We consider the difference schemes for the one-dimensional versions of nonlinear dispersive shallow water models. We analyse the dissipative and dispersive properties and give the results of numerical calculations.Nonlinear dispersive models hold an intermediate position between the complete equations of fluid motion (the Euler equations of motion or the Cauchy-Poisson problem) and the nonlinear and linear shallow water equations [12]. On the one hand, they are approximate models and may result from an asymptotic expansion of the complete model equations in terms of the nonlineary parameter α=^4 0 /// 0 and the variance β = (// 0 /A 0 ) 2 , where A Q is the mean value of the amplitude, // 0 is the average depth, A O is the wavelength [1,16,17,20]. On the other hand, unlike the nonlinear and linear shallow water equations these equations take into account the relationship between the phase velocities of wave motion and the wave number. This allows us to describe the effects of the wave motion which cannot be described in the framework of the nonlinear and linear shallow water equations.In spite of great interest in the nonlinear dispersive models there are not many results of an investigation of these equations as equations of mathematical physics.Note that there exist exact analytic solutions of a travelling wave type for some nonlinear dispersive models (the Green -Naghdi model [19], the Korteweg-de Vries model [10], the regularized long-wave equation [5], the Kim-Reid-Whittaker model [7]). The continuous dependence of the solution on the initial data is investigated for the Bonna-Smith model in [4].Therefore the method of numerical calculations is one of the most frequently used methods of obtaining the solution.The nonlinear dispersive models can be divided into several groups by the dispersive relation. The first model in [16] and the Peregrine second model in [17] have the same dispersive relation which allows imaginary values of the frequency. Stable difference algorithms are not known for them [8].The stable difference algorithms for nonlinear dispersive models, in which the frequency ω is a real function of the wave number K, are described in the literature. These are nonlinear dispersive models by Korteweg-de Vries [3], Long [11], Green-Naghdi [6], Pelinovsky-Zheleznyak [20], Aleshkov [13], Peregrine [17], Kim-Reid-Whittaker [15], the regularized long-wave equation [5]. The nonlinear dispersive models by Boussinesq, Kim-Reid-Whittaker, Peregrine (first model), Green-Naghdi, Bazdenkov-Morozov-Pogutstse [2],

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The problem of large-density variations in supercooled and ambient water has been widely discussed in the past years. Recent studies have indicated the possibility of nanometer-sized density variations on the subpicosecond and picosecond time scales. The nature of fluctuating density heterogeneities remains a highly debated issue. In the present work, we address the problem of possible association of such density variations with the dynamics of terahertz longitudinal acoustic-like modes in liquid water. Our study is based on the fact that the subpicosecond dynamics of liquid water are essentially governed by the structural relaxation. Using a mode coupling theory approach, we found that for typical values of parameters of liquid water, the dynamic mechanism coming from the combination of the structural relaxation process and the finiteness of the amplitude of terahertz longitudinal acoustic-like mode gives rise to a soliton-like collective mode on a temperature-dependent nanometer length scale. The characteristics of this mode are consistent with the estimates of the amplitudes and temperature-dependent correlation lengths of density fluctuations in liquid water obtained in experiments and simulations. Thus, the fully dynamic mechanism could contribute to the formation and dynamics of fluctuating density heterogeneities. The soliton-like collective excitations suggested by our analysis may be relevant to different phenomena connected with supercooled water and can be expected to be associated with some ultrafast biological processes.

Получено 28 июня 2009 г.Найдено точное решение стационарной задачи ветрового движения вязкой двухслойной жидкости для двумерного в вертикальной плоскости течения и для дрейфовой составляющей трехмерного течения. На дне бассейна ставится условие проскальзывания, на вертикальных боковых стенкахусловие непротекания. Приводятся примеры расчетов конкретных течений и сравнение полученных результатов с решениями аналогичной задачи по модели Экмана (без учета горизонтальной вязкости).Ключевые слова: вязкая жидкость, ветровое движение жидкости, модель Экмана Abstract. -Exact solution of the stationary problem of wind-induced flow of two-layered viscous fluid is found for two-dimensional in vertical plane current and for drift current in three-dimensional case. The condition of sliding is set on the bottom of a water body. The condition of nonpassage is set on the lateral surface. Results of some calculations are given in comparison with those obtained using Ekman's model (which does not take into account horizontal viscosity of the layer). ВведениеПроблема определения ветрового движения жидкости в замкнутом водоеме очень сложная, но практически важная задача [5]. В общей постановке она сводится к решению нестационарной начально-краевой задачи для системы нелинейных уравнений, решение которой возможно только численными методами.Большую помощь при этом оказывают аналитические решения упрощенных задач, получающихся в результате правдоподобных предположений. При этом выстраивается некоторая иерархия используемых моделей: достаточно простые модели двумерного (в вертикальной плоскости) стационарного течения однородной жидкости без учета горизонтальной вязкости, модели с учетом горизонтальной вязкости, трехмерные течения без учета членов горизонтальной вязкости, с учетом этих членов, нестационарные течения [1][2][3][4]. Чем проще модель, тем больше аналитических решений удается получить и проанализировать. Так, для первой из указанных моделей известно аналитическое решение для случая неровного дна и переменного коэффициента вертикального турбулентного обмена. Чем сложнее модель, тем аналитических результатов, как правило, меньше, и задача решается численными методами.

In this paper we consider various difference algorithms for nonlinear dispersive shallow-water models for the two-dimensional case. We analyse the dissipative and dispersive properties of the difference schemes, give the results of the numerical calculations of model problems.