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Slope distribution in front-back asymmetric stochastic Lagrange time waves

Part of:Stochastic processesGeophysics (general)Incompressible inviscid fluidsGeophysics Waves

Published online by Cambridge University Press:  01 July 2016

G. Lindgren
G. Lindgren*
Affiliation:
Lund University
*
∗ Postal address: Mathematical Statistics, Lund University, Box 118, SE-221 00 Lund, Sweden. Email address: georg@maths.lth.se
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Abstract

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The stochastic Lagrange wave model is a realistic alternative to the Gaussian linear wave model, which has been successfully used in ocean engineering for more than half a century. In this paper we present the slope distributions and other characteristic distributions at level crossings for asymmetric Lagrange time waves, i.e. what can be observed at a fixed measuring station, thereby extending results previously given for space waves. The distributions are given as expectations in a multivariate normal distribution, and they have to be evaluated by simulation or numerical integration. Interesting characteristic variables are the slope in time, the slope in space, and the vertical particle velocity when the waves are observed close to instances when the water level crosses a predetermined level. The theory has been made possible by recent generalizations of Rice's formula for the expected number of marked crossings in random fields.

Keywords

Crossing theoryGaussian processPalm distributionRice formulaSlepian modelwave steepness

MSC classification

Primary:60G15: Gaussian processes
Secondary:60G70: Extreme value theory; extremal processes74J05: Linear waves74J30: Nonlinear waves76B15: Water waves, gravity waves; dispersion and scattering, nonlinear interaction86A05: Hydrology, hydrography, oceanography

Information

Type
General Applied Probability
Information
Advances in Applied Probability , Volume 42 , Issue 2 , June 2010, pp. 489 - 508
Copyright
Copyright © Applied Probability Trust 2010 

References

Åberg, S. (2007). Wave intensities and slopes in Lagrangian seas. Adv. Appl. Prob. 39, 1020–1035.Google Scholar
Åberg, S. and Lindgren, G. (2008). Height distribution of stochastic Lagrange ocean waves. Prob. Eng. Mechanics 23, 359–363.Google Scholar
Azaïs, J.-M. and Wschebor, M. (2009). Level Sets and Extrema of Random Processes and Fields. John Wiley, Hoboken.CrossRefGoogle Scholar
Elgar, S. (1987). Relationships involving third moments and bispectra of a harmonic process. IEEE Trans. Acoust. Speech Signal Process. 35, 1725–1726.Google Scholar
Fouques, S., Krogstad, H. E. and Myrhaug, D. (2006). A second-order Lagrangian model for irregular ocean waves. Trans. ASME J. Offshore Mechanics Arctic Eng. 128, 177–183.CrossRefGoogle Scholar
Gerstner, F. J. (1809). Theorie der Wellen. Ann. Phys. 32, 412–445.Google Scholar
Gjøsund, S. H. (2003). A Lagrangian model for irregular waves and wave kinematics. Trans. ASME J. Offshore Mechanics Arctic Eng. 125, 94–102.Google Scholar
Li, W. V. and Wei, A. (2009). Gaussian integrals involving absolute value functions. In High Dimensional Probability V. The Luminy Volume (Beachwood, Ohio, USA), Institute of Mathematical Statistics, pp. 43–59.Google Scholar
Lindgren, G. (2006). Slepian models for the stochastic shape of individual Lagrange sea waves. Adv. Appl. Prob. 38, 430–450.Google Scholar
Lindgren, G. (2009). Exact asymmetric slope distributions in stochastic Gauss–Lagrange ocean waves. Appl. Ocean Res. 31, 65–73.Google Scholar
Lindgren, G. and Åberg, S. (2009). First order stochastic Lagrange models for front-back asymmetric ocean waves. J. Offshore Mechanics Arctic Eng. 131, 031602-1–031602-8.Google Scholar
Mercadier, C. (2006). Numerical bounds for the distributions of the maxima of some one- and two-parameter Gaussian processes. Adv. Appl. Prob. 38, 149–170.Google Scholar
Miche, M. (1944). Mouvements ondulatoires de la mer en profondeur constante ou décroissante. Forme limit de la houle lors de son déferlement. Application aux digues marines. Ann. Ponts Chassées 1944, 25–78.Google Scholar
Socquet-Juglard, H. et al. (2004). Spatial extremes, shape of large waves, and Lagrangian models. In Proc. Rogue Waves 2004, eds Olagnon, M. and Prevosto, M., IFREMER. Available at http://www.ifremer.fr/web-com/stw2004/rw/fullpapers/krogstad.pdf.Google Scholar