stochastic models for ocean waves gaussian fields … seminar asymmetric lagrange ... flection and...

Stochastic models for ocean wavesGaussian elds made more realisticby Rice formula and some physics

Georg Lindgren, Marc Prevosto, Soa berg, . . .

Lund University, IFREMER Brest, Lund University

Bocconi University of Milan, January 21, 2016Statistics seminar

Asymmetric Lagrange waves January 2016 1 / 41

Introduction A long long time ago

An experiment 64 years ago the real thing

JOURNAL OF THE OPTICAL SOCIETY OF AMERICAV

Measurement of the Roughness of the Sea Surface from Photographsof the Sun's Glitter

CIIARLES COX AND WALTER MUNKScripps Institution of Oceanography,* La Jolla, California

(Received April 28, 1954)

A method is developed for interpreting the statistics of thesun's glitter on the sea surface in terms of the statistics of theslope distribution. The method consists of two principal phases:(1) of identifying, from geometric considerations, any point onthe surface with the particular slope required for the reflection ofthe sun's rays toward the observer; and (2) of interpreting theaverage brightness of the sea surface in the vicinity of this pointin terms of the frequency with which this particular slope occurs.The computation of the probability of large (and infrequent)slopes is limited by the disappearance of the glitter into a back-ground consisting of (1) the sunlight scattered from particlesbeneath the sea surface, and (2) the skylight reflected by the seasurface.

The method has been applied to aerial photographs taken undercarefully chosen conditions in the Hawaiian area. Winds weremeasured from a vessel at the time and place of the aerial photo-graphs, and cover a range from 1 to 14 m sec~'. The effect of

surface slicks, laid by the vessel, are included in the study.A two-dimensional Gram-Charlier series is fitted to the data.As a first approximation the distribution is Gaussian and isotropicwith respect to direction. The mean square slope (regardless ofdirection) increases linearly with the wind speed, reaching a valueof (tanl6 0 )2 for a wind speed of 14 m sec-'. The ratio of the up/downwind to the crosswind component of mean square slopevaries from 1.0 to 1.9. There is some up/downwind skewnesswhich increases with increasing wind speed. As a result the mostprobable slope at high winds is not zero but a few degrees, withthe azimuth of ascent pointing downwind. The measured peaked-ness which is barely above the limit of observational error, is suchas to make the probability of very large and very small slopesgreater than Gaussian. The effect of oil slicks covering an area ofone-quarter square mile is to reduce the mean square slopes bya factor of two or three, to eliminate skewness, but to leave peaked-ness unchanged.

1. INTRODUCTION

THE purpose of this study was to make quantita-T tive measurements pertaining to the roughness ofthe sea surface; in particular, to learn something con-cerning the distribution of slope at various wind speeds.This distribution plays an important part in the re-flection and refraction of acoustic and electromagneticradiation, and in the complex problem of wind stress onthe water surface.

Our method consists in photographing from a planethe sun's glitter pattern on the sea surface, and trans-lating the statistics of the glitter into the statistics ofthe slope distribution. Winds were measured from avessel at the time and place the photographs were taken.They ranged from 1 to 14 m sec-'.

If the sea surface were absolutely calm, a single,mirror-like reflection of the sun would be seen at thehorizontal specular point. In the usual case there arethousands of "dancing" highlights. At each highlightthere must be a water facet, possibly quite small, whichis so inclined as to reflect an incoming ray from the suntowards the observer. The farther the highlighted facetis from the horizontal specular point, the larger mustbe this inclination. The width of the glitter pattern istherefore an indication of the maximum slope of thesea surface.

Spooner' in a letter to Baron de Zach reports fourmeasurements by this method in the Tyrrhenian Sea,

* Scripps Institution contribution No. 737. This work has beensupported by the Geophysical Research Directorate of the AirForce Cambridge Research Center, AMC, under contract No. AF19(122)-413.

1 J. Spooner, Corresp. Astro. du Baron de Zach, 6 (1822).

all yielding maximum slopes of 250. Hulbert2 demon-strates by this method that the maximum slope in theNorth Atlantic increased from 150 at a 3-knot wind to250 at an 18-knot wind. Shuleikin3 took a long seriesof measurements of the width of the "road to happi-ness" over the Black Sea (a Russian synonym for theglitter pattern from the setting sun), and deduced thatslopes up to 300 were not uncommon.

These measurements of maximum slope, so widelyseparated in space and time, are reasonably consistent.They do depend, however, on the manner in which theouter boundary of the glitter pattern is selected. Thisselection is apparently influenced by the brightness ofthe light source relative to the sky, and by the sensi-tivity of the eye. For otherwise the moon's glitterwould not appear narrower than the sun's glitter underotherwise identical conditions. We have avoided thisdifficulty by computing the distribution of slopes fromthe measured variation of brightness within the glitterpattern (rather than computing maximum slopes fromthe outer boundaries). Our method gives more infor-mation-and requires much more work.

The two principal phases are (1) to identify, fromgeometric considerations, a point on the sea surface(as it appears on the photographs) with the particularslope required at this point for the reflection of sun-light into the camera, and (2) interpret the averagebrightness of the sea surface (or darkening of the nega-tive) at this point in terms of the frequency with whichthis particular slope occurs. By choosing many suchpoints we derive the frequency distribution of slopes.

2 E. 0. Hulbert, J. Opt. Soc. Am. 24, 35 (1934).3 V. V. Shuleikin, Fizika Moria (Physics of the Sea) (Izdatelstvo

Akad. Nauk. U.S.S.R., Moscow, 1941).

838

VOLUME 44, NUMBER 11 NOVEMBER, 954

November 1954 839MEASUREMENT OF THE SEA SURFACE

FIG. 1. Glitter patterns photographed by aerial camera pointing vertically downward at solar elevation of k=700.

The superimposed grids consist of lines of constant slope azimuth a (radial) drawn for every 30, and of constant tilt /3

(closed) for every 5. Grids have been translated and rotated to allow for roll, pitch, and yaw of plane. Shadow of plane

can barely be seen along a= 1800 within white cross. White arrow shows wind direction. Left: water surface covered by

natural slick, wind 1.8 m sec', rms tilt o= 0.0022. Right: clean surface, wind 8.6 m sec', =-0.045. The vessel Reverie iswithin white circle.

2. THE OBSERVATIONS

Aerial Observations

The derivation of Sec. 4 will show that the radiance

of reflected sunlight from the sea surface is determinedby the probability distribution of slopes provided thelight is reflected only once. To avoid multiple reflec-

tions we have made measurements only when the sunwas high (only slopes greater than about one-half the

angle of sun elevation can cause a second reflection).For a high sun the glitter pattern covers the surface toall sides of a point directly beneath the observer, andaerial observation is indicated.

A B-17G airplane was made available from the 3171stElectronics Research and Development Group, GriffissAir Force Base, Rome, New York. Four K-17 (six-inchfocal length) aerial cameras were mounted on a framewhich could be lowered through the bomb bay andleveled during flight. They were wired for synchronous

exposures. Two cameras pointed vertically downward,the other two pointed to port and were inclined down-ward at an angle of 300 with the horizontal. Thisallowed for a 25 percent overlap between the verticaland tilted photographs. One of the vertical cameras andone of the tilted cameras took ordinary in-focus orimage photographs (see Fig. 1) using "variable densityminus blue" filters. At an altitude of 2000 feet twopoints on the sea surface separated by more than 40 cmare resolved on the image photographs. The two re-maining cameras took photometric photographs. Fromthese cameras the lens systems had been removed, andglass sandwich filters containing Wratten gelatin A-25absorbers installed.

During the photographic runs, the plane was steeredby sun compass so that the azimuth of the tilted cameras

was toward the sun. An attempt was made to avoidcloud shadows and atmospheric haze. In most cases thefield of the cameras was sufficiently restricted to avoidthese effects when the plane was flying at an altitudeof 2000 feet.

Observations at Sea

In order to correlate measurements of wind speedwith slope distribution free from modifying effects ofland it was necessary to have meteorological recordsfrom a vessel near the location of the photographs.For this purpose a 58-foot schooner, the Reverie, waschartered and equipped with anemometers on the fore-masthead (41 feet above sea level) and the bowsprit(9 feet). The signal from the anemometers was smoothedwith an electrical low-pass filter having an 18-secondtime constant, then recorded. Wind direction was esti-mated by eye. Other measurements included the airand water temperatures, and the wet and dry bulbtemperatures.

One of the objects of this investigation was to studythe effect of surface slicks. First we attempted to spraypowdered detergent from the vessel and later from theplane, but the slicks thus produced did not persistsufficiently. A satisfactory solution was to pump oil onthe water, using a mixture consisting of 40 percent usedcrankcase oil, 40 percent Diesel oil, and 20 percentfish oil. With 200 gallons of this mixture a coherentslick 2000 feet by 200 feet could be laid in 25 minutes,provided the wind did not exceed 20 mph.

Location of Observations

During July, 1951, observations were taken offshorefrom Monterey, California, where a variety of wind


Introduction Very recent movie

A Lagrange crest-trough asymmetric wave

A Lagrange simulation based on directional energy spectrum:



The Cox/Munk experiment 1951-1954 not Gaussian!

Slope asymmetry; left Cox/Munk, right Lagrangetop across waves bottom along waves

2 0 2



What is a Lagrange wave model?

Wikipedia:

An Eulerian specication of the ow eld is a way of looking at uidmotion that focuses on specic locations in the space through whichthe uid ows as time passes

A Lagrangian specication of the (wave) eld is a way of looking atuid motion where the observer follows an individual uid parcel

What is a stochastic Gauss-Lagrange wave model?

A stochastic model for a uid where the vertical and horizontalmovements of a uid particle is described by correlated (Gaussian)stochastic processes a 3D Gaussian space-time eld


Introduction A long time ago

How do statisticians think?

Nature is modeled by a systems equation

Nature is observed through an observation equationImportant observables:

I crest height, height of local maximaI steepness = wave height/wave lengthI front and back steepness

One goal: to nd the distribution of the observables from the model



The correct frequency interpretation

What is the height of an arbitrary local maximum in a stationary(dierentiable) stochastic process X (t) and what is its distribution?

Answer: The limit of the empirical distribution function as observationinterval grows. If the process is ergodic!

P(height of a local maximum u)

= limT

T1#local maxima in [0,T ] with height less than uT1#local maxima in [0,T ]

=E (#{t [0, 1],X (t) = 0, downcrossing, AND X (t) u})

E (#{t [0, 1],X (t) = 0, downcrossing})

Also David Slepian 1962.



Rice formula for crossing intensity in a stationary process

0 5 10 15 20 25 30 35 40 45 503

2

1

0

1

2

3

4

Slope at upcrossings are not halfnormal but Rayleigh

Rice formula for the expected number of upcrossings of a level u per timeunit Derivative X (t) acts as a slope-bias factor:

E [N+u ] = E[X (0)+ | X (0) = u

]fX (0)(u) =

0

z fX (0),X (0)(u, z) dz

For a Gaussian process X (0),X (0) are independent and normal!


Gauss versus Lagrange More recent

The Gaussian wave model (1952)

The height W (t, s) of the water surface at location s at time t is aGaussian stationary (homogeneous) random process, expressed as a sum(integral !)

W (t, s) =k

Ak cos(ks kt + k)

of moving cosines with

random amplitudes Akrandom phases k

and with xed frequencies (1/wave period) and wave numbers (1/wavelength).



Gaussian characteristics

A Gaussian sea is statistically symmetric:

the sea surface can be turned upside down

a wave movie can be run backwards

and you don't see any dierence

It needs to be combined with physics/hydrodynamics gives the Lagrange model



Why using a stochastic Lagrange model?

The (linear) Gaussian wave model allows exact computation ofwave characteristic distributions (the WAFO toolbox)

produces crest-trough and front-back stochastically symmetric waves

The modied stochastic Lagrange model can produce(2006) crest-trough asymmetric waves (2D)(2009) front-back asymmetric waves (2D)(2011) front-back asymmetric waves (3D)

still allows for exact computation of wavecharacteristic distributions

Software exists for calculation of statistical wave characteristicdistributions (crest, trough heights, period, steepness etc)


The stochastic 3D Lagrange model The Gaussian generator

Waves according to Google search

Circles in deep water, ellipses in shallow water

TASK: Combine Gauss with Google


The stochastic 3D Lagrange model The Gaussian generator

Gaussian generator and the orbital spectrum

In the Gaussian model the vertical height W (t, s) of a particle at the freesurface at time t and location s = (u, v) is an integral of harmonics withrandom phases and amplitudes:

= (x , y ) = (cos , sin )

= () =g tanhh

W (t, s) =

=0

=

e i(st) d(, )

with S(, ) = the orbital spectrum and (, ) is a Gaussian complexspectral process.

Wave direction =


The stochastic 3D Lagrange model The free Lagrange model

The stochastic Lagrange model

Describes horizontal and vertical movements of individual surface waterparticles. Use

W (t, s) =

e i(st) d(, )

for the vertical movement of a particle with (intitial) reference coordinates = (u, v) and write

(t, s) =

(X (t, s)Y (t, s)

)= horizontal location at time t

(t, s) s is the horizontal discplacement of a particle from its originallocation s



with horizontal Gaussian movements

Use the same (vertical) Gaussian spectral process as in W (t, s) to generatealso the horizontal variation

Fouques,Krogstad,Myrhaug,Socquet-Juglard (2004), Gjsund (2000,2003)Aberg, Lindgren, Lindgren (2006,2007,2008,2009,2011), Guerrin (2009),Nouguier (2015),

(t, s) =

(X (t, s)Y (t, s)

)= s +

H(, ) e i(st) d(, )

where the lter function H depends on water depth h:

H(, ) = icoshhsinhh

(cos sin

)



The stochastic Lagrange model

The 3D stochastic rst order Lagrange wave model is the triple of Gaussianprocesses

(W (t, s),(t, s)) = (W (t, s),X (t, s),Y (t, s))

All covariance functions and auto-spectral and cross-spectral densityfunctions for (t, s) follow from the orbital spectrum S(, ) and the lterequation.

Space wave : keep time coordinate xed

Time wave : keep space coordinate (X (t, s),Y (t, s)) xed

This will (later) lead us to a Palm type problem what are thedistributions of (W (t, s),1(t, (0, 0))) when (X (t, s),Y (t, s)) = (0, 0).



Lagrange 2D space waves, time xed

A Lagrange space wave at time t0 is the parametric curve (2D) or surface(3D) L(x) : s (XM(t0, s),W (t0, s))

0 50 100 150 200 250 300 350 400 450 500

5

0

5Depth = 4m

0 50 100 150 200 250 300 350 400 450 500

5

0

5Depth = 8m

0 50 100 150 200 250 300 350 400 450 500

5

0

5Depth = 16m

0 50 100 150 200 250 300 350 400 450 500

5

0

5Depth = 32m

0 50 100 150 200 250 300 350 400 450 500

5

0

5Depth =



Ocean waves need a direction

The free Lagrange waves are peaked

but they have no preferred direction!


Front-back asymmetry The modied Lagrange model

Diresction gives front-back asymmetric Lagrange waves

To get realistic front-back asymmetry one needs a model with externalinput from wind, for example, by a parameter :

2X (t, s)

t2=2XM(t, s)

t2+ W (t, s)

The lter function from vertical W (t, u) to horizontal X (t, u) is then

H() = icoshhsinhh

+

(i)2= () e i(),

Implies an extra phase shift ( = /2 in the free model)

X (t, u) = u +

e i(ut+())() d(, )



Space and time waves

0 100 200 300 400 5005

0

5

A Lagrange space wave

x (m)

0 10 20 30 40 505

0

5

A Lagrange time wave

t (s)



For 3D waves the lter function needs to take directionalspreading into account

Not much theory available.

An example:

H(, ) =

(i)2(

cos2() | cos()|cos2() sin() sign(cos )

)+ i

coshhsinhh

(cos sin

).

to take care of wind blowing in positive x-direction.

Front-back asymmetry depends on directional spreading in the orbitalspectrum


The 3D-model and the generalized Rice formula Explicit denitions

Explicit denitions I

The model

(W (t, s),(t, s)) = (W (t, s),X (t, s),Y (t, s))

is an implicitly dened model. The space and time models can be madeexplicit by

s = 1(t, (x , y))

equal to the reference point that is mapped to the observation point (x , y).

Note: There may be many solutions to (t, s) = (x , y).



Explicit denitions II

Space wave L(t0, (x , y)) = W (t0,1(t0, (x , y)))= photo of the surface

Time wave L(t, (x0, y0)) is the parametric curve:

t 7W (t,1(t, (x0, y0)))= measured at a wave pole



Wave slopes in space (Cox/Munk) and time

The Cox/Munk experiment observed wave slopes at xed time needsspace derivatives: (

WuWv

)=

(Xu YuXv Yv

)(LxLy

)(1)

An oil platform experiences wave slopes at xed location needs timederivatives

Wt = Lt +(Xt Yt

)(LxLy

)Lt = Wt

(Wu Wv

)(Xu XvYu Yv

)1(XtYt

)(2)


The 3D-model and the generalized Rice formula Pependiularitybias

Asynchronous sampling in space = Cox/Munch, t0 = 0

To nd the distributions of the slopes Lx , Ly at a xed point, say (0, 0) ,invert (1) and nd

the distribution of the mark (Wu,Wv ,Xu,Xv ,Yu,Yv ) at reference point s

under the condition that (X (s),Y (s)) = (0, 0)

Contour lines prefer to cross understraight angles;perpendicularitybias

0 50 100 150 200 2500

50

100

150

200

250



Some old literature still very useful

Proc, Camb. Phil,. Soc. (1960), 62, 283PCPS 62-39

Pril..ted i,n Greqt Bri.bdn

Local maxima of stationary processes

Bv M. R,. LEADBETTER,Research Trdangl,e Insti,tute, Drnhnm, N, Carol,i,na, a.S.A.

(Recei,ueil, 15 July 1965)

Abstract. Two natural definitions for the distribution function of the height of an'arbitrary local maximum' of a stationary process are given and shown to be equiva-lent. It is further shown that the distribution function so defiaed has the correctfrequency interpretation, for an ergodic process. Explicit results are obtained in thenormal case.

L lntrod,ucti,on Throughout this pa/per we shall be concerned with the localmaxima of a given stationary stochastic process r(f) which will be assumed to possessa continuous one-dimensional distribution and, with probability one, &n evory-where continuous sample derivative n'(tl.Itwill be convenient to describe the problemsconcerning local maxima in terms of 'upcrossings' and 'downcrossings' of certainlevels by r(f) and a'(l). Specifically rpe shall say that o(f) has arLupuossirq of the leveluat,toif for some I > 0, tr(r) < trin (fo-d,fo) and n(tl > uin(to,h+d). A,itrotntcrossinyis de6.ned by reversing the inequalities. Further c(f) will be said to have a uossi,n4of the level u at to if there are two points in any neighbourhood of fo at which r(t) -uhas opposite sign.

It is not our intention to discuss here the implications of these deGnitions, beyondnoting that they are &ppropriate and sensible within the present framework. (Fullerdescriptions of properties of upcrossings and downcrossings have been given byYlvisaker ((z)), Leadbetter ((e)).)

We shall say that c(f) has a local maximum at toif, c'(f) has a downcrossing of zeroat to. The instants at which local maxima occur then form a stati,onary stream ofevents in the sense of Khintchine ((s)), as also do the upcrossings and the downcrossingsof the level a by c(f). Let us further &ssume throughout that the means of the numberof crossings of the level a by r(t), and of zero by n'(t) in rrnit time are ffnite. Then itcan be shown by a lemma of Dobrushin (Volkonski ((ol)) that each of these streans isalso ord,erlg in the sense of Khintchine (ta)).

Sufficient conditions for r(tl to satisfy all the above requirements have been givenby Leadbetter((+1). For a (separable) rwrmal stationary process, it follows from thepa,per of Ylvisaker ((7)) that the requirements will be satisfied if and only if the spectrum

X(i.) of a(f) has a finitrs fourth moment f-,foaf1,f1.JO

The occurrence of a local maximum at a point fo has probability zero. tr\rrther,in defining the conditional distribution for the ordinate 'at a point where a localm&ximum occurs ' it is not, in general, sensible to use the oldins,ry (i.e. Radon Nikodym

263



Lesson to be learned

The ratio between the expected valuesof marked and unmarked crossings

is equal to the empirical observable distribution


The 3D-model and the generalized Rice formula Generalized Rice formula

Standard Rice formula

The original Rice formula gives the expected number of level crossings oflevel u per time unit in a stationary process:

n+u = E (#{t [0, 1];X (t) = u, upcrossing})= E (X (0)+ | X (0) = u) fX (0)(u)



Marked Rice formula

Derivative at level crossing is Rayleigh.

The Rice formula for marked crossing gives the expectede number of levelcrossings per time unit in a stationary process, at which a mark has acertain characteristic. The mark can be e.g. the derivative at theupcrossing. We seek Prob(X (t) z | X (t) = u, upcrossing):

n+u (z) = E (#{t [0, 1];X (t) = u, upcrossing, and X (t) z})= E (X (0)+ 1X (0)z | X (0) = u) fX (0)(u)

Prob(X (t) z | X (t) = u, upcrossing) = n+u (z)/n+u

=

z0

m2e

2/m2d, m2 = V (X (t)), a Rayleigh distribution.



Multivariate Rice formula

A generalized (multivariate) Rice formula gives the expected number ofsimultaneous crossings with marks, for example

mark:(Wu,Wv ,Xu,Xv ,Yu,Yv )]s

withsimultanous crossings: (X (s),Y (s)) = (0, 0)

Reference: Azas and Wschebor: Level sets and extrema of randomprocesses and elds, Wiley, 2009.



Application for slope distribution with asynchronoussampling

Consider the slope at (0, 0) in a specied direction. Slope is dened as afunction of (Wu,Wv ,Xu,Xv ,Yu,Yv )]s . With

N0(A) = #{s R2;(s) = (0, 0), slope A}

E (N0(A)) =

R2

E

(| det(s)| 1slopeA | (s) = (0, 0)

)f(s)(0, 0) ds

perpendicularity-bias-correction

The expectation is simply obtained by simulation.


Example Six directional spectra

Example specta - Pierson-Moskowitz with dierent spreading

1

2

30

210

60

240

90

270

120

300

150

330

180 0

Directional Spectrum

Level curves at:

0.20.40.60.811.2

1

2

30

210

60

240

90

270

120

300

150

330

180 0


Level curves at:

0.511.522.53

1

2

30

210

60

240

90

270

120

300

150

330

180 0


Level curves at:

0.511.522.533.544.5

1

2

30

210

60

240

90

270

120

300

150

330

180 0


Level curves at:

123456

1

2

30

210

60

240

90

270

120

300

150

330

180 0


Level curves at:

123456789

1

2

30

210

60

240

90

270

120

300

150

330

180 0


Level curves at:

5101520


Example Asynchrone sampling

Slope PDF across wind and along wind, asynchrone

Top: Large spreading Bottom: Little spreadingLeft: Along wind Right: Across wind; cf. Cox and Munk

0.4 0.2 0 0.2 0.40

1

2

3

4

slope

pd

f

0.2 0.1 0 0.1 0.2 0.30

2

4

6

8

10

slope

pd

f

0.4 0.2 0 0.2 0.40

1

2

3

4

slope

pd

f

0.2 0.1 0 0.1 0.2 0.30

2

4

6

8

10

slope

pd

f


Example Asynchrone sampling

The Cox and Munk experiment 19511954 on waveasymmetry

compared to Lagrange asymmetry

2 0 2


Example Synchrone sampling at crossings

Slope CDF for dierent directional spreading

Left: slope at mean-level upcrossings Right: Slope CDF at downcrossingsModerate linkage parameter: = 0.4

0 0.2 0.4 0.60

0.2

0.4

0.6

0.8

1

slope

cdf

PM spectrum, slope at upcrossings

= 0.4, h = 32m

solid curves: m = 0, 2, 5, 10, 20, 120dashed curve: unidirectional spectrum

0 0.2 0.4 0.60

0.2

0.4

0.6

0.8

1

slope

cdf

PM spectrum, slope at downcrossings

= 0.4, h=32m

solid curves: m=0, 2, 5, 10, 20, 120dashed curve: unidirectional spectrum


Example Slope depends on main heading

Slope CDF for dierent heading; Strong linkage: = 2

Left: slope at mean-level upcrossings Right: Slope CDF at downcrossings

0 0.2 0.4 0.60

0.2

0.4

0.6

0.8

1

slope

cdf

0 =

0 = 3/4

CDF for slope at upcrossingsPM spectrum with m = 10

= 2.0, h = 32m

0 = 0, ...,

dashed curve: unidirectional spectrum

0 = /2

0 = /4

0 = 0

0 0.2 0.4 0.60

0.2

0.4

0.6

0.8

1

slope

cdf

CDF for slope at downcrossingsPM spectrum with m = 10

= 2.0, h = 32m

0 = 0, ...,

dashed curve: unidirectional spectrum

0 =

0 = /4

0 = 0

0 = 3/4

0 = /2


2nd order waves with interaction

From Gauss to Stokes waves

In the Gauss-Lagrange model there is no interaction betweenelementary waves with dierent frequencies the result is just the sumof independent waves

W (t, s) =k

Ak cos(ks kt + k)

The 2nd order Stokes model contains also terms of the type

AkAjHkj cos(... (k + j)t...)AkA

j Hkj cos(... (k j)t...)

and Stokes drift

Vertical and horizontal; Marc Prevosto, Nouguier, and others


2nd order waves with interaction Asymmetric movie

Will be more peaked and can be made asymmetric

A front-back Lagrange simulation based on directional energy spectrum:


2nd order waves with interaction Asymmetric movie

Software

Wafo available on GitHubhttps://github.com/wafo-project

WafoL available athttp://www.maths.lth.se/matstat/wafo/

WafoL a Wafo module for

Analysis of Random Lagrange Waves

Tutorial for WafoL version 1.2

by Georg Lindgren and Marc Prevosto

Lund, January 2016

Faculty of Engineering

Centre for Mathematical Sciences

Mathematical Statistics


Summary and references

Summary

A rst order Lagrange model for individual water particles modelsvertical and horizontal movements as correlated Gaussian processes

The Lagrange model with frequency dependent phase shift gives bothcrest-trough and front/back asymmetry

Distributions of slopes and other wave characteristcs can be found bythe generalized Rice formula - software WafoL

Second order Stokes-Lagrange includes interaction waves



References

Aberg, S. (2007). Wave intensities and slopes in Lagrangian seas. Adv. Appl. Prob.

Lindgren, G. (2006). Slepian models for the stochastic shape of individualLagrange sea waves. Adv. Appl. Prob.

Lindgren, G. (2009). Exact asymmetric slope distributions in stochasticGauss-Lagrange ocean waves. Appl. Ocean Res

Lindgren, G. (2010). Slope distribution in front-back asymmetric stochasticLagrange time waves. Adv. Appl. Prob.

Lindgren, G. and Aberg, S. (2008). First order stochastic Lagrange models forfront-back asymmetric ocean waves. J. OMAE

Lindgren, G. and Lindgren, F. (2011). Stochastic asymmetry properties of 3D-Gauss-Lagrange ocean waves with directional spreading. Stochastic Models

Lindgren, G. (2015). Horseshoe-like patterns in rst-order 3D randomGauss-Lagrange waves with directional spreading. Waves in random and complexmedia

Nouguier, F, Chapron, B., Gurin, C-A. (2015). Second-order Lagrangiandescription of tri-dimensional gravity wave interactions. J. Fluid Mech


IntroductionA long long time agoVery recent movieA long long time agoA long time ago

Gauss versus LagrangeMore recent

The stochastic 3D Lagrange modelThe Gaussian generatorThe free Lagrange model

Front-back asymmetryThe modified Lagrange model

The 3D-model and the generalized Rice formulaExplicit definitionsPependiularitybiasGeneralized Rice formula

ExampleSix directional spectraAsynchrone samplingSynchrone sampling at crossingsSlope depends on main heading

2nd order waves with interactionAsymmetric movie


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stochastic models for ocean waves gaussian fields … seminar asymmetric lagrange ... flection and...

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