סמינר מחלקתי בי"ס להנדסה מכאנית Prof. Victor Shrira

KINETIC EQUATIONS VS DIRECT NUMERICAL SIMULATIONS OF WEAKLY NONLINEAR RANDOM WAVE FIELDS:
WHAT IS WRONG WITH THE KINETIC EQUATIONS?
 

The challenge of describing   evolution of random weakly nonlinear dispersive waves in fluids and solids in various contexts is a major open fundamental problem despite being intensively studied theoretically and experimentally for more than fifty years. In contrast to the classical hydrodynamic turbulence, there is a well-established general formalism for treating weakly nonlinear wave fields that exploits smallness of nonlinearity and subtle assumptions about quasi-Gaussianity of a statistically homogeneous wave field. This approach leads to a closed equation for the second statistical momenta of the field which called the kinetic equation (KE). Although the theory based upon the KE has been able to predict the major features of wave field evolution and is widely used.  However the basic question --- to what extent the theory captures the actual behavior of the wave field --- remains open.   Here we address it  by performing a detailed comparison of predictions of the KE and its generalization (gKE) with the results of direct numerical simulations (DNS) employing  the algorithm  specially designed for  long term evolution of random weakly nonlinear wave fields. For certainty and without much loss of generality we perform these comparisons for weakly nonlinear water waves.

To make the comparisons maximally clean and simple and retain as much generality as possible we do the following. We take as the starting point the equations of motion in the form of the "four-wave" Zakharov equation without forcing. The KE and gKE  are derived from the this Zakharov equation under an assumption of weak non-Gausianity of the wave field and a closure hypothesis for the field higher statistical moments. We simulate numerically long-term evolution of  wave spectra without  forcing using three different models: (i)  the classical kinetic equation (KE); (ii)  the generalized kinetic equation (gKE)  valid also  when the wave spectrum is changing rapidly; (iii)  the DNS based on the Zakharov  integrodifferential equation for water wave which  does not rely on any statistical assumptions. As the initial conditions we choose two spectra with the same frequency distribution and different degrees of directionality. All three approaches demonstrate very close evolution of integral characteristics of spectra. Theoretically predicted regimes and asympotics do occur. However, there are substantial systematic differences (e.g. the broadening of angular spectra is much faster for the kinetic equations, the shape of the spectra are also noticeably different), which suggests the presence and significance of coherent interactions not accounted for by the established closure for the kinetic equations. This implies that the fundamental issue of closure for random wave fields has to be revisited.
 

INERTIAL WAVES AND DEEP OCEAN MIXING

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