We consider highorder strongly nonlinear long wave models expanded in a single small parameter measuring the ratio of the water depth to the characteristic wavelength. By examining its dispersion relation, the highorder system for the bottom velocity is found stable to all disturbances at any order of approximation. On the other hand, systems for other velocities can be unstable and even illposed, as signified by the unbounded maximum growth. Under the steady assumption, a new thirdorder solitary wave solution of the Euler equations is obtained using the highorder strongly nonlinear system and is expanded in an amplitude parameter, which is different from that used in weakly nonlinear theory. The thirdorder solution is shown to well describe various physical quantities induced by a finiteamplitude solitary wave, including the wave profile, horizontal velocity profile, particle velocity at the crest and bottom pressure. For numerical computations, the first and secondorder strongly nonlinear systems for the bottom velocity are considered. It is shown that finite difference schemes are unstable due to truncation errors introduced in approximating highorder spatial derivatives and, therefore, a more accurate spatial discretization scheme is necessary. Using a pseudospectral method based on finite Fourier series combined with an iterative scheme formore »
Limits on Mode Coherence Due to a Nonstatic Convection Zone
The standard theory of pulsations deals with the frequencies and growth rates of infinitesimal perturbations in a stellar model. Modes which are calculated to be linearly driven should increase their amplitudes exponentially with time; the fact that nearly constant amplitudes are usually observed is evidence that nonlinear mechanisms inhibit the growth of finite amplitude pulsations. Models predict that the mass of DAV convection zones is very sensitive to temperature (i.e., MCZ∝T−90eff) leading to the possibility that even "small amplitude" pulsators may experience significant nonlinear effects. In particular, the outer turning point of finiteamplitude gmode pulsations can vary with the local surface temperature, producing a reflected wave that is slightly out of phase with that required for a standing wave. This can lead to a lack of coherence of the mode and a reduction in its global amplitude. We compute the size of this effect for specific examples and discuss the results in the context of Kepler and K2 observations.
 Award ID(s):
 1707419
 Publication Date:
 NSFPAR ID:
 10111448
 Journal Name:
 21st European Workshop on White Dwarfs
 Sponsoring Org:
 National Science Foundation
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