[707] | 1 | % ================================================================ |
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[4147] | 2 | % Chapter 1 � Ocean Tracers (TRA) |
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[707] | 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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[817] | 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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[707] | 12 | |
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[1224] | 13 | %\newpage |
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| 14 | \vspace{2.cm} |
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| 15 | %$\ $\newline % force a new ligne |
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[707] | 16 | |
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[817] | 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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[707] | 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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[817] | 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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[707] | 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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[817] | 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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[1224] | 37 | BBC, BBL and DMP are optional. The external forcings and parameterisations |
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[817] | 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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[707] | 47 | |
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[817] | 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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[2285] | 50 | freezing point with associated modules \mdl{eosbn2} and \mdl{phycst}). |
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[707] | 51 | |
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[2282] | 52 | The different options available to the user are managed by namelist logicals or |
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[1224] | 53 | CPP keys. For each equation term \textit{ttt}, the namelist logicals are \textit{ln\_trattt\_xxx}, |
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[2282] | 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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[817] | 56 | found in the \textit{trattt} or \textit{trattt\_xxx} module, in the NEMO/OPA/TRA directory. |
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[707] | 57 | |
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[817] | 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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[707] | 60 | |
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[1224] | 61 | $\ $\newline % force a new ligne |
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[707] | 62 | % ================================================================ |
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| 63 | % Tracer Advection |
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| 64 | % ================================================================ |
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[817] | 65 | \section [Tracer Advection (\textit{traadv})] |
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| 66 | {Tracer Advection (\mdl{traadv})} |
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[707] | 67 | \label{TRA_adv} |
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[2282] | 68 | %------------------------------------------namtra_adv----------------------------------------------------- |
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| 69 | \namdisplay{namtra_adv} |
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[707] | 70 | %------------------------------------------------------------------------------------------------------------- |
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| 71 | |
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[817] | 72 | The advection tendency of a tracer in flux form is the divergence of the advective |
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[707] | 73 | fluxes. Its discrete expression is given by : |
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| 74 | \begin{equation} \label{Eq_tra_adv} |
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[2282] | 75 | ADV_\tau =-\frac{1}{b_t} \left( |
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[1224] | 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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[2282] | 78 | -\frac{1}{e_{3t}} \;\delta _k \left[ w\; \tau _w \right] |
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[707] | 79 | \end{equation} |
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[2282] | 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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[2285] | 81 | The flux form in \eqref{Eq_tra_adv} |
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[2282] | 82 | implicitly requires the use of the continuity equation. Indeed, it is obtained |
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[1224] | 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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[2286] | 85 | $ \partial _t e_3 + e_3\;\nabla \cdot \vect{U}=0$ in constant volume or variable volume case, respectively. |
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[1224] | 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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[817] | 88 | enforce the conservation properties of the continuous equations. In other words, |
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[2282] | 89 | by replacing $\tau$ by the number 1 in (\ref{Eq_tra_adv}) we recover the discrete form of |
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[817] | 90 | the continuity equation which is used to calculate the vertical velocity. |
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[707] | 91 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
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[2376] | 92 | \begin{figure}[!t] \begin{center} |
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[998] | 93 | \includegraphics[width=0.9\textwidth]{./TexFiles/Figures/Fig_adv_scheme.pdf} |
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[2376] | 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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[817] | 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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[707] | 104 | \end{center} \end{figure} |
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| 105 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
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[817] | 106 | |
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[2282] | 107 | The key difference between the advection schemes available in \NEMO is the choice |
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[817] | 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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[2282] | 111 | Along solid lateral and bottom boundaries a zero tracer flux is automatically |
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[707] | 112 | specified, since the normal velocity is zero there. At the sea surface the |
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[817] | 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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[2282] | 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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[817] | 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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[2282] | 132 | and also \citet{Roullet_Madec_JGR00,Griffies_al_MWR01,Campin2004}). |
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[707] | 133 | |
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[817] | 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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[2282] | 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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[707] | 138 | |
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[4147] | 139 | The choice of an advection scheme is made in the \textit{\ngn{nam\_traadv}} namelist, by |
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[707] | 140 | setting to \textit{true} one and only one of the logicals \textit{ln\_traadv\_xxx}. The |
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[817] | 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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[707] | 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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[817] | 147 | Note that |
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| 148 | (1) cen2, cen4 and TVD schemes require an explicit diffusion |
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[707] | 149 | operator while the other schemes are diffusive enough so that they do not |
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[817] | 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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[2286] | 155 | (3) It is recommended that the same advection-diffusion scheme is |
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[817] | 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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[2286] | 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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[707] | 163 | |
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| 164 | % ------------------------------------------------------------------------------------------------------------- |
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| 165 | % 2nd order centred scheme |
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| 166 | % ------------------------------------------------------------------------------------------------------------- |
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[817] | 167 | \subsection [$2^{nd}$ order centred scheme (cen2) (\np{ln\_traadv\_cen2})] |
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[2282] | 168 | {$2^{nd}$ order centred scheme (cen2) (\np{ln\_traadv\_cen2}=true)} |
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[707] | 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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[817] | 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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[707] | 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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[817] | 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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[1224] | 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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[2282] | 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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[1224] | 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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[2286] | 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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[707] | 193 | |
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[817] | 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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[2282] | 196 | have this order of accuracy. \gmcomment{Note also that ... blah, blah} |
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[707] | 197 | |
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| 198 | % ------------------------------------------------------------------------------------------------------------- |
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| 199 | % 4nd order centred scheme |
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| 200 | % ------------------------------------------------------------------------------------------------------------- |
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[817] | 201 | \subsection [$4^{nd}$ order centred scheme (cen4) (\np{ln\_traadv\_cen4})] |
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[2282] | 202 | {$4^{nd}$ order centred scheme (cen4) (\np{ln\_traadv\_cen4}=true)} |
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[707] | 203 | \label{TRA_adv_cen4} |
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| 204 | |
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[817] | 205 | In the $4^{th}$ order formulation (to be implemented), tracer values are |
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[2282] | 206 | evaluated at velocity points as a $4^{th}$ order interpolation, and thus depend on |
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[817] | 207 | the four neighbouring $T$-points. For example, in the $i$-direction: |
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[707] | 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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[817] | 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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[707] | 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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[817] | 220 | continuity equation and the momentum advection terms. |
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[707] | 221 | |
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| 222 | A direct consequence of the pseudo-fourth order nature of the scheme is that |
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[1224] | 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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[707] | 228 | |
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[817] | 229 | At a $T$-grid cell adjacent to a boundary (coastline, bottom and surface), an |
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[707] | 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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[817] | 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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[707] | 235 | |
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| 236 | % ------------------------------------------------------------------------------------------------------------- |
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| 237 | % TVD scheme |
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| 238 | % ------------------------------------------------------------------------------------------------------------- |
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[817] | 239 | \subsection [Total Variance Dissipation scheme (TVD) (\np{ln\_traadv\_tvd})] |
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[2282] | 240 | {Total Variance Dissipation scheme (TVD) (\np{ln\_traadv\_tvd}=true)} |
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[707] | 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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[1224] | 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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[707] | 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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[1224] | 256 | where $c_u$ is a flux limiter function taking values between 0 and 1. |
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[2282] | 257 | There exist many ways to define $c_u$, each corresponding to a different |
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[1224] | 258 | total variance decreasing scheme. The one chosen in \NEMO is described in |
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[2282] | 259 | \citet{Zalesak_JCP79}. $c_u$ only departs from $1$ when the advective term |
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[817] | 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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[2282] | 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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[707] | 265 | |
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[3764] | 266 | For stability reasons (see \S\ref{STP}), |
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[2282] | 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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[817] | 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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[707] | 271 | |
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| 272 | % ------------------------------------------------------------------------------------------------------------- |
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| 273 | % MUSCL scheme |
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| 274 | % ------------------------------------------------------------------------------------------------------------- |
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[817] | 275 | \subsection[MUSCL scheme (\np{ln\_traadv\_muscl})] |
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[707] | 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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[2282] | 280 | implemented by \citet{Levy_al_GRL01}. In its formulation, the tracer at velocity points |
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[817] | 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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[707] | 283 | \begin{equation} \label{Eq_tra_adv_muscl} |
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| 284 | \tau _u^{mus} = \left\{ \begin{aligned} |
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[2282] | 285 | &\tau _i &+ \frac{1}{2} \;\left( 1-\frac{u_{i+1/2} \;\rdt}{e_{1u}} \right) |
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[707] | 286 | &\ \widetilde{\partial _i \tau} & \quad \text{if }\;u_{i+1/2} \geqslant 0 \\ |
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[2282] | 287 | &\tau _{i+1/2} &+\frac{1}{2}\;\left( 1+\frac{u_{i+1/2} \;\rdt}{e_{1u} } \right) |
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[707] | 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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[817] | 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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[707] | 293 | |
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[817] | 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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[707] | 296 | |
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[817] | 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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[2282] | 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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[817] | 301 | the \textit{positive} character of the scheme. Only the former can be used |
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[2282] | 302 | on both active and passive tracers. The two MUSCL schemes are implemented |
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[1224] | 303 | in the \mdl{traadv\_tvd} and \mdl{traadv\_tvd2} modules. |
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[707] | 304 | |
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| 305 | % ------------------------------------------------------------------------------------------------------------- |
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| 306 | % UBS scheme |
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| 307 | % ------------------------------------------------------------------------------------------------------------- |
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[817] | 308 | \subsection [Upstream-Biased Scheme (UBS) (\np{ln\_traadv\_ubs})] |
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[2282] | 309 | {Upstream-Biased Scheme (UBS) (\np{ln\_traadv\_ubs}=true)} |
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[707] | 310 | \label{TRA_adv_ubs} |
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| 311 | |
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[817] | 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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[707] | 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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[2282] | 326 | error \citep{Shchepetkin_McWilliams_OM05}. The overall performance of the advection |
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[1224] | 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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[707] | 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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[1224] | 336 | Therefore the vertical flux is evaluated using the TVD scheme when |
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[2282] | 337 | \np{ln\_traadv\_ubs}=true. |
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[707] | 338 | |
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[3764] | 339 | For stability reasons (see \S\ref{STP}), |
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[2282] | 340 | the first term in \eqref{Eq_tra_adv_ubs} (which corresponds to a second order centred scheme) |
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[817] | 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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[2282] | 344 | This choice is discussed by \citet{Webb_al_JAOT98} in the context of the |
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[1224] | 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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[2282] | 347 | leads to the QUICK advection scheme \citep{Webb_al_JAOT98}. |
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[1224] | 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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[707] | 351 | |
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[2282] | 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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[817] | 361 | Note that : |
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| 362 | |
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[1224] | 363 | (1) When a high vertical resolution $O(1m)$ is used, the model stability can |
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[707] | 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 |
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[1224] | 366 | time-splitting technique on vertical advection. Such a technique has been |
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| 367 | implemented and validated in ORCA05 with 301 levels. It is not available |
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| 368 | in the current reference version. |
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[707] | 369 | |
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[2282] | 370 | (2) It is straightforward to rewrite \eqref{Eq_tra_adv_ubs} as follows: |
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[1224] | 371 | \begin{equation} \label{Eq_traadv_ubs2} |
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| 372 | \tau _u^{ubs} = \tau _u^{cen4} + \frac{1}{12} \left\{ |
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| 373 | \begin{aligned} |
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| 374 | & + \tau"_i & \quad \text{if }\ u_{i+1/2} \geqslant 0 \\ |
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| 375 | & - \tau"_{i+1} & \quad \text{if }\ u_{i+1/2} < 0 |
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| 376 | \end{aligned} \right. |
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[707] | 377 | \end{equation} |
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| 378 | or equivalently |
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[1224] | 379 | \begin{equation} \label{Eq_traadv_ubs2b} |
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[707] | 380 | u_{i+1/2} \ \tau _u^{ubs} |
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| 381 | =u_{i+1/2} \ \overline{ T - \frac{1}{6}\,\delta _i\left[ \delta_{i+1/2}[T] \,\right] }^{\,i+1/2} |
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| 382 | - \frac{1}{2} |u|_{i+1/2} \;\frac{1}{6} \;\delta_{i+1/2}[\tau"_i] |
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| 383 | \end{equation} |
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[1224] | 384 | |
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| 385 | \eqref{Eq_traadv_ubs2} has several advantages. Firstly, it clearly reveals |
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[817] | 386 | that the UBS scheme is based on the fourth order scheme to which an |
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| 387 | upstream-biased diffusion term is added. Secondly, this emphasises that the |
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| 388 | $4^{th}$ order part (as well as the $2^{nd}$ order part as stated above) has |
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| 389 | to be evaluated at the \emph{now} time step using \eqref{Eq_tra_adv_ubs}. |
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| 390 | Thirdly, the diffusion term is in fact a biharmonic operator with an eddy |
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| 391 | coefficient which is simply proportional to the velocity: |
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[3308] | 392 | $A_u^{lm}= - \frac{1}{12}\,{e_{1u}}^3\,|u|$. Note that NEMO v3.4 still uses |
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[2282] | 393 | \eqref{Eq_tra_adv_ubs}, not \eqref{Eq_traadv_ubs2}. |
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[817] | 394 | %%% |
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| 395 | \gmcomment{the change in UBS scheme has to be done} |
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| 396 | %%% |
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[707] | 397 | |
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| 398 | % ------------------------------------------------------------------------------------------------------------- |
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| 399 | % QCK scheme |
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| 400 | % ------------------------------------------------------------------------------------------------------------- |
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[817] | 401 | \subsection [QUICKEST scheme (QCK) (\np{ln\_traadv\_qck})] |
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[2282] | 402 | {QUICKEST scheme (QCK) (\np{ln\_traadv\_qck}=true)} |
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[707] | 403 | \label{TRA_adv_qck} |
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| 404 | |
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| 405 | The Quadratic Upstream Interpolation for Convective Kinematics with |
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[817] | 406 | Estimated Streaming Terms (QUICKEST) scheme proposed by \citet{Leonard1979} |
---|
| 407 | is the third order Godunov scheme. It is associated with the ULTIMATE QUICKEST |
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[707] | 408 | limiter \citep{Leonard1991}. It has been implemented in NEMO by G. Reffray |
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[1224] | 409 | (MERCATOR-ocean) and can be found in the \mdl{traadv\_qck} module. |
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| 410 | The resulting scheme is quite expensive but \emph{positive}. |
---|
| 411 | It can be used on both active and passive tracers. |
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[2282] | 412 | However, the intrinsic diffusion of QCK makes its use risky in the vertical |
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[1224] | 413 | direction where the control of artificial diapycnal fluxes is of paramount importance. |
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| 414 | Therefore the vertical flux is evaluated using the CEN2 scheme. |
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[2282] | 415 | This no longer guarantees the positivity of the scheme. The use of TVD in the vertical |
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| 416 | direction (as for the UBS case) should be implemented to restore this property. |
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[707] | 417 | |
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| 418 | |
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| 419 | % ------------------------------------------------------------------------------------------------------------- |
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| 420 | % PPM scheme |
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| 421 | % ------------------------------------------------------------------------------------------------------------- |
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[817] | 422 | \subsection [Piecewise Parabolic Method (PPM) (\np{ln\_traadv\_ppm})] |
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[2282] | 423 | {Piecewise Parabolic Method (PPM) (\np{ln\_traadv\_ppm}=true)} |
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[707] | 424 | \label{TRA_adv_ppm} |
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| 425 | |
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| 426 | The Piecewise Parabolic Method (PPM) proposed by Colella and Woodward (1984) |
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[2282] | 427 | \sgacomment{reference?} |
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| 428 | is based on a quadradic piecewise construction. Like the QCK scheme, it is associated |
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[817] | 429 | with the ULTIMATE QUICKEST limiter \citep{Leonard1991}. It has been implemented |
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| 430 | in \NEMO by G. Reffray (MERCATOR-ocean) but is not yet offered in the reference |
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[2282] | 431 | version 3.3. |
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[707] | 432 | |
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| 433 | % ================================================================ |
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| 434 | % Tracer Lateral Diffusion |
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| 435 | % ================================================================ |
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[817] | 436 | \section [Tracer Lateral Diffusion (\textit{traldf})] |
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| 437 | {Tracer Lateral Diffusion (\mdl{traldf})} |
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[707] | 438 | \label{TRA_ldf} |
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| 439 | %-----------------------------------------nam_traldf------------------------------------------------------ |
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[2282] | 440 | \namdisplay{namtra_ldf} |
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[707] | 441 | %------------------------------------------------------------------------------------------------------------- |
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| 442 | |
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[4147] | 443 | Options are defined through the \ngn{namtra\_ldf} namelist variables. |
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[817] | 444 | The options available for lateral diffusion are a laplacian (rotated or not) |
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| 445 | or a biharmonic operator, the latter being more scale-selective (more |
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[707] | 446 | diffusive at small scales). The specification of eddy diffusivity |
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[817] | 447 | coefficients (either constant or variable in space and time) as well as the |
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| 448 | computation of the slope along which the operators act, are performed in the |
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[1224] | 449 | \mdl{ldftra} and \mdl{ldfslp} modules, respectively. This is described in Chap.~\ref{LDF}. |
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| 450 | The lateral diffusion of tracers is evaluated using a forward scheme, |
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[817] | 451 | $i.e.$ the tracers appearing in its expression are the \textit{before} tracers in time, |
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| 452 | except for the pure vertical component that appears when a rotation tensor |
---|
| 453 | is used. This latter term is solved implicitly together with the |
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[3764] | 454 | vertical diffusion term (see \S\ref{STP}). |
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[707] | 455 | |
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| 456 | % ------------------------------------------------------------------------------------------------------------- |
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| 457 | % Iso-level laplacian operator |
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| 458 | % ------------------------------------------------------------------------------------------------------------- |
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[1224] | 459 | \subsection [Iso-level laplacian operator (lap) (\np{ln\_traldf\_lap})] |
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[2282] | 460 | {Iso-level laplacian operator (lap) (\np{ln\_traldf\_lap}=true) } |
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[707] | 461 | \label{TRA_ldf_lap} |
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| 462 | |
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[1224] | 463 | A laplacian diffusion operator ($i.e.$ a harmonic operator) acting along the model |
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[817] | 464 | surfaces is given by: |
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[707] | 465 | \begin{equation} \label{Eq_tra_ldf_lap} |
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[2282] | 466 | D_T^{lT} =\frac{1}{b_tT} \left( \; |
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[1224] | 467 | \delta _{i}\left[ A_u^{lT} \; \frac{e_{2u}\,e_{3u}}{e_{1u}} \;\delta _{i+1/2} [T] \right] |
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| 468 | + \delta _{j}\left[ A_v^{lT} \; \frac{e_{1v}\,e_{3v}}{e_{2v}} \;\delta _{j+1/2} [T] \right] \;\right) |
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[707] | 469 | \end{equation} |
---|
[2282] | 470 | where $b_t$=$e_{1t}\,e_{2t}\,e_{3t}$ is the volume of $T$-cells. |
---|
| 471 | It is implemented in the \mdl{traadv\_lap} module. |
---|
[707] | 472 | |
---|
[1224] | 473 | This lateral operator is computed in \mdl{traldf\_lap}. It is a \emph{horizontal} |
---|
| 474 | operator ($i.e.$ acting along geopotential surfaces) in the $z$-coordinate with |
---|
[2282] | 475 | or without partial steps, but is simply an iso-level operator in the $s$-coordinate. |
---|
| 476 | It is thus used when, in addition to \np{ln\_traldf\_lap}=true, we have |
---|
| 477 | \np{ln\_traldf\_level}=true or \np{ln\_traldf\_hor}=\np{ln\_zco}=true. |
---|
[1224] | 478 | In both cases, it significantly contributes to diapycnal mixing. |
---|
| 479 | It is therefore not recommended. |
---|
[707] | 480 | |
---|
[2285] | 481 | Note that in the partial step $z$-coordinate (\np{ln\_zps}=true), tracers in horizontally |
---|
[817] | 482 | adjacent cells are located at different depths in the vicinity of the bottom. |
---|
| 483 | In this case, horizontal derivatives in (\ref{Eq_tra_ldf_lap}) at the bottom level |
---|
| 484 | require a specific treatment. They are calculated in the \mdl{zpshde} module, |
---|
| 485 | described in \S\ref{TRA_zpshde}. |
---|
[707] | 486 | |
---|
| 487 | % ------------------------------------------------------------------------------------------------------------- |
---|
| 488 | % Rotated laplacian operator |
---|
| 489 | % ------------------------------------------------------------------------------------------------------------- |
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[1224] | 490 | \subsection [Rotated laplacian operator (iso) (\np{ln\_traldf\_lap})] |
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[2282] | 491 | {Rotated laplacian operator (iso) (\np{ln\_traldf\_lap}=true)} |
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[707] | 492 | \label{TRA_ldf_iso} |
---|
| 493 | |
---|
[3294] | 494 | If the Griffies trad scheme is not employed |
---|
| 495 | (\np{ln\_traldf\_grif}=true; see App.\ref{sec:triad}) the general form of the second order lateral tracer subgrid scale physics |
---|
[817] | 496 | (\ref{Eq_PE_zdf}) takes the following semi-discrete space form in $z$- and |
---|
| 497 | $s$-coordinates: |
---|
[707] | 498 | \begin{equation} \label{Eq_tra_ldf_iso} |
---|
| 499 | \begin{split} |
---|
[2282] | 500 | D_T^{lT} = \frac{1}{b_t} & \left\{ \,\;\delta_i \left[ A_u^{lT} \left( |
---|
| 501 | \frac{e_{2u}\,e_{3u}}{e_{1u}} \,\delta_{i+1/2}[T] |
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[1224] | 502 | - e_{2u}\;r_{1u} \,\overline{\overline{ \delta_{k+1/2}[T] }}^{\,i+1/2,k} |
---|
| 503 | \right) \right] \right. \\ |
---|
| 504 | & +\delta_j \left[ A_v^{lT} \left( |
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| 505 | \frac{e_{1v}\,e_{3v}}{e_{2v}} \,\delta_{j+1/2} [T] |
---|
| 506 | - e_{1v}\,r_{2v} \,\overline{\overline{ \delta_{k+1/2} [T] }}^{\,j+1/2,k} |
---|
| 507 | \right) \right] \\ |
---|
| 508 | & +\delta_k \left[ A_w^{lT} \left( |
---|
| 509 | -\;e_{2w}\,r_{1w} \,\overline{\overline{ \delta_{i+1/2} [T] }}^{\,i,k+1/2} |
---|
| 510 | \right. \right. \\ |
---|
[707] | 511 | & \qquad \qquad \quad |
---|
[1224] | 512 | - e_{1w}\,r_{2w} \,\overline{\overline{ \delta_{j+1/2} [T] }}^{\,j,k+1/2} \\ |
---|
| 513 | & \left. {\left. { \qquad \qquad \ \ \ \left. { |
---|
| 514 | +\;\frac{e_{1w}\,e_{2w}}{e_{3w}} \,\left( r_{1w}^2 + r_{2w}^2 \right) |
---|
| 515 | \,\delta_{k+1/2} [T] } \right) } \right] \quad } \right\} |
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[707] | 516 | \end{split} |
---|
| 517 | \end{equation} |
---|
[2282] | 518 | where $b_t$=$e_{1t}\,e_{2t}\,e_{3t}$ is the volume of $T$-cells, |
---|
[1224] | 519 | $r_1$ and $r_2$ are the slopes between the surface of computation |
---|
[817] | 520 | ($z$- or $s$-surfaces) and the surface along which the diffusion operator |
---|
| 521 | acts ($i.e.$ horizontal or iso-neutral surfaces). It is thus used when, |
---|
[2282] | 522 | in addition to \np{ln\_traldf\_lap}= true, we have \np{ln\_traldf\_iso}=true, |
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| 523 | or both \np{ln\_traldf\_hor}=true and \np{ln\_zco}=true. The way these |
---|
[817] | 524 | slopes are evaluated is given in \S\ref{LDF_slp}. At the surface, bottom |
---|
| 525 | and lateral boundaries, the turbulent fluxes of heat and salt are set to zero |
---|
| 526 | using the mask technique (see \S\ref{LBC_coast}). |
---|
[707] | 527 | |
---|
[817] | 528 | The operator in \eqref{Eq_tra_ldf_iso} involves both lateral and vertical |
---|
| 529 | derivatives. For numerical stability, the vertical second derivative must |
---|
| 530 | be solved using the same implicit time scheme as that used in the vertical |
---|
| 531 | physics (see \S\ref{TRA_zdf}). For computer efficiency reasons, this term |
---|
[1224] | 532 | is not computed in the \mdl{traldf\_iso} module, but in the \mdl{trazdf} module |
---|
[817] | 533 | where, if iso-neutral mixing is used, the vertical mixing coefficient is simply |
---|
| 534 | increased by $\frac{e_{1w}\,e_{2w} }{e_{3w} }\ \left( {r_{1w} ^2+r_{2w} ^2} \right)$. |
---|
[707] | 535 | |
---|
[817] | 536 | This formulation conserves the tracer but does not ensure the decrease |
---|
| 537 | of the tracer variance. Nevertheless the treatment performed on the slopes |
---|
| 538 | (see \S\ref{LDF}) allows the model to run safely without any additional |
---|
[2282] | 539 | background horizontal diffusion \citep{Guilyardi_al_CD01}. An alternative scheme |
---|
[3294] | 540 | developed by \cite{Griffies_al_JPO98} which ensures tracer variance decreases |
---|
[2282] | 541 | is also available in \NEMO (\np{ln\_traldf\_grif}=true). A complete description of |
---|
[3294] | 542 | the algorithm is given in App.\ref{sec:triad}. |
---|
[707] | 543 | |
---|
[2282] | 544 | Note that in the partial step $z$-coordinate (\np{ln\_zps}=true), the horizontal |
---|
[817] | 545 | derivatives at the bottom level in \eqref{Eq_tra_ldf_iso} require a specific |
---|
| 546 | treatment. They are calculated in module zpshde, described in \S\ref{TRA_zpshde}. |
---|
[707] | 547 | |
---|
| 548 | % ------------------------------------------------------------------------------------------------------------- |
---|
| 549 | % Iso-level bilaplacian operator |
---|
| 550 | % ------------------------------------------------------------------------------------------------------------- |
---|
[1224] | 551 | \subsection [Iso-level bilaplacian operator (bilap) (\np{ln\_traldf\_bilap})] |
---|
[2282] | 552 | {Iso-level bilaplacian operator (bilap) (\np{ln\_traldf\_bilap}=true)} |
---|
[707] | 553 | \label{TRA_ldf_bilap} |
---|
| 554 | |
---|
[817] | 555 | The lateral fourth order bilaplacian operator on tracers is obtained by |
---|
[2282] | 556 | applying (\ref{Eq_tra_ldf_lap}) twice. The operator requires an additional assumption |
---|
| 557 | on boundary conditions: both first and third derivative terms normal to the |
---|
| 558 | coast are set to zero. It is used when, in addition to \np{ln\_traldf\_bilap}=true, |
---|
| 559 | we have \np{ln\_traldf\_level}=true, or both \np{ln\_traldf\_hor}=true and |
---|
| 560 | \np{ln\_zco}=false. In both cases, it can contribute diapycnal mixing, |
---|
[817] | 561 | although less than in the laplacian case. It is therefore not recommended. |
---|
[707] | 562 | |
---|
[817] | 563 | Note that in the code, the bilaplacian routine does not call the laplacian |
---|
[1224] | 564 | routine twice but is rather a separate routine that can be found in the |
---|
| 565 | \mdl{traldf\_bilap} module. This is due to the fact that we introduce the |
---|
| 566 | eddy diffusivity coefficient, A, in the operator as: |
---|
| 567 | $\nabla \cdot \nabla \left( {A\nabla \cdot \nabla T} \right)$, |
---|
| 568 | instead of |
---|
| 569 | $-\nabla \cdot a\nabla \left( {\nabla \cdot a\nabla T} \right)$ |
---|
| 570 | where $a=\sqrt{|A|}$ and $A<0$. This was a mistake: both formulations |
---|
| 571 | ensure the total variance decrease, but the former requires a larger |
---|
[2282] | 572 | number of code-lines. |
---|
[707] | 573 | |
---|
| 574 | % ------------------------------------------------------------------------------------------------------------- |
---|
| 575 | % Rotated bilaplacian operator |
---|
| 576 | % ------------------------------------------------------------------------------------------------------------- |
---|
[1224] | 577 | \subsection [Rotated bilaplacian operator (bilapg) (\np{ln\_traldf\_bilap})] |
---|
[2282] | 578 | {Rotated bilaplacian operator (bilapg) (\np{ln\_traldf\_bilap}=true)} |
---|
[707] | 579 | \label{TRA_ldf_bilapg} |
---|
| 580 | |
---|
[817] | 581 | The lateral fourth order operator formulation on tracers is obtained by |
---|
| 582 | applying (\ref{Eq_tra_ldf_iso}) twice. It requires an additional assumption |
---|
| 583 | on boundary conditions: first and third derivative terms normal to the |
---|
[2282] | 584 | coast, normal to the bottom and normal to the surface are set to zero. It can be found in the |
---|
[1224] | 585 | \mdl{traldf\_bilapg}. |
---|
[707] | 586 | |
---|
[2282] | 587 | It is used when, in addition to \np{ln\_traldf\_bilap}=true, we have |
---|
| 588 | \np{ln\_traldf\_iso}= .true, or both \np{ln\_traldf\_hor}=true and \np{ln\_zco}=true. |
---|
| 589 | This rotated bilaplacian operator has never been seriously |
---|
| 590 | tested. There are no guarantees that it is either free of bugs or correctly formulated. |
---|
[817] | 591 | Moreover, the stability range of such an operator will be probably quite |
---|
[2282] | 592 | narrow, requiring a significantly smaller time-step than the one used with an |
---|
[817] | 593 | unrotated operator. |
---|
[707] | 594 | |
---|
| 595 | % ================================================================ |
---|
| 596 | % Tracer Vertical Diffusion |
---|
| 597 | % ================================================================ |
---|
[817] | 598 | \section [Tracer Vertical Diffusion (\textit{trazdf})] |
---|
| 599 | {Tracer Vertical Diffusion (\mdl{trazdf})} |
---|
[707] | 600 | \label{TRA_zdf} |
---|
| 601 | %--------------------------------------------namzdf--------------------------------------------------------- |
---|
| 602 | \namdisplay{namzdf} |
---|
| 603 | %-------------------------------------------------------------------------------------------------------------- |
---|
| 604 | |
---|
[4147] | 605 | Options are defined through the \ngn{namzdf} namelist variables. |
---|
[817] | 606 | The formulation of the vertical subgrid scale tracer physics is the same |
---|
| 607 | for all the vertical coordinates, and is based on a laplacian operator. |
---|
| 608 | The vertical diffusion operator given by (\ref{Eq_PE_zdf}) takes the |
---|
| 609 | following semi-discrete space form: |
---|
[707] | 610 | \begin{equation} \label{Eq_tra_zdf} |
---|
| 611 | \begin{split} |
---|
[2282] | 612 | D^{vT}_T &= \frac{1}{e_{3t}} \; \delta_k \left[ \;\frac{A^{vT}_w}{e_{3w}} \delta_{k+1/2}[T] \;\right] |
---|
[707] | 613 | \\ |
---|
[2282] | 614 | D^{vS}_T &= \frac{1}{e_{3t}} \; \delta_k \left[ \;\frac{A^{vS}_w}{e_{3w}} \delta_{k+1/2}[S] \;\right] |
---|
[707] | 615 | \end{split} |
---|
| 616 | \end{equation} |
---|
| 617 | where $A_w^{vT}$ and $A_w^{vS}$ are the vertical eddy diffusivity |
---|
[1224] | 618 | coefficients on temperature and salinity, respectively. Generally, |
---|
[817] | 619 | $A_w^{vT}=A_w^{vS}$ except when double diffusive mixing is |
---|
[1224] | 620 | parameterised ($i.e.$ \key{zdfddm} is defined). The way these coefficients |
---|
[817] | 621 | are evaluated is given in \S\ref{ZDF} (ZDF). Furthermore, when |
---|
| 622 | iso-neutral mixing is used, both mixing coefficients are increased |
---|
| 623 | by $\frac{e_{1w}\,e_{2w} }{e_{3w} }\ \left( {r_{1w} ^2+r_{2w} ^2} \right)$ |
---|
| 624 | to account for the vertical second derivative of \eqref{Eq_tra_ldf_iso}. |
---|
[707] | 625 | |
---|
[817] | 626 | At the surface and bottom boundaries, the turbulent fluxes of |
---|
[1224] | 627 | heat and salt must be specified. At the surface they are prescribed |
---|
| 628 | from the surface forcing and added in a dedicated routine (see \S\ref{TRA_sbc}), |
---|
[817] | 629 | whilst at the bottom they are set to zero for heat and salt unless |
---|
| 630 | a geothermal flux forcing is prescribed as a bottom boundary |
---|
[1224] | 631 | condition (see \S\ref{TRA_bbc}). |
---|
[707] | 632 | |
---|
| 633 | The large eddy coefficient found in the mixed layer together with high |
---|
[817] | 634 | vertical resolution implies that in the case of explicit time stepping |
---|
[2282] | 635 | (\np{ln\_zdfexp}=true) there would be too restrictive a constraint on |
---|
[817] | 636 | the time step. Therefore, the default implicit time stepping is preferred |
---|
| 637 | for the vertical diffusion since it overcomes the stability constraint. |
---|
[2282] | 638 | A forward time differencing scheme (\np{ln\_zdfexp}=true) using a time |
---|
| 639 | splitting technique (\np{nn\_zdfexp} $> 1$) is provided as an alternative. |
---|
| 640 | Namelist variables \np{ln\_zdfexp} and \np{nn\_zdfexp} apply to both |
---|
[817] | 641 | tracers and dynamics. |
---|
[707] | 642 | |
---|
| 643 | % ================================================================ |
---|
| 644 | % External Forcing |
---|
| 645 | % ================================================================ |
---|
| 646 | \section{External Forcing} |
---|
| 647 | \label{TRA_sbc_qsr_bbc} |
---|
| 648 | |
---|
| 649 | % ------------------------------------------------------------------------------------------------------------- |
---|
| 650 | % surface boundary condition |
---|
| 651 | % ------------------------------------------------------------------------------------------------------------- |
---|
[817] | 652 | \subsection [Surface boundary condition (\textit{trasbc})] |
---|
| 653 | {Surface boundary condition (\mdl{trasbc})} |
---|
[707] | 654 | \label{TRA_sbc} |
---|
| 655 | |
---|
| 656 | The surface boundary condition for tracers is implemented in a separate |
---|
| 657 | module (\mdl{trasbc}) instead of entering as a boundary condition on the vertical |
---|
| 658 | diffusion operator (as in the case of momentum). This has been found to |
---|
| 659 | enhance readability of the code. The two formulations are completely |
---|
| 660 | equivalent; the forcing terms in trasbc are the surface fluxes divided by |
---|
[2282] | 661 | the thickness of the top model layer. |
---|
[707] | 662 | |
---|
[2282] | 663 | Due to interactions and mass exchange of water ($F_{mass}$) with other Earth system components ($i.e.$ atmosphere, sea-ice, land), |
---|
| 664 | the change in the heat and salt content of the surface layer of the ocean is due both |
---|
| 665 | to the heat and salt fluxes crossing the sea surface (not linked with $F_{mass}$) |
---|
| 666 | and to the heat and salt content of the mass exchange. |
---|
| 667 | \sgacomment{ the following does not apply to the release to which this documentation is |
---|
| 668 | attached and so should not be included .... |
---|
| 669 | In a forthcoming release, these two parts, computed in the surface module (SBC), will be included directly |
---|
| 670 | in $Q_{ns}$, the surface heat flux and $F_{salt}$, the surface salt flux. |
---|
| 671 | The specification of these fluxes is further detailed in the SBC chapter (see \S\ref{SBC}). |
---|
| 672 | This change will provide a forcing formulation which is the same for any tracer (including temperature and salinity). |
---|
| 673 | |
---|
| 674 | In the current version, the situation is a little bit more complicated. } |
---|
[2286] | 675 | |
---|
[2282] | 676 | The surface module (\mdl{sbcmod}, see \S\ref{SBC}) provides the following |
---|
| 677 | forcing fields (used on tracers): |
---|
[817] | 678 | |
---|
[2282] | 679 | $\bullet$ $Q_{ns}$, the non-solar part of the net surface heat flux that crosses the sea surface |
---|
| 680 | (i.e. the difference between the total surface heat flux and the fraction of the short wave flux that |
---|
| 681 | penetrates into the water column, see \S\ref{TRA_qsr}) |
---|
| 682 | |
---|
| 683 | $\bullet$ \textit{emp}, the mass flux exchanged with the atmosphere (evaporation minus precipitation) |
---|
| 684 | |
---|
| 685 | $\bullet$ $\textit{emp}_S$, an equivalent mass flux taking into account the effect of ice-ocean mass exchange |
---|
| 686 | |
---|
| 687 | $\bullet$ \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) |
---|
| 688 | |
---|
| 689 | The $\textit{emp}_S$ field is not simply the budget of evaporation-precipitation+freezing-melting because |
---|
| 690 | the sea-ice is not currently embedded in the ocean but levitates above it. There is no mass |
---|
| 691 | exchanged between the sea-ice and the ocean. Instead we only take into account the salt |
---|
| 692 | flux associated with the non-zero salinity of sea-ice, and the concentration/dilution effect |
---|
| 693 | due to the freezing/melting (F/M) process. These two parts of the forcing are then converted into |
---|
| 694 | an equivalent mass flux given by $\textit{emp}_S - \textit{emp}$. As a result of this mess, |
---|
| 695 | the surface boundary condition on temperature and salinity is applied as follows: |
---|
| 696 | |
---|
| 697 | In the nonlinear free surface case (\key{vvl} is defined): |
---|
| 698 | \begin{equation} \label{Eq_tra_sbc} |
---|
[707] | 699 | \begin{aligned} |
---|
[2282] | 700 | &F^T = \frac{ 1 }{\rho _o \;C_p \,\left. e_{3t} \right|_{k=1} } |
---|
| 701 | &\overline{ \left( Q_{ns} - \textit{emp}\;C_p\,\left. T \right|_{k=1} \right) }^t & \\ |
---|
| 702 | % |
---|
| 703 | & F^S =\frac{ 1 }{\rho _o \,\left. e_{3t} \right|_{k=1} } |
---|
| 704 | &\overline{ \left( (\textit{emp}_S - \textit{emp})\;\left. S \right|_{k=1} \right) }^t & \\ |
---|
[707] | 705 | \end{aligned} |
---|
| 706 | \end{equation} |
---|
| 707 | |
---|
[2282] | 708 | In the linear free surface case (\key{vvl} not defined): |
---|
| 709 | \begin{equation} \label{Eq_tra_sbc_lin} |
---|
| 710 | \begin{aligned} |
---|
| 711 | &F^T = \frac{ 1 }{\rho _o \;C_p \,\left. e_{3t} \right|_{k=1} } &\overline{ Q_{ns} }^t & \\ |
---|
| 712 | % |
---|
| 713 | & F^S =\frac{ 1 }{\rho _o \,\left. e_{3t} \right|_{k=1} } |
---|
| 714 | &\overline{ \left( \textit{emp}_S\;\left. S \right|_{k=1} \right) }^t & \\ |
---|
| 715 | \end{aligned} |
---|
[707] | 716 | \end{equation} |
---|
[2282] | 717 | where $\overline{x }^t$ means that $x$ is averaged over two consecutive time steps |
---|
| 718 | ($t-\rdt/2$ and $t+\rdt/2$). Such time averaging prevents the |
---|
| 719 | divergence of odd and even time step (see \S\ref{STP}). |
---|
[707] | 720 | |
---|
[2282] | 721 | The two set of equations, \eqref{Eq_tra_sbc} and \eqref{Eq_tra_sbc_lin}, are obtained |
---|
| 722 | by assuming that the temperature of precipitation and evaporation are equal to |
---|
| 723 | the ocean surface temperature and that their salinity is zero. Therefore, the heat content |
---|
| 724 | of the \textit{emp} budget must be added to the temperature equation in the variable volume case, |
---|
| 725 | while it does not appear in the constant volume case. Similarly, the \textit{emp} budget affects |
---|
| 726 | the ocean surface salinity in the constant volume case (through the concentration dilution effect) |
---|
| 727 | while it does not appears explicitly in the variable volume case since salinity change will be |
---|
| 728 | induced by volume change. In both constant and variable volume cases, surface salinity |
---|
| 729 | will change with ice-ocean salt flux and F/M flux (both contained in $\textit{emp}_S - \textit{emp}$) without mass exchanges. |
---|
[707] | 730 | |
---|
[2282] | 731 | Note that the concentration/dilution effect due to F/M is computed using |
---|
| 732 | a constant ice salinity as well as a constant ocean salinity. |
---|
| 733 | This approximation suppresses the correlation between \textit{SSS} |
---|
| 734 | and F/M flux, allowing the ice-ocean salt exchanges to be conservative. |
---|
| 735 | Indeed, if this approximation is not made, even if the F/M budget is zero |
---|
| 736 | on average over the whole ocean domain and over the seasonal cycle, |
---|
| 737 | the associated salt flux is not zero, since sea-surface salinity and F/M flux are |
---|
| 738 | intrinsically correlated (high \textit{SSS} are found where freezing is |
---|
| 739 | strong whilst low \textit{SSS} is usually associated with high melting areas). |
---|
[707] | 740 | |
---|
[2282] | 741 | Even using this approximation, an exact conservation of heat and salt content |
---|
| 742 | is only achieved in the variable volume case. In the constant volume case, |
---|
| 743 | there is a small imbalance associated with the product $(\partial_t\eta - \textit{emp}) * \textit{SSS}$. |
---|
| 744 | Nevertheless, the salt content variation is quite small and will not induce |
---|
| 745 | a long term drift as there is no physical reason for $(\partial_t\eta - \textit{emp})$ |
---|
| 746 | and \textit{SSS} to be correlated \citep{Roullet_Madec_JGR00}. |
---|
| 747 | Note that, while quite small, the imbalance in the constant volume case is larger |
---|
| 748 | than the imbalance associated with the Asselin time filter \citep{Leclair_Madec_OM09}. |
---|
| 749 | This is the reason why the modified filter is not applied in the constant volume case. |
---|
[707] | 750 | |
---|
| 751 | % ------------------------------------------------------------------------------------------------------------- |
---|
| 752 | % Solar Radiation Penetration |
---|
| 753 | % ------------------------------------------------------------------------------------------------------------- |
---|
[817] | 754 | \subsection [Solar Radiation Penetration (\textit{traqsr})] |
---|
| 755 | {Solar Radiation Penetration (\mdl{traqsr})} |
---|
[707] | 756 | \label{TRA_qsr} |
---|
[817] | 757 | %--------------------------------------------namqsr-------------------------------------------------------- |
---|
[2282] | 758 | \namdisplay{namtra_qsr} |
---|
[707] | 759 | %-------------------------------------------------------------------------------------------------------------- |
---|
| 760 | |
---|
[4147] | 761 | Options are defined through the \ngn{namtra\_qsr} namelist variables. |
---|
[2282] | 762 | When the penetrative solar radiation option is used (\np{ln\_flxqsr}=true), |
---|
| 763 | the solar radiation penetrates the top few tens of meters of the ocean. If it is not used |
---|
| 764 | (\np{ln\_flxqsr}=false) all the heat flux is absorbed in the first ocean level. |
---|
[817] | 765 | Thus, in the former case a term is added to the time evolution equation of |
---|
[2282] | 766 | temperature \eqref{Eq_PE_tra_T} and the surface boundary condition is |
---|
[817] | 767 | modified to take into account only the non-penetrative part of the surface |
---|
| 768 | heat flux: |
---|
[707] | 769 | \begin{equation} \label{Eq_PE_qsr} |
---|
| 770 | \begin{split} |
---|
| 771 | \frac{\partial T}{\partial t} &= {\ldots} + \frac{1}{\rho_o\, C_p \,e_3} \; \frac{\partial I}{\partial k} \\ |
---|
| 772 | Q_{ns} &= Q_\text{Total} - Q_{sr} |
---|
| 773 | \end{split} |
---|
| 774 | \end{equation} |
---|
[2282] | 775 | where $Q_{sr}$ is the penetrative part of the surface heat flux ($i.e.$ the shortwave radiation) |
---|
| 776 | and $I$ is the downward irradiance ($\left. I \right|_{z=\eta}=Q_{sr}$). |
---|
| 777 | The additional term in \eqref{Eq_PE_qsr} is discretized as follows: |
---|
[707] | 778 | \begin{equation} \label{Eq_tra_qsr} |
---|
[2282] | 779 | \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] |
---|
[707] | 780 | \end{equation} |
---|
| 781 | |
---|
[2282] | 782 | The shortwave radiation, $Q_{sr}$, consists of energy distributed across a wide spectral range. |
---|
| 783 | The ocean is strongly absorbing for wavelengths longer than 700~nm and these |
---|
| 784 | wavelengths contribute to heating the upper few tens of centimetres. The fraction of $Q_{sr}$ |
---|
| 785 | that resides in these almost non-penetrative wavebands, $R$, is $\sim 58\%$ (specified |
---|
| 786 | through namelist parameter \np{rn\_abs}). It is assumed to penetrate the ocean |
---|
| 787 | with a decreasing exponential profile, with an e-folding depth scale, $\xi_0$, |
---|
| 788 | of a few tens of centimetres (typically $\xi_0=0.35~m$ set as \np{rn\_si0} in the namtra\_qsr namelist). |
---|
| 789 | For shorter wavelengths (400-700~nm), the ocean is more transparent, and solar energy |
---|
| 790 | propagates to larger depths where it contributes to |
---|
| 791 | local heating. |
---|
| 792 | The way this second part of the solar energy penetrates into the ocean depends on |
---|
| 793 | which formulation is chosen. In the simple 2-waveband light penetration scheme (\np{ln\_qsr\_2bd}=true) |
---|
| 794 | a chlorophyll-independent monochromatic formulation is chosen for the shorter wavelengths, |
---|
| 795 | leading to the following expression \citep{Paulson1977}: |
---|
[707] | 796 | \begin{equation} \label{Eq_traqsr_iradiance} |
---|
[2282] | 797 | I(z) = Q_{sr} \left[Re^{-z / \xi_0} + \left( 1-R\right) e^{-z / \xi_1} \right] |
---|
[707] | 798 | \end{equation} |
---|
[2282] | 799 | where $\xi_1$ is the second extinction length scale associated with the shorter wavelengths. |
---|
| 800 | It is usually chosen to be 23~m by setting the \np{rn\_si0} namelist parameter. |
---|
| 801 | The set of default values ($\xi_0$, $\xi_1$, $R$) corresponds to a Type I water in |
---|
| 802 | Jerlov's (1968) classification (oligotrophic waters). |
---|
| 803 | |
---|
| 804 | Such assumptions have been shown to provide a very crude and simplistic |
---|
| 805 | representation of observed light penetration profiles (\cite{Morel_JGR88}, see also |
---|
| 806 | Fig.\ref{Fig_traqsr_irradiance}). Light absorption in the ocean depends on |
---|
| 807 | particle concentration and is spectrally selective. \cite{Morel_JGR88} has shown |
---|
| 808 | that an accurate representation of light penetration can be provided by a 61 waveband |
---|
| 809 | formulation. Unfortunately, such a model is very computationally expensive. |
---|
| 810 | Thus, \cite{Lengaigne_al_CD07} have constructed a simplified version of this |
---|
| 811 | formulation in which visible light is split into three wavebands: blue (400-500 nm), |
---|
| 812 | green (500-600 nm) and red (600-700nm). For each wave-band, the chlorophyll-dependent |
---|
| 813 | attenuation coefficient is fitted to the coefficients computed from the full spectral model |
---|
| 814 | of \cite{Morel_JGR88} (as modified by \cite{Morel_Maritorena_JGR01}), assuming |
---|
| 815 | the same power-law relationship. As shown in Fig.\ref{Fig_traqsr_irradiance}, |
---|
| 816 | this formulation, called RGB (Red-Green-Blue), reproduces quite closely |
---|
| 817 | the light penetration profiles predicted by the full spectal model, but with much greater |
---|
| 818 | computational efficiency. The 2-bands formulation does not reproduce the full model very well. |
---|
| 819 | |
---|
| 820 | The RGB formulation is used when \np{ln\_qsr\_rgb}=true. The RGB attenuation coefficients |
---|
| 821 | ($i.e.$ the inverses of the extinction length scales) are tabulated over 61 nonuniform |
---|
| 822 | chlorophyll classes ranging from 0.01 to 10 g.Chl/L (see the routine \rou{trc\_oce\_rgb} |
---|
| 823 | in \mdl{trc\_oce} module). Three types of chlorophyll can be chosen in the RGB formulation: |
---|
| 824 | (1) a constant 0.05 g.Chl/L value everywhere (\np{nn\_chdta}=0) ; (2) an observed |
---|
| 825 | time varying chlorophyll (\np{nn\_chdta}=1) ; (3) simulated time varying chlorophyll |
---|
| 826 | by TOP biogeochemical model (\np{ln\_qsr\_bio}=true). In the latter case, the RGB |
---|
| 827 | formulation is used to calculate both the phytoplankton light limitation in PISCES |
---|
| 828 | or LOBSTER and the oceanic heating rate. |
---|
| 829 | |
---|
| 830 | The trend in \eqref{Eq_tra_qsr} associated with the penetration of the solar radiation |
---|
| 831 | is added to the temperature trend, and the surface heat flux is modified in routine \mdl{traqsr}. |
---|
| 832 | |
---|
| 833 | When the $z$-coordinate is preferred to the $s$-coordinate, the depth of $w-$levels does |
---|
| 834 | not significantly vary with location. The level at which the light has been totally |
---|
| 835 | absorbed ($i.e.$ it is less than the computer precision) is computed once, |
---|
| 836 | and the trend associated with the penetration of the solar radiation is only added down to that level. |
---|
| 837 | Finally, note that when the ocean is shallow ($<$ 200~m), part of the |
---|
[817] | 838 | solar radiation can reach the ocean floor. In this case, we have |
---|
| 839 | chosen that all remaining radiation is absorbed in the last ocean |
---|
[2282] | 840 | level ($i.e.$ $I$ is masked). |
---|
[707] | 841 | |
---|
[2282] | 842 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
---|
[2376] | 843 | \begin{figure}[!t] \begin{center} |
---|
[2282] | 844 | \includegraphics[width=1.0\textwidth]{./TexFiles/Figures/Fig_TRA_Irradiance.pdf} |
---|
[2376] | 845 | \caption{ \label{Fig_traqsr_irradiance} |
---|
| 846 | Penetration profile of the downward solar irradiance calculated by four models. |
---|
| 847 | Two waveband chlorophyll-independent formulation (blue), a chlorophyll-dependent |
---|
| 848 | monochromatic formulation (green), 4 waveband RGB formulation (red), |
---|
[2282] | 849 | 61 waveband Morel (1988) formulation (black) for a chlorophyll concentration of |
---|
| 850 | (a) Chl=0.05 mg/m$^3$ and (b) Chl=0.5 mg/m$^3$. From \citet{Lengaigne_al_CD07}.} |
---|
| 851 | \end{center} \end{figure} |
---|
| 852 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
---|
[707] | 853 | |
---|
| 854 | % ------------------------------------------------------------------------------------------------------------- |
---|
| 855 | % Bottom Boundary Condition |
---|
| 856 | % ------------------------------------------------------------------------------------------------------------- |
---|
[2349] | 857 | \subsection [Bottom Boundary Condition (\textit{trabbc})] |
---|
| 858 | {Bottom Boundary Condition (\mdl{trabbc})} |
---|
[707] | 859 | \label{TRA_bbc} |
---|
| 860 | %--------------------------------------------nambbc-------------------------------------------------------- |
---|
[2282] | 861 | \namdisplay{namtra_bbc} |
---|
[707] | 862 | %-------------------------------------------------------------------------------------------------------------- |
---|
| 863 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
---|
[2376] | 864 | \begin{figure}[!t] \begin{center} |
---|
[998] | 865 | \includegraphics[width=1.0\textwidth]{./TexFiles/Figures/Fig_TRA_geoth.pdf} |
---|
[2376] | 866 | \caption{ \label{Fig_geothermal} |
---|
| 867 | Geothermal Heat flux (in $mW.m^{-2}$) used by \cite{Emile-Geay_Madec_OS09}. |
---|
[2282] | 868 | It is inferred from the age of the sea floor and the formulae of \citet{Stein_Stein_Nat92}.} |
---|
[707] | 869 | \end{center} \end{figure} |
---|
| 870 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
---|
| 871 | |
---|
[817] | 872 | Usually it is assumed that there is no exchange of heat or salt through |
---|
| 873 | the ocean bottom, $i.e.$ a no flux boundary condition is applied on active |
---|
| 874 | tracers at the bottom. This is the default option in \NEMO, and it is |
---|
[1224] | 875 | implemented using the masking technique. However, there is a |
---|
[817] | 876 | non-zero heat flux across the seafloor that is associated with solid |
---|
| 877 | earth cooling. This flux is weak compared to surface fluxes (a mean |
---|
[2286] | 878 | global value of $\sim0.1\;W/m^2$ \citep{Stein_Stein_Nat92}), but it warms |
---|
| 879 | systematically the ocean and acts on the densest water masses. |
---|
[1224] | 880 | Taking this flux into account in a global ocean model increases |
---|
| 881 | the deepest overturning cell ($i.e.$ the one associated with the Antarctic |
---|
[2282] | 882 | Bottom Water) by a few Sverdrups \citep{Emile-Geay_Madec_OS09}. |
---|
[707] | 883 | |
---|
[4147] | 884 | Options are defined through the \ngn{namtra\_bbc} namelist variables. |
---|
[2349] | 885 | The presence of geothermal heating is controlled by setting the namelist |
---|
| 886 | parameter \np{ln\_trabbc} to true. Then, when \np{nn\_geoflx} is set to 1, |
---|
| 887 | a constant geothermal heating is introduced whose value is given by the |
---|
| 888 | \np{nn\_geoflx\_cst}, which is also a namelist parameter. |
---|
| 889 | When \np{nn\_geoflx} is set to 2, a spatially varying geothermal heat flux is |
---|
| 890 | introduced which is provided in the \ifile{geothermal\_heating} NetCDF file |
---|
| 891 | (Fig.\ref{Fig_geothermal}) \citep{Emile-Geay_Madec_OS09}. |
---|
[707] | 892 | |
---|
| 893 | % ================================================================ |
---|
| 894 | % Bottom Boundary Layer |
---|
| 895 | % ================================================================ |
---|
[2282] | 896 | \section [Bottom Boundary Layer (\mdl{trabbl} - \key{trabbl})] |
---|
| 897 | {Bottom Boundary Layer (\mdl{trabbl} - \key{trabbl})} |
---|
[707] | 898 | \label{TRA_bbl} |
---|
| 899 | %--------------------------------------------nambbl--------------------------------------------------------- |
---|
| 900 | \namdisplay{nambbl} |
---|
| 901 | %-------------------------------------------------------------------------------------------------------------- |
---|
| 902 | |
---|
[4147] | 903 | Options are defined through the \ngn{nambbl} namelist variables. |
---|
[817] | 904 | In a $z$-coordinate configuration, the bottom topography is represented by a |
---|
[707] | 905 | series of discrete steps. This is not adequate to represent gravity driven |
---|
[2282] | 906 | downslope flows. Such flows arise either downstream of sills such as the Strait of |
---|
| 907 | Gibraltar or Denmark Strait, where dense water formed in marginal seas flows |
---|
| 908 | into a basin filled with less dense water, or along the continental slope when dense |
---|
| 909 | water masses are formed on a continental shelf. The amount of entrainment |
---|
| 910 | that occurs in these gravity plumes is critical in determining the density |
---|
| 911 | and volume flux of the densest waters of the ocean, such as Antarctic Bottom Water, |
---|
| 912 | or North Atlantic Deep Water. $z$-coordinate models tend to overestimate the |
---|
| 913 | entrainment, because the gravity flow is mixed vertically by convection |
---|
| 914 | as it goes ''downstairs'' following the step topography, sometimes over a thickness |
---|
| 915 | much larger than the thickness of the observed gravity plume. A similar problem |
---|
| 916 | occurs in the $s$-coordinate when the thickness of the bottom level varies rapidly |
---|
| 917 | downstream of a sill \citep{Willebrand_al_PO01}, and the thickness |
---|
| 918 | of the plume is not resolved. |
---|
[707] | 919 | |
---|
[2282] | 920 | The idea of the bottom boundary layer (BBL) parameterisation, first introduced by |
---|
| 921 | \citet{Beckmann_Doscher1997}, is to allow a direct communication between |
---|
[817] | 922 | two adjacent bottom cells at different levels, whenever the densest water is |
---|
[2282] | 923 | located above the less dense water. The communication can be by a diffusive flux |
---|
| 924 | (diffusive BBL), an advective flux (advective BBL), or both. In the current |
---|
[817] | 925 | implementation of the BBL, only the tracers are modified, not the velocities. |
---|
| 926 | Furthermore, it only connects ocean bottom cells, and therefore does not include |
---|
[2286] | 927 | all the improvements introduced by \citet{Campin_Goosse_Tel99}. |
---|
[707] | 928 | |
---|
| 929 | % ------------------------------------------------------------------------------------------------------------- |
---|
| 930 | % Diffusive BBL |
---|
| 931 | % ------------------------------------------------------------------------------------------------------------- |
---|
[2282] | 932 | \subsection{Diffusive Bottom Boundary layer (\np{nn\_bbl\_ldf}=1)} |
---|
[707] | 933 | \label{TRA_bbl_diff} |
---|
| 934 | |
---|
[2282] | 935 | When applying sigma-diffusion (\key{trabbl} defined and \np{nn\_bbl\_ldf} set to 1), |
---|
| 936 | the diffusive flux between two adjacent cells at the ocean floor is given by |
---|
[817] | 937 | \begin{equation} \label{Eq_tra_bbl_diff} |
---|
| 938 | {\rm {\bf F}}_\sigma=A_l^\sigma \; \nabla_\sigma T |
---|
| 939 | \end{equation} |
---|
| 940 | with $\nabla_\sigma$ the lateral gradient operator taken between bottom cells, |
---|
[2282] | 941 | and $A_l^\sigma$ the lateral diffusivity in the BBL. Following \citet{Beckmann_Doscher1997}, |
---|
| 942 | the latter is prescribed with a spatial dependence, $i.e.$ in the conditional form |
---|
[817] | 943 | \begin{equation} \label{Eq_tra_bbl_coef} |
---|
[707] | 944 | A_l^\sigma (i,j,t)=\left\{ {\begin{array}{l} |
---|
[817] | 945 | A_{bbl} \quad \quad \mbox{if} \quad \nabla_\sigma \rho \cdot \nabla H<0 \\ |
---|
[707] | 946 | \\ |
---|
| 947 | 0\quad \quad \;\,\mbox{otherwise} \\ |
---|
| 948 | \end{array}} \right. |
---|
| 949 | \end{equation} |
---|
[817] | 950 | where $A_{bbl}$ is the BBL diffusivity coefficient, given by the namelist |
---|
[2282] | 951 | parameter \np{rn\_ahtbbl} and usually set to a value much larger |
---|
| 952 | than the one used for lateral mixing in the open ocean. The constraint in \eqref{Eq_tra_bbl_coef} |
---|
| 953 | implies that sigma-like diffusion only occurs when the density above the sea floor, at the top of |
---|
| 954 | the slope, is larger than in the deeper ocean (see green arrow in Fig.\ref{Fig_bbl}). |
---|
| 955 | In practice, this constraint is applied separately in the two horizontal directions, |
---|
| 956 | and the density gradient in \eqref{Eq_tra_bbl_coef} is evaluated with the log gradient formulation: |
---|
| 957 | \begin{equation} \label{Eq_tra_bbl_Drho} |
---|
| 958 | \nabla_\sigma \rho / \rho = \alpha \,\nabla_\sigma T + \beta \,\nabla_\sigma S |
---|
| 959 | \end{equation} |
---|
| 960 | where $\rho$, $\alpha$ and $\beta$ are functions of $\overline{T}^\sigma$, |
---|
| 961 | $\overline{S}^\sigma$ and $\overline{H}^\sigma$, the along bottom mean temperature, |
---|
| 962 | salinity and depth, respectively. |
---|
[707] | 963 | |
---|
| 964 | % ------------------------------------------------------------------------------------------------------------- |
---|
| 965 | % Advective BBL |
---|
| 966 | % ------------------------------------------------------------------------------------------------------------- |
---|
[2282] | 967 | \subsection {Advective Bottom Boundary Layer (\np{nn\_bbl\_adv}= 1 or 2)} |
---|
[707] | 968 | \label{TRA_bbl_adv} |
---|
| 969 | |
---|
[2282] | 970 | \sgacomment{"downsloping flow" has been replaced by "downslope flow" in the following |
---|
| 971 | if this is not what is meant then "downwards sloping flow" is also a possibility"} |
---|
[707] | 972 | |
---|
[817] | 973 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
---|
[2376] | 974 | \begin{figure}[!t] \begin{center} |
---|
[2282] | 975 | \includegraphics[width=0.7\textwidth]{./TexFiles/Figures/Fig_BBL_adv.pdf} |
---|
[2376] | 976 | \caption{ \label{Fig_bbl} |
---|
| 977 | Advective/diffusive Bottom Boundary Layer. The BBL parameterisation is |
---|
[2282] | 978 | activated when $\rho^i_{kup}$ is larger than $\rho^{i+1}_{kdnw}$. |
---|
| 979 | Red arrows indicate the additional overturning circulation due to the advective BBL. |
---|
| 980 | The transport of the downslope flow is defined either as the transport of the bottom |
---|
| 981 | ocean cell (black arrow), or as a function of the along slope density gradient. |
---|
| 982 | The green arrow indicates the diffusive BBL flux directly connecting $kup$ and $kdwn$ |
---|
| 983 | ocean bottom cells. |
---|
| 984 | connection} |
---|
[817] | 985 | \end{center} \end{figure} |
---|
| 986 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
---|
[707] | 987 | |
---|
[2282] | 988 | |
---|
| 989 | %!! nn_bbl_adv = 1 use of the ocean velocity as bbl velocity |
---|
| 990 | %!! nn_bbl_adv = 2 follow Campin and Goosse (1999) implentation |
---|
| 991 | %!! i.e. transport proportional to the along-slope density gradient |
---|
| 992 | |
---|
[817] | 993 | %%%gmcomment : this section has to be really written |
---|
| 994 | |
---|
[2282] | 995 | When applying an advective BBL (\np{nn\_bbl\_adv} = 1 or 2), an overturning |
---|
| 996 | circulation is added which connects two adjacent bottom grid-points only if dense |
---|
| 997 | water overlies less dense water on the slope. The density difference causes dense |
---|
| 998 | water to move down the slope. |
---|
[817] | 999 | |
---|
[2282] | 1000 | \np{nn\_bbl\_adv} = 1 : the downslope velocity is chosen to be the Eulerian |
---|
| 1001 | ocean velocity just above the topographic step (see black arrow in Fig.\ref{Fig_bbl}) |
---|
| 1002 | \citep{Beckmann_Doscher1997}. It is a \textit{conditional advection}, that is, advection |
---|
| 1003 | is allowed only if dense water overlies less dense water on the slope ($i.e.$ |
---|
| 1004 | $\nabla_\sigma \rho \cdot \nabla H<0$) and if the velocity is directed towards |
---|
| 1005 | greater depth ($i.e.$ $\vect{U} \cdot \nabla H>0$). |
---|
[817] | 1006 | |
---|
[2282] | 1007 | \np{nn\_bbl\_adv} = 2 : the downslope velocity is chosen to be proportional to $\Delta \rho$, |
---|
| 1008 | the density difference between the higher cell and lower cell densities \citep{Campin_Goosse_Tel99}. |
---|
| 1009 | The advection is allowed only if dense water overlies less dense water on the slope ($i.e.$ |
---|
| 1010 | $\nabla_\sigma \rho \cdot \nabla H<0$). For example, the resulting transport of the |
---|
| 1011 | downslope flow, here in the $i$-direction (Fig.\ref{Fig_bbl}), is simply given by the |
---|
| 1012 | following expression: |
---|
| 1013 | \begin{equation} \label{Eq_bbl_Utr} |
---|
| 1014 | u^{tr}_{bbl} = \gamma \, g \frac{\Delta \rho}{\rho_o} e_{1u} \; min \left( {e_{3u}}_{kup},{e_{3u}}_{kdwn} \right) |
---|
| 1015 | \end{equation} |
---|
| 1016 | where $\gamma$, expressed in seconds, is the coefficient of proportionality |
---|
| 1017 | provided as \np{rn\_gambbl}, a namelist parameter, and \textit{kup} and \textit{kdwn} |
---|
| 1018 | are the vertical index of the higher and lower cells, respectively. |
---|
| 1019 | The parameter $\gamma$ should take a different value for each bathymetric |
---|
| 1020 | step, but for simplicity, and because no direct estimation of this parameter is |
---|
| 1021 | available, a uniform value has been assumed. The possible values for $\gamma$ |
---|
| 1022 | range between 1 and $10~s$ \citep{Campin_Goosse_Tel99}. |
---|
| 1023 | |
---|
| 1024 | Scalar properties are advected by this additional transport $( u^{tr}_{bbl}, v^{tr}_{bbl} )$ |
---|
| 1025 | using the upwind scheme. Such a diffusive advective scheme has been chosen |
---|
| 1026 | to mimic the entrainment between the downslope plume and the surrounding |
---|
| 1027 | water at intermediate depths. The entrainment is replaced by the vertical mixing |
---|
| 1028 | implicit in the advection scheme. Let us consider as an example the |
---|
| 1029 | case displayed in Fig.\ref{Fig_bbl} where the density at level $(i,kup)$ is |
---|
| 1030 | larger than the one at level $(i,kdwn)$. The advective BBL scheme |
---|
| 1031 | modifies the tracer time tendency of the ocean cells near the |
---|
| 1032 | topographic step by the downslope flow \eqref{Eq_bbl_dw}, |
---|
| 1033 | the horizontal \eqref{Eq_bbl_hor} and the upward \eqref{Eq_bbl_up} |
---|
| 1034 | return flows as follows: |
---|
| 1035 | \begin{align} |
---|
| 1036 | \partial_t T^{do}_{kdw} &\equiv \partial_t T^{do}_{kdw} |
---|
| 1037 | + \frac{u^{tr}_{bbl}}{{b_t}^{do}_{kdw}} \left( T^{sh}_{kup} - T^{do}_{kdw} \right) \label{Eq_bbl_dw} \\ |
---|
| 1038 | % |
---|
| 1039 | \partial_t T^{sh}_{kup} &\equiv \partial_t T^{sh}_{kup} |
---|
| 1040 | + \frac{u^{tr}_{bbl}}{{b_t}^{sh}_{kup}} \left( T^{do}_{kup} - T^{sh}_{kup} \right) \label{Eq_bbl_hor} \\ |
---|
| 1041 | % |
---|
| 1042 | \intertext{and for $k =kdw-1,\;..., \; kup$ :} |
---|
| 1043 | % |
---|
| 1044 | \partial_t T^{do}_{k} &\equiv \partial_t S^{do}_{k} |
---|
| 1045 | + \frac{u^{tr}_{bbl}}{{b_t}^{do}_{k}} \left( T^{do}_{k+1} - T^{sh}_{k} \right) \label{Eq_bbl_up} |
---|
| 1046 | \end{align} |
---|
| 1047 | where $b_t$ is the $T$-cell volume. |
---|
| 1048 | |
---|
| 1049 | Note that the BBL transport, $( u^{tr}_{bbl}, v^{tr}_{bbl} )$, is available in |
---|
| 1050 | the model outputs. It has to be used to compute the effective velocity |
---|
| 1051 | as well as the effective overturning circulation. |
---|
| 1052 | |
---|
[707] | 1053 | % ================================================================ |
---|
| 1054 | % Tracer damping |
---|
| 1055 | % ================================================================ |
---|
[817] | 1056 | \section [Tracer damping (\textit{tradmp})] |
---|
| 1057 | {Tracer damping (\mdl{tradmp})} |
---|
[707] | 1058 | \label{TRA_dmp} |
---|
[2282] | 1059 | %--------------------------------------------namtra_dmp------------------------------------------------- |
---|
| 1060 | \namdisplay{namtra_dmp} |
---|
[707] | 1061 | %-------------------------------------------------------------------------------------------------------------- |
---|
| 1062 | |
---|
[817] | 1063 | In some applications it can be useful to add a Newtonian damping term |
---|
| 1064 | into the temperature and salinity equations: |
---|
[707] | 1065 | \begin{equation} \label{Eq_tra_dmp} |
---|
| 1066 | \begin{split} |
---|
| 1067 | \frac{\partial T}{\partial t}=\;\cdots \;-\gamma \,\left( {T-T_o } \right) \\ |
---|
| 1068 | \frac{\partial S}{\partial t}=\;\cdots \;-\gamma \,\left( {S-S_o } \right) |
---|
| 1069 | \end{split} |
---|
| 1070 | \end{equation} |
---|
[817] | 1071 | where $\gamma$ is the inverse of a time scale, and $T_o$ and $S_o$ |
---|
| 1072 | are given temperature and salinity fields (usually a climatology). |
---|
[4147] | 1073 | Options are defined through the \ngn{namtra\_dmp} namelist variables. |
---|
[3294] | 1074 | The restoring term is added when the namelist parameter \np{ln\_tradmp} is set to true. |
---|
| 1075 | It also requires that both \np{ln\_tsd\_init} and \np{ln\_tsd\_tradmp} are set to true |
---|
| 1076 | in \textit{namtsd} namelist as well as \np{sn\_tem} and \np{sn\_sal} structures are |
---|
| 1077 | correctly set ($i.e.$ that $T_o$ and $S_o$ are provided in input files and read |
---|
| 1078 | using \mdl{fldread}, see \S\ref{SBC_fldread}). |
---|
| 1079 | The restoring coefficient $\gamma$ is a three-dimensional array initialized by the |
---|
| 1080 | user in routine \rou{dtacof} also located in module \mdl{tradmp}. |
---|
[707] | 1081 | |
---|
[817] | 1082 | The two main cases in which \eqref{Eq_tra_dmp} is used are \textit{(a)} |
---|
| 1083 | the specification of the boundary conditions along artificial walls of a |
---|
| 1084 | limited domain basin and \textit{(b)} the computation of the velocity |
---|
| 1085 | field associated with a given $T$-$S$ field (for example to build the |
---|
| 1086 | initial state of a prognostic simulation, or to use the resulting velocity |
---|
| 1087 | field for a passive tracer study). The first case applies to regional |
---|
| 1088 | models that have artificial walls instead of open boundaries. |
---|
[2282] | 1089 | In the vicinity of these walls, $\gamma$ takes large values (equivalent to |
---|
[817] | 1090 | a time scale of a few days) whereas it is zero in the interior of the |
---|
| 1091 | model domain. The second case corresponds to the use of the robust |
---|
| 1092 | diagnostic method \citep{Sarmiento1982}. It allows us to find the velocity |
---|
[2282] | 1093 | field consistent with the model dynamics whilst having a $T$, $S$ field |
---|
| 1094 | close to a given climatological field ($T_o$, $S_o$). The time scale |
---|
[817] | 1095 | associated with $S_o$ is generally not a constant but spatially varying |
---|
| 1096 | in order to respect other properties. For example, it is usually set to zero |
---|
| 1097 | in the mixed layer (defined either on a density or $S_o$ criterion) |
---|
[2282] | 1098 | \citep{Madec_al_JPO96} and in the equatorial region |
---|
| 1099 | \citep{Reverdin1991, Fujio1991, Marti_PhD92} since these two regions |
---|
| 1100 | have a short time scale of adjustment; while smaller $\gamma$ are used |
---|
[817] | 1101 | in the deep ocean where the typical time scale is long \citep{Sarmiento1982}. |
---|
| 1102 | In addition the time scale is reduced (even to zero) along the western |
---|
| 1103 | boundary to allow the model to reconstruct its own western boundary |
---|
[2282] | 1104 | structure in equilibrium with its physics. |
---|
| 1105 | The choice of the shape of the Newtonian damping is controlled by two |
---|
[2285] | 1106 | namelist parameters \np{nn\_hdmp} and \np{nn\_zdmp}. The former allows us to specify: the |
---|
[2282] | 1107 | width of the equatorial band in which no damping is applied; a decrease |
---|
| 1108 | in the vicinity of the coast; and a damping everywhere in the Red and Med Seas. |
---|
| 1109 | The latter sets whether damping should act in the mixed layer or not. |
---|
| 1110 | The time scale associated with the damping depends on the depth as |
---|
| 1111 | a hyperbolic tangent, with \np{rn\_surf} as surface value, \np{rn\_bot} as |
---|
| 1112 | bottom value and a transition depth of \np{rn\_dep}. |
---|
[707] | 1113 | |
---|
[817] | 1114 | The robust diagnostic method is very efficient in preventing temperature |
---|
| 1115 | drift in intermediate waters but it produces artificial sources of heat and salt |
---|
| 1116 | within the ocean. It also has undesirable effects on the ocean convection. |
---|
| 1117 | It tends to prevent deep convection and subsequent deep-water formation, |
---|
| 1118 | by stabilising the water column too much. |
---|
[707] | 1119 | |
---|
[2282] | 1120 | An example of the computation of $\gamma$ for a robust diagnostic experiment |
---|
[817] | 1121 | with the ORCA2 model is provided in the \mdl{tradmp} module |
---|
| 1122 | (subroutines \rou{dtacof} and \rou{cofdis} which compute the coefficient |
---|
| 1123 | and the distance to the bathymetry, respectively). These routines are |
---|
| 1124 | provided as examples and can be customised by the user. |
---|
[707] | 1125 | |
---|
| 1126 | % ================================================================ |
---|
| 1127 | % Tracer time evolution |
---|
| 1128 | % ================================================================ |
---|
[817] | 1129 | \section [Tracer time evolution (\textit{tranxt})] |
---|
| 1130 | {Tracer time evolution (\mdl{tranxt})} |
---|
[707] | 1131 | \label{TRA_nxt} |
---|
| 1132 | %--------------------------------------------namdom----------------------------------------------------- |
---|
| 1133 | \namdisplay{namdom} |
---|
| 1134 | %-------------------------------------------------------------------------------------------------------------- |
---|
| 1135 | |
---|
[4147] | 1136 | Options are defined through the \ngn{namdom} namelist variables. |
---|
[2282] | 1137 | The general framework for tracer time stepping is a modified leap-frog scheme |
---|
| 1138 | \citep{Leclair_Madec_OM09}, $i.e.$ a three level centred time scheme associated |
---|
| 1139 | with a Asselin time filter (cf. \S\ref{STP_mLF}): |
---|
[707] | 1140 | \begin{equation} \label{Eq_tra_nxt} |
---|
[2282] | 1141 | \begin{aligned} |
---|
| 1142 | (e_{3t}T)^{t+\rdt} &= (e_{3t}T)_f^{t-\rdt} &+ 2 \, \rdt \,e_{3t}^t\ \text{RHS}^t & \\ |
---|
[707] | 1143 | \\ |
---|
[2282] | 1144 | (e_{3t}T)_f^t \;\ \quad &= (e_{3t}T)^t \;\quad |
---|
| 1145 | &+\gamma \,\left[ {(e_{3t}T)_f^{t-\rdt} -2(e_{3t}T)^t+(e_{3t}T)^{t+\rdt}} \right] & \\ |
---|
| 1146 | & &- \gamma\,\rdt \, \left[ Q^{t+\rdt/2} - Q^{t-\rdt/2} \right] & |
---|
| 1147 | \end{aligned} |
---|
[707] | 1148 | \end{equation} |
---|
[2282] | 1149 | where RHS is the right hand side of the temperature equation, |
---|
| 1150 | the subscript $f$ denotes filtered values, $\gamma$ is the Asselin coefficient, |
---|
| 1151 | and $S$ is the total forcing applied on $T$ ($i.e.$ fluxes plus content in mass exchanges). |
---|
| 1152 | $\gamma$ is initialized as \np{rn\_atfp} (\textbf{namelist} parameter). |
---|
| 1153 | Its default value is \np{rn\_atfp}=$10^{-3}$. Note that the forcing correction term in the filter |
---|
| 1154 | is not applied in linear free surface (\jp{lk\_vvl}=false) (see \S\ref{TRA_sbc}. |
---|
| 1155 | Not also that in constant volume case, the time stepping is performed on $T$, |
---|
| 1156 | not on its content, $e_{3t}T$. |
---|
[707] | 1157 | |
---|
[817] | 1158 | When the vertical mixing is solved implicitly, the update of the \textit{next} tracer |
---|
| 1159 | fields is done in module \mdl{trazdf}. In this case only the swapping of arrays |
---|
| 1160 | and the Asselin filtering is done in the \mdl{tranxt} module. |
---|
[707] | 1161 | |
---|
[817] | 1162 | In order to prepare for the computation of the \textit{next} time step, |
---|
[2282] | 1163 | a swap of tracer arrays is performed: $T^{t-\rdt} = T^t$ and $T^t = T_f$. |
---|
[707] | 1164 | |
---|
| 1165 | % ================================================================ |
---|
| 1166 | % Equation of State (eosbn2) |
---|
| 1167 | % ================================================================ |
---|
[817] | 1168 | \section [Equation of State (\textit{eosbn2}) ] |
---|
| 1169 | {Equation of State (\mdl{eosbn2}) } |
---|
[707] | 1170 | \label{TRA_eosbn2} |
---|
| 1171 | %--------------------------------------------nameos----------------------------------------------------- |
---|
| 1172 | \namdisplay{nameos} |
---|
| 1173 | %-------------------------------------------------------------------------------------------------------------- |
---|
| 1174 | |
---|
| 1175 | % ------------------------------------------------------------------------------------------------------------- |
---|
| 1176 | % Equation of State |
---|
| 1177 | % ------------------------------------------------------------------------------------------------------------- |
---|
[2282] | 1178 | \subsection{Equation of State (\np{nn\_eos} = 0, 1 or 2)} |
---|
[707] | 1179 | \label{TRA_eos} |
---|
| 1180 | |
---|
[817] | 1181 | It is necessary to know the equation of state for the ocean very accurately |
---|
| 1182 | to determine stability properties (especially the Brunt-Vais\"{a}l\"{a} frequency), |
---|
[2282] | 1183 | particularly in the deep ocean. The ocean seawater volumic mass, $\rho$, |
---|
| 1184 | abusively called density, is a non linear empirical function of \textit{in situ} |
---|
| 1185 | temperature, salinity and pressure. The reference equation of state is that |
---|
| 1186 | defined by the Joint Panel on Oceanographic Tables and Standards |
---|
| 1187 | \citep{UNESCO1983}. It was the standard equation of state used in early |
---|
| 1188 | releases of OPA. However, even though this computation is fully vectorised, |
---|
| 1189 | it is quite time consuming ($15$ to $20${\%} of the total CPU time) since |
---|
| 1190 | it requires the prior computation of the \textit{in situ} temperature from the |
---|
| 1191 | model \textit{potential} temperature using the \citep{Bryden1973} polynomial |
---|
| 1192 | for adiabatic lapse rate and a $4^th$ order Runge-Kutta integration scheme. |
---|
| 1193 | Since OPA6, we have used the \citet{JackMcD1995} equation of state for |
---|
| 1194 | seawater instead. It allows the computation of the \textit{in situ} ocean density |
---|
| 1195 | directly as a function of \textit{potential} temperature relative to the surface |
---|
| 1196 | (an \NEMO variable), the practical salinity (another \NEMO variable) and the |
---|
| 1197 | pressure (assuming no pressure variation along geopotential surfaces, $i.e.$ |
---|
| 1198 | the pressure in decibars is approximated by the depth in meters). |
---|
| 1199 | Both the \citet{UNESCO1983} and \citet{JackMcD1995} equations of state |
---|
| 1200 | have exactly the same except that the values of the various coefficients have |
---|
| 1201 | been adjusted by \citet{JackMcD1995} in order to directly use the \textit{potential} |
---|
| 1202 | temperature instead of the \textit{in situ} one. This reduces the CPU time of the |
---|
| 1203 | \textit{in situ} density computation to about $3${\%} of the total CPU time, |
---|
| 1204 | while maintaining a quite accurate equation of state. |
---|
[707] | 1205 | |
---|
[2282] | 1206 | In the computer code, a \textit{true} density anomaly, $d_a= \rho / \rho_o - 1$, |
---|
| 1207 | is computed, with $\rho_o$ a reference volumic mass. Called \textit{rau0} |
---|
| 1208 | in the code, $\rho_o$ is defined in \mdl{phycst}, and a value of $1,035~Kg/m^3$. |
---|
| 1209 | This is a sensible choice for the reference density used in a Boussinesq ocean |
---|
| 1210 | climate model, as, with the exception of only a small percentage of the ocean, |
---|
| 1211 | density in the World Ocean varies by no more than 2$\%$ from $1,035~kg/m^3$ |
---|
| 1212 | \citep{Gill1982}. |
---|
| 1213 | |
---|
[4147] | 1214 | Options are defined through the \ngn{nameos} namelist variables. |
---|
[2282] | 1215 | The default option (namelist parameter \np{nn\_eos}=0) is the \citet{JackMcD1995} |
---|
| 1216 | equation of state. Its use is highly recommended. However, for process studies, |
---|
| 1217 | it is often convenient to use a linear approximation of the density. |
---|
| 1218 | With such an equation of state there is no longer a distinction between |
---|
| 1219 | \textit{in situ} and \textit{potential} density and both cabbeling and thermobaric |
---|
| 1220 | effects are removed. |
---|
| 1221 | Two linear formulations are available: a function of $T$ only (\np{nn\_eos}=1) |
---|
| 1222 | and a function of both $T$ and $S$ (\np{nn\_eos}=2): |
---|
[707] | 1223 | \begin{equation} \label{Eq_tra_eos_linear} |
---|
[2282] | 1224 | \begin{split} |
---|
| 1225 | d_a(T) &= \rho (T) / \rho_o - 1 = \ 0.0285 - \alpha \;T \\ |
---|
| 1226 | d_a(T,S) &= \rho (T,S) / \rho_o - 1 = \ \beta \; S - \alpha \;T |
---|
| 1227 | \end{split} |
---|
[707] | 1228 | \end{equation} |
---|
[817] | 1229 | where $\alpha$ and $\beta$ are the thermal and haline expansion |
---|
| 1230 | coefficients, and $\rho_o$, the reference volumic mass, $rau0$. |
---|
[2282] | 1231 | ($\alpha$ and $\beta$ can be modified through the \np{rn\_alpha} and |
---|
[4147] | 1232 | \np{rn\_beta} namelist variables). Note that when $d_a$ is a function |
---|
[2282] | 1233 | of $T$ only (\np{nn\_eos}=1), the salinity is a passive tracer and can be |
---|
[817] | 1234 | used as such. |
---|
[707] | 1235 | |
---|
| 1236 | % ------------------------------------------------------------------------------------------------------------- |
---|
| 1237 | % Brunt-Vais\"{a}l\"{a} Frequency |
---|
| 1238 | % ------------------------------------------------------------------------------------------------------------- |
---|
[2282] | 1239 | \subsection{Brunt-Vais\"{a}l\"{a} Frequency (\np{nn\_eos} = 0, 1 or 2)} |
---|
[707] | 1240 | \label{TRA_bn2} |
---|
| 1241 | |
---|
[817] | 1242 | An accurate computation of the ocean stability (i.e. of $N$, the brunt-Vais\"{a}l\"{a} |
---|
| 1243 | frequency) is of paramount importance as it is used in several ocean |
---|
| 1244 | parameterisations (namely TKE, KPP, Richardson number dependent |
---|
| 1245 | vertical diffusion, enhanced vertical diffusion, non-penetrative convection, |
---|
| 1246 | iso-neutral diffusion). In particular, one must be aware that $N^2$ has to |
---|
| 1247 | be computed with an \textit{in situ} reference. The expression for $N^2$ |
---|
[2282] | 1248 | depends on the type of equation of state used (\np{nn\_eos} namelist parameter). |
---|
[707] | 1249 | |
---|
[2282] | 1250 | For \np{nn\_eos}=0 (\citet{JackMcD1995} equation of state), the \citet{McDougall1987} |
---|
[817] | 1251 | polynomial expression is used (with the pressure in decibar approximated by |
---|
| 1252 | the depth in meters): |
---|
| 1253 | \begin{equation} \label{Eq_tra_bn2} |
---|
| 1254 | N^2 = \frac{g}{e_{3w}} \; \beta \ |
---|
| 1255 | \left( \alpha / \beta \ \delta_{k+1/2}[T] - \delta_{k+1/2}[S] \right) |
---|
| 1256 | \end{equation} |
---|
[2282] | 1257 | where $\alpha$ and $\beta$ are the thermal and haline expansion coefficients. |
---|
| 1258 | They are a function of $\overline{T}^{\,k+1/2},\widetilde{S}=\overline{S}^{\,k+1/2} - 35.$, |
---|
| 1259 | and $z_w$, with $T$ the \textit{potential} temperature and $\widetilde{S}$ a salinity anomaly. |
---|
[817] | 1260 | Note that both $\alpha$ and $\beta$ depend on \textit{potential} |
---|
| 1261 | temperature and salinity which are averaged at $w$-points prior |
---|
| 1262 | to the computation instead of being computed at $T$-points and |
---|
| 1263 | then averaged to $w$-points. |
---|
[707] | 1264 | |
---|
[2282] | 1265 | When a linear equation of state is used (\np{nn\_eos}=1 or 2, |
---|
[817] | 1266 | \eqref{Eq_tra_bn2} reduces to: |
---|
[707] | 1267 | \begin{equation} \label{Eq_tra_bn2_linear} |
---|
| 1268 | N^2 = \frac{g}{e_{3w}} \left( \beta \;\delta_{k+1/2}[S] - \alpha \;\delta_{k+1/2}[T] \right) |
---|
| 1269 | \end{equation} |
---|
[817] | 1270 | where $\alpha$ and $\beta $ are the constant coefficients used to |
---|
| 1271 | defined the linear equation of state \eqref{Eq_tra_eos_linear}. |
---|
[707] | 1272 | |
---|
| 1273 | % ------------------------------------------------------------------------------------------------------------- |
---|
| 1274 | % Specific Heat |
---|
| 1275 | % ------------------------------------------------------------------------------------------------------------- |
---|
[2282] | 1276 | \subsection [Specific Heat (\textit{phycst})] |
---|
[817] | 1277 | {Specific Heat (\mdl{phycst})} |
---|
[707] | 1278 | \label{TRA_adv_ldf} |
---|
| 1279 | |
---|
[817] | 1280 | The specific heat of sea water, $C_p$, is a function of temperature, salinity |
---|
| 1281 | and pressure \citep{UNESCO1983}. It is only used in the model to convert |
---|
| 1282 | surface heat fluxes into surface temperature increase and so the pressure |
---|
| 1283 | dependence is neglected. The dependence on $T$ and $S$ is weak. |
---|
| 1284 | For example, with $S=35~psu$, $C_p$ increases from $3989$ to $4002$ |
---|
| 1285 | when $T$ varies from -2~\degres C to 31~\degres C. Therefore, $C_p$ has |
---|
| 1286 | been chosen as a constant: $C_p=4.10^3~J\,Kg^{-1}\,\degres K^{-1}$. |
---|
| 1287 | Its value is set in \mdl{phycst} module. |
---|
[707] | 1288 | |
---|
[817] | 1289 | |
---|
[707] | 1290 | % ------------------------------------------------------------------------------------------------------------- |
---|
| 1291 | % Freezing Point of Seawater |
---|
| 1292 | % ------------------------------------------------------------------------------------------------------------- |
---|
[2282] | 1293 | \subsection [Freezing Point of Seawater] |
---|
| 1294 | {Freezing Point of Seawater} |
---|
[707] | 1295 | \label{TRA_fzp} |
---|
| 1296 | |
---|
| 1297 | The freezing point of seawater is a function of salinity and pressure \citep{UNESCO1983}: |
---|
| 1298 | \begin{equation} \label{Eq_tra_eos_fzp} |
---|
| 1299 | \begin{split} |
---|
[1224] | 1300 | T_f (S,p) = \left( -0.0575 + 1.710523 \;10^{-3} \, \sqrt{S} |
---|
[707] | 1301 | - 2.154996 \;10^{-4} \,S \right) \ S \\ |
---|
[1224] | 1302 | - 7.53\,10^{-3} \ \ p |
---|
[707] | 1303 | \end{split} |
---|
| 1304 | \end{equation} |
---|
| 1305 | |
---|
[817] | 1306 | \eqref{Eq_tra_eos_fzp} is only used to compute the potential freezing point of |
---|
| 1307 | sea water ($i.e.$ referenced to the surface $p=0$), thus the pressure dependent |
---|
[2282] | 1308 | terms in \eqref{Eq_tra_eos_fzp} (last term) have been dropped. The freezing |
---|
| 1309 | point is computed through \textit{tfreez}, a \textsc{Fortran} function that can be found |
---|
| 1310 | in \mdl{eosbn2}. |
---|
[707] | 1311 | |
---|
| 1312 | % ================================================================ |
---|
| 1313 | % Horizontal Derivative in zps-coordinate |
---|
| 1314 | % ================================================================ |
---|
[817] | 1315 | \section [Horizontal Derivative in \textit{zps}-coordinate (\textit{zpshde})] |
---|
| 1316 | {Horizontal Derivative in \textit{zps}-coordinate (\mdl{zpshde})} |
---|
[707] | 1317 | \label{TRA_zpshde} |
---|
| 1318 | |
---|
[817] | 1319 | \gmcomment{STEVEN: to be consistent with earlier discussion of differencing and averaging operators, I've changed "derivative" to "difference" and "mean" to "average"} |
---|
| 1320 | |
---|
[2282] | 1321 | With partial bottom cells (\np{ln\_zps}=true), in general, tracers in horizontally |
---|
[817] | 1322 | adjacent cells live at different depths. Horizontal gradients of tracers are needed |
---|
| 1323 | for horizontal diffusion (\mdl{traldf} module) and for the hydrostatic pressure |
---|
| 1324 | gradient (\mdl{dynhpg} module) to be active. |
---|
| 1325 | \gmcomment{STEVEN from gm : question: not sure of what -to be active- means} |
---|
| 1326 | Before taking horizontal gradients between the tracers next to the bottom, a linear |
---|
| 1327 | interpolation in the vertical is used to approximate the deeper tracer as if it actually |
---|
| 1328 | lived at the depth of the shallower tracer point (Fig.~\ref{Fig_Partial_step_scheme}). |
---|
| 1329 | For example, for temperature in the $i$-direction the needed interpolated |
---|
| 1330 | temperature, $\widetilde{T}$, is: |
---|
| 1331 | |
---|
[707] | 1332 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
---|
[2376] | 1333 | \begin{figure}[!p] \begin{center} |
---|
[998] | 1334 | \includegraphics[width=0.9\textwidth]{./TexFiles/Figures/Partial_step_scheme.pdf} |
---|
[2376] | 1335 | \caption{ \label{Fig_Partial_step_scheme} |
---|
| 1336 | Discretisation of the horizontal difference and average of tracers in the $z$-partial |
---|
| 1337 | step coordinate (\np{ln\_zps}=true) in the case $( e3w_k^{i+1} - e3w_k^i )>0$. |
---|
| 1338 | A linear interpolation is used to estimate $\widetilde{T}_k^{i+1}$, the tracer value |
---|
| 1339 | at the depth of the shallower tracer point of the two adjacent bottom $T$-points. |
---|
| 1340 | The horizontal difference is then given by: $\delta _{i+1/2} T_k= \widetilde{T}_k^{\,i+1} -T_k^{\,i}$ |
---|
| 1341 | and the average by: $\overline{T}_k^{\,i+1/2}= ( \widetilde{T}_k^{\,i+1/2} - T_k^{\,i} ) / 2$. } |
---|
[707] | 1342 | \end{center} \end{figure} |
---|
| 1343 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
---|
| 1344 | \begin{equation*} |
---|
| 1345 | \widetilde{T}= \left\{ \begin{aligned} |
---|
| 1346 | &T^{\,i+1} -\frac{ \left( e_{3w}^{i+1} -e_{3w}^i \right)}{ e_{3w}^{i+1} }\;\delta _k T^{i+1} |
---|
| 1347 | && \quad\text{if $\ e_{3w}^{i+1} \geq e_{3w}^i$ } \\ |
---|
| 1348 | \\ |
---|
| 1349 | &T^{\,i} \ \ \ \,+\frac{ \left( e_{3w}^{i+1} -e_{3w}^i \right) }{e_{3w}^i }\;\delta _k T^{i+1} |
---|
| 1350 | && \quad\text{if $\ e_{3w}^{i+1} < e_{3w}^i$ } |
---|
| 1351 | \end{aligned} \right. |
---|
| 1352 | \end{equation*} |
---|
[817] | 1353 | and the resulting forms for the horizontal difference and the horizontal average |
---|
| 1354 | value of $T$ at a $U$-point are: |
---|
[707] | 1355 | \begin{equation} \label{Eq_zps_hde} |
---|
| 1356 | \begin{aligned} |
---|
| 1357 | \delta _{i+1/2} T= \begin{cases} |
---|
| 1358 | \ \ \ \widetilde {T}\quad\ -T^i & \ \ \quad\quad\text{if $\ e_{3w}^{i+1} \geq e_{3w}^i$ } \\ |
---|
| 1359 | \\ |
---|
| 1360 | \ \ \ T^{\,i+1}-\widetilde{T} & \ \ \quad\quad\text{if $\ e_{3w}^{i+1} < e_{3w}^i$ } |
---|
| 1361 | \end{cases} \\ |
---|
| 1362 | \\ |
---|
| 1363 | \overline {T}^{\,i+1/2} \ = \begin{cases} |
---|
| 1364 | ( \widetilde {T}\ \ \;\,-T^{\,i}) / 2 & \;\ \ \quad\text{if $\ e_{3w}^{i+1} \geq e_{3w}^i$ } \\ |
---|
| 1365 | \\ |
---|
| 1366 | ( T^{\,i+1}-\widetilde{T} ) / 2 & \;\ \ \quad\text{if $\ e_{3w}^{i+1} < e_{3w}^i$ } |
---|
| 1367 | \end{cases} |
---|
| 1368 | \end{aligned} |
---|
| 1369 | \end{equation} |
---|
| 1370 | |
---|
[817] | 1371 | The computation of horizontal derivative of tracers as well as of density is |
---|
| 1372 | performed once for all at each time step in \mdl{zpshde} module and stored |
---|
| 1373 | in shared arrays to be used when needed. It has to be emphasized that the |
---|
| 1374 | procedure used to compute the interpolated density, $\widetilde{\rho}$, is not |
---|
| 1375 | the same as that used for $T$ and $S$. Instead of forming a linear approximation |
---|
| 1376 | of density, we compute $\widetilde{\rho }$ from the interpolated values of $T$ |
---|
| 1377 | and $S$, and the pressure at a $u$-point (in the equation of state pressure is |
---|
| 1378 | approximated by depth, see \S\ref{TRA_eos} ) : |
---|
[707] | 1379 | \begin{equation} \label{Eq_zps_hde_rho} |
---|
| 1380 | \widetilde{\rho } = \rho ( {\widetilde{T},\widetilde {S},z_u }) |
---|
| 1381 | \quad \text{where }\ z_u = \min \left( {z_T^{i+1} ,z_T^i } \right) |
---|
| 1382 | \end{equation} |
---|
| 1383 | |
---|
[817] | 1384 | This is a much better approximation as the variation of $\rho$ with depth (and |
---|
| 1385 | thus pressure) is highly non-linear with a true equation of state and thus is badly |
---|
| 1386 | approximated with a linear interpolation. This approximation is used to compute |
---|
| 1387 | both the horizontal pressure gradient (\S\ref{DYN_hpg}) and the slopes of neutral |
---|
| 1388 | surfaces (\S\ref{LDF_slp}) |
---|
[707] | 1389 | |
---|
[817] | 1390 | Note that in almost all the advection schemes presented in this Chapter, both |
---|
| 1391 | averaging and differencing operators appear. Yet \eqref{Eq_zps_hde} has not |
---|
| 1392 | been used in these schemes: in contrast to diffusion and pressure gradient |
---|
| 1393 | computations, no correction for partial steps is applied for advection. The main |
---|
| 1394 | motivation is to preserve the domain averaged mean variance of the advected |
---|
| 1395 | field when using the $2^{nd}$ order centred scheme. Sensitivity of the advection |
---|
| 1396 | schemes to the way horizontal averages are performed in the vicinity of partial |
---|
| 1397 | cells should be further investigated in the near future. |
---|
| 1398 | %%% |
---|
| 1399 | \gmcomment{gm : this last remark has to be done} |
---|
| 1400 | %%% |
---|