source: NEMO/trunk/doc/latex/NEMO/subfiles/chap_DIU.tex @ 11577

Last change on this file since 11577 was 11577, checked in by nicolasmartin, 13 months ago

New LaTeX commands \nam and \np to mention namelist content
(Partial commit to serve as a backup before other large edits)
In order to benefit of the syntax highlighting and to have a simpler syntax for
citing namelist block (\nam) and parameter (\np) with an optional variable assignment (\forcode{…}),
at this time the only viable solution I found is to require a double marker for
what it looks like the same item:

  1. Marker with the real name: 'tra_adv' block or 'ln_flx' parameter
  2. Marker with underscore character escaping: 'tra\_adv' block or 'ln\_flx' parameter

Despite many searches and attempts, I did not find a workaround to edit on-the-fly one or
the other marker.
In fact, the problem is on one side that the LaTeX index interprets '_' as a switch for lowering like
in math mode while on the other hand the backslash is considered for Pygments as a typo in Fortran
(red box).

For instance, \nam and \np have as of now the aforementioned 2 mandatory arguments in
the previous order (between braces) + an optional argument for \np when the parameter is defined
(between brackets at the first position):

  • \nam: LaTeX code in the \nam{tra_adv}{tra\_adv} → PDF ' in the &namtra_adv (namelist X.X) ' with syntax highlighting, the hyperlink and the index entry
  • \np: LaTeX code \np[=.true.]{ln_flx}{ln\_flx} → PDF ln_flux=.true. with syntax highlighting for the whole string and the entry in the 'parameters' index
File size: 8.0 KB
Line 
1\documentclass[../main/NEMO_manual]{subfiles}
2
3\begin{document}
4% ================================================================
5% Diurnal SST models (DIU)
6% Edited by James While
7% ================================================================
8\chapter{Diurnal SST Models (DIU)}
9\label{chap:DIU}
10
11\chaptertoc
12
13
14\newpage
15$\ $\newline % force a new line
16
17Code to produce an estimate of the diurnal warming and cooling of the sea surface skin
18temperature (skin SST) is found in the DIU directory.
19The skin temperature can be split into three parts:
20\begin{itemize}
21\item
22  A foundation SST which is free from diurnal warming.
23\item
24  A warm layer, typically ~3\,m thick,
25  where heating from solar radiation can cause a warm stably stratified layer during the daytime
26\item
27  A cool skin, a thin layer, approximately ~1\, mm thick,
28  where long wave cooling is dominant and cools the immediate ocean surface.
29\end{itemize}
30
31Models are provided for both the warm layer, \mdl{diurnal\_bulk}, and the cool skin, \mdl{cool\_skin}.
32Foundation SST is not considered as it can be obtained either from the main \NEMO\ model
33(\ie\ from the temperature of the top few model levels) or from some other source.
34It must be noted that both the cool skin and warm layer models produce estimates of the change in temperature
35($\Delta T_{\mathrm{cs}}$ and $\Delta T_{\mathrm{wl}}$) and
36both must be added to a foundation SST to obtain the true skin temperature.
37
38Both the cool skin and warm layer models are controlled through the namelist \nam{diu}:
39
40\begin{listing}
41  \nlst{namdiu}
42  \caption{\forcode{&namdiu}}
43  \label{lst:namdiu}
44\end{listing}
45
46This namelist contains only two variables:
47\begin{description}
48\item[\np{ln_diurnal}{ln\_diurnal}]
49  A logical switch for turning on/off both the cool skin and warm layer.
50\item[\np{ln_diurnal_only}{ln\_diurnal\_only}]
51  A logical switch which if \forcode{.true.} will run the diurnal model without the other dynamical parts of \NEMO.
52  \np{ln_diurnal_only}{ln\_diurnal\_only} must be \forcode{.false.} if \np{ln_diurnal}{ln\_diurnal} is \forcode{.false.}.
53\end{description}
54
55Output for the diurnal model is through the variables `sst\_wl' (warm\_layer) and `sst\_cs' (cool skin).
56These are 2-D variables which will be included in the model output if they are specified in the iodef.xml file.
57
58Initialisation is through the restart file.
59Specifically the code will expect the presence of the 2-D variable ``Dsst'' to initialise the warm layer.
60The cool skin model, which is determined purely by the instantaneous fluxes, has no initialisation variable.
61
62%===============================================================
63\section{Warm layer model}
64\label{sec:DIU_warm_layer_sec}
65%===============================================================
66
67The warm layer is calculated using the model of \citet{takaya.bidlot.ea_JGR10} (TAKAYA10 model hereafter).
68This is a simple flux based model that is defined by the equations
69\begin{align}
70\frac{\partial{\Delta T_{\mathrm{wl}}}}{\partial{t}}&=&\frac{Q(\nu+1)}{D_T\rho_w c_p
71\nu}-\frac{(\nu+1)ku^*_{w}f(L_a)\Delta T}{D_T\Phi\!\left(\frac{D_T}{L}\right)} \mbox{,}
72\label{eq:DIU_ecmwf1} \\
73L&=&\frac{\rho_w c_p u^{*^3}_{w}}{\kappa g \alpha_w Q }\mbox{,}\label{eq:DIU_ecmwf2}
74\end{align}
75where $\Delta T_{\mathrm{wl}}$ 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.
76In equation (\autoref{eq:DIU_ecmwf1}) $\alpha_w=2\times10^{-4}$ is the thermal expansion coefficient of water,
77$\kappa=0.4$ is von K\'{a}rm\'{a}n's constant, $c_p$ is the heat capacity at constant pressure of sea water,
78$\rho_w$ is the water density, and $L$ is the Monin-Obukhov length.
79The tunable variable $\nu$ is a shape parameter that defines the expected subskin temperature profile via
80$T(z) = T(0) - \left( \frac{z}{D_T} \right)^\nu \Delta T_{\mathrm{wl}}$,
81where $T$ is the absolute temperature and $z\le D_T$ is the depth below the top of the warm layer.
82The influence of wind on TAKAYA10 comes through the magnitude of the friction velocity of the water $u^*_{w}$,
83which can be related to the 10\,m wind speed $u_{10}$ through
84the relationship $u^*_{w} = u_{10}\sqrt{\frac{C_d\rho_a}{\rho_w}}$, where $C_d$ is the drag coefficient,
85and $\rho_a$ is the density of air.
86The symbol $Q$ in equation (\autoref{eq:DIU_ecmwf1}) is the instantaneous total thermal energy flux into
87the diurnal layer, \ie
88\[
89  Q = Q_{\mathrm{sol}} + Q_{\mathrm{lw}} + Q_{\mathrm{h}}\mbox{,}
90  % \label{eq:DIU_e_flux_eqn}
91\]
92where $Q_{\mathrm{h}}$ is the sensible and latent heat flux, $Q_{\mathrm{lw}}$ is the long wave flux,
93and $Q_{\mathrm{sol}}$ is the solar flux absorbed within the diurnal warm layer.
94For $Q_{\mathrm{sol}}$ the 9 term representation of \citet{gentemann.minnett.ea_JGR09} is used.
95In equation \autoref{eq:DIU_ecmwf1} the function $f(L_a)=\max(1,L_a^{\frac{2}{3}})$,
96where $L_a=0.3$\footnote{
97  This is a global average value, more accurately $L_a$ could be computed as $L_a=(u^*_{w}/u_s)^{\frac{1}{2}}$,
98  where $u_s$ is the stokes drift, but this is not currently done
99} is the turbulent Langmuir number and is a parametrization of the effect of waves.
100The function $\Phi\!\left(\frac{D_T}{L}\right)$ is the similarity function that
101parametrizes the stability of the water column and is given by:
102\begin{equation}
103\Phi(\zeta) = \left\{ \begin{array}{cc} 1 + \frac{5\zeta +
1044\zeta^2}{1+3\zeta+0.25\zeta^2} &(\zeta \ge 0) \\
105                                    (1 - 16\zeta)^{-\frac{1}{2}} & (\zeta < 0) \mbox{,}
106                                    \end{array} \right. \label{eq:DIU_stab_func_eqn}
107\end{equation}
108where $\zeta=\frac{D_T}{L}$.  It is clear that the first derivative of (\autoref{eq:DIU_stab_func_eqn}),
109and thus of (\autoref{eq:DIU_ecmwf1}), is discontinuous at $\zeta=0$ (\ie\ $Q\rightarrow0$ in
110equation (\autoref{eq:DIU_ecmwf2})).
111
112The two terms on the right hand side of (\autoref{eq:DIU_ecmwf1}) represent different processes.
113The first term is simply the diabatic heating or cooling of the diurnal warm layer due to
114thermal energy fluxes into and out of the layer.
115The second term parametrizes turbulent fluxes of heat out of the diurnal warm layer due to wind induced mixing.
116In practice the second term acts as a relaxation on the temperature.
117
118%===============================================================
119
120\section{Cool skin model}
121\label{sec:DIU_cool_skin_sec}
122
123%===============================================================
124
125The cool skin is modelled using the framework of \citet{saunders_JAS67} who used a formulation of the near surface temperature difference based upon the heat flux and the friction velocity $u^*_{w}$.
126As 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
127\[
128  % \label{eq:DIU_sunders_eqn}
129  \Delta T_{\mathrm{cs}}=\frac{Q_{\mathrm{ns}}\delta}{k_t} \mbox{,}
130\]
131where $Q_{\mathrm{ns}}$ is the, usually negative, non-solar heat flux into the ocean and
132$k_t$ is the thermal conductivity of sea water.
133$\delta$ is the thickness of the skin layer and is given by
134\begin{equation}
135\label{eq:DIU_sunders_thick_eqn}
136\delta=\frac{\lambda \mu}{u^*_{w}} \mbox{,}
137\end{equation}
138where $\mu$ is the kinematic viscosity of sea water and $\lambda$ is a constant of proportionality which
139\citet{saunders_JAS67} suggested varied between 5 and 10.
140
141The value of $\lambda$ used in equation (\autoref{eq:DIU_sunders_thick_eqn}) is that of \citet{artale.iudicone.ea_JGR02},
142which is shown in \citet{tu.tsuang_GRL05} to outperform a number of other parametrisations at
143both low and high wind speeds.
144Specifically,
145\[
146  % \label{eq:DIU_artale_lambda_eqn}
147  \lambda = \frac{ 8.64\times10^4 u^*_{w} k_t }{ \rho c_p h \mu \gamma }\mbox{,}
148\]
149where $h=10$\,m is a reference depth and
150$\gamma$ is a dimensionless function of wind speed $u$:
151\[
152  % \label{eq:DIU_artale_gamma_eqn}
153  \gamma =
154  \begin{cases}
155    0.2u+0.5\mbox{,} & u \le 7.5\,\mbox{ms}^{-1} \\
156    1.6u-10\mbox{,} & 7.5 < u < 10\,\mbox{ms}^{-1} \\
157    6\mbox{,} & u \ge 10\,\mbox{ms}^{-1} \\
158  \end{cases}
159\]
160
161\biblio
162
163\pindex
164
165\end{document}
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