1 | % ================================================================ |
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2 | % Chapter 1 Ñ Ocean Tracers (TRA) |
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3 | % ================================================================ |
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4 | \chapter{Ocean Tracers (TRA)} |
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5 | \label{TRA} |
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6 | \minitoc |
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7 | |
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8 | % missing/update |
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9 | % traqsr: need to coordinate with SBC module |
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10 | |
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11 | %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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12 | |
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13 | %\newpage |
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14 | \vspace{2.cm} |
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15 | %$\ $\newline % force a new ligne |
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16 | |
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17 | Using the representation described in Chap.~\ref{DOM}, several semi-discrete |
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18 | space forms of the tracer equations are available depending on the vertical |
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19 | coordinate used and on the physics used. In all the equations presented |
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20 | here, the masking has been omitted for simplicity. One must be aware that |
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21 | all the quantities are masked fields and that each time a mean or difference |
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22 | operator is used, the resulting field is multiplied by a mask. |
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23 | |
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24 | The two active tracers are potential temperature and salinity. Their prognostic |
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25 | equations can be summarized as follows: |
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26 | \begin{equation*} |
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27 | \text{NXT} = \text{ADV}+\text{LDF}+\text{ZDF}+\text{SBC} |
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28 | \ (+\text{QSR})\ (+\text{BBC})\ (+\text{BBL})\ (+\text{DMP}) |
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29 | \end{equation*} |
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30 | |
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31 | NXT stands for next, referring to the time-stepping. From left to right, the terms |
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32 | on the rhs of the tracer equations are the advection (ADV), the lateral diffusion |
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33 | (LDF), the vertical diffusion (ZDF), the contributions from the external forcings |
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34 | (SBC: Surface Boundary Condition, QSR: penetrative Solar Radiation, and BBC: |
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35 | Bottom Boundary Condition), the contribution from the bottom boundary Layer |
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36 | (BBL) parametrisation, and an internal damping (DMP) term. The terms QSR, |
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37 | BBC, BBL and DMP are optional. The external forcings and parameterisations |
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38 | require complex inputs and complex calculations (e.g. bulk formulae, estimation |
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39 | of mixing coefficients) that are carried out in the SBC, LDF and ZDF modules and |
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40 | described in chapters \S\ref{SBC}, \S\ref{LDF} and \S\ref{ZDF}, respectively. |
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41 | Note that \mdl{tranpc}, the non-penetrative convection module, although |
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42 | (temporarily) located in the NEMO/OPA/TRA directory, is described with the |
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43 | model vertical physics (ZDF). |
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44 | %%% |
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45 | \gmcomment{change the position of eosbn2 in the reference code} |
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46 | %%% |
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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-Vais\"{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 |
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53 | CPP keys. 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 (\key{trdtra} is defined), as described in Chap.~\ref{MISC}. |
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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 | The advection tendency of a tracer in flux form is the divergence of the advective |
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73 | fluxes. Its discrete expression is given by : |
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74 | \begin{equation} \label{Eq_tra_adv} |
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75 | ADV_\tau =-\frac{1}{b_t} \left( |
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76 | \;\delta _i \left[ e_{2u}\,e_{3u} \; u\; \tau _u \right] |
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77 | +\delta _j \left[ e_{1v}\,e_{3v} \; v\; \tau _v \right] \; \right) |
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78 | -\frac{1}{e_{3t}} \;\delta _k \left[ w\; \tau _w \right] |
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79 | \end{equation} |
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80 | 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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81 | The flux form in \eqref{Eq_tra_adv} |
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82 | implicitly requires the use of the continuity equation. Indeed, it is obtained |
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83 | by using the following equality : $\nabla \cdot \left( \vect{U}\,T \right)=\vect{U} \cdot \nabla T$ |
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84 | which results from the use of the continuity equation, $\nabla \cdot \vect{U}=0$ or |
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85 | $ \partial _t e_3 + e_3\;\nabla \cdot \vect{U}=0$ in constant volume or variable volume case, respectively. |
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86 | Therefore it is of paramount importance to design the discrete analogue of the |
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87 | advection tendency so that it is consistent with the continuity equation in order to |
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88 | enforce the conservation properties of the continuous equations. In other words, |
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89 | by replacing $\tau$ by the number 1 in (\ref{Eq_tra_adv}) we recover the discrete form of |
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90 | the continuity equation which is used to calculate the vertical velocity. |
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91 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
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92 | \begin{figure}[!t] \begin{center} |
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93 | \includegraphics[width=0.9\textwidth]{./TexFiles/Figures/Fig_adv_scheme.pdf} |
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94 | \caption{ \label{Fig_adv_scheme} |
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95 | Schematic representation of some ways used to evaluate the tracer value |
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96 | at $u$-point and the amount of tracer exchanged between two neighbouring grid |
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97 | points. Upsteam biased scheme (ups): the upstream value is used and the black |
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98 | area is exchanged. Piecewise parabolic method (ppm): a parabolic interpolation |
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99 | is used and the black and dark grey areas are exchanged. Monotonic upstream |
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100 | scheme for conservative laws (muscl): a parabolic interpolation is used and black, |
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101 | dark grey and grey areas are exchanged. Second order scheme (cen2): the mean |
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102 | value is used and black, dark grey, grey and light grey areas are exchanged. Note |
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103 | that this illustration does not include the flux limiter used in ppm and muscl schemes.} |
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104 | \end{center} \end{figure} |
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105 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
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106 | |
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107 | The key difference between the advection schemes available in \NEMO is the choice |
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108 | made in space and time interpolation to define the value of the tracer at the |
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109 | velocity points (Fig.~\ref{Fig_adv_scheme}). |
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110 | |
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111 | Along solid lateral and bottom boundaries a zero tracer flux is automatically |
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112 | specified, since the normal velocity is zero there. At the sea surface the |
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113 | boundary condition depends on the type of sea surface chosen: |
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114 | \begin{description} |
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115 | \item [linear free surface:] the first level thickness is constant in time: |
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116 | the vertical boundary condition is applied at the fixed surface $z=0$ |
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117 | rather than on the moving surface $z=\eta$. There is a non-zero advective |
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118 | flux which is set for all advection schemes as |
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119 | $\left. {\tau _w } \right|_{k=1/2} =T_{k=1} $, $i.e.$ |
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120 | the product of surface velocity (at $z=0$) by the first level tracer value. |
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121 | \item [non-linear free surface:] (\key{vvl} is defined) |
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122 | convergence/divergence in the first ocean level moves the free surface |
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123 | up/down. There is no tracer advection through it so that the advective |
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124 | fluxes through the surface are also zero |
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125 | \end{description} |
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126 | In all cases, this boundary condition retains local conservation of tracer. |
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127 | Global conservation is obtained in both rigid-lid and non-linear free surface |
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128 | cases, but not in the linear free surface case. Nevertheless, in the latter |
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129 | case, it is achieved to a good approximation since the non-conservative |
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130 | term is the product of the time derivative of the tracer and the free surface |
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131 | height, two quantities that are not correlated (see \S\ref{PE_free_surface}, |
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132 | and also \citet{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{eulerian} ocean velocity (see Chap.~\ref{DYN}). |
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136 | When eddy induced velocity (\textit{eiv}) parameterisation is used it is the \textit{now} |
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137 | \textit{effective} velocity ($i.e.$ the sum of the eulerian and eiv velocities) which is used. |
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138 | |
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139 | The choice of an advection scheme is made in the \textit{nam\_traadv} namelist, by |
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140 | setting to \textit{true} one and only one of the logicals \textit{ln\_traadv\_xxx}. The |
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141 | corresponding code can be found in the \textit{traadv\_xxx.F90} module, where |
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142 | \textit{xxx} is a 3 or 4 letter acronym corresponding to each scheme. Details |
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143 | of the advection schemes are given below. The choice of an advection scheme |
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144 | is a complex matter which depends on the model physics, model resolution, |
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145 | type of tracer, as well as the issue of numerical cost. |
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146 | |
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147 | Note that |
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148 | (1) cen2, cen4 and TVD schemes require an explicit diffusion |
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149 | operator while the other schemes are diffusive enough so that they do not |
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150 | require additional diffusion ; |
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151 | (2) cen2, cen4, MUSCL2, and UBS are not \textit{positive} schemes |
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152 | \footnote{negative values can appear in an initially strictly positive tracer field |
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153 | which is advected} |
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154 | , implying that false extrema are permitted. Their use is not recommended on passive tracers ; |
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155 | (3) It is recommended that the same advection-diffusion scheme is |
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156 | used on both active and passive tracers. Indeed, if a source or sink of a |
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157 | passive tracer depends on an active one, the difference of treatment of |
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158 | active and passive tracers can create very nice-looking frontal structures |
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159 | that are pure numerical artefacts. Nevertheless, most of our users set a different |
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160 | treatment on passive and active tracers, that's the reason why this possibility |
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161 | is offered. We strongly suggest them to perform a sensitivity experiment |
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162 | using a same treatment to assess the robustness of their results. |
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163 | |
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164 | % ------------------------------------------------------------------------------------------------------------- |
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165 | % 2nd order centred scheme |
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166 | % ------------------------------------------------------------------------------------------------------------- |
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167 | \subsection [$2^{nd}$ order centred scheme (cen2) (\np{ln\_traadv\_cen2})] |
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168 | {$2^{nd}$ order centred scheme (cen2) (\np{ln\_traadv\_cen2}=true)} |
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169 | \label{TRA_adv_cen2} |
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170 | |
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171 | In the centred second order formulation, the tracer at velocity points is |
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172 | evaluated as the mean of the two neighbouring $T$-point values. |
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173 | For example, in the $i$-direction : |
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174 | \begin{equation} \label{Eq_tra_adv_cen2} |
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175 | \tau _u^{cen2} =\overline T ^{i+1/2} |
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176 | \end{equation} |
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177 | |
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178 | The scheme is non diffusive ($i.e.$ it conserves the tracer variance, $\tau^2)$ |
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179 | but dispersive ($i.e.$ it may create false extrema). It is therefore notoriously |
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180 | noisy and must be used in conjunction with an explicit diffusion operator to |
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181 | produce a sensible solution. The associated time-stepping is performed using |
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182 | a leapfrog scheme in conjunction with an Asselin time-filter, so $T$ in |
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183 | (\ref{Eq_tra_adv_cen2}) is the \textit{now} tracer value. The centered second |
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184 | order advection is computed in the \mdl{traadv\_cen2} module. In this module, |
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185 | it is advantageous to combine the \textit{cen2} scheme with an upstream scheme |
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186 | in specific areas which require a strong diffusion in order to avoid the generation |
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187 | of false extrema. These areas are the vicinity of large river mouths, some straits |
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188 | with coarse resolution, and the vicinity of ice cover area ($i.e.$ when the ocean |
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189 | temperature is close to the freezing point). |
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190 | This combined scheme has been included for specific grid points in the ORCA2 |
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191 | and ORCA4 configurations only. This is an obsolescent feature as the recommended |
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192 | advection scheme for the ORCA configuration is TVD (see \S\ref{TRA_adv_tvd}). |
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193 | |
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194 | Note that using the cen2 scheme, the overall tracer advection is of second |
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195 | order accuracy since both (\ref{Eq_tra_adv}) and (\ref{Eq_tra_adv_cen2}) |
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196 | have this order of accuracy. \gmcomment{Note also that ... blah, blah} |
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197 | |
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198 | % ------------------------------------------------------------------------------------------------------------- |
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199 | % 4nd order centred scheme |
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200 | % ------------------------------------------------------------------------------------------------------------- |
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201 | \subsection [$4^{nd}$ order centred scheme (cen4) (\np{ln\_traadv\_cen4})] |
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202 | {$4^{nd}$ order centred scheme (cen4) (\np{ln\_traadv\_cen4}=true)} |
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203 | \label{TRA_adv_cen4} |
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204 | |
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205 | In the $4^{th}$ order formulation (to be implemented), tracer values are |
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206 | evaluated at velocity points as a $4^{th}$ order interpolation, and thus depend on |
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207 | the four neighbouring $T$-points. For example, in the $i$-direction: |
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208 | \begin{equation} \label{Eq_tra_adv_cen4} |
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209 | \tau _u^{cen4} |
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210 | =\overline{ T - \frac{1}{6}\,\delta _i \left[ \delta_{i+1/2}[T] \,\right] }^{\,i+1/2} |
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211 | \end{equation} |
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212 | |
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213 | Strictly speaking, the cen4 scheme is not a $4^{th}$ order advection scheme |
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214 | but a $4^{th}$ order evaluation of advective fluxes, since the divergence of |
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215 | advective fluxes \eqref{Eq_tra_adv} is kept at $2^{nd}$ order. The phrase ``$4^{th}$ |
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216 | order scheme'' used in oceanographic literature is usually associated |
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217 | with the scheme presented here. Introducing a \textit{true} $4^{th}$ order advection |
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218 | scheme is feasible but, for consistency reasons, it requires changes in the |
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219 | discretisation of the tracer advection together with changes in both the |
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220 | continuity equation and the momentum advection terms. |
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221 | |
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222 | A direct consequence of the pseudo-fourth order nature of the scheme is that |
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223 | it is not non-diffusive, i.e. the global variance of a tracer is not preserved using |
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224 | \textit{cen4}. Furthermore, it must be used in conjunction with an explicit |
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225 | diffusion operator to produce a sensible solution. The time-stepping is also |
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226 | performed using a leapfrog scheme in conjunction with an Asselin time-filter, |
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227 | so $T$ in (\ref{Eq_tra_adv_cen4}) is the \textit{now} tracer. |
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228 | |
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229 | At a $T$-grid cell adjacent to a boundary (coastline, bottom and surface), an |
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230 | additional hypothesis must be made to evaluate $\tau _u^{cen4}$. This |
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231 | hypothesis usually reduces the order of the scheme. Here we choose to set |
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232 | the gradient of $T$ across the boundary to zero. Alternative conditions can be |
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233 | specified, such as a reduction to a second order scheme for these near boundary |
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234 | grid points. |
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235 | |
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236 | % ------------------------------------------------------------------------------------------------------------- |
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237 | % TVD scheme |
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238 | % ------------------------------------------------------------------------------------------------------------- |
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239 | \subsection [Total Variance Dissipation scheme (TVD) (\np{ln\_traadv\_tvd})] |
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240 | {Total Variance Dissipation scheme (TVD) (\np{ln\_traadv\_tvd}=true)} |
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241 | \label{TRA_adv_tvd} |
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242 | |
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243 | In the Total Variance Dissipation (TVD) formulation, the tracer at velocity |
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244 | points is evaluated using a combination of an upstream and a centred scheme. |
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245 | For example, in the $i$-direction : |
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246 | \begin{equation} \label{Eq_tra_adv_tvd} |
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247 | \begin{split} |
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248 | \tau _u^{ups}&= \begin{cases} |
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249 | T_{i+1} & \text{if $\ u_{i+1/2} < 0$} \hfill \\ |
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250 | T_i & \text{if $\ u_{i+1/2} \geq 0$} \hfill \\ |
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251 | \end{cases} \\ |
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252 | \\ |
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253 | \tau _u^{tvd}&=\tau _u^{ups} +c_u \;\left( {\tau _u^{cen2} -\tau _u^{ups} } \right) |
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254 | \end{split} |
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255 | \end{equation} |
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256 | where $c_u$ is a flux limiter function taking values between 0 and 1. |
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257 | There exist many ways to define $c_u$, each corresponding to a different |
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258 | total variance decreasing scheme. The one chosen in \NEMO is described in |
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259 | \citet{Zalesak_JCP79}. $c_u$ only departs from $1$ when the advective term |
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260 | produces a local extremum in the tracer field. The resulting scheme is quite |
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261 | expensive but \emph{positive}. It can be used on both active and passive tracers. |
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262 | This scheme is tested and compared with MUSCL and the MPDATA scheme in |
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263 | \citet{Levy_al_GRL01}; note that in this paper it is referred to as "FCT" (Flux corrected |
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264 | transport) rather than TVD. The TVD scheme is implemented in the \mdl{traadv\_tvd} module. |
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265 | |
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266 | For stability reasons (see \S\ref{STP}), |
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267 | $\tau _u^{cen2}$ is evaluated in (\ref{Eq_tra_adv_tvd}) using the \textit{now} tracer while $\tau _u^{ups}$ |
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268 | is evaluated using the \textit{before} tracer. In other words, the advective part of |
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269 | the scheme is time stepped with a leap-frog scheme while a forward scheme is |
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270 | used for the diffusive part. |
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271 | |
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272 | % ------------------------------------------------------------------------------------------------------------- |
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273 | % MUSCL scheme |
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274 | % ------------------------------------------------------------------------------------------------------------- |
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275 | \subsection[MUSCL scheme (\np{ln\_traadv\_muscl})] |
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276 | {Monotone Upstream Scheme for Conservative Laws (MUSCL) (\np{ln\_traadv\_muscl}=T)} |
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277 | \label{TRA_adv_muscl} |
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278 | |
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279 | The Monotone Upstream Scheme for Conservative Laws (MUSCL) has been |
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280 | implemented by \citet{Levy_al_GRL01}. In its formulation, the tracer at velocity points |
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281 | is evaluated assuming a linear tracer variation between two $T$-points |
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282 | (Fig.\ref{Fig_adv_scheme}). For example, in the $i$-direction : |
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283 | \begin{equation} \label{Eq_tra_adv_muscl} |
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284 | \tau _u^{mus} = \left\{ \begin{aligned} |
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285 | &\tau _i &+ \frac{1}{2} \;\left( 1-\frac{u_{i+1/2} \;\rdt}{e_{1u}} \right) |
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286 | &\ \widetilde{\partial _i \tau} & \quad \text{if }\;u_{i+1/2} \geqslant 0 \\ |
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287 | &\tau _{i+1/2} &+\frac{1}{2}\;\left( 1+\frac{u_{i+1/2} \;\rdt}{e_{1u} } \right) |
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288 | &\ \widetilde{\partial_{i+1/2} \tau } & \text{if }\;u_{i+1/2} <0 |
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289 | \end{aligned} \right. |
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290 | \end{equation} |
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291 | where $\widetilde{\partial _i \tau}$ is the slope of the tracer on which a limitation |
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292 | is imposed to ensure the \textit{positive} character of the scheme. |
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293 | |
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294 | The time stepping is performed using a forward scheme, that is the \textit{before} |
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295 | tracer field is used to evaluate $\tau _u^{mus}$. |
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296 | |
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297 | For an ocean grid point adjacent to land and where the ocean velocity is |
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298 | directed toward land, two choices are available: an upstream flux |
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299 | (\np{ln\_traadv\_muscl}=true) or a second order flux |
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300 | (\np{ln\_traadv\_muscl2}=true). Note that the latter choice does not ensure |
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301 | the \textit{positive} character of the scheme. Only the former can be used |
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302 | on both active and passive tracers. The two MUSCL schemes are implemented |
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303 | in the \mdl{traadv\_tvd} and \mdl{traadv\_tvd2} modules. |
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304 | |
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305 | % ------------------------------------------------------------------------------------------------------------- |
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306 | % UBS scheme |
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307 | % ------------------------------------------------------------------------------------------------------------- |
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308 | \subsection [Upstream-Biased Scheme (UBS) (\np{ln\_traadv\_ubs})] |
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309 | {Upstream-Biased Scheme (UBS) (\np{ln\_traadv\_ubs}=true)} |
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310 | \label{TRA_adv_ubs} |
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311 | |
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312 | The UBS advection scheme is an upstream-biased third order scheme based on |
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313 | an upstream-biased parabolic interpolation. It is also known as the Cell |
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314 | Averaged QUICK scheme (Quadratic Upstream Interpolation for Convective |
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315 | Kinematics). For example, in the $i$-direction : |
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316 | \begin{equation} \label{Eq_tra_adv_ubs} |
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317 | \tau _u^{ubs} =\overline T ^{i+1/2}-\;\frac{1}{6} \left\{ |
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318 | \begin{aligned} |
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319 | &\tau"_i & \quad \text{if }\ u_{i+1/2} \geqslant 0 \\ |
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320 | &\tau"_{i+1} & \quad \text{if }\ u_{i+1/2} < 0 |
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321 | \end{aligned} \right. |
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322 | \end{equation} |
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323 | where $\tau "_i =\delta _i \left[ {\delta _{i+1/2} \left[ \tau \right]} \right]$. |
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324 | |
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325 | This results in a dissipatively dominant (i.e. hyper-diffusive) truncation |
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326 | error \citep{Shchepetkin_McWilliams_OM05}. The overall performance of the advection |
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327 | scheme is similar to that reported in \cite{Farrow1995}. |
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328 | It is a relatively good compromise between accuracy and smoothness. |
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329 | It is not a \emph{positive} scheme, meaning that false extrema are permitted, |
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330 | but the amplitude of such are significantly reduced over the centred second |
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331 | order method. Nevertheless it is not recommended that it should be applied |
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332 | to a passive tracer that requires positivity. |
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333 | |
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334 | The intrinsic diffusion of UBS makes its use risky in the vertical direction |
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335 | where the control of artificial diapycnal fluxes is of paramount importance. |
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336 | Therefore the vertical flux is evaluated using the TVD scheme when |
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337 | \np{ln\_traadv\_ubs}=true. |
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338 | |
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339 | For stability reasons (see \S\ref{STP}), |
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340 | the first term in \eqref{Eq_tra_adv_ubs} (which corresponds to a second order centred scheme) |
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341 | is evaluated using the \textit{now} tracer (centred in time) while the |
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342 | second term (which is the diffusive part of the scheme), is |
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343 | evaluated using the \textit{before} tracer (forward in time). |
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344 | This choice is discussed by \citet{Webb_al_JAOT98} in the context of the |
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345 | QUICK advection scheme. UBS and QUICK schemes only differ |
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346 | by one coefficient. Replacing 1/6 with 1/8 in \eqref{Eq_tra_adv_ubs} |
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347 | leads to the QUICK advection scheme \citep{Webb_al_JAOT98}. |
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348 | This option is not available through a namelist parameter, since the |
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349 | 1/6 coefficient is hard coded. Nevertheless it is quite easy to make the |
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350 | substitution in the \mdl{traadv\_ubs} module and obtain a QUICK scheme. |
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351 | |
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352 | Four different options are possible for the vertical |
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353 | component used in the UBS scheme. $\tau _w^{ubs}$ can be evaluated |
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354 | using either \textit{(a)} a centred $2^{nd}$ order scheme, or \textit{(b)} |
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355 | a TVD scheme, or \textit{(c)} an interpolation based on conservative |
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356 | parabolic splines following the \citet{Shchepetkin_McWilliams_OM05} |
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357 | implementation of UBS in ROMS, or \textit{(d)} a UBS. The $3^{rd}$ case |
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358 | has dispersion properties similar to an eighth-order accurate conventional scheme. |
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359 | The current reference version uses method b) |
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360 | |
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361 | Note that : |
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362 | |
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363 | (1) When a high vertical resolution $O(1m)$ is used, the model stability can |
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364 | be controlled by vertical advection (not vertical diffusion which is usually |
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365 | solved using an implicit scheme). Computer time can be saved by using a |
---|
366 | time-splitting technique on vertical advection. Such a technique has been |
---|
367 | implemented and validated in ORCA05 with 301 levels. It is not available |
---|
368 | in the current reference version. |
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369 | |
---|
370 | (2) It is straightforward to rewrite \eqref{Eq_tra_adv_ubs} as follows: |
---|
371 | \begin{equation} \label{Eq_traadv_ubs2} |
---|
372 | \tau _u^{ubs} = \tau _u^{cen4} + \frac{1}{12} \left\{ |
---|
373 | \begin{aligned} |
---|
374 | & + \tau"_i & \quad \text{if }\ u_{i+1/2} \geqslant 0 \\ |
---|
375 | & - \tau"_{i+1} & \quad \text{if }\ u_{i+1/2} < 0 |
---|
376 | \end{aligned} \right. |
---|
377 | \end{equation} |
---|
378 | or equivalently |
---|
379 | \begin{equation} \label{Eq_traadv_ubs2b} |
---|
380 | u_{i+1/2} \ \tau _u^{ubs} |
---|
381 | =u_{i+1/2} \ \overline{ T - \frac{1}{6}\,\delta _i\left[ \delta_{i+1/2}[T] \,\right] }^{\,i+1/2} |
---|
382 | - \frac{1}{2} |u|_{i+1/2} \;\frac{1}{6} \;\delta_{i+1/2}[\tau"_i] |
---|
383 | \end{equation} |
---|
384 | |
---|
385 | \eqref{Eq_traadv_ubs2} has several advantages. Firstly, it clearly reveals |
---|
386 | that the UBS scheme is based on the fourth order scheme to which an |
---|
387 | upstream-biased diffusion term is added. Secondly, this emphasises that the |
---|
388 | $4^{th}$ order part (as well as the $2^{nd}$ order part as stated above) has |
---|
389 | to be evaluated at the \emph{now} time step using \eqref{Eq_tra_adv_ubs}. |
---|
390 | Thirdly, the diffusion term is in fact a biharmonic operator with an eddy |
---|
391 | coefficient which is simply proportional to the velocity: |
---|
392 | $A_u^{lm}= - \frac{1}{12}\,{e_{1u}}^3\,|u|$. Note that NEMO v3.4 still uses |
---|
393 | \eqref{Eq_tra_adv_ubs}, not \eqref{Eq_traadv_ubs2}. |
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394 | %%% |
---|
395 | \gmcomment{the change in UBS scheme has to be done} |
---|
396 | %%% |
---|
397 | |
---|
398 | % ------------------------------------------------------------------------------------------------------------- |
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399 | % QCK scheme |
---|
400 | % ------------------------------------------------------------------------------------------------------------- |
---|
401 | \subsection [QUICKEST scheme (QCK) (\np{ln\_traadv\_qck})] |
---|
402 | {QUICKEST scheme (QCK) (\np{ln\_traadv\_qck}=true)} |
---|
403 | \label{TRA_adv_qck} |
---|
404 | |
---|
405 | The Quadratic Upstream Interpolation for Convective Kinematics with |
---|
406 | Estimated Streaming Terms (QUICKEST) scheme proposed by \citet{Leonard1979} |
---|
407 | is the third order Godunov scheme. It is associated with the ULTIMATE QUICKEST |
---|
408 | limiter \citep{Leonard1991}. It has been implemented in NEMO by G. Reffray |
---|
409 | (MERCATOR-ocean) and can be found in the \mdl{traadv\_qck} module. |
---|
410 | The resulting scheme is quite expensive but \emph{positive}. |
---|
411 | It can be used on both active and passive tracers. |
---|
412 | However, the intrinsic diffusion of QCK makes its use risky in the vertical |
---|
413 | direction where the control of artificial diapycnal fluxes is of paramount importance. |
---|
414 | Therefore the vertical flux is evaluated using the CEN2 scheme. |
---|
415 | This no longer guarantees the positivity of the scheme. The use of TVD in the vertical |
---|
416 | direction (as for the UBS case) should be implemented to restore this property. |
---|
417 | |
---|
418 | |
---|
419 | % ------------------------------------------------------------------------------------------------------------- |
---|
420 | % PPM scheme |
---|
421 | % ------------------------------------------------------------------------------------------------------------- |
---|
422 | \subsection [Piecewise Parabolic Method (PPM) (\np{ln\_traadv\_ppm})] |
---|
423 | {Piecewise Parabolic Method (PPM) (\np{ln\_traadv\_ppm}=true)} |
---|
424 | \label{TRA_adv_ppm} |
---|
425 | |
---|
426 | The Piecewise Parabolic Method (PPM) proposed by Colella and Woodward (1984) |
---|
427 | \sgacomment{reference?} |
---|
428 | is based on a quadradic piecewise construction. Like the QCK scheme, it is associated |
---|
429 | with the ULTIMATE QUICKEST limiter \citep{Leonard1991}. It has been implemented |
---|
430 | in \NEMO by G. Reffray (MERCATOR-ocean) but is not yet offered in the reference |
---|
431 | version 3.3. |
---|
432 | |
---|
433 | % ================================================================ |
---|
434 | % Tracer Lateral Diffusion |
---|
435 | % ================================================================ |
---|
436 | \section [Tracer Lateral Diffusion (\textit{traldf})] |
---|
437 | {Tracer Lateral Diffusion (\mdl{traldf})} |
---|
438 | \label{TRA_ldf} |
---|
439 | %-----------------------------------------nam_traldf------------------------------------------------------ |
---|
440 | \namdisplay{namtra_ldf} |
---|
441 | %------------------------------------------------------------------------------------------------------------- |
---|
442 | |
---|
443 | The options available for lateral diffusion are a laplacian (rotated or not) |
---|
444 | or a biharmonic operator, the latter being more scale-selective (more |
---|
445 | diffusive at small scales). The specification of eddy diffusivity |
---|
446 | coefficients (either constant or variable in space and time) as well as the |
---|
447 | computation of the slope along which the operators act, are performed in the |
---|
448 | \mdl{ldftra} and \mdl{ldfslp} modules, respectively. This is described in Chap.~\ref{LDF}. |
---|
449 | The lateral diffusion of tracers is evaluated using a forward scheme, |
---|
450 | $i.e.$ the tracers appearing in its expression are the \textit{before} tracers in time, |
---|
451 | except for the pure vertical component that appears when a rotation tensor |
---|
452 | is used. This latter term is solved implicitly together with the |
---|
453 | vertical diffusion term (see \S\ref{STP}). |
---|
454 | |
---|
455 | % ------------------------------------------------------------------------------------------------------------- |
---|
456 | % Iso-level laplacian operator |
---|
457 | % ------------------------------------------------------------------------------------------------------------- |
---|
458 | \subsection [Iso-level laplacian operator (lap) (\np{ln\_traldf\_lap})] |
---|
459 | {Iso-level laplacian operator (lap) (\np{ln\_traldf\_lap}=true) } |
---|
460 | \label{TRA_ldf_lap} |
---|
461 | |
---|
462 | A laplacian diffusion operator ($i.e.$ a harmonic operator) acting along the model |
---|
463 | surfaces is given by: |
---|
464 | \begin{equation} \label{Eq_tra_ldf_lap} |
---|
465 | D_T^{lT} =\frac{1}{b_tT} \left( \; |
---|
466 | \delta _{i}\left[ A_u^{lT} \; \frac{e_{2u}\,e_{3u}}{e_{1u}} \;\delta _{i+1/2} [T] \right] |
---|
467 | + \delta _{j}\left[ A_v^{lT} \; \frac{e_{1v}\,e_{3v}}{e_{2v}} \;\delta _{j+1/2} [T] \right] \;\right) |
---|
468 | \end{equation} |
---|
469 | where $b_t$=$e_{1t}\,e_{2t}\,e_{3t}$ is the volume of $T$-cells. |
---|
470 | It is implemented in the \mdl{traadv\_lap} module. |
---|
471 | |
---|
472 | This lateral operator is computed in \mdl{traldf\_lap}. It is a \emph{horizontal} |
---|
473 | operator ($i.e.$ acting along geopotential surfaces) in the $z$-coordinate with |
---|
474 | or without partial steps, but is simply an iso-level operator in the $s$-coordinate. |
---|
475 | It is thus used when, in addition to \np{ln\_traldf\_lap}=true, we have |
---|
476 | \np{ln\_traldf\_level}=true or \np{ln\_traldf\_hor}=\np{ln\_zco}=true. |
---|
477 | In both cases, it significantly contributes to diapycnal mixing. |
---|
478 | It is therefore not recommended. |
---|
479 | |
---|
480 | Note that in the partial step $z$-coordinate (\np{ln\_zps}=true), tracers in horizontally |
---|
481 | adjacent cells are located at different depths in the vicinity of the bottom. |
---|
482 | In this case, horizontal derivatives in (\ref{Eq_tra_ldf_lap}) at the bottom level |
---|
483 | require a specific treatment. They are calculated in the \mdl{zpshde} module, |
---|
484 | described in \S\ref{TRA_zpshde}. |
---|
485 | |
---|
486 | % ------------------------------------------------------------------------------------------------------------- |
---|
487 | % Rotated laplacian operator |
---|
488 | % ------------------------------------------------------------------------------------------------------------- |
---|
489 | \subsection [Rotated laplacian operator (iso) (\np{ln\_traldf\_lap})] |
---|
490 | {Rotated laplacian operator (iso) (\np{ln\_traldf\_lap}=true)} |
---|
491 | \label{TRA_ldf_iso} |
---|
492 | |
---|
493 | If the Griffies trad scheme is not employed |
---|
494 | (\np{ln\_traldf\_grif}=true; see App.\ref{sec:triad}) the general form of the second order lateral tracer subgrid scale physics |
---|
495 | (\ref{Eq_PE_zdf}) takes the following semi-discrete space form in $z$- and |
---|
496 | $s$-coordinates: |
---|
497 | \begin{equation} \label{Eq_tra_ldf_iso} |
---|
498 | \begin{split} |
---|
499 | D_T^{lT} = \frac{1}{b_t} & \left\{ \,\;\delta_i \left[ A_u^{lT} \left( |
---|
500 | \frac{e_{2u}\,e_{3u}}{e_{1u}} \,\delta_{i+1/2}[T] |
---|
501 | - e_{2u}\;r_{1u} \,\overline{\overline{ \delta_{k+1/2}[T] }}^{\,i+1/2,k} |
---|
502 | \right) \right] \right. \\ |
---|
503 | & +\delta_j \left[ A_v^{lT} \left( |
---|
504 | \frac{e_{1v}\,e_{3v}}{e_{2v}} \,\delta_{j+1/2} [T] |
---|
505 | - e_{1v}\,r_{2v} \,\overline{\overline{ \delta_{k+1/2} [T] }}^{\,j+1/2,k} |
---|
506 | \right) \right] \\ |
---|
507 | & +\delta_k \left[ A_w^{lT} \left( |
---|
508 | -\;e_{2w}\,r_{1w} \,\overline{\overline{ \delta_{i+1/2} [T] }}^{\,i,k+1/2} |
---|
509 | \right. \right. \\ |
---|
510 | & \qquad \qquad \quad |
---|
511 | - e_{1w}\,r_{2w} \,\overline{\overline{ \delta_{j+1/2} [T] }}^{\,j,k+1/2} \\ |
---|
512 | & \left. {\left. { \qquad \qquad \ \ \ \left. { |
---|
513 | +\;\frac{e_{1w}\,e_{2w}}{e_{3w}} \,\left( r_{1w}^2 + r_{2w}^2 \right) |
---|
514 | \,\delta_{k+1/2} [T] } \right) } \right] \quad } \right\} |
---|
515 | \end{split} |
---|
516 | \end{equation} |
---|
517 | where $b_t$=$e_{1t}\,e_{2t}\,e_{3t}$ is the volume of $T$-cells, |
---|
518 | $r_1$ and $r_2$ are the slopes between the surface of computation |
---|
519 | ($z$- or $s$-surfaces) and the surface along which the diffusion operator |
---|
520 | acts ($i.e.$ horizontal or iso-neutral surfaces). It is thus used when, |
---|
521 | in addition to \np{ln\_traldf\_lap}= true, we have \np{ln\_traldf\_iso}=true, |
---|
522 | or both \np{ln\_traldf\_hor}=true and \np{ln\_zco}=true. The way these |
---|
523 | slopes are evaluated is given in \S\ref{LDF_slp}. At the surface, bottom |
---|
524 | and lateral boundaries, the turbulent fluxes of heat and salt are set to zero |
---|
525 | using the mask technique (see \S\ref{LBC_coast}). |
---|
526 | |
---|
527 | The operator in \eqref{Eq_tra_ldf_iso} involves both lateral and vertical |
---|
528 | derivatives. For numerical stability, the vertical second derivative must |
---|
529 | be solved using the same implicit time scheme as that used in the vertical |
---|
530 | physics (see \S\ref{TRA_zdf}). For computer efficiency reasons, this term |
---|
531 | is not computed in the \mdl{traldf\_iso} module, but in the \mdl{trazdf} module |
---|
532 | where, if iso-neutral mixing is used, the vertical mixing coefficient is simply |
---|
533 | increased by $\frac{e_{1w}\,e_{2w} }{e_{3w} }\ \left( {r_{1w} ^2+r_{2w} ^2} \right)$. |
---|
534 | |
---|
535 | This formulation conserves the tracer but does not ensure the decrease |
---|
536 | of the tracer variance. Nevertheless the treatment performed on the slopes |
---|
537 | (see \S\ref{LDF}) allows the model to run safely without any additional |
---|
538 | background horizontal diffusion \citep{Guilyardi_al_CD01}. An alternative scheme |
---|
539 | developed by \cite{Griffies_al_JPO98} which ensures tracer variance decreases |
---|
540 | is also available in \NEMO (\np{ln\_traldf\_grif}=true). A complete description of |
---|
541 | the algorithm is given in App.\ref{sec:triad}. |
---|
542 | |
---|
543 | Note that in the partial step $z$-coordinate (\np{ln\_zps}=true), the horizontal |
---|
544 | derivatives at the bottom level in \eqref{Eq_tra_ldf_iso} require a specific |
---|
545 | treatment. They are calculated in module zpshde, described in \S\ref{TRA_zpshde}. |
---|
546 | |
---|
547 | % ------------------------------------------------------------------------------------------------------------- |
---|
548 | % Iso-level bilaplacian operator |
---|
549 | % ------------------------------------------------------------------------------------------------------------- |
---|
550 | \subsection [Iso-level bilaplacian operator (bilap) (\np{ln\_traldf\_bilap})] |
---|
551 | {Iso-level bilaplacian operator (bilap) (\np{ln\_traldf\_bilap}=true)} |
---|
552 | \label{TRA_ldf_bilap} |
---|
553 | |
---|
554 | The lateral fourth order bilaplacian operator on tracers is obtained by |
---|
555 | applying (\ref{Eq_tra_ldf_lap}) twice. The operator requires an additional assumption |
---|
556 | on boundary conditions: both first and third derivative terms normal to the |
---|
557 | coast are set to zero. It is used when, in addition to \np{ln\_traldf\_bilap}=true, |
---|
558 | we have \np{ln\_traldf\_level}=true, or both \np{ln\_traldf\_hor}=true and |
---|
559 | \np{ln\_zco}=false. In both cases, it can contribute diapycnal mixing, |
---|
560 | although less than in the laplacian case. It is therefore not recommended. |
---|
561 | |
---|
562 | Note that in the code, the bilaplacian routine does not call the laplacian |
---|
563 | routine twice but is rather a separate routine that can be found in the |
---|
564 | \mdl{traldf\_bilap} module. This is due to the fact that we introduce the |
---|
565 | eddy diffusivity coefficient, A, in the operator as: |
---|
566 | $\nabla \cdot \nabla \left( {A\nabla \cdot \nabla T} \right)$, |
---|
567 | instead of |
---|
568 | $-\nabla \cdot a\nabla \left( {\nabla \cdot a\nabla T} \right)$ |
---|
569 | where $a=\sqrt{|A|}$ and $A<0$. This was a mistake: both formulations |
---|
570 | ensure the total variance decrease, but the former requires a larger |
---|
571 | number of code-lines. |
---|
572 | |
---|
573 | % ------------------------------------------------------------------------------------------------------------- |
---|
574 | % Rotated bilaplacian operator |
---|
575 | % ------------------------------------------------------------------------------------------------------------- |
---|
576 | \subsection [Rotated bilaplacian operator (bilapg) (\np{ln\_traldf\_bilap})] |
---|
577 | {Rotated bilaplacian operator (bilapg) (\np{ln\_traldf\_bilap}=true)} |
---|
578 | \label{TRA_ldf_bilapg} |
---|
579 | |
---|
580 | The lateral fourth order operator formulation on tracers is obtained by |
---|
581 | applying (\ref{Eq_tra_ldf_iso}) twice. It requires an additional assumption |
---|
582 | on boundary conditions: first and third derivative terms normal to the |
---|
583 | coast, normal to the bottom and normal to the surface are set to zero. It can be found in the |
---|
584 | \mdl{traldf\_bilapg}. |
---|
585 | |
---|
586 | It is used when, in addition to \np{ln\_traldf\_bilap}=true, we have |
---|
587 | \np{ln\_traldf\_iso}= .true, or both \np{ln\_traldf\_hor}=true and \np{ln\_zco}=true. |
---|
588 | This rotated bilaplacian operator has never been seriously |
---|
589 | tested. There are no guarantees that it is either free of bugs or correctly formulated. |
---|
590 | Moreover, the stability range of such an operator will be probably quite |
---|
591 | narrow, requiring a significantly smaller time-step than the one used with an |
---|
592 | unrotated operator. |
---|
593 | |
---|
594 | % ================================================================ |
---|
595 | % Tracer Vertical Diffusion |
---|
596 | % ================================================================ |
---|
597 | \section [Tracer Vertical Diffusion (\textit{trazdf})] |
---|
598 | {Tracer Vertical Diffusion (\mdl{trazdf})} |
---|
599 | \label{TRA_zdf} |
---|
600 | %--------------------------------------------namzdf--------------------------------------------------------- |
---|
601 | \namdisplay{namzdf} |
---|
602 | %-------------------------------------------------------------------------------------------------------------- |
---|
603 | |
---|
604 | The formulation of the vertical subgrid scale tracer physics is the same |
---|
605 | for all the vertical coordinates, and is based on a laplacian operator. |
---|
606 | The vertical diffusion operator given by (\ref{Eq_PE_zdf}) takes the |
---|
607 | following semi-discrete space form: |
---|
608 | \begin{equation} \label{Eq_tra_zdf} |
---|
609 | \begin{split} |
---|
610 | D^{vT}_T &= \frac{1}{e_{3t}} \; \delta_k \left[ \;\frac{A^{vT}_w}{e_{3w}} \delta_{k+1/2}[T] \;\right] |
---|
611 | \\ |
---|
612 | D^{vS}_T &= \frac{1}{e_{3t}} \; \delta_k \left[ \;\frac{A^{vS}_w}{e_{3w}} \delta_{k+1/2}[S] \;\right] |
---|
613 | \end{split} |
---|
614 | \end{equation} |
---|
615 | where $A_w^{vT}$ and $A_w^{vS}$ are the vertical eddy diffusivity |
---|
616 | coefficients on temperature and salinity, respectively. Generally, |
---|
617 | $A_w^{vT}=A_w^{vS}$ except when double diffusive mixing is |
---|
618 | parameterised ($i.e.$ \key{zdfddm} is defined). The way these coefficients |
---|
619 | are evaluated is given in \S\ref{ZDF} (ZDF). Furthermore, when |
---|
620 | iso-neutral mixing is used, both mixing coefficients are increased |
---|
621 | by $\frac{e_{1w}\,e_{2w} }{e_{3w} }\ \left( {r_{1w} ^2+r_{2w} ^2} \right)$ |
---|
622 | to account for the vertical second derivative of \eqref{Eq_tra_ldf_iso}. |
---|
623 | |
---|
624 | At the surface and bottom boundaries, the turbulent fluxes of |
---|
625 | heat and salt must be specified. At the surface they are prescribed |
---|
626 | from the surface forcing and added in a dedicated routine (see \S\ref{TRA_sbc}), |
---|
627 | whilst at the bottom they are set to zero for heat and salt unless |
---|
628 | a geothermal flux forcing is prescribed as a bottom boundary |
---|
629 | condition (see \S\ref{TRA_bbc}). |
---|
630 | |
---|
631 | The large eddy coefficient found in the mixed layer together with high |
---|
632 | vertical resolution implies that in the case of explicit time stepping |
---|
633 | (\np{ln\_zdfexp}=true) there would be too restrictive a constraint on |
---|
634 | the time step. Therefore, the default implicit time stepping is preferred |
---|
635 | for the vertical diffusion since it overcomes the stability constraint. |
---|
636 | A forward time differencing scheme (\np{ln\_zdfexp}=true) using a time |
---|
637 | splitting technique (\np{nn\_zdfexp} $> 1$) is provided as an alternative. |
---|
638 | Namelist variables \np{ln\_zdfexp} and \np{nn\_zdfexp} apply to both |
---|
639 | tracers and dynamics. |
---|
640 | |
---|
641 | % ================================================================ |
---|
642 | % External Forcing |
---|
643 | % ================================================================ |
---|
644 | \section{External Forcing} |
---|
645 | \label{TRA_sbc_qsr_bbc} |
---|
646 | |
---|
647 | % ------------------------------------------------------------------------------------------------------------- |
---|
648 | % surface boundary condition |
---|
649 | % ------------------------------------------------------------------------------------------------------------- |
---|
650 | \subsection [Surface boundary condition (\textit{trasbc})] |
---|
651 | {Surface boundary condition (\mdl{trasbc})} |
---|
652 | \label{TRA_sbc} |
---|
653 | |
---|
654 | The surface boundary condition for tracers is implemented in a separate |
---|
655 | module (\mdl{trasbc}) instead of entering as a boundary condition on the vertical |
---|
656 | diffusion operator (as in the case of momentum). This has been found to |
---|
657 | enhance readability of the code. The two formulations are completely |
---|
658 | equivalent; the forcing terms in trasbc are the surface fluxes divided by |
---|
659 | the thickness of the top model layer. |
---|
660 | |
---|
661 | Due to interactions and mass exchange of water ($F_{mass}$) with other Earth system components ($i.e.$ atmosphere, sea-ice, land), |
---|
662 | the change in the heat and salt content of the surface layer of the ocean is due both |
---|
663 | to the heat and salt fluxes crossing the sea surface (not linked with $F_{mass}$) |
---|
664 | and to the heat and salt content of the mass exchange. |
---|
665 | \sgacomment{ the following does not apply to the release to which this documentation is |
---|
666 | attached and so should not be included .... |
---|
667 | In a forthcoming release, these two parts, computed in the surface module (SBC), will be included directly |
---|
668 | in $Q_{ns}$, the surface heat flux and $F_{salt}$, the surface salt flux. |
---|
669 | The specification of these fluxes is further detailed in the SBC chapter (see \S\ref{SBC}). |
---|
670 | This change will provide a forcing formulation which is the same for any tracer (including temperature and salinity). |
---|
671 | |
---|
672 | In the current version, the situation is a little bit more complicated. } |
---|
673 | |
---|
674 | The surface module (\mdl{sbcmod}, see \S\ref{SBC}) provides the following |
---|
675 | forcing fields (used on tracers): |
---|
676 | |
---|
677 | $\bullet$ $Q_{ns}$, the non-solar part of the net surface heat flux that crosses the sea surface |
---|
678 | (i.e. the difference between the total surface heat flux and the fraction of the short wave flux that |
---|
679 | penetrates into the water column, see \S\ref{TRA_qsr}) |
---|
680 | |
---|
681 | $\bullet$ \textit{emp}, the mass flux exchanged with the atmosphere (evaporation minus precipitation) |
---|
682 | |
---|
683 | $\bullet$ $\textit{emp}_S$, an equivalent mass flux taking into account the effect of ice-ocean mass exchange |
---|
684 | |
---|
685 | $\bullet$ \textit{rnf}, the mass flux associated with runoff (see \S\ref{SBC_rnf} for further detail of how it acts on temperature and salinity tendencies) |
---|
686 | |
---|
687 | The $\textit{emp}_S$ field is not simply the budget of evaporation-precipitation+freezing-melting because |
---|
688 | the sea-ice is not currently embedded in the ocean but levitates above it. There is no mass |
---|
689 | exchanged between the sea-ice and the ocean. Instead we only take into account the salt |
---|
690 | flux associated with the non-zero salinity of sea-ice, and the concentration/dilution effect |
---|
691 | due to the freezing/melting (F/M) process. These two parts of the forcing are then converted into |
---|
692 | an equivalent mass flux given by $\textit{emp}_S - \textit{emp}$. As a result of this mess, |
---|
693 | the surface boundary condition on temperature and salinity is applied as follows: |
---|
694 | |
---|
695 | In the nonlinear free surface case (\key{vvl} is defined): |
---|
696 | \begin{equation} \label{Eq_tra_sbc} |
---|
697 | \begin{aligned} |
---|
698 | &F^T = \frac{ 1 }{\rho _o \;C_p \,\left. e_{3t} \right|_{k=1} } |
---|
699 | &\overline{ \left( Q_{ns} - \textit{emp}\;C_p\,\left. T \right|_{k=1} \right) }^t & \\ |
---|
700 | % |
---|
701 | & F^S =\frac{ 1 }{\rho _o \,\left. e_{3t} \right|_{k=1} } |
---|
702 | &\overline{ \left( (\textit{emp}_S - \textit{emp})\;\left. S \right|_{k=1} \right) }^t & \\ |
---|
703 | \end{aligned} |
---|
704 | \end{equation} |
---|
705 | |
---|
706 | In the linear free surface case (\key{vvl} not defined): |
---|
707 | \begin{equation} \label{Eq_tra_sbc_lin} |
---|
708 | \begin{aligned} |
---|
709 | &F^T = \frac{ 1 }{\rho _o \;C_p \,\left. e_{3t} \right|_{k=1} } &\overline{ Q_{ns} }^t & \\ |
---|
710 | % |
---|
711 | & F^S =\frac{ 1 }{\rho _o \,\left. e_{3t} \right|_{k=1} } |
---|
712 | &\overline{ \left( \textit{emp}_S\;\left. S \right|_{k=1} \right) }^t & \\ |
---|
713 | \end{aligned} |
---|
714 | \end{equation} |
---|
715 | where $\overline{x }^t$ means that $x$ is averaged over two consecutive time steps |
---|
716 | ($t-\rdt/2$ and $t+\rdt/2$). Such time averaging prevents the |
---|
717 | divergence of odd and even time step (see \S\ref{STP}). |
---|
718 | |
---|
719 | The two set of equations, \eqref{Eq_tra_sbc} and \eqref{Eq_tra_sbc_lin}, are obtained |
---|
720 | by assuming that the temperature of precipitation and evaporation are equal to |
---|
721 | the ocean surface temperature and that their salinity is zero. Therefore, the heat content |
---|
722 | of the \textit{emp} budget must be added to the temperature equation in the variable volume case, |
---|
723 | while it does not appear in the constant volume case. Similarly, the \textit{emp} budget affects |
---|
724 | the ocean surface salinity in the constant volume case (through the concentration dilution effect) |
---|
725 | while it does not appears explicitly in the variable volume case since salinity change will be |
---|
726 | induced by volume change. In both constant and variable volume cases, surface salinity |
---|
727 | will change with ice-ocean salt flux and F/M flux (both contained in $\textit{emp}_S - \textit{emp}$) without mass exchanges. |
---|
728 | |
---|
729 | Note that the concentration/dilution effect due to F/M is computed using |
---|
730 | a constant ice salinity as well as a constant ocean salinity. |
---|
731 | This approximation suppresses the correlation between \textit{SSS} |
---|
732 | and F/M flux, allowing the ice-ocean salt exchanges to be conservative. |
---|
733 | Indeed, if this approximation is not made, even if the F/M budget is zero |
---|
734 | on average over the whole ocean domain and over the seasonal cycle, |
---|
735 | the associated salt flux is not zero, since sea-surface salinity and F/M flux are |
---|
736 | intrinsically correlated (high \textit{SSS} are found where freezing is |
---|
737 | strong whilst low \textit{SSS} is usually associated with high melting areas). |
---|
738 | |
---|
739 | Even using this approximation, an exact conservation of heat and salt content |
---|
740 | is only achieved in the variable volume case. In the constant volume case, |
---|
741 | there is a small imbalance associated with the product $(\partial_t\eta - \textit{emp}) * \textit{SSS}$. |
---|
742 | Nevertheless, the salt content variation is quite small and will not induce |
---|
743 | a long term drift as there is no physical reason for $(\partial_t\eta - \textit{emp})$ |
---|
744 | and \textit{SSS} to be correlated \citep{Roullet_Madec_JGR00}. |
---|
745 | Note that, while quite small, the imbalance in the constant volume case is larger |
---|
746 | than the imbalance associated with the Asselin time filter \citep{Leclair_Madec_OM09}. |
---|
747 | This is the reason why the modified filter is not applied in the constant volume case. |
---|
748 | |
---|
749 | % ------------------------------------------------------------------------------------------------------------- |
---|
750 | % Solar Radiation Penetration |
---|
751 | % ------------------------------------------------------------------------------------------------------------- |
---|
752 | \subsection [Solar Radiation Penetration (\textit{traqsr})] |
---|
753 | {Solar Radiation Penetration (\mdl{traqsr})} |
---|
754 | \label{TRA_qsr} |
---|
755 | %--------------------------------------------namqsr-------------------------------------------------------- |
---|
756 | \namdisplay{namtra_qsr} |
---|
757 | %-------------------------------------------------------------------------------------------------------------- |
---|
758 | |
---|
759 | When the penetrative solar radiation option is used (\np{ln\_flxqsr}=true), |
---|
760 | the solar radiation penetrates the top few tens of meters of the ocean. If it is not used |
---|
761 | (\np{ln\_flxqsr}=false) all the heat flux is absorbed in the first ocean level. |
---|
762 | Thus, in the former case a term is added to the time evolution equation of |
---|
763 | temperature \eqref{Eq_PE_tra_T} and the surface boundary condition is |
---|
764 | modified to take into account only the non-penetrative part of the surface |
---|
765 | heat flux: |
---|
766 | \begin{equation} \label{Eq_PE_qsr} |
---|
767 | \begin{split} |
---|
768 | \frac{\partial T}{\partial t} &= {\ldots} + \frac{1}{\rho_o\, C_p \,e_3} \; \frac{\partial I}{\partial k} \\ |
---|
769 | Q_{ns} &= Q_\text{Total} - Q_{sr} |
---|
770 | \end{split} |
---|
771 | \end{equation} |
---|
772 | where $Q_{sr}$ is the penetrative part of the surface heat flux ($i.e.$ the shortwave radiation) |
---|
773 | and $I$ is the downward irradiance ($\left. I \right|_{z=\eta}=Q_{sr}$). |
---|
774 | The additional term in \eqref{Eq_PE_qsr} is discretized as follows: |
---|
775 | \begin{equation} \label{Eq_tra_qsr} |
---|
776 | \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] |
---|
777 | \end{equation} |
---|
778 | |
---|
779 | The shortwave radiation, $Q_{sr}$, consists of energy distributed across a wide spectral range. |
---|
780 | The ocean is strongly absorbing for wavelengths longer than 700~nm and these |
---|
781 | wavelengths contribute to heating the upper few tens of centimetres. The fraction of $Q_{sr}$ |
---|
782 | that resides in these almost non-penetrative wavebands, $R$, is $\sim 58\%$ (specified |
---|
783 | through namelist parameter \np{rn\_abs}). It is assumed to penetrate the ocean |
---|
784 | with a decreasing exponential profile, with an e-folding depth scale, $\xi_0$, |
---|
785 | of a few tens of centimetres (typically $\xi_0=0.35~m$ set as \np{rn\_si0} in the namtra\_qsr namelist). |
---|
786 | For shorter wavelengths (400-700~nm), the ocean is more transparent, and solar energy |
---|
787 | propagates to larger depths where it contributes to |
---|
788 | local heating. |
---|
789 | The way this second part of the solar energy penetrates into the ocean depends on |
---|
790 | which formulation is chosen. In the simple 2-waveband light penetration scheme (\np{ln\_qsr\_2bd}=true) |
---|
791 | a chlorophyll-independent monochromatic formulation is chosen for the shorter wavelengths, |
---|
792 | leading to the following expression \citep{Paulson1977}: |
---|
793 | \begin{equation} \label{Eq_traqsr_iradiance} |
---|
794 | I(z) = Q_{sr} \left[Re^{-z / \xi_0} + \left( 1-R\right) e^{-z / \xi_1} \right] |
---|
795 | \end{equation} |
---|
796 | where $\xi_1$ is the second extinction length scale associated with the shorter wavelengths. |
---|
797 | It is usually chosen to be 23~m by setting the \np{rn\_si0} namelist parameter. |
---|
798 | The set of default values ($\xi_0$, $\xi_1$, $R$) corresponds to a Type I water in |
---|
799 | Jerlov's (1968) classification (oligotrophic waters). |
---|
800 | |
---|
801 | Such assumptions have been shown to provide a very crude and simplistic |
---|
802 | representation of observed light penetration profiles (\cite{Morel_JGR88}, see also |
---|
803 | Fig.\ref{Fig_traqsr_irradiance}). Light absorption in the ocean depends on |
---|
804 | particle concentration and is spectrally selective. \cite{Morel_JGR88} has shown |
---|
805 | that an accurate representation of light penetration can be provided by a 61 waveband |
---|
806 | formulation. Unfortunately, such a model is very computationally expensive. |
---|
807 | Thus, \cite{Lengaigne_al_CD07} have constructed a simplified version of this |
---|
808 | formulation in which visible light is split into three wavebands: blue (400-500 nm), |
---|
809 | green (500-600 nm) and red (600-700nm). For each wave-band, the chlorophyll-dependent |
---|
810 | attenuation coefficient is fitted to the coefficients computed from the full spectral model |
---|
811 | of \cite{Morel_JGR88} (as modified by \cite{Morel_Maritorena_JGR01}), assuming |
---|
812 | the same power-law relationship. As shown in Fig.\ref{Fig_traqsr_irradiance}, |
---|
813 | this formulation, called RGB (Red-Green-Blue), reproduces quite closely |
---|
814 | the light penetration profiles predicted by the full spectal model, but with much greater |
---|
815 | computational efficiency. The 2-bands formulation does not reproduce the full model very well. |
---|
816 | |
---|
817 | The RGB formulation is used when \np{ln\_qsr\_rgb}=true. The RGB attenuation coefficients |
---|
818 | ($i.e.$ the inverses of the extinction length scales) are tabulated over 61 nonuniform |
---|
819 | chlorophyll classes ranging from 0.01 to 10 g.Chl/L (see the routine \rou{trc\_oce\_rgb} |
---|
820 | in \mdl{trc\_oce} module). Three types of chlorophyll can be chosen in the RGB formulation: |
---|
821 | (1) a constant 0.05 g.Chl/L value everywhere (\np{nn\_chdta}=0) ; (2) an observed |
---|
822 | time varying chlorophyll (\np{nn\_chdta}=1) ; (3) simulated time varying chlorophyll |
---|
823 | by TOP biogeochemical model (\np{ln\_qsr\_bio}=true). In the latter case, the RGB |
---|
824 | formulation is used to calculate both the phytoplankton light limitation in PISCES |
---|
825 | or LOBSTER and the oceanic heating rate. |
---|
826 | |
---|
827 | The trend in \eqref{Eq_tra_qsr} associated with the penetration of the solar radiation |
---|
828 | is added to the temperature trend, and the surface heat flux is modified in routine \mdl{traqsr}. |
---|
829 | |
---|
830 | When the $z$-coordinate is preferred to the $s$-coordinate, the depth of $w-$levels does |
---|
831 | not significantly vary with location. The level at which the light has been totally |
---|
832 | absorbed ($i.e.$ it is less than the computer precision) is computed once, |
---|
833 | and the trend associated with the penetration of the solar radiation is only added down to that level. |
---|
834 | Finally, note that when the ocean is shallow ($<$ 200~m), part of the |
---|
835 | solar radiation can reach the ocean floor. In this case, we have |
---|
836 | chosen that all remaining radiation is absorbed in the last ocean |
---|
837 | level ($i.e.$ $I$ is masked). |
---|
838 | |
---|
839 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
---|
840 | \begin{figure}[!t] \begin{center} |
---|
841 | \includegraphics[width=1.0\textwidth]{./TexFiles/Figures/Fig_TRA_Irradiance.pdf} |
---|
842 | \caption{ \label{Fig_traqsr_irradiance} |
---|
843 | Penetration profile of the downward solar irradiance calculated by four models. |
---|
844 | Two waveband chlorophyll-independent formulation (blue), a chlorophyll-dependent |
---|
845 | monochromatic formulation (green), 4 waveband RGB formulation (red), |
---|
846 | 61 waveband Morel (1988) formulation (black) for a chlorophyll concentration of |
---|
847 | (a) Chl=0.05 mg/m$^3$ and (b) Chl=0.5 mg/m$^3$. From \citet{Lengaigne_al_CD07}.} |
---|
848 | \end{center} \end{figure} |
---|
849 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
---|
850 | |
---|
851 | % ------------------------------------------------------------------------------------------------------------- |
---|
852 | % Bottom Boundary Condition |
---|
853 | % ------------------------------------------------------------------------------------------------------------- |
---|
854 | \subsection [Bottom Boundary Condition (\textit{trabbc})] |
---|
855 | {Bottom Boundary Condition (\mdl{trabbc})} |
---|
856 | \label{TRA_bbc} |
---|
857 | %--------------------------------------------nambbc-------------------------------------------------------- |
---|
858 | \namdisplay{namtra_bbc} |
---|
859 | %-------------------------------------------------------------------------------------------------------------- |
---|
860 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
---|
861 | \begin{figure}[!t] \begin{center} |
---|
862 | \includegraphics[width=1.0\textwidth]{./TexFiles/Figures/Fig_TRA_geoth.pdf} |
---|
863 | \caption{ \label{Fig_geothermal} |
---|
864 | Geothermal Heat flux (in $mW.m^{-2}$) used by \cite{Emile-Geay_Madec_OS09}. |
---|
865 | It is inferred from the age of the sea floor and the formulae of \citet{Stein_Stein_Nat92}.} |
---|
866 | \end{center} \end{figure} |
---|
867 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
---|
868 | |
---|
869 | Usually it is assumed that there is no exchange of heat or salt through |
---|
870 | the ocean bottom, $i.e.$ a no flux boundary condition is applied on active |
---|
871 | tracers at the bottom. This is the default option in \NEMO, and it is |
---|
872 | implemented using the masking technique. However, there is a |
---|
873 | non-zero heat flux across the seafloor that is associated with solid |
---|
874 | earth cooling. This flux is weak compared to surface fluxes (a mean |
---|
875 | global value of $\sim0.1\;W/m^2$ \citep{Stein_Stein_Nat92}), but it warms |
---|
876 | systematically the ocean and acts on the densest water masses. |
---|
877 | Taking this flux into account in a global ocean model increases |
---|
878 | the deepest overturning cell ($i.e.$ the one associated with the Antarctic |
---|
879 | Bottom Water) by a few Sverdrups \citep{Emile-Geay_Madec_OS09}. |
---|
880 | |
---|
881 | The presence of geothermal heating is controlled by setting the namelist |
---|
882 | parameter \np{ln\_trabbc} to true. Then, when \np{nn\_geoflx} is set to 1, |
---|
883 | a constant geothermal heating is introduced whose value is given by the |
---|
884 | \np{nn\_geoflx\_cst}, which is also a namelist parameter. |
---|
885 | When \np{nn\_geoflx} is set to 2, a spatially varying geothermal heat flux is |
---|
886 | introduced which is provided in the \ifile{geothermal\_heating} NetCDF file |
---|
887 | (Fig.\ref{Fig_geothermal}) \citep{Emile-Geay_Madec_OS09}. |
---|
888 | |
---|
889 | % ================================================================ |
---|
890 | % Bottom Boundary Layer |
---|
891 | % ================================================================ |
---|
892 | \section [Bottom Boundary Layer (\mdl{trabbl} - \key{trabbl})] |
---|
893 | {Bottom Boundary Layer (\mdl{trabbl} - \key{trabbl})} |
---|
894 | \label{TRA_bbl} |
---|
895 | %--------------------------------------------nambbl--------------------------------------------------------- |
---|
896 | \namdisplay{nambbl} |
---|
897 | %-------------------------------------------------------------------------------------------------------------- |
---|
898 | |
---|
899 | In a $z$-coordinate configuration, the bottom topography is represented by a |
---|
900 | series of discrete steps. This is not adequate to represent gravity driven |
---|
901 | downslope flows. Such flows arise either downstream of sills such as the Strait of |
---|
902 | Gibraltar or Denmark Strait, where dense water formed in marginal seas flows |
---|
903 | into a basin filled with less dense water, or along the continental slope when dense |
---|
904 | water masses are formed on a continental shelf. The amount of entrainment |
---|
905 | that occurs in these gravity plumes is critical in determining the density |
---|
906 | and volume flux of the densest waters of the ocean, such as Antarctic Bottom Water, |
---|
907 | or North Atlantic Deep Water. $z$-coordinate models tend to overestimate the |
---|
908 | entrainment, because the gravity flow is mixed vertically by convection |
---|
909 | as it goes ''downstairs'' following the step topography, sometimes over a thickness |
---|
910 | much larger than the thickness of the observed gravity plume. A similar problem |
---|
911 | occurs in the $s$-coordinate when the thickness of the bottom level varies rapidly |
---|
912 | downstream of a sill \citep{Willebrand_al_PO01}, and the thickness |
---|
913 | of the plume is not resolved. |
---|
914 | |
---|
915 | The idea of the bottom boundary layer (BBL) parameterisation, first introduced by |
---|
916 | \citet{Beckmann_Doscher1997}, is to allow a direct communication between |
---|
917 | two adjacent bottom cells at different levels, whenever the densest water is |
---|
918 | located above the less dense water. The communication can be by a diffusive flux |
---|
919 | (diffusive BBL), an advective flux (advective BBL), or both. In the current |
---|
920 | implementation of the BBL, only the tracers are modified, not the velocities. |
---|
921 | Furthermore, it only connects ocean bottom cells, and therefore does not include |
---|
922 | all the improvements introduced by \citet{Campin_Goosse_Tel99}. |
---|
923 | |
---|
924 | % ------------------------------------------------------------------------------------------------------------- |
---|
925 | % Diffusive BBL |
---|
926 | % ------------------------------------------------------------------------------------------------------------- |
---|
927 | \subsection{Diffusive Bottom Boundary layer (\np{nn\_bbl\_ldf}=1)} |
---|
928 | \label{TRA_bbl_diff} |
---|
929 | |
---|
930 | When applying sigma-diffusion (\key{trabbl} defined and \np{nn\_bbl\_ldf} set to 1), |
---|
931 | the diffusive flux between two adjacent cells at the ocean floor is given by |
---|
932 | \begin{equation} \label{Eq_tra_bbl_diff} |
---|
933 | {\rm {\bf F}}_\sigma=A_l^\sigma \; \nabla_\sigma T |
---|
934 | \end{equation} |
---|
935 | with $\nabla_\sigma$ the lateral gradient operator taken between bottom cells, |
---|
936 | and $A_l^\sigma$ the lateral diffusivity in the BBL. Following \citet{Beckmann_Doscher1997}, |
---|
937 | the latter is prescribed with a spatial dependence, $i.e.$ in the conditional form |
---|
938 | \begin{equation} \label{Eq_tra_bbl_coef} |
---|
939 | A_l^\sigma (i,j,t)=\left\{ {\begin{array}{l} |
---|
940 | A_{bbl} \quad \quad \mbox{if} \quad \nabla_\sigma \rho \cdot \nabla H<0 \\ |
---|
941 | \\ |
---|
942 | 0\quad \quad \;\,\mbox{otherwise} \\ |
---|
943 | \end{array}} \right. |
---|
944 | \end{equation} |
---|
945 | where $A_{bbl}$ is the BBL diffusivity coefficient, given by the namelist |
---|
946 | parameter \np{rn\_ahtbbl} and usually set to a value much larger |
---|
947 | than the one used for lateral mixing in the open ocean. The constraint in \eqref{Eq_tra_bbl_coef} |
---|
948 | implies that sigma-like diffusion only occurs when the density above the sea floor, at the top of |
---|
949 | the slope, is larger than in the deeper ocean (see green arrow in Fig.\ref{Fig_bbl}). |
---|
950 | In practice, this constraint is applied separately in the two horizontal directions, |
---|
951 | and the density gradient in \eqref{Eq_tra_bbl_coef} is evaluated with the log gradient formulation: |
---|
952 | \begin{equation} \label{Eq_tra_bbl_Drho} |
---|
953 | \nabla_\sigma \rho / \rho = \alpha \,\nabla_\sigma T + \beta \,\nabla_\sigma S |
---|
954 | \end{equation} |
---|
955 | where $\rho$, $\alpha$ and $\beta$ are functions of $\overline{T}^\sigma$, |
---|
956 | $\overline{S}^\sigma$ and $\overline{H}^\sigma$, the along bottom mean temperature, |
---|
957 | salinity and depth, respectively. |
---|
958 | |
---|
959 | % ------------------------------------------------------------------------------------------------------------- |
---|
960 | % Advective BBL |
---|
961 | % ------------------------------------------------------------------------------------------------------------- |
---|
962 | \subsection {Advective Bottom Boundary Layer (\np{nn\_bbl\_adv}= 1 or 2)} |
---|
963 | \label{TRA_bbl_adv} |
---|
964 | |
---|
965 | \sgacomment{"downsloping flow" has been replaced by "downslope flow" in the following |
---|
966 | if this is not what is meant then "downwards sloping flow" is also a possibility"} |
---|
967 | |
---|
968 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
---|
969 | \begin{figure}[!t] \begin{center} |
---|
970 | \includegraphics[width=0.7\textwidth]{./TexFiles/Figures/Fig_BBL_adv.pdf} |
---|
971 | \caption{ \label{Fig_bbl} |
---|
972 | Advective/diffusive Bottom Boundary Layer. The BBL parameterisation is |
---|
973 | activated when $\rho^i_{kup}$ is larger than $\rho^{i+1}_{kdnw}$. |
---|
974 | Red arrows indicate the additional overturning circulation due to the advective BBL. |
---|
975 | The transport of the downslope flow is defined either as the transport of the bottom |
---|
976 | ocean cell (black arrow), or as a function of the along slope density gradient. |
---|
977 | The green arrow indicates the diffusive BBL flux directly connecting $kup$ and $kdwn$ |
---|
978 | ocean bottom cells. |
---|
979 | connection} |
---|
980 | \end{center} \end{figure} |
---|
981 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
---|
982 | |
---|
983 | |
---|
984 | %!! nn_bbl_adv = 1 use of the ocean velocity as bbl velocity |
---|
985 | %!! nn_bbl_adv = 2 follow Campin and Goosse (1999) implentation |
---|
986 | %!! i.e. transport proportional to the along-slope density gradient |
---|
987 | |
---|
988 | %%%gmcomment : this section has to be really written |
---|
989 | |
---|
990 | When applying an advective BBL (\np{nn\_bbl\_adv} = 1 or 2), an overturning |
---|
991 | circulation is added which connects two adjacent bottom grid-points only if dense |
---|
992 | water overlies less dense water on the slope. The density difference causes dense |
---|
993 | water to move down the slope. |
---|
994 | |
---|
995 | \np{nn\_bbl\_adv} = 1 : the downslope velocity is chosen to be the Eulerian |
---|
996 | ocean velocity just above the topographic step (see black arrow in Fig.\ref{Fig_bbl}) |
---|
997 | \citep{Beckmann_Doscher1997}. It is a \textit{conditional advection}, that is, advection |
---|
998 | is allowed only if dense water overlies less dense water on the slope ($i.e.$ |
---|
999 | $\nabla_\sigma \rho \cdot \nabla H<0$) and if the velocity is directed towards |
---|
1000 | greater depth ($i.e.$ $\vect{U} \cdot \nabla H>0$). |
---|
1001 | |
---|
1002 | \np{nn\_bbl\_adv} = 2 : the downslope velocity is chosen to be proportional to $\Delta \rho$, |
---|
1003 | the density difference between the higher cell and lower cell densities \citep{Campin_Goosse_Tel99}. |
---|
1004 | The advection is allowed only if dense water overlies less dense water on the slope ($i.e.$ |
---|
1005 | $\nabla_\sigma \rho \cdot \nabla H<0$). For example, the resulting transport of the |
---|
1006 | downslope flow, here in the $i$-direction (Fig.\ref{Fig_bbl}), is simply given by the |
---|
1007 | following expression: |
---|
1008 | \begin{equation} \label{Eq_bbl_Utr} |
---|
1009 | u^{tr}_{bbl} = \gamma \, g \frac{\Delta \rho}{\rho_o} e_{1u} \; min \left( {e_{3u}}_{kup},{e_{3u}}_{kdwn} \right) |
---|
1010 | \end{equation} |
---|
1011 | where $\gamma$, expressed in seconds, is the coefficient of proportionality |
---|
1012 | provided as \np{rn\_gambbl}, a namelist parameter, and \textit{kup} and \textit{kdwn} |
---|
1013 | are the vertical index of the higher and lower cells, respectively. |
---|
1014 | The parameter $\gamma$ should take a different value for each bathymetric |
---|
1015 | step, but for simplicity, and because no direct estimation of this parameter is |
---|
1016 | available, a uniform value has been assumed. The possible values for $\gamma$ |
---|
1017 | range between 1 and $10~s$ \citep{Campin_Goosse_Tel99}. |
---|
1018 | |
---|
1019 | Scalar properties are advected by this additional transport $( u^{tr}_{bbl}, v^{tr}_{bbl} )$ |
---|
1020 | using the upwind scheme. Such a diffusive advective scheme has been chosen |
---|
1021 | to mimic the entrainment between the downslope plume and the surrounding |
---|
1022 | water at intermediate depths. The entrainment is replaced by the vertical mixing |
---|
1023 | implicit in the advection scheme. Let us consider as an example the |
---|
1024 | case displayed in Fig.\ref{Fig_bbl} where the density at level $(i,kup)$ is |
---|
1025 | larger than the one at level $(i,kdwn)$. The advective BBL scheme |
---|
1026 | modifies the tracer time tendency of the ocean cells near the |
---|
1027 | topographic step by the downslope flow \eqref{Eq_bbl_dw}, |
---|
1028 | the horizontal \eqref{Eq_bbl_hor} and the upward \eqref{Eq_bbl_up} |
---|
1029 | return flows as follows: |
---|
1030 | \begin{align} |
---|
1031 | \partial_t T^{do}_{kdw} &\equiv \partial_t T^{do}_{kdw} |
---|
1032 | + \frac{u^{tr}_{bbl}}{{b_t}^{do}_{kdw}} \left( T^{sh}_{kup} - T^{do}_{kdw} \right) \label{Eq_bbl_dw} \\ |
---|
1033 | % |
---|
1034 | \partial_t T^{sh}_{kup} &\equiv \partial_t T^{sh}_{kup} |
---|
1035 | + \frac{u^{tr}_{bbl}}{{b_t}^{sh}_{kup}} \left( T^{do}_{kup} - T^{sh}_{kup} \right) \label{Eq_bbl_hor} \\ |
---|
1036 | % |
---|
1037 | \intertext{and for $k =kdw-1,\;..., \; kup$ :} |
---|
1038 | % |
---|
1039 | \partial_t T^{do}_{k} &\equiv \partial_t S^{do}_{k} |
---|
1040 | + \frac{u^{tr}_{bbl}}{{b_t}^{do}_{k}} \left( T^{do}_{k+1} - T^{sh}_{k} \right) \label{Eq_bbl_up} |
---|
1041 | \end{align} |
---|
1042 | where $b_t$ is the $T$-cell volume. |
---|
1043 | |
---|
1044 | Note that the BBL transport, $( u^{tr}_{bbl}, v^{tr}_{bbl} )$, is available in |
---|
1045 | the model outputs. It has to be used to compute the effective velocity |
---|
1046 | as well as the effective overturning circulation. |
---|
1047 | |
---|
1048 | % ================================================================ |
---|
1049 | % Tracer damping |
---|
1050 | % ================================================================ |
---|
1051 | \section [Tracer damping (\textit{tradmp})] |
---|
1052 | {Tracer damping (\mdl{tradmp})} |
---|
1053 | \label{TRA_dmp} |
---|
1054 | %--------------------------------------------namtra_dmp------------------------------------------------- |
---|
1055 | \namdisplay{namtra_dmp} |
---|
1056 | %-------------------------------------------------------------------------------------------------------------- |
---|
1057 | |
---|
1058 | In some applications it can be useful to add a Newtonian damping term |
---|
1059 | into the temperature and salinity equations: |
---|
1060 | \begin{equation} \label{Eq_tra_dmp} |
---|
1061 | \begin{split} |
---|
1062 | \frac{\partial T}{\partial t}=\;\cdots \;-\gamma \,\left( {T-T_o } \right) \\ |
---|
1063 | \frac{\partial S}{\partial t}=\;\cdots \;-\gamma \,\left( {S-S_o } \right) |
---|
1064 | \end{split} |
---|
1065 | \end{equation} |
---|
1066 | where $\gamma$ is the inverse of a time scale, and $T_o$ and $S_o$ |
---|
1067 | are given temperature and salinity fields (usually a climatology). |
---|
1068 | The restoring term is added when the namelist parameter \np{ln\_tradmp} is set to true. |
---|
1069 | It also requires that both \np{ln\_tsd\_init} and \np{ln\_tsd\_tradmp} are set to true |
---|
1070 | in \textit{namtsd} namelist as well as \np{sn\_tem} and \np{sn\_sal} structures are |
---|
1071 | correctly set ($i.e.$ that $T_o$ and $S_o$ are provided in input files and read |
---|
1072 | using \mdl{fldread}, see \S\ref{SBC_fldread}). |
---|
1073 | The restoring coefficient $\gamma$ is a three-dimensional array initialized by the |
---|
1074 | user in routine \rou{dtacof} also located in module \mdl{tradmp}. |
---|
1075 | |
---|
1076 | The two main cases in which \eqref{Eq_tra_dmp} is used are \textit{(a)} |
---|
1077 | the specification of the boundary conditions along artificial walls of a |
---|
1078 | limited domain basin and \textit{(b)} the computation of the velocity |
---|
1079 | field associated with a given $T$-$S$ field (for example to build the |
---|
1080 | initial state of a prognostic simulation, or to use the resulting velocity |
---|
1081 | field for a passive tracer study). The first case applies to regional |
---|
1082 | models that have artificial walls instead of open boundaries. |
---|
1083 | In the vicinity of these walls, $\gamma$ takes large values (equivalent to |
---|
1084 | a time scale of a few days) whereas it is zero in the interior of the |
---|
1085 | model domain. The second case corresponds to the use of the robust |
---|
1086 | diagnostic method \citep{Sarmiento1982}. It allows us to find the velocity |
---|
1087 | field consistent with the model dynamics whilst having a $T$, $S$ field |
---|
1088 | close to a given climatological field ($T_o$, $S_o$). The time scale |
---|
1089 | associated with $S_o$ is generally not a constant but spatially varying |
---|
1090 | in order to respect other properties. For example, it is usually set to zero |
---|
1091 | in the mixed layer (defined either on a density or $S_o$ criterion) |
---|
1092 | \citep{Madec_al_JPO96} and in the equatorial region |
---|
1093 | \citep{Reverdin1991, Fujio1991, Marti_PhD92} since these two regions |
---|
1094 | have a short time scale of adjustment; while smaller $\gamma$ are used |
---|
1095 | in the deep ocean where the typical time scale is long \citep{Sarmiento1982}. |
---|
1096 | In addition the time scale is reduced (even to zero) along the western |
---|
1097 | boundary to allow the model to reconstruct its own western boundary |
---|
1098 | structure in equilibrium with its physics. |
---|
1099 | The choice of the shape of the Newtonian damping is controlled by two |
---|
1100 | namelist parameters \np{nn\_hdmp} and \np{nn\_zdmp}. The former allows us to specify: the |
---|
1101 | width of the equatorial band in which no damping is applied; a decrease |
---|
1102 | in the vicinity of the coast; and a damping everywhere in the Red and Med Seas. |
---|
1103 | The latter sets whether damping should act in the mixed layer or not. |
---|
1104 | The time scale associated with the damping depends on the depth as |
---|
1105 | a hyperbolic tangent, with \np{rn\_surf} as surface value, \np{rn\_bot} as |
---|
1106 | bottom value and a transition depth of \np{rn\_dep}. |
---|
1107 | |
---|
1108 | The robust diagnostic method is very efficient in preventing temperature |
---|
1109 | drift in intermediate waters but it produces artificial sources of heat and salt |
---|
1110 | within the ocean. It also has undesirable effects on the ocean convection. |
---|
1111 | It tends to prevent deep convection and subsequent deep-water formation, |
---|
1112 | by stabilising the water column too much. |
---|
1113 | |
---|
1114 | An example of the computation of $\gamma$ for a robust diagnostic experiment |
---|
1115 | with the ORCA2 model is provided in the \mdl{tradmp} module |
---|
1116 | (subroutines \rou{dtacof} and \rou{cofdis} which compute the coefficient |
---|
1117 | and the distance to the bathymetry, respectively). These routines are |
---|
1118 | provided as examples and can be customised by the user. |
---|
1119 | |
---|
1120 | % ================================================================ |
---|
1121 | % Tracer time evolution |
---|
1122 | % ================================================================ |
---|
1123 | \section [Tracer time evolution (\textit{tranxt})] |
---|
1124 | {Tracer time evolution (\mdl{tranxt})} |
---|
1125 | \label{TRA_nxt} |
---|
1126 | %--------------------------------------------namdom----------------------------------------------------- |
---|
1127 | \namdisplay{namdom} |
---|
1128 | %-------------------------------------------------------------------------------------------------------------- |
---|
1129 | |
---|
1130 | The general framework for tracer time stepping is a modified leap-frog scheme |
---|
1131 | \citep{Leclair_Madec_OM09}, $i.e.$ a three level centred time scheme associated |
---|
1132 | with a Asselin time filter (cf. \S\ref{STP_mLF}): |
---|
1133 | \begin{equation} \label{Eq_tra_nxt} |
---|
1134 | \begin{aligned} |
---|
1135 | (e_{3t}T)^{t+\rdt} &= (e_{3t}T)_f^{t-\rdt} &+ 2 \, \rdt \,e_{3t}^t\ \text{RHS}^t & \\ |
---|
1136 | \\ |
---|
1137 | (e_{3t}T)_f^t \;\ \quad &= (e_{3t}T)^t \;\quad |
---|
1138 | &+\gamma \,\left[ {(e_{3t}T)_f^{t-\rdt} -2(e_{3t}T)^t+(e_{3t}T)^{t+\rdt}} \right] & \\ |
---|
1139 | & &- \gamma\,\rdt \, \left[ Q^{t+\rdt/2} - Q^{t-\rdt/2} \right] & |
---|
1140 | \end{aligned} |
---|
1141 | \end{equation} |
---|
1142 | where RHS is the right hand side of the temperature equation, |
---|
1143 | the subscript $f$ denotes filtered values, $\gamma$ is the Asselin coefficient, |
---|
1144 | and $S$ is the total forcing applied on $T$ ($i.e.$ fluxes plus content in mass exchanges). |
---|
1145 | $\gamma$ is initialized as \np{rn\_atfp} (\textbf{namelist} parameter). |
---|
1146 | Its default value is \np{rn\_atfp}=$10^{-3}$. Note that the forcing correction term in the filter |
---|
1147 | is not applied in linear free surface (\jp{lk\_vvl}=false) (see \S\ref{TRA_sbc}. |
---|
1148 | Not also that in constant volume case, the time stepping is performed on $T$, |
---|
1149 | not on its content, $e_{3t}T$. |
---|
1150 | |
---|
1151 | When the vertical mixing is solved implicitly, the update of the \textit{next} tracer |
---|
1152 | fields is done in module \mdl{trazdf}. In this case only the swapping of arrays |
---|
1153 | and the Asselin filtering is done in the \mdl{tranxt} module. |
---|
1154 | |
---|
1155 | In order to prepare for the computation of the \textit{next} time step, |
---|
1156 | a swap of tracer arrays is performed: $T^{t-\rdt} = T^t$ and $T^t = T_f$. |
---|
1157 | |
---|
1158 | % ================================================================ |
---|
1159 | % Equation of State (eosbn2) |
---|
1160 | % ================================================================ |
---|
1161 | \section [Equation of State (\textit{eosbn2}) ] |
---|
1162 | {Equation of State (\mdl{eosbn2}) } |
---|
1163 | \label{TRA_eosbn2} |
---|
1164 | %--------------------------------------------nameos----------------------------------------------------- |
---|
1165 | \namdisplay{nameos} |
---|
1166 | %-------------------------------------------------------------------------------------------------------------- |
---|
1167 | |
---|
1168 | % ------------------------------------------------------------------------------------------------------------- |
---|
1169 | % Equation of State |
---|
1170 | % ------------------------------------------------------------------------------------------------------------- |
---|
1171 | \subsection{Equation of State (\np{nn\_eos} = 0, 1 or 2)} |
---|
1172 | \label{TRA_eos} |
---|
1173 | |
---|
1174 | It is necessary to know the equation of state for the ocean very accurately |
---|
1175 | to determine stability properties (especially the Brunt-Vais\"{a}l\"{a} frequency), |
---|
1176 | particularly in the deep ocean. The ocean seawater volumic mass, $\rho$, |
---|
1177 | abusively called density, is a non linear empirical function of \textit{in situ} |
---|
1178 | temperature, salinity and pressure. The reference equation of state is that |
---|
1179 | defined by the Joint Panel on Oceanographic Tables and Standards |
---|
1180 | \citep{UNESCO1983}. It was the standard equation of state used in early |
---|
1181 | releases of OPA. However, even though this computation is fully vectorised, |
---|
1182 | it is quite time consuming ($15$ to $20${\%} of the total CPU time) since |
---|
1183 | it requires the prior computation of the \textit{in situ} temperature from the |
---|
1184 | model \textit{potential} temperature using the \citep{Bryden1973} polynomial |
---|
1185 | for adiabatic lapse rate and a $4^th$ order Runge-Kutta integration scheme. |
---|
1186 | Since OPA6, we have used the \citet{JackMcD1995} equation of state for |
---|
1187 | seawater instead. It allows the computation of the \textit{in situ} ocean density |
---|
1188 | directly as a function of \textit{potential} temperature relative to the surface |
---|
1189 | (an \NEMO variable), the practical salinity (another \NEMO variable) and the |
---|
1190 | pressure (assuming no pressure variation along geopotential surfaces, $i.e.$ |
---|
1191 | the pressure in decibars is approximated by the depth in meters). |
---|
1192 | Both the \citet{UNESCO1983} and \citet{JackMcD1995} equations of state |
---|
1193 | have exactly the same except that the values of the various coefficients have |
---|
1194 | been adjusted by \citet{JackMcD1995} in order to directly use the \textit{potential} |
---|
1195 | temperature instead of the \textit{in situ} one. This reduces the CPU time of the |
---|
1196 | \textit{in situ} density computation to about $3${\%} of the total CPU time, |
---|
1197 | while maintaining a quite accurate equation of state. |
---|
1198 | |
---|
1199 | In the computer code, a \textit{true} density anomaly, $d_a= \rho / \rho_o - 1$, |
---|
1200 | is computed, with $\rho_o$ a reference volumic mass. Called \textit{rau0} |
---|
1201 | in the code, $\rho_o$ is defined in \mdl{phycst}, and a value of $1,035~Kg/m^3$. |
---|
1202 | This is a sensible choice for the reference density used in a Boussinesq ocean |
---|
1203 | climate model, as, with the exception of only a small percentage of the ocean, |
---|
1204 | density in the World Ocean varies by no more than 2$\%$ from $1,035~kg/m^3$ |
---|
1205 | \citep{Gill1982}. |
---|
1206 | |
---|
1207 | The default option (namelist parameter \np{nn\_eos}=0) is the \citet{JackMcD1995} |
---|
1208 | equation of state. Its use is highly recommended. However, for process studies, |
---|
1209 | it is often convenient to use a linear approximation of the density. |
---|
1210 | With such an equation of state there is no longer a distinction between |
---|
1211 | \textit{in situ} and \textit{potential} density and both cabbeling and thermobaric |
---|
1212 | effects are removed. |
---|
1213 | Two linear formulations are available: a function of $T$ only (\np{nn\_eos}=1) |
---|
1214 | and a function of both $T$ and $S$ (\np{nn\_eos}=2): |
---|
1215 | \begin{equation} \label{Eq_tra_eos_linear} |
---|
1216 | \begin{split} |
---|
1217 | d_a(T) &= \rho (T) / \rho_o - 1 = \ 0.0285 - \alpha \;T \\ |
---|
1218 | d_a(T,S) &= \rho (T,S) / \rho_o - 1 = \ \beta \; S - \alpha \;T |
---|
1219 | \end{split} |
---|
1220 | \end{equation} |
---|
1221 | where $\alpha$ and $\beta$ are the thermal and haline expansion |
---|
1222 | coefficients, and $\rho_o$, the reference volumic mass, $rau0$. |
---|
1223 | ($\alpha$ and $\beta$ can be modified through the \np{rn\_alpha} and |
---|
1224 | \np{rn\_beta} namelist parameters). Note that when $d_a$ is a function |
---|
1225 | of $T$ only (\np{nn\_eos}=1), the salinity is a passive tracer and can be |
---|
1226 | used as such. |
---|
1227 | |
---|
1228 | % ------------------------------------------------------------------------------------------------------------- |
---|
1229 | % Brunt-Vais\"{a}l\"{a} Frequency |
---|
1230 | % ------------------------------------------------------------------------------------------------------------- |
---|
1231 | \subsection{Brunt-Vais\"{a}l\"{a} Frequency (\np{nn\_eos} = 0, 1 or 2)} |
---|
1232 | \label{TRA_bn2} |
---|
1233 | |
---|
1234 | An accurate computation of the ocean stability (i.e. of $N$, the brunt-Vais\"{a}l\"{a} |
---|
1235 | frequency) is of paramount importance as it is used in several ocean |
---|
1236 | parameterisations (namely TKE, KPP, Richardson number dependent |
---|
1237 | vertical diffusion, enhanced vertical diffusion, non-penetrative convection, |
---|
1238 | iso-neutral diffusion). In particular, one must be aware that $N^2$ has to |
---|
1239 | be computed with an \textit{in situ} reference. The expression for $N^2$ |
---|
1240 | depends on the type of equation of state used (\np{nn\_eos} namelist parameter). |
---|
1241 | |
---|
1242 | For \np{nn\_eos}=0 (\citet{JackMcD1995} equation of state), the \citet{McDougall1987} |
---|
1243 | polynomial expression is used (with the pressure in decibar approximated by |
---|
1244 | the depth in meters): |
---|
1245 | \begin{equation} \label{Eq_tra_bn2} |
---|
1246 | N^2 = \frac{g}{e_{3w}} \; \beta \ |
---|
1247 | \left( \alpha / \beta \ \delta_{k+1/2}[T] - \delta_{k+1/2}[S] \right) |
---|
1248 | \end{equation} |
---|
1249 | where $\alpha$ and $\beta$ are the thermal and haline expansion coefficients. |
---|
1250 | They are a function of $\overline{T}^{\,k+1/2},\widetilde{S}=\overline{S}^{\,k+1/2} - 35.$, |
---|
1251 | and $z_w$, with $T$ the \textit{potential} temperature and $\widetilde{S}$ a salinity anomaly. |
---|
1252 | Note that both $\alpha$ and $\beta$ depend on \textit{potential} |
---|
1253 | temperature and salinity which are averaged at $w$-points prior |
---|
1254 | to the computation instead of being computed at $T$-points and |
---|
1255 | then averaged to $w$-points. |
---|
1256 | |
---|
1257 | When a linear equation of state is used (\np{nn\_eos}=1 or 2, |
---|
1258 | \eqref{Eq_tra_bn2} reduces to: |
---|
1259 | \begin{equation} \label{Eq_tra_bn2_linear} |
---|
1260 | N^2 = \frac{g}{e_{3w}} \left( \beta \;\delta_{k+1/2}[S] - \alpha \;\delta_{k+1/2}[T] \right) |
---|
1261 | \end{equation} |
---|
1262 | where $\alpha$ and $\beta $ are the constant coefficients used to |
---|
1263 | defined the linear equation of state \eqref{Eq_tra_eos_linear}. |
---|
1264 | |
---|
1265 | % ------------------------------------------------------------------------------------------------------------- |
---|
1266 | % Specific Heat |
---|
1267 | % ------------------------------------------------------------------------------------------------------------- |
---|
1268 | \subsection [Specific Heat (\textit{phycst})] |
---|
1269 | {Specific Heat (\mdl{phycst})} |
---|
1270 | \label{TRA_adv_ldf} |
---|
1271 | |
---|
1272 | The specific heat of sea water, $C_p$, is a function of temperature, salinity |
---|
1273 | and pressure \citep{UNESCO1983}. It is only used in the model to convert |
---|
1274 | surface heat fluxes into surface temperature increase and so the pressure |
---|
1275 | dependence is neglected. The dependence on $T$ and $S$ is weak. |
---|
1276 | For example, with $S=35~psu$, $C_p$ increases from $3989$ to $4002$ |
---|
1277 | when $T$ varies from -2~\degres C to 31~\degres C. Therefore, $C_p$ has |
---|
1278 | been chosen as a constant: $C_p=4.10^3~J\,Kg^{-1}\,\degres K^{-1}$. |
---|
1279 | Its value is set in \mdl{phycst} module. |
---|
1280 | |
---|
1281 | |
---|
1282 | % ------------------------------------------------------------------------------------------------------------- |
---|
1283 | % Freezing Point of Seawater |
---|
1284 | % ------------------------------------------------------------------------------------------------------------- |
---|
1285 | \subsection [Freezing Point of Seawater] |
---|
1286 | {Freezing Point of Seawater} |
---|
1287 | \label{TRA_fzp} |
---|
1288 | |
---|
1289 | The freezing point of seawater is a function of salinity and pressure \citep{UNESCO1983}: |
---|
1290 | \begin{equation} \label{Eq_tra_eos_fzp} |
---|
1291 | \begin{split} |
---|
1292 | T_f (S,p) = \left( -0.0575 + 1.710523 \;10^{-3} \, \sqrt{S} |
---|
1293 | - 2.154996 \;10^{-4} \,S \right) \ S \\ |
---|
1294 | - 7.53\,10^{-3} \ \ p |
---|
1295 | \end{split} |
---|
1296 | \end{equation} |
---|
1297 | |
---|
1298 | \eqref{Eq_tra_eos_fzp} is only used to compute the potential freezing point of |
---|
1299 | sea water ($i.e.$ referenced to the surface $p=0$), thus the pressure dependent |
---|
1300 | terms in \eqref{Eq_tra_eos_fzp} (last term) have been dropped. The freezing |
---|
1301 | point is computed through \textit{tfreez}, a \textsc{Fortran} function that can be found |
---|
1302 | in \mdl{eosbn2}. |
---|
1303 | |
---|
1304 | % ================================================================ |
---|
1305 | % Horizontal Derivative in zps-coordinate |
---|
1306 | % ================================================================ |
---|
1307 | \section [Horizontal Derivative in \textit{zps}-coordinate (\textit{zpshde})] |
---|
1308 | {Horizontal Derivative in \textit{zps}-coordinate (\mdl{zpshde})} |
---|
1309 | \label{TRA_zpshde} |
---|
1310 | |
---|
1311 | \gmcomment{STEVEN: to be consistent with earlier discussion of differencing and averaging operators, I've changed "derivative" to "difference" and "mean" to "average"} |
---|
1312 | |
---|
1313 | With partial bottom cells (\np{ln\_zps}=true), in general, tracers in horizontally |
---|
1314 | adjacent cells live at different depths. Horizontal gradients of tracers are needed |
---|
1315 | for horizontal diffusion (\mdl{traldf} module) and for the hydrostatic pressure |
---|
1316 | gradient (\mdl{dynhpg} module) to be active. |
---|
1317 | \gmcomment{STEVEN from gm : question: not sure of what -to be active- means} |
---|
1318 | Before taking horizontal gradients between the tracers next to the bottom, a linear |
---|
1319 | interpolation in the vertical is used to approximate the deeper tracer as if it actually |
---|
1320 | lived at the depth of the shallower tracer point (Fig.~\ref{Fig_Partial_step_scheme}). |
---|
1321 | For example, for temperature in the $i$-direction the needed interpolated |
---|
1322 | temperature, $\widetilde{T}$, is: |
---|
1323 | |
---|
1324 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
---|
1325 | \begin{figure}[!p] \begin{center} |
---|
1326 | \includegraphics[width=0.9\textwidth]{./TexFiles/Figures/Partial_step_scheme.pdf} |
---|
1327 | \caption{ \label{Fig_Partial_step_scheme} |
---|
1328 | Discretisation of the horizontal difference and average of tracers in the $z$-partial |
---|
1329 | step coordinate (\np{ln\_zps}=true) in the case $( e3w_k^{i+1} - e3w_k^i )>0$. |
---|
1330 | A linear interpolation is used to estimate $\widetilde{T}_k^{i+1}$, the tracer value |
---|
1331 | at the depth of the shallower tracer point of the two adjacent bottom $T$-points. |
---|
1332 | The horizontal difference is then given by: $\delta _{i+1/2} T_k= \widetilde{T}_k^{\,i+1} -T_k^{\,i}$ |
---|
1333 | and the average by: $\overline{T}_k^{\,i+1/2}= ( \widetilde{T}_k^{\,i+1/2} - T_k^{\,i} ) / 2$. } |
---|
1334 | \end{center} \end{figure} |
---|
1335 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
---|
1336 | \begin{equation*} |
---|
1337 | \widetilde{T}= \left\{ \begin{aligned} |
---|
1338 | &T^{\,i+1} -\frac{ \left( e_{3w}^{i+1} -e_{3w}^i \right)}{ e_{3w}^{i+1} }\;\delta _k T^{i+1} |
---|
1339 | && \quad\text{if $\ e_{3w}^{i+1} \geq e_{3w}^i$ } \\ |
---|
1340 | \\ |
---|
1341 | &T^{\,i} \ \ \ \,+\frac{ \left( e_{3w}^{i+1} -e_{3w}^i \right) }{e_{3w}^i }\;\delta _k T^{i+1} |
---|
1342 | && \quad\text{if $\ e_{3w}^{i+1} < e_{3w}^i$ } |
---|
1343 | \end{aligned} \right. |
---|
1344 | \end{equation*} |
---|
1345 | and the resulting forms for the horizontal difference and the horizontal average |
---|
1346 | value of $T$ at a $U$-point are: |
---|
1347 | \begin{equation} \label{Eq_zps_hde} |
---|
1348 | \begin{aligned} |
---|
1349 | \delta _{i+1/2} T= \begin{cases} |
---|
1350 | \ \ \ \widetilde {T}\quad\ -T^i & \ \ \quad\quad\text{if $\ e_{3w}^{i+1} \geq e_{3w}^i$ } \\ |
---|
1351 | \\ |
---|
1352 | \ \ \ T^{\,i+1}-\widetilde{T} & \ \ \quad\quad\text{if $\ e_{3w}^{i+1} < e_{3w}^i$ } |
---|
1353 | \end{cases} \\ |
---|
1354 | \\ |
---|
1355 | \overline {T}^{\,i+1/2} \ = \begin{cases} |
---|
1356 | ( \widetilde {T}\ \ \;\,-T^{\,i}) / 2 & \;\ \ \quad\text{if $\ e_{3w}^{i+1} \geq e_{3w}^i$ } \\ |
---|
1357 | \\ |
---|
1358 | ( T^{\,i+1}-\widetilde{T} ) / 2 & \;\ \ \quad\text{if $\ e_{3w}^{i+1} < e_{3w}^i$ } |
---|
1359 | \end{cases} |
---|
1360 | \end{aligned} |
---|
1361 | \end{equation} |
---|
1362 | |
---|
1363 | The computation of horizontal derivative of tracers as well as of density is |
---|
1364 | performed once for all at each time step in \mdl{zpshde} module and stored |
---|
1365 | in shared arrays to be used when needed. It has to be emphasized that the |
---|
1366 | procedure used to compute the interpolated density, $\widetilde{\rho}$, is not |
---|
1367 | the same as that used for $T$ and $S$. Instead of forming a linear approximation |
---|
1368 | of density, we compute $\widetilde{\rho }$ from the interpolated values of $T$ |
---|
1369 | and $S$, and the pressure at a $u$-point (in the equation of state pressure is |
---|
1370 | approximated by depth, see \S\ref{TRA_eos} ) : |
---|
1371 | \begin{equation} \label{Eq_zps_hde_rho} |
---|
1372 | \widetilde{\rho } = \rho ( {\widetilde{T},\widetilde {S},z_u }) |
---|
1373 | \quad \text{where }\ z_u = \min \left( {z_T^{i+1} ,z_T^i } \right) |
---|
1374 | \end{equation} |
---|
1375 | |
---|
1376 | This is a much better approximation as the variation of $\rho$ with depth (and |
---|
1377 | thus pressure) is highly non-linear with a true equation of state and thus is badly |
---|
1378 | approximated with a linear interpolation. This approximation is used to compute |
---|
1379 | both the horizontal pressure gradient (\S\ref{DYN_hpg}) and the slopes of neutral |
---|
1380 | surfaces (\S\ref{LDF_slp}) |
---|
1381 | |
---|
1382 | Note that in almost all the advection schemes presented in this Chapter, both |
---|
1383 | averaging and differencing operators appear. Yet \eqref{Eq_zps_hde} has not |
---|
1384 | been used in these schemes: in contrast to diffusion and pressure gradient |
---|
1385 | computations, no correction for partial steps is applied for advection. The main |
---|
1386 | motivation is to preserve the domain averaged mean variance of the advected |
---|
1387 | field when using the $2^{nd}$ order centred scheme. Sensitivity of the advection |
---|
1388 | schemes to the way horizontal averages are performed in the vicinity of partial |
---|
1389 | cells should be further investigated in the near future. |
---|
1390 | %%% |
---|
1391 | \gmcomment{gm : this last remark has to be done} |
---|
1392 | %%% |
---|