Changeset 4644 for branches/2014/dev_r4642_WavesWG/DOC/TexFiles/Chapters/Chap_DYN.tex

Timestamp:

2014-05-15T15:56:53+02:00 (10 years ago)

Author:

acc

Message:

Branch 2014/dev_r4642_WavesWG #1323. Import of surface wave components from the 2013/dev_ECMWF_waves branch + a few compatability changes and some mislaid documentation

File:

• branches/2014/dev_r4642_WavesWG/DOC/TexFiles/Chapters/Chap_DYN.tex (modified) (3 diffs)

branches/2014/dev_r4642_WavesWG/DOC/TexFiles/Chapters/Chap_DYN.tex

@@ -1 +1 @@
 % ================================================================
-% Chapter � Ocean Dynamics (DYN)
+% Chapter Ocean Dynamics (DYN)
 % ================================================================
 \chapter{Ocean Dynamics (DYN)}
@@ -795 +795 @@
 %>   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >
 \begin{figure}[!t]    \begin{center}
-\includegraphics[width=0.7\textwidth]{./TexFiles/Figures/Fig_time_split.pdf}
+\includegraphics[width=0.7\textwidth]{./TexFiles/Figures/Fig_DYN_dynspg_ts.pdf}
 \caption{  \label{Fig_DYN_dynspg_ts}
 Schematic of the split-explicit time stepping scheme for the external 
@@ -1293 +1293 @@
 
 % ================================================================
+% Coriolis-Stokes force
+% ================================================================
+\section  [Coriolis-Stokes Force (\textit{dynstcor})]
+                {Coriolis-Stokes Force (\mdl{dynstcor})}
+\label{DYN_stcor}
+Waves set up a Lagrangian drift in the down-wave direction known
+as the Stokes drift \citep{Stokes_TCPS47}. Although its drift speed
+$\mathbf{v}_\mathrm{s}$ decays rapidly with depth, it can be substantial
+near the surface ($v_\mathrm{s} {\sim}0.7\, \mathrm{m/s}$). In combination
+with the earth's rotation it adds an additional veering to the upper-ocean
+currents known as the Coriolis-Stokes force \citep{Hasselmann_GAFD70},
+\begin{equation}
+   \frac{D\mathbf{u}}{Dt} = -\frac{1}{\rho} \nabla p 
+  + (\mathbf{u} + \mathbf{v}_\mathrm{s}) \times f\hat{\mathbf{z}}
+  + \frac{1}{\rho} \frac{\partial \tau}{\partial z}.
+  \label{Eq_dynstcor_stcor}
+\end{equation}
+It requires integration of the full two-dimensional spectrum to get the
+Stokes profile \citep{Janssen_AH04,Janssen_JGR12},
+\begin{equation}
+   \mathbf{v}_\mathrm{s}(z) = 4\pi \int_0^{2\pi} \int_0^{\infty} 
+                              f \mathbf{k} e^{2kz} F(f,\theta) \, df\, d\theta,
+   \label{Eq_dynstcor_uvfth}
+\end{equation}
+This is computationally demanding and requires access to the full
+two-dimensional wave spectra from a numerical wave model (see e.g. the
+ECMWF WAM implementation, ECWAM, \citealt{wam38r1}), so
+we introduce a parameterized Stokes drift velocity profile
+\citep{Janssen_ECMWF13,Breivik_ECMWF13},
+\begin{equation}
+   \mathbf{v}_\mathrm{e} = \mathbf{v}_0 
+   \frac{e^{2k_\mathrm{e}z}}{1-8k_\mathrm{e}z}.
+   \label{Eq_dynstcor_uve1}
+\end{equation}
+The surface velocity vector $\mathbf{v}_0$ is computed by ECWAM and is
+available both in ERA-Interim \citep{Dee_QJRMS11} and operationally.
+
+The transport under such a profile involves the exponential integral $E_1$ and 
+can be solved analytically \citep{Breivik_ECMWF13} to yield
+\begin{equation}
+   {T}_\mathrm{s} = \frac{{v}_0 e^{1/4} E_1(1/4)}{8 k_\mathrm{e}}.
+   \label{Eq_dynstcor_UVe} 
+\end{equation}
+This imposes the following constraint on the wavenumber,
+\begin{equation}
+   k_\mathrm{e} = \frac{{v}_0 e^{1/4} E_1(1/4)}{8
+   {T}_\mathrm{s}}.
+   \label{Eq_dynstcor_ke} 
+\end{equation}
+Here $E_1(1/4) \approx 1.34$, thus
+\begin{equation}
+   k_\mathrm{e} \approx \frac{{v}_0}{5.97{T}_\mathrm{s}}.
+   \label{Eq_dynstcor_keapprox} 
+\end{equation}
+The $n$-th order spectral moment is defined as
+\begin{equation}
+   m_{n} = \int_0^{2\pi} \int_0^{\infty} 
+           f^{n} F(f,\theta) \, df\, d\theta.
+   \label{Eq_dynstcor_moment}
+\end{equation}
+The mean frequency is defined as $\overline{f} = m_1/m_0$
+\citep{wmo98,Holthuijsen07} and the significant wave height $H_{m_0} =
+4\sqrt{m_0}$.  We can derive the first moment from the integrated parameters
+of a wave model or from wave observations and find an estimate for the
+Stokes transport,
+\begin{equation}
+  \mathbf{T}_\mathrm{s} \approx \frac{2\pi}{16} \overline{f} H_{m_0}^2
+  \hat{\mathbf{k}}_\mathrm{s}.
+  \label{Eq_dynstcor_UVHsf}
+\end{equation}
+Here $\hat{\mathbf{k}}_\mathrm{s} = (\sin \theta_\mathrm{s},
+\cos \theta_\mathrm{s})$ is the unit vector in the
+direction $\theta_\mathrm{s}$ of the Stokes transport.  From
+Eqs~(\ref{Eq_dynstcor_keapprox})-(\ref{Eq_dynstcor_UVHsf}) it is clear that
+in order to compute the Stokes drift velocity profile at the desired vertical
+levels we need $H_\mathrm{s}$, $\overline{f}$ and $\mathbf{v}_0$.
+
+The Coriolis-Stokes effect is enabled when \np{ln\_stcor} = true (default =
+false). All wave-related switches are found in \ngn{namsbc}.
+The surface Stokes drift velocity vectors (east and north components) are
+archived in ERA-Interim as GRIB parameters 215 and 216 respectively (table
+140).
+%\smallskip
+%%----------------------------------------------namsbc----------------------------------------------------
+%\namdisplay{namsbc}
+%%--------------------------------------------------------------------------------------------------------
+%\smallskip
+%
+% ================================================================