errata.tex

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\begin{document}
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\LARGE 
\begin{center} 
\centering
\textbf {Corrigendum}
\end{center}

\normalsize
\noindent

\noindent
In Chapter 2, Equations (2.3) and (2.4) should be
\begin{align}
 \partial_t U + (\nabla \cdot {\bf V} u )
- \partial_x (p\phi_\eta)
+ \partial_\eta (p\phi_x) &= F_U
\tag{2.3}
\\
%
 \partial_t V + (\nabla \cdot {\bf V} v )
- \partial_y (p\phi_\eta)
+ \partial_\eta (p\phi_y) &= F_V 
\tag{2.4}
%
\end{align}

\noindent
In Chapter 2, Governing Equations, the correct equation (2.33) should be:
\setlength{\parskip}{0.00in}

\begin{align}
%
F_{V_{cor}} & = - \biggl(f + u {\partial m \over
\partial y} - v {\partial m \over \partial x}\biggr) U
+ e W \sin \alpha_r - {vW \over r_e}
\tag{2.33}
\end{align}

\noindent
In Chapter 3, Model Discretization, the coefficient $C$ on page 12
should be defined as 

$$C=c_s^2/\mu^{t^*}{\alpha^{t^*}_d}^2$$.

\noindent
The correct equation 3.11 is
\begin{align}
 \delta_\tau W'' 
- m^{-1} g\overline{\left[(\alpha/\alpha_d)^{t^*} \biggl[
     \partial_\eta (C \partial_\eta \phi'') 
   + \partial_\eta\biggl({c_s^2\over\alpha^{t^*}}{\Theta''\over\Theta^{t^*}}\biggr)\biggr]
- \mu_d''\right]}^\tau
                                                   &= {R_W}^{t^*}  
\tag{3.11}
\end{align}


\noindent
The correct equation (3.26) are:
%%\begin{align}
%% \delta_\tau \phi'' + {1\over\mu_d^{t^*}}
%%[
%%m \Omega''^{\tau+\Delta \tau}\phi_\eta^{t^*}  - \overline{g W''}^\tau ]
%%&= {R_\phi}^{t^*}
%%\tag{3.12}
%%\end{align}

\begin{align}
 \delta_\tau \phi'' + {1\over\mu_d^{t^*}}
[
 m \Omega''^{\tau+\Delta \tau} \delta_\eta \overline{\phi^{t^*}}^\eta  - m_y\overline{g W''}^\tau ]
&= {R_\phi}^{t^*}
\tag{3.26}
\end{align}

\noindent
For equations (3.13), (3.14), (3.15), and (3.17), the correct ones are:

%
\begin{align}
R_U^{t^*} = &
- m[\partial_x(Uu) + \partial_y(Vu)] - \partial_\eta (\Omega u)
- ({\mu}_d \alpha \partial_x p'   
- {\mu}_d \alpha' \partial_x \bar{p}) ~~~~~~~~ \notag
\\
& - (\alpha/\alpha_d) ( {\mu}_d \partial_x \phi' 
  -  \partial_\eta p' \partial_x \phi
  + {\mu}_d' \partial_x \phi ) + F_U
\tag{3.13}
\\
%
R_V^{t^*} = &
- m[\partial_x (Uv) + \partial_y (Vv)] - \partial_\eta (\Omega v)
- ({\mu}_d \alpha \partial_y p'
- {\mu}_d \alpha' \partial_y \bar{p}) ~~~~~~~~ \notag
\\
& - (\alpha/\alpha_d) ( {\mu}_d \partial_y \phi'
  - \partial_\eta p' \partial_y \phi
  + {\mu}_d' \partial_y \phi ) + F_V
\tag{3.14}
\\
%
R_{\mu_d}^{t^*} = &
- m^2[\partial_x U + \partial_y V] - m \partial_\eta \Omega
\tag{3.15}
\\
%
R_W^{t^*} = &
- m[\partial_x (Uw) + \partial_y (Vw)] - \partial_\eta
(\Omega w)  ~~~~~~~~ ~~~~~~~~ ~~~~~~~~ ~~~~~~~~ ~~~ \notag
\\
& + m^{-1} g (\alpha/\alpha_d) [\partial_\eta p'
  - {\bar{\mu}}_d (q_v + q_c +q_r)]
  - m^{-1} {\mu}_d'g + F_W
\tag{3.17}
%
\end{align}

\vfill\eject
\noindent
At the top of page 14 in Chapter 3, we have corrected the statement
concerning the lower boundary condition for the vertically-implicit
solution for $W''$, the correct lower boundary condition being
$\Omega = \Omega'' = 0$.

Also in Chapter 3, page 18, the correct equation for $V$ component of 
the Coriolis and curvature term is

\begin{align}
F_{V_{cor}} & =  -
\bigl({\overline f}^y +
{\overline{
{\overline u}^x 
{\delta_y m}
- {\overline v}^y \delta_x m}}^y \bigr)
{\overline U}^{xy}
+ {\overline e}^y
{\overline W}^{y\eta}\, {\overline{\sin \alpha_r}}^y
- {v {\overline W}^{y\eta} \over r_e}
\notag
\end{align}

\noindent
In Chapter 4, Turbulent Mixing and Model Filters, equation (4.6) has been
corrected to:

\begin{equation}
K_{h,v} = C_s^2 \, l_{h,v}^2 \,\hbox{max}
\biggl[ 0., \bigl( D^2 - P_r^{-1} N^2 \bigr)^{1/2}\biggr],
\tag{4.6}
\end{equation}

\begin{equation}
D^2 =
{1 \over 2} \biggl[
D^2_{11} +
D^2_{22} +
D^2_{33} \biggr] +
\bigl({{\overline{D_{12}}}^{xy}}\bigr)^2 +
\bigl({{\overline{D_{13}}}^{x\eta}}\bigr)^2 +
\bigl({{\overline{D_{23}}}^{y\eta}}\bigr)^2
\notag
\end{equation}


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