TY - JOUR

T1 - Statistical dynamical model to predict extreme events and anomalous features in shallow water waves with abrupt depth change

AU - Majda, Andrew J.

AU - Moore, M. N.J.

AU - Qi, Di

N1 - Funding Information:
ACKNOWLEDGMENTS. The research of A.J.M. is partially supported by Office of Naval Research Grant MURI N00014-16-1-2161, and D.Q. is supported as a postdoctoral fellow on the grant. M.N.J.M. acknowledges support from Simons Grant 524259.
Publisher Copyright:
© 2019 National Academy of Sciences. All Rights Reserved.

PY - 2019

Y1 - 2019

N2 - Understanding and predicting extreme events and their anomalous statistics in complex nonlinear systems are a grand challenge in climate, material, and neuroscience as well as for engineering design. Recent laboratory experiments in weakly turbulent shallow water reveal a remarkable transition from Gaussian to anomalous behavior as surface waves cross an abrupt depth change (ADC). Downstream of the ADC, probability density functions of surface displacement exhibit strong positive skewness accompanied by an elevated level of extreme events. Here, we develop a statistical dynamical model to explain and quantitatively predict the above anomalous statistical behavior as experimental control parameters are varied. The first step is to use incoming and outgoing truncated Korteweg–de Vries (TKdV) equations matched in time at the ADC. The TKdV equation is a Hamiltonian system, which induces incoming and outgoing statistical Gibbs invariant measures. The statistical matching of the known nearly Gaussian incoming Gibbs state at the ADC completely determines the predicted anomalous outgoing Gibbs state, which can be calculated by a simple sampling algorithm verified by direct numerical simulations, and successfully captures key features of the experiment. There is even an analytic formula for the anomalous outgoing skewness. The strategy here should be useful for predicting extreme anomalous statistical behavior in other dispersive media.

AB - Understanding and predicting extreme events and their anomalous statistics in complex nonlinear systems are a grand challenge in climate, material, and neuroscience as well as for engineering design. Recent laboratory experiments in weakly turbulent shallow water reveal a remarkable transition from Gaussian to anomalous behavior as surface waves cross an abrupt depth change (ADC). Downstream of the ADC, probability density functions of surface displacement exhibit strong positive skewness accompanied by an elevated level of extreme events. Here, we develop a statistical dynamical model to explain and quantitatively predict the above anomalous statistical behavior as experimental control parameters are varied. The first step is to use incoming and outgoing truncated Korteweg–de Vries (TKdV) equations matched in time at the ADC. The TKdV equation is a Hamiltonian system, which induces incoming and outgoing statistical Gibbs invariant measures. The statistical matching of the known nearly Gaussian incoming Gibbs state at the ADC completely determines the predicted anomalous outgoing Gibbs state, which can be calculated by a simple sampling algorithm verified by direct numerical simulations, and successfully captures key features of the experiment. There is even an analytic formula for the anomalous outgoing skewness. The strategy here should be useful for predicting extreme anomalous statistical behavior in other dispersive media.

KW - Extreme anomalous event

KW - Matching Gibbs measures

KW - Statistical TKdV model

KW - Surface wave displacement

KW - Surface wave slope

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U2 - 10.1073/pnas.1820467116

DO - 10.1073/pnas.1820467116

M3 - Article

C2 - 30760588

AN - SCOPUS:85062545501

SN - 0027-8424

VL - 116

SP - 3982

EP - 3987

JO - Proceedings of the National Academy of Sciences of the United States of America

JF - Proceedings of the National Academy of Sciences of the United States of America

IS - 10

ER -