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1\documentclass[NEMO_book]{subfiles}
2\begin{document}
3% ================================================================
4% Diurnal SST models (DIU)
5% Edited by James While
6% ================================================================
7\chapter{Diurnal SST models (DIU)}
8\label{DIU}
9
10\minitoc
11
12
13\newpage
14$\$\newline % force a new line
15
16Code to produce an estimate of the diurnal warming and cooling of the sea surface skin
17temperature (skin SST) is found in the DIU directory.
18The skin temperature can be split into three parts:
19\begin{itemize}
20\item A foundation SST which is free from diurnal warming.
21\item A warm layer, typically ~3\,m thick, where heating from solar radiation can
22cause a warm stably stratified layer during the daytime
23\item A cool skin, a thin layer, approximately ~1\,mm thick, where long wave cooling
24is dominant and cools the immediate ocean surface.
25\end{itemize}
26
27Models are provided for both the warm layer, diurnal\_bulk.F90, and the cool skin,
28cool\_skin.F90.  Foundation SST is not considered as it can be obtained
29either from the main NEMO model ($i.e.$ from the temperature of the top few model levels)
30or from some other source.
31It must be noted that both the cool skin and warm layer models produce estimates of
32the change in temperature ($\Delta T_{\rm{cs}}$ and $\Delta T_{\rm{wl}}$)
33and both must be added to a foundation SST to obtain the true skin temperature.
34
35Both the cool skin and warm layer models are controlled through the namelist \ngn{namdiu}:
36\namdisplay{namdiu}
37This namelist contains only two variables:
38\begin{description}
39\item[\np{ln\_diurnal}] A logical switch for turning on/off both the cool skin and warm layer.
40\item[\np{ln\_diurnal\_only}] A logical switch which if .TRUE. will run the diurnal model
41without the other dynamical parts of NEMO.
42\np{ln\_diurnal\_only} must be .FALSE. if \np{ln\_diurnal} is .FALSE.
43\end{description}
44
45Output for the diurnal model is through the variables sst\_wl' (warm\_layer) and
46sst\_cs' (cool skin).  These are 2-D variables which will be included in the model
47output if they are specified in the iodef.xml file.
48
49Initialisation is through the restart file.  Specifically the code will expect
50the presence of the 2-D variable Dsst'' to initialise the warm layer.
51The cool skin model, which is determined purely by the instantaneous fluxes,
52has no initialisation variable.
53
54%===============================================================
55\section{Warm Layer model}
56\label{warm_layer_sec}
57%===============================================================
58
59The warm layer is calculated using the model of \citet{Takaya_al_JGR10} (TAKAYA10 model
60hereafter).  This is a simple flux based model that is defined by the equations
61\begin{eqnarray}
62\frac{\partial{\Delta T_{\rm{wl}}}}{\partial{t}}&=&\frac{Q(\nu+1)}{D_T\rho_w c_p
63\nu}-\frac{(\nu+1)ku^*_{w}f(L_a)\Delta T}{D_T\Phi\!\left(\frac{D_T}{L}\right)} \mbox{,}
64\label{ecmwf1} \\
65L&=&\frac{\rho_w c_p u^{*^3}_{w}}{\kappa g \alpha_w Q }\mbox{,}\label{ecmwf2}
66\end{eqnarray}
67where $\Delta T_{\rm{wl}}$ is the temperature difference between the top of the warm
68layer and the depth $D_T=3$\,m at which there is assumed to be no diurnal signal. In
69equation (\ref{ecmwf1}) $\alpha_w=2\times10^{-4}$ is the thermal expansion
70coefficient of water, $\kappa=0.4$ is von K\'{a}rm\'{a}n's constant, $c_p$ is the heat
71capacity at constant pressure of sea water, $\rho_w$ is the
72water density, and $L$ is the Monin-Obukhov length. The tunable
73variable $\nu$ is a shape parameter that defines the expected
74subskin temperature profile via $T(z)=T(0)-\left(\frac{z}{D_T}\right)^\nu\Delta 75T_{\rm{wl}}$,
76where $T$ is the absolute temperature and $z\le D_T$ is the depth
77below the top of the warm layer.
78The influence of wind on TAKAYA10 comes through the magnitude of the friction velocity
79of the water
80$u^*_{w}$, which can be related to the 10\,m wind speed $u_{10}$ through the relationship
81$u^*_{w} = u_{10}\sqrt{\frac{C_d\rho_a}{\rho_w}}$, where $C_d$ is
82the drag coefficient, and $\rho_a$ is the density of air.  The symbol $Q$ in equation
83(\ref{ecmwf1}) is the instantaneous total thermal energy
84flux into
85the diurnal layer, $i.e.$
86\begin{equation}
87Q = Q_{\rm{sol}} + Q_{\rm{lw}} + Q_{\rm{h}}\mbox{,} \label{e_flux_eqn}
88\end{equation}
89where $Q_{\rm{h}}$ is the sensible and latent heat flux, $Q_{\rm{lw}}$ is the long
90wave flux, and $Q_{\rm{sol}}$ is the solar flux absorbed
91within the diurnal warm layer. For $Q_{\rm{sol}}$ the 9 term
92representation of \citet{Gentemann_al_JGR09} is used.  In equation \ref{ecmwf1}
93the function $f(L_a)=\max(1,L_a^{\frac{2}{3}})$, where $L_a=0.3$\footnote{This
94is a global average value, more accurately $L_a$ could be computed as
95$L_a=(u^*_{w}/u_s)^{\frac{1}{2}}$, where $u_s$ is the stokes drift, but this is not
96currently done} is the turbulent Langmuir number and is a
97parametrization of the effect of waves.
98The function $\Phi\!\left(\frac{D_T}{L}\right)$ is the similarity function that
99parametrizes the stability of the water column and
100is given by:
101\begin{equation}
102\Phi(\zeta) = \left\{ \begin{array}{cc} 1 + \frac{5\zeta +
1034\zeta^2}{1+3\zeta+0.25\zeta^2} &(\zeta \ge 0) \\
104                                    (1 - 16\zeta)^{-\frac{1}{2}} & (\zeta < 0) \mbox{,}
105                                    \end{array} \right. \label{stab_func_eqn}
106\end{equation}
107where $\zeta=\frac{D_T}{L}$.  It is clear that the first derivative of
108(\ref{stab_func_eqn}), and thus of (\ref{ecmwf1}),
109is discontinuous at $\zeta=0$ ($i.e.$ $Q\rightarrow0$ in equation (\ref{ecmwf2})).
110
111The two terms on the right hand side of (\ref{ecmwf1}) represent different processes.
112The first term is simply the diabatic heating or cooling of the
113diurnal warm
114layer due to thermal energy
115fluxes into and out of the layer.  The second term
116parametrizes turbulent fluxes of heat out of the diurnal warm layer due to wind
117induced mixing. In practice the second term acts as a relaxation
118on the temperature.
119
120%===============================================================
121
122\section{Cool Skin model}
123\label{cool_skin_sec}
124
125%===============================================================
126
127The cool skin is modelled using the framework of \citet{Saunders_JAS82} who used a
128formulation of the near surface temperature difference based upon the heat flux and
129the friction velocity $u^*_{w}$.  As the cool skin
130is so thin (~1\,mm) we ignore the solar flux component to the heat flux and the
131Saunders equation for the cool skin temperature difference $\Delta T_{\rm{cs}}$ becomes
132\begin{equation}
133\label{sunders_eqn}
134\Delta T_{\rm{cs}}=\frac{Q_{\rm{ns}}\delta}{k_t} \mbox{,}
135\end{equation}
136where $Q_{\rm{ns}}$ is the, usually negative, non-solar heat flux into the ocean and
137$k_t$ is the thermal conductivity of sea water. $\delta$ is the thickness of the
138skin layer and is given by
139\begin{equation}
140\label{sunders_thick_eqn}
141\delta=\frac{\lambda \mu}{u^*_{w}} \mbox{,}
142\end{equation}
143where $\mu$ is the kinematic viscosity of sea water and $\lambda$ is a constant of
144proportionality which \citet{Saunders_JAS82} suggested varied between 5 and 10.
145
146The value of $\lambda$ used in equation (\ref{sunders_thick_eqn}) is that of
147\citet{Artale_al_JGR02},
148which is shown in \citet{Tu_Tsuang_GRL05} to outperform a number of other
149parametrisations at both low and high wind speeds. Specifically,
150\begin{equation}
151\label{artale_lambda_eqn}
152\lambda = \frac{ 8.64\times10^4 u^*_{w} k_t }{ \rho c_p h \mu \gamma }\mbox{,}
153\end{equation}
154where $h=10$\,m is a reference depth and
155$\gamma$ is a dimensionless function of wind speed $u$:
156\begin{equation}
157\label{artale_gamma_eqn}
158\gamma = \left\{ \begin{matrix}
159                     0.2u+0.5\mbox{,} & u \le 7.5\,\mbox{ms}^{-1} \\
160                     1.6u-10\mbox{,} & 7.5 < u < 10\,\mbox{ms}^{-1} \\
161                     6\mbox{,} & \ge 10\,\mbox{ms}^{-1} \\
162                 \end{matrix}
163          \right.
164\end{equation}
165
166\end{document}
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