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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\minitoc
12
13
14\newpage
15$\ $\newline % force a new line
16
17Code to produce an estimate of the diurnal warming and cooling of the sea surface skin
18temperature (skin SST) is found in the DIU directory. 
19The skin temperature can be split into three parts:
20\begin{itemize}
21\item
22  A foundation SST which is free from diurnal warming.
23\item
24  A warm layer, typically ~3\,m thick,
25  where heating from solar radiation can cause a warm stably stratified layer during the daytime
26\item
27  A cool skin, a thin layer, approximately ~1\, mm thick,
28  where long wave cooling is dominant and cools the immediate ocean surface.
29\end{itemize}
30
31Models are provided for both the warm layer, \mdl{diurnal\_bulk}, and the cool skin, \mdl{cool\_skin}.
32Foundation SST is not considered as it can be obtained either from the main NEMO model
33($i.e.$ 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($\Delta T_{\rm{cs}}$ and $\Delta T_{\rm{wl}}$) 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 \ngn{namdiu}:
39
40\nlst{namdiu}
41This namelist contains only two variables:
42\begin{description}
43\item[\np{ln\_diurnal}]
44  A logical switch for turning on/off both the cool skin and warm layer.
45\item[\np{ln\_diurnal\_only}]
46  A logical switch which if \forcode{.true.} will run the diurnal model without the other dynamical parts of NEMO.
47  \np{ln\_diurnal\_only} must be \forcode{.false.} if \np{ln\_diurnal} is \forcode{.false.}.
48\end{description}
49
50Output for the diurnal model is through the variables `sst\_wl' (warm\_layer) and `sst\_cs' (cool skin).
51These are 2-D variables which will be included in the model output if they are specified in the iodef.xml file.
52
53Initialisation is through the restart file.
54Specifically the code will expect the presence of the 2-D variable ``Dsst'' to initialise the warm layer.
55The cool skin model, which is determined purely by the instantaneous fluxes, has no initialisation variable. 
56
57%===============================================================
58\section{Warm layer model}
59\label{sec:warm_layer_sec}
60%===============================================================
61
62The warm layer is calculated using the model of \citet{Takaya_al_JGR10} (TAKAYA10 model hereafter).
63This is a simple flux based model that is defined by the equations
64\begin{align}
65\frac{\partial{\Delta T_{\rm{wl}}}}{\partial{t}}&=&\frac{Q(\nu+1)}{D_T\rho_w c_p
66\nu}-\frac{(\nu+1)ku^*_{w}f(L_a)\Delta T}{D_T\Phi\!\left(\frac{D_T}{L}\right)} \mbox{,}
67\label{eq:ecmwf1} \\
68L&=&\frac{\rho_w c_p u^{*^3}_{w}}{\kappa g \alpha_w Q }\mbox{,}\label{eq:ecmwf2}
69\end{align}
70where $\Delta T_{\rm{wl}}$ 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.
71In equation (\autoref{eq:ecmwf1}) $\alpha_w=2\times10^{-4}$ is the thermal expansion coefficient of water,
72$\kappa=0.4$ is von K\'{a}rm\'{a}n's constant, $c_p$ is the heat capacity at constant pressure of sea water,
73$\rho_w$ is the water density, and $L$ is the Monin-Obukhov length.
74The tunable variable $\nu$ is a shape parameter that defines the expected subskin temperature profile via
75$T(z) = T(0) - \left( \frac{z}{D_T} \right)^\nu \Delta T_{\rm{wl}}$,
76where $T$ is the absolute temperature and $z\le D_T$ is the depth below the top of the warm layer.
77The influence of wind on TAKAYA10 comes through the magnitude of the friction velocity of the water $u^*_{w}$,
78which can be related to the 10\,m wind speed $u_{10}$ through
79the relationship $u^*_{w} = u_{10}\sqrt{\frac{C_d\rho_a}{\rho_w}}$, where $C_d$ is the drag coefficient,
80and $\rho_a$ is the density of air.
81The symbol $Q$ in equation (\autoref{eq:ecmwf1}) is the instantaneous total thermal energy flux into
82the diurnal layer, $i.e.$
83\[
84  Q = Q_{\rm{sol}} + Q_{\rm{lw}} + Q_{\rm{h}}\mbox{,}
85  % \label{eq:e_flux_eqn}
86\]
87where $Q_{\rm{h}}$ is the sensible and latent heat flux, $Q_{\rm{lw}}$ is the long wave flux,
88and $Q_{\rm{sol}}$ is the solar flux absorbed within the diurnal warm layer.
89For $Q_{\rm{sol}}$ the 9 term representation of \citet{Gentemann_al_JGR09} is used.
90In equation \autoref{eq:ecmwf1} the function $f(L_a)=\max(1,L_a^{\frac{2}{3}})$,
91where $L_a=0.3$\footnote{
92  This is a global average value, more accurately $L_a$ could be computed as $L_a=(u^*_{w}/u_s)^{\frac{1}{2}}$,
93  where $u_s$ is the stokes drift, but this is not currently done
94} is the turbulent Langmuir number and is a parametrization of the effect of waves.
95The function $\Phi\!\left(\frac{D_T}{L}\right)$ is the similarity function that
96parametrizes the stability of the water column and is given by:
97\begin{equation}
98\Phi(\zeta) = \left\{ \begin{array}{cc} 1 + \frac{5\zeta +
994\zeta^2}{1+3\zeta+0.25\zeta^2} &(\zeta \ge 0) \\
100                                    (1 - 16\zeta)^{-\frac{1}{2}} & (\zeta < 0) \mbox{,}
101                                    \end{array} \right. \label{eq:stab_func_eqn}
102\end{equation}
103where $\zeta=\frac{D_T}{L}$.  It is clear that the first derivative of (\autoref{eq:stab_func_eqn}),
104and thus of (\autoref{eq:ecmwf1}), is discontinuous at $\zeta=0$ ($i.e.$ $Q\rightarrow0$ in
105equation (\autoref{eq:ecmwf2})).
106
107The two terms on the right hand side of (\autoref{eq:ecmwf1}) represent different processes.
108The first term is simply the diabatic heating or cooling of the diurnal warm layer due to
109thermal energy fluxes into and out of the layer.
110The second term parametrizes turbulent fluxes of heat out of the diurnal warm layer due to wind induced mixing.
111In practice the second term acts as a relaxation on the temperature.
112
113%===============================================================
114
115\section{Cool skin model}
116\label{sec:cool_skin_sec}
117
118%===============================================================
119
120The cool skin is modelled using the framework of \citet{Saunders_JAS82} who used a formulation of the near surface temperature difference based upon the heat flux and the friction velocity $u^*_{w}$.
121As 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 $\Delta T_{\rm{cs}}$ becomes
122\[
123  % \label{eq:sunders_eqn}
124  \Delta T_{\rm{cs}}=\frac{Q_{\rm{ns}}\delta}{k_t} \mbox{,}
125\]
126where $Q_{\rm{ns}}$ is the, usually negative, non-solar heat flux into the ocean and
127$k_t$ is the thermal conductivity of sea water.
128$\delta$ is the thickness of the skin layer and is given by
129\begin{equation}
130\label{eq:sunders_thick_eqn}
131\delta=\frac{\lambda \mu}{u^*_{w}} \mbox{,}
132\end{equation}
133where $\mu$ is the kinematic viscosity of sea water and $\lambda$ is a constant of proportionality which
134\citet{Saunders_JAS82} suggested varied between 5 and 10.
135
136The value of $\lambda$ used in equation (\autoref{eq:sunders_thick_eqn}) is that of \citet{Artale_al_JGR02},
137which is shown in \citet{Tu_Tsuang_GRL05} to outperform a number of other parametrisations at
138both low and high wind speeds.
139Specifically,
140\[
141  % \label{eq:artale_lambda_eqn}
142  \lambda = \frac{ 8.64\times10^4 u^*_{w} k_t }{ \rho c_p h \mu \gamma }\mbox{,}
143\]
144where $h=10$\,m is a reference depth and
145$\gamma$ is a dimensionless function of wind speed $u$:
146\[
147  % \label{eq:artale_gamma_eqn}
148  \gamma =
149  \begin{cases}
150    0.2u+0.5\mbox{,} & u \le 7.5\,\mbox{ms}^{-1} \\
151    1.6u-10\mbox{,} & 7.5 < u < 10\,\mbox{ms}^{-1} \\
152    6\mbox{,} & u \ge 10\,\mbox{ms}^{-1} \\
153  \end{cases}
154\]
155
156\biblio
157
158\end{document}
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