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

Last change on this file was 11693, checked in by nicolasmartin, 13 months ago

Macros renaming

File size: 8.1 KB
Line 
1\documentclass[../main/NEMO_manual]{subfiles}
2
3\begin{document}
4
5\chapter{Diurnal SST Models (DIU)}
6\label{chap:DIU}
7
8\thispagestyle{plain}
9
10\chaptertoc
11
12\paragraph{Changes record} ~\\
13
14{\footnotesize
15  \begin{tabularx}{\textwidth}{l||X|X}
16    Release & Author(s) & Modifications \\
17    \hline
18    {\em   4.0} & {\em ...} & {\em ...} \\
19    {\em   3.6} & {\em ...} & {\em ...} \\
20    {\em   3.4} & {\em ...} & {\em ...} \\
21    {\em <=3.4} & {\em ...} & {\em ...}
22  \end{tabularx}
23}
24
25\clearpage
26
27Code to produce an estimate of the diurnal warming and cooling of the sea surface skin
28temperature (skin SST) is found in the DIU directory.
29The skin temperature can be split into three parts:
30\begin{itemize}
31\item A foundation SST which is free from diurnal warming.
32\item A warm layer, typically ~3\,m thick,
33  where heating from solar radiation can cause a warm stably stratified layer during the daytime
34\item A cool skin, a thin layer, approximately ~1\, mm thick,
35  where long wave cooling is dominant and cools the immediate ocean surface.
36\end{itemize}
37
38Models are provided for both the warm layer, \mdl{diurnal\_bulk}, and the cool skin, \mdl{cool\_skin}.
39Foundation SST is not considered as it can be obtained either from the main \NEMO\ model
40(\ie\ from the temperature of the top few model levels) or from some other source.
41It must be noted that both the cool skin and warm layer models produce estimates of the change in temperature
42($\Delta T_{\mathrm{cs}}$ and $\Delta T_{\mathrm{wl}}$) and
43both must be added to a foundation SST to obtain the true skin temperature.
44
45Both the cool skin and warm layer models are controlled through the namelist \nam{diu}{diu}:
46
47\begin{listing}
48  \nlst{namdiu}
49  \caption{\forcode{&namdiu}}
50  \label{lst:namdiu}
51\end{listing}
52
53This namelist contains only two variables:
54\begin{description}
55\item [{\np{ln_diurnal}{ln\_diurnal}}] A logical switch for turning on/off both the cool skin and warm layer.
56\item [{\np{ln_diurnal_only}{ln\_diurnal\_only}}] A logical switch which if \forcode{.true.} will run the diurnal model without the other dynamical parts of \NEMO.
57  \np{ln_diurnal_only}{ln\_diurnal\_only} must be \forcode{.false.} if \np{ln_diurnal}{ln\_diurnal} is \forcode{.false.}.
58\end{description}
59
60Output for the diurnal model is through the variables `sst\_wl' (warm\_layer) and `sst\_cs' (cool skin).
61These are 2-D variables which will be included in the model output if they are specified in the iodef.xml file.
62
63Initialisation is through the restart file.
64Specifically the code will expect the presence of the 2-D variable ``Dsst'' to initialise the warm layer.
65The cool skin model, which is determined purely by the instantaneous fluxes, has no initialisation variable.
66
67%% =================================================================================================
68\section{Warm layer model}
69\label{sec:DIU_warm_layer_sec}
70
71The warm layer is calculated using the model of \citet{takaya.bidlot.ea_JGR10} (TAKAYA10 model hereafter).
72This is a simple flux based model that is defined by the equations
73\begin{align}
74\frac{\partial{\Delta T_{\mathrm{wl}}}}{\partial{t}}&=&\frac{Q(\nu+1)}{D_T\rho_w c_p
75\nu}-\frac{(\nu+1)ku^*_{w}f(L_a)\Delta T}{D_T\Phi\!\left(\frac{D_T}{L}\right)} \mbox{,}
76\label{eq:DIU_ecmwf1} \\
77L&=&\frac{\rho_w c_p u^{*^3}_{w}}{\kappa g \alpha_w Q }\mbox{,}\label{eq:DIU_ecmwf2}
78\end{align}
79where $\Delta T_{\mathrm{wl}}$ 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.
80In equation (\autoref{eq:DIU_ecmwf1}) $\alpha_w=2\times10^{-4}$ is the thermal expansion coefficient of water,
81$\kappa=0.4$ is von K\'{a}rm\'{a}n's constant, $c_p$ is the heat capacity at constant pressure of sea water,
82$\rho_w$ is the water density, and $L$ is the Monin-Obukhov length.
83The tunable variable $\nu$ is a shape parameter that defines the expected subskin temperature profile via
84$T(z) = T(0) - \left( \frac{z}{D_T} \right)^\nu \Delta T_{\mathrm{wl}}$,
85where $T$ is the absolute temperature and $z\le D_T$ is the depth below the top of the warm layer.
86The influence of wind on TAKAYA10 comes through the magnitude of the friction velocity of the water $u^*_{w}$,
87which can be related to the 10\,m wind speed $u_{10}$ through
88the relationship $u^*_{w} = u_{10}\sqrt{\frac{C_d\rho_a}{\rho_w}}$, where $C_d$ is the drag coefficient,
89and $\rho_a$ is the density of air.
90The symbol $Q$ in equation (\autoref{eq:DIU_ecmwf1}) is the instantaneous total thermal energy flux into
91the diurnal layer, \ie
92\[
93  Q = Q_{\mathrm{sol}} + Q_{\mathrm{lw}} + Q_{\mathrm{h}}\mbox{,}
94  % \label{eq:DIU_e_flux_eqn}
95\]
96where $Q_{\mathrm{h}}$ is the sensible and latent heat flux, $Q_{\mathrm{lw}}$ is the long wave flux,
97and $Q_{\mathrm{sol}}$ is the solar flux absorbed within the diurnal warm layer.
98For $Q_{\mathrm{sol}}$ the 9 term representation of \citet{gentemann.minnett.ea_JGR09} is used.
99In equation \autoref{eq:DIU_ecmwf1} the function $f(L_a)=\max(1,L_a^{\frac{2}{3}})$,
100where $L_a=0.3$\footnote{
101  This is a global average value, more accurately $L_a$ could be computed as $L_a=(u^*_{w}/u_s)^{\frac{1}{2}}$,
102  where $u_s$ is the stokes drift, but this is not currently done
103} is the turbulent Langmuir number and is a parametrization of the effect of waves.
104The function $\Phi\!\left(\frac{D_T}{L}\right)$ is the similarity function that
105parametrizes the stability of the water column and is given by:
106\begin{equation}
107\Phi(\zeta) = \left\{ \begin{array}{cc} 1 + \frac{5\zeta +
1084\zeta^2}{1+3\zeta+0.25\zeta^2} &(\zeta \ge 0) \\
109                                    (1 - 16\zeta)^{-\frac{1}{2}} & (\zeta < 0) \mbox{,}
110                                    \end{array} \right. \label{eq:DIU_stab_func_eqn}
111\end{equation}
112where $\zeta=\frac{D_T}{L}$.  It is clear that the first derivative of (\autoref{eq:DIU_stab_func_eqn}),
113and thus of (\autoref{eq:DIU_ecmwf1}), is discontinuous at $\zeta=0$ (\ie\ $Q\rightarrow0$ in
114equation (\autoref{eq:DIU_ecmwf2})).
115
116The two terms on the right hand side of (\autoref{eq:DIU_ecmwf1}) represent different processes.
117The first term is simply the diabatic heating or cooling of the diurnal warm layer due to
118thermal energy fluxes into and out of the layer.
119The second term parametrizes turbulent fluxes of heat out of the diurnal warm layer due to wind induced mixing.
120In practice the second term acts as a relaxation on the temperature.
121
122%% =================================================================================================
123\section{Cool skin model}
124\label{sec:DIU_cool_skin_sec}
125
126The 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 $u^*_{w}$.
127As 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
128\[
129  % \label{eq:DIU_sunders_eqn}
130  \Delta T_{\mathrm{cs}}=\frac{Q_{\mathrm{ns}}\delta}{k_t} \mbox{,}
131\]
132where $Q_{\mathrm{ns}}$ is the, usually negative, non-solar heat flux into the ocean and
133$k_t$ is the thermal conductivity of sea water.
134$\delta$ is the thickness of the skin layer and is given by
135\begin{equation}
136\label{eq:DIU_sunders_thick_eqn}
137\delta=\frac{\lambda \mu}{u^*_{w}} \mbox{,}
138\end{equation}
139where $\mu$ is the kinematic viscosity of sea water and $\lambda$ is a constant of proportionality which
140\citet{saunders_JAS67} suggested varied between 5 and 10.
141
142The value of $\lambda$ used in equation (\autoref{eq:DIU_sunders_thick_eqn}) is that of \citet{artale.iudicone.ea_JGR02},
143which is shown in \citet{tu.tsuang_GRL05} to outperform a number of other parametrisations at
144both low and high wind speeds.
145Specifically,
146\[
147  % \label{eq:DIU_artale_lambda_eqn}
148  \lambda = \frac{ 8.64\times10^4 u^*_{w} k_t }{ \rho c_p h \mu \gamma }\mbox{,}
149\]
150where $h=10$\,m is a reference depth and
151$\gamma$ is a dimensionless function of wind speed $u$:
152\[
153  % \label{eq:DIU_artale_gamma_eqn}
154  \gamma =
155  \begin{cases}
156    0.2u+0.5\mbox{,} & u \le 7.5\,\mbox{ms}^{-1} \\
157    1.6u-10\mbox{,} & 7.5 < u < 10\,\mbox{ms}^{-1} \\
158    6\mbox{,} & u \ge 10\,\mbox{ms}^{-1} \\
159  \end{cases}
160\]
161
162\subinc{\input{../../global/epilogue}}
163
164\end{document}
Note: See TracBrowser for help on using the repository browser.