commit 70d4cd983be8a044c0180250a9842e1c548b672b
parent eba73f69e728b6abd663a48b0d1cf5c93975f973
Author: Ivan Gankevich <igankevich@ya.ru>
Date: Fri, 19 Jul 2019 21:29:41 +0300
Finish model.
Diffstat:
| main.tex | | | 39 | +++++++++++++++++++++++++++++++++------ |
1 file changed, 33 insertions(+), 6 deletions(-)
diff --git a/main.tex b/main.tex
@@ -113,6 +113,13 @@ wave with steepness \(\approx{}1/4\) as well as 30 degree slope limit for
progressive (traveling) wave with steepness \(\approx{}1/7\) are not correctly
captured by the model.
+State of the art mathematical and computational models do not simulate wave
+groups that are integral part of ocean wavy surface motion. Our model, which is
+a modified version of Gertner wave, includes wave groups. They are described as
+a dependency between fluid parcel trajectory radius and instantaneous
+displacement of the parcel with respect to calm sea level
+(fig.~\ref{fig-trochoidal}).
+
\begin{figure}
\centering
\includegraphics[width=\textwidth]{graphics/03-wind-wave.png}
@@ -123,9 +130,26 @@ captured by the model.
(bottom).\label{fig-trochoidal}}
\end{figure}
-\(^{A}R_W=^{A}Kr_W\left(\cos{}x_W-1\right)\), where \(^{A}K=[1,0..\sqrt{2}]\)
-\(^{W}R_W=^{W}Kr_W\left(\sin{}x_W-1\right)\), where \(^{W}K=[0..1]\)
-
+We write adjusted radius as \(^{A}R_W={}^{A}K{}r_W\left(\cos{}x_W-1\right)\), where
+\(^{A}K=[1,0..\sqrt{2}]\) is radius coefficient that makes wave crests cnoidal
+and raises mean sea level. We choose \(^{A}K\) to be slightly less than 1 to
+reduce the effect of gusty winds on the wave form and prevent forming of
+cycloidal loops in wave crests, that appear for waves with overly large
+amplitude, that may occur as a result of the interference with waves
+heading from the opposite direction.
+
+The pressure on windward slope of the wave is smaller, because wind slides on
+the surface of the wave at high speed and makes the slope more flat, while on
+the leeward slope of the wave wind speed drops significantly or even goes to
+nought and creates vorticity.
+
+The coefficient of wind stress \(^{W}K\) (the parameter that was shifted by one
+fourth of the phase) determines the assymmetry of steepness of windward and
+leeward slopes of the wave (fig.~\ref{fig-trochoidal}):
+\(^{W}R_W={}^{W}Kr_W\left(\sin{}x_W-1\right)\), where \(^{W}K=[0..1]\). The
+coefficient is close to unity for fresh wind waves and close to nought for
+swell. In two-dimensional sea surface \(^{W}K\) is used in dot product
+between wind and wave direction vectors:
\begin{equation*}
\begin{aligned}
\zeta_Z &= r_W \cos x_W \exp\left(
@@ -137,16 +161,19 @@ captured by the model.
2\pi \left[
-z_W + r_W ^{A} K \left( \cos x_W - 1\right) + r_W ^{W} K\sin x_W
\right] / \lambda_W
- \right)
+ \right).
\end{aligned}
\end{equation*}
-
+Then energy conservation is defined by Bernoulli's principle
\begin{equation*}
\frac{\rho V^2}{2} + \rho g \zeta_W = \text{const},
\qquad \left[\text{N}/\text{m}^2\right]
\end{equation*}
+where particle velocity \(V\) contributes the most to balancing
+the pressure \(\rho{}g\zeta_W\) on the wavy surface down to nought
+for breaking waves.
-\section{Group structures of trochoidal waves}
+\section{Trochoidal wave groups}
\begin{figure}
\centering
