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mode_split1.lyx
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mode_split1.lyx
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#LyX 1.5.2 created this file. For more info see http://www.lyx.org/
\lyxformat 276
\begin_document
\begin_header
\textclass article
\language english
\inputencoding auto
\font_roman default
\font_sans default
\font_typewriter default
\font_default_family default
\font_sc false
\font_osf false
\font_sf_scale 100
\font_tt_scale 100
\graphics default
\paperfontsize default
\papersize default
\use_geometry false
\use_amsmath 1
\use_esint 1
\cite_engine basic
\use_bibtopic false
\paperorientation portrait
\secnumdepth 3
\tocdepth 3
\paragraph_separation indent
\defskip medskip
\quotes_language english
\papercolumns 1
\papersides 1
\paperpagestyle default
\tracking_changes false
\output_changes false
\author ""
\end_header
\begin_body
\begin_layout Standard
Mode-splitting error
\end_layout
\begin_layout Standard
In the split-explicit method, the shallow water equations (SWE) are advanced
in time, using the smaller CFL-limited time step, to obtain the depth-integrate
d mode,
\emph on
e.g.
\emph default
,
\end_layout
\begin_layout Standard
\begin_inset Formula $\frac{\partial\overline{U}}{\partial t}+\ldots=-gD\nabla_{x}\zeta+\{gD\nabla_{x}\zeta+f\}$
\end_inset
.
\end_layout
\begin_layout Standard
Here,
\begin_inset Formula $g$
\end_inset
is the acceleration of gravity;
\begin_inset Formula $D=h+\zeta$
\end_inset
is total depth;
\begin_inset Formula $\overline{U}\equiv D\overline{u}$
\end_inset
is the depth-integrated velocity (barotropic mass flux);
\begin_inset Formula $\nabla_{x}\zeta$
\end_inset
is a shorthand for
\begin_inset Formula $\partial\zeta/\partial x$
\end_inset
; and
\end_layout
\begin_layout Standard
\begin_inset Formula $f=-\frac{1}{\rho_{0}}\int_{-h}^{\\zeta}\frac{\partial P}{\partial x}dz$
\end_inset
\end_layout
\begin_layout Standard
is the vertically integrated pressure gradient.
The latter is a functional of the topography, free-surface gradient, and
the free surface itself, as well as the vertical distribution of density
and its gradient,
\end_layout
\begin_layout Standard
\end_layout
\end_body
\end_document