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appenb.tex
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\chapter{List of Symbols}
\label{List of Symbols}
Symbols used in this document are listed in alphabatical order in this appendix.
\\[2ex]
\begin{tabular}{ l p{5.5in} }
{\em Symbols} & {\em Definition} \\
\\
$a$ & generic variable \\
$A$ & coefficient (Chapter \ref{filter_chap}), base-state lapse rate constant (Chapter \ref{initialization_chap}) \\
{\bf B} & background error covariance matrix \\
$c$ & scalar coefficient \\
$c_s$ & speed of sound \\
$C_k$ & a constant used in TKE closure \\
$Cr$ & Courant number \\
$Cr_{max}$ & maximum Courant number \\
$Cr_{theory}$ & Courant number from Table 3.1 \\
$Cr_{\beta}$ & activation Courant number in vertical velocity damping \\
$C_s$ & a constant used in eddy viscosity calculation \\
$D$ & deformation \\
$D_{nm}$ & deformation tensor, where $n, m$ = 1, 2 and 3 \\
$e$ & cosine component of the Coriolis term (Chapters \ref{equation_chap}, \ref{discretization_chap}); turbulent kinetic energy (Chapter \ref{filter_chap}) \\
{\bf E} & observation error covariance matrix \\
$f$ & sine component of the Coriolis term \\
$F$ & forcing terms for $U$, $V$, $W$, $\Theta$ and $Q_m$ \\
{\bf F} & representivity error covariance matrix \\
$F_{X_{cor}}$ & Coriolis forcing terms for $X = $ $U$, $V$, and $W$ \\
$F_{1,2}$ & coefficients for weighting functions in specified boundary condition \\
$g$ & acceleration due to gravity \\
$G_k$ & regression coefficient \\
$H$ & observation operator \\
$J$ & cost function \\
$K_{dh,dv}$ & horizontal and vertical eddy viscosity for gravity wave absorbing layer \\
$K_{h,v}$ & horizontal and vertical eddy viscosities \\
\end{tabular}
\newpage
\vskip 5pt
\begin{tabular}{ l p{5.5in} }
{\em Symbols} & {\em Definition} \\
\\
$l_0$ & minimum length scale for dissipation \\
$l_{h,v}$ & horizontal and vertical length scales for turbulence \\
$l_{cr}$ & critical length scale for dissipation \\
$L$ & latent heat of condensation \\
$L_{x,y}$ & periodicity length in $x$ and $y$ \\
$m$ & map scale factor \\
$n_s$ & ratio of the RK3 time step to the acoustic time step \\
$N$ & Brunt-V\"ais\"al\"a frequency \\
$p$ & pressure \\
$p'$ & perturbation pressure \\
$p_0$ & reference sea-level pressure \\
$p_h$ & hydrostatic pressure \\
$p_{ht,hs}$ & hydrostatic pressure at the top and surface of the model \\
$p_{dht,dhs}$ & dry hydrostatic pressure at the top and surface of the model \\
$p_s$ & surface pressure \\
$P_r$ & Prandtl number \\
$q$ & generic scalar \\
$q_{c,i,r,s}$ & mixing ratios for cloud water, ice, rain water and snow \\
$q_m$ & generic mixing ratios for moisture \\
$q_v$ & mixing ratio for water vapor \\
$q_{vs}$ & saturation mixing ratio for water vapor \\
$Q_m$ & generic coupled moisture variable \\
$r$ & relative humidity \\
$r_e$ & radius of earth \\
$R$ & remaining terms in equations \\
$R_d$ & gas constant for dry air \\
$R_v$ & gas constant for water vapor \\
$t$ & time \\
$\Delta t$ & a full model time step \\
$T$ & temperature \\
$T_0$ & reference sea-level temperature \\
$u$ & horizontal component of velocity in $x$-direction \\
$U$ & coupled horizontal component of velocity in $x$-direction (Chapters \ref{equation_chap}, \ref{discretization_chap}, \ref{lbc_chap}, \ref{nesting_chap}); control variable transform (Chapter \ref{var_chap}) \\
$U_h$ & horizontal correlation \\
$U_p$ & multivariate covariance \\
$U_v$ & vertical covariance \\
$v$ & horizontal component of velocity in $y$-direction \\
$\bf v$ & three dimensional vector velocity \\
$V$ & coupled horizontal component of velocity in $y$-direction \\
$\bf V$ & three dimensional coupled vector velocity \\
$w$ & vertical component of velocity \\
$W$ & coupled vertical component of velocity \\
\end{tabular}
\newpage
\begin{tabular}{ l p{5.5in} }
{\em Symbols} & {\em Definition} \\
\\
$W_k$ & regression coefficient \\
$z$ & height \\
$z_d$ & depth of damping layer \\
$z_{top}$ & height of model top \\
$\alpha$ & inverse density of air \\
$\alpha'$ & perturbation inverse density of air \\
$\bar\alpha$ & inverse density of air for the reference state \\
$\alpha_d$ & inverse density of dry air \\
$\alpha_r$ & local rotation angle between $y$-axis and the meridian \\
$\beta$ & off-centering coefficient for semi-implicit acoustic step \\
$\gamma$ & ratio of heat capacities for dry air at constant pressure and volume \\
$\gamma_d$ & divergence damping coefficient \\
$\gamma_e$ & external mode damping coefficient \\
$\gamma_g$ & damping coefficient for upper boundary gravity wave absorbing layer \\
$\gamma_r$ & Rayleigh damping coefficient \\
$\epsilon$ & molecular weight of water over the molecular weight of dry air (Chapter \ref{filter_chap}); true background error (Chapter \ref{var_chap}) \\
$\eta$ & terrain-following hydrostatic-pressure vertical coordinate \\
$\dot\eta$ & contravariant `vertical' velocity or coordinate velocity \\
$\theta$ & potential temperature \\
$\theta_e$ & equivalent potential temperature \\
$\theta_m$ & moist potential temperature \\
$\Theta$ & coupled potential temperature \\
$\mu$ & hydrostatic pressure difference between surface and top of the model \\
$\bar\mu$ & reference state hydrostatic pressure difference between surface and top of the model \\
$\mu_d$ & dry hydrostatic pressure difference between surface and top of the model \\
$\tau$ & acoustic time (Chapter \ref{discretization_chap}), vertical structure function for Rayleigh damping (Chapter \ref{filter_chap}) \\
$\tau_{nm}$ & stress tensor (Chapter \ref{filter_chap}) where $n. m$ = 1, 2 and 3 \\
$\Delta \tau$ & acoustic time step \\
$\phi$ & geopotential (Chapters \ref{equation_chap}, \ref{discretization_chap}, \ref{initialization_chap}); latitude (Chapter \ref{var_chap}) \\
$\bar \phi$ & geopotential for reference state \\
$\phi'$ & perturbation geopotential \\
$\Phi$ & generic prognostic variable (coupled) \\
$\psi$ & generic variable (Chapter \ref{lbc_chap}) \\
$\psi'$ & streamfunction increment \\
$\chi'$ & velocity potential increment \\
$\omega$ & same as $\dot\eta$ \\
$\Omega$ & coupled coordinate velocity \\
$\Omega_e$ & angular rotation rate of the earth \\
\\
\end{tabular}
\newpage
\normalsize
\begin{tabular}{ l p{5.5in} }
{\em Subscripts/Superscripts} & {\em Definition} \\
\\
$()_d$ & dry \\
$()_h$ & hydrostatic \\
$()_0$ & base state sea-level constant \\
$\overline{()}$ & reference state \\
$()'$ & perturbation from reference state \\
$()^{t^*}$ & full value at a Runge-Kutta step \\
$()''$ & perturbation from Runge-Kutta step value in acoustic steps \\
\end{tabular}