The refraction, shoaling and structure of non linear internal waves at a continental shelf margin

University of Southampton Research Repository

ePrints Soton

Copyright © and Moral Rights for this thesis are retained by the author and/or other

copyright owners. A copy can be downloaded for personal non-commercial

research or study, without prior permission or charge. This thesis cannot be

reproduced or quoted extensively from without first obtaining permission in writing

from the copyright holder/s. The content must not be changed in any way or sold

commercially in any format or medium without the formal permission of the

copyright holders.

When referring to this work, full bibliographic details including the author, title,

awarding institution and date of the thesis must be given e.g.

AUTHOR (year of submission) "Full thesis title", University of Southampton, name

of the University School or Department, PhD Thesis, pagination

UNIVERSITY OF

SOUTHAMPTON

THE

REFRACTION,

SHOALING AND STRUCTURE OF

NON-LINEAR INTERNAL WAVES

AT

A

CONTINENTAL SHELF MARGIN

by

Richard

Justin

Orford Small

Thesis submitted in

partial

fulfilment of the

requirements

for the

degree

of Doctor of

Philisophy

DEPARTMENT OF

OCEANOGRAPHY

FACULTY OF

SCIENCE

UNIVERSITYOFSOUTHAMPTON ABSTRACT

FACULTY OF SCIENCE

OCEANOGRAPHY

Doctor ofPhilosophy

The

refraction, shoaling

and structureofnon-linearinternalwavesatacontinentalshelf

margin

by

RichardJustinOrford Small

Observationsofinternalwavesnearthe Continental

shelf-edge

are

generally

ascribedto

generation by

oscillating

tidal flow overthe local

bathymetry,

in the_presence ofastratified water column,

giving

risetothe internal tide.Inthis thesis observationsare

presented

which demonstrate that internalwaves

attheMalin

shelf-edge

comprise

of boththe

locally generated

internaltide, andwavesfromadistant source. This thesis focuses on_processes

affecting

the latter

phenomenon

at the continental

slope.

A

comprehensive

collection of in-situ and satellite data from the Shelf

Edge Study (SES)

and the Shelf

Edge Study

AcousticMeasurement

Experiment (SESAME)

from

August-September

1995 and

August

1996 is used to describe the internal wave characteristics.

During

a

period

of neap tides a set of

internal

solitary

waves wastrackedacross thecontinental

slope

_everytidal

cycle

for three

days.

The

measurementsindicate that the wavesevolved from an initial

drop

in thethermocline, andwere not

significantly

refracted as

they

crossed the

slope,

due to the small

change

in

phase speed

across the

slope,

fromaround 0.8to0.6

ms"1.

The internal waves

depressed

the thermocline

by

between30 and 50mand had

particle speeds

of 0.4to 0.8

ms"1.

Thestructure of theinternal wavesis examined and

compared

to

weakly

non-linear

theory,

and it is found that firstorder

theory

adequately

describes the waves over the

slope

but thata second order

theory

is

required

to model the internal waves on the shelf. A non-linear refraction model is

developed

to simulate the internal wave

propagation

and

evolution. Initial tests of the model for the refraction and

shoaling

of interfacial

solitary

waves

propagating

in

simple

environments show _agreement with

analytical

results. The model is then

extended to simulate the refraction and transformation ofthe internal waves observed

during

SES,

using

realistic

density

stratification and

bathymetry.

When realistic initial conditions derived from

measurements are used, it is found that the model

reproduces

the

phase

speeds

and refraction characteristics verywell, but overestimateswave

amplitudes

atthe

shelf-edge

and theshelf.

Analysis

ofthe simulated internalwaves _suggests thatthe waves would become unstable at these

amplitudes

and wouldin

reality

be

damped.

Infactit isshown from the observationsthatinstabilitiesinthewave are

likely

to occurdue to the

high

shear and

high particle speed

relative to the

phase speed,

and an

example

of

possible breaking

internal waves is illustrated. The

likely regions

of non-linear internal

wave

dissipation

areconsidered in theDiscussion,

together

withthe local

generation

ofinternaltides,

In memory

of

Phil

Acknowledgements

A number of

people

have

provided

advice and

help

towards this work. Thanks toSteve

Thorpe

for his

steering

atcrucial times. The thesis wasmade

possible

dueto

funding

by

the Defence Evaluation Research

Agency (DERA).

John Scott

provided inspiration

for the

analysis

of SESAME data

by

providing

mewitha 'backhander' of_temperaturedatafrom ZackHallock,who is also thanked. The enthusiasm and interestof Efim

Pelinovsky

and

Tatyana

Talipova

provided

a

background

for the theoretical

analysis. Toby

Sherwin, MarkInall, Gus Jeans and other SES

participants

have

always

been

helpful.

Kevin Lamb and Theo Gerkema

generously

allowed useof their models. The_support and offbeat humour of _my DERA

colleagues

in the

Oceanography

_{group has}

given

me a

happy

working

environment. The

experience

gained

whilst

working

on this thesis has

provided

me an

opportunity

to

participate

infurtherocean

going experiments

invarious_partsoftheworld,forwhich

I'm very

grateful.

Contents

Abstract ₂

Acknowledgements

3

Contents ₄

List of

symbols

"

7

Abbreviations ₉

1.

Introduction,

Thesis

Objectives,

and

aReview of Internal WaveObservations 10

1.1 Introduction ₁₀

1.2

Objectives

₁₁

1.3 Structure of thethesis 11

1.4 Areview of ObservationsofInternal Wavesnear

Varying Topography

13

1.5

Summary

of observations ₂₂

1.4

Figures

₂₃

2. AReview of the

Theory

of Oceanic Internal Waves 26

2.1 Introduction ₂₆

2.2

Equations

ofmotion ₂₆

2.3 Linear Internal Waves ₂₇

2.4 The

development

of

weakly

non-linear

equations

33

2.5

Algebraic

solutionstotheKdV

equation

37

2.6 ModificationstoKdV

theory

₄₂

2.7

Comparisons

of

weakly

non-linear

solitary

wavesolutions,

laboratory

experiments,

fully

non-linear

theory,

andoceanicobservations 51

2.8 Evolution of non-linearwaves 52

2.9

Fully

non-linear models ₅₅

2.10

Summary

56

2.11 Tables ₅₉

2.12

Figures

₆₁

3. Overviewof

background oceanography

andtheinternalwavefield

during

the

SES/SESAME

experiments

₆₈

3.1 SESAMEandSES ₆₈

3.2 OverviewofThe

background

environment 70

3.3 Overviewofthe internal tide and internal wavefield 73

3.4

Summary

79

3.5 Tables

81

3.6

Figures

4 _{Observations of the evolutionandrefraction of}

anon-linear internal tideacrossthe

continental

slope

94

4.1 Introduction ₉₄

4.2

Propagation

andTransformation of the internalwave

packet

overa_{sequence of}six tidal

cycles

95

4.3 Refraction and

speed

of

propagation

of the internal waves 97

4.4 _{Variations in Wave Packet Transformation between Tidal}

cycles

102

4.5 Instabilities oftheinternal waves 109

4.6

Summary

112

4.7 Tables ₁₁₄

4.8

Figures

117

5.

Analysis

of thestructureof internal

solitary

wavesmeasured

during

the

n>^2184

August

1995: observationsand

theory

₁₃₉

5.1 Introduction ₁₃₉

5.2 Observed internal

solitary

wavestructurefromthermistor

string

and ADCP 139 5.3 _{Environmental parameters}derivedfromthe SES stratification 143

5.4 Theoretical

predictions

ofinternal

solitary

wave

shape

146

5.5

Comparisons

ofobservedandtheoretical internal

solitary

wave

shapes

and

speeds

151

5.6

Summary

154

5.7 Tables

157

5.8

Figures

159 6. Anon-linearrefraction model of interfacial

solitary

wavesin theocean 179

6.1 Introduction ₁₇₉

6.2 Therefraction of linear and non-linearwaves 181

6.3

Development

ofanon-linear refraction model 182

6.4 The

shoaling

of internal

solitary

waves 189

6.5 Radial

spreading

ofinternal

solitary

waves 194

6.6 Refractionofa

planar

linearwave

obliquely

incidenton acontinental

slope

196 6.7 Refractionofa

planar

internal

solitary

waves

obliquely

incidenton aContinental

slope

197

6.8

Summary

197

6.9

Figures

7. Simulationof therefractionandtransformation of internal

solitary

wavesatthe

Malin shelf break 210

7.1 Introduction ₂₁₀

7.2

Description

and

inputs

ofthe numerical model 210

7.3 Simulation of theextentof refraction of theinternal waves 212

7.4 Adiabatic

predictions

of the evolutionof

purely solitary

internal wavesintheSES

environment ₂₁₅

7.5 Evolution of observed waveforms 216

7.6 Predictedcurrentsandcurrentstrain 219

7.7 Predictionsofshear

instability

and

gravitational

instability

221

7.8

Summary

₂₂₃

7.9

Figures

₂₂₆

8. Conclusions and Discussion 243

8.1 Conclusions ₂₄₃

8.2 Discussion ₂₄₇

8.3 Future Work ₂₄₉

8.4

Figures

251

Annex A Coefficients oftheFEKdV

equation

foratwo

layer

environment 253 Annex B Calculationof

higher

order modes and coefficients of the EKdV

equation

254

Annex C Numerical

algorithm

for

solving

the EKdV

equation

259

Annex D

Analysis

of the

barotropic

andbaroclinic tide

using

harmonic

decomposition

265

Annex E A non-linear modelofinternal tide

generation.

266

Listof

Symbols

The

following

isalist of the

symbols commonly

referredto

throughout

the paper.Insome

equations,

which arereferredto

just,

oncea

symbol

_may havea different

meaning

butthis will be

explicitly

described inthe

text.

Greek

Symbols

a Non-linear coefficient ofEKdV

equation,

see

[2-28]

a. Cubic Non-linear coefficientofEKdV

equation,

see

[2-47]

ß

Dispersive

parameterof B-0

equation [2-43c]

8

Dispersive

_parameterofKdV

equation,

see[2-23]

e _{Non-linear parameter}ofKdV

equation,

see

[2-23]

(|)

Modal function ofz,see

[2-6]

<|>p

amplitude

of

(j>,

see

[2-9]

(|>lj

Non-linear modal functionof

streamfunction,

foretothe_powerI,8tothe_power

j

see [2-24]

y

Dispersive

coefficient of EKdV

equation,

see

[2-28]

r\ Waveform of

displacement,

see

[2-27]

x\q

Amplitude

of

displacement

ofaninternal wave

r)c

Limiting amplitude

of

displacement

ofaninternalwave

predicted

by

EKdV, [2-51]

A, Characteristic

slope

ofinternalwaverays, see

[2-7]

v

Variable-depth

Coefficient of vEKdV

equation,

see

[6-11]

p

Density

p*

Potential

density

Po Reference

Density

usedin

Boussinesq

approximation

pi

Density

of upper

layer

of

two-layer

_system

p2

Density

oflower

layer

of

two-layer

_system

,

Displacement,

see

[2-22]

and

[2-29].

a>

Frequency

\|/

Streamfunction,

see

[2-2]

A Wavefront

length

Ap

Density

differenceacross aninterface

AP

pycnocline

thickness

LowerRoman Case

amax

Limiting

amplitude

of

displacement

ofaninternalwavefrom

fully

non-linear

theory,

[2-53]

b

Buoyancy

(see

[2-22]

Co Linear

phase speed

u)/k

c

Solitary

wave

phase speed

cn Cnoidal

function,

see

[2-31]

cmax

Limiting

phase

speed

ofaninternalwave from

fully

non-linear

theory,

[2-53]

f Coriolis acceleration

g Accelerationdueto

gravity

hc

Critical

depth

of

two-layer

_systemfor EKdV

dynamics,

see

[2-49]

h] _{Thickness of upper}

layer

of

two-layer

_system Ii2 Thickness of lower

layer

of

two-layer

_system h

separation

ofinterface and

hc, [2-49]

k horizontal wavenumber

ko

horizontal wavenumberofcnoidal and dnoidal functions

m verticalwavenumber

p Pressure

r Radial coordinate of cKdV

equation

u Internal

velocity

indirectionofwave

propagation

v Internal

velocity

perpendicular

todirection ofwave

propagation

v0 CharacteristicInternalwave

velocity

,see

[2-22]

w Internal vertical

velocity

x

Range along

direction ofwave

propagation

y

Range perpendicular

todirectionofwave

propagation

z

Height

fromsea

surface,

measured

positive upwards,

sothat seabedisatz=-H, surfaceatz=0

Upper

Roman Case

A,,

A2 parameterofFEKdV

solitary

_wave,seeAnnexA B Instantaneous

buoyancy frequency,

see

[2-2lb]

C _parameterofFEKdV

solitary

_wave,seeAnnexA

C3,

C4,C5 Parameters of FEKdV

equation, [2-45]

D A

depth

scale,

see

[2-22]

D'J Non-linear modal function of

buoyancy

b,foreto_{the power}i,8to_{the power}

j

see

[2-24]

E

Energy

of

solitary

_waves,see

[6-19]

E(s)

complete

elliptic integral

ofsecond kind F

Energy

fluxof

solitary

waves,see

[6-4]

F,,

Spectral

energy

density

of

displacement,

[3-1]

G Kernel of Whitham evolution

equation [2-39]

H Totalwater

depth

L

Length-scale

of internal

solitary

waves M Termused for vKdV

equation,

see [6-8]

N

Buoyancy

(Brunt-Vaisala)

frequency

,see

[2-4]

Ri Richardson_number,see

[2-21]

Ro

Rossby

number,see

[2-54]

S

Spreading

Coefficient of vEKdV

equation,

see

[6-12]

U

Background

shear

velocity

Abbreviations ADCP CKdV CTD DABS DERA-DR EKdV ERS ESA EKdV FEKdV GEBCO GM GPS HNLMS KB Kl KdV LOIS MB M2 M4 NERC NRL OAT 01 OS RCM rEKdV rKdV S2 SARSEX SeaSoar SES SESAME SAR UTC Ur VEKdV VKdV WAVEX XBT

Acoustic

Doppler

Current Profiler

Cylindrical

KdV

Conductivity,

temperature,

depth

Digital

Acquisition Buoy System

Defence Evaluation Research

Agency

Djordevic

and

Redekopp

(1978)

Extended

Korteweg-de-Vries

European

Research satellite

European Space

Agency

Extended

Korteweg-de-Vries

Fully

Extended

Korteweg-de-Vries

General

bathymetric

chart oftheoceans Garrett andMunk

Geographical positioning

system Her

Royal

Netherlands

Majesty's

ship

Koop

and Butler 1981 Luni-solartide

Korteweg-de-Vries

Land-Ocean Interaction

Study

Michallet and

Bartholemey

(1998)

lunar semi-diurnal tide

lunar

quarter-diurnal

tide

Natural Environment Research Council Naval Research

Laboratory

Ocean Acoustic

Tomography

Principal

lunar tide

Ostrovsky

and

Stepanyants

Recording

Current Meter

Extended

Korteweg-de-Vries

equation

withrotation

Korteweg-de-Vries

equation

with rotation solar semi-diurnal tide

SAR

experiment, Gasporovich

_etal 1988

Undulating

towed

body,

measureswater

properties.

Shelf

Edge

Study

Shelf

Edge

Study

Acoustic Measurement

Experiment

Synthetic

Aperture

Radar Universal Time Constant

(=Zulu)

Ursell number

Variable-coefficientEKdV Variable-coefficient KdV

Digital

X-band radar

Chapter

1.

INTRODUCTION,

THESIS

OBJECTIVES,

AND A REVIEW OF

|INTERNAL

WAVE

OBSERVATIONS

1.1 Introduction

Internal waves are

generated by

disturbances to a stratified ocean at

frequencies

between the inertial and the

buoyancy frequency.

Acommon sourceis the

oscillating barotropic

tidal flowover

varying

topography,

which

gives

rise to _{the internal tide. Under certain circumstances the internal tide}

can transform into a set of

high

frequency,

non-linear internal waves. This thesis

investigates

oneof_{the processes}

whereby

this transformation

occurs,

namely

the

shoaling

ofaninternal tideatacontinental

slope.

Internalwaves arestudiedfora

variety

ofreasons.

They provide

amechanism for the removal of_energyfrom the

barotropic

tide, via the internal tide, into the

high frequency

internal waves,

eventually

to be

dissipated

into turbulent motions when the waves break

(Kantha

and

Tierney

1997).

The observations

presented

in _{this paper}

give

informationon

possible

_{mechanisms for the}

breaking

process. Internal waves _may also transportnutrients towards thesurface,

particularly

when

mixing

occurs

(Haury

etal 1979, New

1988),

and also maycauselateral transport of nutrients (Lamb

1997).

The

displacement

of the thermocline

by

internal waves can affect the refraction of sound andhas

importance

foracoustic studies

(Headrick

et al

2000)

of relevance to the

design

of sonar_{systems, and acoustic underwater communication. The}

strongcurrentsandcurrentshear in internalwaves,

combined with variable

buoyancy

effects across interfaces, _may also destabilise underwater

platforms

and

drilling

operations

for oil

exploration (Bole

etal

1994).

This

study

utilises in-water and

Synthetic

Aperture

Radar

(SAR)

data fromthe recent Shelf

Edge Study

(SES)

and Shelf

Edge Study

Acoustic Measurement

Experiment

(SESAME)

trialsatthe Malin shelf

edge

in

August

to

September

1995 and

August

1996. A

comprehensive

collection of_temperature and current time-series from

moorings

and _{towed surveys, and}

profiles

from

Conductivity-Temperature-Depth

probes (CTDs),

_{has been}

analysed

in

conjunction

with the

synoptic

SAR

images

to

provide

an

interpretation

of the

background

1.2

Objectives

The

objectives

of this researchareto:

1)

Analyse

data from the SESAME and SES

projects

to

identify

the characteristics of internal tides and

waves,andthe

background

environmentinthe Malin

shelf-edge

region,

and any

relationships

between these features.

2)

Interpret

observations ofthe

propagation

ofinternal

solitary

waves across the Continental

slope.

3)

Compare

observed internal

solitary

waveswith theoretical

predictions.

4)

Develop

anon-linearrefraction modelofinternalwaves.

5)

Simulate the observed refractionandtransformation of internal waves across the continental

slope

using

the non-linear refraction model.

1.3Structureof the thesis

The thesis is

presented

as a series of

chapters,

each with an introduction and a _summary to

help

the reader.

Analyses

of dataor

theory

for which the

only

the conclusionsare

important

tothe

study

arecontained in Annexes

and referredtointhe maintext.Tablesand

Figures

are

placed

atthe end of each

chapter.

Chapter

1 introduces the

study

ofinternal waves. The

objectives

and structureofthe

study

arethen

presented.

Theremainder of the

chapter

provides

areview of the observation of oceanic internal waves. Thefocus of the review isoninternal tides and associated non-linear internal waves. In additionto a

description

of

important

in-situ

observations,

theremote

sensing

of internal wavesand internaltidesis introduced. In

particular

the

analysis

by

satellite SAR of non-linear internal waves is

discussed, together

with measurements

by

altimeter ofthe

internal tide. The

long

distance

propagation

_{of internal tides andwavesis}

emphasised.

Chapter

2 _presentsareviewofthe

theory

of oceanic internal waves. The focus ison non-linear internal wave

theory,

as this will bemost

pertinent

to the

study,

butashort

description

of relevant_aspects of linear

theory

is

also

given.

The

paths

of_energy

propagation

in theocean,

along

the interfacein _strong

pycnocline

_{environments,}

wave

theory

presented.

Also includedisareview of

previous

comparisons

of non-linear theories with

laboratory

experiments

and with

oceanographic

observationsofinternal

solitary

waves.

In

Chapter

3,the SES and SESAME

experiments

and dataare introduced.

Analysis

is done ofthe

relationship

between the

background

environment,

including barotropic

tides,stratification and

background

_{currents, and the} internal waves. One ofthe

important

conclusions of the

study

is

presented:

that internal waves from distant sources canbeas

large

asandoften

larger

than

locally generated

internalwaves.

In

Chapter

4the behaviour ofa

particular

feature observed between the

19*

and

21st

August

1995isdescribedin

detail.

Every

tidal

cycle

a

drop

of the

pycnocline

in

deep

watertransformsintoasetof internal

solitary

wavesas

it propagates on-shelf. The waves are

hypothesised

to be from a

distantly

generated

internal

tide,

and the refraction and transformationofthewaves across the

slope

isexamined. Possible

breaking

eventsareshownand the observationsare

analysed

toseeif the criteria for internal wave

instability

aremet.

In

Chapter

5 the observationsofinternal

solitary

wavesfrom this

period

are

interpreted

intermsof the non-linear theories of internal waves

presented

in

Chapter

2. Adetailed

analysis

ofthe structureofthe observed waves is

madeand

compared against predictions

from

weakly

non-linear

theory.

Peculiar features of thecurrentand the

displacement

profiles

inthewaves aredescribed.

In

Chapter

6a non-linear refraction model is

developed.

This is doneto

investigate

how non-linear effects on

phase speed

affect internalwaverefractionacross

bathymetric

featuressuchascontinental

slopes.

The model also

predicts

the

shoaling

and

spreading

of internalwaves.The model istested ina

two-layer

environment in idealised situations where

comparison

can be made with

analytical

_{results. Good agreement with}

analytical predictions

_is obtained for:

shoaling

_upa

slope:

radial

spreading:

andthe refraction ofa

planar

wave asit

obliquely

_passes a

wedge-shaped slope.

Chapter

7

applies

the non-linearrefraction model to simulate the observed internal wave behaviour covered in

Chapter

4.

Using

realistic

density

stratification and

bathymetry,

a

description

of the

propagation

ofafirstmode

featureacrossthe

slope

is described. Simulationsofthe evolution ofan idealised initial waveform and ofareal initialcondition are

performed.

The

predicted

_{refraction and transformation of the featureis}

compared

withthe observations. The

predictions

ofsurfacecurrentstrainareillustratedtodetermine whether thewaves are

likely

to

be

imaged by

SAR.Themodel

predictions

of when theonsetof

instability

_mayoccur arealso

investigated.

Finally

Chapter

8 presentstheConclusions of the

study,

a

Discussion,

and

proposed

futurework.The discussion considers

possible

source

regions

for the internal

solitary

wavesdescribedin

chapter

4,and

1.4.Areview of observationsofinternalwaves near

varying

topography

Observations of internal waves

generated by

tidal flownear

varying

topography

have identifiedtwo_typesof

phenomenon:

a

long

low

frequency

waveof tidal

periodicity

and

O(10 km) wavelength, generally

referred to

asthe internal tide, and,

superimposed

onthis,

higher

frequency

0(5 cph)

waves with short

wavelengths

of

0(100

m) (Huthnance

1989), variously

classified assolitons, undular bores,

hydraulic jumps

or

just

internal waves. These observations are discussed below. This is not intended to be a

complete

list of such observations,but instead shows selected observationstoillustrate

particular

mechanisms of interest.

1.4.1 Internal tides

The internal tideissetup

by

periodic

tidal flow

parallel

to

gradients

in

topography

in_{stratified water, where} the forced vertical currents cause an oscillation of the stratification with

corresponding

tidal

period.

Associated withthe

oscillating

thermoclinearebaroclinic velocities.

Internal tides have been identified and _{characterised in many}

regions

of

varying

topography

on the

globe.

These

regions

include

shelf-breaks,

sills

(regions

of vertical

constriction),

straits

(regions

of horizontal

constriction),

seamounts, and

ridges.

For

illustration,

a few of these observations are detailed below. For

more

comprehensive discussions,

Wunsch

(1975)

and Huthnance

(1989)

reviewed the observations and

theory

of internal tides_up tothattime.More

recently

reviews havebeenmade of the

high

frequency

waves

arising

fromthe internal tide(suchas

Ostrovskly

and

Stepanyants

_1989,Jeans

1998).

Halpern (1971a)

was oneofthefirst to

identify

the internal tide,

by

measuring

the thermocline variation in

Massachusetts

Bay

9 km from a submarine sill

(Stellwagen

Bank) using

a moored thermistor

string,

observing

M2 oscillations in _temperature of the order of 1-2 C in the thermocline, dominated

by

the first internal mode. Thewaves

explained

50 % of thetotal_temperaturevarianceatthatsite. Itwas

speculated

that

theinternal tides here

dissipate

around 8% of the

barotropic

_{tidal energy.}

Pingree

etal

(1983)

observed the M2 internal tideon the Celtic Shelf break

using

moored thermistor chain and found

peak

to

trough amplitudes

of _up to 50 m in the seasonal thermocline. Evidence was found for waves

propagating

both on and off-shelfin

Pingree

and Mardell

(1984).

_{The internal tide showed distinct} non-linear characteristicsofaflattenedcrestand

relatively

narrow

trough

(Figure 1-1).

Pingree

etal

(1986)

foundadominance

by

the first internal modeon theshelf,in the_presenceofaseasonal

thermocline,

and

by

mode 3 in

deep

water, where the

deeper

stratification is

significant.

New

(1988)

also

speculated

that

mixing

in

large amplitude,

_{low Richardson number internal tides}

explained

observed cold surface

patches

nearthe

Sherwin

(1988) reported

and

analysed

observations from the Malin Shelf

(north

of

Ireland)

inthesummerof 1983, in the _presence ofa _strong seasonal thermocline. Current meter and thermistor chain measurements

indicatedanM2internal tide

propagating

from

315T,

inducing

thermocline oscillations of_upto20 mModal

decomposition

ofthe internal tide showed that the first internal modedominated witha

wavelength

of 38

km,

and associated internal currents of 15cm/s near the surface and 6cm/s near _{the bottom. The energy}

decay

length

was estimatedto bearound 80 km, with atimescale of about4

days.

A

mooring

on the shelf

(100

m

depth)

showed a clear

spring-neap

cycle

of internal tide

amplitude,

but this was not _apparent at the shelf break(200m

depth.)

Further, the internal tide

amplitudes

were strong

despite relatively

weak

(M2

13

cm/s)

tidalcurrents.

De Witt et al

(1986)

measured internal tides in the Rockall

Trough,

north-east of Rockall Bank. A

high

degree

of

temporal

and

spatial

variance was observed in the internal tide

amplitude. During

a week of

energetic

internal tides in

August

1978, 20 m

peak-trough

amplitude

waves were observed on the seasonal

thermocline,

and the direction of

propagation,

resolved from a

triangular

array of

moorings

indicated

propagation

from the Rockall Bank. The

suggested explanation

for the

temporal

and

spatial variability

was

fluctuation in_{the energy}

paths

of the internal tide

emanating

from the Bank. These

fluctuating

_energy

paths

mayinturnhave been dueto

changes

in the

density

field caused

by

mesoscale

(eddy)

motions.

Recently

Farmer and Armi

(1999)

made _{very detailed}measurementsof the internal tideover asill in

Knight

Inlet, British

Colombia,

using

echo-sounder and hull-mounted Acoustic

Doppler

Current Profiler

(ADCP).

The stratificationinthis

fjord

is_strongdueto freshriver andice

run-off,

which collects inthe_upper5 m or so.

Ebbing

tidal currents over the _steeper side of the sill were found to, induce strong internal tides. Downstream of thesillthe

layer

interfacewasfoundtobifurcatewithone

portion plummeting

tothe

depth

of

the sill

(60

m) or more.

Along

the

plummeting

interface instabilities were observed to

form,

which were

speculated

toevolveintoasetof

solitary

wavesobservedsome30 minutes later. Thesewaves wereobserved

to _{propagate upstream}

along

the shallow interface. Evidence was also found for

solitary

wave

generation

during

flood tide over_{the shallow side of the sill: here the}

waves

appeared

tobe

trapped

overthesill

edge.

Solitary

waves arediscussed inmoredetail inthe

following

section. 1.4.2 Non-linearwaves

It has been

speculated (Baines 1984)

that

large

amplitude

_{internal tides will transform from smooth,}

long

undulations into

higher

frequency

_{non-linear waves such as solitons and undular}

bores, or

just

sharp

are often associated with

rapidly changing

high amplitude

_currents.

Examples

_of

some of the well-documented observationsaredescribed below.

Haury

et al 1979

presented

a detailed set of observations of non-linear internal waves made via acoustic

echo-sounder, thermistors and CTDs at

Stellwagen

Bank inMassachusetts

Bay,

which rises to 30 m

depth

from

surrounding

depths

of90-100m. Thewaves wereobservedto_{propagateaway from}thecrestofthebank towards thecoast. A

hypothesis

wasmade thatinternal leewaves wereformedontheoceanside ofthebank

during

ebb-tide, which werereleased

during

theslackwater andflood tideto _propagateacross the bankand

on-shore,

becoming

more

solitary

as

they propagated.

Vertical velocities of0.2to0.4

ms'1

wereinferred from the descentrateof isothermsin the waves.Thewavesreached

amplitudes

ofatleast30min 90m

depth

and on morethanoneoccasion wereobservedtobreak. The acoustic echo-soundertraceof theeventis shown in

Figure

1-2. Here thedirection of

propagation

was

right

to left whilstthe

overturning

occurred from leftto

right, suggesting

a shear

opposed

to the wave

propagation

direction

(Thorpe 1978a).

Associated with the

mixing

was averticalredistribution of

chlorophyll

and othernutrients.

Pingree

and Mardell

(1985)

identified internal waves25km shelf-wards from the

Bay

of

Biscay

shelfbreak,

in athermistorchain recordover

springs

and neaps

during

thesummer.

They

found

typical

phase

speeds

of 0.7

ms"',

withhalf widths of-100m. Peakto

trough amplitudes

were_up to50 mat

springs

and between

13-17mat_neaps.

Separations

betweenwaves weremeasured around 1100mand 1800m,

increasing

the further from the shelf break that thewave

packet

was

observed, suggesting

dispersion.

The waves wereobservedto

takeon a

solitary

_appearanceas

they

propagated

onshore

Pingree

and Mardell

(1985)

further found that these waves were

mainly

generated

during

maximumebb tide (the

typical

tidal

velocity

on the shelfwas 0.7

ms"1

at

springs

and 0.3

ms"1

at

neaps)

atthe shelf break and

propagated

onshelfas thetide turned. However, some wave

packets

did not_{follow this pattern and} it was

suggested

that

generation

might

also be

occurring

along irregular

canyons and

ridges

along

the shelf break.

This

possibility

was

supported

by

New

(1988)

who showed a SEASAT SAR

image

(see

section

1.4)

for

discussion of

SAR)

indicating

waveforms

apparently spreading

_out

radially

_{from these}

irregular

features.

Holloway

(1987)

observed solitons and boresatthe north-west Australian shelf breakin

depths

around 100m

using

moored thermistors andcurrent meters. Internal bores and

solitary

waves were found to occur at the

back and sometimes the front of the internal tide

trough,

with

typical

timescales of 5-10 minutes, and

amplitudes

of upto50_m,

equivalent

to6 C_temperature

changes.

surface

pycnocline

was

displaced by

_upto afactor offour, with maximal 25 m waves of

depression.

The wavesoccurredatthe frontofan internal tide

trough

during

the

period

_{of strongest tidal}

forcing.

Sandstrom

andElliott

(1984)

also observed

large

non-linear internal wavesof 50 m

amplitude

on a

deeper

thermocline

(between

20 and 50m

depth)

onthe Scotian shelf.

Brandtetal

(1999a)

measured

large

internal waves closetothe Straits of Messina with towed

conductivity-thermistor chain and hull-mounted ADCP. Waves were observed both north and south of the Strait, in October 1995. These

high

resolution measurements, similar to the _types of observations

presented

in this thesis, showed the detailed structureof the _waves, which were up to 70 m

(peak-trough)

in

amplitude,

and often

solitary

in character. Baroclinic current

jets

of _up to 0.9

ms"1

were associated with the waves. The waves are also clear on SAR

images

of the area

(Brandt

et al 1997: see section

1.4.3).

Variations of the waveforms were often found and were linked to variations in the tidal

strength

and the

underlying

general

circulation. In

particular

Brandt et al

postulated

thatthe anomalous intrusion ofthe Atlantic-Ionian stream

intothe_{Strait may have increased the}

amplitude

_ofthe

solitary

waves.

Internal

solitary

wavesin

deep

waterhave also been

reported.

Apel

etal

(1985)

conducteda

comprehensive

satellite and sea _surveyof

regular

packets

of non-linear internal waves inthe Sulu Sea.

Up

to5

packets

of

tidally

generated waves werevisibleintheSeaat_anyonetime,

emanating

fromasillatthe southernentrance to the sea and

propagating

cylindrically

outwards into the

deep

ocean in

depths

_up to 3300 m. From the satellite

imagery

(visible

sunglint:

see section _{1.4.3) strong}

dispersion

of the waves was noted asthe waves travelled across the sea, with up to 16 km

separating

_waves, and

long

crest

lengths

up to 350 km were observed. The in-situ data indicated wave

amplitudes (of

depression)

up to 90 m on a

strong

thermocline

extending

fromthesurfaceto 150m,andsurfacecurrent

pulses

_upto 1

ms"1.

Thewaves

propagated

intotal a distance of almost 500 kmover2Y2

days.

Deep-sea

internal wavesinthe South China Sea have been observedto _{have many}

interesting

characteristics. Hsuetal

(2000)

found thatin the north-eastofthe Sea internal waves were

generated

atLuzonStrait,south ofTaiwan,

propagated

westsome200 to 300 km across the

abyssal

plain,

then on

reaching

the

shelf-edge

they

appeared

to be

magnified.

Significant

_{refraction and diffraction}

was seen to occur around an island located in the middle ofthecontinental

slope.

Bole et al

(1994)

showedcurrent meter measurements of the internal waves nearthe island which indicated maximumcurrent

speeds

of1.5

ms"1,

located sub-surfaceat20

to40m

depth,

and lower

layer

currents_upto 1

ms"1

flowing

in the

opposite

direction. The interface between the two currentdirections occurred around

mid-depth

(thewater

depth

ofmeasurementwas around 300

m).

Pinkeletal

(1997)

discussed internal

solitary

wavesof

peak-trough

amplitudes

upto60m, in

packets

of2-3, measuredat a

mooring

in the western

equatorial

Pacific Ocean. The

packets

occurred at

spring

tides for six

successive tidal

cycles.

Thewater

depth

was over3000mand the

generation region

was

speculated

tobethe

Nugarba

islandchain, some200kmorsodistant. Wavefronts of_upto50 km

long

wereobserved,

indicating

the

large

horizontal coherence. It was believed that the enhanced shear associated with the wave was sufficientto

trigger

instabilitiesinthe

background

shearflow.

New and

Pingree (1990

and

1992)

observed

high frequency

solitonsin the central

Bay

of

Biscay

some 150 kmfromtheCeltic shelf

edge using

thermistorchain,ADCPandSeaSoar, in waterover3000m

deep.

These waves had

typical periods

of 30 minutes, with horizontal currents of0.4-0.5

ms"1

in the _upper

layer

and vertical velocities of up to 0.16

ms"',

and with thermocline

displacements

of 50 m.

Rays

of internal wave energy, which

originated

atthe shelfbreak,travelled into the

deep

ocean,and werereflected off thebottom, were

hypothesised

tointersect the seasonal

thermocline,

and

give

rise tothe

highly

energetic

internal waves

there.

Ray propagation

wasalso

quoted

asthereasonfor internal wavesobservednearthe Mascarene

Ridge

(Indian

Ocean)

closetoasill

(Konyaev

etal

1995).

Internalwaves wereobserved above the

ridge,

with

amplitudes

of

upto40m,some

elevating

the thermocline. A

large

internal waveof elevationwasfollowed later

by

a train ofsmaller, narrower solitons. Features were also observed some90 km from the sill. These features were

again

attributedtoray

propagation

from the sill into the

deep

ocean,with reflection off the floor backtothe surfacenearthe observations.

These are

just

a few of the _many

reported

observations of internal

solitary

waves and bores. From the extensive_{range of}observationsofthesewaves(see

Miyata 2000)

itseemsclear that non-linearwaves occur

at mostshelf-breaks andother

topographic

features

during

times of strong stratification andatleast moderate tidal

strength.

One of the most

striking

points

that arises from

comparison

of the observations is the

predominance

of internal

solitary

waves of

depression

(shown

schematically

in

Figure 1-3).

These waves are

theoretically

predicted

to occur when the upper

layer

is thinner, as often occurs in the summer

when,

in fact, most

observations are made. This

theory

is discussedin section 2.5. (Note thatin timeseries _{of temperature} at a fixed

depth,

thesewaves will

usually

appear as

peaks

of

higher

temperature).

Of the observations discussed above,

only

Hollo_way et al (1997,

1999)

have shown waves of elevation (as well as

depression

_waves). Waves ofelevation have also been observed

by

Ivanov and

Konyaev (1976)

in the

Caspian

Sea where a

thinner lower

layer

(of

high

salinity)

exists. Liu

(1998)

inferred the existence ofwaves ofelevation in the

Recently

studies have been made of the conversionofinternal

solitary

waves as

they

pass

through

the critical

point

where the_upper

layer

thickness becomes

equal

tothelower

layer

thickness

(e.g.

Liu 1998,Grimshawet

al 1999, and Saffarinia and Kao

1996).

Weakly

non-linear EKdV models

predict

atransformation of

solitary

waves of

depression

intoa setofwavesof elevation (Grimshawet al 1999,Liu

1998)

and thereare limited observationsto_supportthis

(Liu 1998).

Other_papershave

suggested

thewaves

dissipate

atthis

point

(Porter

and

Thompson

1998).

This_aspectwillnotbe dealt with hereindetailasthe observations ofthisthesisdonot

extend into such shallow water. However, the _{process is}

important

in terms of the eventual

dissipation

of energy of internal

solitary

waves as

they

approach

the coast. Whether the waves

dissipate

at that critical

depth,

or ata front with mixedwater in tidal _zones, or

through shoaling

and

breaking

(see

chapters

4 and 5 fordiscussionsof

this),

itis

probable

thatthecoastal zoneisoneof the sinks for internal

solitary

_wave,and hence internal tide _{energy, and} of

barotropic

tidal _energy

through

baroclinic conversion. (To the authors

knowledge

therearenorecorded instances of internal

solitary

wavereflection from thecoast).

1.4.3 Remote Observations

of

internalwaves

In addition to the in-situ measurements of internal waves described above, further

possibilities

of internal wavedetection

using

space-borne

radar or

photography, altimetry,

and remote acoustic measurements have

recently

been

reported.

_{These observations have the}

advantage

_of

being

synoptic,

but somewhatlack

spatial

continuity

in some cases

(e.g.

altimetry

data lies on

ground

tracks

widely spaced apart)

and/or

temporal

continuity (e.g

SAR swaths are

repeated typically

on O

(days)).

That said,

they

can

provide

valuable and at

times themost

comprehensive

dataonoceanic internalwaves.

The radarand

sun-glint

photography

techniques

rely

on the modulation of sea-surface

roughness by

_internal waves. The detection of internal waves is somewhat

weather-dependent, being

most effective at low to

moderate wind

speeds.

The mechanism of the modulation ofseasurface

roughness

by

internal wavehas been

studied since _{the time of the Earth's Resources}

Technology

_Satellite

(Apel

etal

(1975))

and has becomean

essential element ofinternal waveresearch. The two main

mechanisms,

which have been

distinguished,

are the so-called

hydrodynamic

mechanism,

due to surface

straining,

and the slick mechanism, due to

redistribution of surfactants

(biological

or

mineral).

The former mechanism tendsto _{induce the presence of}

dual-polarity

signatures

onthesea-surface, dueto_{the alternate effects of convergence and}

divergence

(due

to

internal _waves) on the surface

roughness.

This is illustrated in

Figure

1-3. The latter mechanism tends to

favour

single-sign signatures,

due to the concentration and advection of surfactants in the

regions

of maximum surfacecurrent,

coinciding

with the

trough

of themost

commonly

occurring

internal waves (non¬ linearwavesof

depression,

asdiscussedinthe

previous

_section).

connection with internal waves. Osborne and Burch

(1980) investigated

internal

solitary

waves in the AndamanSea,

showing

afinesetof

photographs

ofthe sea-surface

roughness

coincident with internal waves of upto90min

amplitude.

_{Bands of increased}

roughness,

_with

wavesof upto2m

high

ina

background

level

of0.1 m were observed ahead ofthe

trough

of the waves, in the surface_convergent

region.

These surface effectswerebelievedto createthe banded

signatures

of wavefronts seenin satellite

sun-glint photographs

of

the Andaman Sea from Landsat and

Apollo-Soyez

missions

(where roughly

speaking

bright regions

imply

calmseas,dark bands

imply rough

seasandless

reflection).

Apel

etal

(1985)

showed similar results from the internalwaves in theSulu Seawhere

heightened

roughness

occurred

just

before or

during

the

troughs

of the internal wavesof

depression.

Hydrodynamic

signatures

_on

SAR were first attributed

solely

to the

Bragg-scale

waves

(Alpers 1985).

More recent and

sophisticated

modelsof the

signatures

take into accountthe influence of the wholewavespectrum

(Holliday

etal

1986),

or divide the_{spectrum into} a short

wavelength Bragg

_{component, short waves tilted}

by

_the

long

waves

(tilted

Bragg),

anda

longer

wavelength

specular point

component

(Lyzenga

and Bennett

1988,

Romeiser and

Alpers

1997).

These models

performed

better than

simple

Bragg

at

simulating

the_strong observed radar

signatures

in the short

wavelength

(X

or

C)

band

(Holliday

et al

1986).

Hughes

1978 has also discussed the enhancementof

signatures

duetothe

possible

blocking

ofsurface waves

by

the internal currents

(when

the internalwavesurface

particle speed

+ surfacewavegroup

speed

=internal_wave

phase

speed).

Further

important

studies of internal wave SAR

signatures

were

performed by

Apel

and coworkers

(SARSEX)

and

Hughes

and coworkers

(JOWIPS)

in 1983-1984 in the New York

Bight

and the

Georgia

Strait(all

reported

in J

Geophys.

Res,

Special

Issue, 93,

(CIO),

12217-14164,

1988).

More

recently Kropfli

et

al

(1999) presented

_{another detailed survey off the}

Oregon

coast where

strongly

non-linear waves were

measured.These werethewavesdiscussed

by

Stanton and

Ostrovsky

(seesection

1.4.2),

with

amplitudes

(of

depression)

as

high

as 25 m on a

pycnocline

centred at 5 m

depth.

Shore-based radar modulations here indicated

suppression

ofsurface waves over the

troughs

_{(maximum currents) of the internal}

waves. The

suggested

mechanisms for this included both the surface wave

blocking

effect discussed above, and the influenceofsurfactants

Slick modulated internal wave

signatures

have alsobeen studied

by,

for

example,

Ewing

(1950),

Ermakovet

al

(1992),

DaSilva et al 1998. Here the surfactants are

proposed

to concentrate in the internal wave and

propagate with the wave, with

highest

concentration over the

region

of maximum surface current

(Ewing

1950).

For the

commonly

occurring

internal waves of

depression,

this occursover the

trough

of the wave (DaSilvaetal

1998).

Thesurfactants act to

damp

the small wavesdetected

by

SAR,

leaving regions

oflow backscatter return. In some

exceptional

circumstances

single

high

backscatter bands are visible in

background

environment oflow wind

speed

and_strongsurface slicks. InthiscaseDaSilvaet al

(1998)

have

which are

imaged by

radar. Recent studies have also indicated that

long

wavelength

_{internal tides may also} leave

signatures

duetothe concentrationof surfactants

(Ermakov

etal

1998).

Thereare now_{many papers} available, which make

analysis

of internal wave

fields,

and

properties

based on observations of internal waves from SAR and otherremote measures of sea-surface

roughness.

Zheng

et al

1993, 1995 have used

Space

Shuttle

photographs

toinfer characteristics of internal waves onthe shelfwaters

oftheMid-American

Bight

and the

deep

water of the Indian Ocean

respectively.

Brandtet al

(1996,1997)

have used

large

numbers ofERSSAR

images

toinfer the variation

(with

thetide, and with the

seasons)

of internal

solitary

waves

generated

at the Strait of Gibraltar and Strait ofMessina

receptively.

Watson 1994 also studiedthe Straitof Gibraltar

phenomenon using

land-based radar.

Apel

and Gonzales

(1983)

and

Apel

et al 1998 have

attempted

to describe wave

packets

as undular bore solutions to the KdV

equation

_(see

section

2.5),

where the variationof_separationofthewaves in a

packet (due

tonon-linear

dispersion)

isused

to _{infer the model parameters. Small et al}

(1999a)

has made a

study

of the

possibility

of

inferring

internal

solitary

wave

amplitude

from the SAR

image presented

in

Chapter

4 of this thesis,

using

the in-situ observationsto validate the method. Brandtet al

(1999b)

then showed that the success orotherwise of this

methodwas_{very sensitive}tothenatureof the surface wind conditions.

Inthis thesis, _many of the

proposed

results

rely

on information from satellite SAR(see

Chapters

3 and

4).

The informationis

mainly

in theform of the

position

ofwavefrontsat

particular

timesin thetidal

cycle.

This thesis will not

investigate

in detail the method

by

which the observed

signatures

appear, for two reasons.

Firstly,

no

exactly

coincident SAR and in-situ dataisavailable

(the

closesttimesofmeasurement

by

the two

methodsareoftheorderofhours

apart),

and

secondly,

itis

beyond

the_{scope of the}

study

to

investigate

this aspect.

However,

some

simple interpretations

will be

presented

_{which appear} to confirm that the internal waves measured in-situ and simulated

by

a refraction model are of sufficient

amplitude

to cause internal wave

signatures.

Internal tides and waves have

recently

been detected

by

satellite

altimetry.

Ray

and Mitchum

(1996)

have described measurements ofthe internal tide off theHawaiian Islands

by

altimetry,

which revealed surface

height

fluctuationsofa fewcmsorless. It should be noted that

altimetry, by

measuring

surface wave

height,

ratherthan

roughness,

measuresthe direct effect of internal waves onthe sea-surface: but this surface effect issmall

compared

with the

displacements

inside theoceanwhich,wehaveseenabove,are

O(10 m).

Due to the restricted

spatial

natureof

altimetry

observations(measurements

along

a line

only:

with across-track

spacing

of 100 kmormore_{and repeat}

periods

ofafew

days),

itis

only

possible

to_{compute internal tide}

effects from

long

records

(Ray

_{and Mitchum quote 3}

years worth ofdata

required

to resolve M2 and S2 internal tides).

Using Topex

Poseidon data,

Ray

and Mitchum succeeded in

computing

the

amplitude

and

amplitude

and

phase

overshort

wavelengths

of -100 km,

clearly

toosmall for the

barotropic

tide. Further, the results

suggested

that the internal tide was coherent and

propagated

over

huge

- 1000 km -distances away from the islands.

(Recently

Morozov et al

(1999)

showed even

longer,

basin scale (2000-3000

km)

propagation

ranges oftheinternal tideinthe Indian Ocean from the Mascarene

Ridge,

derived from

mooring

data. This

possibility

was

supported by

the non-linear model of Vlasenko et al

(1996)

which estimated

propagation

ofover 1

km).

Kantha and

Tierney (1997)

performed

asimilar

analysis

tothat of

Ray

and Mitchumto

investigate

global

M2

baroclinic tides from

altimetry.

Inordertoextendthe

analysis

to other tidal_components, which could notbe resolved from

altimetry,

the M2 altimetric results were used tocalibrate a

simple two-layer

model

(Kantha

and

Tiurney 1997). Using

this calibrationanestimate wasmadethat internal tide_energywasaround 16%of the

barotropic

tide_energy.Further itwasestimated that energy

dissipation

frominternal tideswasabout 15% of the total

input

into the

barotropic

tide

by

lunisolar forces. This_suggeststhat internal tide

dissipation

isan

important

sink for _{tidal energy. Global}

maps of baroclinic M2 energy

density

indicated that

island-ridge

chains

(e.g.

Hawaii, Melanesia and Micronesia in the western

equatorial Pacific),

and mid-ocean

ridges

(Mascarene,

Ninety

East)

were

important

internal tide

producers

(Kantha and

Tierney 1997).

However

Ray

and Mitchum noted thatcareshould be takenin

interpreting

altimetric

analysis

of internal tides: ifthe tracks arenot

parallel

tothe

expected

internal tide

propagation (e.g.

Ninety

East

ridge

for

Topex Poseidon),

orthere are _many close sources of internal tides

(e.g.

Bay

of

Biscay shelf-edge),

then the

analysis

can be inconclusive.

Internal

solitary

waves can also be visible from

altimetry.

Picaut et al

(1995)

found sea surface

height

displacements

of_upto 30

dyn.

cm associated with internal wavesatthe TOGA sitein the western

equatorial

Pacific

(the

site where Pinkel et al _{made the in-situ observations described in section}

1.4.2).

Although

the

surface

displacements

were small

compared

with the interfacial

displacements

of _up to 100 m,

they

were

significant enough

tobedetected

by

_{altimeter. Altimeter tracks areone-dimensional and}

so do not

give

the

synoptic

two-dimensional fields derivablefromSAR.

Finally

we noteanotherremotemeans of

detecting

internal fluctuations in theocean,

namely

thatof ocean-acoustic

tomography (OAT).

Essentially

_{this consists of}

measuring

fluctuations in arrival time and

phase

between acousticsourcesand receiversatknown

positions (moored,

or

towed)

and

depths.

These fluctuations

areduein_partto

changes

ofthesound

speed

_{structure, and hencetemperature}

structure, in thewatercolumn.

Inversion ofthe travel time fluctuations is used to _compute the internal temperature field

(Dushaw

et al

1995).

This methodwas

originally applied

to

long

_{range, basin scale}

propagation

_to

measure,amongstother

things,

slow

changes

in _temperaturein theocean

possibly

due to

global warming

_(Munk and Forbes

1989).

More

recently

the method has been

applied

to shorter acoustic

propagation

distances to

investigate

smaller

Dushawetal 1995 detected internal tides

by

OAT in the North Pacific and from their

predicted

direction of

propagation,

identifieda source ontheHawaiian

Ridge

some2000 kmtothe south. Thiswas consistent with the

findings

of

Ray

and Mitchum

(1997)

from

altimetry.

Dushaw and Worcester

(1998)

investigated

OAT

signals

from an _array in between Puerto Rico and Bermuda, and found evidence for

resonantly

trapped

diurnal internal tides. Thecauseof thisresonance was

hypothesised

tobe thatwhenthe diurnal internal tides

approached

the critical latitude for diurnal internal tides

(where

the tidal

frequency

is

equal

to the inertial

frequency),

the waves areforcedtoreflectbacktowards thesourcelatitudes. This is

explained

inmoredetail

in section2.3.

1.5

Summary

of observations

Observationshaveshown that internal waves are

prominent

in

regions

of

varying topography

such as shelf

breaks,

ridges,

seamounts, and sills. The

generation

mechanism is

commonly

found to be tidal flow across the

topography,

in_{the presence of}a

pycnocline.

These internal tide

phenomena

are

quite

often found to be associatedwith

high

frequency

non-linearwaves,which_mayarise

directly

inalee-wavemechanism, orfrom non-linear

disintegration

ofthe internaltide,discussed laterinsection 2.8. The non-linearwaves can takethe form of sudden

jumps

orbores, undularbores, or

solitary

_waves, and

generally

have the_property of

rank-ordering

in

amplitude.

Internal waves have been detected from their in-situ

density

and current

fluctuations,

and also from remote

sensing

of the sea surface

roughness

which is modulated

by

internal wave currents, and altimetric

measurementsofthe smallseasurface

height

fluctuationsduetothewaves.Thesemeasurementshave shown another

important

property of the waves,

namely

that internal tides can_propagate 1000s of kms from their

22-7-82

80-Figure

1-1. Isotherms

from

a thermistor chain

mooring

on the Celtic

shelf-break,

with some

smoothing, from

Pingree

etal

(1983), from 22"rf July

1982to

23rd July

1982. The horizontalaxis is

the totaltimeduration

of

48 hours

(with

12

hourly

tick marksatthe

top).

Themeasurementswere

taken

during

spring

tideswith tidalcurrents

typically

0.8 to0.9

ms'1,

and the internal tide shows clear non-linear

deformation

with

steepened troughs

_{and broad crests.}

Courtesy

_Journal

of

the

23.0 23.4 23 8 ~ir

24.2 24.6 250 23 0 23.4 23.8

firae (mini

Figure

1-2. Acoustic record

of

breaking

internal waves

from Haury

et al

(1979).

A 200 KHz

acoustic record

of

an internal wave

packet (propagating from right

to

left)

over 11 minutes

duration. Thedirection

of

_overturning

(left

to

right)

is_oppositetothat

of

the

packet

_propagation.

Superimposed

are the

corroborating density profiles

obtained

by

CTD. The instrument's

path

is seen asthe

oblique

traces in theacoustic record. The

ship

was

drifting during

the observations, and the windswere <

2ms'1. Reprinted by

_permission

from

Nature, Volume278,p 315,

copyright

ROUGH

SMOOTH

ROUGH

Surface

Convergence Divergence

\

Thermocline

Figure

1-3. Schematic

of

the circulation within non-linear internal

solitary

waves,

moving

with the

phase speed

c,

showing

the orbital circulation around thethermocline

forcing

_convergence and

Chapter

2.

AREVIEW OF THE THEORY OFOCEANICINTERNAL WAVES

2.1 Introduction

First, areview is

given

ofthe

theory

of internal waves in the ocean, bothlinear and non-linear. As the observations

presented

laterin the thesis areconcerned with

strongly

non-linear features, the

emphasis

here is on non-linear

theory.

Solutions to the

weakly

non-linear

equations

are

presented,

and

compared

against

theories withnorestrictionson wave

amplitude.

Previousassessmentsof the

ability

of the theories

todescribe

laboratory

_{and oceanic observations of internal}

solitary

waves arereviewed. A

description

of thenatureof the evolution of

long

wavesinto

solitary

wavesis also

given.

2.2

Equations

ofmotion

Ocean

dynamics

is

governed by

the

equations

ofmomentum,

continuity

and

incompressibility.

Underthe

Boussinesq

approximation,

where

density

anomalies are

only

considered

important

when

multiplied by

the accelerationdueto

gravity

_g,the

equations

are

given

by:

[2-1]

Momentum:

ut

+uux+vuy+wuz-

fv

=

(a)

Po

v +uv +vv +_wv, +

fu

=

(b)

Po

w+uw+vw+ww- _g

(c)

Po

Po

Continuity:

Incompressibility:

pt

+

upx

+

vpy

+wpz=0

(e)

[2-1] where x,yarethe horizontalco-ordinates,zis theverticalco-ordinate,

positive upwards

from theseasurface,t

istime,

subscripts

denotedifferentiation,andthe field variablesare

velocity (u,

v,

w),

density

p, pressurep.f isthe Coriolisacceleration,andpo isthe

Boussinesq

approximation

tothe

density.

The Coriolis accelerationis

given by

f=2Q,sin_{cp, where (pisthe latitudeand Q isthe rotation}

Insomecircumstancesof

particularly

_{strong features}the

Boussinesq

approximation

may be invalid. Here the

values_{of po}wouldhavetobe

replaced

by

the real

density

_{p.The consequences}for

non-Boussinesq

fluidsare discussedin section2.5.4

2.3 Linear internalwaves

Linearwave

theory

uses the

approximation

thatthewaves are

'infinitesimally'

small. Inother words,the waves have a small

enough amplitude

that

product

terms in the

equations

of motion can be

neglected.

Starting by considering

2-D flow in the

(x,

z)

plane,

so

9/9y=0,

and

consequently

simplifying

[2-1],

the

equation

of

continuity [2-Id]

implies

thatwe candefineastreamfunction

by

u

=VZ

w =

~VX

[2-2]

Under the linear

approximation, equation [2-1]

_{can be reducedto a}

single equation

forthe streamfunction

[Phillips

1977]

[2-3a]

where

V2

hereis

(92/3x2+32/3z2).

The

boundary

_conditions,underthe

approximation

ofa

rigid

lid,and aflat bottom,are

Ip

=0 on z =

0,

_{z= -H}

[2-3b]

where Histhe

depth

of thewatercolumn and

N(z)

is the

buoyancy frequency

defined

by

po{dz

c2)

[2-4a]

where

p(z)

is the

temporal

meanof the

density

at

depth

zandcis the

speed

of soundatthat

depth.

Intermsof the

potential density p*(z),

whichisthe

density

the

particle

would have if moved

adiabatically

tothe surface

(i.e. removing compression effects),

the

buoyancy

frequency

is

given by

[2-4b]

wavesfar fromthe

generation

site,haveamodal structure,

usually

of low

order).

Each modeioftheinternal wave

frequency

cohasitsownhorizontalwavenumber

ki,

so welook for solutions of the form:

[2-5]

where

(j>j(z)

is the vertical structure. The solutions arethen

basically

eigenfunctions

ornormal modes of the

equation

[2-6a]

with

boundary

conditions

</>,(z)

=

0

_{on z}=

0,

_z=-H

[2-6b]

whereXisthe characteristic

slope

defined

by

[2-7]

and

{kJX}

is the set of

eigenvalues relating

to the

eigenfunctions

{<j)j (z)}.

This

equation only

has real

solutionswhen f< co <N

(where

A2

is

positive),

which defines the range of

frequencies

ofrealinternalwaves

'.

Whenthewave

frequency

lies towardsthe middleoftheinternalwavecontinuum,sothat

fcoN,

[2-6a]

becomes

[2-8]

2.3.1

Examples

Two

simple

cases _{may be considered} as illustration. For

simplicity

we consider the situation where rotation is

unimportant

(f~0).

Physically

this

corresponds

towherethewave

frequency

is

high compared

As_longinternal tidewavesarelimitedto_{frequenciesgreater thanthe local value off,}it followsthatinternal tides ofagiventidal component(e.g. M2, OlorKl)aretrappedatlower latitudesthan_{the critical latitude (pcgiven by}o=f=2 Q sin (pc, where_aisthetidal

frequency.

DushawandWorcester(1998)determinedresonanceof diurnal internal tides below their critical_amplitudesof 25-30, whilstthe

critical_amplitudesfor M2 is78. At_higherlatitudes than the criticallatitude,itisspeculatedthattheinternal tides becomeevanescentand

to f

(cof),

_{or cases near}the _{equator. (Rotation}effects arediscussedbelow). This doesnot include the internal tide at mid-latitudes as the M2 and other dominant tidal

frequencies

areoften

comparable

with

the inertial

frequency.

Constant

Density

Gradient

In thecaseofconstantstratification,

N(z)=N0,

themodesare

simple

sine waves,with avertical wavenumber

m,whichisdefined

by

the

boundary

conditions:

</>i (z)

=

fa sin(mz)

=

</>0

sin

-[2-9]

whereiisthemode number and

()>o

isaconstant.The

dispersion

relationis then

given by equation [2-3a]:

(m2+k2)

[2-10]

An

important

_aspect of_{the ray like}

propagation

_{of internal waves in this environment}

can be seen

by

expressing

onemodeof

[2-5]

asthe

following

=cos(~kx

+ mz +

cat)cos(fcc

+

mz~cot)

=cos(fcc

-mz-cot)

cos(fcc

+ mz

-ox)

[2-11]

where the

trigonometrical

identities

sin(a)sin(b)=

V/2(cos(a-b)-cos(a+b))

_and

cos(-a)=cos(a)

_{have been}

used. This shows thatthemodal

description

_may be writtenasthe difference oftwo_rays, with

equal

and

opposite

vertical wavenumbers (m and -m). The

phase velocity

_Co of each _{ray is} now a vector in the direction of the wavenumber vector (k, m) with

angle

0 to the

horizontal,

so

tan(0)=m/k,

where + refers to_{\j/2 and}-to\]/i

dco

[2-12a]

Which from

[2-10]

are

given by

dco

_

N0m2

dco

_

Nom

k

~dk~(e

+

m2)

~fai~(k2

+

m2)

[2-12b]

andsothe

angle <|>

that_cg makes withthe horizontalis

given

by

tan(<|))=(-/+k/m).

In otherwords,for both \|/i and \|/2,

c^is

perpendicular

to _eg.

Energy

moves with the _group

velocity

of linear internal waves. As the vertical_componentsof_care

equal

and

opposite

_{for \|/i and \y2, this}

implies

_{that there is}

nonetvertical flux of

energy

(but

thereisahorizontal

flux).

The

peculiar

feature_{of group}

velocity perpendicular

to

phase velocity,

first shown in tank

experiments

by Mowbary

and

Rarity

(1967),

isincontrasttothe situation for surfaceand

interfacial waves _{where group}

velocity

is in the samedirectionas the

phase velocity.

Mowbary

and

Rarity

showed that foraconstantNsituation,wavecrestan