<p><b>dudhia</b> 2008-06-04 18:15:17 -0600 (Wed, 04 Jun 2008)</p><p>fixes from Joe Klemp based on reviewer comments<br>
</p><hr noshade><pre><font color="gray">Modified: trunk/wrf/technote/discretization.tex
===================================================================
--- trunk/wrf/technote/discretization.tex 2008-06-05 00:14:20 UTC (rev 82)
+++ trunk/wrf/technote/discretization.tex 2008-06-05 00:15:17 UTC (rev 83)
@@ -172,7 +172,8 @@
\label{w-small-step}
\\
\delta_\tau \phi'' + {1\over\mu_d^{t^*}}
-[ m_y \Omega''^{\tau+\Delta \tau}\phi_\eta^{t^*} - m_y \overline{g W''}^\tau ]
+%% [ m_y \Omega''^{\tau+\Delta \tau}\phi_\eta^{t^*} - m_y \overline{g W''}^\tau ]
+[ m_y \Omega''^{\tau+\Delta \tau}\partial_\eta\phi^{t^*} - m_y \overline{g W''}^\tau ]
&= {R_\phi}^{t^*}.
\label{geo-small-step}
\end{align}
@@ -228,8 +229,10 @@
%
R_\phi^{t^*} = &
- \mu_d^{-1}
-[m_x m_y (U\phi_x + V\phi_y) + m_y
-\Omega\phi_\eta - m_y gW ],
+%% [m_x m_y (U\phi_x + V\phi_y) + m_y
+%% \Omega\phi_\eta - m_y gW ],
+[m_x m_y (U\partial_x\phi + V\partial_y\phi) + m_y
+\Omega\partial_\eta\phi - m_y gW ],
\label{geo-rhs}
%
\end{align}
@@ -320,7 +323,8 @@
mixing). \smallskip \hfill \break
%
\hphantom{BeginBegin}
-(2) Compute $R_\Phi^{t^*}$,
+%% (2) Compute $R_\Phi^{t^*}$,
+(2) Compute $R^{t^*}$,
\eqref{u-rhs}--\eqref{geo-rhs}
\medskip \hfill \break
%
@@ -761,7 +765,8 @@
\overline{U}^{xy}
- {\overline{u}^{xy} \overline{U}^{xy} \over r_e} \tan \psi
+ \overline{e}^y \overline{W}^{y\eta}\overline{\sin \alpha_r}^y
-+ {v\overline{W}^{y\eta} \over r_e}
+%% + {v\overline{W}^{y\eta} \over r_e}
+- {v\overline{W}^{y\eta} \over r_e}
\biggr],
\label{coriolis-v-global}
\\
@@ -795,7 +800,8 @@
\\
%
R_{\mu_{adv}}^{t^*} = &
-- m_x m_y[U_x + V_y] + m_y \Omega_\eta
+%% - m_x m_y[U_x + V_y] + m_y \Omega_\eta
+- m_x m_y[\partial_x U + \partial_y V] + m_y \partial_\eta\Omega
\\
R_{\Theta_{adv}}^{t^*} = &
- m_x m_y [\partial_x (U\theta) + \partial_y (V\theta)] - m_y \partial_\eta
@@ -809,8 +815,10 @@
%
R_{\phi_{adv}}^{t^*} = &
- \mu_d^{-1}
-[m_x m_y (U\phi_x + V\phi_y) + m_y
-\Omega\phi_\eta].
+%% [m_x m_y (U\phi_x + V\phi_y) + m_y
+%% \Omega\phi_\eta].
+[m_x m_y (U\partial_x \phi + V\partial_y \phi) + m_y
+\Omega\partial_\eta\phi].
\end{align}
%
</font>
<font color="gray">oindent
@@ -938,7 +946,8 @@
</font>
<font color="red">otag
\\
= & \, {\Delta t} {\delta (Uq)|}^{6th} - {Cr \over 60} \Delta x^6 {\partial^6
-q \over \partial x^6} + \, H.O.T.
+%% q \over \partial x^6} + \, H.O.T.
+q \over \partial x^6} + \, {\rm higher\ order\ terms}
</font>
<font color="gray">otag
\end{align}
%
@@ -948,7 +957,8 @@
\begin{align}
\Delta t \delta_x (U \overline{q}^{x_{adv}})
= & \, {\Delta t} {\delta (Uq)|}^{4th} + {Cr \over 12} \Delta x^4 {\partial^4
-q \over \partial x^4} + \, H.O.T.
+%% q \over \partial x^4} + \, H.O.T.
+q \over \partial x^4} + \, {\rm higher\ order\ terms}
</font>
<font color="gray">otag
\end{align}
%
Modified: trunk/wrf/technote/equation.tex
===================================================================
--- trunk/wrf/technote/equation.tex 2008-06-05 00:14:20 UTC (rev 82)
+++ trunk/wrf/technote/equation.tex 2008-06-05 00:15:17 UTC (rev 83)
@@ -72,16 +72,20 @@
\partial_t U + (</font>
<font color="red">abla \cdot {\bf V} u )
%+ \mu \alpha \partial_x p
%+ \partial_\eta p \partial_x \phi &= F_U
-- \partial_x (p\phi_\eta)
-+ \partial_\eta (p\phi_x) &= F_U
+%% - \partial_x (p\phi_\eta)
+%% + \partial_\eta (p\phi_x) &= F_U
+- \partial_x (p\partial_\eta \phi)
++ \partial_\eta (p\partial_x \phi) &= F_U
\label{cartesian_begin}
\\
%
\partial_t V + (</font>
<font color="red">abla \cdot {\bf V} v )
%+ \mu \alpha \partial_y p
%+ \partial_\eta p \partial_y \phi &= F_V \\
-- \partial_y (p\phi_\eta)
-+ \partial_\eta (p\phi_y) &= F_V \\
+%% - \partial_y (p\phi_\eta)
+%% + \partial_\eta (p\phi_y) &= F_V \\
+- \partial_y (p\partial_\eta \phi)
++ \partial_\eta (p\partial_y \phi) &= F_V \\
%
\partial_t W + (</font>
<font color="gray">abla \cdot {\bf V} w )
- g (\partial_\eta p - \mu) &= F_W
@@ -318,8 +322,10 @@
\label{mass-cons-full}
\\
%
-\partial_t \phi + \mu_d^{-1} [m_x m_y (U\phi_x + V\phi_y) + m_y
-\Omega\phi_\eta - m_y gW ] & = 0
+%% \partial_t \phi + \mu_d^{-1} [m_x m_y (U\phi_x + V\phi_y) + m_y
+%% \Omega\phi_\eta - m_y gW ] & = 0
+\partial_t \phi + \mu_d^{-1} [m_x m_y (U\partial_x\phi + V\partial_y\phi) + m_y
+\Omega\partial_\eta\phi - m_y gW ] & = 0
\label{geo-full}
\\
%
@@ -391,7 +397,8 @@
\\
%
F_{V_{cor}} & = {m_y \over m_x} \biggl[ - fU - {uU \over r_e}
-\tan \psi + {vW \over r_e}
+%% \tan \psi + {vW \over r_e}
+\tan \psi - {vW \over r_e}
+ eW\sin \alpha_r \biggr]
\label{v-mom-rhs-global}
\\
@@ -402,8 +409,10 @@
%
\end{align}
-For idealized cases, the map scale factor $m_x = m_y = 1$,
-$f$ is often taken to be constant, and $e = 0$.
+%% For idealized cases, the map scale factor $m_x = m_y = 1$,
+%% $f$ is often taken to be constant, and $e=0$.
+For idealized cases on a Cartesian grid, the map scale factor $m_x = m_y = 1$,
+$f$ is specified, and $e$ and $r_e^{-1}$ should be zero to remove the curvature terms.
\section{Perturbation Form of the Governing Equations}
@@ -473,8 +482,10 @@
%
\partial_t \phi'
+ \mu_d^{-1}
-[m_x m_y (U\phi_x + V\phi_y) + m_y
-\Omega\phi_\eta -m_y gW ] = 0.
+%% [m_x m_y (U\phi_x + V\phi_y) + m_y
+%% \Omega\phi_\eta -m_y gW ] = 0.
+[m_x m_y (U\partial_x\phi + V\partial_y\phi) + m_y
+\Omega\partial_\eta\phi -m_y gW ] = 0.
%
\end{align}
%
Modified: trunk/wrf/technote/filter.tex
===================================================================
--- trunk/wrf/technote/filter.tex 2008-06-05 00:14:20 UTC (rev 82)
+++ trunk/wrf/technote/filter.tex 2008-06-05 00:15:17 UTC (rev 83)
@@ -107,7 +107,9 @@
used to compute the eddy viscosities and a description of
the prognostic turbulent kinetic energy (TKE) equation used in one set
of these formulations. The sixth order spatial filter is described at
-the end of this section.
+the end of this section. In these formulations, the horizontal ($K_h$) and
+vertical ($K_v$) eddy viscosities are defined at scalar points on the
+staggered model grid.
\subsection{Horizontal and Vertical Diffusion on Coordinate Surfaces}
@@ -135,7 +137,8 @@
\bigl[ \delta_x(m_x K_h \delta_x u)
+ \delta_y({\overline{m_y}}^{xy} {\overline{K_h}}^{xy} \delta_y u) \bigr]
+ m_y^{-1} g^2({\overline{\mu_d}}^x {\overline\alpha}^x)^{-1}
-\delta_\eta(K_v ({\overline \alpha}^{x\eta})^{-1} \delta_\eta u)
+%% \delta_\eta(K_v ({\overline \alpha}^{x\eta})^{-1} \delta_\eta u)
+\delta_\eta({\overline K_v}^{x\eta} ({\overline \alpha}^{x\eta})^{-1} \delta_\eta u)
</font>
<font color="gray">otag \\
%
\partial_t V &= ... \,\, +
@@ -145,7 +148,8 @@
+\delta_y(m_y K_h \delta_y v)
\bigr]
+ m_x^{-1} g^2({\overline{\mu_d}}^y {\overline\alpha}^y)^{-1}
-\delta_\eta(K_v ({\overline \alpha}^{y\eta})^{-1} \delta_\eta v)
+%% \delta_\eta(K_v ({\overline \alpha}^{y\eta})^{-1} \delta_\eta v)
+\delta_\eta({\overline K_v}^{y\eta} ({\overline \alpha}^{y\eta})^{-1} \delta_\eta v)
</font>
<font color="gray">otag \\
%
\partial_t W &= ... \,\, +
@@ -171,7 +175,8 @@
+\delta_y({\overline m_y}^{y} P_r^{-1} {\overline K_h}^{y} \delta_y q)
\bigr]
+ g^2({\mu_d} {\alpha})^{-1}
-\delta_\eta(K_v \alpha^{-1} \delta_\eta q).
+%% \delta_\eta(K_v \alpha^{-1} \delta_\eta q).
+\delta_\eta({\overline K_v}^\eta ({\overline \alpha}^\eta)^{-1} \delta_\eta q).
</font>
<font color="gray">otag
\end{equation}
%
@@ -304,12 +309,15 @@
\begin{align}
\partial_t (\mu_d q) = ...\,\, + \bigl[
& \, m_x \bigl(\partial_x - \partial_z z_x \bigr )
-\bigl( \mu_d m_x K (\partial_x - z_x \partial_z ) \bigr)
+%% \bigl( \mu_d m_x K (\partial_x - z_x \partial_z ) \bigr)
+\bigl( \mu_d m_x K_h (\partial_x - z_x \partial_z ) \bigr)
+
</font>
<font color="gray">otag \\
& \, m_y \bigl(\partial_y - \partial_z z_y \bigr )
-\bigl( \mu_d m_y K (\partial_y - z_y \partial_z ) \bigr)
-+ \partial_z \mu_d K \partial_z
+%% \bigl( \mu_d m_y K (\partial_y - z_y \partial_z ) \bigr)
+\bigl( \mu_d m_y K_h (\partial_y - z_y \partial_z ) \bigr)
+%% + \partial_z \mu_d K \partial_z
++ \partial_z \mu_d K_v \partial_z
\bigr] q.
\label{scalar-disp-phys}
\end{align}
@@ -516,9 +524,13 @@
including moisture effects, is outlined in Section \ref{tke_section}.
Two options are available for calculating the mixing length $l_{h,v}$ in
-\eqref{k_calc}. An isotropic length scale can be chosen where $l_{h,v}
-= (\Delta x \Delta y \Delta z)^{1/3}$ and thus $K_h = K_v = K$. The
-anisotropic option sets the horizontal mixing length $l_h = \sqrt{\Delta
+%% \eqref{k_calc}. An isotropic length scale can be chosen where $l_{h,v}
+%% = (\Delta x \Delta y \Delta z)^{1/3}$ and thus $K_h = K_v = K$. The
+%% anisotropic option sets the horizontal mixing length $l_h = \sqrt{\Delta
+\eqref{k_calc}. An isotropic length scale (appropriate for $\Delta x, \Delta y
+\simeq \Delta z$) can be chosen where $l_{h,v} = (\Delta x \Delta y \Delta z)^{1/3}$
+and thus $K_h = K_v = K$. The anisotropic option (appropriate for $\Delta x,
+\Delta y >> \Delta z$) sets the horizontal mixing length $l_h = \sqrt{\Delta
x \Delta y}$ in the calculation of the horizontal eddy viscosity $K_h$
using \eqref{k_calc}, and $l_v = \Delta z$ for the calculation of the
vertical eddy viscosity $K_v$ using \eqref{k_calc}.
@@ -632,14 +644,20 @@
the formula for a moist saturated or unsaturated environment:
%
\begin{alignat}{2}
-N^2 & = g \biggl[A {\partial \theta_e \over \partial z}
-- {\partial q_w \over \partial z} \biggr] & \qquad
-& \hbox{if $q_v \ge q_s$ or $q_c \ge 0.01$ g/Kg;} </font>
<font color="red">otag \\
-N^2 & = g \biggl[{1 \over \theta}{\partial \theta \over \partial z}
-+ 1.61 {\partial q_v \over \partial z} - {\partial q_w \over \partial z}
+%% N^2 & = g \biggl[A {\partial \theta_e \over \partial z}
+N^2 & = g \bigl[A {\partial_z \theta_e }
+%% - {\partial q_w \over \partial z} \biggr] & \qquad
+- {\partial_z q_w } \bigr] & \qquad
+%% & \hbox{if $q_v \ge q_s$ or $q_c \ge 0.01$ g/Kg;} </font>
<font color="blue">otag \\
+& \hbox{if $q_v \ge q_{vs}$ or $q_c \ge 0.01$ g/Kg;} </font>
<font color="red">otag \\
+%% N^2 & = g \biggl[{1 \over \theta}{\partial \theta \over \partial z}
+N^2 & = g \biggl[{1 \over \theta}{\partial_z \theta}
+%% + 1.61 {\partial q_v \over \partial z} - {\partial q_w \over \partial z}
++ 1.61 {\partial_z q_v } - {\partial_z q_w }
\biggr]
& \qquad
-& \hbox{if $q_v < q_s$ or $q_c < 0.01$ g/Kg}.
+%% & \hbox{if $q_v < q_s$ or $q_c < 0.01$ g/Kg}.
+& \hbox{if $q_v < q_{vs}$ or $q_c < 0.01$ g/Kg}.
</font>
<font color="gray">otag
\end{alignat}
%
@@ -656,12 +674,14 @@
</font>
<font color="red">oindent
where $q_w$ represents the total water (vapor + all liquid species
+ all ice species), $L$ is the latent heat of condensation and
-$\epsilon$ is the molecular weight of water over the molecular weight of
+%% $\epsilon$ is the molecular weight of water over the molecular weight of
+$\epsilon$ is the ratio of the molecular weight of water vapor to the molecular weight of
dry air. $\theta_e$ is the equivalent potential temperature and is
defined as
%
\begin{equation}
-\theta_e = \theta \biggl(1 + {\epsilon L q_{vs} \over C_p T} \biggr),
+%% \theta_e = \theta \biggl(1 + {\epsilon L q_{vs} \over C_p T} \biggr),
+\theta_e = \theta \biggl(1 + { L q_{vs} \over C_p T} \biggr),
</font>
<font color="gray">otag
\end{equation}
%
@@ -738,7 +758,7 @@
divergence damping (an acoustic model filter); an external-mode filter
that damps vertically-integrated horizontal divergence; and off-centering
of the vertically implicit integration of the vertical momentum
-equation and geopoential equation. Each of these is described in the
+equation and geopotential equation. Each of these is described in the
following sections.
\subsection{Three-Dimensional Divergence Damping}
@@ -886,7 +906,6 @@
recovered using \eqref{geo-small-step}. In the solution procedure that
includes the implicit Rayleigh damping for $W$, after the tridiagonal
equation for $W''$ %$W''^{\tau+\Delta \tau}$
-
is solved and before the
geopotential $\phi$ is updated, the implicit Rayleigh damping is
included by calculating $W''^{\tau+\Delta \tau}$ using
@@ -932,10 +951,14 @@
relax the variable back to a predetermined reference state value,
%
\begin{eqnarray*}
- \frac{\partial u }{\partial t} & = & -\tau (z) \left( u - \overline{u} \right) ,\\
- \frac{\partial v }{\partial t} & = & -\tau (z) \left( v - \overline{v} \right) ,\\
- \frac{\partial w }{\partial t} & = & -\tau (z) w ,\\
- \frac{\partial \theta}{\partial t} & = & -\tau (z) \left( \theta - \overline{\theta} \right).
+%% \frac{\partial u }{\partial t} & = & -\tau (z) \left( u - \overline{u} \right) ,\\
+%% \frac{\partial v }{\partial t} & = & -\tau (z) \left( v - \overline{v} \right) ,\\
+%% \frac{\partial w }{\partial t} & = & -\tau (z) w ,\\
+%% \frac{\partial \theta}{\partial t} & = & -\tau (z) \left( \theta - \overline{\theta} \right).
+ {\partial_t u } & = & -\tau (z) \left( u - \overline{u} \right) ,\\
+ {\partial_t v } & = & -\tau (z) \left( v - \overline{v} \right) ,\\
+ {\partial_t w } & = & -\tau (z) w ,\\
+ {\partial_t \theta} & = & -\tau (z) \left( \theta - \overline{\theta} \right).
\end{eqnarray*}
%
Overbars indicate the reference state fields, which are functions of $z$
</font>
</pre>