Surface Marks of the Sea Bed, A hydrodynamic theory for the imaging of bottom topography by X- and K-band radar

surface marks of the sea bed

a hydrodynamic theory for the imaging of

bottom topography by X- and K-band radar

Stéflingen

behorende bij het proefschrift Surface marks of the sea bed

De instelling van veel oceanografen orn kennis van vissers en zéisters over de zee te negeren, betekent het mislopen van een vette vangst.

Omdat onze 'fysische intü!tie' gebaseerd is op lineair denken is

het, vóór het doen van berekeningen aan een niet-lineaire vergelijking, zinvol orn wat te jongleren met centrale aspecten van die vergelijkiñg.

(dit proefschrift)

In het geval dat zowel korte als lange golven op zee aanwezig zijn, geldt enerzijds op theoretische gronden dat de korte golven

geadvecteerd worden met de orbitaalsnelheid van de lange golven, terwijl anderzijds radarbeelden laten zien dat het patroon van

eriergierijke korte golven zich verplaatst met de fasesnelheid van de lange golven. Het is ten onrechte dat nog ved oceanografen in deze

situatie eèn parádox zien, dié zôu stoelen op fouten. Oer8tijging vañ-dït standpüht wòrdt bereikt door te erkennen dat er spràke i

van twee ,erschillehdesnèlheden: de ¿nélhéid van het patrodn en die yan de korte golven (vergeiijk met de ouderwetse wit met rode

draalende kapperszuilen).

(Shemdln et al. 1986, TOWARD field experiment interim report, JPL, California)

4 _{Eén van de basisvragen. betrefiende watergolven, namelijk in hoeverre}

korte fourler-componenten van het wateropperviak vrije golven zijn, is nog niet opgeiost.

(B.M. Lake & H.C. Yuen 1978, JFM 88, pp. 33-62; H. Mitsuyasu, Y.Y. Kuo & A. Masuda 19?9, ,JF1 92,

Het is wenselijk dat

f bewezén wordt dat de continutteitseisen aan

het grensvlak van twee niet-visceuze vÏoeistoffen als limiet volgen

uit de continúlteitneisen voor visceuze vloeiàtoffen, òf dat het

tegendeel bewezen wordt.

De voorwaarde van Banner & Phillips (19714) voor het breken van korte

golven, nanielljk dat locaa]. de snelheid van de waterdeeltjes groter

is dan de snelheid van het golfprofieì, is niet juist. Zij geldt

slechts als noodzakelijke, maar niet voldoende, voorwaardevoor één

manier van breken, nl. die welke vergelijkbaar is met die van lange

golven dIe het strand op lopen.

(M.L. Banner & 0.M. Phillips 19714, JFM65, pp. 6147-56).

Met verstand zó te gebrulken in de wetenschap als daar gebruikelljk

is, is ongeveer even weinig doeltreffend als je magisch zwaard i-n de

schede laten en met een keukenmesje ten strijde trekken.

(Variant op een uitspraak van Benjamin Hoff.

In: Tao of Pooh, Dutton, New Yòrk, 1982).

Voor

nieuwe

inzichten

in

de

niet-lineaire

eigensohappen

van

watergolven is het denken in termen van fourier-componenten

belemmerend.

Het

gebruik

van

mOmenten orn

turbulentie

in

een

stroming.

te

beschrijven is nlet.meer inspirerend. Het is tljd voor een nieuwe

invalshoek, waarbij de structuur van de stronhing voorop staat.

De

rol

van

de

natuur'kunde

als

objectieve waàrheidazoekster

is

uitgespeeld. Haar nieuwe rol zal zij delen met al onze andere kennis

over het zijn op aarde.

Aarinemende dat 'de slag bij Waterloo gewonnen is op de sportvelden

van Eton' (Duke of Wellirigton)o kunnende ver].iezers, orn devOlgende

slag te wnnen, het beste-alle mogelijkhederi en onmogelijkheden

opzij zetten en alsnog gaan sporten op de velden van Eton.

-Golven zijnmooi.

0

150

Ultriodlgthg

voor het bijwonen van mijñ promótie op 3 Juni a.s. orn 1I.3O uur, en de receptie naafloop. Beide vinden plaats in het Academiegeboùw, Domplein

29, Utrecht.. Een roútebeschrijving ¿tàat öp de achterkant _van deze bladzijde..

Klaaz-tje van Gâstel Zandhof'sestraat 11 7 3562 GD Utrecht

surface marks of the

sea bed.

a hydrodynamic theory for the imaging of

bottom topography by X- and Kband radar

tekenen van de zeebodem aan het wateropperviak

(met een inleiding en samenvatting ¡n het Nederlands)

FROEFSCH RIFT

ter verkrijging van de graad van doctor in de wiskunde

en natuurwetenschappen aan de Rijksuniversiteit te

Utrecht, op gezag van de rector magnificus prof.dr. J. A.

van Ginkel, volgens besluit van het college van dekanen

in het openbaarte verdedigen opwoensdag 3juni1987

dès hamiddas te 14.30 uur

door

KLAARTJE VAN GASTEL

promotor : Prof.dr. J.T.F. Zimmerman

co-promotor: Dr. G.J. Komen

Dit onderzoek is uitgevoerd met f inanciële steun van de werkgroep Meteorologie en Fysische Oceanograf ie van de Stichting voor Zuiver Weterischappeiijk Onderzoek (ZWO).

Table of contents

i Introduction 13

On phase velocity and growth rate of 17

wind-induced gravity-capillary waves

Abstract 17

.2.1. Introduction 17

2.2. Theory 20

2.2.1. Methods arid equations 20

2.2.2. Basic flow and Perturbation stream 23

function in water

2.2.3. Basic flow and pérturbation stream 25

function in air

2.2»4 Phase velocity and growth rate 27

2.3. Discussion . 31 2.3.1. Growth rates 31 2.3.2. Phase velocities 37 2.l. Main conclusions 40 Appendix 42 References 44

Nonlinear interactions of gravity-capillary waves: 117

Lagrangian theory and effects on the spectrum

Abstract l7

3.1. Introduction 147

3.2. The interaction equations 50

3.3. Multiwave space . 61

3.1!. Symmetries within the triads 70

3.5. Discretization 71 3.6. Spectral devèlopment 711 3.7. Conclusions 81! Appendix 3.A. 85 Appendix 3.B 86 References 87

14 _{Imaging by X-band radar of bottom topography and}

91

internal waves: a nonlinear phenomenon

Abstract

4.1. Introduction 4.2. Model equations

14.3. _{Quasi-equilibrium spectra at short fetches} Spectra in a slowly-varying current field 4.5. Conclusions References 5. Inleiding en samenvattÏng Curriculum vitae 125 91 91 93 96 102 113 115 119

Chapter 2 has been published as:

Van Gastel, K., Janssen, P.A.E.M. & Kornen, G.J.

_1985.

J. Fluid Méch.

161 pp. 199-216.

2 3 4 g :UNITED . j:KING DOTM l'3D 'E 4.2 A 51'30'N 2E FRANCE

cg

D 230'E

/

I

2E ENGLISH CHANNEL

FIG

I 0.3

'n-0.5

Hl

J

I

?30 E 0.5 DUNKERQUE OOSTENDE

-t

BELGUIM CONTOURS IN METERS CURRENT SPEEDS IN METERS PER SECOND

I I I I I I I I

0.9 3'E

Fig. 1.1: (a) A radar image of the southern bight of the North Sea. The different

grey tones can be related to the local depth. To illustrate this a depth chart of

short

seabed

surface

_{-9 surface -* backscatter}

internal waveJ i

current

i

_waves

I I I _I

I.

measurements

I

van

¿.astel (1986)

I

ampification :1-2

energy balance

I

wind input

nonlinear interactions I

Ch4: solutions of energy.

balance

I

amplificatión:100 .1000

Neasrements

by

Kwoh et al. (1986)

amplification:10

total amplification

1000_10,000

Fig. 1.2: Chain by which modulations of the inteñsity of the radar image can be

retreated to changes in the local depth. Indicated is where and how the links are

treatéd in this thesis, supplemented by the work of Kwoh et al. (1986). and the amplification for each link as found in these studies. The total amplification is in good agreement with data.

Introduction

In

1969

de Loor discovered that on X-hand radar images of the

sea Uem

3 cm) bottom topography could be identified (de Loor

1981).

Since then this phenomenon has received much attention, mainly due to its occurrence on SAR-images (Synthetic Aperture Radar) taken from the

satellite SEASAT (Fu & Holt

1982,

Phillips

1985,

Valenzuela et al.

1985,

Halsema et al.

1986).

Figure 1.1 is an example of this imaging. It was quickly recognised that not only bottom topography could be imaged this

way but also internal waves and large-scale current structures like eddies (Fu & Holt

1982).

The imaging cari be performed by K- to L-band radar, A = 8 to

200

mm.

How this imaging is possible was a mystery. The electromagnetic waves do

not enter the sea but are reflected at the sea surface, thus a direct modulation is impossible. The quantity determining the backscatter at

the sea surface is the energy of surface waves having about the electromagnetic wavelength, thus the radar Images imply an indirect modulation via the short surface waves. However, a modulation of these

short waves was also a surprise, because they are too short to feel the

bottom dirèctly and the current gradients are too small to expect significant refraction effects on the surface waves. Most surprising of

all the modulation of these waves can be extremely large: under light

winds contrasts of a factor 1000 and more have been seen, even when the

change in surface current is lessthan 25% (Valenzuela et al.

1985,

Kwoh et al.

1986,

Halsema et al.

1986).

The modulation chain can hé split into three steps, indicated in figure

1.2.

Elswhere, I have analysed measurements on the first. step, the

modulation of surface current by a change in depth (Van Gastel,

1986).

Two conclusions of this study are important to the imaging mechanism: 1) the surface current can be slightly out of phase with the depth

modulation, arid 2) the arnplication of the modulation factor is small in this step: less than 2.

In this thesis an account is given of my theoretical research on the second step, the modulation of short surface waves by surface current. The parameter space of the second step is generated by the quantities wind speed, surface current speed and wavelength of the surface waves.

Here only low wind speeds, less than 5 m/s, and millimeter and

centimeter wäves are considered. This is the regime for which the very

large contrasts on the radar images occur. A restriction of my analysis is that it is one-dimensional.

The idea behind the treatment of this step is first to find an energy balance for waves of 0.5 to 15 cm and then to solve, this balance.

Effects I want to include in the balance are refraction, input of energy by wind, dissipation and nonlinear interactions between the short waves.

In chapter 2 a term describing the input of energy by wind, in other

words the generation of waves by wind, is found. New s the nearly

analytical description of this term. Chapter 3 deals with the nonlinear interactIons in triads of short surface waves. Many questions and

problems encumbering the incorporation of this term in an energy balance

are being solved here. In chapter I an energy balance is completed, by finding a description of the dissipation, and solved. As not much is

known of the dissipation rough estimates have to be used for this term. New items are the dissipation functions and thé solutions of a nonlinear

balance. Former theories (Hughes

1978,

Lyzenga et al.

1983,

Alpers &

Hennings 19811, Ermakov & Pelinovsky 19811 and Yuen et al..

1986)

all use balances witIiout the nonlinear term.

The solutions of the so-constructed energy balance show modulations of the energy by slowly-varying currents of a factor 100 to 1000. These large modulation factors are only found for k

260

m1. Together with the results of Van Gastel

(1.986)

on the first step of the. modulation mechanism and of Kwoh e.t al.

(1986)

on the third step, from surface

waves to backscatter, this thesis gives a description of the imaging of bottom topography by X- and K-band radar, for low windspeeds and

wavenumbers smaller than 260 m1. It is the first theory to predict correct order of magnitude of the modulation. This agreement with data

is due to the incorporatiòn of the nonlinear interactions in the energy balance.

References

Alpers, W. & Hennings, I. 19811. A theory of the imaging mechanism of underwater bottom topography by real and synthetic aperture radar.

J. of Geophys. Research, 89, C6, pp. 1O.529-1O.5'46.

Ermakov, S.A. & Pelinovsky, E.N. 19814. Variations of the spectrum of wind ripple on coastal waters under the action of internal waves.

Dyn. of Atmospheres and Oceans, 8, pp. 95-lOO.

Fu, L. -L. & Holt, B. 1982. Seasat views oceans and sea ice with

synthetic-aperture radar. JPL Pubi. 81-210.

Gastel, K. van 1986. Velocity profiles of tidal currents over sand

waves. Royal Netherlands Meteorological Institute,

De But, The

Netherlands.

Halsema, D. van, Gray, A.L., Hughes, S.J. & Hughes, B.A. 1986. C- and Ku-band scatterometer results from the scattmod internal wave experiment. IGARSS '86 symposium, ESA SP-254, pp. 311-317.

Hughes, B.A. 1978. The effect of internal waves on surface wind waves.

2. Theoretical analysis. J. of Geophys. Research, 83, Cl, pp. 1155-1165.

Kwoh. D.S.W., Lake, B.M. & Rungaldier, H. 1986. IdentIfication of the

contribution of Bragg-scattering and specular reflection to X-band

microwave backscattering in an ocean experiment. TRW, Space & Techn. Group, California.

Loor, G.P. de 1981. The observation of tidal patterns, currents, and bathymetr' with SLAR imagery of the sea. IEEE journal of oceanic

engineering, 0E-6, no. 14

Lyzenga, D.R., Shuchnan, R.A., Kasischke, E.S. & Meadows, G.A. 1983.

Modeling of bottom-related surface patterns imaged by synthetic

aperture radar. IGARSS'83 symposium.

Phillips, O.M. 19814. On the response of short ocean wave components a

fixed wavenumber to ocean current variations. J. of Physical Ocean.

111, pp. 11425-11433.

Valenzuela, G.R., Plant, W.J., Schuler, D.L., Chen, D.T. & Keller, W.C.

1985.

Microwave probing of shallow water bottom topography in the Nantucket Shoals. J. of GeOphysical Research 9O _{no. C3, pp.}

14931-49142.

Yuen, H.C. Crawford, D.R. & Saffnan, P.G. 1986. _{SAR Imaging of bottom}

topography in the ocean: results from an improved model. IGARSS'86, ESA SP-254, pp.

807-812.

2

On phase velocity and growth rate of:

wind-induced gravity-capillary waves

Abstract

Generation and growth of gravity-capillary waves (À = 1 cm) by wind are

reconsidered using linear Instability theory to describe the process.

For all friction velocities we solve the resulting Orr-Sommerfeld

equation using asymptotic methods. New elements in our theory, compared

with the work of Benjamin (1959) and Miles (1962), are more stress on

mathematical rigour and the incorporation of the wind-induced shear

current. We find that the growth rate of the initial wavelets, the first waves to be generated by the wind, is proportional to u.

We also study the effect of changes in the shape of the profiles of wind

and wind-induced current. In doing this we compare results of Miles

(1962), Larson and Wright (1975), Valenzuela (1976), Kawal (1979a),

Plant & Wright (1980) and our study. We find that the growth rate is very sensitive to the shape of the wind profile while the influence of changes in the current profile is much smaller. To determine correctly the phase velocity,

the value of current and current shear at the

interface are very Important, much more so than the shape of either wind or current profile.

2 1. Introduction

Recently interest in generation, growth and equilibriúm of gravity-capillary waves has been renewed owing to the growing importance of

remote sensing of the sea surface. Microwave-radar backscatter is

determined largely by the energy density of waves with wavelengths of the order of ¿-'l0 cm (Raney et al., 1985). In order to áscertain the

energy density one needs knowledge of sources and sinks of energy and of kinematical quantities like advection and refraction.

y(mm) 0.8- 0.6-y1 0.4-I I I I 0.6 0.8 1.0 1.2

U(m s-')

Fio. 24 Wind speed and current as a function of height. In air a linear-logaritimió profile is drawn; in water , Miles constant profile (Miles took U, = 0, in this figure we took = 0.75 m - - - -. Valènzuela's linear-logarithmic profile; --. Kawai's error-function-like profile; , our

exponential profile. On the vertical scale typical values for various quantities are indicated. À is a wavelength, g,, a thickness of the viscous sublayer in air, y, a critical height, 'i, a wave amplitude and Yiw a thickness of the viscous sublayer in water as assumed in Valenzuela's profile.

In this study önly part of this intricate process is considered. We concentrate on the initial generation and growth of gravity-capillary

waves under the influence of the wind.

Most of the recent studies on this subject use the linear-instability

theory as presented by Miles (1957). In 1959 Benjamin made an analytical

study of the flow over a wavy boundary. He looked at the flow over a

rigid surface, in this way decoupling the flows in the two media. Although he noted the possibility of generalizing this theory to the

0.2-Y0 'io

-0.2_

-0.4- -0.6- -0.8-y -1.0-i 0.2 I 0.4 y(mm)

-3

-4

-5

-6

-7

flow over a fluid and determining growth rates, he did- not càrry out

such a programme. Miles (1962) did apply Benjamin's theory to the growth of gravity-capillary waves by wind. He used a linear-logarithmic flow in air, the profile of which is drawn in figure 2.1. In accordance with

Benjamin he assumed the water to be at rest. It may be noted that these two flows do not satisfy the equation for continuity or shearing stress at the boundary of two fluids.

Valenzuela (1976) numerically solved the equations using a coupled wind-current system satisfying the continuity equations. For the wind as well. as for the current he assumed a linear-logarithmic profile.

Kawai (1979a) extended the research to the generation of

gravity-capillary waves by combining numerical and experimental work. He measured the flow at the moment the initial wavelets- appeared and their growth rate, phase velocity and frequency. His numerical work describes these measurements. He uses a coupled wind-current system, an error-function-like current profile (drawn In figure 2.1) and the usual wind

profile.

In the next section our analytical analysis is presented. We use the linear-logarithmic profile ±n air and an exponential profile in water (figure 2.1). We have chosen this profile because it closely resembles, Kawai's profile and because it allows for an exact solution of the

Rayleigh equation. We briefly discuss the derivation

of the

Orr-Sommerfeld equation plus boundary conditions as a description of the

growth of gravity-capillary waves. We then solve these equatioñs

asymptotically. Asymptotic analysis makes sense becaUse the density of air is small compared to the density of water and because reasonably

large Reynolds numbers can be defined in air and- water. We find expressions for the phase velocity and growth rate of the waves.

It is inherent with the asymptotic methods that we are able to indicate to what order each expression is correct. This is an improvement ori the

wave-growth theories of both Benjamin and Miles. We are even able to Indiôate the order of the errors in Miles' expression for the growth.

Another improvement of our analysis is that the main flows satisfy the

continuity equations. This means that formal justification for our expressions exists, in contrast to the cases of' Benjamin and Miles.

Numerical results of our añalysis are given in section

_2.3.

We also

study how sensitive the phase velocity and growth rate are to changes in

shape of the profiles of wind and current, by comparing the results of Miles, Valenzuela, Kawai and our study: each is based on a different

profile. For verification we use the experimental laboratory results of Kawal (1979a), Larson & Wright (1975) and Plant & Wright (1980).

2.2. Theory

2.2.1.Methods and equations

The growth of waves on the interface of water and air can be seen as the

perturbation of the equilibrium consisting of a plane interface and uniform basic flow in air and water. Physically, the description would

be as follows. The wind sets in and after a few seconds the upper layer

of the water starts to drift with the wind. These flows, both strongly sheared near the interface, are unstable and after another few seconds ripples start to appear (see Kawai 1979a). In this initial stage the growth of the waves is exponential; after a further few seconds other

mechanisms come into effect and saturation sets in. In a final stage the

wind and current profile would be modified by the constant flow of

energy from, the air towards the waves. In this chapter we confine ourselves to the initial stage of wave growth, where instability and viscous damping are the only energy sources. Keeping this in mind a

mathematical description of the growth of the waves can be given. Growth

is then desöribèd as an instability of the equilibrium in the

normal-mode analysis. For simplicity the situation is assumed to be uniform in

one horizontal direction: as we are interested in plane-parallel flow

this does not diminish the possibility of finding growth (Drazin & Reid 1982, p. 155). In a later section the description of the wind and the

to be shear flows satisfying at the interface the usual continuity

equations of normal and tangential velocity, shearing stress and normal pressure (Batchelor i981 pp. 11815O). The equation for the continuity

of shèaring stress will be of importance. It reads:

a 0) 'w U'(0)

, (2.1)

where p is the viscosity U is the velocity of the basic flow in the horizontal direction and a prime denotes differentiation with respect

to , a dimensionless height coordinate equal to the product of

wavenumber (see eq. (2.6)) and height y. The subscripts a and w stand

for air and water respect-ively.

We are interested in the deviation (x,t) of the interface from equilibrium, where x is the non-trivial horizontal coordinate and t the time. To calculate r we introduce a perturbation stream function p. it

is assumed to have a wave-like nature:

= _(2.2)

It will be seen from eq. (2.6) that r has the same x- and t-dependence; thus k is the wavenumber and c the phasè velocity.

The equations for , the height-dependent part of the stream funètion,

can be easily derived from the five conservation laws and the e4uation of state (Batchelor 1981, p. 16l) which together govern the fluid motion. Additional assumptions we made are: temperature and density are

constànt; gravity is the only body force present; and turbulence and other nonlinear features are neglected, although they enter indirectly

through the background profile. Here it

may be ädded

that surface tension, which canñöt be neglected for the short. wavès of intérest hère, is a conservative force like gravity. Thus we know bèforèhand thàt these

two forces influence the frequency and not the growth of the waves. Surface tension is not a body force but enters through the boundary

conditions. The governing equations then become the Orr-Sommerfeld

express the vanishing; of the wave-induced disturbance at large heights and the continuity conditions at the Interface.,

In dimensionless form the equations rad (see Valénzuela 1976; Kàwai 1979à):

(w' +

_W (2.3) (2.1a) (2.lth) (2»c) (2. )4'd) (2.5)

Here u, is the friction vèlocity in air (see seö-tion

2.2.3),

W is a, dimensionless velocity, W = U - c, and W0 is the value of' W at 0,

the air-watér interface. The. gravitational acceleration is g, T is the surface tension and p the density. and c are dimensionless constants:

a'1w an E: = vk/u, trie inverse ofa Reynolds nurnber

Finding the growth of a particular wave is now trañslated into solving

thé problem ôutlinèd abOyé.- To completely determine the probleh we need boundary conditions or iñitial- values i-n x and t-. As we aré interested

in temporal growth rates we take periodic boundary-conditions in x and

initial vaLles for 4, at a given time.

m-la

implies that k is real and that e is salved as a function of k. When one is interested in spatial

growth, i.e. growth with fetch, tlie roles of x and t are reversed and initial valu'es for given xand boundary conditions i t are taken (Kawai 1979a; Drazin & fleid 1982, pp. 152-153). The imaginary part of c determines he growth rate, as can be seen from the equation for the

2 2

-1)

- W'' n, = -1

(-

4,;

=

- w

'

= w'

- w

'

as

o-a

ww

ow

E: W' W'' & [ (i- + '' J = - _{"w +} [

(w' +

't'a - - 4, - ic. 4,'''] = u kW -

(W03)

-

w''

+

ukW

w o

+ O,

4,'. + 0

a°

ik(x-ct)

e o

To find o as a function of k the profiles of wind and current have to be specified; then the problem

Orr-Sömmerfeld equation in

methods (see, för instance,

conditions at infinity are the aid of the continuity

that S, and £ are small

a w

(2.6)

is completely determined. We next solve the

air and water separately using asymptotic

Drazin & Reid 1982, chapter 1!). The boundary

applied and the solutions are coupled with equations. Throughout we will use the fact

parameters:

We take the water flow to be time independent. This is consistent with the experimental results of Kawai (19T9a); he found the flow to depend

on time but on a much larger scale than the growth of the waves.

Substitution of (2.8) in the Orr-Sommerfield equation enables us to find

the perturbation stream function $ in water. We write as a sum öf

two independent solutions. These will be called the inviscid and viscid solution respectively, and $, as one is in first order a solution of the inviscid Rayleigh equation and the other is relatively large in

regions where viscosity is important. We normalize the solution to unity at the surface:

interface (which can derived from the kinematical boundary condition):

«1

a

«1,

£ « 1

(2 . 7a, b, e)

2.2.2 Basic flow & perturbation stream function in water

As mentioned in the introduction the basic flow in the water is taken to

be -U = U (2.8a) w o S u À

eU

(2.8b)

wo

(2.9)

4,(0) 1

Normalization of is possible because the set of equations (2.3)-(2.5) does riot _{depend on the amplitude of the perturbation (substitution} of = a4 yields exactly the same equations). For convenience we also

normalize the two independent solutions:

= .i(0) = 1

The inviscid solution can be found by a formal expansion of and in . To first order this yields:

iw = iwo + iw iwl

U F(p,q;r;) e F(p,q;r;-2) iwl = K . Iwo + 4'iwi + (2.10)

p = 1/A +

(

+ 1/A2),

q =

1/A -

(1 + 1,A2), _{r =} 1 + 2/A

The function is a complicated expression. Upon inspection we find

iw10

= (0) =

which is the only result rêquired below. Here K is an integral over the

total depth of a differential operator working on

jw the exact form

of the operator and an explicit expression för K are given in the

appendix (2.A1-2.A3). F is the hypergeometric function (see Abramowitz & Stegun 1965).

The viscid solution varies on a scale of (Drazin & Reid 1982). It can be found by a WKB approximation:

= exp

fd'

_(2.11)

f =

i)

(-W) - .

+

0(c)

Note that (2.9)-(2.11) give the complete solution of the Orr-Sommerfjeid equation satisfying the boundary conditions at infinite depth.

2.2.3 Basic flow and perturbation stream function in air

We have taken the usual linear-logarithmic wind profile (figure 2.1):

U =+U

_a ca o u*

U =U +U

(-tanh)

a

i

O K sinh = _(2.12) = r e,

K=O»1,

_U1=ru.

The value of r determines the thickness of the viscous _{sublayer. All}

that is known about its value is that it is of the order of unity (Monín & Yagloni 1971 ). In the literature on growth of gravity-capillary waves values of 5 and 8 _{prevail (Miles 1962; Valenzuela 1976; Kawai 1979a). We} have taken r = 5. All the essentials of the analysis _{are independent of} this choice; only the values of the growth rates _{are larger for larger}

r.

From (2.8) and (2.12) it can be deduced that the condition for

continuity of shearing stress at the interface, ie. equation (2.1), _is

fulfilled.

To solve the Orr-Sommerfield equation with (2.12) _{as the basic flow we}

again separate the solution

a into an inviscid and a viscid part:

A

ia + B ,

=

The results of Kawai (1979a) concerning the phase velocity indicate that

it is reasonable to assume that the critical height

c (defined

by Ua(c) = c) is beneath the top of the viscous sublayer:

<

(2.14)

This implies that the Rayleigh equation has no singularities. The zeroth

order expansion of the inviscid solution (which is the solution of the

Rayleigh equation) can then be calculated numerically without

complications. We used the méthod described by Janssen & Peeck (1985).

It will be important in the following to note that:

for n odd

and for n even

0(1/c ) for 0.007 c 0.02

n a a

=

io(i)

for 0.06 _ca

ia

= 0(1).

(2.15)

Note the two ranges for the order of magnitude for when n is odd. For intermediate ca , i.e. 0.02 _ca 0.06, the order of

a is also

intermediate. For u = 0.15 ms the minimum value for ca of 0.007

corresponds to wavelengths where capillary effects become unimportant.

It is the smallest value for c that we have considered.

a

Eq. (2.1) also implies that in the inner viscous layer ( a thin layer

around the critical height; for an exact definition see Drazin & Reid

1982) the profile can be approximated by Ua

= - +

U. In the inner

viscous layer, which includes the interface, the iscid solution varies

2/3 2/3

on a scale of ca . It is given to order Ca by the second integral of the Airy-function (Drazin & Reid 1982):

va = Ai (, 2)/Ai (i13cahh/3Wo2) 2/3 + o a .1/3

Ç -

+ c

W J , phase (i1"3) = 1/6 ir

ao

ca

The function q,'(Ç) has often been tabulated and plotted (e.g. Benjamin (2.16)

1959). It is related to the Tietjens function DT:

2/3 DT

(k01) =

_-

(La _va

2.2.!! Phase velocity and growth rate

Application of the remaining boundary conditions (2.'4b-d) to the stream functioiis in air and water yields expressions for A, B, C1 D and c. We find for 0.007 E 0.02

B=

10(1) for 0.06 £ (2.17)

D=

fo

(6L2"3)

for 0.007 L 0.02 för 0.06

Lp

A=1-B, C=1-D.

The cause of the two distinct ranges in the order of magnitude of A, B and D is their dependence on

ia'

. In principle a third range also occurs when - Wj = O (1/c). However, we have checked that this

does not occur for 0.007.

To find the phase velocity o we introduce an expansion for it. Many

powers and cross-terms of the small parameters ó _La and c appear in

the equations. However, we find that it is possible to define

e=c0+c1

c2 c

--=O(E

J,

_E:),

e a w c2 1/2 2/3

- = O (c

, C1 w a (2.18)

The order estimates depend on

the relative magnitude of the small

parameters. We have indicated several possibilities; the largest of

these determines the accuracy of the expression.

The approximations to the phase velocity cO and cl can be expressed as

follows*:

= U0 A U ((A U0)2 + k + _(2.19a)

icH+ ? i ni75

The terms

h',

, mp

7"

and AVare given in the appendix

(2.A3-2.A8).

H

represents the effects of viscosity and shear flow in water; its value is approximately '.

R

and

T

are respectively the complex

amplitude of normal pressure and shear stress of the air on the surface.

2

-They have been de dimensionless by dividing by _Pak u i- is

dominatéd by the term -i B tp''' and T'by

t

B ''. in, the coefficient

of T,deperlds oh properties df the flow in the water, for the case of

no flow it eqüals 1 . IYalso depends on the water flow and has a value df about 2.

The growth rate of the energy B is given by

B = 2 k Imc . (2.20a)

As c = c0 + c1 and c0 is real this implies

B = 2 k Imc1. (2.20b)

Eq. (2.20a) fòllows immediately from (2.6) and the definition of energy density for surface waves.

* _{We are oñly interested in gravity-capillary waves and neglect the} possibility of finding Tollmien-Schlichting waves, although they are also solutions of the equations (Miles 1962).

cl = u*.

To get some feeling of what (2.19) implies we first consider thé case Ua

= const. Then (2.19) simplifies to

co = Uo +

U =+U

a. _ca o

+ I'2 Ji= i. 2=

g k p , .,

'

2 w ku o 1)W , m=i,T=o andlY=2.

C0 is now the familiar expression for the phase veloóity of free waves. /Xdescribes viscous damping and?/IVthe correction to the phase

velocity due to the renormalization of the gravity force (see Whitham

197k, p. 41l5). Here it may also be noted that when 6 = O growth s

impossible, as C0 is real. This was found numerically by Kawai (1977),.

who studied the possibility of the instability of a sheared current

generating waves.

The effec.t of the shear in the water is to decrease e0 and the damping

due to changes in

H

and IV.. The shear in the air together with its

viscosity have two effects. One is that the pressure is shifted in phase relative to the surface waves, theréby making growth possible (Miles in

1957 was the first to determine the phase shift of the pressure). The

other is that the tangential stress r is now non-zero.

Another interesting simplification is that of a wind profile that is linear up td _{infinity. This is quite realistic}

for the very short

waves (c«1 )

Moreover, this flow allows for an exact solution of the

Orr-Sommerfeld equation in terms of the Airy-function. This was already known by Mises (191.2 a, b) and Hopf (191k) and perhaps the solution is

even of earlier date. For the profile

(2.21)

expression (2.16) for

va becomes valid at all heights and ia becomes

exactly

Equation (2.19) remains the same but the pressure can now be given explicitly: - + ) + 'va (2.22)

kuW0

a o va

Here = U/u; 4j' and

_"V

can be found In the appendix (2.A21-2.A22).

Both the shear stress and the normal pressure appear i-n eq. (2.19b). However, the effect of the stress on the growth is much smaller than the effect of the pressure. This can be deduced by noting that m = 0(1) and

comparing the leading terms of? and

Y:

_IaBcb'

and Then note

that for bth c'' and ''' the real and imaginary part are of the same

order and that i,'j is an order smaller than

Miles (1962) also expressed the growth in terms of approximations to the pressure and shear stress:*

-Ic1I+(

-I

)

w M M

°1M - 2

However, theré are two important differences between Miles' expression

and our's. Firstly, Miles' expression is formally invalid since the

shearing stress of the basic flows is discontinuous at the surface; Miles supposed Uw 0. In the next section it will be seen that the

numerical resülts of (2.23) are, however, satisfactory (this was also

shown by Välenzuela 1976). Secondly, Miles approximated the pressure and shearing stress on physical rather than mathematical grounds (actually, Benjamin (1959) made these approximations and Mlles adopted them).

* _{Miles studied the generalized situation of finite depth.} We have taken the limit of his results for infinite depth and written his

Therefore he was not able to indicate to what order his expressions were

correct. Using our mathematical analysis we can estimate the order of the error in Mlles' expression for the pressure. For the case U 0,

assuming the expressions for the streamfunctions to be exact (though

Miles obtained approximations), it is Ca For comparison, based on the assumption of infinite precision of the c,'s, our expression for the

-1/2 1/3

pressure is correct to order _{ISEw Ca} for 0.007 0.02 and to order . -1/2 2/3 for 0.06

W C a

Another conclusion to be drawn from (2.19) is that, as growth by wind

input and viscous damping nearly cancel, the growth rate of the

gravity-capillary waves is very sensitive to slight changes of the oceanic and

atmospheric Parameters

a'w"'a

and

2.3.Discussion of results

2.3.1. Growth rates

To calculate the growth rate numerically we neglect Y.This

_{can be}

justified by noting that the leading term of

T

is of thé same order as

2/3

the error inr (as

va is correct to order Ca ). We use the following numerical values; all in S.l.: g 9.806, T/Pw = 7.25 10e,

= 1.2 io, "a = 1.510

and =

io6.

In figure 2.2 the curves of the growth rate as a function _{of wavenumber} are shown for several windspeeds. For u _{0.05 rn/s all curves show a} single positive maximum. We f ïnd a critical value for _{wave generation} near u* = 0.05 rn/s. This value is in accordance with Miles (1962) and

Kawai (1979b). For u _{between, roughly, 0.10 and 0.30 mIs the top of the}

curve occurs at such wavenumbers that Ra 36. Growth at a certain

wavenumber strongly increases with windspeed; there is no simple scaling law. _{The growth at the top of the curve increases even faster with} increasing windspeed; this will be discussed later.

3.2

- I I

-.t

I i I I I - lOO 200 300 400 500 600 700 800 900

k(m)

Fm. Z.2. Growth rate as a function of wavenumber for various wind speeds: = 0.248 ms, U5 = 0.102m s'; --. 0.214,0.098;--, 0.170, 0.096; 0.136,0.075; - - - -, 0.050. 0.025. 4.0 3.6

ß(s)

3.2 2.8 2.4 2.0 1.6 1.2 0.8 0.4

/

/

\

-.'

I -- I I I I I I I I lOO ÔÓ 300 400 500 600 700 800 900

k(m)

Fia. t3. Growth rate as a function ofwavenumber; a comparison between different studies, each

using a different flow in water: -, Miles (1962) u. = 0.23 m 5i; Valenzuela (1976)

0.25 m s'; - - - -, Kawai (1979a) 0.248 m s' ...our study 0.248 m s'. Experimental results

of Larson & Wright (1975) at u = 0.27 m s are indiòated with crosses.

2.8

fi (s)

2.4 2.0 1.6 1.2 0.8 0.4

10

fi(s).

10

10-2

I0

l0I

_{u. (ms)}

Fio .. .2.4. Maximum growth rute as a function of frjctioii viIoeitv. 'Ilnuiv . Milis (1962): S. Valenzuela (1976) - - - -. Kawai (1979e): oùrstudy. Measurements: fl. Larson & Wright.

(1975):

0

Kawai (1979e).

To compare our results with those of others we have chosen one specific

friction velocity. The main features are the same for other values of

u*, which we have checked for u 0.14 rn/s. We find the same type of growth curves as Valenzuela (1976) and Kawai (1979a), as can be seen in figure

2.3.

This implies that the decomposition of the stream function into a viscid and an inviscid part, which we have used throughout, is

valid at all friction velocities; this point was left in doubt by Valenzuela (1976). The height of the top of our curve differs by about

10% from that of either Kawal or Valenzuela, in agreement with our estimated accuracy of

c2"3

= 7%. The difference between the öurves of

Kawai and Valenzuela is 20%. As each curve is based on a different

current profile these differences determine the sensitivity of the

growth to small changes in the current profile. Miles (1962) studied smaller wavenumbers than we have. Therefore not much more can be said

than that the results of the two studies do not disagree. The measurements of Larson & wright (1975) on B give values which are

reproduced by our theory within 25%*.

Those waves for which the growth rates are largest are the first to be

generated by the wind (Kawal 1979a). Therefore the maximum growth rates

as a function of u are of interest and we have plotted them in figure

2.11. This plot also offers another method to compare the effects of the various curreit profiles.

As a result of our calculations we find, for the range of friction

velocities 0.05 rn/s u 0.11 m/s (roughly, this range coincides with i rn/s _°io 12 m/s), that

B

max (2.211)

This is a surprisingly simple result considering the intricate expression (2.1gb).

* _{There is some uncertainty concerning the friction velocity; Larson &} Wright give the value u in the steady state while our computations

are for the transient state. According to Kawai (1979a) u in the transient state is considerably less (up to 50%) than in the steady state at the same value of, for instance, U2. Another uncertainty is introduced by the fact that u determinations in the laboratory are,

as a rule, exact up to not more than 5 - 10%. However, we have neglected these complications and simple compared data at the same

4.4 4.0 fi(s) 3.6 3.2 2.8 2.4 2.0 1.6 1.2 0.8 0.4 I I I I I -- I t I I 100 200 300 400 500 600 700 800 900

k(m')

FIG. . 5. Growth rate as a function of wavenumber at u = 0.214 m s. effect ofchanges in profiles s

Our theory: ----. linear wind profile; -.-, wO . (25) as approximation to the

l)'eSSIIre.

In the range considered we again find that our results are close to those of Miles, Valenzuela and Kawai; deviations are within 20%. The

values of the measurements of Kawal and Larson & Wright are higher than the theoretical values; the largest difference is 100% of the

theoretical maximum growth rate. It must be noted that the functional

dependence on u differs in the various theories and experiments; e.g. Kawai finds numerically that

max Thus1 for

given u,

B is

independent of the current profile within 20% but relation (2.21) is different for the various profiles.

,__

To study the effect of the shape of the wind profile we have taken r = 8 (see eq. (2.12)) and we have compared the linear-logarithmic profile with the lineär profile (2.21). Note that these changes occur above the critical height. We have also studied separately the effects of the two

features which distinguish Miles' theory from ours. We took a Benjamin

type of approximation to the pressure - equal to to

order

a - together with the exponential profile in water; that is, we

used eq. (2.19b) with replaced by

/ 2 ku_* W o À

- W C.

[1/c - W0 -o o iwo

W ('-')

o va la (2.25)

Also we took = O together with our expression for the pressure. The results for one value of u are shown in figure 2.5.

When the current is set uniformly equal to zero the growth rate becomes about 15% läwer than when the exponential profile is used. This deviation is in accordance with the sensitivity to the current profile we fàund above. When the linear wind profile is used we still obtain

growth but the growth rates are incorrect. The results are best for high wavenumbers but even then the growth is two times too large. The growth

curves as a function of wavenumber no longer show a maximum. When the

linear-logarithmic profile is used with a thicker viscous sublayer, r = 8 instead of r 5, growth is nearly, though nog quite, 8/5 times as

much. When (2.25) is used for the pressure the growth corresponds within 20% with the valuès found using our full expression for ; the growth is

always tod large.

Our analysis shows that the growth is very sensitive to changes in the

shape of the wind profile, even when these changes occur well above the critical height. This is not as surprising as it may seem. Miles (1957) was able to express first the growth rate in terms of an integral of the stream function and the wind pröfile from the interface up to infinity, and then in terms of values of these functions and their derivatives at

the stream function and its derivatives at 'the critical height still depend on all parts of the wind profile because the stream function is the solution of a differential equation containing this profile at all heights. Note that the second step of Miles as sketched above is not

possible when viscosity is taken into account.

2.3.2 Phase velocities

In principle, the phase speed of the growing waves depends in two ways

on the wind: directly; but also indirectly through the wind-induced

current. First we treat the effect' 'of the current. The current can be

characterized by its value at the surface U0 and by the shape of its

profile. To study the effect of these two we have varied both. First, to

investigate the effect of the shape of the current profile we compared

our results with those of Valenzuela and Kawai using their values of U0

and our profile. We find that the phase speed is insensitive to the

choice of linear-logarithmic, exponential or Kawai's profile. Differences between values of the phase speed are in the order of a few

percent. Using a constant profile, either U O or U U0, leads to

errors of about 20% at u* = 0.136 m/s and 50% at u = 0.6 rn/s.

As Valenzuela uses U0 = O.8u* while Kawai used his measured values, which are near U0 = 0.5u, it is difficult to present illustrations of

the foregoing in a figure. In figure 2.6 we show results for u = 0.136 m/s: apart from Valenzuela's results all theoretical values are based on

Kawai's value for U0. Measurements of' Kawai and Plant & Wright (1980)

are also presented in figure 2.6.' The effect of the value of U0 depends on wavenumber and u. At higher wavenumbers the phase speed becomes less

sensitive to U0. For k = 155 m1 the dependence on U0 for different friction velocities is shown in figure 2.7. At low friction velocities

the influence of U0 on o is small. However, at u = 0.6 m/sValenzuela's value based on U = 0.8 u* is 50% larger than our value, obtained by

using U0 0.65 u. If we use U0 = 0.5 u in our exponential profile we obtain a phase speed '40% lower than that at U0 = 0.65 u. Using U0 =

0.45 0.40

c(m s-')

0.35 0.20 0.15 0.10 I I I I I I I_ I 100 200 300 400 500 600 700 800 900

k (m')

F10. 2.6. Phase velocity as a function of wavenumber at u = 0.136 ms'. Theory:

Valenzuela (1976) . Kawai (1979a):

or study. (19a): -.... our study.

= 0; -.-,

our study, (J 0. Measurement: x. Kawai (1979a): O. Plant & Wright (1980).

Plant & Wright (1980), see figure 2.7. In their experiment U0 was not measured, however, they suggested U0 = 0.6 u. This analysis shows the

importance of U0 in comparing various theoretical methods and in comparing experimental and theoretical data.

Next we studied the direct effect of the wind. The wind has no direct

effect on c0 but has on c1 (see eq. (2.18) and (2.19)); the real part of c1 is the first order correction in

a and on the phase speed.

Thus the maximum possible direct effect is given by

0.I 0.2 0.3 0.4 0.5 0.6

u. (mis)

2.7. Phase velocity as a function of friction velocity for different values of U0. Theory

Valenzuela (1976) U0 = 0.8u; - -: our study: e0, U0=0.65u .c. U0=

O.65u: --.

c.

U0 = O.5v.Measurements: . Plant & Wright (P180).

In practice (1/c) Re e1 is even smaller than this. For example1 for

u

=0.136 rn/sand lOOm1

<k

<500m1,

(1/c) Re c1< 0.01

a"< 0.014),

while for u = 0.6 m/s and k = 700 rn

14/3

and for k = 155 ni1

(1/c) Ree1 0.30 =

Incidentally, Re e1 is always negative (see figure 2.7).

As well as this order analysis we have calculated the effect on the phase speed of changes in the shape of the wind profile by varying the thickness of the viscous sublayer; we compared r = 5 and r = 8. As is expected from the foregoing, for u < 0.25 rn/s the difference is less

than 1% for all wave-ñumbers. For u = 0.6 rn/s the difference is about 15% of e (± 30% of Re e1).

Summarizing, we can say that the wind-induced current has a large effect ori the phase velocity; this can be 50% of the phase speed of free waves.

The exact shape of the profile, linear-logarithmic, exponential or Kawai's, is not of importance, though a constant profile leads to errors. However, the value of the current at the surface certainly is of importance, an error in U0 of 25% can lead to an error in the value of e of 50%. This suggests that the value of the current and its shear at the surface are the two most important features of the current profile.. The

direct effect of the wind on the phase velocity is only noticeable at

larger friction velocities: for u = 0.6 rn/s it can be 15% of e.

2.!!. Main conclusions

It is possible to describe the Initial growth of gravity-capillary waves with asyrnptdtic methods. Our analysis results in expressions for phase

-2/3 2/3

velocity and growth rate accurate to order 6Ca and Ea respectively for 0.007 .02 and to order

-1/3 2/3

6E and c resp. for E 0.06, where c is the inverse of a

a a a a

Reynolds number: E \a'u*. For intermediate Ca the accuracy is also intermediate. Our analysis confirms the validity of Miles' (1962) expression fpr the growth rate.

The wind speed strongly influences their growth; we find that the growth rate of the initial. wavelets is proportional to u3. There is no simple scaling law for the growth rate at fixed wavenumber but it alsò depends strongly on u. Changes in the shape of the wind profile, even above the

critical height, can change the growth rate by a factor of more than a

three. The influence of the current profile on the growth is within ±20%.

The phase velocity is more sensitive to the wind-induced current than to the wind itself; the effect of the current can be 50% of the phase speed while the direct effect of the wind is less than 15%. A linear-logarithmic, exponential or error-function-like profile all lead to the same results. Good agreement with experimental data can be obtaine when

the value of the current at the surface is known; errors in U0 of 25%

lead to errors in c of up to 50%.

Finally we checked that the perturbation normál pressure of the air on the surface causes the growth; the effect of the perturbation shear

stress is ari order Ra213 smaller.

Acknowledgements

This work was partly supported by the Netherlands Organization for the

Advancement of Pure Research (ZWO). Wé would like to thank G.R.

Valenzuela for making his original data available to us, Anne de Baas

and Wim Verkleij for comments on early versions of this paper, and Fred Dobson for an extensive review.

Appendix K

= -

:!-.

-1)2

10d'

A3Uc

14U2

U

-

+ a

(2)m[2++2)F(3,rn+2+2/A;

_m+3+2/A; .2) r(r) 2

a=[r()-t()

a prime stañds for differentiation with respect to the argumnt.

E.

'-w

c

. ' w ' + C

(3!

-

''')

-o 1 iwo o

iwl

i

_w o vwl iwo Iwo vwi

(2.A11) w Terms -(2. A2) U

sas

e

D ' and - D ''' also appear in (2.A14) but these cancel.

lE VWO VWO

w

+ Ati - w0c

Iwo +

3aA

- a (A ''' +

B ''')

(2.A5)

u*

-r=

c.B

(2.A6)

+

(2Aml)

F (, m+3+2/A; m+11+2/A;

2)]

(2.A1)

r(p+j) .r(q+j) i(p+m-j) 1(q+mej) r(r+j)

j!

r(r+m-j) (m-j)!

Here and in the following F has p,

q and r as parameters and

argument when these are not specified.

Ìwo ¡ c F

=

+ __2 r

AU

J=Csiw-Wc_îs t

_{o c iwo} + B = W (, - p L i/c - W _{5 '} + W ₅ o va ja a o ja o Iwo

A =1-B

D =

b+

'

'c

o =

(-i A2t vw w C = C + ic C o

wi

st,

iwo c , (W _{5 '} + W

_5'

+ ic

5ttt)

o iwo o vwi. w vwl w vw A W o

= - -

AU vwl 1E w (

+ Th

22k

p

uW

w o = 0 (1) Co C

=1

C = (5 o 1 1 2 1 It

5''

w vw !

=K

iwl AU , A2U A3I A3 2 t'' (1 + __.2 !_) (1 + ___-..2

+ _._2

______ Iwo e F W W 2 o o W o -W '' vwo _ic AU k? '4vw1 (2.A15) (2.A16) (2.A17) (2.A18) (2.A19) (2.A7) (2.A8) = A o (2. A9) (2.A1 0). -

s)

(2.A11 ) (2.Al2)

10

(2.A13) ic w (2.Ai LI)

A.?'?

_!

_{(at the surface!)} "la -va 2 Eca Wo 'va - 1] 2

i.

'

=__..2r_!LL+

(!.L'2___E]

aclwo

3 F F U F

T

_{M -}

-(1/c- W0

_{W ('}

wo) (1/ca_ + W0)

-ova

ia (2.A20) (2.A22)

This last expression is a translation of Miles' result into our notation; Miles did not substitute U' = Uk/c and 1h his case W _{= - c/ui.}

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awo)

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Mises, R. von, 1912a. Beitrag zum Oszillationsproblem. In: Féstschrift

H. Weber, pp. 252-282. Teubner, Leipzig.

Mises, R. von, 1912b. Kleine Schwingungen und Turbulenz. Jber. Deutsch. Math.-Verein 21, pp. 2141-2148.

Monin, A.S. & Yaglom, A.M., 1971. Statistical Fluid Mechanics: Mechanics of Turbulence. Vol. I, M.I.T. Press, Cambridge, Massachusetts. Plant, W.J. & Wright, J.W., 1980. Phase Speeds of Upwind and Downwind

Travêlliñg Short Gravity Waves. J. of Geophysical Research, 85, no. C6, pp. 33014-3310.

Raney, R.K., Hasselmann, K., Plant, J.W., Alpers, W., Shuchman, R.A. Jam, A. & Shemdin, O.H., 1983. Theory of SCAR Ocean Wave Imaging:

the Marsen Consensus, to be published.

Valenzuela, G.R., 1976. The Growth of Gravity-Capillary Waves in the

Coupled Shear Flow, J. of Fluid Mech., 76, pp. 229-250.

Whitham, G.B., 19714. Linear and Nonlinear Waves, John Wiley & Sons, Néw York.

3

Nonlinear interactions of gravity-capillary waves:

Lagrangian theory and effects on the spectrum

Abstract

A weakly nonlinear inviscid theory describing the interactions within a continuous spectrum of gravity-capillary waves is developed. The theory

is based on the principle of least action and uses a Lagrangian in

wavenumber-time space. Advantages of this approach compared to the method of Valenzuela & Laing (1972) are much simplified mathematics and

final equations and validity on a larger time scale. It is shown that much of the development of the spectrum under influence of nonlinear

terms can be understood without actually having to integrate the

equations. To this end multiwave space, a new concept comparable to phase space, is introduced. Using multiwave space the magnitude of the nonlinear transfer is estimated and it is shown how the energy goes through the spectrum. Also it is predicted that at fixed wavenumbers, the smallest being 520 m , finite peaks will arise in the spectrum. This is confirmed by numerical integrations. From the integrations it is

also deduced that nonlinear interactions are at least as important to

the development of the spectrum as wind growth. Finally it is shown both

analytically and numerically that the near-Gaussian statistics of the

sea surface are unaffected by nonlinear interactions.

3.1. Introduction

Remote sensing of our seas, which has taken such a flight over the last decade, gives a lot of information which the scientific world still has

difficulty in interpreting correctly. One example of this is formed by

the images of bottom topography of shallow waters taken by microwave

radar. An essential link in the imaging process is formed by

bottom topography images is that not all is known about the behaviour of these wavelets while part of what is known is still too intricate to be handled in a compound model (Phillips, 19814).

This paper deals with gravity-capillary waves. _{I will first give a brief}

review of the present level of our knowledge of them. This is done most

easily in the context of the energy balance equation (Willebrand,

_1975):

+5

+S

3t 3k 3x 3x 3k wind vise nl br'

Q = k.0 + w

Here A(k, x, t) denotes the action density, which is defined by

A = E/w; E being the local energy density and w the intrinsic frequency. Also, Q is the apparent frequency, U the surface current and S stands for source or sink.; resp. acting due to wind, viscosity,

nonlinear interactions and breaking events. _{Actually all sources are} coupled, but as an approximation they are dealt with separately (Komen

et al., 19814).

The damping due to viscosity is well known: Sviso = -14vk2A (Phillips,

1977,

_{p. 52).}

The equations governing the energy input by wind are also familiar (Miles, 1962, Valenzuela, 1976), but it is only recently that a quick and accurate way to solve them has been found (Van Gastel et al.,

1985).

We know very little of the causes and frequency of breaking of the waves (Phillips, 19814); Banner & Phillips (19714) have nade some

theoretical predictions on this subject. The nonlinear interactions in a spectrum of gravity-capillary waves have been calculated by Valenzuela & Laing,

(1972).

A drawback of their method which is based on Hasselmann's

(1962)

perturbation analysis, is that the resulting expressions for the interaction coefficients are complicated and cannot be understood

physically; also the numerical computations necessary for quantitative results are delicate and lengthy. This is probably one of the reasons

why the nonlinear interactions have never yet been included when solving the energy balance. As Longuet-Higgins

₍₁₉₇₆₎

puts it: "there is

obviously a need for a much simpler approach, more amenable to physical interpretation".

The aim of this

paper is to raise our knowledge of the nonlinear

interactions in a continuous spectrum of gravity-capillary waves to such

a level that an accurate description of these interactions can be used when solving the energy balance (3.1) for this part of the spectrum.

This work can be divided into three steps: 1) mathematical derivation of

the interaction equations, 2) construction of a physical image of how the wavenumbers are related to each other by the resonance conditions

and

₃₎

study of the symmetries present within a triad. Point 2) helps to make interpretations of numerical integrations possible and greatly

affects their efficiency.

Analytical expressions for the nonlinear interactions in a continuous spectrum of gravity-capillary waves are obtained as the Euler-Lagrange equations for the Lagrangian in wavenumber-time space. An expansion is

ide in powers of the wave steepness. This method can be seen as a

generalization to the continuous case of the method of Simmons

₍₁₉₆₉₎

or

of that described by Whitham

₍₁₉₆₇₎

for resonant interactions. The

advantages of using a Lagrangian or Hamiltonian formalism in this Oase are simpler mathematics, simpler final expressions and validity on a

longer time scale. These advantages have been pointed out frequently in recent years (Miles & Salmon

_1985,

Henyey

1983).

To achieve the second point I introduce a new concept. This is the

multiplet-wavevector space, multiwave space for short (a multiplet is a

set of waves that

together fulfill the resonance conditions). This

concept can be compared to phase space in classical mechanics. Phase

space tells us at a glance how physical space

is interconnected by

trajectories. Similarly, in multiwave space we see the interconnections

in wavevector space due to the nonlinear interactions. The multiwave space is constructed explicitly for triad resonances between parallel

gravity-capillary waves. In this case it is two-dimensional.

The third point., the use of symmetries, is straightforward. It is

comparable to the way Hasselmann & Hasselmann

₍₁₉₈₁₎

use symmetries for four-wave interactions.

One of the gains or using multiwave space and symmetries is that one has to solve resonance conditions and interaction coefficients only for part

of wavevector space, in the present case only for k

I

(g/2T). Another gain is that it allows for a natural construction of a grid for numerical calculations, implying high accuracy. This grid follows the

paths of the energy through the spectrum.

An energy balance containing nonlinear interactions is actually

integrated. As initial states the spectra measured by Liu & Lin (1982) are used. A surprising phenomenon is encountered: the existence of

preferred wavelengths, i.e. peaks in the energy spectrum. I explain the occurrence Of these peaks using multiwave space.

As a sideliné the issue of near-Gaussianity of the seasurface is considered. The Gaussian approximation is essential to the weakly nonlinear theory underlying multiplet interactions. I show analytically

that the Gaussian distribution is a stable equilibrium. In other words,

under influence of nonlinear interactions the third cumulant, commonly

related to phase-locking, is constantly being reduced. This result was

already obtained by Davidson in 1972 for three-wave as well as four-wave interactions in general. Thus also for gravity waves nonlinear interactions drive the spectrum to a Gaussian distribution.

3.2. The interaction equations

The equations for the surface elevation are derived using the

principle of least act-ion. Nonlinear effects are included up to first order. The surface elevation is supposed to consist of a continuum of

free gravity--dapillary waves. Viscosity is neglected.

The action J can be written as

For water waves the Lagrangian L is a function of the elevation and the potential .For infinitely deep water it is given by (Simmons,

_1969,

Luke,

1967):

L + T [(1 +

.C)-1] +

(Vcp.V

+ (3.3)

Here.g is the acceleration of gravity and T the surface tension divided

by the density. The derivatives are defined as follows:

V

(±ì î)

'r

Bt ' Bx' ay' Bz

=(

ax' By

The Lagrariglan as presented in (3.3) is a function and time. To be able to perform the integration

I apply Fourier transformation and substitute the

vertical for each mode:

(x, t)

_j:;

f c

(k, t) e k.x dk 4,

(z, z,

t) = j.- j (k, t) e k.x + kz dk k _Iki

(3.11)

of horizontal space in the Lagrangian dependence on the (3.5)

A new Lagranglan C depending on wavenumber and time is defined by

= ff

L d k d t (3.6)

It can be given explicitly by assuming the waves to be of small, though finite amplitude, or, equivalently:

V k, t c (k,t) = c 4,(k,t) = (3.7)

= 0 (1)

where c is a small parameter proportional to the wave steepness. This

the following. c being small enables the Lagrangian C to be expanded in powers of c . Using the identity

f

eik

dx = 1hr

k)

(3.8)

this expansion becomes (here and in the following the dependence on t of all functions is not stated explicitly when no confusion can arise):

L (k, t) = C2 (g + k2T) (k) (-k) + k4,(k)Q (-k) +

+ C3

ff[(icic'

+ kk')(k);(k) C(k'') + k(k)

(k')(k'')]

& (k+ k'+ k'') dk'dk''

+ O ().

(3.9)

The principle of least actIon states that physical realizations of the system are given by SJ = O. This condition can be transformed into a

condition on the Lagrangian, as in the familiar Euler-Lagrange equations. To illustrate the method I use a simplified Lagrangian t.; for

the actual Lagrangian given by (3.9) the procedure is analogous. Let t.

be a function of (k) and p (-k). Then:

j =f

t {(k)

+ u(k), (-k) + _{u(-k)]dk -}

_f

t. (cp(k), q(-k))

dk.

Here is a small parameter and u is a smooth function of

k

disappearing at the integration boundaries. Continuing:

=

f

k) i ((k), 4(-k))) u (k) +

t. ((k),

(-k)))

u (-k) ] dk = ( t. ((k), k)) +

t. (k), $(k)))]

u (k) dk.

Thus

= ° <>

(k) ((k), 4(-k)) + (q(-k), q(k))} =

o.

Applying this procedure to the present Lagrangian C yields two equations as there are two independent functions _c and .

Using the substitution k -* -k these equations can be written as:

(k) = (g + k2T) - c

_j-

ff[-k'k" + k'k") (k') (k")

+ k" (k") (k')] 6 (k-k'-k")dk'dk"

c (k) = + c j ¡f{(kk' + kk') c

(k')

(k")

(3.10)

-

k (k')

c (k")]

6 (k-k'-k") dk' dk"

The zeroth order or linear solutions are two independent functions oscillating with frequencies

(k)

= I

(gk + k3T). _(3.11)

These functions are the normal modes of the system. By eliminating from (3.10) and transforming to these normal modes according to

c (k,

t) = a (t) iw(k)t + a (t) -iw(k)t

(3.12)

c (k,

t) = i(k) a(t)

i(k)t

-

i(k)

a (t)

equations for the amplitudes of the modes are derived which are uncoupled in lowest order:

-ike

'f

kk' + kk' U)

()

+ U)!!

)

--

-' kk' a' a' a" ot G a 'a" kk" + kk" , ,, -k'k"+ k'k" V' kk" ma" + a' + k'k" + 2(ù') + 2 (w")2 + 2w, i i(w w' ,-w" )t a a a" 6 (k-k'-k")dk'dk" c&', a,, + O (:2)

The following abbreviations are used:

w = w(k),

(3.114)

a = i

ra

(k, t) o = i

w =1

_.0

_-w

_a=-1

_{a a.}

_{(k,t) a=-1}

In the summation a' and a" take on the values -1 and +1, a can be both +1 or -1.

Equation (3.13) governs the nonlinear interactions of water waves. One piece of information has not been used so far and should be added; the elevation and potential are real functions. This leads to a relation

between a+ and a- . Two conventions for stating this relation are in use (Hasselmann, 1962, Davidson, 1972). I will have use for both of them here; the first convention is

Y k a(k) =

-k) , w1 = w ,

w1 =

w , (3.15)

and the other:

Yk, a = -1, 1 ; a(k) (-k) , w(k)

w(k)

(3.16)

k>o

:

'1='

The interactions described by (3.13) are cyclic in time except when the

exponent vanishes, or, more explicitly, when