Rotating gravity current and channel flows

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

ROTATING GRAVITY CURRENT

AND

CHANNEL FLOWS

by

Joan

Rosemary

MARTIN

{nee

de

Lacey)

A

dissertation

submitted

for

the

degree

of

Doctor

ofPhilosophy

in

the

School

of

Ocean and Earth Science

UNIVERSITY OF SOUTHAMPTON

ABSTRACT

FACULTY OF SCIENCE

SCHOOL OF OCEANAND EARTH SCIENCE

Doctor of

Philosophy

ROTATING GRAVITY CURRENTAND CHANNEL FLOWS

by

Joan

Rosemary

Martin

Atheoretical and

laboratory Investigation

of

rotating gravity

currentsand channel flows is

presented.

The

study

is

applicable

to

buoyancy

driven flows

through

straits,

mid ocean

ridge valleys

and fracture_zones,and intermittent

gravity

currents. Inthe theoretical

study

two extensions areachievedto_{the energy}

conserving theory

of Hacker

(1996).

Hacker considered three flow

geometries,

caseA-weak

rotation,

_caseB- intermediate rotation

andcaseC-

strong

rotation.

Firstly,

the

theory

is extendedtoinclude

dissipation.

This is

achieved inasimilarmannertothat used

by

Benjamin (1968)

to _{include energy loss in the}

non-rotating gravity

current

theory.

The

governing equations

and numerical solutions for the three flow

geometries

are

presented.

For shallow currents the energy loss

theory

predicts

that the Froude number tendsto

2* irrespective

of the rotationrate. For

deeper

currentsthe Froude number increases with rotation. The second extensionto_{the energy}

conserving theory

is the inclusion ofanupstream

potential vorticity boundary

condition in thecurrent. The

approach

taken is based on amethod used

by

van

Heijst

(1985).

The

governing equations

and

preliminary

solutions for eachcase are derived. The

potential

vorticity

theory

provides

an

insight

into the circulation that

develops

within thecurrent.

However,

varying

the

pre-set

potential vorticity

in the source

region

doesnot _appearto

havea

significant

effect_upon the front

speed

and the other

principle

variables.

In the

laboratory investigation

the effects of fractional

depth

and rotationrate onthe

velocity

and other _parameters which characterise the flow are

quantified.

For

weakly

rotating

currents, w/R < 0.7

(where

w is the width of the channel and R the

Rossby

radius),

the measured front

speed

is in fair

agreement

_{with the energy loss and}

potential

vorticity

theories. At

higher

rotation rates the front

speed

is lower than

predicted.

However,

the theoriesassumethat the fluid is

inviscid,

the

no-slip

condition isnot

applied

atthe

boundary, potential vorticity

is conserved and that_energy loss is uniformacrossthe channel. The

theory

doesnotinclude factors suchasthe enhanced vertical

mixing

and the

PREFACE

This thesis is the result of my own work and it is not the outcome of work done in collaboration.No_parthas been submitted fora

degree

orsimilar

qualification

at _{any other}

university.

However,

Ido wishto

acknowledge

the contribution

by

_{my PhD}

Supervisor,

Dr.

Gregory

Lane-Serff,

who

proofread

the

manuscript

andalsowroteaFortran programmeto

give

an

indication of the

possible

solutions to the _energy

conserving theory

with

prescribed

potential

vorticity,

as discussedin

§

5.2.1 would also liketo

thank,

Dr. David Smeed for his

guidance

particularly

with the theoretical _{aspectof my research and for}

always finding

timeto discuss and advise me

regarding

the

theory.

Also,

Prof. S.

Thorpe

for his

helpful

suggestions

regarding

the

laboratory investigation particularly during

the initial

stage

_{of my} research.

Finally

I wish tothank_my

family;

_{my father for all the}

physics

bookshe

bought

me; my mother for

keeping

my home in order

enabling

meto

study;

to _my

dear,

patient

husband David for all his love and _support and my sons

Christopher

and Andrew for the

interest

they

have

always

shown in_mywork.

Contents

CONTENTS

Chapter

1

Introduction

1

1.1

Applications

1

1.2 Basic features 3

1.2.1

Non-rotating gravity

currents 3

1.2.2

Rotating gravity

currents 4

1.2.3 Basic scales 5

1.3 Aims and Thesis Outline 6

Chapter

2

Environmental

Applications

8

2.1 Introduction 8

2.2

Atmospheric gravity

currents 8

2.3 Estuarine

gravity

currents 10

2.4 Overflows 12

2.5 Surfacecurrents 14

Chapter

3

Theoretical Framework

17

3.1 Literature review 17

3.1.1

Non-rotating

_energy

conserving

theory

20

3.2

Governing

equations

22

3.2.1 Model

description

22

3.2.2 Basic scales 23

3.2.3

Stage

1 -

Governing equations

24

3.2.3.1 Momentum

equation

24

3.2.3.2

Geostrophic

equation

25

3.2.3.3 _{Across-stream pressure} 26

3.2.3.4 Conservation

ofpotential

vorticity

2 7

3.2.3.5 Flowstructure

equations

27

3.2.4

Stage

2- Conservationof the fundamental

properties

29

3.2.4.1 Conservation

of

_energy 29

3.2.4.2 Conservation

of

volume

flux

30

3.2.4.3 Conservation

of

momentum 31

3.2.5

Stage

3 -Numerical solution 33

3.3 Review of results withno _energylossand

simple

flow 34

3.3.1 Results and discussion 3 4

Chapter

4

Theoretical Results

(I)

36

4.1

Energy

dissipation

36

4.1.1

Background

to _energyloss

theory

3 6

4.1.2

Governing equations quantifying

_energyloss 40

4.1.3

Continuity

of volume flux 42

4.1.4 Conservation ofmomentum 43

4.1.5

Governing equations

45

4.3

Chapter

5 5.1 5.2

Chapter

6 6.1 6.2 6.3

4.2.1 CaseA

4.2.1.1 Solution

for

caseA

4.2.1.2

Asymptotic

solutionas W tendstozero

4.2.1.3 Transition

point

betweencasesA and B

4.2.1.4 Numerical solution

for

caseA

4.2.2 CaseB

4.2.2.1 Solution

for

caseB

4.2.2.2 Transition betweencasesC andB 4.2.2.3 Numerical solution

for

caseB

4.2.3 Case C

4.2.3.1 Solution

for

caseC

4.2.3.2 Numerical solution

for

case C Review of results with_energyloss and

simple

flow

4.3.1

Properties

of the solution 4.3.2

Comparison

with

previous

work

Theoretical Results

(2)

Upstream

potential vorticity

distribution 5.1.1 Model

description

5.1.2 Basic scales

5.1.3

Stage

1 -

Governing equations

5.1.3.1 Conservation

ofpotential

vorticity

5.1.3.2

Geostrophic

equations

andMar

gules relationship

5.1.3.3 Flowstructure

equations

5.1.3.4 Pressure

equations

5.1.4

Stage

2 - Conservation of fundamental

properties

5.1.4.1 Conservation

of

_energy

5.1.4.2 Conservation

of

volume

flux

5.1.4.3 Conservation

of

momentum

5.1.5

Summary

of

general

form of the

governing equations

5.1.6 The

governing equations

for eachcase

5.1.6.1 Case A

5.1.6.2 CaseB

5.1.6.3 CaseC

Numerical solution

5.2.1

Summary

and Discussion

Laboratory Investigation

Aims of

laboratory

investigation

Review of others

experimental findings

Experimental procedure

6.3.1

Apparatus

6.3.2 Method

6.3.2.1 With

permanent

barrier-sets1 &2

6.3.2.2 Without_permanentbarrier-sets

3,

4& 5

6.3.2.3 Shallowcurrentstank

geometry

-set4

6.3.2.4 Full

depth

currents-set5

6.3.2.5 Particle

tracking

-set6

Contents

6.3.2.6

Image processing

109

Chapter

7

Laboratory

Results

112

7.1 Witha

permanent

barrier 113

7.1.1

Qualitative

features - sets 1 &2 113

7.1.2 Inertial-viscous transition 115

7.1.3

Upstream

tail

depth, Tz (preliminary experiments)

117

7.1.4 Head

depth,

Hz

(preliminary experiments)

119

7.1.5 Bore

strength

(preliminary experiments)

120

7.1.6 Front

speed (preliminary experiments)

122

7.2 Without a

permanent

barrier 123

7.2.1

Qualitative

features - Intermediate

depth

gravity

current 123

7.2.2

Qualitative

features- Shallow

depth gravity

current 125

7.2.3

Qualitative

features -Full

depth gravity

current 126

7.2.4

Qualitative

features-Reflected

gravity

current 127

7.2.5

Qualitative

features -Particle

tracking (set

6)

127 7.3

Quantitative

results ofsets

3,

4& 5 128

7.3.1 Current

width,

d 129

7,3.2 Front

speed,

c 130

7.3.3 Froude

number,

Fr 132

7.4

Summary

133

Chapter

8

Conclusion

135

8.1

Summary

and Discussion 135

8.2 Future work 137

CHAPTER 1

Introduction

CHAPTER

1

Introduction

Whenadense fluidis

released

in theabsenceof

background

rotationit

spreads

radially

_until

it reaches a

boundary

which _{prevents further}

spreading.

_{In the}

presence of

background

rotation the fluid

spreads

laterally

_{and motions}

normal to the axis of rotation induce Coriolis forces. TheCoriolis forceisan _{apparent force}thatacts normaltothe directionof

motionandcausesa

deflection

tothe

right

in thenorthern

hemisphere

_{andtothe left in the}

southern

hemisphere.

Eventually

_{in the absence of}

dissipation

astateof

equilibrium

will

develop

wherethe

buoyancy

_{and Coriolis forces}

balance, preventing

further

spreading,

known as a

geostrophic eddy.

_{Plate 1 shows}

a

geostrophic eddy

formed

by

_{the axial release}

ofa

lighter

fluidontoadenser fluid

rotating

_at

f=

1,

where/is

twicethe

angular

frequency

ofrotation. Ifa

boundary

is

introduced,

_{Coriolis forcesnormal}

totheborderare

removed,

since flow _acrossthe

boundary

is

impossible.

A

jet

forms

parallel

to the

boundary

and is

held

against

it

by

Coriolis forces

_{normalto the direction}

of_{flow. The widthto which the}

flow

adjusts

isknownasthe

Rossby

radius of

deformation,

_R =

(g'Hf'

/f,

where

g'

is the

reduced

gravity

and Hthe

depth

(Rossby

1938).

1.1

Applications

In the

atmosphere

_mountain

ranges form these

boundaries.

The occurrence of

topographical

trapped

gravity

currentsarecommon

along

thewest coastof North America

and

against

the _{double barrierof the Coastaland Sierra}

Nevada Mountains. Other flows

with

gravity

_current

characteristics influenced

by

the_{Coriolis force includethe}

Southerly

Buster,

a cold wind which _runs

parallel

to the Australian

Alps

in eastern Australia

(Couquhoun,

1985),

and sea-breezesin northern

Australia, (Physick

1985).

The passage of

atmospheric

phenomena

_are

accompanied

by

changes

in

wind,

temperature

_{and pressure,} which are monitored

using

instrument towers,

balloons,

lidar,

radar or echo sounders. Thesesudden

meteorological changes

_have

important implications

foraviation and

shipping

Plate 1-

Geostrophic

eddy (/=1). Light

fluid is

supplied by

acentral

ring

source.

Initially

the fluid

movesoutwards duetothe

buoyancy

force. An

axisymmetric

flow

develops

inordertoconserve

angular

momentum. The

resulting

Coriolis forcebalances the radial

buoyancy

force

enabling

CHAPTER 1 Introduction

incoastalwaters.

Inthe_ocean, coastlines and sub-marine

ridges

constrain the motion. Numerous

examples

of

density-driven

flows influenced

by

rotation

exist,

they

can be divided in to surface currents, flows

through

estuaries,

overflows and fracturezones. Satellite

images provide

an

insight

in to the instabilities of

boundary

currents. These flows are

responsible

for meridional

transport

of heat and

salinity. They

are

large

scale and

generally

continuous.

Intermittent

gravity

currentsareoften observed in the estuarine environment.

Spencer

Gulf in Australia

(Nunes

et

al.,

1987)

is a

particularly

well documented case ofan inverse

estuary.

Gravity

currents

develop

here dueto the anomalous tidal

cycle,

whichcauses a marked reduction in turbulence at

fortnightly

intervals and is further enhanced

by

light

winds.

Spencer

Gulf also exhibits aseasonal

gravity

current

(Bowers

et

al.,

1987

).

Other overflows include the flow

through

the Denmark Straits

(Swift,

1980)

and the Straits of Gibraltar

(Price

et

al, 1993),

where_strong internal tidescausevariations in the outflow on a

weekly

timescale.

Atmospheric

stormsalsocause _strongfluctuations and

pulsations

in the

flow,

however little interannual

variability

is observed. Adetailed

picture

of the

salinity,

temperature,

depth,

and

dynamics

ofoverflows is built _up

using ship

borne instruments

including:

CTD-

conductivity,

temperatureand

depth instrument;

XCP - extendible current

profiler;

XDP-extendible

dissipation profiler,

which

profiles

the turbulence

intensity.

It is

generally

agreed

that the

improvement

ofoceancirculation models is

dependent

onthetrue

representation

of the_transport and seawater

properties

of

major

overflows.

Boning

in his

convenors_reportfor the_opensession ofocean

circulation, European

Geophysical Society's

General

Assembly

1997,

highlighted

overflows as a

key physical

_{process for which open}

questions

remained. In

referring

to

intercomparison

studies ofoceanmodels he concluded

that,

"... the numerical

representation

of

the

diapycnic

mixing

inthenarrowDenmark Strait

outflow plume

representsadecisive

factorfor

thestructure

of

the

large-scale

circulation

in the Atlantic Ocean. "

The

study

of

rotating gravity

currents has wider

applications, benefiting

other scientific

fields,

such_as,

biology

andmarine

geology.

For

instance,

itis

conjectured

that the life

cycle

of the krill is

dependent

onthe

transport

of its larvae from the Weddell

Sea,

westwardto

their nursery

grounds,

by

the South Shetland Islandsoverflow

(Whitehead,

1989).

Fishing

grounds

are known to be affected

by

_{the passage of}

gravity

_{currents, for}

example,

the

Kyucho

Current which _propagatesaround

bays

in the Sea of

Japan,

causeshavocto the

fishing

grounds

(Yamagata,

1980).

In the

atmosphere

sea breezes are

responsible

for

transporting

insect _pests and

pollution

inland.

Geologically

a

greater

understanding

of sediment _transport and

deposition

both now and in the

past

_may be obtained. Further

applications

are discussed in

chapter

2.

1.2

Basic features

The

anatomy

of the

non-rotating gravity

currentwas

fully

described

by Simpson (1987).

In

plate

2

images

taken from the _present

study

areused toillustrate the basic features of

non-rotating

and

rotating gravity

currents.The

plan

_{view is the upper}

part

_{and the side view} is the lower_part of each

image.

1.2.1

Non-rotating gravity

current

The

non-rotating gravity

current when viewed from the side has awell defined head and

nose

(plate

2a).

For bottom currentsthenoseis

slightly

raised above the tank floor dueto

bottom friction. Surface currents do not

display

this feature if the surface is free from

contaminants. The

following

flowbehindthe head is shallower. This is knownasthe tail.

Immediately

behind the

head,

Kelvin-Helmholtz billows are seen to

peel

off the outer

surface. This isan

instability

that ariseswhentwo fluidsmoverelativeto oneanother. The

profile

of thecurrent_{may be}modified

by

viscous

effects,

fractional

depth

or an

opposing

flowin the ambient fluid. Inthe

plan

view,

the

leading edge

is broken into aseries of lobes andclefts. As thecurrentadvances

along

the tank the lobes swell and

engulf

orare

engulfed

by

other

lobes,

producing

a

shifting

_pattern. The lobes and clefts arecaused

by lighter

fluid

being

over run

by

the denser fluid. The denser fluid is in a

position

where it is

juaxmoaqj jo

q;dap

aqx paAiasqo uibSb3jb_SA\O[[iq

33JOJ sqoiJO3 aqj

Äq XjBpunoq

aqjjsureSepjaqsi_Ji

aqipireAvaiA_{apis aqi}ui

si_{oaiaqAv ggi}oj quiojjsi

padojaAap

sßq

Xppa

oiqdozjsoaS

b_{puB pasBajoui seq uoiSai}aoinos_aq;ye

qidap

pire qjpiAv qjoqin

ÄpuBotfioSis

_pasBaioapseq pßaq aqx Avog sqjjo uopoajtip aipo}[Buiiou pueq

jqSu

aqj spiBAVOj papajjapsijuaxmoaqx

(66Z

=

()//H

'uopejoj

guojjs)

^uaxnio

Äjpvejg ginpio"a (q)

AV3IA

uBjd

_aq}majnpaijs c;ja[O ptro aqo[,

aqjajoj\[ |iei j3A\o[jBqs aqjsi_{qaiqAV puiqaq}

'pßaq

puijsipaqi

Sumoqs (o

=

C^J/Ah)

liiraxmoa^iaboS

§upBjoj-uoj\[ (b)

'OOOC<

^H'PH 'SO~

H/H

3JB

C^)P116 00

JfOJsuopipuoo jbijiui

puee^uaSeiu'anjqcua3j§

cA\ojjaÄ 'paj

_sinojoojo aouanbas aqjoj

spuodsaxioo

pue

X;isua;tiiaqx aSsuii qoBaoj

paijddB

ajjafBd

jnopa asjBjb

Xq

_pajßijsnjfist_{pmgf aqj}

qSnoiq;

SuissBd

iq§Hjo

Ä}isuajuiaqx Suiuado >[oo[ aq; raojj

passajSoad

SBqjuaxmaaqi aouB;sip

xj^

juaxinoAzubisaqjjoq^piAV

^

pBaqjuaxmoA^iabjSaq; jo

qjdap

'fj

Suiuado_{>[3O| aq}}jbpsjiiSBaui \m aqj jo

qjdap

zj

'uoipSijsaAuiAjo^BJioqB[ aqjm_pajtisBsrasajBos

qjSuaj

aqj q^iAv pajBpuoBluaxmoä}iabjSb_jo

Auiojbuv

_Z31Vld

XMiHaiio ÄXiAvao onuvxoh IMISHaD ÄUAVHD9WU.VX0HJU0W

3

a i s

M

V

this forbottom currents.

The

gravity

currentsin this

study

were

generated

using

thelock-releasemethod.

Basically,

this method involves

dividing

a tank into two reservoirs

by

a removable barrier. One

reservoir is filled with salinewaterand the other with fresh water. A

gravity

current is

generated

when thebarrieris removed. Asa

gravity

current_progresses

along

achannel

twodistinct

phases

may beidentified

(Simpson,

1972).

After the removal of the_gateand the initial

collapse

the two _{flows propagate}in

opposite directions,

_{with the upper fluid}

progressing

faster than the lower. This is knownastheconstant

speed

phase.

On

reaching

the end of the tank the

lighter

fluidreflects _{from the end wall and forms a}

hydraulic jump,

which

propagates

along

_the

length

_{of the denser}

gravity

current. The

hydraulic jump

reaches the head of thecurrentwithinadistance

equivalent

to 10 lock

lengths,

where alock

length

is the

length

_{of the lock from which the saline fluid}

originates.

This results in the

constant

speed phase

ending

and

phase

2 where the

speed

of the current decreaseswith t

"3

begins.

If the

experiment

_{involves shallow}

gravity

_{currents in}

deep

water, the situation

is

slightly

different. Phase 1

(the

constant

speed stage)

is

observed,

but in this case atthe fluid interfacea

long

wave of

depression

_propagates. Thisreflects off the end walland in due time overtakes the head. This

again

results in areduction in

velocity, decreasing

with

t"*

.

Eventually

in both cases a third

phase

is reached where viscous effects

become

dominant, causing

a further reduction in

velocity.

In the _present

study phase

2 is not

observed becausethe ratio of the tankto the lock

length

is

only

3:1. Thetransitionfrom

inertialto viscouscurrentsisdiscussedin

chapter

7.

1.2.2

Rotating gravity

_currents

The

characteristics

of

rotating gravity

_currents

(plate 2b)

_are

dependent

on the level of

rotation. The head and nose are observed but theseare modified. The head _decreases in

depth

and width asthe rotationrateincreases. Behind the headKelvin-Helmholtzbillows areobserved on all surfaces incontact with the ambient fluid. The

depth

atthe

right

hand wallincreaseswith

rotation,

asdoes the

slope

of theinterfaceacross stream. The

boundary

between the head and the tail is less distinct thanthat observed for

non-rotating gravity

currents, because it is often obscured

by

billows. At

high

rotation rates thecurrent hasa

CHAPTER 1 Introduction

wedge

like appearance in the sideviewand an

anticyclonic

_{gyre often}

develops

near the

opening.

The two distinct

phases

identified for

non-rotating gravity

currents are

again

observed. After the initial

collapse

the fluid

adjusts

to aconstant

velocity phase.

However,

slight

oscillations in the

velocity

are observed. Griffiths

(1983)

attributed this oscillationto the

growth

and

decay

of the head of the current. The second

decelerating phase

is also

apparent.

Thiswas

quantified by

Griffiths

(1983),

who established that the

velocity

of the head decreased

exponentially

withtime. He

proposed

that the

decay

in the

velocity

was

dependent

onthe rotation rate, aFroude number based onlock_parameters

and,

in certain

cases, Ekman friction.

Only

theconstant

velocity

phase

is observed in this

study.

1.2.3 Basic scales

Plate2is annotated with the basic features of the

non-rotating

and

rotating gravity

currents

measured in this

study.

The

subscripts

refertothe

plane

in which the measurementswere taken.

H2

head

T2

tail measuredat the

opening

Wy

width of the

rotating

current

w width of the tank

Ho

total

depth

of fluid

h0

inflow

depth

U

propagation

speed

ofnose

To enable measurements taken in the

laboratory

to be scaled _up to the real world

non-dimensionalised numbers are calculated which define the flow. These are written as a function of the parameteronwhich

they

arebasedto ease

understanding

in thetext.

Non-dimensionalnumbers

dependent

on_{pre-set parameters}

Front

speed

c(h0)

$

Rossby

radius

R(h0)

=

(ghoflf

Rotationrate W=

w/R(h0)

Reynolds

number

Re(h0)

=

((g7?0)* ^oVv

Fractional

depth

Tzl h0

N.b. The_{non-dimensional numbers above may also be}

expressed

_{as afunction of}

Ho.

Non-dimensional numbers

dependent

onmeasured_parameters

Froude number

Fr{T2)

=

U/(gT2)>

Rossby

radius

R{T2)

=

(gT2flf

Reynolds

number

Re(Hz)

=

({UH2)lv

Fractional

depth

TJH0

1.3

Aims

and

thesis outline

So far I have

briefly

considered the

diversity

of

gravity

currents in the environment and their _anatomy,

(expanded

on in

chapter

2)

but what of the

attempts

to understand the

structure of the

flow,

the

interplay

of the various forces and their mathematical

representation?

Theforemost

objective

of thisthesis isto _{extend the energy}

conserving

theory

of

rotating gravity

currents

(Hacker 1996)

toinclude

dissipation by quantifying

the effect offractional

depth

and rotation rate on the

velocity

and other _parameters which

characterise theflow. Asafirst

step

it was instructiveto consider how this isachieved in the

non-rotating theory

of

Benjamin (1968).

Next,

an

understanding

of how Hacker derived the energy

conserving

solution is necessaryto decide whereit is

appropriate

to introduce

fractional

depth.

A_{summary of}

Benjamin's

_{and Hacker's theories is}

provided

_in

chapter

_3. In

chapter

4_{energy loss is}

incorporated

_{into the}

rotating theory

_{and in}

chapter

_{5 the effect}

of thesource

conditions,

i.e. the

prescribed potential vorticity,

_{is also} included. In

chapters

6 and 7 the method and results of the

laboratory investigation

are

presented.

The

CHAPTER 1 _Introduction

investigation

and

suggestions

are made for furtherwork.

CHAPTER

2

Environmental

Applications

2.1

Introduction

In

chapter

1 the effects _{of rotation upon} a fluid released into a

rotating

_system in the

laboratory

was

briefly

discussed. Furthermoretheintroductionofabarrierwas shownto

generatea

jet-like

currentconstrained

laterally against

the barrier

by

_{the Coriolis force. This}

suggests that if suitable conditionswereto occurin the

atmosphere

or ocean closeto a barrier such as a _{mountain range} or a

coastline,

thenanintrusion could

develop. Indeed,

on

examining

theliterature onefindsthat there are_many

examples

_{of flows which could} be classified as

rotating gravity

currents. The aim of this

chapter

is to _{compare the}

characteristics

they

_{possess with}thosecurrents

generated

in the

laboratory,

_{the arguments} for and

against

classifying

them as

gravity

_{currents, where}

they

occur, what devices are

employed

to

quantify

them and an

example

ofa case

study

for each.

2.2

Atmospheric

Gravity

Currents

On

researching

the

subject

of

atmospheric

gravity

currentsIcame acrossa

variety

offlows whichcould be

possible

_{candidates for theterm}

gravity

current. Before

proceeding

I shall

discussthe nomenclature used in

conjunction

_{with theseflows as defined}

by Egger

et

al,

1991.

Firstly,

orographic adjustment

_{refers to the}

adjustment

ofa

large

scale flow to a barrier. An

orographic

jet

isacoldfront whose nose _propagates

parallel

toa mountain and isconfinedtowithinafewkilometersof the mountain_{range. A}

particularly

_{well researched}

example

is the

Southerly

_{buster whichoccurs}

along

_{the southern}

coast of New South Wales

ineastern Australia.

Colquhoun

et al

(1985)

_{describes the arrival of the}

southerly

_buster

(or

burster)

which is marked

by

a sudden increase in wind

speed

to

15ms"1

sometimes

CHAPTER2 _{Environmental}

Applications

exceeding

37ms"1

and at thesametime a

drop

in

temperature

of10to 15C within afew

minutes.

Cold air surges may also

undergo orographic adjustment,

however the horizontal scale

associated with these is considerablewith widths of_upto several thousand

kilometers,

examples

include _{the cold surges} constrained

by

the Tibetian Plateau

(Orlanski,

1983),

Andes

(Rutlant, 1981)

and the

Rocky

mountains

(Lilly,

1981;

Shipiro

et

al,

1985). Along

the West coastofNorth America

long

_{shore surges marked}

by

coastal stratus

tongues

are observed

frequently.

Mass et

al,

(1987)

describes how the arrival of these events are marked

by changes

inwind

speed,

_{pressure andtemperature}which occur on atimescaleof less thanone hour. Similar

phenomena

are

experienced along

the coast of SouthAfrica where the

topography

consists of an

abrupt

rise from the coastal

plain

to the interior

plateau.

With the aid of surface and_upperair

stations,

and the

comparison

ofmeasurementsfrom

ships,

buoys

and coastal stationsaswellassatellite

imagery,

it is

possible

to obtaindetailed

measurementsofthe extentof these

atmospheric

_{flows. The vertical structurecan also be}

discerned

using

radiosondemeasurements. Anevent for whichthereare

convincing

GOES satellite

images

is the

long

_{shore surge caused}

by

synoptic

scale

changes

which occurred

along

the West coastof North Americabetween 15-17

May 1985,

see

figure

2.1. Mass describes the

propagation

of the _surgethus

-"Moving

up the

Oregon

coast, the surge

brought

not

only

a

change

inwinddirection but

progressively larger

andmore

abrupt

transitions inwind

speed,

_pressureand

temperature.

Maximum winds increased

from

10

ms'1

insouthern

Oregon

to 17ms~l onthe northern

border,

temperature

drops

varied

from

6C h'1 insouthern

Oregon

to 16C h'1 tothe

north,

and_{pressure rises} varied

from

0.2 mbh'1to thesouth to 2nib h'1in the north. It iswell

known

(e.g.,

Charba,

1974;

Griffiths,

1986)

_{that the}

forward leading edge of

a

gravity

currentcan

steepen

in time.

By

00UTC17

May

(when

_{the surge}was

just

south

of

Astoria,

Oregon)

the

southerly

transitioninvolvedaradical and

abrupt change of

airmass

from

warm,

subsiding

continentalairto

cool,

stratus-filled

marine

flow.

"

19 GMT 16 MAY 23 GMT 16 MAY

19 GMT 17 MAY 23 GMT 17 MAY

FIGURE 2.1.

Topographically

trapped gravity

current

along

thewestern coastof the United States and

Canada,

imaged

using

GOES visible satellite between 16-17th

May,

1985. This

phenomenon

is caused

by

asudden

change

in wind direction from northerlies

(0-10

ms1)

tosoutherlies

(15

ms"1),

accompanied by

a _temperature

drop

> 10C and a sudden rise in pressure. The coastal _tongue is

CHAPTER2 Environmental

Applications

Mass

puts

forward anumber of_argumentsfor

classing

_{the surge}as a

gravity

current. He calculated that the

speed closely

resembled that ofa

gravity

currentbasedon the theoretical

predictions

of Seitter and Muench

(1985).

The width of the southerlies and stratus was foundtobe 180km

i.e.,

within one

Rossby

radiusofthe coastal mountains. Mass cites the theoretical work of

Baines,

1980;

Griffiths,

1986 andGriffithsand

Hopfinger,

1983 which

implied

that a

topographically trapped gravity

current should be within this

spatial

scale.

Observations of the verticalstructureof the

phenomenon

showed that the vertical scalewas

very

shallow,

constrained to the lower

troposphere

to below 850

mb,

which

again

is a

characteristic ofa

gravity

current.

In manycase studies of similar

atmospheric phenomena

there is discussionas towhether

the observations

imply

a

gravity

current or should be more

appropriately

classed as a

topographically

trapped

Kelvin wave. AKelvinwave _maybe defined as a unidirectional wavewhose

amplitude

is

only

significant

within adistance of the order ofa

Rossby

radius froma

boundary.

It travels with the

boundary

onits

right

in theNorthern

hemisphere

and onthe left in the Southern

hemisphere.

To

distinguish

betweenaKelvinwaveand a

gravity

current, it is

important

to consider whethertwo

layers

exist. A_stronginversion

capping

a cool marine

layer,

associated with wave-like disturbances would

imply

a

topographically

trapped

Kelvinwave.

However,

if ahead of the surgeatwo

layer

_systemdoesnotexist and instead the vertical structureis created as a_{consequence of the}

propagation

of the cool air

through

the uniform

subsiding

air,

then this would_suggestthe_presence ofa

gravity

current.

These

phenomena

have

important implications

for

shipping

and aviation. Indeed the

'papal

front' gets its nickname because a

helicopter flight

due to take

Pope

John Paul II from Munich to

Augsburg

hadto be canceledbecauseof theintensificationof the front

by

the

Alps.

2.3 Estuarine

Gravity

Currents

Certain conditions are _{necessary for} the

development

of

gravity

currents within the estuarine environment. Themost common

type

is known as a 'salt

wedge'.

Theseoccurin

estuaries whichare

generally

narrowrelativetotheirwater

depth

and receive a

high

volume of fresh water runofffrom rivers.

They

are also associated with aweak tidal

cycle

that enablesstratificationto

develop.

The surfacecurrentstendto be turbulent dueto the

large

input

offresh_{water, whilst the flow in the saline}wateris minimal. Thisexertsashearon the interface

causing

internal waves and entrainment of saline water into the turbulent surfaceflowfurther

increasing

its volume. Echo

sounding

_surveys

(Geyer, 1983),

show that

they

exhibit _many of the features associated with

gravity

currents

including

the

characteristic 'head'.

During

the flood

stage

of the tide the freshwatermay be forced back

into the _estuary. Often the interface between the two fluids on the surface is marked

by

debris and colour

changes, (Simpson

&

Nunes,

1981).

Afurther

example

ofa

gravity

current whichoccurs in the estuarine environment is that whichoccursin shallow estuaries where the local climatic conditions cause

evaporation

in excessofthat lost

through precipitation

and riverrunoff. This combined with weak tidal

currents

produces

what is knownas aninverse_estuary. Well documented

examples

include

Spencer

Gulf

(Bowers

&

Lennon,

1987)

and Gulf St. Vincent

(de

Silva

Samarasinghe,

1987)

both situated on the South Australian Coast. Herethe lunar and semi-diurnal tidal constituents are

equal,

which results in a

large spring-neap

tidal modulation. Thiscauses a considerable reduction in turbulence and

mixing

on a

fortnightly

cycle,

causing

the

development

ofstratificationi.e. abaroclinic

regime.

The switch to stratified conditions

occurs

abruptly

withinonehour and lasts for_many

days, enabling

the Earth's rotationto

cause the

development

ofa

single large

scale

cyclonic

_{gyre after} an ineitial

period

has

passed.

Ineacharmof the flow the_gyreachievesa

velocity

of

approximately

0.2ms"1.

This is

considerably

greater

than the annualmean circulation

speed

of

0.05ms"1,

(Bullock,

1975;

Nunes &

Lennon,

1986).

Atthe mouth of the Gulf

during

the summermonths the

density

contrastbetweenthe Gulf

and Shelfwateris offset

by

the

higher

temperatures

within the Gulf. However

during

the

autumnand winter the Gulfwaterscool andanoutflow of salinewaterforms

along

thesea

floor whilst the less saline shelfwaters flow into the Gulf. The Coriolis forcecauses the

interfacetobe inclined with_respect tothe horizontal. The outflow continues for 150km.

CHAPTER2 _{Environmental}

Applications

knowntolocal

oceanographers

as

'Bonaparte's tongue'.

Lennonet

al,

(1987),

_{present the}

resultsofa_surveycarriedout

by

the

Royal

Australian

Navy

Vessel,

RVHMAS Cook. The

extent _{of the survey}was from the headwaters of the Gulfto

beyond

the shelf break. The

salinewaterwas found to _{emerge from the mouth of the Gulf}on the eastern side. The

salinity

and

density

contoursin

figure

2.2,

clearly

show the

path

of thecurrent asitcrosses the end of

Investigator

Strait and then flows

along Kangaroo

Island and

finally

off the shelf

edge,

close toDu Conedic

Canyon.

The

salinity

contours

imply

a

depth

of 20- 40m. Its widthwas foundto _{vary between 50km} at the Gulf mouthto 20kmat

Cape

Borda. The effect of the earth's rotation should cause the current to turn into

Investigator

Strait. Lennonattributesthis

divergence

totheeffect of

friction,

suggesting

thesince theflow is

relatively

thin itfeelstheeffect of bottomfrictionto a_greaterextent

causing

itto flowat

a

steeper

angle

across

depth

contours. Lennonestimates the

speed

of thecurrent as0.1

ms'1

using

the

Chezy equation.

His estimates of

continuity

of salt and water _suggest thatthe

current would haveto continue for three monthsto restorethe levelsofsalttothat

prior

tothe

high

evaporation

ratesof thesummer.Lennon

hypothesizes

that the

narrowing

of the

current at

Cape

Borda could be duetothe effects of the anomalous tidal

cycle

in the Gulf

causing

pulsations

150km apart.

Theintermittent

gravity

currentswhich

develop

within the Gulf and

seasonally

atits mouth

are

responsible

for the

dispersion

and

exchange

of

properties

between the Gulf and the Shelfwaters.

2.4 Overflows

An

example

ofan environmental

gravity

current which lends itselftothe

study

of

gravity

currentsin the

laboratory

is that of the overflow orcataract. These dense flowsoccurat

great

depth

across sills

linking

oceanbasins andtheir

importance

inthe meridional transfer ofheat and

salinity

should not be underestimated. The Atlantic Ocean boasts several

examples

of

significant

overflows

including

the Iceland Faroe

overflow,

the flows

through

the Denmark

Strait,

the Straits of

Gibraltar,

and in the Southern

hemisphere

the Shetland

3TVDS

02

02 61

31V0S

?E ec ze

ONVISI

OOHVONVXko

ISVOD ISHAV-HIHON Ü0

91 91 II

IIVHIS OIVOIIS3ANI JOAHVCINnog MH31SHM

x-, V' .,'*<!o

oz et

\! _{yn7J"irii,''"itilJ|liyfllJJ/7yjO9}

|a1I SN3O|

j[in.O NyTDNTPTfTS -TO HI nOTAT

U 01 6

CHAPTER2 Environmental

Applications

Brazilian and North Atlantic basinacrossthe Ceara

Abyssal

Plain.

Measurements of these flowsis limited

by

the

depths

at which

they

occurand often the

strength

of the

flows,

for

example, Worthington (1967),

lost20 of30 current meters

during

a survey of the Denmark Strait. Current

speeds

of up to 1.4

ms"1

were recorded in

comparison

to the surface currents of0.1 to 0.5

ms"1.

The_transport of the

salinity

and

density

has since been

quantified using

a

variety

of instruments

including

CTD,

XCP and

XDP. An

interesting

method of

tracing

the

path

of these flows is theuse of radioactive tracers, such as,tritium

(an

isotope

of

hydrogen

withahalf life of12.5

years).

During

the 1950's -

early

6O's the

high atmospheric

nuclear bombtests released

quantities

of tritium intothe

atmosphere

in the northern

hemisphere.

This

effectively

labeled the surface waters

until

testing stopped

in1963. In 1972the results of the Geochemical Oceanic Section

Study

showed that the water of the Denmark Strait overflow reached a

depth

of3500m at the base of the North Atlantic

Deep

_{water, but}notritiumwas detected in the Antarctic bottom

water. Further studies conducted

by

Peterson &

Rooth,

(1976)

discovered that thesource

of the Denmark Strait overflowwasnotthe bottomwaterof the

Norwegian

Sea but instead froma

depth

of theupper 1000m. Meincke &

Kringe

(1978)

conjectured

that

atmospheric

forcing

was

responsible

for

lifting

thewaterbehind the sillto cause anoverflow event. The actual mechanism is due to a storm

generating cyclonic

winds which cause an Ekman

transport

tobeset_up,which inturncauses

upwelling.

The

depression

of thewatersurface and the associated

rising

of the thermocline lifts the waterabove the

depth

of the sill. The

intermittent nature of_many of these flows which are often initiated

by atmospheric

disturbances

implies

that the initial front of the intrusion would have the form ofa

gravity

current.

Atmospheric

fluctuations are

thought

to be

responsible

for the flow acrossthe

Straits of Gibraltar.

The

instability

of these

flows,

their interaction with

topography

and the effects of tides leadsoneto wonderatthe

feasibility

of

modeling

them in the

laboratory.

However much of the

dynamics

atthe sill is

governed

by

simple hydraulic

laws. It has been shown that estimates of volume_transport deduced from

laboratory experiments

_compare

reasonably

well with surveys of environmental overflows. Whiteheadet

al, (1974),

estimated that the volume _transport of the Denmark Strait was between 2.1x

lO6

to 4.8 x

lO6 m3/s.

A

subsequent

survey of the Strait

(Dickson

et

al, 1990),

measured avolume

transport

of2.9 Sv

(lSv

=

106m3/s).

_{Whitehead also modeled the flow}

through

the Straits of Gibraltar and

concluded from his

experiments

that this waterwas drawn from a_great

depth,

an idea which was

previously proposed by

Stommel

(1973).

This too has been confirmed

by

hydrographic

data obtained

during

a _{survey of the Mediterranean outflow in November}

1986

by

Kinderet al

(1987).

However Whitehead views his results with caution.

"

Thereisstillsome

question

however about whether the results

of

theoretical models and

laboratory experiments

-whichare limited

by

thesize

of

turntables-can bescaled_up to

the realoceanwhere turbulence is

likely

to be much

greater

and where the

topography of

the bottom introducesa

largely

unknownamount

ofdrag

and

mixing.

"

Indeed ifone considers the caseof the Arctic bottom waterthat overflows the sill atthe Denmark

Strait,

onefinds that its

dynamics

and

path

aretheresultofacombination of the effects of

atmospheric

influence,

topography,

tides and the Coriolis force. The effect of the Earth's rotation has notbeenmentioned sofar. Itsinfluence is shownin

figure

2.3 where the

overflowing

waterof the Denmark Strait is seen onthe

right

hand side of the passage

in the direction of the flow. The overflowfollows the continental

slope

around Greenland

where itencountersthe outflow _component of the Iceland-Faroe sill and that of the Faroe

bank channel

(fig.

2.4).

Hence it isseenthat these overflows makea

significant

contributionto theocean

budgets

through

the _transport of

salinity,

temperature,

anthropogenic

substances,

inorganic

and

organic

matter.

2.5

Surface Currents

Surfacecurrentsare afurther

example

of

gravity

currentswhichoccurin the environment. However many surfacecurrents _{may be described}ascontinuousand thereforenotexhibit the characteristic head associated with the

gravity

currents described in the

previous

sections.

Despite

this the

dynamics governing

the flow and

large

scale featuresobserved

CHAPTER2 _{Environmental}

Applications

FIGURE 2.3. DenmarkStraitoverflow_temperaturesectionat ~65N. Note the Arctic Bottom Water

overflowonthe r.h.s. of the_channel.

(From

Worthington,

₁₉₆₉₎

50N

40N 30W W0E

FIGURE 2.4.

Map

showing

thecoursetaken

by

the Arctic Bottom Water

overflow,

where 1000mcontour

deep

convection

in the

laboratory

_{may be}

applied

to these flows. Oceanic

examples

include the

Algerian

Current

(Millot

et

al,

1990),

the East Greenland

(Wadhams

et

al, 1979)

and the

Norwegian

Currents

(Me

Climans et

al,

1985)

and the Leeuwin Current south of Australia

(Griffiths

&Pearce, 1985).

The use of satellite

imagery,

such _as, AVHRR

(Advanced Very

High

Resolution

Radiometer),

in the

study

of these flowscanresolve

temperature

differencesof 0.1C over

> lkm2therefore

enabling

real time features of the current to be discerned and tracked. Ocean colour

images

utilize

biological

_tracers, such as,

phytoplankton.

This

provides

information on the first 10 - 20m of the water column

depending

_on the

particular

attenuation

length

of thewatersconsidered. The oceancolour

signature

is often

coupled

with the

physical

_processes associated with the current and therefore can

provide

information on ocean

circulation,

boundaries betweenwatermasses and the evolutionof eddies.

Much

instability

_{has been documented in observations of the Leeuwin Current}

(Griffiths

_et

al, 1975).

The Leeuwin Current

originates

nearthe NorthWest

Cape

of Australiaat 22S

at the south west corner of the continent it rounds the

Cape

Leeuwin before

finally

continuing

acrossthe Great Australian

Bight.

The

growth

of eddieswasfoundto take 10

-15

days,

i.e.,

12to 17inertial

periods.

This is consistent with

experimental

investigation

of

gravity

currents

by

Griffiths &

Linden,(1981, 1982)

_{which found the disturbances took 10}

to 20 inertial

periods

to

develop. Figure

2.5 shows the

eddy

structure of the Leeuwin Current. The eddieswere foundto have zero

phase

speed

relativeto _{the current, also in}

agreement with

laboratory

studies.

Griffiths,

assuming

a Froude number of 1 for the Leeuwin Current in accordance with

laboratory

results,

calculated a

velocity

of0.9 to

1.4ms"1

basedontheparametersof the Leeuwin current. This is consistent with the results ofa_{survey carried}outin

1982,

which measuredamaximum

velocity

of between 0.5 to 1.8

ms"1

atthe

edge

of thecurrent. The_greatest

velocity

was measured at

Cape

Leeuwin where the

topography

_mayhave

imposed

critical conditionsuponthecurrent. The width of the

current wasfound to be four or more deformationradii. This is_greaterthan the widths

observed in the

laboratory by

Griffiths &

Hopfinger

(1983),

which

suggested

that the

currentwidth

approximated

to the deformation radius. However

dissipation

by

interaction

CHAPTER2 Environmental

Applications

S0flfl7>"63

imvv **w'**

47

*_*'%*' "

i

1

2^f^i4 ^'^

82

,.-i

IT/

FIGURE 2.5. Leeuwincurrent

imaged

using

N0AA7

AVHRR,

on30th

Sept

1982. The satellite

image

clearly

distinguishes

the

eddying

motions within thewarmLeeuwincurrent

(darkest water),

with

topography

could have caused an increase in width

through

release of

potential

energy.

Large

scale disturbances are commonfeatures of surface currents, indeed the

boundary

between the

Norwegian

and EastGreenland Current is marked

by

shear

generated

eddies of 10 - 20km in

diameter,

which

again

have a

significant

life span of20 - 30

days.

Instabilitiesarealso observed in the

Algerian

Current

(Arnone

et

al,

1990),

see

figure

2.6. Eddies are

responsible

for the

transport

of nutrients and marine

organisms

as well as

physical properties.

Griffiths

(1985)

established that thewarmeddies within the Leeuwin Current were

responsible

for

removing

10 % of thewarmwater off the south Australian

coastin the

spring

of1982.

CHAPTER2 Environmental

Applications

I

>35.64

mm

288144

CHLOROPHYLL

I

34

(mg/m:

09

I

I

04

FIGURE 2.6.

Algerian

Current

imaged using

Nimbus CZCS between

April

27 and

May

_11, 1986. Beneath the

images

a non-linear calibration

wedge

of

chlorophyll

concentration is shown. The

chlorophyll

acts as atracer

clearly distinguishing

thewater masses and the

dynamics

within the

CHAPTER

3

Theoretical framework

3.1 Literature

review

First in

importance

ina

study

of

gravity

currentsis the

non-rotating theory

of

Benjamin

(1968).

The

beauty

of

Benjamin's theory

is its

simplicity.

He avoids the

non-hydrostatic

region

in the

vicinity

of the noseof the current,

by

equating

the forces

acting

onthe flowat_upstreamanddownstream

sections,

and the fluxes in and outof this

region.

For the

dissipationless

casehe

predicts

a

unique velocity

and

depth

of the flow.

Benjamin's

non-rotating theory

forms the foundation _{upon which the}

subsequent

theories of

rotating

gravity

currents

(Stern

1982,

Nof

1987,

Hacker

1997)

are based. It is examined inmore detail in

§

3.1.1. His extensionto include

dissipation

is discussed

separately

in

§

4.1.1.,

where it forms thelimittothe

rotating

_{energy loss}

theory.

The aim of this literature review isto consider the relevance and limits of

previous

models associated with

rotating

gravity

currents. Authors in the field of

rotating gravity

currentsinclude: Gill

(1976);

van

Heijst

(1985);

Stern

(1982);

Nof

(1987);

and Hacker

(1996).

Gill

basically

extended the

'Adjustment

Problem' to the

rotating

case. The initial conditionwastakenas a

step

function. The

hydrostatic approximation

wasmade and hence the shallowwater

equations

corrected for rotation were

applied.

The

non-rotating

solutionwasassumedtobe

applicable

to avery narrow,

rotating

channel. The

solution

approximated

tothe sumoftwo wavefronts

moving

in

opposite

directionsat

phase

speed

(gH)2,

where//is the total

depth

of fluid. In the

rotating

casethe solution was taken as the sum of Poincare modes andtwo Kelvinwaves. The Kelvin waves

propagated

at a uniform

speed along

opposite

sides of the channel. The

resulting

disturbancewas

strongly asymmetrical.

Gill

subsequently

considered

hydraulic

control ina

rotating

_system

(1977).

CHAPTER3 _Theoretical

framework

Gill's work is not

directly

relevant to the _present

study,

as it is aone

layer

shallow

watermodel. However the actual

geostrophic adjustment

_process is utilised in

§

5.1

where the initial

potential

vorticity

distribution is

incorporated

_{into the energy}

conserving theory.

This method is similar to that used

by

van

Heijst (1985).

He considered the

adjustment

ofa

three-layer

_systemfromrestwithina

rotating

reference frameto astate of

equilibrium,

after the removal ofa

hypothetical

barrier

separating

the

density

fronts. His work formsanextensiontothetwo

layer adjustment

models of

Csanady

(1971, 1978),

Stommel and Veronis

(1980),

Ou

(1983)

and Hsueh and Cushman-Roisin

(1983).

The method

firstly

considers conservation of

potential

vorticity

between the initial and

adjusted

state in each

layer.

Margules' equation

(5.1.13),

is then usedto determine the

relationship

between the

slope

of the interface betweentwo

layers

and the

velocity jump

acrossthe interface. It is assumed that the channel hasa

rigid

lid. From these

relationships

van

Heijst

derives

general

solutionsfor the

depth

of each

layer,

whicharesolved

by

considering boundary

conditions

particular

to the

equilibrium

state and conservation ofmomentum and mass. In

§

5.1 this method is

applied

to

give

the

general

solution for the

depth

of theinterface.

Sternetal

(1982) presented

atheoretical model of

rotating gravity

currentsbased on

laboratory

observations and Stern's

previous

work

(1980).

He

proposed

that the

currentwidth could be divided intoa

steady

and

unsteady region

seperated

by

what he

calleda

'dividing

streamline'. Conservation of volume flux within this

region

wasnot

used,

instead heformulatedan

expression

for what he calledadetrainment

coefficient,

thatwasbasedonthe ratio of the volume_transportto the absolute

transport.

By

using

the shallow water

equations

and a

generalisation

of his

previous

work

(1980),

he derivedasolution for

long

waves on auniform

potential vorticity

current. Stern found that there weretwo

possible

setsof solutions. These he

interpreted

as either a

wedge

orbore

intrusion,

based onthe

shape

of thecurrent. The

wedge

solutionwas discarded as it was not consistent with his

laboratory

observations. He

proceeded

with a

particular

bore solution which he called the

'limiting

bore'. This

predicted

amaximum

width,

L,

for all intrusive bores of0.413 h=

0.516,

whereL =

L*fl(g'h*f

i.e. the

ratio of the measuredcurrentwidth to the

Rossby Radius,

based onthe dimensional

current

depth.

The

width,

nose

speed

and the detrainment coefficient wereall found to

be

essentially independent

of the

prescribed potential vorticity

andthefinite

depth

of

the lower

layer.

Thenon-dimensional nose

speed

orinternal Froude number

Fr(h)

=

c*/(g'h*y=

1.57 and thedetrainment

coefficient,

5 =_{0.32. Stern}

recognised

that the

long

wave

theory

would fail when the first shockformedat thenose and he

envisaged

that the shortwave

theory

would alter and accelerate thenose

region.

He assumed this would have no effect _upstream in the intrusion. Griffiths

(1986)

questions

the

assumption

ofno

upstream

influence. It is

surprising

that Stern

rejected

the

'thinning

wedge'

solution in favour of the

'limiting

bore',

becausein_my

experiments

a

strongly

rotating

current does havea

wedge-like profile

and the

propagation

rate _{does appear} to increase towards the nose. Stern also found that the theoretical nose

speed

was

unaffected

by

the fractional

depth

of the current whereas in the

non-rotating

case

Benjamin

found that there was a

strong

dependence

onthe finite

depth

of the lower

layer.

In

§

7 I

present

the results of_my

laboratory investigation

which covers awide range of

depths

and rotationrates.In contradictionto SternI have found that the nose

speed

is sensitivetothe fractional

depth

of thecurrent. Stern's workis an

interesting

forerunnertothe

contemporary

theories of Hacker and Nof. However its weakness is

that it

only

considers a

shallow, strongly

rotating

current of limited

width,

on which there isno_upstreaminfluence. HereI

develop

a

theory

thatcoversall levels of rotation

and fractional

depths.

Nof

(1987)

criticised theuse of the

long

wave

equations

in Stern's

theory,

since the

current is

non-hydrostatic

in the nose

region

and instead used a similar method to

Benjamin applying

conservation of _energy, momentum and

continuity

to a control volume

connecting

the flow behind and ahead of the nose. His model includesafree surface and considers a current of

wedge-like

cross-section,

with zero

potential

vorticity

and finite

depth.

Hederived five

algebraic equations

intermsof five unknowns

and solved them

using

an

algebraic manipulation

_{programme known} as

'Macsyma'.

Steady

solutions werefound foralimited range of finite

depths

where the fractional

depth

varied between ~ 0.65_to 1

(Nof, 1987).

The

propagation

rateof the nose was foundtobe

approximately

_constant,

increasing slightly

with

increasing

fractional

depth

from 0.81

l(g'hf

to

0.%24(g'hf-

where h is the

depth

of thecurrent. Fora

particular

fractional

depth

a

unique

current width was

predicted

which varied between0.724

CHAPTER3 Theoretical

framework

(g'hf/f

to0.772

(g'hf/f.

Unfortunately,

Nof found that

steady

solutions didnot exist

foran

infinitely deep,

broad ocean, so it isnot

possible

to _{compare his results}

directly

with those of Stern. However his resultscanbe

compared

with those of Hacker

(1996).

Hacker used a similar

approach

to Nof. He assumed that the fluids

(ambient

and

current)

wereimmiscible and

inviscid,

and that the flowwas

steady.

This enabled him

to meetthe conditionsnecessaryto

apply

conservation of momentum, energy, volume

flux and

potential vorticity

between the_upstreamand downstream cross-sections and

hence avoid the

hydrostatic problem.

Hacker did not assume zero

potential vorticity

and he introduced a

rigid

lid. He

applied

the

governing equations describing

the

structure of the flowtothree flow

geometries,

based onthe width of thecurrent asthe

rotation rateincreases. Foreachcase asolutionwas found for the

speed

of thecurrent.

He

provided

a

steady

solution for all levels of

rotation,

with asmooth

progression

between the

non-rotating

and weak

rotating

case. Nofs solutionscan be

compared

with Hacker's solutions for intermediate rotationrateand

despite

the inclusion ofafree surface and the

assumption

ofzero P.V. in Nofs

analysis

the results compare

quite

well. While Hacker's

theory

doesnotinclude

dissipation

and the effect of the

potential

vorticity

distribution at the source

region

onthe

ensuing flow,

it is

by

far the most

comprehensive.

Hence it is Hacker's _energy

conserving

theory

whichIhave chosen

toextend. The inclusion of _energy loss and theP.V. distributionare addressed in

§4

and

§5

respectively.

The framework _upon which these extensions are made is summarised in

§3.2..

Before

proceeding

with the effects of

rotation,

the

non-rotating

energy

conserving

theory

of

Benjamin

is considered in more detail.

Benjamin's

solutions

provide

the end

points

(W=0)

for both Hacker's

rotating

_energy

conserving

theory

and the _energy loss

theory

described in

chapter

4.

Indeed,

it is shown in

§

4.2.1.2 that the

governing equations

of the _energy loss

theory

reduceto those of the

non-rotating

case described in

§

3.1.1.

3.1.1

Non-rotating

_energy

conserving theory

Benjamin's

model ofa

gravity

current considers the

analogous

flow of a

steady,

inviscid fluid of

density

_{p, past} a

cavity (fig.3.1).

The

cavity

maybe filled with airor

empty.

The flow is confined betweentwo horizontal

planes.

_{The pressure} atthe free

surface is takenas zero.

Upstream

the fluid has

depth,

H,

and

propagates

ataconstant

velocity,

cx. Beneath the

cavity

far downstream the flow is uniform with

depth,

A,

and constant

velocity,

_c2.

Fig.

3.1

Benjamin's (1968)

model of the flow ofaninviscid fluid_pasta

cavity

of air.

Benjamin

obtains an

expression

for _c2

by applying

the Bernoulli

equation along

a streamline

connecting

the

stagnation point,

0 and a

point

downstream on the free

surface,

where

c22=2g(H-h)

(3.1.1)

To obtain the_pressureonthe_upper

boundary

the Bernoulli

equation

is

applied

between the

stagnation point

and the flow far

upstream,

hence

(3.1.2)

Since the _pressure in the

liquid

far _up and downstream varies

hydrostatically

with

depth,

conservation ofmomentum between the two

locations,

or the 'flow force balance' as

Benjamin

refersto

it, yields

(3.1.3)

CHAPTER3 Theoretical

framework

themomentumflux. The r.h. is thesamebut

applies

tothe downstream flow.

Applying

the

continuity equation, gives

asecond

equation

for the downstream

velocity,

_c2.

cxH=c2h,

(3.1.4)

This is

equated

with

(3.1)

toobtaina

quadratic equation

for h interms of//. Theroots

of this

equation

areh=H and

h=

IK

(3.1.5)

Therefore the condition of_energy conservation dictates that the

receding

flow must

occupy half the full

depth,

H.

Substituting

(3.5)

and

(3.4)

into

(3.1) gives

the Froude numbers for the_upand downstream

flows,

(3.1.6)

=

(2i

(3.1.7)

where the

receding

flow is

supercritical.

3.2

Governing equations

3.2.1 Model

description

The energy

conserving

model of Hacker is based on three flow

geometries

as the rotation rate is increased

(fig.

3.2).

In case A the current fills the full width of the

channel,

'weakrotation'. In caseB thecurrent detaches from the left hand wall and

outcropsonthe free

surface,

at

position

d. For_{strongrotation rates,} case

C,

thecurrent

fills the full

depth

of the

channel,

outcropping

on the bottom of the tankat

position

b.

CASEA

CASEB

CASEC

UD(y).

y =_b

FIGURE 3.2. Flow

geometries

for3 levels of

background

rotationinthe

steady

reference frame. The surface

gravity

currentis shown in

blue,

whilst the ambient fluid is clear.

Upstream

in the ambient fluid thereis no flow. The interface between thecurrentand ambient fluid_propagates at

speed

c

parallel

tothe channel walls and floor.Hence,the

gravity

current_appears

stationary

toanobserver

moving

with current, whilst the

oncoming

ambient fluid has

speed

c. This is knownasthe

steady

reference frame.

CaseA

(low rotation)

-currentfills the full width of the channel

CaseB_(moderate

rotation)

- current

outcropsonthe upper

boundary

_dXy=d

Case C

(strong

rotation)

-current

outcropsonthe_upper

boundary

and fills the full

depth

of

channel,

intersecting

the bottom

boundary

at_y=b.

Structure of the flow is defined intermsof:

u0

velocity

of the ambient fluidatthe r.h. wall Tlo

depth

of the ambient fluidatthe r.h. wall

speed

of translation of the reference frame downstream

velocity

in the ambient fluid

velocity

of the ambientfluidat_y

depth

of the ambient fluid_aty

CHAPTER3 Theoretical

framework

Themodel considersasurface

gravity

current. The interface of thecurrent

propagates

ataconstant

speed

c. The reference frame is

rotating

at

angular frequency =//2

and

translating

at

speed

_c, hence the

gravity

current_appears

stationary.

Within thecurrent

there is no flow. The role of thecurrent is

merely

to

produce

the_pressure

gradient

necessarytodrive the flow in the ambient fluid beneath the current. There isno

mixing

between thetwo fluids. Downstream the flow in the

gravity

current is

parallel

to the

walls and floor. The channel hasa

rigid

lid and the

Boussinesq

approximation

is made. The derivation of the energy

conserving theory

is broken down into three

stages.

In

Stage

1 the

governing equations

are

derived,

which describe the structureof the flow in the ambient fluid intermsof four_parameters:

o -

velocity

at the

right

hand wall.

r)0 -

depth

atthe

right<