1 | \documentclass[../main/NEMO_manual]{subfiles} |
---|
2 | |
---|
3 | \begin{document} |
---|
4 | |
---|
5 | % ================================================================ |
---|
6 | % Invariant of the Equations |
---|
7 | % ================================================================ |
---|
8 | \chapter{Invariants of the Primitive Equations} |
---|
9 | \label{chap:Invariant} |
---|
10 | \minitoc |
---|
11 | |
---|
12 | The continuous equations of motion have many analytic properties. |
---|
13 | Many quantities (total mass, energy, enstrophy, etc.) are strictly conserved in the inviscid and unforced limit, |
---|
14 | while ocean physics conserve the total quantities on which they act (momentum, temperature, salinity) but |
---|
15 | dissipate their total variance (energy, enstrophy, etc.). |
---|
16 | Unfortunately, the finite difference form of these equations is not guaranteed to |
---|
17 | retain all these important properties. |
---|
18 | In constructing the finite differencing schemes, we wish to ensure that |
---|
19 | certain integral constraints will be maintained. |
---|
20 | In particular, it is desirable to construct the finite difference equations so that |
---|
21 | horizontal kinetic energy and/or potential enstrophy of horizontally non-divergent flow, |
---|
22 | and variance of temperature and salinity will be conserved in the absence of dissipative effects and forcing. |
---|
23 | \citet{Arakawa1966} has first pointed out the advantage of this approach. |
---|
24 | He showed that if integral constraints on energy are maintained, |
---|
25 | the computation will be free of the troublesome "non linear" instability originally pointed out by |
---|
26 | \citet{Phillips1959}. |
---|
27 | A consistent formulation of the energetic properties is also extremely important in carrying out |
---|
28 | long-term numerical simulations for an oceanographic model. |
---|
29 | Such a formulation avoids systematic errors that accumulate with time \citep{Bryan1997}. |
---|
30 | |
---|
31 | The general philosophy of OPA which has led to the discrete formulation presented in {\S}II.2 and II.3 is to |
---|
32 | choose second order non-diffusive scheme for advective terms for both dynamical and tracer equations. |
---|
33 | At this level of complexity, the resulting schemes are dispersive schemes. |
---|
34 | Therefore, they require the addition of a diffusive operator to be stable. |
---|
35 | The alternative is to use diffusive schemes such as upstream or flux corrected schemes. |
---|
36 | This last option was rejected because we prefer a complete handling of the model diffusion, |
---|
37 | i.e. of the model physics rather than letting the advective scheme produces its own implicit diffusion without |
---|
38 | controlling the space and time structure of this implicit diffusion. |
---|
39 | Note that in some very specific cases as passive tracer studies, the positivity of the advective scheme is required. |
---|
40 | In that case, and in that case only, the advective scheme used for passive tracer is a flux correction scheme |
---|
41 | \citep{Marti1992, Levy1996, Levy1998}. |
---|
42 | |
---|
43 | % ------------------------------------------------------------------------------------------------------------- |
---|
44 | % Conservation Properties on Ocean Dynamics |
---|
45 | % ------------------------------------------------------------------------------------------------------------- |
---|
46 | \section{Conservation properties on ocean dynamics} |
---|
47 | \label{sec:Invariant_dyn} |
---|
48 | |
---|
49 | The non linear term of the momentum equations has been split into a vorticity term, |
---|
50 | a gradient of horizontal kinetic energy and a vertical advection term. |
---|
51 | Three schemes are available for the former (see {\S}~II.2) according to the CPP variable defined |
---|
52 | (default option\textbf{?}or \textbf{key{\_}vorenergy} or \textbf{key{\_}vorcombined} defined). |
---|
53 | They differ in their conservative properties (energy or enstrophy conserving scheme). |
---|
54 | The two latter terms preserve the total kinetic energy: |
---|
55 | the large scale kinetic energy is also preserved in practice. |
---|
56 | The remaining non-diffusive terms of the momentum equation |
---|
57 | (namely the hydrostatic and surface pressure gradient terms) also preserve the total kinetic energy and |
---|
58 | have no effect on the vorticity of the flow. |
---|
59 | |
---|
60 | \textbf{* relative, planetary and total vorticity term:} |
---|
61 | |
---|
62 | Let us define as either the relative, planetary and total potential vorticity, i.e. ?, ?, and ?, respectively. |
---|
63 | The continuous formulation of the vorticity term satisfies following integral constraints: |
---|
64 | \[ |
---|
65 | % \label{eq:vor_vorticity} |
---|
66 | \int_D {{\textbf {k}}\cdot \frac{1}{e_3 }\nabla \times \left( {\varsigma |
---|
67 | \;{\rm {\bf k}}\times {\textbf {U}}_h } \right)\;dv} =0 |
---|
68 | \] |
---|
69 | |
---|
70 | \[ |
---|
71 | % \label{eq:vor_enstrophy} |
---|
72 | if\quad \chi =0\quad \quad \int\limits_D {\varsigma \;{\textbf{k}}\cdot |
---|
73 | \frac{1}{e_3 }\nabla \times \left( {\varsigma {\textbf{k}}\times {\textbf{U}}_h } \right)\;dv} =-\int\limits_D {\frac{1}{2}\varsigma ^2\,\chi \;dv} |
---|
74 | =0 |
---|
75 | \] |
---|
76 | |
---|
77 | \[ |
---|
78 | % \label{eq:vor_energy} |
---|
79 | \int_D {{\textbf{U}}_h \times \left( {\varsigma \;{\textbf{k}}\times {\textbf{U}}_h } \right)\;dv} =0 |
---|
80 | \] |
---|
81 | where $dv = e_1\, e_2\, e_3\, di\, dj\, dk$ is the volume element. |
---|
82 | (II.4.1a) means that $\varsigma $ is conserved. (II.4.1b) is obtained by an integration by part. |
---|
83 | It means that $\varsigma^2$ is conserved for a horizontally non-divergent flow. |
---|
84 | (II.4.1c) is even satisfied locally since the vorticity term is orthogonal to the horizontal velocity. |
---|
85 | It means that the vorticity term has no contribution to the evolution of the total kinetic energy. |
---|
86 | (II.4.1a) is obviously always satisfied, but (II.4.1b) and (II.4.1c) cannot be satisfied simultaneously with |
---|
87 | a second order scheme. |
---|
88 | Using the symmetry or anti-symmetry properties of the operators (Eqs II.1.10 and 11), |
---|
89 | it can be shown that the scheme (II.2.11) satisfies (II.4.1b) but not (II.4.1c), |
---|
90 | while scheme (II.2.12) satisfies (II.4.1c) but not (II.4.1b) (see appendix C). |
---|
91 | Note that the enstrophy conserving scheme on total vorticity has been chosen as the standard discrete form of |
---|
92 | the vorticity term. |
---|
93 | |
---|
94 | \textbf{* Gradient of kinetic energy / vertical advection} |
---|
95 | |
---|
96 | In continuous formulation, the gradient of horizontal kinetic energy has no contribution to the evolution of |
---|
97 | the vorticity as the curl of a gradient is zero. |
---|
98 | This property is satisfied locally with the discrete form of both the gradient and the curl operator we have made |
---|
99 | (property (II.1.9)~). |
---|
100 | Another continuous property is that the change of horizontal kinetic energy due to |
---|
101 | vertical advection is exactly balanced by the change of horizontal kinetic energy due to |
---|
102 | the horizontal gradient of horizontal kinetic energy: |
---|
103 | |
---|
104 | \begin{equation} \label{eq:keg_zad} |
---|
105 | \int_D {{\textbf{U}}_h \cdot \nabla _h \left( {1/2\;{\textbf{U}}_h ^2} \right)\;dv} =-\int_D {{\textbf{U}}_h \cdot \frac{w}{e_3 }\;\frac{\partial |
---|
106 | {\textbf{U}}_h }{\partial k}\;dv} |
---|
107 | \end{equation} |
---|
108 | |
---|
109 | Using the discrete form given in {\S}II.2-a and the symmetry or anti-symmetry properties of |
---|
110 | the mean and difference operators, \autoref{eq:keg_zad} is demonstrated in the Appendix C. |
---|
111 | The main point here is that satisfying \autoref{eq:keg_zad} links the choice of the discrete forms of |
---|
112 | the vertical advection and of the horizontal gradient of horizontal kinetic energy. |
---|
113 | Choosing one imposes the other. |
---|
114 | The discrete form of the vertical advection given in {\S}II.2-a is a direct consequence of |
---|
115 | formulating the horizontal kinetic energy as $1/2 \left( \overline{u^2}^i + \overline{v^2}^j \right) $ in |
---|
116 | the gradient term. |
---|
117 | |
---|
118 | \textbf{* hydrostatic pressure gradient term} |
---|
119 | |
---|
120 | In continuous formulation, a pressure gradient has no contribution to the evolution of the vorticity as |
---|
121 | the curl of a gradient is zero. |
---|
122 | This properties is satisfied locally with the choice of discretization we have made (property (II.1.9)~). |
---|
123 | In addition, when the equation of state is linear |
---|
124 | (i.e. when an advective-diffusive equation for density can be derived from those of temperature and salinity) |
---|
125 | the change of horizontal kinetic energy due to the work of pressure forces is balanced by the change of |
---|
126 | potential energy due to buoyancy forces: |
---|
127 | |
---|
128 | \[ |
---|
129 | % \label{eq:hpg_pe} |
---|
130 | \int_D {-\frac{1}{\rho_o }\left. {\nabla p^h} \right|_z \cdot {\textbf {U}}_h \;dv} \;=\;\int_D {\nabla .\left( {\rho \,{\textbf{U}}} \right)\;g\;z\;\;dv} |
---|
131 | \] |
---|
132 | |
---|
133 | Using the discrete form given in {\S}~II.2-a and the symmetry or anti-symmetry properties of |
---|
134 | the mean and difference operators, (II.4.3) is demonstrated in the Appendix C. |
---|
135 | The main point here is that satisfying (II.4.3) strongly constraints the discrete expression of the depth of |
---|
136 | $T$-points and of the term added to the pressure gradient in $s-$coordinates: the depth of a $T$-point, $z_T$, |
---|
137 | is defined as the sum the vertical scale factors at $w$-points starting from the surface. |
---|
138 | |
---|
139 | \textbf{* surface pressure gradient term} |
---|
140 | |
---|
141 | In continuous formulation, the surface pressure gradient has no contribution to the evolution of vorticity. |
---|
142 | This properties is trivially satisfied locally as (II.2.3) |
---|
143 | (the equation verified by $\psi$ has been derived from the discrete formulation of the momentum equations, |
---|
144 | vertical sum and curl). |
---|
145 | Nevertheless, the $\psi$-equation is solved numerically by an iterative solver (see {\S}~III.5), |
---|
146 | thus the property is only satisfied with the accuracy required on the solver. |
---|
147 | In addition, with the rigid-lid approximation, the change of horizontal kinetic energy due to the work of |
---|
148 | surface pressure forces is exactly zero: |
---|
149 | \[ |
---|
150 | % \label{eq:spg} |
---|
151 | \int_D {-\frac{1}{\rho_o }\nabla _h } \left( {p_s } \right)\cdot {\textbf{U}}_h \;dv=0 |
---|
152 | \] |
---|
153 | |
---|
154 | (II.4.4) is satisfied in discrete form only if |
---|
155 | the discrete barotropic streamfunction time evolution equation is given by (II.2.3) (see appendix C). |
---|
156 | This shows that (II.2.3) is the only way to compute the streamfunction, |
---|
157 | otherwise there is no guarantee that the surface pressure force work vanishes. |
---|
158 | |
---|
159 | % ------------------------------------------------------------------------------------------------------------- |
---|
160 | % Conservation Properties on Ocean Thermodynamics |
---|
161 | % ------------------------------------------------------------------------------------------------------------- |
---|
162 | \section{Conservation properties on ocean thermodynamics} |
---|
163 | \label{sec:Invariant_tra} |
---|
164 | |
---|
165 | In continuous formulation, the advective terms of the tracer equations conserve the tracer content and |
---|
166 | the quadratic form of the tracer, $i.e.$ |
---|
167 | \[ |
---|
168 | % \label{eq:tra_tra2} |
---|
169 | \int_D {\nabla .\left( {T\;{\textbf{U}}} \right)\;dv} =0 |
---|
170 | \;\text{and} |
---|
171 | \int_D {T\;\nabla .\left( {T\;{\textbf{U}}} \right)\;dv} =0 |
---|
172 | \] |
---|
173 | |
---|
174 | The numerical scheme used ({\S}II.2-b) (equations in flux form, second order centred finite differences) satisfies |
---|
175 | (II.4.5) (see appendix C). |
---|
176 | Note that in both continuous and discrete formulations, there is generally no strict conservation of mass, |
---|
177 | since the equation of state is non linear with respect to $T$ and $S$. |
---|
178 | In practice, the mass is conserved with a very good accuracy. |
---|
179 | |
---|
180 | % ------------------------------------------------------------------------------------------------------------- |
---|
181 | % Conservation Properties on Momentum Physics |
---|
182 | % ------------------------------------------------------------------------------------------------------------- |
---|
183 | \subsection{Conservation properties on momentum physics} |
---|
184 | \label{subsec:Invariant_dyn_physics} |
---|
185 | |
---|
186 | \textbf{* lateral momentum diffusion term} |
---|
187 | |
---|
188 | The continuous formulation of the horizontal diffusion of momentum satisfies the following integral constraints~: |
---|
189 | \[ |
---|
190 | % \label{eq:dynldf_dyn} |
---|
191 | \int\limits_D {\frac{1}{e_3 }{\rm {\bf k}}\cdot \nabla \times \left[ {\nabla |
---|
192 | _h \left( {A^{lm}\;\chi } \right)-\nabla _h \times \left( {A^{lm}\;\zeta |
---|
193 | \;{\rm {\bf k}}} \right)} \right]\;dv} =0 |
---|
194 | \] |
---|
195 | |
---|
196 | \[ |
---|
197 | % \label{eq:dynldf_div} |
---|
198 | \int\limits_D {\nabla _h \cdot \left[ {\nabla _h \left( {A^{lm}\;\chi } |
---|
199 | \right)-\nabla _h \times \left( {A^{lm}\;\zeta \;{\rm {\bf k}}} \right)} |
---|
200 | \right]\;dv} =0 |
---|
201 | \] |
---|
202 | |
---|
203 | \[ |
---|
204 | % \label{eq:dynldf_curl} |
---|
205 | \int_D {{\rm {\bf U}}_h \cdot \left[ {\nabla _h \left( {A^{lm}\;\chi } |
---|
206 | \right)-\nabla _h \times \left( {A^{lm}\;\zeta \;{\rm {\bf k}}} \right)} |
---|
207 | \right]\;dv} \leqslant 0 |
---|
208 | \] |
---|
209 | |
---|
210 | \[ |
---|
211 | % \label{eq:dynldf_curl2} |
---|
212 | \mbox{if}\quad A^{lm}=cste\quad \quad \int_D {\zeta \;{\rm {\bf k}}\cdot |
---|
213 | \nabla \times \left[ {\nabla _h \left( {A^{lm}\;\chi } \right)-\nabla _h |
---|
214 | \times \left( {A^{lm}\;\zeta \;{\rm {\bf k}}} \right)} \right]\;dv} |
---|
215 | \leqslant 0 |
---|
216 | \] |
---|
217 | |
---|
218 | \[ |
---|
219 | % \label{eq:dynldf_div2} |
---|
220 | \mbox{if}\quad A^{lm}=cste\quad \quad \int_D {\chi \;\nabla _h \cdot \left[ |
---|
221 | {\nabla _h \left( {A^{lm}\;\chi } \right)-\nabla _h \times \left( |
---|
222 | {A^{lm}\;\zeta \;{\rm {\bf k}}} \right)} \right]\;dv} \leqslant 0 |
---|
223 | \] |
---|
224 | |
---|
225 | |
---|
226 | (II.4.6a) and (II.4.6b) means that the horizontal diffusion of momentum conserve both the potential vorticity and |
---|
227 | the divergence of the flow, while Eqs (II.4.6c) to (II.4.6e) mean that it dissipates the energy, the enstrophy and |
---|
228 | the square of the divergence. |
---|
229 | The two latter properties are only satisfied when the eddy coefficients are horizontally uniform. |
---|
230 | |
---|
231 | Using (II.1.8) and (II.1.9), and the symmetry or anti-symmetry properties of the mean and difference operators, |
---|
232 | it is shown that the discrete form of the lateral momentum diffusion given in |
---|
233 | {\S}II.2-c satisfies all the integral constraints (II.4.6) (see appendix C). |
---|
234 | In particular, when the eddy coefficients are horizontally uniform, |
---|
235 | a complete separation of vorticity and horizontal divergence fields is ensured, |
---|
236 | so that diffusion (dissipation) of vorticity (enstrophy) does not generate horizontal divergence |
---|
237 | (variance of the horizontal divergence) and \textit{vice versa}. |
---|
238 | When the vertical curl of the horizontal diffusion of momentum (discrete sense) is taken, |
---|
239 | the term associated to the horizontal gradient of the divergence is zero locally. |
---|
240 | When the horizontal divergence of the horizontal diffusion of momentum (discrete sense) is taken, |
---|
241 | the term associated to the vertical curl of the vorticity is zero locally. |
---|
242 | The resulting term conserves $\chi$ and dissipates $\chi^2$ when the eddy coefficient is horizontally uniform. |
---|
243 | |
---|
244 | \textbf{* vertical momentum diffusion term} |
---|
245 | |
---|
246 | As for the lateral momentum physics, the continuous form of the vertical diffusion of |
---|
247 | momentum satisfies following integral constraints~: |
---|
248 | |
---|
249 | conservation of momentum, dissipation of horizontal kinetic energy |
---|
250 | |
---|
251 | \[ |
---|
252 | % \label{eq:dynzdf_dyn} |
---|
253 | \begin{aligned} |
---|
254 | & \int_D {\frac{1}{e_3 }} \frac{\partial }{\partial k}\left( \frac{A^{vm}}{e_3 }\frac{\partial {\textbf{U}}_h }{\partial k} \right) \;dv = \overrightarrow{\textbf{0}} \\ |
---|
255 | & \int_D \textbf{U}_h \cdot \frac{1}{e_3} \frac{\partial}{\partial k} \left( {\frac{A^{vm}}{e_3 }}{\frac{\partial \textbf{U}_h }{\partial k}} \right) \;dv \leq 0 \\ |
---|
256 | \end{aligned} |
---|
257 | \] |
---|
258 | conservation of vorticity, dissipation of enstrophy |
---|
259 | \[ |
---|
260 | % \label{eq:dynzdf_vor} |
---|
261 | \begin{aligned} |
---|
262 | & \int_D {\frac{1}{e_3 }{\rm {\bf k}}\cdot \nabla \times \left( {\frac{1}{e_3 |
---|
263 | }\frac{\partial }{\partial k}\left( {\frac{A^{vm}}{e_3 }\frac{\partial {\rm |
---|
264 | {\bf U}}_h }{\partial k}} \right)} \right)\;dv} =0 \\ |
---|
265 | & \int_D {\zeta \,{\rm {\bf k}}\cdot \nabla \times \left( {\frac{1}{e_3 |
---|
266 | }\frac{\partial }{\partial k}\left( {\frac{A^{vm}}{e_3 }\frac{\partial {\rm |
---|
267 | {\bf U}}_h }{\partial k}} \right)} \right)\;dv} \leq 0 \\ |
---|
268 | \end{aligned} |
---|
269 | \] |
---|
270 | conservation of horizontal divergence, dissipation of square of the horizontal divergence |
---|
271 | \[ |
---|
272 | % \label{eq:dynzdf_div} |
---|
273 | \begin{aligned} |
---|
274 | &\int_D {\nabla \cdot \left( {\frac{1}{e_3 }\frac{\partial }{\partial |
---|
275 | k}\left( {\frac{A^{vm}}{e_3 }\frac{\partial {\rm {\bf U}}_h }{\partial k}} |
---|
276 | \right)} \right)\;dv} =0 \\ |
---|
277 | & \int_D {\chi \;\nabla \cdot \left( {\frac{1}{e_3 }\frac{\partial }{\partial |
---|
278 | k}\left( {\frac{A^{vm}}{e_3 }\frac{\partial {\rm {\bf U}}_h }{\partial k}} |
---|
279 | \right)} \right)\;dv} \leq 0 \\ |
---|
280 | \end{aligned} |
---|
281 | \] |
---|
282 | |
---|
283 | In discrete form, all these properties are satisfied in $z$-coordinate (see Appendix C). |
---|
284 | In $s$-coordinates, only first order properties can be demonstrated, |
---|
285 | $i.e.$ the vertical momentum physics conserve momentum, potential vorticity, and horizontal divergence. |
---|
286 | |
---|
287 | % ------------------------------------------------------------------------------------------------------------- |
---|
288 | % Conservation Properties on Tracer Physics |
---|
289 | % ------------------------------------------------------------------------------------------------------------- |
---|
290 | \subsection{Conservation properties on tracer physics} |
---|
291 | \label{subsec:Invariant_tra_physics} |
---|
292 | |
---|
293 | The numerical schemes used for tracer subgridscale physics are written in such a way that |
---|
294 | the heat and salt contents are conserved (equations in flux form, second order centred finite differences). |
---|
295 | As a form flux is used to compute the temperature and salinity, |
---|
296 | the quadratic form of these quantities (i.e. their variance) globally tends to diminish. |
---|
297 | As for the advective term, there is generally no strict conservation of mass even if, |
---|
298 | in practice, the mass is conserved with a very good accuracy. |
---|
299 | |
---|
300 | \textbf{* lateral physics: }conservation of tracer, dissipation of tracer |
---|
301 | variance, i.e. |
---|
302 | |
---|
303 | \[ |
---|
304 | % \label{eq:traldf_t_t2} |
---|
305 | \begin{aligned} |
---|
306 | &\int_D \nabla\, \cdot\, \left( A^{lT} \,\Re \,\nabla \,T \right)\;dv = 0 \\ |
---|
307 | &\int_D \,T\, \nabla\, \cdot\, \left( A^{lT} \,\Re \,\nabla \,T \right)\;dv \leq 0 \\ |
---|
308 | \end{aligned} |
---|
309 | \] |
---|
310 | |
---|
311 | \textbf{* vertical physics: }conservation of tracer, dissipation of tracer variance, $i.e.$ |
---|
312 | |
---|
313 | \[ |
---|
314 | % \label{eq:trazdf_t_t2} |
---|
315 | \begin{aligned} |
---|
316 | & \int_D \frac{1}{e_3 } \frac{\partial }{\partial k}\left( \frac{A^{vT}}{e_3 } \frac{\partial T}{\partial k} \right)\;dv = 0 \\ |
---|
317 | & \int_D \,T \frac{1}{e_3 } \frac{\partial }{\partial k}\left( \frac{A^{vT}}{e_3 } \frac{\partial T}{\partial k} \right)\;dv \leq 0 \\ |
---|
318 | \end{aligned} |
---|
319 | \] |
---|
320 | |
---|
321 | Using the symmetry or anti-symmetry properties of the mean and difference operators, |
---|
322 | it is shown that the discrete form of tracer physics given in {\S}~II.2-c satisfies all the integral constraints |
---|
323 | (II.4.8) and (II.4.9) except the dissipation of the square of the tracer when non-geopotential diffusion is used |
---|
324 | (see appendix C). |
---|
325 | A discrete form of the lateral tracer physics can be derived which satisfies these last properties. |
---|
326 | Nevertheless, it requires a horizontal averaging of the vertical component of the lateral physics that |
---|
327 | prevents the use of implicit resolution in the vertical. |
---|
328 | It has not been implemented. |
---|
329 | |
---|
330 | \biblio |
---|
331 | |
---|
332 | \end{document} |
---|