Numerical investigation of breaking waves in a spectral environment Dmitry Chalikov, Alexander Babanin Swinburne University of Technology, Melbourne, Australia, Institute of Oceanography, Russian Acad. Sci.,Saint-Petersburg Branch EGU General Assembly 2011 : S5.2 – Surface Waves and Wave-Coupled Effects in Lower Atmosphere and Upper Ocean Vienna, April 3-8
Numerical investigation of breaking 1.Onset of breaking is connected with appearance of large steepness, hence all simplified models are not suitable. 2. Model should be based on initial nonlinear equations, which are able to reproduce approaching the surface to ‘vertical wall’ with a good conservation of integral invariants. 3. Investigation of breaking should be done for multimode wave field with a broad and dense spectrum 4. Individual breaking in unpredictable, hence main attention should be focused on statistical characteristics of breaking.
Model Model is based on full nonlinear 2-D (x-z) equation of potential flow with a free surface. Conformal transformation of coordinate Reduces the system to two differential equation, which are approximated with highest accuracy by Fourier transform method and integrated in time with 4-th order Runge-Kutta method.
Breaking Breaking is identified as a first appearance of non-single-value piece of surface. Up to this moment the conservation of integral invariant is excellent. After that moment the surface never returns to stability and numerical and ophysica instability closely follows each other.
Numerical experiments Resolution: number of modes M=1000 number of knots N=4000 Peak wave number Kp=10 Initial conditions: superposition of linear modes with random phases corresponding to the JONSWAP spectrum with U/Cp=2 Time step dt=0.0001, runtime t=1000 (503 peak wave periods). Total number of runs – 5,000 Recording of last stage prior breaking with interval DT=0.1
Definitions
Probability of periods of development of breaking
Example of individual wave breaking elevation energy Evolution of wave energy and maximum columnar energy
a - Evolution of the energy of selected wave averaged over wave length prior breaking as function of time t, expressed in peak wave periods. Aggregated grey lines correspond to single cases, solid line represent the averaged over all cases evolution and dotted lines correspond to dispersion. Moment of breaking corresponds to time t=0 ; b – the same as in panel a, but for maximum value of columnar energy in selected window; c – evolution of for idealised initial conditions; d – evolution of in spectral environment.
Probability distribution for Banner’s criterion of breaking.
Trough-to-crest height Length Overall steepness Asymmetry (I.C.) Asymmetry, S.E. Front trough depth Rear trough depth Geometrical characteristics prior breaking
Probability of overall steepness distribution
Differential characteristics of surface Surface kinematic characteristics
Dynamic characteristics
Parameterization of breaking The dissipation is not too sensitive to parameters Do we really need the criterion for breaking onset? The answer is ‘No!’
Local evolution of surface due to breaking
Spectrum of wave dissipation for different steepness
CONCLUSIONS 1. Spectral approach to investigation of breaking waves is senseless, since the breaking occurs in a narrow interval of physical space, and its spectral image is difficult to interpret. 2. Mechanism of breaking in a spectral environment is quite different to that for idealized situations when the wave field is represented by a few modes. 3. The breaking develops very quickly, on average faster than for half of peak wave period. 4. There was found no robust predictor for breaking in the spectral environment. 5. The differential geometrical and kinematical characteristics like first and second derivatives of elevation, surface orbital velocity and individual accelerations indicate a development towards breaking clearer, but they rather describe the process of breaking itself than predict its onset. 6. Differential formulation of breaking is a resolvable problem