1 | \documentclass[NEMO_book]{subfiles} |
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2 | \begin{document} |
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3 | % ================================================================ |
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4 | % Chapter 1 ——— Ocean Tracers (TRA) |
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5 | % ================================================================ |
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6 | \chapter{Ocean Tracers (TRA)} |
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7 | \label{TRA} |
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8 | \minitoc |
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9 | |
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10 | % missing/update |
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11 | % traqsr: need to coordinate with SBC module |
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12 | |
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13 | %STEVEN : is the use of the word "positive" to describe a scheme enough, or should it be "positive definite"? I added a comment to this effect on some instances of this below |
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14 | |
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15 | %\newpage |
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16 | \vspace{2.cm} |
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17 | %$\ $\newline % force a new ligne |
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18 | |
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19 | Using the representation described in Chap.~\ref{DOM}, several semi-discrete |
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20 | space forms of the tracer equations are available depending on the vertical |
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21 | coordinate used and on the physics used. In all the equations presented |
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22 | here, the masking has been omitted for simplicity. One must be aware that |
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23 | all the quantities are masked fields and that each time a mean or difference |
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24 | operator is used, the resulting field is multiplied by a mask. |
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25 | |
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26 | The two active tracers are potential temperature and salinity. Their prognostic |
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27 | equations can be summarized as follows: |
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28 | \begin{equation*} |
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29 | \text{NXT} = \text{ADV}+\text{LDF}+\text{ZDF}+\text{SBC} |
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30 | \ (+\text{QSR})\ (+\text{BBC})\ (+\text{BBL})\ (+\text{DMP}) |
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31 | \end{equation*} |
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32 | |
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33 | NXT stands for next, referring to the time-stepping. From left to right, the terms |
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34 | on the rhs of the tracer equations are the advection (ADV), the lateral diffusion |
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35 | (LDF), the vertical diffusion (ZDF), the contributions from the external forcings |
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36 | (SBC: Surface Boundary Condition, QSR: penetrative Solar Radiation, and BBC: |
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37 | Bottom Boundary Condition), the contribution from the bottom boundary Layer |
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38 | (BBL) parametrisation, and an internal damping (DMP) term. The terms QSR, |
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39 | BBC, BBL and DMP are optional. The external forcings and parameterisations |
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40 | require complex inputs and complex calculations ($e.g.$ bulk formulae, estimation |
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41 | of mixing coefficients) that are carried out in the SBC, LDF and ZDF modules and |
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42 | described in chapters \S\ref{SBC}, \S\ref{LDF} and \S\ref{ZDF}, respectively. |
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43 | Note that \mdl{tranpc}, the non-penetrative convection module, although |
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44 | located in the NEMO/OPA/TRA directory as it directly modifies the tracer fields, |
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45 | is described with the model vertical physics (ZDF) together with other available |
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46 | parameterization of convection. |
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47 | |
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48 | In the present chapter we also describe the diagnostic equations used to compute |
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49 | the sea-water properties (density, Brunt-V\"{a}is\"{a}l\"{a} frequency, specific heat and |
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50 | freezing point with associated modules \mdl{eosbn2} and \mdl{phycst}). |
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51 | |
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52 | The different options available to the user are managed by namelist logicals or CPP keys. |
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53 | For each equation term \textit{TTT}, the namelist logicals are \textit{ln\_traTTT\_xxx}, |
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54 | where \textit{xxx} is a 3 or 4 letter acronym corresponding to each optional scheme. |
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55 | The CPP key (when it exists) is \textbf{key\_traTTT}. The equivalent code can be |
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56 | found in the \textit{traTTT} or \textit{traTTT\_xxx} module, in the NEMO/OPA/TRA directory. |
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57 | |
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58 | The user has the option of extracting each tendency term on the RHS of the tracer |
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59 | equation for output (\np{ln\_tra\_trd} or \np{ln\_tra\_mxl}~=~true), as described in Chap.~\ref{DIA}. |
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60 | |
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61 | $\ $\newline % force a new ligne |
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62 | % ================================================================ |
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63 | % Tracer Advection |
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64 | % ================================================================ |
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65 | \section [Tracer Advection (\textit{traadv})] |
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66 | {Tracer Advection (\mdl{traadv})} |
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67 | \label{TRA_adv} |
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68 | %------------------------------------------namtra_adv----------------------------------------------------- |
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69 | \namdisplay{namtra_adv} |
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70 | %------------------------------------------------------------------------------------------------------------- |
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71 | |
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72 | When considered ($i.e.$ when \np{ln\_traadv\_NONE} is not set to \textit{true}), |
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73 | the advection tendency of a tracer is expressed in flux form, |
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74 | $i.e.$ as the divergence of the advective fluxes. Its discrete expression is given by : |
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75 | \begin{equation} \label{Eq_tra_adv} |
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76 | ADV_\tau =-\frac{1}{b_t} \left( |
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77 | \;\delta _i \left[ e_{2u}\,e_{3u} \; u\; \tau _u \right] |
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78 | +\delta _j \left[ e_{1v}\,e_{3v} \; v\; \tau _v \right] \; \right) |
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79 | -\frac{1}{e_{3t}} \;\delta _k \left[ w\; \tau _w \right] |
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80 | \end{equation} |
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81 | where $\tau$ is either T or S, and $b_t= e_{1t}\,e_{2t}\,e_{3t}$ is the volume of $T$-cells. |
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82 | The flux form in \eqref{Eq_tra_adv} |
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83 | implicitly requires the use of the continuity equation. Indeed, it is obtained |
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84 | by using the following equality : $\nabla \cdot \left( \vect{U}\,T \right)=\vect{U} \cdot \nabla T$ |
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85 | which results from the use of the continuity equation, $\partial _t e_3 + e_3\;\nabla \cdot \vect{U}=0$ |
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86 | (which reduces to $\nabla \cdot \vect{U}=0$ in linear free surface, $i.e.$ \np{ln\_linssh}=true). |
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87 | Therefore it is of paramount importance to design the discrete analogue of the |
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88 | advection tendency so that it is consistent with the continuity equation in order to |
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89 | enforce the conservation properties of the continuous equations. In other words, |
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90 | by setting $\tau = 1$ in (\ref{Eq_tra_adv}) we recover the discrete form of |
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91 | the continuity equation which is used to calculate the vertical velocity. |
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92 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
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93 | \begin{figure}[!t] \begin{center} |
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94 | \includegraphics[width=0.9\textwidth]{Fig_adv_scheme} |
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95 | \caption{ \label{Fig_adv_scheme} |
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96 | Schematic representation of some ways used to evaluate the tracer value |
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97 | at $u$-point and the amount of tracer exchanged between two neighbouring grid |
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98 | points. Upsteam biased scheme (ups): the upstream value is used and the black |
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99 | area is exchanged. Piecewise parabolic method (ppm): a parabolic interpolation |
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100 | is used and the black and dark grey areas are exchanged. Monotonic upstream |
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101 | scheme for conservative laws (muscl): a parabolic interpolation is used and black, |
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102 | dark grey and grey areas are exchanged. Second order scheme (cen2): the mean |
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103 | value is used and black, dark grey, grey and light grey areas are exchanged. Note |
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104 | that this illustration does not include the flux limiter used in ppm and muscl schemes.} |
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105 | \end{center} \end{figure} |
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106 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
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107 | |
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108 | The key difference between the advection schemes available in \NEMO is the choice |
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109 | made in space and time interpolation to define the value of the tracer at the |
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110 | velocity points (Fig.~\ref{Fig_adv_scheme}). |
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111 | |
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112 | Along solid lateral and bottom boundaries a zero tracer flux is automatically |
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113 | specified, since the normal velocity is zero there. At the sea surface the |
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114 | boundary condition depends on the type of sea surface chosen: |
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115 | \begin{description} |
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116 | \item [linear free surface:] (\np{ln\_linssh}=true) the first level thickness is constant in time: |
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117 | the vertical boundary condition is applied at the fixed surface $z=0$ |
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118 | rather than on the moving surface $z=\eta$. There is a non-zero advective |
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119 | flux which is set for all advection schemes as |
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120 | $\left. {\tau _w } \right|_{k=1/2} =T_{k=1} $, $i.e.$ |
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121 | the product of surface velocity (at $z=0$) by the first level tracer value. |
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122 | \item [non-linear free surface:] (\np{ln\_linssh}=false) |
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123 | convergence/divergence in the first ocean level moves the free surface |
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124 | up/down. There is no tracer advection through it so that the advective |
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125 | fluxes through the surface are also zero |
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126 | \end{description} |
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127 | In all cases, this boundary condition retains local conservation of tracer. |
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128 | Global conservation is obtained in non-linear free surface case, |
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129 | but \textit{not} in the linear free surface case. Nevertheless, in the latter case, |
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130 | it is achieved to a good approximation since the non-conservative |
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131 | term is the product of the time derivative of the tracer and the free surface |
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132 | height, two quantities that are not correlated \citep{Roullet_Madec_JGR00,Griffies_al_MWR01,Campin2004}. |
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133 | |
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134 | The velocity field that appears in (\ref{Eq_tra_adv}) and (\ref{Eq_tra_adv_zco}) |
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135 | is the centred (\textit{now}) \textit{effective} ocean velocity, $i.e.$ the \textit{eulerian} velocity |
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136 | (see Chap.~\ref{DYN}) plus the eddy induced velocity (\textit{eiv}) |
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137 | and/or the mixed layer eddy induced velocity (\textit{eiv}) |
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138 | when those parameterisations are used (see Chap.~\ref{LDF}). |
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139 | |
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140 | Several tracer advection scheme are proposed, namely |
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141 | a $2^{nd}$ or $4^{th}$ order centred schemes (CEN), |
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142 | a $2^{nd}$ or $4^{th}$ order Flux Corrected Transport scheme (FCT), |
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143 | a Monotone Upstream Scheme for Conservative Laws scheme (MUSCL), |
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144 | a $3^{rd}$ Upstream Biased Scheme (UBS, also often called UP3), and |
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145 | a Quadratic Upstream Interpolation for Convective Kinematics with |
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146 | Estimated Streaming Terms scheme (QUICKEST). |
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147 | The choice is made in the \textit{\ngn{namtra\_adv}} namelist, by |
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148 | setting to \textit{true} one of the logicals \textit{ln\_traadv\_xxx}. |
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149 | The corresponding code can be found in the \textit{traadv\_xxx.F90} module, |
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150 | where \textit{xxx} is a 3 or 4 letter acronym corresponding to each scheme. |
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151 | By default ($i.e.$ in the reference namelist, \ngn{namelist\_ref}), all the logicals |
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152 | are set to \textit{false}. If the user does not select an advection scheme |
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153 | in the configuration namelist (\ngn{namelist\_cfg}), the tracers will \textit{not} be advected ! |
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154 | |
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155 | Details of the advection schemes are given below. The choosing an advection scheme |
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156 | is a complex matter which depends on the model physics, model resolution, |
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157 | type of tracer, as well as the issue of numerical cost. In particular, we note that |
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158 | (1) CEN and FCT schemes require an explicit diffusion operator |
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159 | while the other schemes are diffusive enough so that they do not necessarily need additional diffusion ; |
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160 | (2) CEN and UBS are not \textit{positive} schemes |
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161 | \footnote{negative values can appear in an initially strictly positive tracer field |
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162 | which is advected} |
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163 | , implying that false extrema are permitted. Their use is not recommended on passive tracers ; |
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164 | (3) It is recommended that the same advection-diffusion scheme is |
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165 | used on both active and passive tracers. Indeed, if a source or sink of a |
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166 | passive tracer depends on an active one, the difference of treatment of |
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167 | active and passive tracers can create very nice-looking frontal structures |
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168 | that are pure numerical artefacts. Nevertheless, most of our users set a different |
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169 | treatment on passive and active tracers, that's the reason why this possibility |
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170 | is offered. We strongly suggest them to perform a sensitivity experiment |
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171 | using a same treatment to assess the robustness of their results. |
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172 | |
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173 | % ------------------------------------------------------------------------------------------------------------- |
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174 | % 2nd and 4th order centred schemes |
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175 | % ------------------------------------------------------------------------------------------------------------- |
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176 | \subsection [Centred schemes (CEN) (\np{ln\_traadv\_cen})] |
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177 | {Centred schemes (CEN) (\np{ln\_traadv\_cen}=true)} |
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178 | \label{TRA_adv_cen} |
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179 | |
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180 | % 2nd order centred scheme |
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181 | |
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182 | The centred advection scheme (CEN) is used when \np{ln\_traadv\_cen}~=~\textit{true}. |
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183 | Its order ($2^{nd}$ or $4^{th}$) can be chosen independently on horizontal (iso-level) |
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184 | and vertical direction by setting \np{nn\_cen\_h} and \np{nn\_cen\_v} to $2$ or $4$. |
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185 | CEN implementation can be found in the \mdl{traadv\_cen} module. |
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186 | |
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187 | In the $2^{nd}$ order centred formulation (CEN2), the tracer at velocity points |
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188 | is evaluated as the mean of the two neighbouring $T$-point values. |
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189 | For example, in the $i$-direction : |
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190 | \begin{equation} \label{Eq_tra_adv_cen2} |
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191 | \tau _u^{cen2} =\overline T ^{i+1/2} |
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192 | \end{equation} |
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193 | |
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194 | CEN2 is non diffusive ($i.e.$ it conserves the tracer variance, $\tau^2)$ |
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195 | but dispersive ($i.e.$ it may create false extrema). It is therefore notoriously |
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196 | noisy and must be used in conjunction with an explicit diffusion operator to |
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197 | produce a sensible solution. The associated time-stepping is performed using |
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198 | a leapfrog scheme in conjunction with an Asselin time-filter, so $T$ in |
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199 | (\ref{Eq_tra_adv_cen2}) is the \textit{now} tracer value. |
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200 | |
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201 | Note that using the CEN2, the overall tracer advection is of second |
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202 | order accuracy since both (\ref{Eq_tra_adv}) and (\ref{Eq_tra_adv_cen2}) |
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203 | have this order of accuracy. |
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204 | |
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205 | % 4nd order centred scheme |
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206 | |
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207 | In the $4^{th}$ order formulation (CEN4), tracer values are evaluated at u- and v-points as |
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208 | a $4^{th}$ order interpolation, and thus depend on the four neighbouring $T$-points. |
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209 | For example, in the $i$-direction: |
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210 | \begin{equation} \label{Eq_tra_adv_cen4} |
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211 | \tau _u^{cen4} |
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212 | =\overline{ T - \frac{1}{6}\,\delta _i \left[ \delta_{i+1/2}[T] \,\right] }^{\,i+1/2} |
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213 | \end{equation} |
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214 | In the vertical direction (\np{nn\_cen\_v}=$4$), a $4^{th}$ COMPACT interpolation |
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215 | has been prefered \citep{Demange_PhD2014}. |
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216 | In the COMPACT scheme, both the field and its derivative are interpolated, |
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217 | which leads, after a matrix inversion, spectral characteristics |
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218 | similar to schemes of higher order \citep{Lele_JCP1992}. |
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219 | |
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220 | |
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221 | Strictly speaking, the CEN4 scheme is not a $4^{th}$ order advection scheme |
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222 | but a $4^{th}$ order evaluation of advective fluxes, since the divergence of |
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223 | advective fluxes \eqref{Eq_tra_adv} is kept at $2^{nd}$ order. |
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224 | The expression \textit{$4^{th}$ order scheme} used in oceanographic literature |
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225 | is usually associated with the scheme presented here. |
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226 | Introducing a \textit{true} $4^{th}$ order advection scheme is feasible but, |
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227 | for consistency reasons, it requires changes in the discretisation of the tracer |
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228 | advection together with changes in the continuity equation, |
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229 | and the momentum advection and pressure terms. |
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230 | |
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231 | A direct consequence of the pseudo-fourth order nature of the scheme is that |
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232 | it is not non-diffusive, $i.e.$ the global variance of a tracer is not preserved using CEN4. |
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233 | Furthermore, it must be used in conjunction with an explicit diffusion operator |
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234 | to produce a sensible solution. |
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235 | As in CEN2 case, the time-stepping is performed using a leapfrog scheme in conjunction |
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236 | with an Asselin time-filter, so $T$ in (\ref{Eq_tra_adv_cen4}) is the \textit{now} tracer. |
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237 | |
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238 | At a $T$-grid cell adjacent to a boundary (coastline, bottom and surface), |
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239 | an additional hypothesis must be made to evaluate $\tau _u^{cen4}$. |
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240 | This hypothesis usually reduces the order of the scheme. |
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241 | Here we choose to set the gradient of $T$ across the boundary to zero. |
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242 | Alternative conditions can be specified, such as a reduction to a second order scheme |
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243 | for these near boundary grid points. |
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244 | |
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245 | % ------------------------------------------------------------------------------------------------------------- |
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246 | % FCT scheme |
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247 | % ------------------------------------------------------------------------------------------------------------- |
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248 | \subsection [Flux Corrected Transport schemes (FCT) (\np{ln\_traadv\_fct})] |
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249 | {Flux Corrected Transport schemes (FCT) (\np{ln\_traadv\_fct}=true)} |
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250 | \label{TRA_adv_tvd} |
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251 | |
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252 | The Flux Corrected Transport schemes (FCT) is used when \np{ln\_traadv\_fct}~=~\textit{true}. |
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253 | Its order ($2^{nd}$ or $4^{th}$) can be chosen independently on horizontal (iso-level) |
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254 | and vertical direction by setting \np{nn\_fct\_h} and \np{nn\_fct\_v} to $2$ or $4$. |
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255 | FCT implementation can be found in the \mdl{traadv\_fct} module. |
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256 | |
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257 | In FCT formulation, the tracer at velocity points is evaluated using a combination of |
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258 | an upstream and a centred scheme. For example, in the $i$-direction : |
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259 | \begin{equation} \label{Eq_tra_adv_fct} |
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260 | \begin{split} |
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261 | \tau _u^{ups}&= \begin{cases} |
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262 | T_{i+1} & \text{if $\ u_{i+1/2} < 0$} \hfill \\ |
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263 | T_i & \text{if $\ u_{i+1/2} \geq 0$} \hfill \\ |
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264 | \end{cases} \\ |
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265 | \\ |
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266 | \tau _u^{fct}&=\tau _u^{ups} +c_u \;\left( {\tau _u^{cen} -\tau _u^{ups} } \right) |
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267 | \end{split} |
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268 | \end{equation} |
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269 | where $c_u$ is a flux limiter function taking values between 0 and 1. |
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270 | The FCT order is the one of the centred scheme used ($i.e.$ it depends on the setting of |
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271 | \np{nn\_fct\_h} and \np{nn\_fct\_v}. |
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272 | There exist many ways to define $c_u$, each corresponding to a different |
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273 | FCT scheme. The one chosen in \NEMO is described in \citet{Zalesak_JCP79}. |
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274 | $c_u$ only departs from $1$ when the advective term produces a local extremum in the tracer field. |
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275 | The resulting scheme is quite expensive but \emph{positive}. |
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276 | It can be used on both active and passive tracers. |
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277 | A comparison of FCT-2 with MUSCL and a MPDATA scheme can be found in \citet{Levy_al_GRL01}. |
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278 | |
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279 | An additional option has been added controlled by \np{nn\_fct\_zts}. By setting this integer to |
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280 | a value larger than zero, a $2^{nd}$ order FCT scheme is used on both horizontal and vertical direction, |
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281 | but on the latter, a split-explicit time stepping is used, with a number of sub-timestep equals |
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282 | to \np{nn\_fct\_zts}. This option can be useful when the size of the timestep is limited |
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283 | by vertical advection \citep{Lemarie_OM2015}. Note that in this case, a similar split-explicit |
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284 | time stepping should be used on vertical advection of momentum to insure a better stability |
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285 | (see \S\ref{DYN_zad}). |
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286 | |
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287 | For stability reasons (see \S\ref{STP}), $\tau _u^{cen}$ is evaluated in (\ref{Eq_tra_adv_fct}) |
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288 | using the \textit{now} tracer while $\tau _u^{ups}$ is evaluated using the \textit{before} tracer. In other words, |
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289 | the advective part of the scheme is time stepped with a leap-frog scheme |
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290 | while a forward scheme is used for the diffusive part. |
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291 | |
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292 | % ------------------------------------------------------------------------------------------------------------- |
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293 | % MUSCL scheme |
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294 | % ------------------------------------------------------------------------------------------------------------- |
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295 | \subsection[MUSCL scheme (\np{ln\_traadv\_mus})] |
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296 | {Monotone Upstream Scheme for Conservative Laws (MUSCL) (\np{ln\_traadv\_mus}=T)} |
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297 | \label{TRA_adv_mus} |
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298 | |
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299 | The Monotone Upstream Scheme for Conservative Laws (MUSCL) is used when \np{ln\_traadv\_mus}~=~\textit{true}. |
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300 | MUSCL implementation can be found in the \mdl{traadv\_mus} module. |
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301 | |
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302 | MUSCL has been first implemented in \NEMO by \citet{Levy_al_GRL01}. In its formulation, the tracer at velocity points |
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303 | is evaluated assuming a linear tracer variation between two $T$-points |
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304 | (Fig.\ref{Fig_adv_scheme}). For example, in the $i$-direction : |
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305 | \begin{equation} \label{Eq_tra_adv_mus} |
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306 | \tau _u^{mus} = \left\{ \begin{aligned} |
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307 | &\tau _i &+ \frac{1}{2} \;\left( 1-\frac{u_{i+1/2} \;\rdt}{e_{1u}} \right) |
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308 | &\ \widetilde{\partial _i \tau} & \quad \text{if }\;u_{i+1/2} \geqslant 0 \\ |
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309 | &\tau _{i+1/2} &+\frac{1}{2}\;\left( 1+\frac{u_{i+1/2} \;\rdt}{e_{1u} } \right) |
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310 | &\ \widetilde{\partial_{i+1/2} \tau } & \text{if }\;u_{i+1/2} <0 |
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311 | \end{aligned} \right. |
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312 | \end{equation} |
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313 | where $\widetilde{\partial _i \tau}$ is the slope of the tracer on which a limitation |
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314 | is imposed to ensure the \textit{positive} character of the scheme. |
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315 | |
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316 | The time stepping is performed using a forward scheme, that is the \textit{before} |
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317 | tracer field is used to evaluate $\tau _u^{mus}$. |
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318 | |
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319 | For an ocean grid point adjacent to land and where the ocean velocity is |
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320 | directed toward land, an upstream flux is used. This choice ensure |
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321 | the \textit{positive} character of the scheme. |
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322 | In addition, fluxes round a grid-point where a runoff is applied can optionally be |
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323 | computed using upstream fluxes (\np{ln\_mus\_ups}~=~\textit{true}). |
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324 | |
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325 | % ------------------------------------------------------------------------------------------------------------- |
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326 | % UBS scheme |
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327 | % ------------------------------------------------------------------------------------------------------------- |
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328 | \subsection [Upstream-Biased Scheme (UBS) (\np{ln\_traadv\_ubs})] |
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329 | {Upstream-Biased Scheme (UBS) (\np{ln\_traadv\_ubs}=true)} |
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330 | \label{TRA_adv_ubs} |
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331 | |
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332 | The Upstream-Biased Scheme (UBS) is used when \np{ln\_traadv\_ubs}~=~\textit{true}. |
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333 | UBS implementation can be found in the \mdl{traadv\_mus} module. |
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334 | |
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335 | The UBS scheme, often called UP3, is also known as the Cell Averaged QUICK scheme |
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336 | (Quadratic Upstream Interpolation for Convective Kinematics). It is an upstream-biased |
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337 | third order scheme based on an upstream-biased parabolic interpolation. |
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338 | For example, in the $i$-direction : |
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339 | \begin{equation} \label{Eq_tra_adv_ubs} |
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340 | \tau _u^{ubs} =\overline T ^{i+1/2}-\;\frac{1}{6} \left\{ |
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341 | \begin{aligned} |
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342 | &\tau"_i & \quad \text{if }\ u_{i+1/2} \geqslant 0 \\ |
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343 | &\tau"_{i+1} & \quad \text{if }\ u_{i+1/2} < 0 |
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344 | \end{aligned} \right. |
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345 | \end{equation} |
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346 | where $\tau "_i =\delta _i \left[ {\delta _{i+1/2} \left[ \tau \right]} \right]$. |
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347 | |
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348 | This results in a dissipatively dominant (i.e. hyper-diffusive) truncation |
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349 | error \citep{Shchepetkin_McWilliams_OM05}. The overall performance of |
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350 | the advection scheme is similar to that reported in \cite{Farrow1995}. |
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351 | It is a relatively good compromise between accuracy and smoothness. |
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352 | Nevertheless the scheme is not \emph{positive}, meaning that false extrema are permitted, |
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353 | but the amplitude of such are significantly reduced over the centred second |
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354 | or fourth order method. therefore it is not recommended that it should be |
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355 | applied to a passive tracer that requires positivity. |
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356 | |
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357 | The intrinsic diffusion of UBS makes its use risky in the vertical direction |
---|
358 | where the control of artificial diapycnal fluxes is of paramount importance \citep{Shchepetkin_McWilliams_OM05, Demange_PhD2014}. |
---|
359 | Therefore the vertical flux is evaluated using either a $2^nd$ order FCT scheme |
---|
360 | or a $4^th$ order COMPACT scheme (\np{nn\_cen\_v}=2 or 4). |
---|
361 | |
---|
362 | For stability reasons (see \S\ref{STP}), |
---|
363 | the first term in \eqref{Eq_tra_adv_ubs} (which corresponds to a second order |
---|
364 | centred scheme) is evaluated using the \textit{now} tracer (centred in time) |
---|
365 | while the second term (which is the diffusive part of the scheme), is |
---|
366 | evaluated using the \textit{before} tracer (forward in time). |
---|
367 | This choice is discussed by \citet{Webb_al_JAOT98} in the context of the |
---|
368 | QUICK advection scheme. UBS and QUICK schemes only differ |
---|
369 | by one coefficient. Replacing 1/6 with 1/8 in \eqref{Eq_tra_adv_ubs} |
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370 | leads to the QUICK advection scheme \citep{Webb_al_JAOT98}. |
---|
371 | This option is not available through a namelist parameter, since the |
---|
372 | 1/6 coefficient is hard coded. Nevertheless it is quite easy to make the |
---|
373 | substitution in the \mdl{traadv\_ubs} module and obtain a QUICK scheme. |
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374 | |
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375 | Note that it is straightforward to rewrite \eqref{Eq_tra_adv_ubs} as follows: |
---|
376 | \begin{equation} \label{Eq_traadv_ubs2} |
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377 | \tau _u^{ubs} = \tau _u^{cen4} + \frac{1}{12} \left\{ |
---|
378 | \begin{aligned} |
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379 | & + \tau"_i & \quad \text{if }\ u_{i+1/2} \geqslant 0 \\ |
---|
380 | & - \tau"_{i+1} & \quad \text{if }\ u_{i+1/2} < 0 |
---|
381 | \end{aligned} \right. |
---|
382 | \end{equation} |
---|
383 | or equivalently |
---|
384 | \begin{equation} \label{Eq_traadv_ubs2b} |
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385 | u_{i+1/2} \ \tau _u^{ubs} |
---|
386 | =u_{i+1/2} \ \overline{ T - \frac{1}{6}\,\delta _i\left[ \delta_{i+1/2}[T] \,\right] }^{\,i+1/2} |
---|
387 | - \frac{1}{2} |u|_{i+1/2} \;\frac{1}{6} \;\delta_{i+1/2}[\tau"_i] |
---|
388 | \end{equation} |
---|
389 | |
---|
390 | \eqref{Eq_traadv_ubs2} has several advantages. Firstly, it clearly reveals |
---|
391 | that the UBS scheme is based on the fourth order scheme to which an |
---|
392 | upstream-biased diffusion term is added. Secondly, this emphasises that the |
---|
393 | $4^{th}$ order part (as well as the $2^{nd}$ order part as stated above) has |
---|
394 | to be evaluated at the \emph{now} time step using \eqref{Eq_tra_adv_ubs}. |
---|
395 | Thirdly, the diffusion term is in fact a biharmonic operator with an eddy |
---|
396 | coefficient which is simply proportional to the velocity: |
---|
397 | $A_u^{lm}= \frac{1}{12}\,{e_{1u}}^3\,|u|$. Note the current version of NEMO uses |
---|
398 | the computationally more efficient formulation \eqref{Eq_tra_adv_ubs}. |
---|
399 | |
---|
400 | % ------------------------------------------------------------------------------------------------------------- |
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401 | % QCK scheme |
---|
402 | % ------------------------------------------------------------------------------------------------------------- |
---|
403 | \subsection [QUICKEST scheme (QCK) (\np{ln\_traadv\_qck})] |
---|
404 | {QUICKEST scheme (QCK) (\np{ln\_traadv\_qck}=true)} |
---|
405 | \label{TRA_adv_qck} |
---|
406 | |
---|
407 | The Quadratic Upstream Interpolation for Convective Kinematics with |
---|
408 | Estimated Streaming Terms (QUICKEST) scheme proposed by \citet{Leonard1979} |
---|
409 | is used when \np{ln\_traadv\_qck}~=~\textit{true}. |
---|
410 | QUICKEST implementation can be found in the \mdl{traadv\_qck} module. |
---|
411 | |
---|
412 | QUICKEST is the third order Godunov scheme which is associated with the ULTIMATE QUICKEST |
---|
413 | limiter \citep{Leonard1991}. It has been implemented in NEMO by G. Reffray |
---|
414 | (MERCATOR-ocean) and can be found in the \mdl{traadv\_qck} module. |
---|
415 | The resulting scheme is quite expensive but \emph{positive}. |
---|
416 | It can be used on both active and passive tracers. |
---|
417 | However, the intrinsic diffusion of QCK makes its use risky in the vertical |
---|
418 | direction where the control of artificial diapycnal fluxes is of paramount importance. |
---|
419 | Therefore the vertical flux is evaluated using the CEN2 scheme. |
---|
420 | This no longer guarantees the positivity of the scheme. |
---|
421 | The use of FCT in the vertical direction (as for the UBS case) should be implemented |
---|
422 | to restore this property. |
---|
423 | |
---|
424 | %%%gmcomment : Cross term are missing in the current implementation.... |
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425 | |
---|
426 | |
---|
427 | % ================================================================ |
---|
428 | % Tracer Lateral Diffusion |
---|
429 | % ================================================================ |
---|
430 | \section [Tracer Lateral Diffusion (\textit{traldf})] |
---|
431 | {Tracer Lateral Diffusion (\mdl{traldf})} |
---|
432 | \label{TRA_ldf} |
---|
433 | %-----------------------------------------nam_traldf------------------------------------------------------ |
---|
434 | \namdisplay{namtra_ldf} |
---|
435 | %------------------------------------------------------------------------------------------------------------- |
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436 | |
---|
437 | Options are defined through the \ngn{namtra\_ldf} namelist variables. |
---|
438 | They are regrouped in four items, allowing to specify |
---|
439 | $(i)$ the type of operator used (none, laplacian, bilaplacian), |
---|
440 | $(ii)$ the direction along which the operator acts (iso-level, horizontal, iso-neutral), |
---|
441 | $(iii)$ some specific options related to the rotated operators ($i.e.$ non-iso-level operator), and |
---|
442 | $(iv)$ the specification of eddy diffusivity coefficient (either constant or variable in space and time). |
---|
443 | Item $(iv)$ will be described in Chap.\ref{LDF} . |
---|
444 | The direction along which the operators act is defined through the slope between this direction and the iso-level surfaces. |
---|
445 | The slope is computed in the \mdl{ldfslp} module and will also be described in Chap.~\ref{LDF}. |
---|
446 | |
---|
447 | The lateral diffusion of tracers is evaluated using a forward scheme, |
---|
448 | $i.e.$ the tracers appearing in its expression are the \textit{before} tracers in time, |
---|
449 | except for the pure vertical component that appears when a rotation tensor is used. |
---|
450 | This latter component is solved implicitly together with the vertical diffusion term (see \S\ref{STP}). |
---|
451 | When \np{ln\_traldf\_msc}~=~\textit{true}, a Method of Stabilizing Correction is used in which |
---|
452 | the pure vertical component is split into an explicit and an implicit part \citep{Lemarie_OM2012}. |
---|
453 | |
---|
454 | % ------------------------------------------------------------------------------------------------------------- |
---|
455 | % Type of operator |
---|
456 | % ------------------------------------------------------------------------------------------------------------- |
---|
457 | \subsection [Type of operator (\np{ln\_traldf\_NONE}, \np{ln\_traldf\_lap}, \np{ln\_traldf\_blp})] |
---|
458 | {Type of operator (\np{ln\_traldf\_NONE}, \np{ln\_traldf\_lap}, or \np{ln\_traldf\_blp} = true) } |
---|
459 | \label{TRA_ldf_op} |
---|
460 | |
---|
461 | Three operator options are proposed and, one and only one of them must be selected: |
---|
462 | \begin{description} |
---|
463 | \item [\np{ln\_traldf\_NONE}] = true : no operator selected, the lateral diffusive tendency will not be |
---|
464 | applied to the tracer equation. This option can be used when the selected advection scheme |
---|
465 | is diffusive enough (MUSCL scheme for example). |
---|
466 | \item [ \np{ln\_traldf\_lap}] = true : a laplacian operator is selected. This harmonic operator |
---|
467 | takes the following expression: $\mathpzc{L}(T)=\nabla \cdot A_{ht}\;\nabla T $, |
---|
468 | where the gradient operates along the selected direction (see \S\ref{TRA_ldf_dir}), |
---|
469 | and $A_{ht}$ is the eddy diffusivity coefficient expressed in $m^2/s$ (see Chap.~\ref{LDF}). |
---|
470 | \item [\np{ln\_traldf\_blp}] = true : a bilaplacian operator is selected. This biharmonic operator |
---|
471 | takes the following expression: |
---|
472 | $\mathpzc{B}=- \mathpzc{L}\left(\mathpzc{L}(T) \right) = -\nabla \cdot b\nabla \left( {\nabla \cdot b\nabla T} \right)$ |
---|
473 | where the gradient operats along the selected direction, |
---|
474 | and $b^2=B_{ht}$ is the eddy diffusivity coefficient expressed in $m^4/s$ (see Chap.~\ref{LDF}). |
---|
475 | In the code, the bilaplacian operator is obtained by calling the laplacian twice. |
---|
476 | \end{description} |
---|
477 | |
---|
478 | Both laplacian and bilaplacian operators ensure the total tracer variance decrease. |
---|
479 | Their primary role is to provide strong dissipation at the smallest scale supported |
---|
480 | by the grid while minimizing the impact on the larger scale features. |
---|
481 | The main difference between the two operators is the scale selectiveness. |
---|
482 | The bilaplacian damping time ($i.e.$ its spin down time) scales like $\lambda^{-4}$ |
---|
483 | for disturbances of wavelength $\lambda$ (so that short waves damped more rapidelly than long ones), |
---|
484 | whereas the laplacian damping time scales only like $\lambda^{-2}$. |
---|
485 | |
---|
486 | |
---|
487 | % ------------------------------------------------------------------------------------------------------------- |
---|
488 | % Direction of action |
---|
489 | % ------------------------------------------------------------------------------------------------------------- |
---|
490 | \subsection [Direction of action (\np{ln\_traldf\_lev}, \np{ln\_traldf\_hor}, \np{ln\_traldf\_iso}, \np{ln\_traldf\_triad})] |
---|
491 | {Direction of action (\np{ln\_traldf\_lev}, \textit{...\_hor}, \textit{...\_iso}, or \textit{...\_triad} = true) } |
---|
492 | \label{TRA_ldf_dir} |
---|
493 | |
---|
494 | The choice of a direction of action determines the form of operator used. |
---|
495 | The operator is a simple (re-entrant) laplacian acting in the (\textbf{i},\textbf{j}) plane |
---|
496 | when iso-level option is used (\np{ln\_traldf\_lev}~=~\textit{true}) |
---|
497 | or when a horizontal ($i.e.$ geopotential) operator is demanded in \textit{z}-coordinate |
---|
498 | (\np{ln\_traldf\_hor} and \np{ln\_zco} equal \textit{true}). |
---|
499 | The associated code can be found in the \mdl{traldf\_lap\_blp} module. |
---|
500 | The operator is a rotated (re-entrant) laplacian when the direction along which it acts |
---|
501 | does not coincide with the iso-level surfaces, |
---|
502 | that is when standard or triad iso-neutral option is used (\np{ln\_traldf\_iso} or |
---|
503 | \np{ln\_traldf\_triad} equals \textit{true}, see \mdl{traldf\_iso} or \mdl{traldf\_triad} module, resp.), |
---|
504 | or when a horizontal ($i.e.$ geopotential) operator is demanded in \textit{s}-coordinate |
---|
505 | (\np{ln\_traldf\_hor} and \np{ln\_sco} equal \textit{true}) |
---|
506 | \footnote{In this case, the standard iso-neutral operator will be automatically selected}. |
---|
507 | In that case, a rotation is applied to the gradient(s) that appears in the operator |
---|
508 | so that diffusive fluxes acts on the three spatial direction. |
---|
509 | |
---|
510 | The resulting discret form of the three operators (one iso-level and two rotated one) |
---|
511 | is given in the next two sub-sections. |
---|
512 | |
---|
513 | |
---|
514 | % ------------------------------------------------------------------------------------------------------------- |
---|
515 | % iso-level operator |
---|
516 | % ------------------------------------------------------------------------------------------------------------- |
---|
517 | \subsection [Iso-level (bi-)laplacian operator ( \np{ln\_traldf\_iso})] |
---|
518 | {Iso-level (bi-)laplacian operator ( \np{ln\_traldf\_iso}) } |
---|
519 | \label{TRA_ldf_lev} |
---|
520 | |
---|
521 | The laplacian diffusion operator acting along the model (\textit{i,j})-surfaces is given by: |
---|
522 | \begin{equation} \label{Eq_tra_ldf_lap} |
---|
523 | D_t^{lT} =\frac{1}{b_t} \left( \; |
---|
524 | \delta _{i}\left[ A_u^{lT} \; \frac{e_{2u}\,e_{3u}}{e_{1u}} \;\delta _{i+1/2} [T] \right] |
---|
525 | + \delta _{j}\left[ A_v^{lT} \; \frac{e_{1v}\,e_{3v}}{e_{2v}} \;\delta _{j+1/2} [T] \right] \;\right) |
---|
526 | \end{equation} |
---|
527 | where $b_t$=$e_{1t}\,e_{2t}\,e_{3t}$ is the volume of $T$-cells |
---|
528 | and where zero diffusive fluxes is assumed across solid boundaries, |
---|
529 | first (and third in bilaplacian case) horizontal tracer derivative are masked. |
---|
530 | It is implemented in the \rou{traldf\_lap} subroutine found in the \mdl{traldf\_lap} module. |
---|
531 | The module also contains \rou{traldf\_blp}, the subroutine calling twice \rou{traldf\_lap} |
---|
532 | in order to compute the iso-level bilaplacian operator. |
---|
533 | |
---|
534 | It is a \emph{horizontal} operator ($i.e.$ acting along geopotential surfaces) in the $z$-coordinate |
---|
535 | with or without partial steps, but is simply an iso-level operator in the $s$-coordinate. |
---|
536 | It is thus used when, in addition to \np{ln\_traldf\_lap} or \np{ln\_traldf\_blp}~=~\textit{true}, |
---|
537 | we have \np{ln\_traldf\_lev}~=~\textit{true} or \np{ln\_traldf\_hor}~=~\np{ln\_zco}~=~\textit{true}. |
---|
538 | In both cases, it significantly contributes to diapycnal mixing. |
---|
539 | It is therefore never recommended, even when using it in the bilaplacian case. |
---|
540 | |
---|
541 | Note that in the partial step $z$-coordinate (\np{ln\_zps}=true), tracers in horizontally |
---|
542 | adjacent cells are located at different depths in the vicinity of the bottom. |
---|
543 | In this case, horizontal derivatives in (\ref{Eq_tra_ldf_lap}) at the bottom level |
---|
544 | require a specific treatment. They are calculated in the \mdl{zpshde} module, |
---|
545 | described in \S\ref{TRA_zpshde}. |
---|
546 | |
---|
547 | |
---|
548 | % ------------------------------------------------------------------------------------------------------------- |
---|
549 | % Rotated laplacian operator |
---|
550 | % ------------------------------------------------------------------------------------------------------------- |
---|
551 | \subsection [Standard and triad rotated (bi-)laplacian operator (\mdl{traldf\_iso}, \mdl{traldf\_triad})] |
---|
552 | {Standard and triad (bi-)laplacian operator (\mdl{traldf\_iso}, \mdl{traldf\_triad}))} |
---|
553 | \label{TRA_ldf_iso_triad} |
---|
554 | |
---|
555 | %&& Standard rotated (bi-)laplacian operator |
---|
556 | %&& ---------------------------------------------- |
---|
557 | \subsubsection [Standard rotated (bi-)laplacian operator (\mdl{traldf\_iso})] |
---|
558 | {Standard rotated (bi-)laplacian operator (\mdl{traldf\_iso})} |
---|
559 | \label{TRA_ldf_iso} |
---|
560 | The general form of the second order lateral tracer subgrid scale physics |
---|
561 | (\ref{Eq_PE_zdf}) takes the following semi-discrete space form in $z$- and $s$-coordinates: |
---|
562 | \begin{equation} \label{Eq_tra_ldf_iso} |
---|
563 | \begin{split} |
---|
564 | D_T^{lT} = \frac{1}{b_t} & \left\{ \,\;\delta_i \left[ A_u^{lT} \left( |
---|
565 | \frac{e_{2u}\,e_{3u}}{e_{1u}} \,\delta_{i+1/2}[T] |
---|
566 | - e_{2u}\;r_{1u} \,\overline{\overline{ \delta_{k+1/2}[T] }}^{\,i+1/2,k} |
---|
567 | \right) \right] \right. \\ |
---|
568 | & +\delta_j \left[ A_v^{lT} \left( |
---|
569 | \frac{e_{1v}\,e_{3v}}{e_{2v}} \,\delta_{j+1/2} [T] |
---|
570 | - e_{1v}\,r_{2v} \,\overline{\overline{ \delta_{k+1/2} [T] }}^{\,j+1/2,k} |
---|
571 | \right) \right] \\ |
---|
572 | & +\delta_k \left[ A_w^{lT} \left( |
---|
573 | -\;e_{2w}\,r_{1w} \,\overline{\overline{ \delta_{i+1/2} [T] }}^{\,i,k+1/2} |
---|
574 | \right. \right. \\ |
---|
575 | & \qquad \qquad \quad |
---|
576 | - e_{1w}\,r_{2w} \,\overline{\overline{ \delta_{j+1/2} [T] }}^{\,j,k+1/2} \\ |
---|
577 | & \left. {\left. { \qquad \qquad \ \ \ \left. { |
---|
578 | +\;\frac{e_{1w}\,e_{2w}}{e_{3w}} \,\left( r_{1w}^2 + r_{2w}^2 \right) |
---|
579 | \,\delta_{k+1/2} [T] } \right) } \right] \quad } \right\} |
---|
580 | \end{split} |
---|
581 | \end{equation} |
---|
582 | where $b_t$=$e_{1t}\,e_{2t}\,e_{3t}$ is the volume of $T$-cells, |
---|
583 | $r_1$ and $r_2$ are the slopes between the surface of computation |
---|
584 | ($z$- or $s$-surfaces) and the surface along which the diffusion operator |
---|
585 | acts ($i.e.$ horizontal or iso-neutral surfaces). It is thus used when, |
---|
586 | in addition to \np{ln\_traldf\_lap}= true, we have \np{ln\_traldf\_iso}=true, |
---|
587 | or both \np{ln\_traldf\_hor}=true and \np{ln\_zco}=true. The way these |
---|
588 | slopes are evaluated is given in \S\ref{LDF_slp}. At the surface, bottom |
---|
589 | and lateral boundaries, the turbulent fluxes of heat and salt are set to zero |
---|
590 | using the mask technique (see \S\ref{LBC_coast}). |
---|
591 | |
---|
592 | The operator in \eqref{Eq_tra_ldf_iso} involves both lateral and vertical |
---|
593 | derivatives. For numerical stability, the vertical second derivative must |
---|
594 | be solved using the same implicit time scheme as that used in the vertical |
---|
595 | physics (see \S\ref{TRA_zdf}). For computer efficiency reasons, this term |
---|
596 | is not computed in the \mdl{traldf\_iso} module, but in the \mdl{trazdf} module |
---|
597 | where, if iso-neutral mixing is used, the vertical mixing coefficient is simply |
---|
598 | increased by $\frac{e_{1w}\,e_{2w} }{e_{3w} }\ \left( {r_{1w} ^2+r_{2w} ^2} \right)$. |
---|
599 | |
---|
600 | This formulation conserves the tracer but does not ensure the decrease |
---|
601 | of the tracer variance. Nevertheless the treatment performed on the slopes |
---|
602 | (see \S\ref{LDF}) allows the model to run safely without any additional |
---|
603 | background horizontal diffusion \citep{Guilyardi_al_CD01}. |
---|
604 | |
---|
605 | Note that in the partial step $z$-coordinate (\np{ln\_zps}=true), the horizontal derivatives |
---|
606 | at the bottom level in \eqref{Eq_tra_ldf_iso} require a specific treatment. |
---|
607 | They are calculated in module zpshde, described in \S\ref{TRA_zpshde}. |
---|
608 | |
---|
609 | %&& Triad rotated (bi-)laplacian operator |
---|
610 | %&& ------------------------------------------- |
---|
611 | \subsubsection [Triad rotated (bi-)laplacian operator (\np{ln\_traldf\_triad})] |
---|
612 | {Triad rotated (bi-)laplacian operator (\np{ln\_traldf\_triad})} |
---|
613 | \label{TRA_ldf_triad} |
---|
614 | |
---|
615 | If the Griffies triad scheme is employed (\np{ln\_traldf\_triad}=true ; see App.\ref{sec:triad}) |
---|
616 | |
---|
617 | An alternative scheme developed by \cite{Griffies_al_JPO98} which ensures tracer variance decreases |
---|
618 | is also available in \NEMO (\np{ln\_traldf\_grif}=true). A complete description of |
---|
619 | the algorithm is given in App.\ref{sec:triad}. |
---|
620 | |
---|
621 | The lateral fourth order bilaplacian operator on tracers is obtained by |
---|
622 | applying (\ref{Eq_tra_ldf_lap}) twice. The operator requires an additional assumption |
---|
623 | on boundary conditions: both first and third derivative terms normal to the |
---|
624 | coast are set to zero. |
---|
625 | |
---|
626 | The lateral fourth order operator formulation on tracers is obtained by |
---|
627 | applying (\ref{Eq_tra_ldf_iso}) twice. It requires an additional assumption |
---|
628 | on boundary conditions: first and third derivative terms normal to the |
---|
629 | coast, normal to the bottom and normal to the surface are set to zero. |
---|
630 | |
---|
631 | %&& Option for the rotated operators |
---|
632 | %&& ---------------------------------------------- |
---|
633 | \subsubsection [Option for the rotated operators] |
---|
634 | {Option for the rotated operators} |
---|
635 | \label{TRA_ldf_options} |
---|
636 | |
---|
637 | \np{ln\_traldf\_msc} = Method of Stabilizing Correction (both operators) |
---|
638 | |
---|
639 | \np{rn\_slpmax} = slope limit (both operators) |
---|
640 | |
---|
641 | \np{ln\_triad\_iso} = pure horizontal mixing in ML (triad only) |
---|
642 | |
---|
643 | \np{rn\_sw\_triad} =1 switching triad ; =0 all 4 triads used (triad only) |
---|
644 | |
---|
645 | \np{ln\_botmix\_triad} = lateral mixing on bottom (triad only) |
---|
646 | |
---|
647 | % ================================================================ |
---|
648 | % Tracer Vertical Diffusion |
---|
649 | % ================================================================ |
---|
650 | \section [Tracer Vertical Diffusion (\textit{trazdf})] |
---|
651 | {Tracer Vertical Diffusion (\mdl{trazdf})} |
---|
652 | \label{TRA_zdf} |
---|
653 | %--------------------------------------------namzdf--------------------------------------------------------- |
---|
654 | \namdisplay{namzdf} |
---|
655 | %-------------------------------------------------------------------------------------------------------------- |
---|
656 | |
---|
657 | Options are defined through the \ngn{namzdf} namelist variables. |
---|
658 | The formulation of the vertical subgrid scale tracer physics is the same |
---|
659 | for all the vertical coordinates, and is based on a laplacian operator. |
---|
660 | The vertical diffusion operator given by (\ref{Eq_PE_zdf}) takes the |
---|
661 | following semi-discrete space form: |
---|
662 | \begin{equation} \label{Eq_tra_zdf} |
---|
663 | \begin{split} |
---|
664 | D^{vT}_T &= \frac{1}{e_{3t}} \; \delta_k \left[ \;\frac{A^{vT}_w}{e_{3w}} \delta_{k+1/2}[T] \;\right] |
---|
665 | \\ |
---|
666 | D^{vS}_T &= \frac{1}{e_{3t}} \; \delta_k \left[ \;\frac{A^{vS}_w}{e_{3w}} \delta_{k+1/2}[S] \;\right] |
---|
667 | \end{split} |
---|
668 | \end{equation} |
---|
669 | where $A_w^{vT}$ and $A_w^{vS}$ are the vertical eddy diffusivity |
---|
670 | coefficients on temperature and salinity, respectively. Generally, |
---|
671 | $A_w^{vT}=A_w^{vS}$ except when double diffusive mixing is |
---|
672 | parameterised ($i.e.$ \key{zdfddm} is defined). The way these coefficients |
---|
673 | are evaluated is given in \S\ref{ZDF} (ZDF). Furthermore, when |
---|
674 | iso-neutral mixing is used, both mixing coefficients are increased |
---|
675 | by $\frac{e_{1w}\,e_{2w} }{e_{3w} }\ \left( {r_{1w} ^2+r_{2w} ^2} \right)$ |
---|
676 | to account for the vertical second derivative of \eqref{Eq_tra_ldf_iso}. |
---|
677 | |
---|
678 | At the surface and bottom boundaries, the turbulent fluxes of |
---|
679 | heat and salt must be specified. At the surface they are prescribed |
---|
680 | from the surface forcing and added in a dedicated routine (see \S\ref{TRA_sbc}), |
---|
681 | whilst at the bottom they are set to zero for heat and salt unless |
---|
682 | a geothermal flux forcing is prescribed as a bottom boundary |
---|
683 | condition (see \S\ref{TRA_bbc}). |
---|
684 | |
---|
685 | The large eddy coefficient found in the mixed layer together with high |
---|
686 | vertical resolution implies that in the case of explicit time stepping |
---|
687 | (\np{ln\_zdfexp}=true) there would be too restrictive a constraint on |
---|
688 | the time step. Therefore, the default implicit time stepping is preferred |
---|
689 | for the vertical diffusion since it overcomes the stability constraint. |
---|
690 | A forward time differencing scheme (\np{ln\_zdfexp}=true) using a time |
---|
691 | splitting technique (\np{nn\_zdfexp} $> 1$) is provided as an alternative. |
---|
692 | Namelist variables \np{ln\_zdfexp} and \np{nn\_zdfexp} apply to both |
---|
693 | tracers and dynamics. |
---|
694 | |
---|
695 | % ================================================================ |
---|
696 | % External Forcing |
---|
697 | % ================================================================ |
---|
698 | \section{External Forcing} |
---|
699 | \label{TRA_sbc_qsr_bbc} |
---|
700 | |
---|
701 | % ------------------------------------------------------------------------------------------------------------- |
---|
702 | % surface boundary condition |
---|
703 | % ------------------------------------------------------------------------------------------------------------- |
---|
704 | \subsection [Surface boundary condition (\textit{trasbc})] |
---|
705 | {Surface boundary condition (\mdl{trasbc})} |
---|
706 | \label{TRA_sbc} |
---|
707 | |
---|
708 | The surface boundary condition for tracers is implemented in a separate |
---|
709 | module (\mdl{trasbc}) instead of entering as a boundary condition on the vertical |
---|
710 | diffusion operator (as in the case of momentum). This has been found to |
---|
711 | enhance readability of the code. The two formulations are completely |
---|
712 | equivalent; the forcing terms in trasbc are the surface fluxes divided by |
---|
713 | the thickness of the top model layer. |
---|
714 | |
---|
715 | Due to interactions and mass exchange of water ($F_{mass}$) with other Earth system components |
---|
716 | ($i.e.$ atmosphere, sea-ice, land), the change in the heat and salt content of the surface layer |
---|
717 | of the ocean is due both to the heat and salt fluxes crossing the sea surface (not linked with $F_{mass}$) |
---|
718 | and to the heat and salt content of the mass exchange. They are both included directly in $Q_{ns}$, |
---|
719 | the surface heat flux, and $F_{salt}$, the surface salt flux (see \S\ref{SBC} for further details). |
---|
720 | By doing this, the forcing formulation is the same for any tracer (including temperature and salinity). |
---|
721 | |
---|
722 | The surface module (\mdl{sbcmod}, see \S\ref{SBC}) provides the following |
---|
723 | forcing fields (used on tracers): |
---|
724 | |
---|
725 | $\bullet$ $Q_{ns}$, the non-solar part of the net surface heat flux that crosses the sea surface |
---|
726 | (i.e. the difference between the total surface heat flux and the fraction of the short wave flux that |
---|
727 | penetrates into the water column, see \S\ref{TRA_qsr}) plus the heat content associated with |
---|
728 | of the mass exchange with the atmosphere and lands. |
---|
729 | |
---|
730 | $\bullet$ $\textit{sfx}$, the salt flux resulting from ice-ocean mass exchange (freezing, melting, ridging...) |
---|
731 | |
---|
732 | $\bullet$ \textit{emp}, the mass flux exchanged with the atmosphere (evaporation minus precipitation) |
---|
733 | and possibly with the sea-ice and ice-shelves. |
---|
734 | |
---|
735 | $\bullet$ \textit{rnf}, the mass flux associated with runoff |
---|
736 | (see \S\ref{SBC_rnf} for further detail of how it acts on temperature and salinity tendencies) |
---|
737 | |
---|
738 | $\bullet$ \textit{fwfisf}, the mass flux associated with ice shelf melt, |
---|
739 | (see \S\ref{SBC_isf} for further details on how the ice shelf melt is computed and applied). |
---|
740 | |
---|
741 | The surface boundary condition on temperature and salinity is applied as follows: |
---|
742 | \begin{equation} \label{Eq_tra_sbc} |
---|
743 | \begin{aligned} |
---|
744 | &F^T = \frac{ 1 }{\rho _o \;C_p \,\left. e_{3t} \right|_{k=1} } &\overline{ Q_{ns} }^t & \\ |
---|
745 | & F^S =\frac{ 1 }{\rho _o \, \left. e_{3t} \right|_{k=1} } &\overline{ \textit{sfx} }^t & \\ |
---|
746 | \end{aligned} |
---|
747 | \end{equation} |
---|
748 | where $\overline{x }^t$ means that $x$ is averaged over two consecutive time steps |
---|
749 | ($t-\rdt/2$ and $t+\rdt/2$). Such time averaging prevents the |
---|
750 | divergence of odd and even time step (see \S\ref{STP}). |
---|
751 | |
---|
752 | In the linear free surface case (\np{ln\_linssh}~=~\textit{true}), |
---|
753 | an additional term has to be added on both temperature and salinity. |
---|
754 | On temperature, this term remove the heat content associated with mass exchange |
---|
755 | that has been added to $Q_{ns}$. On salinity, this term mimics the concentration/dilution effect that |
---|
756 | would have resulted from a change in the volume of the first level. |
---|
757 | The resulting surface boundary condition is applied as follows: |
---|
758 | \begin{equation} \label{Eq_tra_sbc_lin} |
---|
759 | \begin{aligned} |
---|
760 | &F^T = \frac{ 1 }{\rho _o \;C_p \,\left. e_{3t} \right|_{k=1} } |
---|
761 | &\overline{ \left( Q_{ns} - \textit{emp}\;C_p\,\left. T \right|_{k=1} \right) }^t & \\ |
---|
762 | % |
---|
763 | & F^S =\frac{ 1 }{\rho _o \,\left. e_{3t} \right|_{k=1} } |
---|
764 | &\overline{ \left( \;\textit{sfx} - \textit{emp} \;\left. S \right|_{k=1} \right) }^t & \\ |
---|
765 | \end{aligned} |
---|
766 | \end{equation} |
---|
767 | Note that an exact conservation of heat and salt content is only achieved with non-linear free surface. |
---|
768 | In the linear free surface case, there is a small imbalance. The imbalance is larger |
---|
769 | than the imbalance associated with the Asselin time filter \citep{Leclair_Madec_OM09}. |
---|
770 | This is the reason why the modified filter is not applied in the linear free surface case (see \S\ref{STP}). |
---|
771 | |
---|
772 | % ------------------------------------------------------------------------------------------------------------- |
---|
773 | % Solar Radiation Penetration |
---|
774 | % ------------------------------------------------------------------------------------------------------------- |
---|
775 | \subsection [Solar Radiation Penetration (\textit{traqsr})] |
---|
776 | {Solar Radiation Penetration (\mdl{traqsr})} |
---|
777 | \label{TRA_qsr} |
---|
778 | %--------------------------------------------namqsr-------------------------------------------------------- |
---|
779 | \namdisplay{namtra_qsr} |
---|
780 | %-------------------------------------------------------------------------------------------------------------- |
---|
781 | |
---|
782 | Options are defined through the \ngn{namtra\_qsr} namelist variables. |
---|
783 | When the penetrative solar radiation option is used (\np{ln\_flxqsr}=true), |
---|
784 | the solar radiation penetrates the top few tens of meters of the ocean. If it is not used |
---|
785 | (\np{ln\_flxqsr}=false) all the heat flux is absorbed in the first ocean level. |
---|
786 | Thus, in the former case a term is added to the time evolution equation of |
---|
787 | temperature \eqref{Eq_PE_tra_T} and the surface boundary condition is |
---|
788 | modified to take into account only the non-penetrative part of the surface |
---|
789 | heat flux: |
---|
790 | \begin{equation} \label{Eq_PE_qsr} |
---|
791 | \begin{split} |
---|
792 | \frac{\partial T}{\partial t} &= {\ldots} + \frac{1}{\rho_o\, C_p \,e_3} \; \frac{\partial I}{\partial k} \\ |
---|
793 | Q_{ns} &= Q_\text{Total} - Q_{sr} |
---|
794 | \end{split} |
---|
795 | \end{equation} |
---|
796 | where $Q_{sr}$ is the penetrative part of the surface heat flux ($i.e.$ the shortwave radiation) |
---|
797 | and $I$ is the downward irradiance ($\left. I \right|_{z=\eta}=Q_{sr}$). |
---|
798 | The additional term in \eqref{Eq_PE_qsr} is discretized as follows: |
---|
799 | \begin{equation} \label{Eq_tra_qsr} |
---|
800 | \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] |
---|
801 | \end{equation} |
---|
802 | |
---|
803 | The shortwave radiation, $Q_{sr}$, consists of energy distributed across a wide spectral range. |
---|
804 | The ocean is strongly absorbing for wavelengths longer than 700~nm and these |
---|
805 | wavelengths contribute to heating the upper few tens of centimetres. The fraction of $Q_{sr}$ |
---|
806 | that resides in these almost non-penetrative wavebands, $R$, is $\sim 58\%$ (specified |
---|
807 | through namelist parameter \np{rn\_abs}). It is assumed to penetrate the ocean |
---|
808 | with a decreasing exponential profile, with an e-folding depth scale, $\xi_0$, |
---|
809 | of a few tens of centimetres (typically $\xi_0=0.35~m$ set as \np{rn\_si0} in the namtra\_qsr namelist). |
---|
810 | For shorter wavelengths (400-700~nm), the ocean is more transparent, and solar energy |
---|
811 | propagates to larger depths where it contributes to |
---|
812 | local heating. |
---|
813 | The way this second part of the solar energy penetrates into the ocean depends on |
---|
814 | which formulation is chosen. In the simple 2-waveband light penetration scheme (\np{ln\_qsr\_2bd}=true) |
---|
815 | a chlorophyll-independent monochromatic formulation is chosen for the shorter wavelengths, |
---|
816 | leading to the following expression \citep{Paulson1977}: |
---|
817 | \begin{equation} \label{Eq_traqsr_iradiance} |
---|
818 | I(z) = Q_{sr} \left[Re^{-z / \xi_0} + \left( 1-R\right) e^{-z / \xi_1} \right] |
---|
819 | \end{equation} |
---|
820 | where $\xi_1$ is the second extinction length scale associated with the shorter wavelengths. |
---|
821 | It is usually chosen to be 23~m by setting the \np{rn\_si0} namelist parameter. |
---|
822 | The set of default values ($\xi_0$, $\xi_1$, $R$) corresponds to a Type I water in |
---|
823 | Jerlov's (1968) classification (oligotrophic waters). |
---|
824 | |
---|
825 | Such assumptions have been shown to provide a very crude and simplistic |
---|
826 | representation of observed light penetration profiles (\cite{Morel_JGR88}, see also |
---|
827 | Fig.\ref{Fig_traqsr_irradiance}). Light absorption in the ocean depends on |
---|
828 | particle concentration and is spectrally selective. \cite{Morel_JGR88} has shown |
---|
829 | that an accurate representation of light penetration can be provided by a 61 waveband |
---|
830 | formulation. Unfortunately, such a model is very computationally expensive. |
---|
831 | Thus, \cite{Lengaigne_al_CD07} have constructed a simplified version of this |
---|
832 | formulation in which visible light is split into three wavebands: blue (400-500 nm), |
---|
833 | green (500-600 nm) and red (600-700nm). For each wave-band, the chlorophyll-dependent |
---|
834 | attenuation coefficient is fitted to the coefficients computed from the full spectral model |
---|
835 | of \cite{Morel_JGR88} (as modified by \cite{Morel_Maritorena_JGR01}), assuming |
---|
836 | the same power-law relationship. As shown in Fig.\ref{Fig_traqsr_irradiance}, |
---|
837 | this formulation, called RGB (Red-Green-Blue), reproduces quite closely |
---|
838 | the light penetration profiles predicted by the full spectal model, but with much greater |
---|
839 | computational efficiency. The 2-bands formulation does not reproduce the full model very well. |
---|
840 | |
---|
841 | The RGB formulation is used when \np{ln\_qsr\_rgb}=true. The RGB attenuation coefficients |
---|
842 | ($i.e.$ the inverses of the extinction length scales) are tabulated over 61 nonuniform |
---|
843 | chlorophyll classes ranging from 0.01 to 10 g.Chl/L (see the routine \rou{trc\_oce\_rgb} |
---|
844 | in \mdl{trc\_oce} module). Four types of chlorophyll can be chosen in the RGB formulation: |
---|
845 | \begin{description} |
---|
846 | \item[\np{nn\_chdta}=0] |
---|
847 | a constant 0.05 g.Chl/L value everywhere ; |
---|
848 | \item[\np{nn\_chdta}=1] |
---|
849 | an observed time varying chlorophyll deduced from satellite surface ocean color measurement |
---|
850 | spread uniformly in the vertical direction ; |
---|
851 | \item[\np{nn\_chdta}=2] |
---|
852 | same as previous case except that a vertical profile of chlorophyl is used. |
---|
853 | Following \cite{Morel_Berthon_LO89}, the profile is computed from the local surface chlorophyll value ; |
---|
854 | \item[\np{ln\_qsr\_bio}=true] |
---|
855 | simulated time varying chlorophyll by TOP biogeochemical model. |
---|
856 | In this case, the RGB formulation is used to calculate both the phytoplankton |
---|
857 | light limitation in PISCES or LOBSTER and the oceanic heating rate. |
---|
858 | \end{description} |
---|
859 | The trend in \eqref{Eq_tra_qsr} associated with the penetration of the solar radiation |
---|
860 | is added to the temperature trend, and the surface heat flux is modified in routine \mdl{traqsr}. |
---|
861 | |
---|
862 | When the $z$-coordinate is preferred to the $s$-coordinate, the depth of $w-$levels does |
---|
863 | not significantly vary with location. The level at which the light has been totally |
---|
864 | absorbed ($i.e.$ it is less than the computer precision) is computed once, |
---|
865 | and the trend associated with the penetration of the solar radiation is only added down to that level. |
---|
866 | Finally, note that when the ocean is shallow ($<$ 200~m), part of the |
---|
867 | solar radiation can reach the ocean floor. In this case, we have |
---|
868 | chosen that all remaining radiation is absorbed in the last ocean |
---|
869 | level ($i.e.$ $I$ is masked). |
---|
870 | |
---|
871 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
---|
872 | \begin{figure}[!t] \begin{center} |
---|
873 | \includegraphics[width=1.0\textwidth]{Fig_TRA_Irradiance} |
---|
874 | \caption{ \label{Fig_traqsr_irradiance} |
---|
875 | Penetration profile of the downward solar irradiance calculated by four models. |
---|
876 | Two waveband chlorophyll-independent formulation (blue), a chlorophyll-dependent |
---|
877 | monochromatic formulation (green), 4 waveband RGB formulation (red), |
---|
878 | 61 waveband Morel (1988) formulation (black) for a chlorophyll concentration of |
---|
879 | (a) Chl=0.05 mg/m$^3$ and (b) Chl=0.5 mg/m$^3$. From \citet{Lengaigne_al_CD07}.} |
---|
880 | \end{center} \end{figure} |
---|
881 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
---|
882 | |
---|
883 | % ------------------------------------------------------------------------------------------------------------- |
---|
884 | % Bottom Boundary Condition |
---|
885 | % ------------------------------------------------------------------------------------------------------------- |
---|
886 | \subsection [Bottom Boundary Condition (\textit{trabbc})] |
---|
887 | {Bottom Boundary Condition (\mdl{trabbc})} |
---|
888 | \label{TRA_bbc} |
---|
889 | %--------------------------------------------nambbc-------------------------------------------------------- |
---|
890 | \namdisplay{nambbc} |
---|
891 | %-------------------------------------------------------------------------------------------------------------- |
---|
892 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
---|
893 | \begin{figure}[!t] \begin{center} |
---|
894 | \includegraphics[width=1.0\textwidth]{Fig_TRA_geoth} |
---|
895 | \caption{ \label{Fig_geothermal} |
---|
896 | Geothermal Heat flux (in $mW.m^{-2}$) used by \cite{Emile-Geay_Madec_OS09}. |
---|
897 | It is inferred from the age of the sea floor and the formulae of \citet{Stein_Stein_Nat92}.} |
---|
898 | \end{center} \end{figure} |
---|
899 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
---|
900 | |
---|
901 | Usually it is assumed that there is no exchange of heat or salt through |
---|
902 | the ocean bottom, $i.e.$ a no flux boundary condition is applied on active |
---|
903 | tracers at the bottom. This is the default option in \NEMO, and it is |
---|
904 | implemented using the masking technique. However, there is a |
---|
905 | non-zero heat flux across the seafloor that is associated with solid |
---|
906 | earth cooling. This flux is weak compared to surface fluxes (a mean |
---|
907 | global value of $\sim0.1\;W/m^2$ \citep{Stein_Stein_Nat92}), but it warms |
---|
908 | systematically the ocean and acts on the densest water masses. |
---|
909 | Taking this flux into account in a global ocean model increases |
---|
910 | the deepest overturning cell ($i.e.$ the one associated with the Antarctic |
---|
911 | Bottom Water) by a few Sverdrups \citep{Emile-Geay_Madec_OS09}. |
---|
912 | |
---|
913 | Options are defined through the \ngn{namtra\_bbc} namelist variables. |
---|
914 | The presence of geothermal heating is controlled by setting the namelist |
---|
915 | parameter \np{ln\_trabbc} to true. Then, when \np{nn\_geoflx} is set to 1, |
---|
916 | a constant geothermal heating is introduced whose value is given by the |
---|
917 | \np{nn\_geoflx\_cst}, which is also a namelist parameter. |
---|
918 | When \np{nn\_geoflx} is set to 2, a spatially varying geothermal heat flux is |
---|
919 | introduced which is provided in the \ifile{geothermal\_heating} NetCDF file |
---|
920 | (Fig.\ref{Fig_geothermal}) \citep{Emile-Geay_Madec_OS09}. |
---|
921 | |
---|
922 | % ================================================================ |
---|
923 | % Bottom Boundary Layer |
---|
924 | % ================================================================ |
---|
925 | \section [Bottom Boundary Layer (\mdl{trabbl} - \key{trabbl})] |
---|
926 | {Bottom Boundary Layer (\mdl{trabbl} - \key{trabbl})} |
---|
927 | \label{TRA_bbl} |
---|
928 | %--------------------------------------------nambbl--------------------------------------------------------- |
---|
929 | \namdisplay{nambbl} |
---|
930 | %-------------------------------------------------------------------------------------------------------------- |
---|
931 | |
---|
932 | Options are defined through the \ngn{nambbl} namelist variables. |
---|
933 | In a $z$-coordinate configuration, the bottom topography is represented by a |
---|
934 | series of discrete steps. This is not adequate to represent gravity driven |
---|
935 | downslope flows. Such flows arise either downstream of sills such as the Strait of |
---|
936 | Gibraltar or Denmark Strait, where dense water formed in marginal seas flows |
---|
937 | into a basin filled with less dense water, or along the continental slope when dense |
---|
938 | water masses are formed on a continental shelf. The amount of entrainment |
---|
939 | that occurs in these gravity plumes is critical in determining the density |
---|
940 | and volume flux of the densest waters of the ocean, such as Antarctic Bottom Water, |
---|
941 | or North Atlantic Deep Water. $z$-coordinate models tend to overestimate the |
---|
942 | entrainment, because the gravity flow is mixed vertically by convection |
---|
943 | as it goes ''downstairs'' following the step topography, sometimes over a thickness |
---|
944 | much larger than the thickness of the observed gravity plume. A similar problem |
---|
945 | occurs in the $s$-coordinate when the thickness of the bottom level varies rapidly |
---|
946 | downstream of a sill \citep{Willebrand_al_PO01}, and the thickness |
---|
947 | of the plume is not resolved. |
---|
948 | |
---|
949 | The idea of the bottom boundary layer (BBL) parameterisation, first introduced by |
---|
950 | \citet{Beckmann_Doscher1997}, is to allow a direct communication between |
---|
951 | two adjacent bottom cells at different levels, whenever the densest water is |
---|
952 | located above the less dense water. The communication can be by a diffusive flux |
---|
953 | (diffusive BBL), an advective flux (advective BBL), or both. In the current |
---|
954 | implementation of the BBL, only the tracers are modified, not the velocities. |
---|
955 | Furthermore, it only connects ocean bottom cells, and therefore does not include |
---|
956 | all the improvements introduced by \citet{Campin_Goosse_Tel99}. |
---|
957 | |
---|
958 | % ------------------------------------------------------------------------------------------------------------- |
---|
959 | % Diffusive BBL |
---|
960 | % ------------------------------------------------------------------------------------------------------------- |
---|
961 | \subsection{Diffusive Bottom Boundary layer (\np{nn\_bbl\_ldf}=1)} |
---|
962 | \label{TRA_bbl_diff} |
---|
963 | |
---|
964 | When applying sigma-diffusion (\key{trabbl} defined and \np{nn\_bbl\_ldf} set to 1), |
---|
965 | the diffusive flux between two adjacent cells at the ocean floor is given by |
---|
966 | \begin{equation} \label{Eq_tra_bbl_diff} |
---|
967 | {\rm {\bf F}}_\sigma=A_l^\sigma \; \nabla_\sigma T |
---|
968 | \end{equation} |
---|
969 | with $\nabla_\sigma$ the lateral gradient operator taken between bottom cells, |
---|
970 | and $A_l^\sigma$ the lateral diffusivity in the BBL. Following \citet{Beckmann_Doscher1997}, |
---|
971 | the latter is prescribed with a spatial dependence, $i.e.$ in the conditional form |
---|
972 | \begin{equation} \label{Eq_tra_bbl_coef} |
---|
973 | A_l^\sigma (i,j,t)=\left\{ {\begin{array}{l} |
---|
974 | A_{bbl} \quad \quad \mbox{if} \quad \nabla_\sigma \rho \cdot \nabla H<0 \\ |
---|
975 | \\ |
---|
976 | 0\quad \quad \;\,\mbox{otherwise} \\ |
---|
977 | \end{array}} \right. |
---|
978 | \end{equation} |
---|
979 | where $A_{bbl}$ is the BBL diffusivity coefficient, given by the namelist |
---|
980 | parameter \np{rn\_ahtbbl} and usually set to a value much larger |
---|
981 | than the one used for lateral mixing in the open ocean. The constraint in \eqref{Eq_tra_bbl_coef} |
---|
982 | implies that sigma-like diffusion only occurs when the density above the sea floor, at the top of |
---|
983 | the slope, is larger than in the deeper ocean (see green arrow in Fig.\ref{Fig_bbl}). |
---|
984 | In practice, this constraint is applied separately in the two horizontal directions, |
---|
985 | and the density gradient in \eqref{Eq_tra_bbl_coef} is evaluated with the log gradient formulation: |
---|
986 | \begin{equation} \label{Eq_tra_bbl_Drho} |
---|
987 | \nabla_\sigma \rho / \rho = \alpha \,\nabla_\sigma T + \beta \,\nabla_\sigma S |
---|
988 | \end{equation} |
---|
989 | where $\rho$, $\alpha$ and $\beta$ are functions of $\overline{T}^\sigma$, |
---|
990 | $\overline{S}^\sigma$ and $\overline{H}^\sigma$, the along bottom mean temperature, |
---|
991 | salinity and depth, respectively. |
---|
992 | |
---|
993 | % ------------------------------------------------------------------------------------------------------------- |
---|
994 | % Advective BBL |
---|
995 | % ------------------------------------------------------------------------------------------------------------- |
---|
996 | \subsection {Advective Bottom Boundary Layer (\np{nn\_bbl\_adv}= 1 or 2)} |
---|
997 | \label{TRA_bbl_adv} |
---|
998 | |
---|
999 | \sgacomment{"downsloping flow" has been replaced by "downslope flow" in the following |
---|
1000 | if this is not what is meant then "downwards sloping flow" is also a possibility"} |
---|
1001 | |
---|
1002 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
---|
1003 | \begin{figure}[!t] \begin{center} |
---|
1004 | \includegraphics[width=0.7\textwidth]{Fig_BBL_adv} |
---|
1005 | \caption{ \label{Fig_bbl} |
---|
1006 | Advective/diffusive Bottom Boundary Layer. The BBL parameterisation is |
---|
1007 | activated when $\rho^i_{kup}$ is larger than $\rho^{i+1}_{kdnw}$. |
---|
1008 | Red arrows indicate the additional overturning circulation due to the advective BBL. |
---|
1009 | The transport of the downslope flow is defined either as the transport of the bottom |
---|
1010 | ocean cell (black arrow), or as a function of the along slope density gradient. |
---|
1011 | The green arrow indicates the diffusive BBL flux directly connecting $kup$ and $kdwn$ |
---|
1012 | ocean bottom cells. |
---|
1013 | connection} |
---|
1014 | \end{center} \end{figure} |
---|
1015 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
---|
1016 | |
---|
1017 | |
---|
1018 | %!! nn_bbl_adv = 1 use of the ocean velocity as bbl velocity |
---|
1019 | %!! nn_bbl_adv = 2 follow Campin and Goosse (1999) implentation |
---|
1020 | %!! i.e. transport proportional to the along-slope density gradient |
---|
1021 | |
---|
1022 | %%%gmcomment : this section has to be really written |
---|
1023 | |
---|
1024 | When applying an advective BBL (\np{nn\_bbl\_adv} = 1 or 2), an overturning |
---|
1025 | circulation is added which connects two adjacent bottom grid-points only if dense |
---|
1026 | water overlies less dense water on the slope. The density difference causes dense |
---|
1027 | water to move down the slope. |
---|
1028 | |
---|
1029 | \np{nn\_bbl\_adv} = 1 : the downslope velocity is chosen to be the Eulerian |
---|
1030 | ocean velocity just above the topographic step (see black arrow in Fig.\ref{Fig_bbl}) |
---|
1031 | \citep{Beckmann_Doscher1997}. It is a \textit{conditional advection}, that is, advection |
---|
1032 | is allowed only if dense water overlies less dense water on the slope ($i.e.$ |
---|
1033 | $\nabla_\sigma \rho \cdot \nabla H<0$) and if the velocity is directed towards |
---|
1034 | greater depth ($i.e.$ $\vect{U} \cdot \nabla H>0$). |
---|
1035 | |
---|
1036 | \np{nn\_bbl\_adv} = 2 : the downslope velocity is chosen to be proportional to $\Delta \rho$, |
---|
1037 | the density difference between the higher cell and lower cell densities \citep{Campin_Goosse_Tel99}. |
---|
1038 | The advection is allowed only if dense water overlies less dense water on the slope ($i.e.$ |
---|
1039 | $\nabla_\sigma \rho \cdot \nabla H<0$). For example, the resulting transport of the |
---|
1040 | downslope flow, here in the $i$-direction (Fig.\ref{Fig_bbl}), is simply given by the |
---|
1041 | following expression: |
---|
1042 | \begin{equation} \label{Eq_bbl_Utr} |
---|
1043 | u^{tr}_{bbl} = \gamma \, g \frac{\Delta \rho}{\rho_o} e_{1u} \; min \left( {e_{3u}}_{kup},{e_{3u}}_{kdwn} \right) |
---|
1044 | \end{equation} |
---|
1045 | where $\gamma$, expressed in seconds, is the coefficient of proportionality |
---|
1046 | provided as \np{rn\_gambbl}, a namelist parameter, and \textit{kup} and \textit{kdwn} |
---|
1047 | are the vertical index of the higher and lower cells, respectively. |
---|
1048 | The parameter $\gamma$ should take a different value for each bathymetric |
---|
1049 | step, but for simplicity, and because no direct estimation of this parameter is |
---|
1050 | available, a uniform value has been assumed. The possible values for $\gamma$ |
---|
1051 | range between 1 and $10~s$ \citep{Campin_Goosse_Tel99}. |
---|
1052 | |
---|
1053 | Scalar properties are advected by this additional transport $( u^{tr}_{bbl}, v^{tr}_{bbl} )$ |
---|
1054 | using the upwind scheme. Such a diffusive advective scheme has been chosen |
---|
1055 | to mimic the entrainment between the downslope plume and the surrounding |
---|
1056 | water at intermediate depths. The entrainment is replaced by the vertical mixing |
---|
1057 | implicit in the advection scheme. Let us consider as an example the |
---|
1058 | case displayed in Fig.\ref{Fig_bbl} where the density at level $(i,kup)$ is |
---|
1059 | larger than the one at level $(i,kdwn)$. The advective BBL scheme |
---|
1060 | modifies the tracer time tendency of the ocean cells near the |
---|
1061 | topographic step by the downslope flow \eqref{Eq_bbl_dw}, |
---|
1062 | the horizontal \eqref{Eq_bbl_hor} and the upward \eqref{Eq_bbl_up} |
---|
1063 | return flows as follows: |
---|
1064 | \begin{align} |
---|
1065 | \partial_t T^{do}_{kdw} &\equiv \partial_t T^{do}_{kdw} |
---|
1066 | + \frac{u^{tr}_{bbl}}{{b_t}^{do}_{kdw}} \left( T^{sh}_{kup} - T^{do}_{kdw} \right) \label{Eq_bbl_dw} \\ |
---|
1067 | % |
---|
1068 | \partial_t T^{sh}_{kup} &\equiv \partial_t T^{sh}_{kup} |
---|
1069 | + \frac{u^{tr}_{bbl}}{{b_t}^{sh}_{kup}} \left( T^{do}_{kup} - T^{sh}_{kup} \right) \label{Eq_bbl_hor} \\ |
---|
1070 | % |
---|
1071 | \intertext{and for $k =kdw-1,\;..., \; kup$ :} |
---|
1072 | % |
---|
1073 | \partial_t T^{do}_{k} &\equiv \partial_t S^{do}_{k} |
---|
1074 | + \frac{u^{tr}_{bbl}}{{b_t}^{do}_{k}} \left( T^{do}_{k+1} - T^{sh}_{k} \right) \label{Eq_bbl_up} |
---|
1075 | \end{align} |
---|
1076 | where $b_t$ is the $T$-cell volume. |
---|
1077 | |
---|
1078 | Note that the BBL transport, $( u^{tr}_{bbl}, v^{tr}_{bbl} )$, is available in |
---|
1079 | the model outputs. It has to be used to compute the effective velocity |
---|
1080 | as well as the effective overturning circulation. |
---|
1081 | |
---|
1082 | % ================================================================ |
---|
1083 | % Tracer damping |
---|
1084 | % ================================================================ |
---|
1085 | \section [Tracer damping (\textit{tradmp})] |
---|
1086 | {Tracer damping (\mdl{tradmp})} |
---|
1087 | \label{TRA_dmp} |
---|
1088 | %--------------------------------------------namtra_dmp------------------------------------------------- |
---|
1089 | \namdisplay{namtra_dmp} |
---|
1090 | %-------------------------------------------------------------------------------------------------------------- |
---|
1091 | |
---|
1092 | In some applications it can be useful to add a Newtonian damping term |
---|
1093 | into the temperature and salinity equations: |
---|
1094 | \begin{equation} \label{Eq_tra_dmp} |
---|
1095 | \begin{split} |
---|
1096 | \frac{\partial T}{\partial t}=\;\cdots \;-\gamma \,\left( {T-T_o } \right) \\ |
---|
1097 | \frac{\partial S}{\partial t}=\;\cdots \;-\gamma \,\left( {S-S_o } \right) |
---|
1098 | \end{split} |
---|
1099 | \end{equation} |
---|
1100 | where $\gamma$ is the inverse of a time scale, and $T_o$ and $S_o$ |
---|
1101 | are given temperature and salinity fields (usually a climatology). |
---|
1102 | Options are defined through the \ngn{namtra\_dmp} namelist variables. |
---|
1103 | The restoring term is added when the namelist parameter \np{ln\_tradmp} is set to true. |
---|
1104 | It also requires that both \np{ln\_tsd\_init} and \np{ln\_tsd\_tradmp} are set to true |
---|
1105 | in \textit{namtsd} namelist as well as \np{sn\_tem} and \np{sn\_sal} structures are |
---|
1106 | correctly set ($i.e.$ that $T_o$ and $S_o$ are provided in input files and read |
---|
1107 | using \mdl{fldread}, see \S\ref{SBC_fldread}). |
---|
1108 | The restoring coefficient $\gamma$ is a three-dimensional array read in during the \rou{tra\_dmp\_init} routine. The file name is specified by the namelist variable \np{cn\_resto}. The DMP\_TOOLS tool is provided to allow users to generate the netcdf file. |
---|
1109 | |
---|
1110 | The two main cases in which \eqref{Eq_tra_dmp} is used are \textit{(a)} |
---|
1111 | the specification of the boundary conditions along artificial walls of a |
---|
1112 | limited domain basin and \textit{(b)} the computation of the velocity |
---|
1113 | field associated with a given $T$-$S$ field (for example to build the |
---|
1114 | initial state of a prognostic simulation, or to use the resulting velocity |
---|
1115 | field for a passive tracer study). The first case applies to regional |
---|
1116 | models that have artificial walls instead of open boundaries. |
---|
1117 | In the vicinity of these walls, $\gamma$ takes large values (equivalent to |
---|
1118 | a time scale of a few days) whereas it is zero in the interior of the |
---|
1119 | model domain. The second case corresponds to the use of the robust |
---|
1120 | diagnostic method \citep{Sarmiento1982}. It allows us to find the velocity |
---|
1121 | field consistent with the model dynamics whilst having a $T$, $S$ field |
---|
1122 | close to a given climatological field ($T_o$, $S_o$). |
---|
1123 | |
---|
1124 | The robust diagnostic method is very efficient in preventing temperature |
---|
1125 | drift in intermediate waters but it produces artificial sources of heat and salt |
---|
1126 | within the ocean. It also has undesirable effects on the ocean convection. |
---|
1127 | It tends to prevent deep convection and subsequent deep-water formation, |
---|
1128 | by stabilising the water column too much. |
---|
1129 | |
---|
1130 | The namelist parameter \np{nn\_zdmp} sets whether the damping should be applied in the whole water column or only below the mixed layer (defined either on a density or $S_o$ criterion). It is common to set the damping to zero in the mixed layer as the adjustment time scale is short here \citep{Madec_al_JPO96}. |
---|
1131 | |
---|
1132 | \subsection[DMP\_TOOLS]{Generating resto.nc using DMP\_TOOLS} |
---|
1133 | |
---|
1134 | DMP\_TOOLS can be used to generate a netcdf file containing the restoration coefficient $\gamma$. |
---|
1135 | Note that in order to maintain bit comparison with previous NEMO versions DMP\_TOOLS must be compiled |
---|
1136 | and run on the same machine as the NEMO model. A mesh\_mask.nc file for the model configuration is required as an input. |
---|
1137 | This can be generated by carrying out a short model run with the namelist parameter \np{nn\_msh} set to 1. |
---|
1138 | The namelist parameter \np{ln\_tradmp} will also need to be set to .false. for this to work. |
---|
1139 | The \nl{nam\_dmp\_create} namelist in the DMP\_TOOLS directory is used to specify options for the restoration coefficient. |
---|
1140 | |
---|
1141 | %--------------------------------------------nam_dmp_create------------------------------------------------- |
---|
1142 | \namtools{namelist_dmp} |
---|
1143 | %------------------------------------------------------------------------------------------------------- |
---|
1144 | |
---|
1145 | \np{cp\_cfg}, \np{cp\_cpz}, \np{jp\_cfg} and \np{jperio} specify the model configuration being used and should be the same as specified in \nl{namcfg}. The variable \nl{lzoom} is used to specify that the damping is being used as in case \textit{a} above to provide boundary conditions to a zoom configuration. In the case of the arctic or antarctic zoom configurations this includes some specific treatment. Otherwise damping is applied to the 6 grid points along the ocean boundaries. The open boundaries are specified by the variables \np{lzoom\_n}, \np{lzoom\_e}, \np{lzoom\_s}, \np{lzoom\_w} in the \nl{nam\_zoom\_dmp} name list. |
---|
1146 | |
---|
1147 | The remaining switch namelist variables determine the spatial variation of the restoration coefficient in non-zoom configurations. |
---|
1148 | \np{ln\_full\_field} specifies that newtonian damping should be applied to the whole model domain. |
---|
1149 | \np{ln\_med\_red\_seas} specifies grid specific restoration coefficients in the Mediterranean Sea |
---|
1150 | for the ORCA4, ORCA2 and ORCA05 configurations. |
---|
1151 | If \np{ln\_old\_31\_lev\_code} is set then the depth variation of the coeffients will be specified as |
---|
1152 | a function of the model number. This option is included to allow backwards compatability of the ORCA2 reference |
---|
1153 | configurations with previous model versions. |
---|
1154 | \np{ln\_coast} specifies that the restoration coefficient should be reduced near to coastlines. |
---|
1155 | This option only has an effect if \np{ln\_full\_field} is true. |
---|
1156 | \np{ln\_zero\_top\_layer} specifies that the restoration coefficient should be zero in the surface layer. |
---|
1157 | Finally \np{ln\_custom} specifies that the custom module will be called. |
---|
1158 | This module is contained in the file custom.F90 and can be edited by users. For example damping could be applied in a specific region. |
---|
1159 | |
---|
1160 | The restoration coefficient can be set to zero in equatorial regions by specifying a positive value of \np{nn\_hdmp}. |
---|
1161 | Equatorward of this latitude the restoration coefficient will be zero with a smooth transition to |
---|
1162 | the full values of a 10\deg latitud band. |
---|
1163 | This is often used because of the short adjustment time scale in the equatorial region |
---|
1164 | \citep{Reverdin1991, Fujio1991, Marti_PhD92}. The time scale associated with the damping depends on the depth as a |
---|
1165 | hyperbolic tangent, with \np{rn\_surf} as surface value, \np{rn\_bot} as bottom value and a transition depth of \np{rn\_dep}. |
---|
1166 | |
---|
1167 | % ================================================================ |
---|
1168 | % Tracer time evolution |
---|
1169 | % ================================================================ |
---|
1170 | \section [Tracer time evolution (\textit{tranxt})] |
---|
1171 | {Tracer time evolution (\mdl{tranxt})} |
---|
1172 | \label{TRA_nxt} |
---|
1173 | %--------------------------------------------namdom----------------------------------------------------- |
---|
1174 | \namdisplay{namdom} |
---|
1175 | %-------------------------------------------------------------------------------------------------------------- |
---|
1176 | |
---|
1177 | Options are defined through the \ngn{namdom} namelist variables. |
---|
1178 | The general framework for tracer time stepping is a modified leap-frog scheme |
---|
1179 | \citep{Leclair_Madec_OM09}, $i.e.$ a three level centred time scheme associated |
---|
1180 | with a Asselin time filter (cf. \S\ref{STP_mLF}): |
---|
1181 | \begin{equation} \label{Eq_tra_nxt} |
---|
1182 | \begin{aligned} |
---|
1183 | (e_{3t}T)^{t+\rdt} &= (e_{3t}T)_f^{t-\rdt} &+ 2 \, \rdt \,e_{3t}^t\ \text{RHS}^t & \\ |
---|
1184 | \\ |
---|
1185 | (e_{3t}T)_f^t \;\ \quad &= (e_{3t}T)^t \;\quad |
---|
1186 | &+\gamma \,\left[ {(e_{3t}T)_f^{t-\rdt} -2(e_{3t}T)^t+(e_{3t}T)^{t+\rdt}} \right] & \\ |
---|
1187 | & &- \gamma\,\rdt \, \left[ Q^{t+\rdt/2} - Q^{t-\rdt/2} \right] & |
---|
1188 | \end{aligned} |
---|
1189 | \end{equation} |
---|
1190 | where RHS is the right hand side of the temperature equation, |
---|
1191 | the subscript $f$ denotes filtered values, $\gamma$ is the Asselin coefficient, |
---|
1192 | and $S$ is the total forcing applied on $T$ ($i.e.$ fluxes plus content in mass exchanges). |
---|
1193 | $\gamma$ is initialized as \np{rn\_atfp} (\textbf{namelist} parameter). |
---|
1194 | Its default value is \np{rn\_atfp}=$10^{-3}$. Note that the forcing correction term in the filter |
---|
1195 | is not applied in linear free surface (\jp{lk\_vvl}=false) (see \S\ref{TRA_sbc}. |
---|
1196 | Not also that in constant volume case, the time stepping is performed on $T$, |
---|
1197 | not on its content, $e_{3t}T$. |
---|
1198 | |
---|
1199 | When the vertical mixing is solved implicitly, the update of the \textit{next} tracer |
---|
1200 | fields is done in module \mdl{trazdf}. In this case only the swapping of arrays |
---|
1201 | and the Asselin filtering is done in the \mdl{tranxt} module. |
---|
1202 | |
---|
1203 | In order to prepare for the computation of the \textit{next} time step, |
---|
1204 | a swap of tracer arrays is performed: $T^{t-\rdt} = T^t$ and $T^t = T_f$. |
---|
1205 | |
---|
1206 | % ================================================================ |
---|
1207 | % Equation of State (eosbn2) |
---|
1208 | % ================================================================ |
---|
1209 | \section [Equation of State (\textit{eosbn2}) ] |
---|
1210 | {Equation of State (\mdl{eosbn2}) } |
---|
1211 | \label{TRA_eosbn2} |
---|
1212 | %--------------------------------------------nameos----------------------------------------------------- |
---|
1213 | \namdisplay{nameos} |
---|
1214 | %-------------------------------------------------------------------------------------------------------------- |
---|
1215 | |
---|
1216 | % ------------------------------------------------------------------------------------------------------------- |
---|
1217 | % Equation of State |
---|
1218 | % ------------------------------------------------------------------------------------------------------------- |
---|
1219 | \subsection{Equation Of Seawater (\np{nn\_eos} = -1, 0, or 1)} |
---|
1220 | \label{TRA_eos} |
---|
1221 | |
---|
1222 | The Equation Of Seawater (EOS) is an empirical nonlinear thermodynamic relationship |
---|
1223 | linking seawater density, $\rho$, to a number of state variables, |
---|
1224 | most typically temperature, salinity and pressure. |
---|
1225 | Because density gradients control the pressure gradient force through the hydrostatic balance, |
---|
1226 | the equation of state provides a fundamental bridge between the distribution of active tracers |
---|
1227 | and the fluid dynamics. Nonlinearities of the EOS are of major importance, in particular |
---|
1228 | influencing the circulation through determination of the static stability below the mixed layer, |
---|
1229 | thus controlling rates of exchange between the atmosphere and the ocean interior \citep{Roquet_JPO2015}. |
---|
1230 | Therefore an accurate EOS based on either the 1980 equation of state (EOS-80, \cite{UNESCO1983}) |
---|
1231 | or TEOS-10 \citep{TEOS10} standards should be used anytime a simulation of the real |
---|
1232 | ocean circulation is attempted \citep{Roquet_JPO2015}. |
---|
1233 | The use of TEOS-10 is highly recommended because |
---|
1234 | \textit{(i)} it is the new official EOS, |
---|
1235 | \textit{(ii)} it is more accurate, being based on an updated database of laboratory measurements, and |
---|
1236 | \textit{(iii)} it uses Conservative Temperature and Absolute Salinity (instead of potential temperature |
---|
1237 | and practical salinity for EOS-980, both variables being more suitable for use as model variables |
---|
1238 | \citep{TEOS10, Graham_McDougall_JPO13}. |
---|
1239 | EOS-80 is an obsolescent feature of the NEMO system, kept only for backward compatibility. |
---|
1240 | For process studies, it is often convenient to use an approximation of the EOS. To that purposed, |
---|
1241 | a simplified EOS (S-EOS) inspired by \citet{Vallis06} is also available. |
---|
1242 | |
---|
1243 | In the computer code, a density anomaly, $d_a= \rho / \rho_o - 1$, |
---|
1244 | is computed, with $\rho_o$ a reference density. Called \textit{rau0} |
---|
1245 | in the code, $\rho_o$ is set in \mdl{phycst} to a value of $1,026~Kg/m^3$. |
---|
1246 | This is a sensible choice for the reference density used in a Boussinesq ocean |
---|
1247 | climate model, as, with the exception of only a small percentage of the ocean, |
---|
1248 | density in the World Ocean varies by no more than 2$\%$ from that value \citep{Gill1982}. |
---|
1249 | |
---|
1250 | Options are defined through the \ngn{nameos} namelist variables, and in particular \np{nn\_eos} |
---|
1251 | which controls the EOS used (=-1 for TEOS10 ; =0 for EOS-80 ; =1 for S-EOS). |
---|
1252 | \begin{description} |
---|
1253 | |
---|
1254 | \item[\np{nn\_eos}$=-1$] the polyTEOS10-bsq equation of seawater \citep{Roquet_OM2015} is used. |
---|
1255 | The accuracy of this approximation is comparable to the TEOS-10 rational function approximation, |
---|
1256 | but it is optimized for a boussinesq fluid and the polynomial expressions have simpler |
---|
1257 | and more computationally efficient expressions for their derived quantities |
---|
1258 | which make them more adapted for use in ocean models. |
---|
1259 | Note that a slightly higher precision polynomial form is now used replacement of the TEOS-10 |
---|
1260 | rational function approximation for hydrographic data analysis \citep{TEOS10}. |
---|
1261 | A key point is that conservative state variables are used: |
---|
1262 | Absolute Salinity (unit: g/kg, notation: $S_A$) and Conservative Temperature (unit: \degC, notation: $\Theta$). |
---|
1263 | The pressure in decibars is approximated by the depth in meters. |
---|
1264 | With TEOS10, the specific heat capacity of sea water, $C_p$, is a constant. It is set to |
---|
1265 | $C_p=3991.86795711963~J\,Kg^{-1}\,^{\circ}K^{-1}$, according to \citet{TEOS10}. |
---|
1266 | |
---|
1267 | Choosing polyTEOS10-bsq implies that the state variables used by the model are |
---|
1268 | $\Theta$ and $S_A$. In particular, the initial state deined by the user have to be given as |
---|
1269 | \textit{Conservative} Temperature and \textit{Absolute} Salinity. |
---|
1270 | In addition, setting \np{ln\_useCT} to \textit{true} convert the Conservative SST to potential SST |
---|
1271 | prior to either computing the air-sea and ice-sea fluxes (forced mode) |
---|
1272 | or sending the SST field to the atmosphere (coupled mode). |
---|
1273 | |
---|
1274 | \item[\np{nn\_eos}$=0$] the polyEOS80-bsq equation of seawater is used. |
---|
1275 | It takes the same polynomial form as the polyTEOS10, but the coefficients have been optimized |
---|
1276 | to accurately fit EOS80 (Roquet, personal comm.). The state variables used in both the EOS80 |
---|
1277 | and the ocean model are: |
---|
1278 | the Practical Salinity ((unit: psu, notation: $S_p$)) and Potential Temperature (unit: $^{\circ}C$, notation: $\theta$). |
---|
1279 | The pressure in decibars is approximated by the depth in meters. |
---|
1280 | With thsi EOS, the specific heat capacity of sea water, $C_p$, is a function of temperature, |
---|
1281 | salinity and pressure \citep{UNESCO1983}. Nevertheless, a severe assumption is made in order to |
---|
1282 | have a heat content ($C_p T_p$) which is conserved by the model: $C_p$ is set to a constant |
---|
1283 | value, the TEOS10 value. |
---|
1284 | |
---|
1285 | \item[\np{nn\_eos}$=1$] a simplified EOS (S-EOS) inspired by \citet{Vallis06} is chosen, |
---|
1286 | the coefficients of which has been optimized to fit the behavior of TEOS10 (Roquet, personal comm.) |
---|
1287 | (see also \citet{Roquet_JPO2015}). It provides a simplistic linear representation of both |
---|
1288 | cabbeling and thermobaricity effects which is enough for a proper treatment of the EOS |
---|
1289 | in theoretical studies \citep{Roquet_JPO2015}. |
---|
1290 | With such an equation of state there is no longer a distinction between |
---|
1291 | \textit{conservative} and \textit{potential} temperature, as well as between \textit{absolute} |
---|
1292 | and \textit{practical} salinity. |
---|
1293 | S-EOS takes the following expression: |
---|
1294 | \begin{equation} \label{Eq_tra_S-EOS} |
---|
1295 | \begin{split} |
---|
1296 | d_a(T,S,z) = ( & - a_0 \; ( 1 + 0.5 \; \lambda_1 \; T_a + \mu_1 \; z ) * T_a \\ |
---|
1297 | & + b_0 \; ( 1 - 0.5 \; \lambda_2 \; S_a - \mu_2 \; z ) * S_a \\ |
---|
1298 | & - \nu \; T_a \; S_a \; ) \; / \; \rho_o \\ |
---|
1299 | with \ \ T_a = T-10 \; ; & \; S_a = S-35 \; ;\; \rho_o = 1026~Kg/m^3 |
---|
1300 | \end{split} |
---|
1301 | \end{equation} |
---|
1302 | where the computer name of the coefficients as well as their standard value are given in \ref{Tab_SEOS}. |
---|
1303 | In fact, when choosing S-EOS, various approximation of EOS can be specified simply by changing |
---|
1304 | the associated coefficients. |
---|
1305 | Setting to zero the two thermobaric coefficients ($\mu_1$, $\mu_2$) remove thermobaric effect from S-EOS. |
---|
1306 | setting to zero the three cabbeling coefficients ($\lambda_1$, $\lambda_2$, $\nu$) remove cabbeling effect from S-EOS. |
---|
1307 | Keeping non-zero value to $a_0$ and $b_0$ provide a linear EOS function of T and S. |
---|
1308 | |
---|
1309 | \end{description} |
---|
1310 | |
---|
1311 | |
---|
1312 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
---|
1313 | \begin{table}[!tb] |
---|
1314 | \begin{center} \begin{tabular}{|p{26pt}|p{72pt}|p{56pt}|p{136pt}|} |
---|
1315 | \hline |
---|
1316 | coeff. & computer name & S-EOS & description \\ \hline |
---|
1317 | $a_0$ & \np{rn\_a0} & 1.6550 $10^{-1}$ & linear thermal expansion coeff. \\ \hline |
---|
1318 | $b_0$ & \np{rn\_b0} & 7.6554 $10^{-1}$ & linear haline expansion coeff. \\ \hline |
---|
1319 | $\lambda_1$ & \np{rn\_lambda1}& 5.9520 $10^{-2}$ & cabbeling coeff. in $T^2$ \\ \hline |
---|
1320 | $\lambda_2$ & \np{rn\_lambda2}& 5.4914 $10^{-4}$ & cabbeling coeff. in $S^2$ \\ \hline |
---|
1321 | $\nu$ & \np{rn\_nu} & 2.4341 $10^{-3}$ & cabbeling coeff. in $T \, S$ \\ \hline |
---|
1322 | $\mu_1$ & \np{rn\_mu1} & 1.4970 $10^{-4}$ & thermobaric coeff. in T \\ \hline |
---|
1323 | $\mu_2$ & \np{rn\_mu2} & 1.1090 $10^{-5}$ & thermobaric coeff. in S \\ \hline |
---|
1324 | \end{tabular} |
---|
1325 | \caption{ \label{Tab_SEOS} |
---|
1326 | Standard value of S-EOS coefficients. } |
---|
1327 | \end{center} |
---|
1328 | \end{table} |
---|
1329 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
---|
1330 | |
---|
1331 | |
---|
1332 | % ------------------------------------------------------------------------------------------------------------- |
---|
1333 | % Brunt-V\"{a}is\"{a}l\"{a} Frequency |
---|
1334 | % ------------------------------------------------------------------------------------------------------------- |
---|
1335 | \subsection{Brunt-V\"{a}is\"{a}l\"{a} Frequency (\np{nn\_eos} = 0, 1 or 2)} |
---|
1336 | \label{TRA_bn2} |
---|
1337 | |
---|
1338 | An accurate computation of the ocean stability (i.e. of $N$, the brunt-V\"{a}is\"{a}l\"{a} |
---|
1339 | frequency) is of paramount importance as determine the ocean stratification and |
---|
1340 | is used in several ocean parameterisations (namely TKE, GLS, Richardson number dependent |
---|
1341 | vertical diffusion, enhanced vertical diffusion, non-penetrative convection, tidal mixing |
---|
1342 | parameterisation, iso-neutral diffusion). In particular, $N^2$ has to be computed at the local pressure |
---|
1343 | (pressure in decibar being approximated by the depth in meters). The expression for $N^2$ |
---|
1344 | is given by: |
---|
1345 | \begin{equation} \label{Eq_tra_bn2} |
---|
1346 | N^2 = \frac{g}{e_{3w}} \left( \beta \;\delta_{k+1/2}[S] - \alpha \;\delta_{k+1/2}[T] \right) |
---|
1347 | \end{equation} |
---|
1348 | where $(T,S) = (\Theta, S_A)$ for TEOS10, $= (\theta, S_p)$ for TEOS-80, or $=(T,S)$ for S-EOS, |
---|
1349 | and, $\alpha$ and $\beta$ are the thermal and haline expansion coefficients. |
---|
1350 | The coefficients are a polynomial function of temperature, salinity and depth which expression |
---|
1351 | depends on the chosen EOS. They are computed through \textit{eos\_rab}, a \textsc{Fortran} |
---|
1352 | function that can be found in \mdl{eosbn2}. |
---|
1353 | |
---|
1354 | % ------------------------------------------------------------------------------------------------------------- |
---|
1355 | % Freezing Point of Seawater |
---|
1356 | % ------------------------------------------------------------------------------------------------------------- |
---|
1357 | \subsection [Freezing Point of Seawater] |
---|
1358 | {Freezing Point of Seawater} |
---|
1359 | \label{TRA_fzp} |
---|
1360 | |
---|
1361 | The freezing point of seawater is a function of salinity and pressure \citep{UNESCO1983}: |
---|
1362 | \begin{equation} \label{Eq_tra_eos_fzp} |
---|
1363 | \begin{split} |
---|
1364 | T_f (S,p) = \left( -0.0575 + 1.710523 \;10^{-3} \, \sqrt{S} |
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1365 | - 2.154996 \;10^{-4} \,S \right) \ S \\ |
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1366 | - 7.53\,10^{-3} \ \ p |
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1367 | \end{split} |
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1368 | \end{equation} |
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1369 | |
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1370 | \eqref{Eq_tra_eos_fzp} is only used to compute the potential freezing point of |
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1371 | sea water ($i.e.$ referenced to the surface $p=0$), thus the pressure dependent |
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1372 | terms in \eqref{Eq_tra_eos_fzp} (last term) have been dropped. The freezing |
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1373 | point is computed through \textit{eos\_fzp}, a \textsc{Fortran} function that can be found |
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1374 | in \mdl{eosbn2}. |
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1375 | |
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1376 | |
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1377 | % ------------------------------------------------------------------------------------------------------------- |
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1378 | % Potential Energy |
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1379 | % ------------------------------------------------------------------------------------------------------------- |
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1380 | %\subsection{Potential Energy anomalies} |
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1381 | %\label{TRA_bn2} |
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1382 | |
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1383 | % =====>>>>> TO BE written |
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1384 | % |
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1385 | |
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1386 | |
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1387 | % ================================================================ |
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1388 | % Horizontal Derivative in zps-coordinate |
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1389 | % ================================================================ |
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1390 | \section [Horizontal Derivative in \textit{zps}-coordinate (\textit{zpshde})] |
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1391 | {Horizontal Derivative in \textit{zps}-coordinate (\mdl{zpshde})} |
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1392 | \label{TRA_zpshde} |
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1393 | |
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1394 | \gmcomment{STEVEN: to be consistent with earlier discussion of differencing and averaging operators, |
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1395 | I've changed "derivative" to "difference" and "mean" to "average"} |
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1396 | |
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1397 | With partial cells (\np{ln\_zps}=true) at bottom and top (\np{ln\_isfcav}=true), in general, |
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1398 | tracers in horizontally adjacent cells live at different depths. |
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1399 | Horizontal gradients of tracers are needed for horizontal diffusion (\mdl{traldf} module) |
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1400 | and the hydrostatic pressure gradient calculations (\mdl{dynhpg} module). |
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1401 | The partial cell properties at the top (\np{ln\_isfcav}=true) are computed in the same way as for the bottom. |
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1402 | So, only the bottom interpolation is explained below. |
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1403 | |
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1404 | Before taking horizontal gradients between the tracers next to the bottom, a linear |
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1405 | interpolation in the vertical is used to approximate the deeper tracer as if it actually |
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1406 | lived at the depth of the shallower tracer point (Fig.~\ref{Fig_Partial_step_scheme}). |
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1407 | For example, for temperature in the $i$-direction the needed interpolated |
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1408 | temperature, $\widetilde{T}$, is: |
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1409 | |
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1410 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
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1411 | \begin{figure}[!p] \begin{center} |
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1412 | \includegraphics[width=0.9\textwidth]{Partial_step_scheme} |
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1413 | \caption{ \label{Fig_Partial_step_scheme} |
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1414 | Discretisation of the horizontal difference and average of tracers in the $z$-partial |
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1415 | step coordinate (\np{ln\_zps}=true) in the case $( e3w_k^{i+1} - e3w_k^i )>0$. |
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1416 | A linear interpolation is used to estimate $\widetilde{T}_k^{i+1}$, the tracer value |
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1417 | at the depth of the shallower tracer point of the two adjacent bottom $T$-points. |
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1418 | The horizontal difference is then given by: $\delta _{i+1/2} T_k= \widetilde{T}_k^{\,i+1} -T_k^{\,i}$ |
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1419 | and the average by: $\overline{T}_k^{\,i+1/2}= ( \widetilde{T}_k^{\,i+1/2} - T_k^{\,i} ) / 2$. } |
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1420 | \end{center} \end{figure} |
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1421 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
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1422 | \begin{equation*} |
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1423 | \widetilde{T}= \left\{ \begin{aligned} |
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1424 | &T^{\,i+1} -\frac{ \left( e_{3w}^{i+1} -e_{3w}^i \right)}{ e_{3w}^{i+1} }\;\delta _k T^{i+1} |
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1425 | && \quad\text{if $\ e_{3w}^{i+1} \geq e_{3w}^i$ } \\ |
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1426 | \\ |
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1427 | &T^{\,i} \ \ \ \,+\frac{ \left( e_{3w}^{i+1} -e_{3w}^i \right) }{e_{3w}^i }\;\delta _k T^{i+1} |
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1428 | && \quad\text{if $\ e_{3w}^{i+1} < e_{3w}^i$ } |
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1429 | \end{aligned} \right. |
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1430 | \end{equation*} |
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1431 | and the resulting forms for the horizontal difference and the horizontal average |
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1432 | value of $T$ at a $U$-point are: |
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1433 | \begin{equation} \label{Eq_zps_hde} |
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1434 | \begin{aligned} |
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1435 | \delta _{i+1/2} T= \begin{cases} |
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1436 | \ \ \ \widetilde {T}\quad\ -T^i & \ \ \quad\quad\text{if $\ e_{3w}^{i+1} \geq e_{3w}^i$ } \\ |
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1437 | \\ |
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1438 | \ \ \ T^{\,i+1}-\widetilde{T} & \ \ \quad\quad\text{if $\ e_{3w}^{i+1} < e_{3w}^i$ } |
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1439 | \end{cases} \\ |
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1440 | \\ |
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1441 | \overline {T}^{\,i+1/2} \ = \begin{cases} |
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1442 | ( \widetilde {T}\ \ \;\,-T^{\,i}) / 2 & \;\ \ \quad\text{if $\ e_{3w}^{i+1} \geq e_{3w}^i$ } \\ |
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1443 | \\ |
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1444 | ( T^{\,i+1}-\widetilde{T} ) / 2 & \;\ \ \quad\text{if $\ e_{3w}^{i+1} < e_{3w}^i$ } |
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1445 | \end{cases} |
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1446 | \end{aligned} |
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1447 | \end{equation} |
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1448 | |
---|
1449 | The computation of horizontal derivative of tracers as well as of density is |
---|
1450 | performed once for all at each time step in \mdl{zpshde} module and stored |
---|
1451 | in shared arrays to be used when needed. It has to be emphasized that the |
---|
1452 | procedure used to compute the interpolated density, $\widetilde{\rho}$, is not |
---|
1453 | the same as that used for $T$ and $S$. Instead of forming a linear approximation |
---|
1454 | of density, we compute $\widetilde{\rho }$ from the interpolated values of $T$ |
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1455 | and $S$, and the pressure at a $u$-point (in the equation of state pressure is |
---|
1456 | approximated by depth, see \S\ref{TRA_eos} ) : |
---|
1457 | \begin{equation} \label{Eq_zps_hde_rho} |
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1458 | \widetilde{\rho } = \rho ( {\widetilde{T},\widetilde {S},z_u }) |
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1459 | \quad \text{where }\ z_u = \min \left( {z_T^{i+1} ,z_T^i } \right) |
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1460 | \end{equation} |
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1461 | |
---|
1462 | This is a much better approximation as the variation of $\rho$ with depth (and |
---|
1463 | thus pressure) is highly non-linear with a true equation of state and thus is badly |
---|
1464 | approximated with a linear interpolation. This approximation is used to compute |
---|
1465 | both the horizontal pressure gradient (\S\ref{DYN_hpg}) and the slopes of neutral |
---|
1466 | surfaces (\S\ref{LDF_slp}) |
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1467 | |
---|
1468 | Note that in almost all the advection schemes presented in this Chapter, both |
---|
1469 | averaging and differencing operators appear. Yet \eqref{Eq_zps_hde} has not |
---|
1470 | been used in these schemes: in contrast to diffusion and pressure gradient |
---|
1471 | computations, no correction for partial steps is applied for advection. The main |
---|
1472 | motivation is to preserve the domain averaged mean variance of the advected |
---|
1473 | field when using the $2^{nd}$ order centred scheme. Sensitivity of the advection |
---|
1474 | schemes to the way horizontal averages are performed in the vicinity of partial |
---|
1475 | cells should be further investigated in the near future. |
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1476 | %%% |
---|
1477 | \gmcomment{gm : this last remark has to be done} |
---|
1478 | %%% |
---|
1479 | \end{document} |
---|