ r ( 1 _ d . _ . _ \ 1 dp . .' ^ ( r t f r ) + — (B .l)d t \ r 2 dr r ) p d X
We take the mean of each term in (B .l) in the azimuthal direction by integrating one
with respect to A from 0 to 2n and then dividing by 2ir. This gives
d'V 1 d 1r - - 2 ^ ! - ( r t f r ) + - t f r ¥ rA (B.2)dt r2 dr r
where the bar denotes the azim uthal mean. Using the fact th a t
1 T 9 / T . 1 d . T T . 1 T T“2 ^ A« - ( r ^ r ) = U r W r ^ r^A r (B.3)r 2 dr r l dr r
128
AP PEN D IX B. NO NLINEAR E V O LU TIO N OF TH E M E A N F LO W 129
we can rewrite equation (B.2) in the form
dt r2 dr
The mean of -4 fr \I/rA
d ^ r 1 dr'^r'Hx 2 T T _ .A + - * r t t rA. (B.4)
i r i T T „ i i , r .2 „ , i /■ i T x » / x , x- 1 = - - (*.) IS' - - 1 -Wr dx. ( )
The first term on the right-hand side of (B.5) is zero. So
f 27r 12 / - m ry rXd \ = 0, (B.6)
Jo r
and we can rewrite equation (B.4) in the form
dt r2 dr
Recall from (3.49) tha t the stream function consists of 2 parts
(B-7)
^ ( r ,X , t ) = ip(r) + eip(r, A,<). (B.8)
Let ifi(r, A, t ) be the wave
ip(r, A, t) = 0 ( r ,t )e ifeA + c.c. = <f>(r,t)eikX + 4>*(r,t)e~ikX. (B.9)
Derivatives of ip(r, A, t) with respect to r and A are
ipr(r, A, t ) = 0 r (r, t)etfcA + </>*(r, f)e~tfcA (B.10)
APPEND IX B. NO N LIN EAR E V O LU TIO N OF TH E M E A N F L O W 130
and
A, t) = ikcf)(r, t)elkX — ikcf)*(r, t)e~lkX. (B .ll)
Hence,
iprtpa = (4>v{r,t)elkX + 4>*(r,t)e~lkX) (ik(f>{r,t)elkX - ik<t>*(r,t)e~lkX)
= ik - <j>*r<f>) = - 2 k \ m ( M l ) . (B.12)
In section 6.4 we discussed the time evolution of the mean flow by representing the
mean streamfunction as a sum
4>totai(r,t) = $ (r ) +ipo(r,t) (B.13)
where 4>(r) is the initial mean flow, and ipo(r, t) is the tim e-dependent mean stream
function induced by nonlinearity. The corresponding azim uthal velocity is
Vtotai(r,t) = v(r) + v0(r,t) (B.14)
where, .v d^toha{r,t)vtotai{r,t) = ------—---------. (B.15)
So from (B.7) and (B.12) the equation for the evolution of the m ean azimuthal velocity
is
^ Ira(^ :)) (B 16)
or, for the mean angular velocity,
^ = £2M ( 2 t r Im ('w ’; , ) - (B 17)
A ppendix C
T he phase shift for th e stead y
solution
Here we will show tha t the phase shift of the steady solution (4.59) in the critical layer
is —7r. The steady-state solution (4.59), which is valid in the interval r\ < r < y/2rc,
r > rc, 9 = ±7r, and following Miles (1961), we can determine th e sign by considering
the inverse Laplace transform in the tim e-dependent problem th a t was considered in
chapter 5.
We can write the logarithmic term in the tim e-dependent solution (5.26) in the
s-plane in the form
r rc, or 1 — ^ < 1 , includes the term (1 — r 2/ r 2)lo g (l — r 2/ r 2). This term is2
singular close to the critical radius where r = rc. We observe th a t log(l — h ) =r c
2log [ 1 — + i&, where 0 is the a r g u m e n t F o r r < rc, 9 = 0. For
r cFor r < rc, 9 = 0. For
131
APPEND IX C. THE PH ASE SH IFT FOR TH E S T E A D Y SO L U T IO N 132
log(l - 7 (s )r2)
= log
, „ ikDoar2 \ , / ikQ o& ^iReis) — zlm (s) — ikD0)log 11 - r r - n = r I = log ( 1 ------------------ + (im(s) + knay
kQ, o«r2R e(s)1 +
s ikD0 j yA:Q0o^2(Ini(s) + kQ0)
Re(s)2 + (Im(s) + kQo)2 Re(s)2 4- (Im (s) + kQ0)2( C . l )
Consider the complex-valued function
c w = 1 +M2o(*r2(Im(s) + kQ0)
Re(s)2 + (Im(s) + k f l0)2 _fcfl0o:r2Re(s)
Re(s)2 4 (Im (s) 4- kD0)2
We note tha t Re(s) > 0 along the contour of integration (a — zoo, a 4- zoo) since a > 0.
So Im(£(s)) < 0. This means th a t
—7r < arg{Q < 0.
The steady solution corresponds to the pole s = — iui. We consider the limit s —» —iu,
which means tha t Re(s) —> 0 and Im(s) -» — to. As s —> — iuj,
k t t0a r 2(— to + ktto) _ kQ0a r 2 _Q[S) ( - u + kQ0)2 “ “ kD0a r 2 ~
This means tha t log(£(s)) —> log ^1 — which can be w ritten as
r 2 *c
log|C(s)l + *arg(C(s)) log + zarg 0 4 )
(C.2)
(C.3)
Taking this limit allows us to determ ine the correct sign of th e argument of ^1 — ^ j
APPEND IX C. THE PH ASE SH IFT FOR TH E S T E A D Y SO LU T IO N 133
in the steady-state solution (4.59). We noted in chapter 4 th a t
r 2\ I 0, if r < rc, a r g ( l - i i ) = < (C.4)
±7r = 9, if r > rc.
Since —7r < arg(£) < 0, we conclude th a t as s —> —iou, arg(C(s)) —>■ 0 if r < rc and
arg(£(s)) —> —7r if r > rc. This means th a t the phase shift 9 = — it.
Bibliography
[1] Abramowitz, M., Stegun, I.A., 1964, Handbook o f mathematical functions with
formulas, graphs and mathematical tables, Nat. Bur. Stands, 1046 pp.
[2] Abarca, S.F., Corbosiero, K.L., 2011, Secondary eyewall form ation in W RF sim
ulations of Hurricanes R ita and Katrina(2005), Geophys. Res. Lett., 38, L07802,
doi: 10.1029/2011GL047015.
[3] Abdullah, A.J., 1966, The spiral bands of a hurricane: A possible dynamic
explanation, J. Atmos. Sci., 23, 365-375.
[4] Booker, J.R., Bretherton, F.P., 1967, The critical layer for gravity waves in a
shear flow, J. Fluid Mech., 27, 513-539.
[5] Brunet, G., Montgomery, T.M ., 2002, Vortex Rossby waves on smooth circular
vortices, Part 1, theory, Dyn. Atmos. Oceans, 35, 135-177.
[6] Caillol, P., 2012, Multiple vortices induced by a threedimensional critical layer
in a rapidly rotating vortex, J. Appl. Math., 77, 282-292.
[7] Campbell, L.J., 2004, Wave mean-flow interactions in a forced Rossby wave
packet, Stud. Appl. Math, 112, 39-85.
[8] Campbell, L.J., Maslowe, S.A., 1998, Forced Rossby wave packets in barotropic
shear flows with critical layers, Dyn. Atmos. Oceans, 2 8 , 9-37.
134
B IBLIO G RAPH Y 135
[9] Dickinson, R.E., 1970, Development of a Rossby wave critical level, J. Atmos.
Sci., 27, 627-633.
[10] Emanuel, K., 2003, Tropical cyclones, Annu. Rev. Earth PI. Sc., 31, 75-104.
[11] Erdelyi, A., 1953, Higher Transcendental Functions, Me Graw-Hill, New York,
Volume 1, 302 pp.
[12] Fortner, L.E., 1958, Typhoon Sarah, 1956, B. Am. Meteorol. Soc. 30, 633639.
[13] Gill, A.E., 1982, Atmosphere-ocean dynamics, Academic Press, 662 pp.
[14] Guinn, T.A., Shubert, W .H., 1993, Hurricane spiral bands, J. Atmos. Sci., 50,
3380-3403.
[15] Hawkins, J.D., Helveston, M., 2008, Tropical cyclone m ultiple eyewall character
istics, 28th Conference of Hurricanes and Tropical Meteorology, Amer. Meteor.
Soc., Orlando, FL, 28 April - 2 May.
[16] Hendricks, E.A., Schubert, W .H., Fulton, S.R., McNoldy, B.D., 2010,
Spontaneous-adjustment emission of inertia-gravity waves by unsteady vorti
cal motion in the hurricane core, Q. J. R. Meteorol. Soc., 136, 537-548.
[17] Holton, J.R., 1992, A n introduction to dynamic meteorology, th ird edition, El
sevier Academic Press, 511 pp.
[18] Holton, J.R ., Lindzen, R.S., 1972, An updated theory for the Quasi-Biennial
cycle of the tropical stratosphere, J. Atmos. Sci., 29, 1076-1080.
[19] Houze Jr., R.A., Shuyi, S.C., Bradley, F.S., W en-Chau, L., Bell, M.M., 2007,
Hurricane intensity and eyewall replacement, Science, 315 , 1235-1239.
B IB LIO G RAPH Y 136
[20] Irish, J.L., Resio, D.T., Ratcliff, J .J ., 2008, The influence of storm size on
hurricane surge, J. Phys. Oceanogr., 38, 2003-2013.
[21] Judt, F., Chen, S.S., 2010, Convectively generated poten tial vorticity in rain-
bands and formation of the secondary eyewall in H urricane R ita of 2005, J.
Atmos. Sci., 67, 3581-3599.
[22] Kossin, J .R , Sitkowski, M., 2009, An objective model for identifying secondary
eyewall formation in hurricanes, Mon. Weather Rev., 137 , 876-892.
[23] Kundu, P.K., Cohen, I.M., 2004, Fluid Mechanics, th ird edition, Elsevier Aca
demic Press, 759 pp.
[24] Kurihara, Y., 1976, On the development of spiral bands in a tropical cyclone,
J. Atmos. Sci., 33, 940-958.
[25] Kurihara, Y., Tuleya, R.E., Bender, M.A., 1998, The G FD L hurricane predic
tion system and its performance in the 1995 hurricane season, Mon. Weather
Rev., 126, 1306-1322.
[26] Kuo, H.-C., Schubert, W.H., Tsai, C.-L., Kuo Y.F., 2008, Vortex interactions
and barotropic aspects of concentric eyewall form ation, Mon. Weather Rev.,
136, 5183-5198.
[27] Landau, L.D., Lifshitz, E.M., 1953, Course of theoretical physics, Fluid Mechan
ics, Nauka, 630 pp.
[28] MacDonald, N.J., 1968, The evidence for the existence of Rossby-like waves in
the hurricane vortex, Tellus, 20, 138-150.
[29] Maclay, K.S., DeMaria, M., Vonder Haar, T.H., 2008, Tropical Cyclone Inner-
Core Kinetic Energy Evolution, Mon. Wea. Rev., 136, 48824898.
B IBLIO G RAPH Y 137
[30] Martinez, Y., Brunet, G., Yau, M.K., Wang, X, 2011, On the dynamics of
concentric eyewall genesis: space-time empirical norm al modes diagnosis, J.
Atmos. Sci., 68, 457-476.
[31] Martinez, Y., Brunet, G., Yau, M.K., 2010a, On th e dynamics of two-
dimensional hurricane-like vortex symmetrization, J. Atmos. Sci., 67, 3559-
3580.
[32] Martinez, Y., Brunet, G., Yau, M.K., 2010b, On th e dynamics of two-
dimensional hurricane-like concentric rings vortex formation, J. Atmos. Sci.,
67, 3253-3268.
[33] Miles, J.W ., 1961, On the stability of heterogeneous flows, J. Fluid Mech., 10,
496-508.
[34] Montgomery, M.T., Kallenbach, R .J., 1997, A theory o f vortex Rossby-waves
and its application to spiral bands and intensity changes in hurricanes, Q. J. R.
Meteorol. Soc., 123, 435-465.
[35] Nong, S., Emanuel, K., 2003, A numerical study of th e genesis of concentric
eyewalls in hurricanes, Q. J. R. Meteorol. Soc., 129, 3323-3338.
[36] Ogawa, A., 1992, Vortex flow, CRC Press, 311 pp.
[37] Schecter, D.A., Montgomery, M.T., 2004. Damping and pumping of a vortex
Rossby wave in a monotonic cyclone: Critical layer stirring versus inertia-
buoyancy emission, Phys. Fluids, 16, 1334-1348.
[38] Sitkowski, M., Kossin, J.P., Rozoff, C.M., 2011, Intensity and structure changes
during hurricane eyewall replacement cycles, Mon. Weather Rev., 139, 3829-
3847.
B IBLIO G RAPH Y 138
[39] Stewartson, K., 1978. The evolution of the critical layer of a Rossby wave,
Geophys. Astro. Fluid, 9, 185-200.
[40] Terwey, W.D., Montgomery, M .T., 2008. Secondary eyewall formation in two
idealized, full-physics modeled hurricanes, J. Geophys. Res., 113, D12112,
doi: 10.1029/2007JD008897.
[41] Terwey, W.D., Abarca, S.F., Montgomery, M .T., 2012. Question on “Convec
tive generated potential vorticity in rainbands and form ation of the secondary
eyewall in hurricane R ita of 2005” , to appear in J. A tmos. Sci., early-online
release, e-View, doi: http://dx.doi.org/10.1175/JA S-D -12-015Ll
[42] Warn, T., Warn, H., 1976. On the development of a Rossby wave critical level.
J. Atmos. Sci., 33, 2021-2024.
[43] Warn, T., Warn, H., 1978. The evolution of a nonlinear critical level. Stud. Appl.
Math., 59, 37-71.
[44] Willoughby, H.E., Black, P.G., 1996, Hurricane Andrew in Florida: dynamics
of a disaster, B. Am. Meteorol. Soc., 77, 543-549.
[45] Willoughby, H.E., Clos, J.A ., Shoreibah, M.G., 1982. Concentric eyewalls, sec
ondary wind maxima, and the evolution of the hurricane vortex, J. Atmos. Sci.,
39, 395-411.
[46] Willoughby, H.E., Jin, H.-L., Lord, S.J., Piotrowicz, J.M ., 1984, Hurricane
structure and evolution as simulated by an axisym metric nonhydrostatic nu
merical model, J. Atmos. Sci., 41, 1169-1186.
[47] Willoughby, H.E., Shapiro, L .J., 1982, The response o f balanced hurricanes to
local sources of heat and momentum, J. Atmos. Sci., 3 9 , 378-394.