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branches/DEV_r1826_DOC/DOC/TexFiles/Chapters/Chap_TRA.tex
r2164 r2195 666 666 to the heat and salt fluxes crossing the sea surface and not linked with $F_{mass}$, the water 667 667 exchange with the other media, and to the heat and salt content of this water exchange. 668 These two parts are included in $Q_{ns}$, the surface heat flux and $F_{salt}$, the surface salt flux. 669 The specification of those fluxes is further detail in the SBC chapter (see \S\ref{SBC}). 670 671 In the nonlinear free surface case (\key{vvl} is defined), the forcing term of $T$ and $S$ 672 conservation equations is simply given by: 668 In a forcoming release, these two parts, computed in the surface module (SBC), will included directly 669 in $Q_{ns}$, the surface heat flux and $F_{salt}$, the surface salt flux. 670 \gmcomment{ The specification of those fluxes is further detail in the SBC chapter (see \S\ref{SBC}). } 671 This change will provide a same forcing formulation for any tracers (including temperature and salinity). 672 673 In the current version, the situation is a little bit more complicated. 674 The surface module (\mdl{sbcmod}, see \S\ref{SBC}) provides the following 675 forcing fields (used on tracers): 676 677 $\bullet$ $Q_{ns}$,the non solar part of the net surface heat flux that cross the sea surface 678 (difference between the total surface heat flux and the fraction of the short wave flux that 679 penetrates into the water column, see \S\ref{TRA_qsr}) 680 681 $\bullet$ \textit{emp}, the mass flux exchanged with the atmosphere (evaporation minus precipitation) 682 683 $\bullet$ $\textit{emp}_S$, an equivalent mass flux taking into account the effect of ice-ocean mass exchanged 684 685 $\bullet$ \textit{rnf}, the mass flux associated with runoff (see \S\ref{SBC_rnf} for further detail of how it acts on temperature and salinity tendencies) 686 687 The $\textit{emp}_S$ field is not simply the budget evaporation-precipitation+freezing-melting because 688 the sea-ice is not currently embedded in the ocean but levitates above it. There is not mass 689 exchanged between the sea-ice and the ocean. Instead we only take into account the salt 690 flux link to the fact that sea-ice has a non-sero salinity, and the concentration/dilution effect 691 due to the freezing/melting (F/M) process. These two parts of the forcing are then converted into 692 a equivalent mass flux given by $\textit{emp}_S - \textit{emp}$. As a result of this mess, 693 the surface boundary condition on temperature and salinity is applied as follows: 694 695 In the nonlinear free surface case (\key{vvl} is defined, \jp{lk\_vvl}=true): 673 696 \begin{equation} \label{Eq_tra_sbc} 674 697 \begin{aligned} 675 F^T &=\frac{ Q_{ns} }{\rho _o \;C_p \,\left. e_{3t} \right|_{k=1} } \\ 698 &F^T = \frac{ 1 }{\rho _o \;C_p \,\left. e_{3t} \right|_{k=1} } 699 &\overline{ \left( Q_{ns} - \textit{emp}\;C_p\,\left. T \right|_{k=1} \right) }^t & \\ 676 700 % 677 F^S &=\frac{ F_{salt} }{\rho _o \; \left. e_{3t} \right|_{k=1} } 701 & F^S =\frac{ 1 }{\rho _o \,\left. e_{3t} \right|_{k=1} } 702 &\overline{ \left( (\textit{emp}_S - \textit{emp})\;\left. S \right|_{k=1} \right) }^t & \\ 678 703 \end{aligned} 679 704 \end{equation} 680 where both $Q_{ns}$ and $F_{salt}$ are averaged over two consecutive time step (before and now) 681 in order to prevent the exitation of the divergence of odd and even time step (see \S\ref{STP}). 682 683 In the linear free surface case (\key{vvl} is not defined), the ocean volume is assumed 684 to be fixed in time, and so are the vertical scale factors. In order to compensate the 685 volume change due to water exchange with the other media, an extra term must be added, 686 the so-called concentration/dilution effect. The forcing term becomes: 687 \begin{equation} \label{Eq_tra_sbc_linssh} 705 706 In the linear free surface case (\key{vvl} not defined, , \jp{lk\_vvl}=false): 707 \begin{equation} \label{Eq_tra_sbc_lin} 688 708 \begin{aligned} 689 &F^T = \frac{ \overline{Q_{ns}}^t }{\rho _o \;C_p \,\left. e_{3t} \right|_{k=1} } 690 - \frac{ \overline{F_{mass}}^t\;\left. T \right|_{k=1} }{\rho _o \; \left. e_{3t} \right|_{k=1} } \\ 709 &F^T = \frac{ 1 }{\rho _o \;C_p \,\left. e_{3t} \right|_{k=1} } &\overline{ Q_{ns} }^t & \\ 691 710 % 692 & F^S =\frac{ \overline{F_{salt}}^t}{\rho _o \,\left. e_{3t} \right|_{k=1} }693 - \frac{ \overline{F_{mass}}^t\;\left. S \right|_{k=1} }{\rho _o \; \left. e_{3t} \right|_{k=1} }\\711 & F^S =\frac{ 1 }{\rho _o \,\left. e_{3t} \right|_{k=1} } 712 &\overline{ \left( \textit{emp}_S\;\left. S \right|_{k=1} \right) }^t & \\ 694 713 \end{aligned} 695 714 \end{equation} 696 The total salt content is no more exactly conserved (\citet{Roullet_Madec_JGR00}. 697 See also \S\ref{PE_free_surface}). 698 699 In the case of the rigid lid approximation, the surface salinity forcing $F^s$ 700 is also expressed by \eqref{Eq_tra_forcing}, but now the global integral of 701 the product of \textit{emp} and S, is not compensated by the advection of fluid 702 through the top level: this is because in the rigid lid case \textit{w(k=1) = 0} 703 (in contrast to the linear free surface case). As a result, even if the budget 704 of \textit{emp} is zero on average over the whole ocean domain, the 705 associated salt flux is not, since sea-surface salinity and \textit{emp} are 706 intrinsically correlated (high \textit{SSS} are found where evaporation is 707 strong whilst low \textit{SSS} is usually associated with high precipitation 708 or river runoff). 709 710 711 712 The $Q_{ns} $ and \textit{EMP} fields are defined and updated in the 713 \mdl{sbcmod} module (see \S\ref{SBC}). 714 715 where \textit{emp} is the freshwater budget (evaporation minus precipitation 716 minus river runoff) which forces the ocean volume, $Q_{ns}$ is the 717 non-penetrative part of the net surface heat flux (difference between 718 the total surface heat flux and the fraction of the short wave flux that 719 penetrates into the water column), the product $\textit{emp}_S\;.\left. S \right|_{k=1}$ 720 is the ice-ocean salt flux, and $\left. S\right|_{k=1}$ is the sea surface 721 salinity (\textit{SSS}). The total salt content is conserved in this formulation 722 (except for the effect of the Asselin filter). 723 724 %AMT note: the ice-ocean flux had been forgotten in the first release of the key_vvl option, has this been corrected in the code? ===> gm : NO to be added at NOCS 715 where $\overline{x }^t$ means that $x$ is averaged over two consecutive time step 716 ($t-\rdt/2$ and $t+\rdt/2$). Such a time averaged prevents the excitation of the 717 divergence of odd and even time step (see \S\ref{STP}). 718 719 The two set of equations, \eqref{Eq_tra_sbc} and \eqref{Eq_tra_sbc_lin}, are obtained 720 by assuming that the temperature of precipitation and evaporation are equal to 721 the ocean surface temperature while their salinity is zero. Therefore, the heat content 722 of \textit{emp} budget must be added to the temperature equation in variable volume case, 723 while it does not appear in constant volume. Similarly, the \textit{emp} budget affects 724 the ocean surface salinity in constant volume case (through the concentration dilution effect) 725 while it does not appears explicitly in variable volume as salinity change will be 726 induced by volume change. In both constant and variable volume, surface salinity 727 will change with ice-ocean salt flux and F/M flux without mass exchanges 728 ($\textit{emp}_S - \textit{emp}$). 729 730 Note that concentration/dilution effect due to F/M is computed using 731 a constant ice salinity as well as a constant ocean salinity. 732 This approximation suppresses the correlation between \textit{SSS} 733 and F/M flux, allowing the ice-ocean salt exchanges to be conservative. 734 Indeed, if this approximation is not made, even if the F/M budget is zero 735 on average over the whole ocean domain and over the seasonal cycle, 736 the associated salt flux is not, since sea-surface salinity and F/M flux are 737 intrinsically correlated (high \textit{SSS} are found where freezing is 738 strong whilst low \textit{SSS} is usually associated with high melting areas. 739 740 Even using this approximation, an exact conservation of heat and salt content 741 is only achieved in the variable volume case. In the constant volume case, 742 there is a small unbalance associated with the product $(\partial_t\eta - \textit{emp}) * \textit{SSS}$. 743 Nevertheless, the salt content variation is quite small and will not induce 744 a long term drift as there is no physical reason that $(\partial_t\eta - \textit{emp})$ 745 and \textit{SSS} are correlated \citep{Roullet_Madec_JGR00}. 746 Note that, while quite small, the unbalance in constant volume case is larger 747 than the unbalance associated with the Asselin time filter \citep{Leclair_Madec_OM09}. 748 This is the reason why the modified filter is not applied in constant volume case. 725 749 726 750 % ------------------------------------------------------------------------------------------------------------- … … 735 759 736 760 When the penetrative solar radiation option is used (\np{ln\_flxqsr}=true), 737 the solar radiation penetrates the top few meters of the ocean, otherwise761 the solar radiation penetrates the top few 10 meters of the ocean, otherwise 738 762 all the heat flux is absorbed in the first ocean level (\np{ln\_flxqsr}=false). 739 763 Thus, in the former case a term is added to the time evolution equation of … … 747 771 \end{split} 748 772 \end{equation} 749 750 where $I$ is the downward irradiance. The additional term in \eqref{Eq_PE_qsr}751 is discretized as follows:773 where $Q_{sr}$ is the penetrative part of the surface heat flux ($i.e.$ the shortwave radiation) 774 and $I$ is the downward irradiance ($\left. I \right|_{z=\eta}=Q_{sr}$). 775 The additional term in \eqref{Eq_PE_qsr} is discretized as follows: 752 776 \begin{equation} \label{Eq_tra_qsr} 753 777 \frac{1}{\rho_o\, C_p \,e_3} \; \frac{\partial I}{\partial k} \equiv \frac{1}{\rho_o\, C_p\, e_{3t}} \delta_k \left[ I_w \right] 754 778 \end{equation} 755 779 756 A formulation involving two extinction coefficients is assumed for the 757 downward irradiance $I$ \citep{Paulson1977}: 780 The shortwave radiation, $Q_{sr}$, consists of energy distributed across a wide spectral range. 781 The ocean is strongly absorbing for wavelengths longer than 700~nm and these 782 wavelengths contribute to heating the upper few 10 centimetres. The fraction of $Q_{sr}$ 783 that resides in these almost non-penetrative wavebands, $R$, is $\sim 58\%$ (specified 784 through namelist parameter \np{rn\_abs}). It is assumed to penetrate the ocean 785 following a decreasing exponential profile, with an e-folding depth scale, $\xi_0$, 786 of a few 10 centimetres (typically $\xi_0=0.35~m$ set as \np{rn\_si0} in the namtra\_qsr namlist). 787 For shorter wavelengths (400-700~nm), the ocean is more transparent, and solar energy 788 propagates to depths where it contributes to a penetrating flux of solar energy and thus 789 to local heating below the surface. 790 The way this second part of the solar energy penetrates in the ocean depends on 791 which formulation is chosen. In the simple 2-wavebands light penetration (\np{ln\_qsr\_2bd}=true) 792 a chlorophyll-independent monochromatic formulation is also chosen for the shorter wavelengths, 793 leading to the following expression \citep{Paulson1977}: 758 794 \begin{equation} \label{Eq_traqsr_iradiance} 759 I(z) = Q_{sr} \left[Re^{-z / \xi_1} + \left( 1-R\right) e^{-z / \xi_2} \right] 760 \end{equation} 761 where $Q_{sr}$ is the penetrative part of the surface heat flux, 762 $\xi_1$ and $\xi_2$ are two extinction length scales and $R$ 763 determines the relative contribution of the two terms. 764 The default values used correspond to a Type I water in Jerlov's [1968] 765 % 766 \gmcomment : Jerlov reference to be added 767 % 768 classification: $\xi_1 = 0.35~m$, $\xi_2 = 23~m$ and $R = 0.58$ 769 (corresponding to \np{rn\_si1}, \np{rn\_si2} and \np{rn\_abs} namelist parameters, 770 respectively). $I$ is masked (no flux through the ocean bottom), 771 so all the solar radiation that reaches the last ocean level is absorbed 772 in that level. The trend in \eqref{Eq_tra_qsr} associated with the 773 penetration of the solar radiation is added to the temperature trend, 774 and the surface heat flux is modified in routine \mdl{traqsr}. 775 Note that in the $z$-coordinate, the depth of $T-$levels depends 776 on the single variable $k$. A one dimensional array of the coefficients 777 $gdsr(k) = Re^{-z_w (k)/\xi_1} + (1-R)e^{-z_w (k)/\xi_2}$ can then 778 be computed once and saved in memory. Moreover \textit{nksr}, 779 the level at which $gdrs$ becomes negligible (less than the 780 computer precision) is computed once, and the trend associated 781 with the penetration of the solar radiation is only added until that level. 782 Finally, note that when the ocean is shallow (< 200~m), part of the 795 I(z) = Q_{sr} \left[Re^{-z / \xi_0} + \left( 1-R\right) e^{-z / \xi_1} \right] 796 \end{equation} 797 where $\xi_1$ is the second extinction length scales associated with the shorter wavebands. 798 It is usually chosen to be 23~m through \np{rn\_si0} namelist parameter. 799 The set of default values ($\xi_0$, $\xi_1$, $R$) corresponds to a Type I water in 800 Jerlov's (1968) classification (oligotrophic waters). 801 802 Such assumptions have been shown to provide a very crude and simplistic 803 representation of observed light penetration profiles (\cite{Morel_JGR88}, see also 804 Fig.\ref{Fig_traqsr_irradiance}). Light absorption in the ocean depends on the 805 particules concentration and it is spectrally selective. \cite{Morel_JGR88} has shown 806 that an accurate representation of light penetration can be provided by a 61 waveband 807 formulation. Unfortunately, such a model is very computationally expensive. 808 Thus, \cite{Lengaigne_al_CD07} have constructed a simplified version of this 809 formulation in which visible light is splitted into three wavebands: blue (400-500 nm), 810 green (500-600 nm) and red (600-700nm). For each wave-band, the chlorophyll-dependant 811 attenuation coefficient is fitted to the coefficients computed from the full spectral model 812 of \cite{Morel_JGR88} (as modified by \cite{Morel_Maritorena_JGR01}) assuming 813 the same power-law expression. As shown on Fig.\ref{Fig_traqsr_irradiance}, 814 this formulation, called RGB (Reed-Green-Blue), reproduces quite closely 815 the light penetration profiles predicted by the full spectal model with much faster 816 computing efficiently, in contrast with the 2-bands formulation. 817 818 The RGB formulation is used when \np{ln\_qsr\_rgb}=true. The RGB attenuation coefficients 819 ($i.e.$ the inverse of the extinction length scales) are tabulated over 61 nonuniform 820 chlorophyll classes ranging from 0.01 to 10 g.Chl/L (see the routine \rou{trc\_oce\_rgb} 821 in \mdl{trc\_oce} module). Three type of chlorophyll can be used in the RGB formulation: 822 (1) a constant 0.05 g.Chl/L value everywhere (\np{nn\_chdta}=0) ; (2) observed 823 time varying chlorophyll (\np{nn\_chdta}=0) ; (3) simulated time varying chlorophyll 824 by TOP biogeochemical model (\np{ln\_qsr\_bio}=true). In the later case, the RGB 825 formulation is used to calculated both the phytoplankton light limitation in PISCES 826 or LOBSTER and the oceanic heating rate. 827 828 The trend in \eqref{Eq_tra_qsr} associated with the penetration of the solar radiation 829 is added to the temperature trend, and the surface heat flux is modified in routine \mdl{traqsr}. 830 831 When $z$-coordinate is preferred to $s$-coordinate, the depth of $w-$levels does 832 not significantly vary with location. The level at which the light has been totally 833 absorbed ($i.e.$ it is less than the computer precision) is computed once, 834 and the trend associated with the penetration of the solar radiation is only added until that level. 835 Finally, note that when the ocean is shallow ($<$ 200~m), part of the 783 836 solar radiation can reach the ocean floor. In this case, we have 784 837 chosen that all remaining radiation is absorbed in the last ocean 785 level ($i.e.$ $I_w$ is masked). 786 787 When coupling with a biological model (for example PISCES or LOBSTER), 788 it is possible to calculate the light attenuation using information from 789 the biology model. Without biological model, it is still possible to introduce 790 a horizontal variation of the light attenuation by using the observed ocean 791 surface color. At the time of writing, the latter has not been implemented 792 in the reference version. 793 % 794 \gmcomment{ {yellow}{case 4 bands and bio-coupling to add !!!} } 795 % 838 level ($i.e.$ $I$ is masked). 839 840 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 841 \begin{figure}[!t] \label{Fig_traqsr_irradiance} \begin{center} 842 \includegraphics[width=1.0\textwidth]{./TexFiles/Figures/Fig_TRA_Irradiance.pdf} 843 \caption{Penetration profile of the Downward solar irradiance 844 calculated by four models. Two wavebands chlorophyll-independant formulation (blue), 845 a chlorophyll-dependant monochromatic formulation (green), 4 waveband RGB formulation (red), 846 61 waveband Morel (1988) formulation (black) for a chlorophyll concentration of 847 (a) Chl=0.05 mg/m$^3$ and (b) Chl=0.5 mg/m$^3$. From \citet{Lengaigne_al_CD07}.} 848 \end{center} \end{figure} 849 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 796 850 797 851 % ------------------------------------------------------------------------------------------------------------- … … 1067 1121 %-------------------------------------------------------------------------------------------------------------- 1068 1122 1069 The general framework for tracer time stepping is a leap-frog scheme,1070 $i.e.$ a three level centred time scheme associated with a Asselin time1071 filter (cf. \S\ref{DOM_nxt}):1123 The general framework for tracer time stepping is a modified leap-frog scheme 1124 \citep{Leclair_Madec_OM09}, $i.e.$ a three level centred time scheme associated 1125 with a Asselin time filter (cf. \S\ref{STP_mLF}): 1072 1126 \begin{equation} \label{Eq_tra_nxt} 1073 \begin{ split}1074 T^{t+\rdt} &= T^{t-\rdt} + 2 \, \rdt \ \text{RHS}_T^t\\1127 \begin{aligned} 1128 (e_{3t}T)^{t+\rdt} &= (e_{3t}T)_f^{t-\rdt} &+ 2 \, \rdt \,e_{3t}^t\ \text{RHS}^t & \\ 1075 1129 \\ 1076 T_f^t \;\ \quad &= T^t \;\quad +\gamma \,\left[ {T_f^{t-\rdt} -2t^t+T^{t+\rdt}} \right] 1077 \end{split} 1130 (e_{3t}T)_f^t \;\ \quad &= (e_{3t}T)^t \;\quad 1131 &+\gamma \,\left[ {(e_{3t}T)_f^{t-\rdt} -2(e_{3t}T)^t+(e_{3t}T)^{t+\rdt}} \right] & \\ 1132 & &- \gamma\,\rdt \, \left[ Q^{t+\rdt/2} - Q^{t-\rdt/2} \right] & 1133 \end{aligned} 1078 1134 \end{equation} 1079 where $\text{RHS}_T$ is the right hand side of the temperature equation, 1080 the subscript $f$ denotes filtered values and $\gamma$ is the Asselin 1081 coefficient. $\gamma$ is initialized as \np{rn\_atfp} (\textbf{namelist} parameter). 1082 Its default value is \np{rn\_atfp=0.1}. 1135 where RHS is the right hand side of the temperature equation, 1136 the subscript $f$ denotes filtered values, $\gamma$ is the Asselin coefficient, 1137 and $S$ is the total forcing applied on $T$ ($i.e.$ fluxes plus content in mass exchanges). 1138 $\gamma$ is initialized as \np{rn\_atfp} (\textbf{namelist} parameter). 1139 Its default value is \np{rn\_atfp}=$10^{-3}$. Note that the forcing correction term in the filter 1140 is not applied in linear free surface (\jp{lk\_vvl}=false) (see \S\ref{TRA_sbc}. 1141 Not also that in constant volume case, the time stepping is performed on $T$, 1142 not on its content, $e_{3t}T$. 1083 1143 1084 1144 When the vertical mixing is solved implicitly, the update of the \textit{next} tracer
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