 |
| Titel |
Spatial and temporal compact equations for water waves |
| VerfasserIn |
Alexander Dyachenko, Dmitriy Kachulin, Vladimir Zakharov |
| Konferenz |
EGU General Assembly 2016
|
| Medientyp |
Artikel
|
| Sprache |
en
|
| Digitales Dokument |
PDF |
| Erschienen |
In: GRA - Volume 18 (2016) |
| Datensatznummer |
250125005
|
| Publikation (Nr.) |
 EGU/EGU2016-4527.pdf |
|
|
|
|
|
| Zusammenfassung |
| %%%%%%%%%%%%%%%%%%%%%%% file template.tex %%%%%%%%%%%%%%%%%%%%%%%%%
%
% This is a general template file for the LaTeX package SVJour3
% for Springer journals. Springer Heidelberg 2010/09/16
%
% Copy it to a new file with a new name and use it as the basis
% for your article. Delete % signs as needed.
%
% This template includes a few options for different layouts and
% content for various journals. Please consult a previous issue of
% your journal as needed.
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% First comes an example EPS file -- just ignore it and
% proceed on the \documentclass line
% your LaTeX will extract the file if required
%
\RequirePackage{fix-cm}
%
\documentclass{svjour3} % onecolumn (standard format)
%\documentclass[smallcondensed]{svjour3} % onecolumn (ditto)
%\documentclass[smallextended]{svjour3} % onecolumn (second format)
%\documentclass[twocolumn]{svjour3} % twocolumn
%
%\smartqed % flush right qed marks, e.g. at end of proof
%
\usepackage{graphicx}
%
% \usepackage{mathptmx} % use Times fonts if available on your TeX system
%
% insert here the call for the packages your document requires
%\usepackage{latexsym}
% etc.
%
% please place your own definitions here and don't use \def but
% \newcommand{}{}
%
% Insert the name of "your journal" with
% \journalname{myjournal}
%
\begin{document}
\newcommand\PW{120mm}
%\title{Insert your title here%\thanks{Grants or other notes
%about the article that should go on the front page should be
%placed here. General acknowledgments should be placed at the end of the article.}
%}
\title{Spatial and temporal compact equations for water waves}
%\titlerunning{Short form of title} % if too long for running head
\author{A.I. Dyachenko, D.I. Kachulin and V.E.~Zakharov} %etc.
%\authorrunning{Short form of author list} % if too long for running head
\institute{A.I Dyachenko \at
Landau Institute for Theoretical Physics, 142432, Chernogolovka, Russia \\
Novosibirsk State University, 630090, Novosibirsk-90, Russia\\
\email{alexd@itp.ac.ru} % \\
% \emph{Present address:} of F. Author % if needed
\and
D.I. Kachulin \at
Novosibirsk State University, 630090, Novosibirsk-90, Russia
\and
V.E. Zakharov \at
Novosibirsk State University, 630090, Novosibirsk-90, Russia\\
Department of Mathematics, University of Arizona, Tucson, AZ, 857201, USA\\
Physical Institute of RAS, Leninskiy prospekt, 53, Moscow, 119991, Russia
}
\date{\ }
% The correct dates will be entered by the editor
\maketitle
A one-dimensional potential flow of an ideal incompressible
fluid with a free surface in a gravity field is the Hamiltonian system with the Hamiltonian:
\begin{eqnarray}\nonumber
H &=& \frac{1}{2}\int\!dx\!\int_{-\infty}^\eta |\nabla\phi|^2dz + \frac{g}{2}\int \eta^2\!dx\cr
\end{eqnarray}
$\phi(x,z,t)$ - is the potential of the fluid, $g$ - gravity acceleration, $\eta(x,t)$ - surface profile
Hamiltonian can be expanded as infinite series of steepness:
\begin{eqnarray} \label{Ham4}
H &=& H_2 + H_3 + H_4 + \dots\cr
H_2 &=& \frac{1}{2}\int (g\eta^2 + \psi \hat k\psi) dx, \cr
H_3 &=& -\frac{1}{2}\int \{(\hat k\psi)^2 -(\psi_x)^2\}\eta dx,\cr
H_4 &=&\frac{1}{2}\int \{\psi_{xx} \eta^2 \hat k\psi + \psi \hat k(\eta \hat k(\eta \hat k\psi))\} dx.
\end{eqnarray}
%\begin{eqnarray*}
%H \!&=& \!\frac{1}{2}\!\!\int \!\!{g\eta^2 \!\!+ \!\psi \hat k\psi} dx
% -\frac{1}{2}\!\!\int \!\!\{(\hat k\psi)^2 \!-(\psi_x)^2\}\eta dx +\cr
%&+&\frac{1}{2}\!\!\int\!\! \{\psi_{xx} \eta^2 \hat k\psi \!+ \!\psi \hat k(\eta \hat k(\eta \hat k\psi))\} dx +\dots
%\end{eqnarray*}
where $\hat k$ corresponds to the multiplication by $|k|$ in Fourier space, $\psi(x,t)= \phi(x,\eta(x,t),t)$.
This truncated Hamiltonian is enough for gravity waves of moderate amplitudes and can not be reduced.
We have derived self-consistent compact equations, both spatial and temporal, for unidirectional water waves.
Equations are written for normal complex variable $c(x,t)$, not for $\psi(x,t)$ and $\eta(x,t)$.
Hamiltonian for temporal compact equation can be written in $x$-space as following:
\begin{equation}\label{SPACE_C}
H = \int\!c^*\hat V c \hspace{2pt}dx + \frac{1}{2}\int\!\left [
\frac{i}{4}(c^2 \frac{\partial}{\partial x} {c^*}^2 - {c^*}^2 \frac{\partial}{\partial x} c^2 )-
|c|^2 \hat K(|c|^2)\right ]dx
\end{equation}
Here operator $\hat V$ in K-space is so that $V_k = \frac{\omega_k}{k}$. If along with this to introduce Gardner-Zakharov-Faddeev bracket (for the analytic in the upper half-plane function)
\begin{equation}\label{GZF}
\partial^+_x \Leftrightarrow ik\theta_k
\end{equation}
Hamiltonian for spatial compact equation is the following:
\begin{eqnarray}\label{H24}
&&H=\frac{1}{g}\int\frac{1}{\omega}|c_{\omega}|^2 d\omega +\cr
&+&\frac{1}{2g^3}\int|c|^2(\ddot c^*c + \ddot c c^*)dt +
\frac{i}{g^2}\int |c|^2\hat\omega(\dot c c^* - c\dot c^*)dt.
\end{eqnarray}
equation of motion is:
\begin{eqnarray}\label{t-space}
&&\frac{\partial }{\partial x}c +\frac{i}{g} \frac{\partial^2}{\partial t^2}c =\cr
&=& \frac{1}{2g^3}\frac{\partial^3}{\partial t^3}
\left[
\frac{\partial^2}{\partial t^2}\left(|c|^2c\right) +2 |c|^2\ddot c +\ddot c^*c^2
\right]+\cr
&+&\frac{i}{g^3} \frac{\partial^3}{\partial t^3}
\left[
\frac{\partial }{\partial t}\left( c\hat\omega |c|^2\right) + \dot c \hat\omega |c|^2 +
c \hat\omega\left(\dot c c^* - c\dot c^*\right)
\right].
\end{eqnarray}
It solves the spatial Cauchy problem for surface gravity wave on the deep water.
Main features of the equations are:
\begin{itemize}
\item Equations are written for complex normal variable $c(x,t)$ which is analytic function in the upper half-plane
\item Hamiltonians both for temporal and spatial equations are very simple
\item It can be easily implemented for numerical simulation
\end{itemize}
The equations can be generalized for "almost" 2-D waves like KdV is generalized to KP.
This work was supported by was Grant "Wave turbulence: theory, numerical simulation, experiment" \#14-22-00174 of Russian Science Foundation.
\end{document} |
|
|
|
|
|
|