From 8669ce938a8c6345d0ce8390bf40db6b3ceeb7a0 Mon Sep 17 00:00:00 2001
From: Sara Pavan <sara.pavan@edf.fr>
Date: Mon, 10 Jul 2023 14:26:59 +0200
Subject: [PATCH] [doc][gaia] Minor corrections and homogenization for density
parameters
---
...ortProcessesBedAndStratigraphy_bedload.tex | 22 ++++++++++++-------
documentation/gaia/user/latex/HowTos.tex | 7 +++---
2 files changed, 18 insertions(+), 11 deletions(-)
diff --git a/documentation/gaia/user/latex/3_1-SedimentTransportProcessesBedAndStratigraphy_bedload.tex b/documentation/gaia/user/latex/3_1-SedimentTransportProcessesBedAndStratigraphy_bedload.tex
index f6b631fbcf..08d16e9a12 100644
--- a/documentation/gaia/user/latex/3_1-SedimentTransportProcessesBedAndStratigraphy_bedload.tex
+++ b/documentation/gaia/user/latex/3_1-SedimentTransportProcessesBedAndStratigraphy_bedload.tex
@@ -16,9 +16,9 @@ Above, $Q_b$ is the bedload transport rate per unit width, computed as a functio
As presented and discussed in \S~\ref{sec:bedevol}, \gaia{} solves the Exner equation as a function of the mass transport flux rate and not in terms of volumetric transport rate, as follows:
\begin{equation}
-\mathbf Q_{mb}=\rho \mathbf Q_b,
+\mathbf Q_{mb}=\rho_s \mathbf Q_b,
\end{equation}
-where $\mathbf Q_{mb}$ is the vector of mass transport rate per unit width without pores (kg/(m~s)), with $\rho$ the density.
+where $\mathbf Q_{mb}$ is the vector of mass transport rate per unit width without pores (kg/(m~s)), with $\rho_s$ the sediment density.
%The equation is obtained writing the conservative law equation for sediment and integrating along the vertical, keeping density in the equation:
%\begin{align}
@@ -115,7 +115,13 @@ f(\theta)=\left\{\begin{array}{ll}
\end{array}
\right.
\end{equation*}
-where the non-dimensional diameter $D_*=d[(\rho_s/\rho-1)g/\nu^2]^{1/3}$, with $\nu$ the water viscosity (keyword \telkey{WATER VISCOSITY}, equal to $10^{-6}$ m/s$^2$ by default).
+where $D_*=d[(\rho_s/\rho-1)g/\nu^2]^{1/3}$ is the non-dimensional diameter,
+with $\rho_s$ the sediment density (keyword \telkey{CLASSES SEDIMENT DENSITY}, equal to
+2650~kg/m$^3$ by default),
+$\rho$ the water density (keyword \telkey{WATER DENSITY}, equal to 1000~kg/m$^3$ by default in
+\telemac{2D} or \telkey{AVERAGE WATER DENSITY}, equal to 1025~kg/m$^3$ by default in
+\telemac{3D}),
+$\nu$ the water viscosity (keyword \telkey{WATER VISCOSITY}, equal to $10^{-6}$ m/s$^2$ by default).
\item Fortran subroutine {\ttfamily bedload\_einst\_gaia.f}.
\end{itemize}
@@ -324,7 +330,7 @@ where $\theta_{\beta cr}$ is the corrected Shields number for a sloping bed, $\
\medskip
\item Apsely and Stansby~\cite{ApsleyStansby2008} which is based on the modification of the critical Shields parameter $\theta_{\beta cr}$ and on the calculation of a effective dimensionless shear stress $\theta_{beta}$.
-The modification of the critical shear stress can only be done for threshold bedload formulas. The modification of the Shields parameter $\theta_{\beta}$ could be done for all bedload formulas which are using the shear stress and accordingly the Shields parameter. But it is not recommended.
+The modification of the critical shear stress can only be done for threshold bedload formulas. The modification of the Shields parameter $\theta_{\beta}$ could be done for all bedload formulas which are using the shear stress and accordingly the Shields parameter. But it is not recommended.
\begin{equation*}
\frac{\theta_{\beta cr}}{\theta_{cr}} = \cos\chi = 1 / \sqrt{1 + (\partial z /\partial x)^2 + (\partial z/ \partial y)^2 }
\end{equation*}
@@ -341,7 +347,7 @@ where $\gamma$ is the direction of the bed shear stress.
%-------------------------------------------------------------------------------
\subsubsection{Sediment sliding}
%-------------------------------------------------------------------------------
-If the bottom slope is higher than a critical slope (typically the angle of repose) sediments can be moved due to geomechanical processes. In \gaia{} two formulas are implemented to calculate sediment displacement in down-slope direction only driven by the bottom slope.
+If the bottom slope is higher than a critical slope (typically the angle of repose) sediments can be moved due to geomechanical processes. In \gaia{} two formulas are implemented to calculate sediment displacement in down-slope direction only driven by the bottom slope.
\begin{itemize}
\item The first method is a simple and mass conservative smoothing of the bottom slopes. The changed masses are balanced separately in the mass-balance (MASS EVOLUTION DUE TO SLIDING FORMULA 1).
@@ -374,7 +380,7 @@ The correction of the intensity of bedload transport rate can be done by :
\begin{itemize}
\item the Koch and Flokstra formulation \telkey{FORMULA FOR SLOPE EFFECT} (integer type variable, set to {\ttfamily = 1} by default). This keyword has the associated keyword \telkey{BETA} (real type variable, set to {\ttfamily = 1.30} by default)
\item the Soulsby formulation \telkey{FORMULA FOR SLOPE EFFECT = 2}. This keyword has the associated keyword \telkey{FRICTION ANGLE OF THE SEDIMENT} (real type variable, set to {\ttfamily = 40.} by default) and the critical Shields parameter which is calculated by default or can be set with the keyword \telkey{CLASSES SHIELDS PARAMETERS}.
-\item the Apsley and Stansby formulation \telkey{FORMULA FOR SLOPE EFFECT = 3}. This keyword has the associated keyword \telkey{FRICTION ANGLE OF THE SEDIMENT} (real type variable, set to {\ttfamily = 40} by default) and the critical Shields parameter which is calculated by default or can be set with the keyword \telkey{CLASSES SHIELDS PARAMETERS}.
+\item the Apsley and Stansby formulation \telkey{FORMULA FOR SLOPE EFFECT = 3}. This keyword has the associated keyword \telkey{FRICTION ANGLE OF THE SEDIMENT} (real type variable, set to {\ttfamily = 40} by default) and the critical Shields parameter which is calculated by default or can be set with the keyword \telkey{CLASSES SHIELDS PARAMETERS}.
\end{itemize}
%-------------------------------------------------------------------------------
@@ -385,10 +391,10 @@ The keyword \telkey{SECONDARY CURRENTS} (logical type variable, set to {\ttfamil
%-------------------------------------------------------------------------------
\subsubsection{Correction due to sediment sliding}
%-------------------------------------------------------------------------------
-Sediment sliding will be taken into account if the keyword \telkey{SEDIMENT SLIDE}
+Sediment sliding will be taken into account if the keyword \telkey{SEDIMENT SLIDE}
(integer type variable, set to {\ttfamily = 0} by default) is higher zero:
\begin{itemize}
-\item \telkey{SEDIMENT SLIDE = 1} smoothes the bottom slopes up to the angle of repose.
+\item \telkey{SEDIMENT SLIDE = 1} smoothes the bottom slopes up to the angle of repose.
\item \telkey{SEDIMENT SLIDE = 2} calculates the sediment sliding with the avalanching formula from Apsley and Stansby.
\end{itemize}
This keyword has the associated keyword \telkey{FRICTION ANGLE OF THE SEDIMENT} (real type variable, set to {\ttfamily = 40} by default)
diff --git a/documentation/gaia/user/latex/HowTos.tex b/documentation/gaia/user/latex/HowTos.tex
index fe905bc633..66b50bf2f1 100644
--- a/documentation/gaia/user/latex/HowTos.tex
+++ b/documentation/gaia/user/latex/HowTos.tex
@@ -90,7 +90,7 @@ MASS-BALANCE = YES
\begin{itemize}
\item Sediment diameters, defined by the keyword \telkey{CLASSES SEDIMENT DIAMETERS} (real list, {\ttfamily = 0.01} m by default)
\item Sediment density, defined by the keyword \telkey{CLASSES SEDIMENT DENSITY} (real type, {\ttfamily = 2650.0} kg$/$m$^3$ by default)
-\item Shields parameter $\tau_c$ [N\,m$^{-2}$], defined by the keyword \telkey{CLASSES SHIELDS PARAMETERS} (real list, = -9 by default). If it is not known, it is necessary to give a negative value to let \gaia{} compute it as a function of the non-dimensional grain diameter $D_*=d_{50}[(\rho_s/\rho-1)g/\nu^2]^{1/3}$ in the subroutine \texttt{shields.f}:
+\item Critical Shields parameter, $\theta_{cr}$ ($-$), defined by the keyword \telkey{CLASSES SHIELDS PARAMETERS} (real list, = -9 by default). If it is not known, it is necessary to give a negative value to let \gaia{} compute it as a function of the non-dimensional grain diameter $D_*=d_{50}[(\rho_s/\rho-1)g/\nu^2]^{1/3}$ in the subroutine \texttt{shields.f}:
\begin{equation*}
\frac{\tau_c}{g(\rho_s -\rho)d_{50}}=\left\{\begin{array}{ll}
0.24 D_*^{-1}, & D_* \leq 4 \\
@@ -101,7 +101,8 @@ MASS-BALANCE = YES
\end{array}
\right.
\end{equation*}
-with $d_{50}$ the median sand grain diameter (m), $\rho$ the water density $=1000$ kg$/$m$^3$ by default, $\rho_s$ the sediment density $=2650$kg$/$m$^3$ by default, and $\nu$ the kinematic viscosity $=1.0\times 10^{-6}$m$^2$s$^{-1}$ by default.
+with $d_{50}$ the median sand grain diameter (m), $\rho$ the water density =1000~kg/m$^3$ by default
+in \telemac{2D} and =1025~kg/m$^3$, $\rho_s$ the sediment density =2650~kg/m$^3$ by default, and $\nu$ the kinematic viscosity $=1.0\times 10^{-6}$m$^2$s$^{-1}$ by default.
\item Settling velocity, it can be specified by the user or calculated by the model as a function of grain diameter, keyword \telkey{SETTLING VELOCITIES } (real list, =-9 by default). If a negative value is given, \gaia{} will compute it as function of grain diameter (cf. chapter \ref{Chap:Sed:Ex:cohesive}):
\begin{equation*}
@@ -115,7 +116,7 @@ w_{s} = \left\{\begin{array}{ll}
\end{array}
\right.
\end{equation*}
-with $s=\rho_{s}/\rho_0$ is the relative density and $g$ is the acceleration of the gravity.%
+with $s=\rho_{s}/\rho$ is the relative density and $g$ is the acceleration of the gravity.%
\item Bed porosity, keyword \telkey{LAYERS NON COHESIVE BED POROSITY} (real type, {\ttfamily = 0.40} by default)
\end{itemize}
--
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