[10414] | 1 | \documentclass[../main/NEMO_manual]{subfiles} |
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| 2 | |
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[6997] | 3 | \begin{document} |
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[5859] | 4 | % ================================================================ |
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| 5 | % Diurnal SST models (DIU) |
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| 6 | % Edited by James While |
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| 7 | % ================================================================ |
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[9393] | 8 | \chapter{Diurnal SST Models (DIU)} |
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[9407] | 9 | \label{chap:DIU} |
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[5859] | 10 | |
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| 11 | \minitoc |
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| 12 | |
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| 13 | |
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| 14 | \newpage |
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| 15 | $\ $\newline % force a new line |
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| 16 | |
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| 17 | Code to produce an estimate of the diurnal warming and cooling of the sea surface skin |
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[6289] | 18 | temperature (skin SST) is found in the DIU directory. |
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| 19 | The skin temperature can be split into three parts: |
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[5859] | 20 | \begin{itemize} |
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[10354] | 21 | \item |
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| 22 | A foundation SST which is free from diurnal warming. |
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| 23 | \item |
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| 24 | A warm layer, typically ~3\,m thick, |
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| 25 | where heating from solar radiation can cause a warm stably stratified layer during the daytime |
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| 26 | \item |
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| 27 | A cool skin, a thin layer, approximately ~1\, mm thick, |
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| 28 | where long wave cooling is dominant and cools the immediate ocean surface. |
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[5859] | 29 | \end{itemize} |
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| 30 | |
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[10414] | 31 | Models are provided for both the warm layer, \mdl{diurnal\_bulk}, and the cool skin, \mdl{cool\_skin}. |
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[10354] | 32 | Foundation SST is not considered as it can be obtained either from the main NEMO model |
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[10442] | 33 | (\ie from the temperature of the top few model levels) or from some other source. |
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[10354] | 34 | It must be noted that both the cool skin and warm layer models produce estimates of the change in temperature |
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[11263] | 35 | ($\Delta T_{\mathrm{cs}}$ and $\Delta T_{\mathrm{wl}}$) and |
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[10354] | 36 | both must be added to a foundation SST to obtain the true skin temperature. |
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[5859] | 37 | |
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[6289] | 38 | Both the cool skin and warm layer models are controlled through the namelist \ngn{namdiu}: |
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[10146] | 39 | |
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| 40 | \nlst{namdiu} |
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[5859] | 41 | This namelist contains only two variables: |
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| 42 | \begin{description} |
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[10354] | 43 | \item[\np{ln\_diurnal}] |
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| 44 | A logical switch for turning on/off both the cool skin and warm layer. |
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| 45 | \item[\np{ln\_diurnal\_only}] |
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| 46 | A logical switch which if \forcode{.true.} will run the diurnal model without the other dynamical parts of NEMO. |
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| 47 | \np{ln\_diurnal\_only} must be \forcode{.false.} if \np{ln\_diurnal} is \forcode{.false.}. |
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[5859] | 48 | \end{description} |
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| 49 | |
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[10354] | 50 | Output for the diurnal model is through the variables `sst\_wl' (warm\_layer) and `sst\_cs' (cool skin). |
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| 51 | These are 2-D variables which will be included in the model output if they are specified in the iodef.xml file. |
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[5859] | 52 | |
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[10354] | 53 | Initialisation is through the restart file. |
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| 54 | Specifically the code will expect the presence of the 2-D variable ``Dsst'' to initialise the warm layer. |
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| 55 | The cool skin model, which is determined purely by the instantaneous fluxes, has no initialisation variable. |
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[5859] | 56 | |
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| 57 | %=============================================================== |
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[9393] | 58 | \section{Warm layer model} |
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[9407] | 59 | \label{sec:warm_layer_sec} |
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[5859] | 60 | %=============================================================== |
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| 61 | |
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[11263] | 62 | The warm layer is calculated using the model of \citet{takaya.bidlot.ea_JGR10} (TAKAYA10 model hereafter). |
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[10354] | 63 | This is a simple flux based model that is defined by the equations |
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[10406] | 64 | \begin{align} |
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[11263] | 65 | \frac{\partial{\Delta T_{\mathrm{wl}}}}{\partial{t}}&=&\frac{Q(\nu+1)}{D_T\rho_w c_p |
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[5859] | 66 | \nu}-\frac{(\nu+1)ku^*_{w}f(L_a)\Delta T}{D_T\Phi\!\left(\frac{D_T}{L}\right)} \mbox{,} |
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[9407] | 67 | \label{eq:ecmwf1} \\ |
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| 68 | L&=&\frac{\rho_w c_p u^{*^3}_{w}}{\kappa g \alpha_w Q }\mbox{,}\label{eq:ecmwf2} |
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[10406] | 69 | \end{align} |
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[11263] | 70 | 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. |
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[10354] | 71 | In equation (\autoref{eq:ecmwf1}) $\alpha_w=2\times10^{-4}$ is the thermal expansion coefficient of water, |
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| 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, |
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| 73 | $\rho_w$ is the water density, and $L$ is the Monin-Obukhov length. |
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| 74 | The tunable variable $\nu$ is a shape parameter that defines the expected subskin temperature profile via |
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[11263] | 75 | $T(z) = T(0) - \left( \frac{z}{D_T} \right)^\nu \Delta T_{\mathrm{wl}}$, |
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[10354] | 76 | where $T$ is the absolute temperature and $z\le D_T$ is the depth below the top of the warm layer. |
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| 77 | The influence of wind on TAKAYA10 comes through the magnitude of the friction velocity of the water $u^*_{w}$, |
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| 78 | which can be related to the 10\,m wind speed $u_{10}$ through |
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| 79 | the relationship $u^*_{w} = u_{10}\sqrt{\frac{C_d\rho_a}{\rho_w}}$, where $C_d$ is the drag coefficient, |
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| 80 | and $\rho_a$ is the density of air. |
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| 81 | The symbol $Q$ in equation (\autoref{eq:ecmwf1}) is the instantaneous total thermal energy flux into |
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[10442] | 82 | the diurnal layer, \ie |
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[10414] | 83 | \[ |
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[11263] | 84 | Q = Q_{\mathrm{sol}} + Q_{\mathrm{lw}} + Q_{\mathrm{h}}\mbox{,} |
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[10414] | 85 | % \label{eq:e_flux_eqn} |
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| 86 | \] |
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[11263] | 87 | where $Q_{\mathrm{h}}$ is the sensible and latent heat flux, $Q_{\mathrm{lw}}$ is the long wave flux, |
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| 88 | and $Q_{\mathrm{sol}}$ is the solar flux absorbed within the diurnal warm layer. |
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| 89 | For $Q_{\mathrm{sol}}$ the 9 term representation of \citet{gentemann.minnett.ea_JGR09} is used. |
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[10354] | 90 | In equation \autoref{eq:ecmwf1} the function $f(L_a)=\max(1,L_a^{\frac{2}{3}})$, |
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| 91 | where $L_a=0.3$\footnote{ |
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| 92 | This is a global average value, more accurately $L_a$ could be computed as $L_a=(u^*_{w}/u_s)^{\frac{1}{2}}$, |
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| 93 | where $u_s$ is the stokes drift, but this is not currently done |
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| 94 | } is the turbulent Langmuir number and is a parametrization of the effect of waves. |
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[5859] | 95 | The function $\Phi\!\left(\frac{D_T}{L}\right)$ is the similarity function that |
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[10354] | 96 | parametrizes the stability of the water column and is given by: |
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[5859] | 97 | \begin{equation} |
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| 98 | \Phi(\zeta) = \left\{ \begin{array}{cc} 1 + \frac{5\zeta + |
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| 99 | 4\zeta^2}{1+3\zeta+0.25\zeta^2} &(\zeta \ge 0) \\ |
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| 100 | (1 - 16\zeta)^{-\frac{1}{2}} & (\zeta < 0) \mbox{,} |
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[9407] | 101 | \end{array} \right. \label{eq:stab_func_eqn} |
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[5859] | 102 | \end{equation} |
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[10354] | 103 | where $\zeta=\frac{D_T}{L}$. It is clear that the first derivative of (\autoref{eq:stab_func_eqn}), |
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[10442] | 104 | and thus of (\autoref{eq:ecmwf1}), is discontinuous at $\zeta=0$ (\ie $Q\rightarrow0$ in |
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[10354] | 105 | equation (\autoref{eq:ecmwf2})). |
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[5859] | 106 | |
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[9407] | 107 | The two terms on the right hand side of (\autoref{eq:ecmwf1}) represent different processes. |
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[10354] | 108 | The first term is simply the diabatic heating or cooling of the diurnal warm layer due to |
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| 109 | thermal energy fluxes into and out of the layer. |
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| 110 | The second term parametrizes turbulent fluxes of heat out of the diurnal warm layer due to wind induced mixing. |
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| 111 | In practice the second term acts as a relaxation on the temperature. |
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[5859] | 112 | |
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| 113 | %=============================================================== |
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| 114 | |
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[9393] | 115 | \section{Cool skin model} |
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[9407] | 116 | \label{sec:cool_skin_sec} |
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[5859] | 117 | |
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| 118 | %=============================================================== |
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| 119 | |
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[11263] | 120 | The 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 $u^*_{w}$. |
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| 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 |
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[10414] | 122 | \[ |
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| 123 | % \label{eq:sunders_eqn} |
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[11263] | 124 | \Delta T_{\mathrm{cs}}=\frac{Q_{\mathrm{ns}}\delta}{k_t} \mbox{,} |
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[10414] | 125 | \] |
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[11263] | 126 | where $Q_{\mathrm{ns}}$ is the, usually negative, non-solar heat flux into the ocean and |
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[10354] | 127 | $k_t$ is the thermal conductivity of sea water. |
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| 128 | $\delta$ is the thickness of the skin layer and is given by |
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[5859] | 129 | \begin{equation} |
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[9407] | 130 | \label{eq:sunders_thick_eqn} |
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[5859] | 131 | \delta=\frac{\lambda \mu}{u^*_{w}} \mbox{,} |
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| 132 | \end{equation} |
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[10354] | 133 | where $\mu$ is the kinematic viscosity of sea water and $\lambda$ is a constant of proportionality which |
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[11263] | 134 | \citet{saunders_JAS67} suggested varied between 5 and 10. |
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[5859] | 135 | |
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[11263] | 136 | The value of $\lambda$ used in equation (\autoref{eq:sunders_thick_eqn}) is that of \citet{artale.iudicone.ea_JGR02}, |
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| 137 | which is shown in \citet{tu.tsuang_GRL05} to outperform a number of other parametrisations at |
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[10354] | 138 | both low and high wind speeds. |
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| 139 | Specifically, |
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[10414] | 140 | \[ |
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| 141 | % \label{eq:artale_lambda_eqn} |
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| 142 | \lambda = \frac{ 8.64\times10^4 u^*_{w} k_t }{ \rho c_p h \mu \gamma }\mbox{,} |
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| 143 | \] |
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[5859] | 144 | where $h=10$\,m is a reference depth and |
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| 145 | $\gamma$ is a dimensionless function of wind speed $u$: |
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[10414] | 146 | \[ |
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| 147 | % \label{eq:artale_gamma_eqn} |
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| 148 | \gamma = |
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| 149 | \begin{cases} |
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| 150 | 0.2u+0.5\mbox{,} & u \le 7.5\,\mbox{ms}^{-1} \\ |
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| 151 | 1.6u-10\mbox{,} & 7.5 < u < 10\,\mbox{ms}^{-1} \\ |
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| 152 | 6\mbox{,} & u \ge 10\,\mbox{ms}^{-1} \\ |
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| 153 | \end{cases} |
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| 154 | \] |
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[5859] | 155 | |
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[10414] | 156 | \biblio |
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| 157 | |
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[10442] | 158 | \pindex |
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| 159 | |
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[6997] | 160 | \end{document} |
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