Anomalous statistics and large deviations of turbulent water waves past a step

A computational strategy based on large deviation theory (LDT) is used to study the anomalous statistical features of turbulent surface waves propagating past an abrupt depth change created via a step in the bottom topography. The dynamics of the outgoing waves past the step are modeled using the truncated Korteweg-de Vries (TKdV) equation with random initial conditions at the step drawn from the system’s Gibbs invariant measure of the incoming waves. Within the LDT framework, the probability distributions of the wave height can be obtained via the solution of a deterministic optimization problem. Detailed numerical tests show that this approach accurately captures the non-Gaussian features of the wave height distributions, in particular their asymmetric tails leading to high skewness. These calculations also give the spatio-temporal pattern of the anomalous waves most responsible for these non-Gaussian features. The strategy shows potential for a general class of nonlinear Hamiltonian systems with highly non-Gaussian statistics.

[1]  E. Vanden-Eijnden,et al.  Extreme event probability estimation using PDE-constrained optimization and large deviation theory, with application to tsunamis , 2020, Communications in Applied Mathematics and Computational Science.

[2]  T. Ohira,et al.  Random Perturbations , 2021, Mathematics as a Laboratory Tool.

[3]  Nicholas J. Moore,et al.  Rigorous criteria for anomalous waves induced by abrupt depth change using truncated KdV statistical mechanics , 2020, 2010.02970.

[4]  Andrew J. Majda,et al.  Anomalous Waves Triggered by Abrupt Depth Changes: Laboratory Experiments and Truncated KdV Statistical Mechanics , 2020, Journal of Nonlinear Science.

[5]  A. Majda,et al.  Statistical Phase Transitions and Extreme Events in Shallow Water Waves with an Abrupt Depth Change , 2020 .

[6]  Eric Vanden-Eijnden,et al.  Experimental Evidence of Hydrodynamic Instantons: The Universal Route to Rogue Waves , 2019, Physical Review X.

[7]  A. Majda,et al.  Statistical dynamical model to predict extreme events and anomalous features in shallow water waves with abrupt depth change , 2019, Proceedings of the National Academy of Sciences.

[8]  Eric Vanden-Eijnden,et al.  Numerical computation of rare events via large deviation theory. , 2018, Chaos.

[9]  Eric Vanden-Eijnden,et al.  Extreme Event Quantification in Dynamical Systems with Random Components , 2018, SIAM/ASA J. Uncertain. Quantification.

[10]  K. Speer,et al.  Anomalous wave statistics induced by abrupt depth change , 2018, Physical Review Fluids.

[11]  Eric Vanden-Eijnden,et al.  Rogue waves and large deviations in deep sea , 2017, Proceedings of the National Academy of Sciences.

[12]  A. Majda,et al.  Predicting extreme events for passive scalar turbulence in two-layer baroclinic flows through reduced-order stochastic models , 2018 .

[13]  P. Suret,et al.  Twenty years of progresses in oceanic rogue waves: the role played by weakly nonlinear models , 2016, Natural Hazards.

[14]  A. Majda,et al.  Predicting fat-tailed intermittent probability distributions in passive scalar turbulence with imperfect models through empirical information theory , 2016 .

[15]  Themistoklis P Sapsis,et al.  Unsteady evolution of localized unidirectional deep-water wave groups. , 2014, Physical review. E, Statistical, nonlinear, and soft matter physics.

[16]  P. Taylor,et al.  The physics of anomalous (‘rogue’) ocean waves , 2014, Reports on progress in physics. Physical Society.

[17]  F. Dias,et al.  Extreme waves induced by strong depth transitions: Fully nonlinear results , 2014 .

[18]  Umberto Bortolozzo,et al.  Rogue waves and their generating mechanisms in different physical contexts , 2013 .

[19]  B. Leimkuhler,et al.  Weakly coupled heat bath models for Gibbs-like invariant states in nonlinear wave equations , 2013 .

[20]  K. Trulsen,et al.  Laboratory evidence of freak waves provoked by non-uniform bathymetry , 2012 .

[21]  A. Majda,et al.  Hamiltonian structure and statistically relevant conserved quantities for the truncated Burgers‐Hopf equation , 2003 .

[22]  D K Smith,et al.  Numerical Optimization , 2001, J. Oper. Res. Soc..

[23]  D. Peregrine.  A Modern Introduction to the Mathematical Theory of Water Waves. By R. S. Johnson. Cambridge University Press, 1997. xiv+445 pp. Hardback ISBN 0 521 59172 4 £55.00; paperback 0 521 59832 X £19.95. , 1998, Journal of Fluid Mechanics.

[24]  R. McLachlan.  Symplectic integration of Hamiltonian wave equations , 1993 .

[25]  Srinivasa R. S. Varadhan,et al.  Asymptotic probabilities and differential equations , 1966 .