1 | % ================================================================ |
---|
2 | % Chapter 1 Ñ Ocean Tracers (TRA) |
---|
3 | % ================================================================ |
---|
4 | \chapter{Ocean Tracers (TRA)} |
---|
5 | \label{TRA} |
---|
6 | \minitoc |
---|
7 | |
---|
8 | % missing/update |
---|
9 | % traqsr: need to coordinate with SBC module |
---|
10 | % trabbl : advective case to be discussed |
---|
11 | % diffusive case : add : only the bottom ocean cell is concerned |
---|
12 | % ==> addfigure on bbl |
---|
13 | |
---|
14 | %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 |
---|
15 | |
---|
16 | \newpage |
---|
17 | $\ $\newline % force a new ligne |
---|
18 | |
---|
19 | Using the representation described in Chap.~\ref{DOM}, several semi-discrete |
---|
20 | space forms of the tracer equations are available depending on the vertical |
---|
21 | coordinate used and on the physics used. In all the equations presented |
---|
22 | here, the masking has been omitted for simplicity. One must be aware that |
---|
23 | all the quantities are masked fields and that each time a mean or difference |
---|
24 | operator is used, the resulting field is multiplied by a mask. |
---|
25 | |
---|
26 | The two active tracers are potential temperature and salinity. Their prognostic |
---|
27 | equations can be summarized as follows: |
---|
28 | \begin{equation*} |
---|
29 | \text{NXT} = \text{ADV}+\text{LDF}+\text{ZDF}+\text{SBC} |
---|
30 | \ (+\text{QSR})\ (+\text{BBC})\ (+\text{BBL})\ (+\text{DMP}) |
---|
31 | \end{equation*} |
---|
32 | |
---|
33 | NXT stands for next, referring to the time-stepping. From left to right, the terms |
---|
34 | on the rhs of the tracer equations are the advection (ADV), the lateral diffusion |
---|
35 | (LDF), the vertical diffusion (ZDF), the contributions from the external forcings |
---|
36 | (SBC: Surface Boundary Condition, QSR: penetrative Solar Radiation, and BBC: |
---|
37 | Bottom Boundary Condition), the contribution from the bottom boundary Layer |
---|
38 | (BBL) parametrisation, and an internal damping (DMP) term. The terms QSR, |
---|
39 | BBC, BBL and DMP are optional. The external forcings and parameterizations |
---|
40 | require complex inputs and complex calculations (e.g. bulk formulae, estimation |
---|
41 | of mixing coefficients) that are carried out in the SBC, LDF and ZDF modules and |
---|
42 | described in chapters \S\ref{SBC}, \S\ref{LDF} and \S\ref{ZDF}, respectively. |
---|
43 | Note that \mdl{tranpc}, the non-penetrative convection module, although |
---|
44 | (temporarily) located in the NEMO/OPA/TRA directory, is described with the |
---|
45 | model vertical physics (ZDF). |
---|
46 | %%% |
---|
47 | \gmcomment{change the position of eosbn2 in the reference code} |
---|
48 | %%% |
---|
49 | |
---|
50 | In the present chapter we also describe the diagnostic equations used to compute |
---|
51 | the sea-water properties (density, Brunt-Vais\"{a}l\"{a} frequency, specific heat and |
---|
52 | freezing point) although the associated modules ($i.e.$ \mdl{eosbn2}, \mdl{ocfzpt} |
---|
53 | and \mdl{phycst}) are (temporarily) located in the NEMO/OPA directory. |
---|
54 | |
---|
55 | The different options available to the user are managed by namelist logical or |
---|
56 | CPP keys. For each equation term ttt, the namelist logicals are \textit{ln\_trattt\_xxx}, |
---|
57 | where \textit{xxx} is a 3 or 4 letter acronym accounting for each optional scheme. |
---|
58 | The CPP key (when it exists) is \textbf{key\_trattt}. The corresponding code can be |
---|
59 | found in the \textit{trattt} or \textit{trattt\_xxx} module, in the NEMO/OPA/TRA directory. |
---|
60 | |
---|
61 | The user has the option of extracting each tendency term on the rhs of the tracer |
---|
62 | equation for output (\key{trdtra} is defined), as described in Chap.~\ref{MISC}. |
---|
63 | |
---|
64 | % ================================================================ |
---|
65 | % Tracer Advection |
---|
66 | % ================================================================ |
---|
67 | \section [Tracer Advection (\textit{traadv})] |
---|
68 | {Tracer Advection (\mdl{traadv})} |
---|
69 | \label{TRA_adv} |
---|
70 | %------------------------------------------nam_traadv----------------------------------------------------- |
---|
71 | \namdisplay{nam_traadv} |
---|
72 | %------------------------------------------------------------------------------------------------------------- |
---|
73 | |
---|
74 | The advection tendency of a tracer in flux form is the divergence of the advective |
---|
75 | fluxes. Its discrete expression is given by : |
---|
76 | \begin{equation} \label{Eq_tra_adv} |
---|
77 | ADV_\tau =-\frac{1}{e_{1T} {\kern 1pt}e_{2T} {\kern 1pt}e_{3T} }\left( |
---|
78 | {\;\delta _i \left[ {e_{2u} {\kern 1pt}e_{3u} {\kern 1pt}\;u\;\tau _u } |
---|
79 | \right]+\delta _j \left[ {e_{1v} {\kern 1pt}e_{3v} {\kern 1pt}v\;\tau _v } |
---|
80 | \right]\;} \right)-\frac{1}{\mathop e\nolimits_{3T} }\delta _k \left[ |
---|
81 | {w\;\tau _w } \right] |
---|
82 | \end{equation} |
---|
83 | where $\tau$ is either T or S. In pure $z$-coordinate (\key{zco} is defined), |
---|
84 | it reduces to : |
---|
85 | \begin{equation} \label{Eq_tra_adv_zco} |
---|
86 | ADV_\tau =-\frac{1}{e_{1T} {\kern 1pt}e_{2T} {\kern 1pt}}\left( {\;\delta _i |
---|
87 | \left[ {e_{2u} {\kern 1pt}{\kern 1pt}\;u\;\tau _u } \right]+\delta _j \left[ |
---|
88 | {e_{1v} {\kern 1pt}v\;\tau _v } \right]\;} \right)-\frac{1}{\mathop |
---|
89 | e\nolimits_{3T} }\delta _k \left[ {w\;\tau _w } \right] |
---|
90 | \end{equation} |
---|
91 | since the vertical scale factors are functions of $k$ only, and thus $e_{3u} |
---|
92 | =e_{3v} =e_{3T} $. |
---|
93 | |
---|
94 | The flux form in \eqref{Eq_tra_adv} requires implicitly the use of the continuity equation: |
---|
95 | $\nabla \cdot \left( \vect{U}\,T \right)=\vect{U} \cdot \nabla T$ |
---|
96 | (using $\nabla \cdot \vect{U}=0)$ or $\partial _t e_3 + e_3\;\nabla \cdot \vect{U}=0$ |
---|
97 | in variable volume case ($i.e.$ \key{vvl} defined). Therefore it is of |
---|
98 | paramount importance to design the discrete analogue of the advection |
---|
99 | tendency so that it is consistent with the continuity equation in order to |
---|
100 | enforce the conservation properties of the continuous equations. In other words, |
---|
101 | by substituting $\tau$ by 1 in (\ref{Eq_tra_adv}) we recover the discrete form of |
---|
102 | the continuity equation which is used to calculate the vertical velocity. |
---|
103 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
---|
104 | \begin{figure}[!t] \label{Fig_adv_scheme} \begin{center} |
---|
105 | \includegraphics[width=0.9\textwidth]{./TexFiles/Figures/Fig_adv_scheme.pdf} |
---|
106 | \caption{Schematic representation of some ways used to evaluate the tracer value |
---|
107 | at $u$-point and the amount of tracer exchanged between two neighbouring grid |
---|
108 | points. Upsteam biased scheme (ups): the upstream value is used and the black |
---|
109 | area is exchanged. Piecewise parabolic method (ppm): a parabolic interpolation |
---|
110 | is used and the black and dark grey areas are exchanged. Monotonic upstream |
---|
111 | scheme for conservative laws (muscl): a parabolic interpolation is used and black, |
---|
112 | dark grey and grey areas are exchanged. Second order scheme (cen2): the mean |
---|
113 | value is used and black, dark grey, grey and light grey areas are exchanged. Note |
---|
114 | that this illustration does not include the flux limiter used in ppm and muscl schemes.} |
---|
115 | \end{center} \end{figure} |
---|
116 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
---|
117 | |
---|
118 | The key difference between the advection schemes used in \NEMO is the choice |
---|
119 | made in space and time interpolation to define the value of the tracer at the |
---|
120 | velocity points (Fig.~\ref{Fig_adv_scheme}). |
---|
121 | |
---|
122 | Along solid lateral and bottom boundaries a zero tracer flux is naturally |
---|
123 | specified, since the normal velocity is zero there. At the sea surface the |
---|
124 | boundary condition depends on the type of sea surface chosen: |
---|
125 | \begin{description} |
---|
126 | \item [rigid-lid formulation:] $w=0$ at the surface, so the advective |
---|
127 | fluxes through the surface are zero. |
---|
128 | \item [linear free surface:] the first level thickness is constant in time: |
---|
129 | the vertical boundary condition is applied at the fixed surface $z=0$ |
---|
130 | rather than on the moving surface $z=\eta$. There is a non-zero advective |
---|
131 | flux which is set for all advection schemes as the product of surface |
---|
132 | velocity (at $z=0$) by the first level tracer value: |
---|
133 | $\left. {\tau _w } \right|_{k=1/2} =T_{k=1} $. |
---|
134 | \item [non-linear free surface:] (\key{vvl} is defined) |
---|
135 | convergence/divergence in the first ocean level moves the free surface |
---|
136 | up/down. There is no tracer advection through it so that the advective |
---|
137 | fluxes through the surface are also zero |
---|
138 | \end{description} |
---|
139 | In all cases, this boundary condition retains local conservation of tracer. |
---|
140 | Global conservation is obtained in both rigid-lid and non-linear free surface |
---|
141 | cases, but not in the linear free surface case. Nevertheless, in the latter |
---|
142 | case, it is achieved to a good approximation since the non-conservative |
---|
143 | term is the product of the time derivative of the tracer and the free surface |
---|
144 | height, two quantities that are not correlated (see \S\ref{PE_free_surface}, |
---|
145 | and also \citet{Roullet2000,Griffies2001,Campin2004}). |
---|
146 | |
---|
147 | The velocity field that appears in (\ref{Eq_tra_adv}) and (\ref{Eq_tra_adv_zco}) |
---|
148 | is the centred (\textit{now}) \textit{eulerian} ocean velocity (see Chap.~\ref{DYN}). |
---|
149 | When advective bottom boundary layer (\textit{bbl}) and/or eddy induced velocity |
---|
150 | (\textit{eiv}) parameterisations are used it is the \textit{now} \textit{effective} |
---|
151 | velocity ($i.e.$ the sum of the eulerian, the bbl and/or the eiv velocities) which is used. |
---|
152 | |
---|
153 | The choice of an advection scheme is made in the \np{nam\_traadv} namelist, by |
---|
154 | setting to \textit{true} one and only one of the logicals \textit{ln\_traadv\_xxx}. The |
---|
155 | corresponding code can be found in the \textit{traadv\_xxx.F90} module, where |
---|
156 | \textit{xxx} is a 3 or 4 letter acronym corresponding to each scheme. Details |
---|
157 | of the advection schemes are given below. The choice of an advection scheme |
---|
158 | is a complex matter which depends on the model physics, model resolution, |
---|
159 | type of tracer, as well as the issue of numerical cost. |
---|
160 | |
---|
161 | Note that |
---|
162 | (1) cen2, cen4 and TVD schemes require an explicit diffusion |
---|
163 | operator while the other schemes are diffusive enough so that they do not |
---|
164 | require additional diffusion ; |
---|
165 | (2) cen2, cen4, MUSCL2, and UBS are not \textit{positive} schemes |
---|
166 | \footnote{negative values can appear in an initially strictly positive tracer field |
---|
167 | which is advected} |
---|
168 | , implying that false extrema are permitted. Their use is not recommended on passive tracers ; |
---|
169 | (3) It is highly recommended that the same advection-diffusion scheme is |
---|
170 | used on both active and passive tracers. Indeed, if a source or sink of a |
---|
171 | passive tracer depends on an active one, the difference of treatment of |
---|
172 | active and passive tracers can create very nice-looking frontal structures |
---|
173 | that are pure numerical artefacts. |
---|
174 | |
---|
175 | % ------------------------------------------------------------------------------------------------------------- |
---|
176 | % 2nd order centred scheme |
---|
177 | % ------------------------------------------------------------------------------------------------------------- |
---|
178 | \subsection [$2^{nd}$ order centred scheme (cen2) (\np{ln\_traadv\_cen2})] |
---|
179 | {$2^{nd}$ order centred scheme (cen2) (\np{ln\_traadv\_cen2}=.true.)} |
---|
180 | \label{TRA_adv_cen2} |
---|
181 | |
---|
182 | In the centred second order formulation, the tracer at velocity points is |
---|
183 | evaluated as the mean of the two neighbouring $T$-point values. |
---|
184 | For example, in the $i$-direction : |
---|
185 | \begin{equation} \label{Eq_tra_adv_cen2} |
---|
186 | \tau _u^{cen2} =\overline T ^{i+1/2} |
---|
187 | \end{equation} |
---|
188 | |
---|
189 | The scheme is non diffusive ($i.e.$ it conserves the tracer variance, $\tau^2)$ |
---|
190 | but dispersive ($i.e.$ it may create false extrema). It is therefore notoriously |
---|
191 | noisy and must be used in conjunction with an explicit diffusion operator to |
---|
192 | produce a sensible solution. The associated time-stepping is performed using |
---|
193 | a leapfrog scheme in conjunction with an Asselin time-filter, so $T$ in |
---|
194 | (\ref{Eq_tra_adv_cen2}) is the \textit{now} tracer value. |
---|
195 | |
---|
196 | Note that using the cen2 scheme, the overall tracer advection is of second |
---|
197 | order accuracy since both (\ref{Eq_tra_adv}) and (\ref{Eq_tra_adv_cen2}) |
---|
198 | have this order of accuracy. |
---|
199 | |
---|
200 | % ------------------------------------------------------------------------------------------------------------- |
---|
201 | % 4nd order centred scheme |
---|
202 | % ------------------------------------------------------------------------------------------------------------- |
---|
203 | \subsection [$4^{nd}$ order centred scheme (cen4) (\np{ln\_traadv\_cen4})] |
---|
204 | {$4^{nd}$ order centred scheme (cen4) (\np{ln\_traadv\_cen4}=.true.)} |
---|
205 | \label{TRA_adv_cen4} |
---|
206 | |
---|
207 | In the $4^{th}$ order formulation (to be implemented), tracer values are |
---|
208 | evaluated at velocity points as a $4^{th}$ order interpolation, and thus uses |
---|
209 | the four neighbouring $T$-points. For example, in the $i$-direction: |
---|
210 | \begin{equation} \label{Eq_tra_adv_cen4} |
---|
211 | \tau _u^{cen4} |
---|
212 | =\overline{ T - \frac{1}{6}\,\delta _i \left[ \delta_{i+1/2}[T] \,\right] }^{\,i+1/2} |
---|
213 | \end{equation} |
---|
214 | |
---|
215 | Strictly speaking, the cen4 scheme is not a $4^{th}$ order advection scheme |
---|
216 | but a $4^{th}$ order evaluation of advective fluxes, since the divergence of |
---|
217 | advective fluxes \eqref{Eq_tra_adv} is kept at $2^{nd}$ order. The phrase ``$4^{th}$ |
---|
218 | order scheme'' used in oceanographic literature is usually associated |
---|
219 | with the scheme presented here. Introducing a \textit{true} $4^{th}$ order advection |
---|
220 | scheme is feasible but, for consistency reasons, it requires changes in the |
---|
221 | discretisation of the tracer advection together with changes in both the |
---|
222 | continuity equation and the momentum advection terms. |
---|
223 | |
---|
224 | A direct consequence of the pseudo-fourth order nature of the scheme is that |
---|
225 | it is not non-diffusive, i.e. the global variance of a tracer is not |
---|
226 | preserved using \textit{cen4}. Furthermore, it must be used in conjunction with an |
---|
227 | explicit diffusion operator to produce a sensible solution. The |
---|
228 | time-stepping is also performed using a leapfrog scheme in conjunction with |
---|
229 | an Asselin time-filter, so $T$ in (\ref{Eq_tra_adv_cen4}) is the \textit{now} tracer. |
---|
230 | |
---|
231 | At a $T$-grid cell adjacent to a boundary (coastline, bottom and surface), an |
---|
232 | additional hypothesis must be made to evaluate $\tau _u^{cen4}$. This |
---|
233 | hypothesis usually reduces the order of the scheme. Here we choose to set |
---|
234 | the gradient of $T$ across the boundary to zero. Alternative conditions can be |
---|
235 | specified, such as a reduction to a second order scheme for these near boundary |
---|
236 | grid points. |
---|
237 | |
---|
238 | % ------------------------------------------------------------------------------------------------------------- |
---|
239 | % TVD scheme |
---|
240 | % ------------------------------------------------------------------------------------------------------------- |
---|
241 | \subsection [Total Variance Dissipation scheme (TVD) (\np{ln\_traadv\_tvd})] |
---|
242 | {Total Variance Dissipation scheme (TVD) (\np{ln\_traadv\_tvd}=.true.)} |
---|
243 | \label{TRA_adv_tvd} |
---|
244 | |
---|
245 | In the Total Variance Dissipation (TVD) formulation, the tracer at velocity |
---|
246 | points is evaluated using a combination of an upstream and a centred scheme. For |
---|
247 | example, in the $i$-direction : |
---|
248 | \begin{equation} \label{Eq_tra_adv_tvd} |
---|
249 | \begin{split} |
---|
250 | \tau _u^{ups}&= \begin{cases} |
---|
251 | T_{i+1} & \text{if $\ u_{i+1/2} < 0$} \hfill \\ |
---|
252 | T_i & \text{if $\ u_{i+1/2} \geq 0$} \hfill \\ |
---|
253 | \end{cases} \\ |
---|
254 | \\ |
---|
255 | \tau _u^{tvd}&=\tau _u^{ups} +c_u \;\left( {\tau _u^{cen2} -\tau _u^{ups} } \right) |
---|
256 | \end{split} |
---|
257 | \end{equation} |
---|
258 | where $c_u$ is a flux limiter function taking values between 0 and 1. There |
---|
259 | exist many ways to define $c_u$, each correcponding to a different total |
---|
260 | variance decreasing scheme. The one chosen in \NEMO is described in |
---|
261 | \citet{Zalesak1979}. $c_u$ only departs from $1$ when the advective term |
---|
262 | produces a local extremum in the tracer field. The resulting scheme is quite |
---|
263 | expensive but \emph{positive}. It can be used on both active and passive tracers. |
---|
264 | This scheme is tested and compared with MUSCL and the MPDATA scheme in |
---|
265 | \citet{Levy2001}; note that in this paper it is referred to as "FCT" (Flux corrected |
---|
266 | transport) rather than TVD. |
---|
267 | |
---|
268 | For stability reasons (see \S\ref{DOM_nxt}), in (\ref{Eq_tra_adv_tvd}) |
---|
269 | $\tau _u^{cen2}$ is evaluated using the \textit{now} tracer while $\tau _u^{ups}$ |
---|
270 | is evaluated using the \textit{before} tracer. In other words, the advective part of |
---|
271 | the scheme is time stepped with a leap-frog scheme while a forward scheme is |
---|
272 | used for the diffusive part. |
---|
273 | |
---|
274 | % ------------------------------------------------------------------------------------------------------------- |
---|
275 | % MUSCL scheme |
---|
276 | % ------------------------------------------------------------------------------------------------------------- |
---|
277 | \subsection[MUSCL scheme (\np{ln\_traadv\_muscl})] |
---|
278 | {Monotone Upstream Scheme for Conservative Laws (MUSCL) (\np{ln\_traadv\_muscl}=T)} |
---|
279 | \label{TRA_adv_muscl} |
---|
280 | |
---|
281 | The Monotone Upstream Scheme for Conservative Laws (MUSCL) has been |
---|
282 | implemented by \citet{Levy2001}. In its formulation, the tracer at velocity points |
---|
283 | is evaluated assuming a linear tracer variation between two $T$-points |
---|
284 | (Fig.\ref{Fig_adv_scheme}). For example, in the $i$-direction : |
---|
285 | \begin{equation} \label{Eq_tra_adv_muscl} |
---|
286 | \tau _u^{mus} = \left\{ \begin{aligned} |
---|
287 | &\tau _i &+ \frac{1}{2} \;\left( 1-\frac{u_{i+1/2} \;\Delta t}{e_{1u}} \right) |
---|
288 | &\ \widetilde{\partial _i \tau} & \quad \text{if }\;u_{i+1/2} \geqslant 0 \\ |
---|
289 | &\tau _{i+1/2} &+\frac{1}{2}\;\left( 1+\frac{u_{i+1/2} \;\Delta t}{e_{1u} } \right) |
---|
290 | &\ \widetilde{\partial_{i+1/2} \tau } & \text{if }\;u_{i+1/2} <0 |
---|
291 | \end{aligned} \right. |
---|
292 | \end{equation} |
---|
293 | where $\widetilde{\partial _i \tau}$ is the slope of the tracer on which a limitation |
---|
294 | is imposed to ensure the \textit{positive} character of the scheme. |
---|
295 | |
---|
296 | The time stepping is performed using a forward scheme, that is the \textit{before} |
---|
297 | tracer field is used to evaluate $\tau _u^{mus}$. |
---|
298 | |
---|
299 | For an ocean grid point adjacent to land and where the ocean velocity is |
---|
300 | directed toward land, two choices are available: an upstream flux |
---|
301 | (\np{ln\_traadv\_muscl}=.true.) or a second order flux |
---|
302 | (\np{ln\_traadv\_muscl2}=.true.). Note that the latter choice does not ensure |
---|
303 | the \textit{positive} character of the scheme. Only the former can be used |
---|
304 | on both active and passive tracers. |
---|
305 | |
---|
306 | % ------------------------------------------------------------------------------------------------------------- |
---|
307 | % UBS scheme |
---|
308 | % ------------------------------------------------------------------------------------------------------------- |
---|
309 | \subsection [Upstream-Biased Scheme (UBS) (\np{ln\_traadv\_ubs})] |
---|
310 | {Upstream-Biased Scheme (UBS) (\np{ln\_traadv\_ubs}=.true.)} |
---|
311 | \label{TRA_adv_ubs} |
---|
312 | |
---|
313 | The UBS advection scheme is an upstream-biased third order scheme based on |
---|
314 | an upstream-biased parabolic interpolation. It is also known as the Cell |
---|
315 | Averaged QUICK scheme (Quadratic Upstream Interpolation for Convective |
---|
316 | Kinematics). For example, in the $i$-direction : |
---|
317 | \begin{equation} \label{Eq_tra_adv_ubs} |
---|
318 | \tau _u^{ubs} =\overline T ^{i+1/2}-\;\frac{1}{6} \left\{ |
---|
319 | \begin{aligned} |
---|
320 | &\tau"_i & \quad \text{if }\ u_{i+1/2} \geqslant 0 \\ |
---|
321 | &\tau"_{i+1} & \quad \text{if }\ u_{i+1/2} < 0 |
---|
322 | \end{aligned} \right. |
---|
323 | \end{equation} |
---|
324 | where $\tau "_i =\delta _i \left[ {\delta _{i+1/2} \left[ \tau \right]} \right]$. |
---|
325 | |
---|
326 | This results in a dissipatively dominant (i.e. hyper-diffusive) truncation |
---|
327 | error \citep{Sacha2005}. The overall performance of the |
---|
328 | advection scheme is similar to that reported in \cite{Farrow1995}. |
---|
329 | It is a relatively good compromise between accuracy and smoothness. It is |
---|
330 | not a \emph{positive} scheme, meaning that false extrema are permitted, but the |
---|
331 | amplitude of such are significantly reduced over the centred second order |
---|
332 | method. Nevertheless it is not recommended that it should be applied to a passive |
---|
333 | tracer that requires positivity. |
---|
334 | |
---|
335 | The intrinsic diffusion of UBS makes its use risky in the vertical direction |
---|
336 | where the control of artificial diapycnal fluxes is of paramount importance. |
---|
337 | Therefore the vertical flux is evaluated using the TVD |
---|
338 | scheme when \np{ln\_traadv\_ubs}=.true.. |
---|
339 | |
---|
340 | For stability reasons (see \S\ref{DOM_nxt}), in \eqref{Eq_tra_adv_ubs}, |
---|
341 | the first term (which corresponds to a second order centred scheme) |
---|
342 | is evaluated using the \textit{now} tracer (centred in time) while the |
---|
343 | second term (which is the diffusive part of the scheme), is |
---|
344 | evaluated using the \textit{before} tracer (forward in time). |
---|
345 | This is discussed by \citet{Webb1998} in the context of the Quick |
---|
346 | advection scheme. UBS and QUICK |
---|
347 | schemes only differ by one coefficient. Replacing 1/6 with 1/8 in |
---|
348 | \eqref{Eq_tra_adv_ubs} leads to the QUICK advection scheme |
---|
349 | \citep{Webb1998}. This option is not available through a namelist |
---|
350 | parameter, since the 1/6 coefficient is hard coded. Nevertheless |
---|
351 | it is quite easy to make the substitution in the \mdl{traadv\_ubs} module |
---|
352 | and obtain a QUICK scheme. |
---|
353 | |
---|
354 | Note that : |
---|
355 | |
---|
356 | (1): When a high vertical resolution $O(1m)$ is used, the model stability can |
---|
357 | be controlled by vertical advection (not vertical diffusion which is usually |
---|
358 | solved using an implicit scheme). Computer time can be saved by using a |
---|
359 | time-splitting technique on vertical advection. This case has been |
---|
360 | implemented and validated in ORCA05 with 301 levels. It is not available in the |
---|
361 | current reference version. |
---|
362 | |
---|
363 | (2) : In a forthcoming release four options will be available for the vertical |
---|
364 | component used in the UBS scheme. $\tau _w^{ubs}$ will be evaluated |
---|
365 | using either \textit{(a)} a centred $2^{nd}$ order scheme , or \textit{(b)} |
---|
366 | a TVD scheme, or \textit{(c)} an interpolation based on conservative |
---|
367 | parabolic splines following the \citet{Sacha2005} implementation of UBS |
---|
368 | in ROMS, or \textit{(d)} a UBS. The $3^{rd}$ case has dispersion properties |
---|
369 | similar to an eighth-order accurate conventional scheme. |
---|
370 | |
---|
371 | following \citet{Sacha2005} implementation of UBS in ROMS, or \textit{(d)} |
---|
372 | an UBS. The $3^{rd}$ case has dispersion properties similar to an |
---|
373 | eight-order accurate conventional scheme. |
---|
374 | |
---|
375 | (3) : It is straightforward to rewrite \eqref{Eq_tra_adv_ubs} as follows: |
---|
376 | \begin{equation} \label{Eq_tra_adv_ubs2} |
---|
377 | \tau _u^{ubs} = \left\{ \begin{aligned} |
---|
378 | & \tau _u^{cen4} + \frac{1}{12} \tau"_i & \quad \text{if }\ u_{i+1/2} \geqslant 0 \\ |
---|
379 | & \tau _u^{cen4} - \frac{1}{12} \tau"_{i+1} & \quad \text{if }\ u_{i+1/2} < 0 |
---|
380 | \end{aligned} \right. |
---|
381 | \end{equation} |
---|
382 | or equivalently |
---|
383 | \begin{equation} \label{Eq_tra_adv_ubs2b} |
---|
384 | u_{i+1/2} \ \tau _u^{ubs} |
---|
385 | =u_{i+1/2} \ \overline{ T - \frac{1}{6}\,\delta _i\left[ \delta_{i+1/2}[T] \,\right] }^{\,i+1/2} |
---|
386 | - \frac{1}{2} |u|_{i+1/2} \;\frac{1}{6} \;\delta_{i+1/2}[\tau"_i] |
---|
387 | \end{equation} |
---|
388 | \eqref{Eq_tra_adv_ubs2} has several advantages. Firstly, it clearly reveals |
---|
389 | that the UBS scheme is based on the fourth order scheme to which an |
---|
390 | upstream-biased diffusion term is added. Secondly, this emphasises that the |
---|
391 | $4^{th}$ order part (as well as the $2^{nd}$ order part as stated above) has |
---|
392 | to be evaluated at the \emph{now} time step using \eqref{Eq_tra_adv_ubs}. |
---|
393 | Thirdly, the diffusion term is in fact a biharmonic operator with an eddy |
---|
394 | coefficient which is simply proportional to the velocity: |
---|
395 | $A_u^{lm}= - \frac{1}{12}\,{e_{1u}}^3\,|u|$. Note that NEMO v2.3 still uses |
---|
396 | \eqref{Eq_tra_adv_ubs}, not \eqref{Eq_tra_adv_ubs2}. This should be |
---|
397 | changed in forthcoming release. |
---|
398 | %%% |
---|
399 | \gmcomment{the change in UBS scheme has to be done} |
---|
400 | %%% |
---|
401 | |
---|
402 | % ------------------------------------------------------------------------------------------------------------- |
---|
403 | % QCK scheme |
---|
404 | % ------------------------------------------------------------------------------------------------------------- |
---|
405 | \subsection [QUICKEST scheme (QCK) (\np{ln\_traadv\_qck})] |
---|
406 | {QUICKEST scheme (QCK) (\np{ln\_traadv\_qck}=.true.)} |
---|
407 | \label{TRA_adv_qck} |
---|
408 | |
---|
409 | The Quadratic Upstream Interpolation for Convective Kinematics with |
---|
410 | Estimated Streaming Terms (QUICKEST) scheme proposed by \citet{Leonard1979} |
---|
411 | is the third order Godunov scheme. It is associated with the ULTIMATE QUICKEST |
---|
412 | limiter \citep{Leonard1991}. It has been implemented in NEMO by G. Reffray |
---|
413 | (MERCATOR-ocean). |
---|
414 | The resulting scheme is quite expensive but \emph{positive}. It can be used on both active and passive tracers. Nevertheless, the intrinsic diffusion of QCK makes its use |
---|
415 | risky in the vertical direction where the control of artificial diapycnal fluxes is of |
---|
416 | paramount importance. Therefore the vertical flux is evaluated using the CEN2 |
---|
417 | scheme. This no more ensure the positivity of the scheme. The use of TVD in the |
---|
418 | vertical direction as for the UBS case should be implemented to maintain the property. |
---|
419 | |
---|
420 | |
---|
421 | % ------------------------------------------------------------------------------------------------------------- |
---|
422 | % PPM scheme |
---|
423 | % ------------------------------------------------------------------------------------------------------------- |
---|
424 | \subsection [Piecewise Parabolic Method (PPM) (\np{ln\_traadv\_ppm})] |
---|
425 | {Piecewise Parabolic Method (PPM) (\np{ln\_traadv\_ppm}=.true.)} |
---|
426 | \label{TRA_adv_ppm} |
---|
427 | |
---|
428 | The Piecewise Parabolic Method (PPM) proposed by Colella and Woodward (1984) |
---|
429 | is based on a quadradic piecewise rebuilding. Like the QCK scheme, it is associated |
---|
430 | with the ULTIMATE QUICKEST limiter \citep{Leonard1991}. It has been implemented |
---|
431 | in \NEMO by G. Reffray (MERCATOR-ocean) but is not yet offered in the reference |
---|
432 | version 2.3. |
---|
433 | |
---|
434 | % ================================================================ |
---|
435 | % Tracer Lateral Diffusion |
---|
436 | % ================================================================ |
---|
437 | \section [Tracer Lateral Diffusion (\textit{traldf})] |
---|
438 | {Tracer Lateral Diffusion (\mdl{traldf})} |
---|
439 | \label{TRA_ldf} |
---|
440 | %-----------------------------------------nam_traldf------------------------------------------------------ |
---|
441 | \namdisplay{nam_traldf} |
---|
442 | %------------------------------------------------------------------------------------------------------------- |
---|
443 | |
---|
444 | The options available for lateral diffusion are a laplacian (rotated or not) |
---|
445 | or a biharmonic operator, the latter being more scale-selective (more |
---|
446 | diffusive at small scales). The specification of eddy diffusivity |
---|
447 | coefficients (either constant or variable in space and time) as well as the |
---|
448 | computation of the slope along which the operators act, are performed in the |
---|
449 | \mdl{ldftra} and \mdl{ldfslp} modules, respectively. This is described in Chap.~\ref{LDF}. The lateral diffusion of tracers is evaluated using a forward scheme, |
---|
450 | $i.e.$ the tracers appearing in its expression are the \textit{before} tracers in time, |
---|
451 | except for the pure vertical component that appears when a rotation tensor |
---|
452 | is used. This latter term is solved implicitly together with the |
---|
453 | vertical diffusion term (see \S\ref{DOM_nxt}). |
---|
454 | |
---|
455 | % ------------------------------------------------------------------------------------------------------------- |
---|
456 | % Iso-level laplacian operator |
---|
457 | % ------------------------------------------------------------------------------------------------------------- |
---|
458 | \subsection [Iso-level laplacian operator (\textit{traldf\_lap} - \np{ln\_traldf\_lap})] |
---|
459 | {Iso-level laplacian operator (\mdl{traldf\_lap} - \np{ln\_traldf\_lap}=.true.) } |
---|
460 | \label{TRA_ldf_lap} |
---|
461 | |
---|
462 | A laplacian diffusion operator (i.e. a harmonic operator) acting along the model |
---|
463 | surfaces is given by: |
---|
464 | \begin{equation} \label{Eq_tra_ldf_lap} |
---|
465 | \begin{split} |
---|
466 | D_T^{lT} =\frac{1}{e_{1T} \; e_{2T}\; e_{3T} } &\left[ {\quad \delta _i |
---|
467 | \left[ {A_u^{lT} \left( {\frac{e_{2u} e_{3u} }{e_{1u} }\;\delta _{i+1/2} |
---|
468 | \left[ T \right]} \right)} \right]} \right. |
---|
469 | \\ |
---|
470 | &\ \left. {+\; \delta _j \left[ |
---|
471 | {A_v^{lT} \left( {\frac{e_{1v} e_{3v} }{e_{2v} }\;\delta _{j+1/2} \left[ T |
---|
472 | \right]} \right)} \right]\quad } \right] |
---|
473 | \end{split} |
---|
474 | \end{equation} |
---|
475 | |
---|
476 | This lateral operator is a \emph{horizontal} one ($i.e.$ acting along |
---|
477 | geopotential surfaces) in the $z$-coordinate with or without partial step, |
---|
478 | but is simply an iso-level operator in the $s$-coordinate. |
---|
479 | It is thus used when, in addition to \np{ln\_traldf\_lap}=.true., we have |
---|
480 | \np{ln\_traldf\_level}=.true., or both \np{ln\_traldf\_hor}=.true. and |
---|
481 | \np{ln\_zco}=.false.. In both cases, it significantly contributes to |
---|
482 | diapycnal mixing. It is therefore not recommended. |
---|
483 | |
---|
484 | Note that |
---|
485 | (1) In pure $z$-coordinate (\key{zco} is defined), $e_{3u}=e_{3v}=e_{3T}$, so |
---|
486 | that the vertical scale factors disappear from (\ref{Eq_tra_ldf_lap}). |
---|
487 | (2) In partial step $z$-coordinate (\np{ln\_zps}=.true.), tracers in horizontally |
---|
488 | adjacent cells are located at different depths in the vicinity of the bottom. |
---|
489 | In this case, horizontal derivatives in (\ref{Eq_tra_ldf_lap}) at the bottom level |
---|
490 | require a specific treatment. They are calculated in the \mdl{zpshde} module, |
---|
491 | described in \S\ref{TRA_zpshde}. |
---|
492 | |
---|
493 | % ------------------------------------------------------------------------------------------------------------- |
---|
494 | % Rotated laplacian operator |
---|
495 | % ------------------------------------------------------------------------------------------------------------- |
---|
496 | \subsection [Rotated laplacian operator (\textit{traldf\_iso} - \np{ln\_traldf\_lap})] |
---|
497 | {Rotated laplacian operator (\mdl{traldf\_iso} - \np{ln\_traldf\_lap}=.true.)} |
---|
498 | \label{TRA_ldf_iso} |
---|
499 | |
---|
500 | The general form of the second order lateral tracer subgrid scale physics |
---|
501 | (\ref{Eq_PE_zdf}) takes the following semi-discrete space form in $z$- and |
---|
502 | $s$-coordinates: |
---|
503 | |
---|
504 | \begin{equation} \label{Eq_tra_ldf_iso} |
---|
505 | \begin{split} |
---|
506 | D_T^{lT} =& \frac{1}{e_{1T}\,e_{2T}\,e_{3T} } |
---|
507 | \\ |
---|
508 | & \left\{ {\delta _i \left[ {A_u^{lT} \left( |
---|
509 | {\frac{e_{2u} \; e_{3u} }{e_{1u} } \,\delta _{i+1/2}[T] |
---|
510 | -e_{2u} \; r_{1u} \,\overline{\overline {\delta _{k+1/2}[T]}}^{\,i+1/2,k}} |
---|
511 | \right)} \right]} \right. |
---|
512 | \\ |
---|
513 | & +\delta |
---|
514 | _j \left[ {A_v^{lT} \left( {\frac{e_{1v}\,e_{3v} }{e_{2v} |
---|
515 | }\,\delta _{j+1/2} \left[ T \right]-e_{1v}\,r_{2v} |
---|
516 | \,\overline{\overline {\delta _{k+1/2} \left[ T \right]}} ^{\,j+1/2,k}} |
---|
517 | \right)} \right] |
---|
518 | \\ |
---|
519 | & +\delta |
---|
520 | _k \left[ {A_w^{lT} \left( |
---|
521 | -e_{2w}\,r_{1w} \,\overline{\overline {\delta _{i+1/2} \left[ T \right]}} ^{\,i,k+1/2} |
---|
522 | \right.} \right. |
---|
523 | \\ |
---|
524 | & \qquad \qquad \quad |
---|
525 | -e_{1w}\,r_{2w} \,\overline{\overline {\delta _{j+1/2} \left[ T \right]}} ^{\,j,k+1/2} |
---|
526 | \\ |
---|
527 | & \left. {\left. { |
---|
528 | \quad \quad \quad \left. {{\kern |
---|
529 | 1pt}+\frac{e_{1w}\,e_{2w} }{e_{3w} }\,\left( {r_{1w} ^2+r_{2w} ^2} |
---|
530 | \right)\,\delta _{k+1/2} \left[ T \right]} \right)} \right]\;\;\;} \right\} |
---|
531 | \end{split} |
---|
532 | \end{equation} |
---|
533 | where $r_1$ and $r_2$ are the slopes between the surface of computation |
---|
534 | ($z$- or $s$-surfaces) and the surface along which the diffusion operator |
---|
535 | acts ($i.e.$ horizontal or iso-neutral surfaces). It is thus used when, |
---|
536 | in addition to \np{ln\_traldf\_lap}=.true., we have \np{ln\_traldf\_iso}=.true., |
---|
537 | or both \np{ln\_traldf\_hor}=.true. and \np{ln\_zco}=.true.. The way these |
---|
538 | slopes are evaluated is given in \S\ref{LDF_slp}. At the surface, bottom |
---|
539 | and lateral boundaries, the turbulent fluxes of heat and salt are set to zero |
---|
540 | using the mask technique (see \S\ref{LBC_coast}). |
---|
541 | |
---|
542 | The operator in \eqref{Eq_tra_ldf_iso} involves both lateral and vertical |
---|
543 | derivatives. For numerical stability, the vertical second derivative must |
---|
544 | be solved using the same implicit time scheme as that used in the vertical |
---|
545 | physics (see \S\ref{TRA_zdf}). For computer efficiency reasons, this term |
---|
546 | is not computed in the \mdl{traldf} module, but in the \mdl{trazdf} module |
---|
547 | where, if iso-neutral mixing is used, the vertical mixing coefficient is simply |
---|
548 | increased by $\frac{e_{1w}\,e_{2w} }{e_{3w} }\ \left( {r_{1w} ^2+r_{2w} ^2} \right)$. |
---|
549 | |
---|
550 | This formulation conserves the tracer but does not ensure the decrease |
---|
551 | of the tracer variance. Nevertheless the treatment performed on the slopes |
---|
552 | (see \S\ref{LDF}) allows the model to run safely without any additional |
---|
553 | background horizontal diffusion \citep{Guily2001}. An alternative scheme |
---|
554 | \citep{Griffies1998} which preserves both tracer and its variance is currently |
---|
555 | been tested in \NEMO. |
---|
556 | |
---|
557 | Note that in the partial step $z$-coordinate (\np{ln\_zps}=.true.), the horizontal |
---|
558 | derivatives at the bottom level in \eqref{Eq_tra_ldf_iso} require a specific |
---|
559 | treatment. They are calculated in module zpshde, described in \S\ref{TRA_zpshde}. |
---|
560 | |
---|
561 | % ------------------------------------------------------------------------------------------------------------- |
---|
562 | % Iso-level bilaplacian operator |
---|
563 | % ------------------------------------------------------------------------------------------------------------- |
---|
564 | \subsection [Iso-level bilaplacian operator (\textit{traldf\_bilap} - \np{ln\_traldf\_bilap})] |
---|
565 | {Iso-level bilaplacian operator (\mdl{traldf\_bilap} - \np{ln\_traldf\_bilap}=.true.)} |
---|
566 | \label{TRA_ldf_bilap} |
---|
567 | |
---|
568 | The lateral fourth order bilaplacian operator on tracers is obtained by |
---|
569 | applying (\ref{Eq_tra_ldf_lap}) twice. It requires an additional assumption |
---|
570 | on boundary conditions: the first and third derivative terms normal to the |
---|
571 | coast are set to zero. |
---|
572 | |
---|
573 | It is used when, in addition to \np{ln\_traldf\_bilap}=.true., we have |
---|
574 | \np{ln\_traldf\_level}=.true., or both \np{ln\_traldf\_hor}=.true. and |
---|
575 | \np{ln\_zco}=.false.. In both cases, it can contribute diapycnal mixing, |
---|
576 | although less than in the laplacian case. It is therefore not recommended. |
---|
577 | |
---|
578 | Note that in the code, the bilaplacian routine does not call the laplacian |
---|
579 | routine twice but is rather a separate routine. This is due to the fact that we |
---|
580 | introduce the eddy diffusivity coefficient, A, in the operator as: $\nabla |
---|
581 | \cdot \nabla \left( {A\nabla \cdot \nabla T} \right)$, instead of |
---|
582 | $-\nabla \cdot a\nabla \left( {\nabla \cdot a\nabla T} \right)$ where |
---|
583 | $a=\sqrt{|A|}$ and $A<0$. This was a mistake: both formulations |
---|
584 | ensure the total variance decrease, but the former requires a larger number |
---|
585 | of code-lines. It will be corrected in a forthcoming release. |
---|
586 | |
---|
587 | % ------------------------------------------------------------------------------------------------------------- |
---|
588 | % Rotated bilaplacian operator |
---|
589 | % ------------------------------------------------------------------------------------------------------------- |
---|
590 | \subsection [Rotated bilaplacian operator (\textit{traldf\_bilapg} - \np{ln\_traldf\_bilap})] |
---|
591 | {Rotated bilaplacian operator (\mdl{traldf\_bilapg} - \np{ln\_traldf\_bilap}=.true.)} |
---|
592 | \label{TRA_ldf_bilapg} |
---|
593 | |
---|
594 | The lateral fourth order operator formulation on tracers is obtained by |
---|
595 | applying (\ref{Eq_tra_ldf_iso}) twice. It requires an additional assumption |
---|
596 | on boundary conditions: first and third derivative terms normal to the |
---|
597 | coast, the bottom and the surface are set to zero. |
---|
598 | |
---|
599 | It is used when, in addition to \np{ln\_traldf\_bilap}=T, we have |
---|
600 | \np{ln\_traldf\_iso}=T, or both \np{ln\_traldf\_hor}=T and \np{ln\_zco}=T. |
---|
601 | Nevertheless, this rotated bilaplacian operator has never been seriously |
---|
602 | tested. No warranties that it is neither free of bugs or correctly formulated. |
---|
603 | Moreover, the stability range of such an operator will be probably quite |
---|
604 | narrow, requiring a significantly smaller time-step than the one used on |
---|
605 | unrotated operator. |
---|
606 | |
---|
607 | % ================================================================ |
---|
608 | % Tracer Vertical Diffusion |
---|
609 | % ================================================================ |
---|
610 | \section [Tracer Vertical Diffusion (\textit{trazdf})] |
---|
611 | {Tracer Vertical Diffusion (\mdl{trazdf})} |
---|
612 | \label{TRA_zdf} |
---|
613 | %--------------------------------------------namzdf--------------------------------------------------------- |
---|
614 | \namdisplay{namzdf} |
---|
615 | %-------------------------------------------------------------------------------------------------------------- |
---|
616 | |
---|
617 | The formulation of the vertical subgrid scale tracer physics is the same |
---|
618 | for all the vertical coordinates, and is based on a laplacian operator. |
---|
619 | The vertical diffusion operator given by (\ref{Eq_PE_zdf}) takes the |
---|
620 | following semi-discrete space form: |
---|
621 | (\ref{Eq_PE_zdf}) takes the following semi-discrete space form: |
---|
622 | \begin{equation} \label{Eq_tra_zdf} |
---|
623 | \begin{split} |
---|
624 | D^{vT}_T &= \frac{1}{e_{3T}} \; \delta_k \left[ |
---|
625 | \frac{A^{vT}_w}{e_{3w}} \delta_{k+1/2}[T] \right] |
---|
626 | \\ |
---|
627 | D^{vS}_T &= \frac{1}{e_{3T}} \; \delta_k \left[ |
---|
628 | \frac{A^{vS}_w}{e_{3w}} \delta_{k+1/2}[S] \right] |
---|
629 | \end{split} |
---|
630 | \end{equation} |
---|
631 | |
---|
632 | where $A_w^{vT}$ and $A_w^{vS}$ are the vertical eddy diffusivity |
---|
633 | coefficients on Temperature and Salinity, respectively. Generally, |
---|
634 | $A_w^{vT}=A_w^{vS}$ except when double diffusive mixing is |
---|
635 | parameterised (\key{zdfddm} is defined). The way these coefficients |
---|
636 | are evaluated is given in \S\ref{ZDF} (ZDF). Furthermore, when |
---|
637 | iso-neutral mixing is used, both mixing coefficients are increased |
---|
638 | by $\frac{e_{1w}\,e_{2w} }{e_{3w} }\ \left( {r_{1w} ^2+r_{2w} ^2} \right)$ |
---|
639 | to account for the vertical second derivative of \eqref{Eq_tra_ldf_iso}. |
---|
640 | |
---|
641 | At the surface and bottom boundaries, the turbulent fluxes of |
---|
642 | momentum, heat and salt must be specified. At the surface they |
---|
643 | are prescribed from the surface forcing (see \S\ref{TRA_sbc}), |
---|
644 | whilst at the bottom they are set to zero for heat and salt unless |
---|
645 | a geothermal flux forcing is prescribed as a bottom boundary |
---|
646 | condition (\S\ref{TRA_bbc}). |
---|
647 | |
---|
648 | The large eddy coefficient found in the mixed layer together with high |
---|
649 | vertical resolution implies that in the case of explicit time stepping |
---|
650 | (\np{ln\_zdfexp}=.true.) there would be too restrictive a constraint on |
---|
651 | the time step. Therefore, the default implicit time stepping is preferred |
---|
652 | for the vertical diffusion since it overcomes the stability constraint. |
---|
653 | A forward time differencing scheme (\np{ln\_zdfexp}=.true.) using a time |
---|
654 | splitting technique (\np{n\_zdfexp} $> 1$) is provided as an alternative. |
---|
655 | Namelist variables \np{ln\_zdfexp} and \np{n\_zdfexp} apply to both |
---|
656 | tracers and dynamics. |
---|
657 | |
---|
658 | % ================================================================ |
---|
659 | % External Forcing |
---|
660 | % ================================================================ |
---|
661 | \section{External Forcing} |
---|
662 | \label{TRA_sbc_qsr_bbc} |
---|
663 | |
---|
664 | % ------------------------------------------------------------------------------------------------------------- |
---|
665 | % surface boundary condition |
---|
666 | % ------------------------------------------------------------------------------------------------------------- |
---|
667 | \subsection [Surface boundary condition (\textit{trasbc})] |
---|
668 | {Surface boundary condition (\mdl{trasbc})} |
---|
669 | \label{TRA_sbc} |
---|
670 | |
---|
671 | The surface boundary condition for tracers is implemented in a separate |
---|
672 | module (\mdl{trasbc}) instead of entering as a boundary condition on the vertical |
---|
673 | diffusion operator (as in the case of momentum). This has been found to |
---|
674 | enhance readability of the code. The two formulations are completely |
---|
675 | equivalent; the forcing terms in trasbc are the surface fluxes divided by |
---|
676 | the thickness of the top model layer. Following \citet{Roullet2000} the |
---|
677 | forcing on an ocean tracer, $c$, can be split into two parts: $F_{ext}$, the |
---|
678 | flux of tracer crossing the sea surface and not linked with the water |
---|
679 | exchange with the atmosphere, $F_{wf}^d$, and $F_{wf}^i$ the forcing |
---|
680 | on the tracer concentration associated with this water exchange. |
---|
681 | The latter forcing has two components: a direct effect of change |
---|
682 | in concentration associated with the tracer carried by the water flux, |
---|
683 | and an indirect concentration/dilution effect : |
---|
684 | \begin{equation*} |
---|
685 | \begin{split} |
---|
686 | F^C &= F_{ext} + F_{wf}^d +F_{wf}^i \\ |
---|
687 | &= F_{ext} - \left( c_E \, E - c_p \,P - c_R \,R \right) +c\left( E-P-R \right) |
---|
688 | \end{split} |
---|
689 | \end{equation*} |
---|
690 | |
---|
691 | \gmcomment{add here a description of the variable names used in the above equation} |
---|
692 | |
---|
693 | Two cases must be distinguished, the nonlinear free surface case |
---|
694 | (\key{vvl} is defined) and the linear free surface case. The first case |
---|
695 | is simpler, because the indirect concentration/dilution effect is naturally |
---|
696 | taken into account by letting the vertical scale factors vary in time. |
---|
697 | The salinity of water exchanged at the surface is assumed to be zero, |
---|
698 | so there is no salt flux at the free surface, except in the presence of sea ice. |
---|
699 | The heat flux at the free surface is the sum of $F_{ext}$, the direct |
---|
700 | heating/cooling (by the total non-penetrative heat flux) and $F_{wf}^e$ |
---|
701 | the heat carried by the water exchanged through the surface (evaporation, |
---|
702 | precipitation, runoff). The temperature of precipitation is not well known. |
---|
703 | In the model we assume that this water has the same temperature as |
---|
704 | the sea surface temperature. The resulting forcing terms for temperature |
---|
705 | T and salinity S are: |
---|
706 | \begin{equation} \label{Eq_tra_forcing} |
---|
707 | \begin{aligned} |
---|
708 | &F^T =\frac{Q_{ns} }{\rho _o \;C_p \,e_{3T} }-\frac{\text{EMP}\;\left. T |
---|
709 | \right|_{k=1} }{e_{3T} } & \\ |
---|
710 | \\ |
---|
711 | & F^S =\frac{\text{EMP}_S\;\left. S \right|_{k=1} }{e_{3T} } & |
---|
712 | \end{aligned} |
---|
713 | \end{equation} |
---|
714 | |
---|
715 | where EMP is the freshwater budget (evaporation minus precipitation |
---|
716 | minus river runoff) which forces the ocean volume, $Q_{ns}$ is the |
---|
717 | non-penetrative part of the net surface heat flux (difference between |
---|
718 | the total surface heat flux and the fraction of the short wave flux that |
---|
719 | penetrates into the water column), the product EMP$_S\;.\left. S \right|_{k=1}$ |
---|
720 | is the ice-ocean salt flux, and $\left. S\right|_{k=1}$ is the sea surface |
---|
721 | salinity (\textit{SSS}). The total salt content is conserved in this formulation |
---|
722 | (except for the effect of the Asselin filter). |
---|
723 | |
---|
724 | %AMT note: the ice-ocean flux had been forgotten in the first release of the key_vvl option, has this been corrected in the code? |
---|
725 | |
---|
726 | In the second case (linear free surface), the vertical scale factors are |
---|
727 | fixed in time so that the concentration/dilution effect must be added in |
---|
728 | the \mdl{trasbc} module. Because of the hypothesis made for the |
---|
729 | temperature of precipitation and runoffs, $F_{wf}^e +F_{wf}^i =0$ |
---|
730 | for temperature. The resulting forcing term for temperature is: |
---|
731 | |
---|
732 | \begin{equation} \label{Eq_tra_forcing_q} |
---|
733 | F^T=\frac{Q_{ns} }{\rho _o \;C_p \,e_{3T} } |
---|
734 | \end{equation} |
---|
735 | |
---|
736 | The salinity forcing is still given by \eqref{Eq_tra_forcing} but the |
---|
737 | definition of EMP$_S$ is different: it is the total surface freshwater |
---|
738 | budget (evaporation minus precipitation minus river runoff plus |
---|
739 | the rate of change of the sea ice thickness). The total salt content |
---|
740 | is not exactly conserved (\citet{Roullet2000}. |
---|
741 | See also \S\ref{PE_free_surface}). |
---|
742 | |
---|
743 | In the case of the rigid lid approximation, the surface salinity forcing $F^s$ |
---|
744 | is also expressed by \eqref{Eq_tra_forcing}, but now the global integral of |
---|
745 | the product of EMP and S, is not compensated by the advection of fluid |
---|
746 | through the top level: this is because in the rigid lid case \textit{w(k=1) = 0} |
---|
747 | (in contrast to the linear free surface case). As a result, even if the budget |
---|
748 | of \textit{EMP} is zero on average over the whole ocean domain, the |
---|
749 | associated salt flux is not, since sea-surface salinity and \textit{EMP} are |
---|
750 | intrinsically correlated (high \textit{SSS} are found where evaporation is |
---|
751 | strong whilst low \textit{SSS} is usually associated with high precipitation |
---|
752 | or river runoff). |
---|
753 | |
---|
754 | The $Q_{ns} $ and \textit{EMP} fields are defined and updated in the |
---|
755 | \mdl{sbcmod} module (see \S\ref{SBC}). |
---|
756 | |
---|
757 | % ------------------------------------------------------------------------------------------------------------- |
---|
758 | % Solar Radiation Penetration |
---|
759 | % ------------------------------------------------------------------------------------------------------------- |
---|
760 | \subsection [Solar Radiation Penetration (\textit{traqsr})] |
---|
761 | {Solar Radiation Penetration (\mdl{traqsr})} |
---|
762 | \label{TRA_qsr} |
---|
763 | %--------------------------------------------namqsr-------------------------------------------------------- |
---|
764 | \namdisplay{namqsr} |
---|
765 | %-------------------------------------------------------------------------------------------------------------- |
---|
766 | |
---|
767 | When the penetrative solar radiation option is used (\np{ln\_flxqsr}=.true.), |
---|
768 | the solar radiation penetrates the top few meters of the ocean, otherwise |
---|
769 | all the heat flux is absorbed in the first ocean level (\np{ln\_flxqsr}=.false.). |
---|
770 | Thus, in the former case a term is added to the time evolution equation of |
---|
771 | temperature \eqref{Eq_PE_tra_T} whilst the surface boundary condition is |
---|
772 | modified to take into account only the non-penetrative part of the surface |
---|
773 | heat flux: |
---|
774 | \begin{equation} \label{Eq_PE_qsr} |
---|
775 | \begin{split} |
---|
776 | \frac{\partial T}{\partial t} &= {\ldots} + \frac{1}{\rho_o\, C_p \,e_3} \; \frac{\partial I}{\partial k} \\ |
---|
777 | Q_{ns} &= Q_\text{Total} - Q_{sr} |
---|
778 | \end{split} |
---|
779 | \end{equation} |
---|
780 | |
---|
781 | where $I$ is the downward irradiance. The additional term in \eqref{Eq_PE_qsr} is discretized as follows: |
---|
782 | \begin{equation} \label{Eq_tra_qsr} |
---|
783 | \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] |
---|
784 | \end{equation} |
---|
785 | |
---|
786 | A formulation involving two extinction coefficients is assumed for the |
---|
787 | downward irradiance $I$ \citep{Paulson1977}: |
---|
788 | \begin{equation} \label{Eq_traqsr_iradiance} |
---|
789 | I(z) = Q_{sr} \left[Re^{-z / \xi_1} + \left( 1-R\right) e^{-z / \xi_2} \right] |
---|
790 | \end{equation} |
---|
791 | where $Q_{sr}$ is the penetrative part of the surface heat flux, |
---|
792 | $\xi_1$ and $\xi_2$ are two extinction length scales and $R$ |
---|
793 | determines the relative contribution of the two terms. |
---|
794 | The default values used correspond to a Type I water in Jerlov's [1968] |
---|
795 | % |
---|
796 | \gmcomment : Jerlov reference to be added |
---|
797 | % |
---|
798 | classification: $\xi_1 = 0.35m$, $\xi_2 = 0.23m$ and $R = 0.58$ |
---|
799 | (corresponding to \np{xsi1}, \np{xsi2} and \np{rabs} namelist parameters, |
---|
800 | respectively). $I$ is masked (no flux through the ocean bottom), |
---|
801 | so all the solar radiation that reaches the last ocean level is absorbed |
---|
802 | in that level. The trend in \eqref{Eq_tra_qsr} associated with the |
---|
803 | penetration of the solar radiation is added to the temperature trend, |
---|
804 | and the surface heat flux is modified in routine \mdl{traqsr}. |
---|
805 | Note that in the $z$-coordinate, the depth of $T-$levels depends |
---|
806 | on the single variable $k$. A one dimensional array of the coefficients |
---|
807 | $gdsr(k) = Re^{-z_w (k)/\xi_1} + (1-R)e^{-z_w (k)/\xi_2}$ can then |
---|
808 | be computed once and saved in memory. Moreover \textit{nksr}, |
---|
809 | the level at which $gdrs$ becomes negligible (less than the |
---|
810 | computer precision) is computed once, and the trend associated |
---|
811 | with the penetration of the solar radiation is only added until that level. |
---|
812 | Finally, note that when the ocean is shallow (< 200~m), part of the |
---|
813 | solar radiation can reach the ocean floor. In this case, we have |
---|
814 | chosen that all remaining radiation is absorbed in the last ocean |
---|
815 | level ($i.e.$ $I_w$ is masked). |
---|
816 | |
---|
817 | When coupling with a biological model (for example PISCES or LOBSTER), |
---|
818 | it is possible to calculate the light attenuation using information from |
---|
819 | the biology model. Without biological model, it is still possible to introduce |
---|
820 | a horizontal variation of the light attenuation by using the observed ocean |
---|
821 | surface color. At the time of writing, the latter has not been implemented |
---|
822 | in the reference version. |
---|
823 | % |
---|
824 | \gmcomment{ {yellow}{case 4 bands and bio-coupling to add !!!} } |
---|
825 | % |
---|
826 | |
---|
827 | % ------------------------------------------------------------------------------------------------------------- |
---|
828 | % Bottom Boundary Condition |
---|
829 | % ------------------------------------------------------------------------------------------------------------- |
---|
830 | \subsection [Bottom Boundary Condition (\textit{trabbc} - \key{bbc})] |
---|
831 | {Bottom Boundary Condition (\mdl{trabbc} - \key{bbc})} |
---|
832 | \label{TRA_bbc} |
---|
833 | %--------------------------------------------nambbc-------------------------------------------------------- |
---|
834 | \namdisplay{nambbc} |
---|
835 | %-------------------------------------------------------------------------------------------------------------- |
---|
836 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
---|
837 | \begin{figure}[!t] \label{Fig_geothermal} \begin{center} |
---|
838 | \includegraphics[width=1.0\textwidth]{./TexFiles/Figures/Fig_TRA_geoth.pdf} |
---|
839 | \caption{Geothermal Heat flux (in $mW.m^{-2}$) as inferred from the age |
---|
840 | of the sea floor and the formulae of \citet{Stein1992}.} |
---|
841 | \end{center} \end{figure} |
---|
842 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
---|
843 | |
---|
844 | Usually it is assumed that there is no exchange of heat or salt through |
---|
845 | the ocean bottom, $i.e.$ a no flux boundary condition is applied on active |
---|
846 | tracers at the bottom. This is the default option in \NEMO, and it is |
---|
847 | implemented using the masking technique. Hoever, there is a |
---|
848 | non-zero heat flux across the seafloor that is associated with solid |
---|
849 | earth cooling. This flux is weak compared to surface fluxes (a mean |
---|
850 | global value of $\sim0.1\;W/m^2$ \citep{Stein1992}), but it is |
---|
851 | systematically positive and acts on the densest water masses. Taking |
---|
852 | this flux into account in a global ocean model increases |
---|
853 | the deepest overturning cell (i.e. the one associated with the Antarctic |
---|
854 | Bottom Water) by a few Sverdrups. |
---|
855 | |
---|
856 | The presence or not of geothermal heating is controlled by the namelist |
---|
857 | parameter \np{ngeo\_flux}. If this parameter is set to 1, a constant |
---|
858 | geothermal heating is introduced whose value is given by the |
---|
859 | \np{ngeo\_flux\_const}, which is also a namelist parameter. If it is set to 2, |
---|
860 | a spatially varying geothermal heat flux is introduced which is provided |
---|
861 | in the geothermal\_heating.nc NetCDF file (Fig.\ref{Fig_geothermal}). |
---|
862 | |
---|
863 | % ================================================================ |
---|
864 | % Bottom Boundary Layer |
---|
865 | % ================================================================ |
---|
866 | \section [Bottom Boundary Layer (\textit{trabbl}, \textit{trabbl\_adv} )] |
---|
867 | {Bottom Boundary Layer (\mdl{trabbl}, \mdl{trabbl\_adv})} |
---|
868 | \label{TRA_bbl} |
---|
869 | %--------------------------------------------nambbl--------------------------------------------------------- |
---|
870 | \namdisplay{nambbl} |
---|
871 | %-------------------------------------------------------------------------------------------------------------- |
---|
872 | |
---|
873 | In a $z$-coordinate configuration, the bottom topography is represented by a |
---|
874 | series of discrete steps. This is not adequate to represent gravity driven |
---|
875 | downslope flows. Such flows arise downstream of sills such as the Strait of |
---|
876 | Gibraltar, Bab El Mandeb, or Denmark Strait, where dense water formed in |
---|
877 | marginal seas flows into a basin filled with less dense water. The amount of |
---|
878 | entrainment that occurs in these gravity plumes is critical in determining the |
---|
879 | density and volume flux of the densest waters of the ocean, such as |
---|
880 | Antarctic Bottom Water, or North Atlantic Deep Water. $z$-coordinate |
---|
881 | models tend to overestimate the entrainment, because the gravity flow is |
---|
882 | mixed down vertically by convection as it goes ``downstairs'' following the |
---|
883 | step topography, sometimes over a thickness much larger than the thickness |
---|
884 | of the observed gravity plume. A similar problem occurs in the $s$-coordinate when |
---|
885 | the thickness of the bottom level varies in large proportions downstream of |
---|
886 | a sill \citep{Willebrand2001}, and the thickness of the plume is not resolved. |
---|
887 | |
---|
888 | The idea of the bottom boundary layer (BBL) parameterization first introduced by |
---|
889 | \citet{BeckDos1998} is to allow a direct communication between |
---|
890 | two adjacent bottom cells at different levels, whenever the densest water is |
---|
891 | located above the less dense water. The communication can be by a diffusive |
---|
892 | (diffusive BBL), advective fluxes (advective BBL), or both. In the current |
---|
893 | implementation of the BBL, only the tracers are modified, not the velocities. |
---|
894 | Furthermore, it only connects ocean bottom cells, and therefore does not include |
---|
895 | the improvment proposed by \citet{Campin_Goosse_Tel99}. |
---|
896 | |
---|
897 | % ------------------------------------------------------------------------------------------------------------- |
---|
898 | % Diffusive BBL |
---|
899 | % ------------------------------------------------------------------------------------------------------------- |
---|
900 | \subsection{Diffusive Bottom Boundary layer (\key{bbl\_diff})} |
---|
901 | \label{TRA_bbl_diff} |
---|
902 | |
---|
903 | When applying sigma-diffusion (\key{trabbl} is defined), the diffusive flux between |
---|
904 | two adjacent cells living at the ocean bottom is given by |
---|
905 | \begin{equation} \label{Eq_tra_bbl_diff} |
---|
906 | {\rm {\bf F}}_\sigma=A_l^\sigma \; \nabla_\sigma T |
---|
907 | \end{equation} |
---|
908 | with $\nabla_\sigma$ the lateral gradient operator taken between bottom cells, |
---|
909 | and $A_l^\sigma $ the lateral diffusivity in the BBL. Following \citet{BeckDos1998}, |
---|
910 | the latter is prescribed with a spatial dependence, $e.g.$ in the conditional form |
---|
911 | \begin{equation} \label{Eq_tra_bbl_coef} |
---|
912 | A_l^\sigma (i,j,t)=\left\{ {\begin{array}{l} |
---|
913 | A_{bbl} \quad \quad \mbox{if} \quad \nabla_\sigma \rho \cdot \nabla H<0 \\ |
---|
914 | \\ |
---|
915 | 0\quad \quad \;\,\mbox{otherwise} \\ |
---|
916 | \end{array}} \right. |
---|
917 | \end{equation} |
---|
918 | where $A_{bbl}$ is the BBL diffusivity coefficient, given by the namelist |
---|
919 | parameter \np{atrbbl}. $A_{bbl}$ is usually set to a value much larger |
---|
920 | than the one used on lateral mixing in open ocean. |
---|
921 | Note that in practice, \eqref{Eq_tra_bbl_coef} constraint is applied |
---|
922 | separately in the two horizontal directions, and the density gradient in |
---|
923 | \eqref{Eq_tra_bbl_coef} is evaluated at $\overline{H}^i$ ($\overline{H}^j$) |
---|
924 | using the along bottom mean temperature and salinity. |
---|
925 | |
---|
926 | % ------------------------------------------------------------------------------------------------------------- |
---|
927 | % Advective BBL |
---|
928 | % ------------------------------------------------------------------------------------------------------------- |
---|
929 | \subsection {Advective Bottom Boundary Layer (\key{bbl\_adv})} |
---|
930 | \label{TRA_bbl_adv} |
---|
931 | |
---|
932 | |
---|
933 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
---|
934 | \begin{figure}[!t] \label{Fig_bbl} \begin{center} |
---|
935 | \includegraphics[width=1.0\textwidth]{./TexFiles/Figures/Fig_BBL_adv.pdf} |
---|
936 | \caption{Advective Bottom Boundary Layer.} |
---|
937 | \end{center} \end{figure} |
---|
938 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
---|
939 | |
---|
940 | %%%gmcomment : this section has to be really written |
---|
941 | |
---|
942 | The advective BBL is in fact not only an advective one but include a diffusive |
---|
943 | component as we chose an upstream scheme to perform the advection within |
---|
944 | the BBL. The associated diffusion only act in the stream direction and is |
---|
945 | proportional to the velocity. |
---|
946 | |
---|
947 | When applying sigma-advection (\key{trabbl\_adv} defined), the advective |
---|
948 | flux between two adjacent cells living at the ocean bottom is given by |
---|
949 | \begin{equation} \label{Eq_tra_bbl_fadv} |
---|
950 | {\rm {\bf F}}_\sigma={\rm {\bf U}}_h^\sigma \; \overline{T}^\sigma |
---|
951 | \end{equation} |
---|
952 | with $\nabla_\sigma$ the lateral gradient operator taken between bottom cells, |
---|
953 | and $A_l^\sigma$ the lateral diffusivity in the BBL. Following \citet{BeckDos1998}, |
---|
954 | the latter is prescribed with a spatial dependence, $e.g.$ in the conditional form |
---|
955 | \begin{equation} \label{Eq_tra_bbl_Aadv} |
---|
956 | A_l^\sigma (i,j,t)=\left\{ {\begin{array}{l} |
---|
957 | A_{bbl} \quad \quad \mbox{if} \quad \nabla_\sigma \rho \cdot \nabla H<0 |
---|
958 | \quad \quad \mbox{and} \quad {\rm {\bf U}}_h \cdot \nabla H<0 \\ |
---|
959 | \\ |
---|
960 | 0\quad \quad \;\,\mbox{otherwise} \\ |
---|
961 | \end{array}} \right. |
---|
962 | \end{equation} |
---|
963 | |
---|
964 | % ================================================================ |
---|
965 | % Tracer damping |
---|
966 | % ================================================================ |
---|
967 | \section [Tracer damping (\textit{tradmp})] |
---|
968 | {Tracer damping (\mdl{tradmp})} |
---|
969 | \label{TRA_dmp} |
---|
970 | %--------------------------------------------namdmp----------------------------------------------------- |
---|
971 | \namdisplay{namdmp} |
---|
972 | %-------------------------------------------------------------------------------------------------------------- |
---|
973 | |
---|
974 | In some applications it can be useful to add a Newtonian damping term |
---|
975 | into the temperature and salinity equations: |
---|
976 | \begin{equation} \label{Eq_tra_dmp} |
---|
977 | \begin{split} |
---|
978 | \frac{\partial T}{\partial t}=\;\cdots \;-\gamma \,\left( {T-T_o } \right) \\ |
---|
979 | \\ |
---|
980 | \frac{\partial S}{\partial t}=\;\cdots \;-\gamma \,\left( {S-S_o } \right) |
---|
981 | \end{split} |
---|
982 | \end{equation} |
---|
983 | where $\gamma$ is the inverse of a time scale, and $T_o$ and $S_o$ |
---|
984 | are given temperature and salinity fields (usually a climatology). |
---|
985 | The restoring term is added when \key{tradmp} is defined. |
---|
986 | It also requires that both \key{temdta} and \key{saldta} are defined |
---|
987 | ($i.e.$ that $T_o$ and $S_o$ are read). The restoring coefficient |
---|
988 | $S_o$ is a three-dimensional array initialized by the user in routine |
---|
989 | \rou{dtacof} also located in module \mdl{tradmp}. |
---|
990 | |
---|
991 | The two main cases in which \eqref{Eq_tra_dmp} is used are \textit{(a)} |
---|
992 | the specification of the boundary conditions along artificial walls of a |
---|
993 | limited domain basin and \textit{(b)} the computation of the velocity |
---|
994 | field associated with a given $T$-$S$ field (for example to build the |
---|
995 | initial state of a prognostic simulation, or to use the resulting velocity |
---|
996 | field for a passive tracer study). The first case applies to regional |
---|
997 | models that have artificial walls instead of open boundaries. |
---|
998 | In the vicinity of these walls, $S_o$ takes large values (equivalent to |
---|
999 | a time scale of a few days) whereas it is zero in the interior of the |
---|
1000 | model domain. The second case corresponds to the use of the robust |
---|
1001 | diagnostic method \citep{Sarmiento1982}. It allows us to find the velocity |
---|
1002 | field consistent with the model dynamics whilst having a $T$-$S$ field |
---|
1003 | close to a given climatological field ($T_o -S_o$). The time scale |
---|
1004 | associated with $S_o$ is generally not a constant but spatially varying |
---|
1005 | in order to respect other properties. For example, it is usually set to zero |
---|
1006 | in the mixed layer (defined either on a density or $S_o$ criterion) |
---|
1007 | \citep{Madec1996} and in the equatorial region |
---|
1008 | \citep{Reverdin1991, Fujio1991, MartiTh1992} since these two regions |
---|
1009 | have a short time scale of adjustment; while smaller $S_o$ are used |
---|
1010 | in the deep ocean where the typical time scale is long \citep{Sarmiento1982}. |
---|
1011 | In addition the time scale is reduced (even to zero) along the western |
---|
1012 | boundary to allow the model to reconstruct its own western boundary |
---|
1013 | structure in equilibrium with its physics. The choice of a |
---|
1014 | Newtonian damping acting in the mixed layer or not is controlled by |
---|
1015 | namelist parameter \np{nmldmp}. |
---|
1016 | |
---|
1017 | The robust diagnostic method is very efficient in preventing temperature |
---|
1018 | drift in intermediate waters but it produces artificial sources of heat and salt |
---|
1019 | within the ocean. It also has undesirable effects on the ocean convection. |
---|
1020 | It tends to prevent deep convection and subsequent deep-water formation, |
---|
1021 | by stabilising the water column too much. |
---|
1022 | |
---|
1023 | An example of the computation of $S_o$ for robust diagnostic experiments |
---|
1024 | with the ORCA2 model is provided in the \mdl{tradmp} module |
---|
1025 | (subroutines \rou{dtacof} and \rou{cofdis} which compute the coefficient |
---|
1026 | and the distance to the bathymetry, respectively). These routines are |
---|
1027 | provided as examples and can be customised by the user. |
---|
1028 | |
---|
1029 | % ================================================================ |
---|
1030 | % Tracer time evolution |
---|
1031 | % ================================================================ |
---|
1032 | \section [Tracer time evolution (\textit{tranxt})] |
---|
1033 | {Tracer time evolution (\mdl{tranxt})} |
---|
1034 | \label{TRA_nxt} |
---|
1035 | %--------------------------------------------namdom----------------------------------------------------- |
---|
1036 | \namdisplay{namdom} |
---|
1037 | %-------------------------------------------------------------------------------------------------------------- |
---|
1038 | |
---|
1039 | The general framework for tracer time stepping is a leap-frog scheme, |
---|
1040 | $i.e.$ a three level centred time scheme associated with a Asselin time |
---|
1041 | filter (cf. \S\ref{DOM_nxt}): |
---|
1042 | \begin{equation} \label{Eq_tra_nxt} |
---|
1043 | \begin{split} |
---|
1044 | T^{t+\Delta t} &= T^{t-\Delta t} + 2 \, \Delta t \ \text{RHS}_T^t \\ |
---|
1045 | \\ |
---|
1046 | T_f^t \;\ \quad &= T^t \;\quad +\gamma \,\left[ {T_f^{t-\Delta t} -2T^t+T^{t+\Delta t}} \right] |
---|
1047 | \end{split} |
---|
1048 | \end{equation} |
---|
1049 | |
---|
1050 | where $\text{RHS}_T$ is the right hand side of the temperature equation, |
---|
1051 | the subscript $f$ denotes filtered values and $\gamma$ is the Asselin |
---|
1052 | coefficient. $\gamma$ is initialized as \np{atfp} (\textbf{namelist} parameter). |
---|
1053 | Its default value is \np{atfp=0.1}. |
---|
1054 | |
---|
1055 | When the vertical mixing is solved implicitly, the update of the \textit{next} tracer |
---|
1056 | fields is done in module \mdl{trazdf}. In this case only the swapping of arrays |
---|
1057 | and the Asselin filtering is done in the \mdl{tranxt} module. |
---|
1058 | |
---|
1059 | In order to prepare for the computation of the \textit{next} time step, |
---|
1060 | a swap of tracer arrays is performed: $T^{t-\Delta t} = T^t$ and $T^t = T_f$. |
---|
1061 | |
---|
1062 | % ================================================================ |
---|
1063 | % Equation of State (eosbn2) |
---|
1064 | % ================================================================ |
---|
1065 | \section [Equation of State (\textit{eosbn2}) ] |
---|
1066 | {Equation of State (\mdl{eosbn2}) } |
---|
1067 | \label{TRA_eosbn2} |
---|
1068 | %--------------------------------------------nameos----------------------------------------------------- |
---|
1069 | \namdisplay{nameos} |
---|
1070 | %-------------------------------------------------------------------------------------------------------------- |
---|
1071 | |
---|
1072 | % ------------------------------------------------------------------------------------------------------------- |
---|
1073 | % Equation of State |
---|
1074 | % ------------------------------------------------------------------------------------------------------------- |
---|
1075 | \subsection{Equation of State (\np{neos} = 0, 1 or 2)} |
---|
1076 | \label{TRA_eos} |
---|
1077 | |
---|
1078 | It is necessary to know the equation of state for the ocean very accurately |
---|
1079 | to determine stability properties (especially the Brunt-Vais\"{a}l\"{a} frequency), |
---|
1080 | particularly in the deep ocean. The ocean density is a non linear empirical |
---|
1081 | function of \textit{in situ }temperature, salinity and pressure. The reference |
---|
1082 | equation of state is that defined by the Joint Panel on Oceanographic Tables |
---|
1083 | and Standards \citep{UNESCO1983}. It was the standard equation of state |
---|
1084 | used in early releases of OPA. However, even though this computation is |
---|
1085 | fully vectorised, it is quite time consuming ($15$ to $20${\%} of the total |
---|
1086 | CPU time) since it requires the prior computation of the \textit{in situ} |
---|
1087 | temperature from the model \textit{potential} temperature using the |
---|
1088 | \citep{Bryden1973} polynomial for adiabatic lapse rate and a $4^th$ order |
---|
1089 | Runge-Kutta integration scheme. Since OPA6, we have used the |
---|
1090 | \citet{JackMcD1995} equation of state for seawater instead. It allows the |
---|
1091 | computation of the \textit{in situ} ocean density directly as a function of |
---|
1092 | \textit{potential} temperature relative to the surface (an \NEMO variable), |
---|
1093 | the practical salinity (another \NEMO variable) and the pressure (assuming no |
---|
1094 | pressure variation along geopotential surfaces, i.e. the pressure in decibars is |
---|
1095 | approximated by the depth in meters). Both the \citet{UNESCO1983} and \citet{JackMcD1995} equations of state have exactly the same except that |
---|
1096 | the values of the various coefficients have been adjusted by \citet{JackMcD1995} |
---|
1097 | in order to directly use the \textit{potential} temperature instead of the |
---|
1098 | \textit{in situ} one. This reduces the CPU time of the in situ density computation |
---|
1099 | to about $3${\%} of the total CPU time, while maintaining a quite accurate |
---|
1100 | equation of state. |
---|
1101 | |
---|
1102 | In the computer code, a \textit{true} density $d$ is computed, $i.e.$ the ratio |
---|
1103 | of seawater volumic mass to $\rho_o$, a reference volumic mass (\textit{rau0} |
---|
1104 | defined in \mdl{phycst}, usually $rau0= 1,020~Kg/m^3$). The default option |
---|
1105 | (namelist prameter \np{neos}=0) is the \citet{JackMcD1995} equation of state. |
---|
1106 | Its use is highly recommended. However, for process studies, it is often |
---|
1107 | convenient to use a linear approximation of the density$^{\ast}$ |
---|
1108 | \footnote{$^{\ast }$ With the linear equation of state there is no longer |
---|
1109 | a distinction between \textit{in situ} and \textit{potential} density. Cabling |
---|
1110 | and thermobaric effects are also removed.}. |
---|
1111 | Two linear formulations are available: a function of $T$ only (\np{neos}=1) |
---|
1112 | and a function of both $T$ and $S$ (\np{neos}=2): |
---|
1113 | \begin{equation} \label{Eq_tra_eos_linear} |
---|
1114 | \begin{aligned} |
---|
1115 | d(T) &= {\rho (T)} / {\rho _0 } &&= 1.028 - \alpha \;T \\ |
---|
1116 | d(T,S) &= {\rho (T,S)} &&= \ \ \ \beta \;S - \alpha \;T |
---|
1117 | \end{aligned} |
---|
1118 | \end{equation} |
---|
1119 | where $\alpha$ and $\beta$ are the thermal and haline expansion |
---|
1120 | coefficients, and $\rho_o$, the reference volumic mass, $rau0$. |
---|
1121 | ($\alpha$ and $\beta$ can be modified through the \np{ralpha} and |
---|
1122 | \np{rbeta} namelist parameters). Note that when $d$ is a function |
---|
1123 | of $T$ only (\np{neos}=1), the salinity is a passive tracer and can be |
---|
1124 | used as such. |
---|
1125 | |
---|
1126 | % ------------------------------------------------------------------------------------------------------------- |
---|
1127 | % Brunt-Vais\"{a}l\"{a} Frequency |
---|
1128 | % ------------------------------------------------------------------------------------------------------------- |
---|
1129 | \subsection{Brunt-Vais\"{a}l\"{a} Frequency (\np{neos} = 0, 1 or 2)} |
---|
1130 | \label{TRA_bn2} |
---|
1131 | |
---|
1132 | An accurate computation of the ocean stability (i.e. of $N$, the brunt-Vais\"{a}l\"{a} |
---|
1133 | frequency) is of paramount importance as it is used in several ocean |
---|
1134 | parameterisations (namely TKE, KPP, Richardson number dependent |
---|
1135 | vertical diffusion, enhanced vertical diffusion, non-penetrative convection, |
---|
1136 | iso-neutral diffusion). In particular, one must be aware that $N^2$ has to |
---|
1137 | be computed with an \textit{in situ} reference. The expression for $N^2$ |
---|
1138 | depends on the type of equation of state used (\np{neos} namelist parameter). |
---|
1139 | |
---|
1140 | For \np{neos}=0 (\citet{JackMcD1995} equation of state), the \citet{McDougall1987} |
---|
1141 | polynomial expression is used (with the pressure in decibar approximated by |
---|
1142 | the depth in meters): |
---|
1143 | \begin{equation} \label{Eq_tra_bn2} |
---|
1144 | N^2 = \frac{g}{e_{3w}} \; \beta \ |
---|
1145 | \left( \alpha / \beta \ \delta_{k+1/2}[T] - \delta_{k+1/2}[S] \right) |
---|
1146 | \end{equation} |
---|
1147 | where $\alpha$ ($\beta$) is the thermal (haline) expansion coefficient. |
---|
1148 | They are a function of |
---|
1149 | $\overline{T}^{\,k+1/2},\widetilde{S}=\overline{S}^{\,k+1/2} - 35.$, |
---|
1150 | and $z_w$, with $T$ the \textit{potential} temperature and |
---|
1151 | $\widetilde{S}$ a salinity anomaly. |
---|
1152 | Note that both $\alpha$ and $\beta$ depend on \textit{potential} |
---|
1153 | temperature and salinity which are averaged at $w$-points prior |
---|
1154 | to the computation instead of being computed at $T$-points and |
---|
1155 | then averaged to $w$-points. |
---|
1156 | |
---|
1157 | When a linear equation of state is used (\np{neos}=1 or 2, |
---|
1158 | \eqref{Eq_tra_bn2} reduces to: |
---|
1159 | \begin{equation} \label{Eq_tra_bn2_linear} |
---|
1160 | N^2 = \frac{g}{e_{3w}} \left( \beta \;\delta_{k+1/2}[S] - \alpha \;\delta_{k+1/2}[T] \right) |
---|
1161 | \end{equation} |
---|
1162 | where $\alpha$ and $\beta $ are the constant coefficients used to |
---|
1163 | defined the linear equation of state \eqref{Eq_tra_eos_linear}. |
---|
1164 | |
---|
1165 | % ------------------------------------------------------------------------------------------------------------- |
---|
1166 | % Specific Heat |
---|
1167 | % ------------------------------------------------------------------------------------------------------------- |
---|
1168 | \subsection [Specific Heat (\textit{phycst})] |
---|
1169 | {Specific Heat (\mdl{phycst})} |
---|
1170 | \label{TRA_adv_ldf} |
---|
1171 | |
---|
1172 | The specific heat of sea water, $C_p$, is a function of temperature, salinity |
---|
1173 | and pressure \citep{UNESCO1983}. It is only used in the model to convert |
---|
1174 | surface heat fluxes into surface temperature increase and so the pressure |
---|
1175 | dependence is neglected. The dependence on $T$ and $S$ is weak. |
---|
1176 | For example, with $S=35~psu$, $C_p$ increases from $3989$ to $4002$ |
---|
1177 | when $T$ varies from -2~\degres C to 31~\degres C. Therefore, $C_p$ has |
---|
1178 | been chosen as a constant: $C_p=4.10^3~J\,Kg^{-1}\,\degres K^{-1}$. |
---|
1179 | Its value is set in \mdl{phycst} module. |
---|
1180 | |
---|
1181 | %%% |
---|
1182 | \gmcomment{ STEVEN: consistency, no other computer variable names are |
---|
1183 | supplied, so why this one} |
---|
1184 | %%% |
---|
1185 | |
---|
1186 | % ------------------------------------------------------------------------------------------------------------- |
---|
1187 | % Freezing Point of Seawater |
---|
1188 | % ------------------------------------------------------------------------------------------------------------- |
---|
1189 | \subsection [Freezing Point of Seawater (\textit{ocfzpt})] |
---|
1190 | {Freezing Point of Seawater (\mdl{ocfzpt})} |
---|
1191 | \label{TRA_fzp} |
---|
1192 | |
---|
1193 | The freezing point of seawater is a function of salinity and pressure \citep{UNESCO1983}: |
---|
1194 | \begin{equation} \label{Eq_tra_eos_fzp} |
---|
1195 | \begin{split} |
---|
1196 | T_f (S,p) &= \left( -0.0575 + 1.710523 \;10^{-3} \, \sqrt{S} |
---|
1197 | - 2.154996 \;10^{-4} \,S \right) \ S \\ |
---|
1198 | & - 7.53\,10^{-3}\,p |
---|
1199 | \end{split} |
---|
1200 | \end{equation} |
---|
1201 | |
---|
1202 | \eqref{Eq_tra_eos_fzp} is only used to compute the potential freezing point of |
---|
1203 | sea water ($i.e.$ referenced to the surface $p=0$), thus the pressure dependent |
---|
1204 | terms in \eqref{Eq_tra_eos_fzp} (last term) have been dropped. The \textit{before} |
---|
1205 | and \textit{now} surface freezing point is introduced in the code as $fzptb$ and |
---|
1206 | $fzptn$ 2D arrays together with a \textit{now} mask (\textit{freezn}) which takes |
---|
1207 | the value 0 or 1 depending on whether the ocean temperature is above or at the |
---|
1208 | freezing point. Caution: do not confuse \textit{freezn} with the fraction of lead |
---|
1209 | (\textit{frld}) defined in LIM. |
---|
1210 | |
---|
1211 | %%% |
---|
1212 | \gmcomment{STEVEN: consistency, not many computer variable names are supplied, so why these ===> gm I agree this should evolve both here and in the code itself} |
---|
1213 | %%% |
---|
1214 | |
---|
1215 | % ================================================================ |
---|
1216 | % Horizontal Derivative in zps-coordinate |
---|
1217 | % ================================================================ |
---|
1218 | \section [Horizontal Derivative in \textit{zps}-coordinate (\textit{zpshde})] |
---|
1219 | {Horizontal Derivative in \textit{zps}-coordinate (\mdl{zpshde})} |
---|
1220 | \label{TRA_zpshde} |
---|
1221 | |
---|
1222 | \gmcomment{STEVEN: to be consistent with earlier discussion of differencing and averaging operators, I've changed "derivative" to "difference" and "mean" to "average"} |
---|
1223 | |
---|
1224 | With partial bottom cells (\np{ln\_zps}=.true.), in general, tracers in horizontally |
---|
1225 | adjacent cells live at different depths. Horizontal gradients of tracers are needed |
---|
1226 | for horizontal diffusion (\mdl{traldf} module) and for the hydrostatic pressure |
---|
1227 | gradient (\mdl{dynhpg} module) to be active. |
---|
1228 | \gmcomment{STEVEN from gm : question: not sure of what -to be active- means} |
---|
1229 | Before taking horizontal gradients between the tracers next to the bottom, a linear |
---|
1230 | interpolation in the vertical is used to approximate the deeper tracer as if it actually |
---|
1231 | lived at the depth of the shallower tracer point (Fig.~\ref{Fig_Partial_step_scheme}). |
---|
1232 | For example, for temperature in the $i$-direction the needed interpolated |
---|
1233 | temperature, $\widetilde{T}$, is: |
---|
1234 | |
---|
1235 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
---|
1236 | \begin{figure}[!p] \label{Fig_Partial_step_scheme} \begin{center} |
---|
1237 | \includegraphics[width=0.9\textwidth]{./TexFiles/Figures/Partial_step_scheme.pdf} |
---|
1238 | \caption{ Discretisation of the horizontal difference and average of tracers in the $z$-partial step coordinate (\np{ln\_zps}=.true.) in the case $( e3w_k^{i+1} - e3w_k^i )>0$. A linear interpolation is used to estimate $\widetilde{T}_k^{i+1}$, the tracer value at the depth of the shallower tracer point of the two adjacent bottom $T$-points. The horizontal difference is then given by: $\delta _{i+1/2} T_k= \widetilde{T}_k^{\,i+1} -T_k^{\,i}$ and the average by: $\overline{T}_k^{\,i+1/2}= ( \widetilde{T}_k^{\,i+1/2} - T_k^{\,i} ) / 2$. } |
---|
1239 | \end{center} \end{figure} |
---|
1240 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
---|
1241 | \begin{equation*} |
---|
1242 | \widetilde{T}= \left\{ \begin{aligned} |
---|
1243 | &T^{\,i+1} -\frac{ \left( e_{3w}^{i+1} -e_{3w}^i \right)}{ e_{3w}^{i+1} }\;\delta _k T^{i+1} |
---|
1244 | && \quad\text{if $\ e_{3w}^{i+1} \geq e_{3w}^i$ } \\ |
---|
1245 | \\ |
---|
1246 | &T^{\,i} \ \ \ \,+\frac{ \left( e_{3w}^{i+1} -e_{3w}^i \right) }{e_{3w}^i }\;\delta _k T^{i+1} |
---|
1247 | && \quad\text{if $\ e_{3w}^{i+1} < e_{3w}^i$ } |
---|
1248 | \end{aligned} \right. |
---|
1249 | \end{equation*} |
---|
1250 | and the resulting forms for the horizontal difference and the horizontal average |
---|
1251 | value of $T$ at a $U$-point are: |
---|
1252 | \begin{equation} \label{Eq_zps_hde} |
---|
1253 | \begin{aligned} |
---|
1254 | \delta _{i+1/2} T= \begin{cases} |
---|
1255 | \ \ \ \widetilde {T}\quad\ -T^i & \ \ \quad\quad\text{if $\ e_{3w}^{i+1} \geq e_{3w}^i$ } \\ |
---|
1256 | \\ |
---|
1257 | \ \ \ T^{\,i+1}-\widetilde{T} & \ \ \quad\quad\text{if $\ e_{3w}^{i+1} < e_{3w}^i$ } |
---|
1258 | \end{cases} \\ |
---|
1259 | \\ |
---|
1260 | \overline {T}^{\,i+1/2} \ = \begin{cases} |
---|
1261 | ( \widetilde {T}\ \ \;\,-T^{\,i}) / 2 & \;\ \ \quad\text{if $\ e_{3w}^{i+1} \geq e_{3w}^i$ } \\ |
---|
1262 | \\ |
---|
1263 | ( T^{\,i+1}-\widetilde{T} ) / 2 & \;\ \ \quad\text{if $\ e_{3w}^{i+1} < e_{3w}^i$ } |
---|
1264 | \end{cases} |
---|
1265 | \end{aligned} |
---|
1266 | \end{equation} |
---|
1267 | |
---|
1268 | The computation of horizontal derivative of tracers as well as of density is |
---|
1269 | performed once for all at each time step in \mdl{zpshde} module and stored |
---|
1270 | in shared arrays to be used when needed. It has to be emphasized that the |
---|
1271 | procedure used to compute the interpolated density, $\widetilde{\rho}$, is not |
---|
1272 | the same as that used for $T$ and $S$. Instead of forming a linear approximation |
---|
1273 | of density, we compute $\widetilde{\rho }$ from the interpolated values of $T$ |
---|
1274 | and $S$, and the pressure at a $u$-point (in the equation of state pressure is |
---|
1275 | approximated by depth, see \S\ref{TRA_eos} ) : |
---|
1276 | \begin{equation} \label{Eq_zps_hde_rho} |
---|
1277 | \widetilde{\rho } = \rho ( {\widetilde{T},\widetilde {S},z_u }) |
---|
1278 | \quad \text{where }\ z_u = \min \left( {z_T^{i+1} ,z_T^i } \right) |
---|
1279 | \end{equation} |
---|
1280 | |
---|
1281 | This is a much better approximation as the variation of $\rho$ with depth (and |
---|
1282 | thus pressure) is highly non-linear with a true equation of state and thus is badly |
---|
1283 | approximated with a linear interpolation. This approximation is used to compute |
---|
1284 | both the horizontal pressure gradient (\S\ref{DYN_hpg}) and the slopes of neutral |
---|
1285 | surfaces (\S\ref{LDF_slp}) |
---|
1286 | |
---|
1287 | Note that in almost all the advection schemes presented in this Chapter, both |
---|
1288 | averaging and differencing operators appear. Yet \eqref{Eq_zps_hde} has not |
---|
1289 | been used in these schemes: in contrast to diffusion and pressure gradient |
---|
1290 | computations, no correction for partial steps is applied for advection. The main |
---|
1291 | motivation is to preserve the domain averaged mean variance of the advected |
---|
1292 | field when using the $2^{nd}$ order centred scheme. Sensitivity of the advection |
---|
1293 | schemes to the way horizontal averages are performed in the vicinity of partial |
---|
1294 | cells should be further investigated in the near future. |
---|
1295 | %%% |
---|
1296 | \gmcomment{gm : this last remark has to be done} |
---|
1297 | %%% |
---|