Changeset 11544 for NEMO/trunk/doc/latex/NEMO/subfiles/chap_DIU.tex

Timestamp:

2019-09-13T16:37:38+02:00 (5 years ago)

Author:

nicolasmartin

Message:

Missing modifications from previous commit

File:

• NEMO/trunk/doc/latex/NEMO/subfiles/chap_DIU.tex (modified) (9 diffs)

NEMO/trunk/doc/latex/NEMO/subfiles/chap_DIU.tex

@@ -57 +57 @@
 %===============================================================
 \section{Warm layer model}
-\label{sec:warm_layer_sec}
+\label{sec:DIU_warm_layer_sec}
 %===============================================================
 
@@ -65 +65 @@
 \frac{\partial{\Delta T_{\mathrm{wl}}}}{\partial{t}}&=&\frac{Q(\nu+1)}{D_T\rho_w c_p
 \nu}-\frac{(\nu+1)ku^*_{w}f(L_a)\Delta T}{D_T\Phi\!\left(\frac{D_T}{L}\right)} \mbox{,}
-\label{eq:ecmwf1} \\
-L&=&\frac{\rho_w c_p u^{*^3}_{w}}{\kappa g \alpha_w Q }\mbox{,}\label{eq:ecmwf2}
+\label{eq:DIU_ecmwf1} \\
+L&=&\frac{\rho_w c_p u^{*^3}_{w}}{\kappa g \alpha_w Q }\mbox{,}\label{eq:DIU_ecmwf2}
 \end{align}
 where $\Delta T_{\mathrm{wl}}$ is the temperature difference between the top of the warm layer and the depth $D_T=3$\,m at which there is assumed to be no diurnal signal.
-In equation (\autoref{eq:ecmwf1}) $\alpha_w=2\times10^{-4}$ is the thermal expansion coefficient of water,
+In equation (\autoref{eq:DIU_ecmwf1}) $\alpha_w=2\times10^{-4}$ is the thermal expansion coefficient of water,
 $\kappa=0.4$ is von K\'{a}rm\'{a}n's constant, $c_p$ is the heat capacity at constant pressure of sea water,
 $\rho_w$ is the water density, and $L$ is the Monin-Obukhov length.
@@ -79 +79 @@
 the relationship $u^*_{w} = u_{10}\sqrt{\frac{C_d\rho_a}{\rho_w}}$, where $C_d$ is the drag coefficient,
 and $\rho_a$ is the density of air.
-The symbol $Q$ in equation (\autoref{eq:ecmwf1}) is the instantaneous total thermal energy flux into
+The symbol $Q$ in equation (\autoref{eq:DIU_ecmwf1}) is the instantaneous total thermal energy flux into
 the diurnal layer, \ie
 \[
   Q = Q_{\mathrm{sol}} + Q_{\mathrm{lw}} + Q_{\mathrm{h}}\mbox{,}
-  % \label{eq:e_flux_eqn}
+  % \label{eq:DIU_e_flux_eqn}
 \]
 where $Q_{\mathrm{h}}$ is the sensible and latent heat flux, $Q_{\mathrm{lw}}$ is the long wave flux,
 and $Q_{\mathrm{sol}}$ is the solar flux absorbed within the diurnal warm layer.
 For $Q_{\mathrm{sol}}$ the 9 term representation of \citet{gentemann.minnett.ea_JGR09} is used.
-In equation \autoref{eq:ecmwf1} the function $f(L_a)=\max(1,L_a^{\frac{2}{3}})$,
+In equation \autoref{eq:DIU_ecmwf1} the function $f(L_a)=\max(1,L_a^{\frac{2}{3}})$,
 where $L_a=0.3$\footnote{
   This is a global average value, more accurately $L_a$ could be computed as $L_a=(u^*_{w}/u_s)^{\frac{1}{2}}$,
@@ -99 +99 @@
 4\zeta^2}{1+3\zeta+0.25\zeta^2} &(\zeta \ge 0) \\
                                     (1 - 16\zeta)^{-\frac{1}{2}} & (\zeta < 0) \mbox{,}
-                                    \end{array} \right. \label{eq:stab_func_eqn}
+                                    \end{array} \right. \label{eq:DIU_stab_func_eqn}
 \end{equation}
-where $\zeta=\frac{D_T}{L}$.  It is clear that the first derivative of (\autoref{eq:stab_func_eqn}),
-and thus of (\autoref{eq:ecmwf1}), is discontinuous at $\zeta=0$ (\ie\ $Q\rightarrow0$ in
-equation (\autoref{eq:ecmwf2})).
+where $\zeta=\frac{D_T}{L}$.  It is clear that the first derivative of (\autoref{eq:DIU_stab_func_eqn}),
+and thus of (\autoref{eq:DIU_ecmwf1}), is discontinuous at $\zeta=0$ (\ie\ $Q\rightarrow0$ in
+equation (\autoref{eq:DIU_ecmwf2})).
 
-The two terms on the right hand side of (\autoref{eq:ecmwf1}) represent different processes.
+The two terms on the right hand side of (\autoref{eq:DIU_ecmwf1}) represent different processes.
 The first term is simply the diabatic heating or cooling of the diurnal warm layer due to
 thermal energy fluxes into and out of the layer.
@@ -114 +114 @@
 
 \section{Cool skin model}
-\label{sec:cool_skin_sec}
+\label{sec:DIU_cool_skin_sec}
 
 %===============================================================
@@ -121 +121 @@
 As the cool skin is so thin (~1\,mm) we ignore the solar flux component to the heat flux and the Saunders equation for the cool skin temperature difference $\Delta T_{\mathrm{cs}}$ becomes
 \[
-  % \label{eq:sunders_eqn}
+  % \label{eq:DIU_sunders_eqn}
   \Delta T_{\mathrm{cs}}=\frac{Q_{\mathrm{ns}}\delta}{k_t} \mbox{,}
 \]
@@ -128 +128 @@
 $\delta$ is the thickness of the skin layer and is given by
 \begin{equation}
-\label{eq:sunders_thick_eqn}
+\label{eq:DIU_sunders_thick_eqn}
 \delta=\frac{\lambda \mu}{u^*_{w}} \mbox{,}
 \end{equation}
@@ -134 +134 @@
 \citet{saunders_JAS67} suggested varied between 5 and 10.
 
-The value of $\lambda$ used in equation (\autoref{eq:sunders_thick_eqn}) is that of \citet{artale.iudicone.ea_JGR02},
+The value of $\lambda$ used in equation (\autoref{eq:DIU_sunders_thick_eqn}) is that of \citet{artale.iudicone.ea_JGR02},
 which is shown in \citet{tu.tsuang_GRL05} to outperform a number of other parametrisations at
 both low and high wind speeds.
 Specifically,
 \[
-  % \label{eq:artale_lambda_eqn}
+  % \label{eq:DIU_artale_lambda_eqn}
   \lambda = \frac{ 8.64\times10^4 u^*_{w} k_t }{ \rho c_p h \mu \gamma }\mbox{,}
 \]
@@ -145 +145 @@
 $\gamma$ is a dimensionless function of wind speed $u$:
 \[
-  % \label{eq:artale_gamma_eqn}
+  % \label{eq:DIU_artale_gamma_eqn}
   \gamma =
   \begin{cases}