1 | \documentclass[NEMO_book]{subfiles} |
---|
2 | \begin{document} |
---|
3 | |
---|
4 | % ================================================================ |
---|
5 | % Invariant of the Equations |
---|
6 | % ================================================================ |
---|
7 | \chapter{Invariants of the Primitive Equations} |
---|
8 | \label{Invariant} |
---|
9 | \minitoc |
---|
10 | |
---|
11 | The continuous equations of motion have many analytic properties. Many |
---|
12 | quantities (total mass, energy, enstrophy, etc.) are strictly conserved in |
---|
13 | the inviscid and unforced limit, while ocean physics conserve the total |
---|
14 | quantities on which they act (momentum, temperature, salinity) but dissipate |
---|
15 | their total variance (energy, enstrophy, etc.). Unfortunately, the finite |
---|
16 | difference form of these equations is not guaranteed to retain all these |
---|
17 | important properties. In constructing the finite differencing schemes, we |
---|
18 | wish to ensure that certain integral constraints will be maintained. In |
---|
19 | particular, it is desirable to construct the finite difference equations so |
---|
20 | that horizontal kinetic energy and/or potential enstrophy of horizontally |
---|
21 | non-divergent flow, and variance of temperature and salinity will be |
---|
22 | conserved in the absence of dissipative effects and forcing. \citet{Arakawa1966} |
---|
23 | has first pointed out the advantage of this approach. He showed that if |
---|
24 | integral constraints on energy are maintained, the computation will be free |
---|
25 | of the troublesome "non linear" instability originally pointed out by |
---|
26 | \citet{Phillips1959}. A consistent formulation of the energetic properties is |
---|
27 | also extremely important in carrying out long-term numerical simulations for |
---|
28 | an oceanographic model. Such a formulation avoids systematic errors that |
---|
29 | accumulate with time \citep{Bryan1997}. |
---|
30 | |
---|
31 | The general philosophy of OPA which has led to the discrete formulation |
---|
32 | presented in {\S}II.2 and II.3 is to choose second order non-diffusive |
---|
33 | scheme for advective terms for both dynamical and tracer equations. At this |
---|
34 | level of complexity, the resulting schemes are dispersive schemes. |
---|
35 | Therefore, they require the addition of a diffusive operator to be stable. |
---|
36 | The alternative is to use diffusive schemes such as upstream or flux |
---|
37 | corrected schemes. This last option was rejected because we prefer a |
---|
38 | complete handling of the model diffusion, i.e. of the model physics rather |
---|
39 | than letting the advective scheme produces its own implicit diffusion |
---|
40 | without controlling the space and time structure of this implicit diffusion. |
---|
41 | Note that in some very specific cases as passive tracer studies, the |
---|
42 | positivity of the advective scheme is required. In that case, and in that |
---|
43 | case only, the advective scheme used for passive tracer is a flux correction |
---|
44 | scheme \citep{Marti1992, Levy1996, Levy1998}. |
---|
45 | |
---|
46 | % ------------------------------------------------------------------------------------------------------------- |
---|
47 | % Conservation Properties on Ocean Dynamics |
---|
48 | % ------------------------------------------------------------------------------------------------------------- |
---|
49 | \section{Conservation Properties on Ocean Dynamics} |
---|
50 | \label{Invariant_dyn} |
---|
51 | |
---|
52 | The non linear term of the momentum equations has been split into a |
---|
53 | vorticity term, a gradient of horizontal kinetic energy and a vertical |
---|
54 | advection term. Three schemes are available for the former (see {\S}~II.2) |
---|
55 | according to the CPP variable defined (default option\textbf{ |
---|
56 | }or \textbf{key{\_}vorenergy } or \textbf{key{\_}vorcombined |
---|
57 | } defined). They differ in their conservative |
---|
58 | properties (energy or enstrophy conserving scheme). The two latter terms |
---|
59 | preserve the total kinetic energy: the large scale kinetic energy is also |
---|
60 | preserved in practice. The remaining non-diffusive terms of the momentum |
---|
61 | equation (namely the hydrostatic and surface pressure gradient terms) also |
---|
62 | preserve the total kinetic energy and have no effect on the vorticity of the |
---|
63 | flow. |
---|
64 | |
---|
65 | \textbf{* relative, planetary and total vorticity term:} |
---|
66 | |
---|
67 | Let us define as either the relative, planetary and total potential |
---|
68 | vorticity, i.e. , , and , respectively. The continuous formulation of the |
---|
69 | vorticity term satisfies following integral constraints: |
---|
70 | \begin{equation} \label{Eq_vor_vorticity} |
---|
71 | \int_D {{\textbf {k}}\cdot \frac{1}{e_3 }\nabla \times \left( {\varsigma |
---|
72 | \;{\rm {\bf k}}\times {\textbf {U}}_h } \right)\;dv} =0 |
---|
73 | \end{equation} |
---|
74 | |
---|
75 | \begin{equation} \label{Eq_vor_enstrophy} |
---|
76 | if\quad \chi =0\quad \quad \int\limits_D {\varsigma \;{\textbf{k}}\cdot |
---|
77 | \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} |
---|
78 | =0 |
---|
79 | \end{equation} |
---|
80 | |
---|
81 | \begin{equation} \label{Eq_vor_energy} |
---|
82 | \int_D {{\textbf{U}}_h \times \left( {\varsigma \;{\textbf{k}}\times {\textbf{U}}_h } \right)\;dv} =0 |
---|
83 | \end{equation} |
---|
84 | where $dv = e_1\, e_2\, e_3\, di\, dj\, dk$ is the volume element. |
---|
85 | (II.4.1a) means that $\varsigma $ is conserved. (II.4.1b) is obtained by an |
---|
86 | integration by part. It means that $\varsigma^2$ is conserved for a horizontally |
---|
87 | non-divergent flow. |
---|
88 | (II.4.1c) is even satisfied locally since the vorticity term is orthogonal |
---|
89 | to the horizontal velocity. It means that the vorticity term has no |
---|
90 | contribution to the evolution of the total kinetic energy. (II.4.1a) is |
---|
91 | obviously always satisfied, but (II.4.1b) and (II.4.1c) cannot be satisfied |
---|
92 | simultaneously with a second order scheme. Using the symmetry or |
---|
93 | anti-symmetry properties of the operators (Eqs II.1.10 and 11), it can be |
---|
94 | shown that the scheme (II.2.11) satisfies (II.4.1b) but not (II.4.1c), while |
---|
95 | scheme (II.2.12) satisfies (II.4.1c) but not (II.4.1b) (see appendix C). |
---|
96 | Note that the enstrophy conserving scheme on total vorticity has been chosen |
---|
97 | as the standard discrete form of the vorticity term. |
---|
98 | |
---|
99 | \textbf{* Gradient of kinetic energy / vertical advection} |
---|
100 | |
---|
101 | In continuous formulation, the gradient of horizontal kinetic energy has no |
---|
102 | contribution to the evolution of the vorticity as the curl of a gradient is |
---|
103 | zero. This property is satisfied locally with the discrete form of both the |
---|
104 | gradient and the curl operator we have made (property (II.1.9)~). Another |
---|
105 | continuous property is that the change of horizontal kinetic energy due to |
---|
106 | vertical advection is exactly balanced by the change of horizontal kinetic |
---|
107 | energy due to the horizontal gradient of horizontal kinetic energy: |
---|
108 | |
---|
109 | \begin{equation} \label{Eq_keg_zad} |
---|
110 | \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 |
---|
111 | {\textbf{U}}_h }{\partial k}\;dv} |
---|
112 | \end{equation} |
---|
113 | |
---|
114 | Using the discrete form given in {\S}II.2-a and the symmetry or |
---|
115 | anti-symmetry properties of the mean and difference operators, \eqref{Eq_keg_zad} is |
---|
116 | demonstrated in the Appendix C. The main point here is that satisfying |
---|
117 | \eqref{Eq_keg_zad} links the choice of the discrete forms of the vertical advection |
---|
118 | and of the horizontal gradient of horizontal kinetic energy. Choosing one |
---|
119 | imposes the other. The discrete form of the vertical advection given in |
---|
120 | {\S}II.2-a is a direct consequence of formulating the horizontal kinetic |
---|
121 | energy as $1/2 \left( \overline{u^2}^i + \overline{v^2}^j \right) $ in the gradient term. |
---|
122 | |
---|
123 | \textbf{* hydrostatic pressure gradient term} |
---|
124 | |
---|
125 | In continuous formulation, a pressure gradient has no contribution to the |
---|
126 | evolution of the vorticity as the curl of a gradient is zero. This |
---|
127 | properties is satisfied locally with the choice of discretization we have |
---|
128 | made (property (II.1.9)~). In addition, when the equation of state is linear |
---|
129 | (i.e. when an advective-diffusive equation for density can be derived from |
---|
130 | those of temperature and salinity) the change of horizontal kinetic energy |
---|
131 | due to the work of pressure forces is balanced by the change of potential |
---|
132 | energy due to buoyancy forces: |
---|
133 | |
---|
134 | \begin{equation} \label{Eq_hpg_pe} |
---|
135 | \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} |
---|
136 | \end{equation} |
---|
137 | |
---|
138 | Using the discrete form given in {\S}~II.2-a and the symmetry or |
---|
139 | anti-symmetry properties of the mean and difference operators, (II.4.3) is |
---|
140 | demonstrated in the Appendix C. The main point here is that satisfying |
---|
141 | (II.4.3) strongly constraints the discrete expression of the depth of |
---|
142 | $T$-points and of the term added to the pressure gradient in $s-$coordinates: the |
---|
143 | depth of a $T$-point, $z_T$, is defined as the sum the vertical scale |
---|
144 | factors at $w$-points starting from the surface. |
---|
145 | |
---|
146 | \textbf{* surface pressure gradient term} |
---|
147 | |
---|
148 | In continuous formulation, the surface pressure gradient has no contribution |
---|
149 | to the evolution of vorticity. This properties is trivially satisfied |
---|
150 | locally as (II.2.3) (the equation verified by $\psi$ has been |
---|
151 | derived from the discrete formulation of the momentum equations, vertical |
---|
152 | sum and curl. Nevertheless, the $\psi$-equation is solved numerically by an |
---|
153 | iterative solver (see {\S}~III.5), thus the property is only satisfied with |
---|
154 | the accuracy required on the solver. In addition, with the rigid-lid |
---|
155 | approximation, the change of horizontal kinetic energy due to the work of |
---|
156 | surface pressure forces is exactly zero: |
---|
157 | \begin{equation} \label{Eq_spg} |
---|
158 | \int_D {-\frac{1}{\rho _o }\nabla _h } \left( {p_s } \right)\cdot {\textbf{U}}_h \;dv=0 |
---|
159 | \end{equation} |
---|
160 | |
---|
161 | (II.4.4) is satisfied in discrete form only if the discrete barotropic |
---|
162 | streamfunction time evolution equation is given by (II.2.3) (see appendix |
---|
163 | C). This shows that (II.2.3) is the only way to compute the streamfunction, |
---|
164 | otherwise there is no guarantee that the surface pressure force work |
---|
165 | vanishes. |
---|
166 | |
---|
167 | % ------------------------------------------------------------------------------------------------------------- |
---|
168 | % Conservation Properties on Ocean Thermodynamics |
---|
169 | % ------------------------------------------------------------------------------------------------------------- |
---|
170 | \section{Conservation Properties on Ocean Thermodynamics} |
---|
171 | \label{Invariant_tra} |
---|
172 | |
---|
173 | In continuous formulation, the advective terms of the tracer equations |
---|
174 | conserve the tracer content and the quadratic form of the tracer, i.e. |
---|
175 | \begin{equation} \label{Eq_tra_tra2} |
---|
176 | \int_D {\nabla .\left( {T\;{\textbf{U}}} \right)\;dv} =0 |
---|
177 | \;\text{and} |
---|
178 | \int_D {T\;\nabla .\left( {T\;{\textbf{U}}} \right)\;dv} =0 |
---|
179 | \end{equation} |
---|
180 | |
---|
181 | The numerical scheme used ({\S}II.2-b) (equations in flux form, second order |
---|
182 | centred finite differences) satisfies (II.4.5) (see appendix C). Note that |
---|
183 | in both continuous and discrete formulations, there is generally no strict |
---|
184 | conservation of mass, since the equation of state is non linear with respect |
---|
185 | to $T$ and $S$. In practice, the mass is conserved with a very good accuracy. |
---|
186 | |
---|
187 | % ------------------------------------------------------------------------------------------------------------- |
---|
188 | % Conservation Properties on Momentum Physics |
---|
189 | % ------------------------------------------------------------------------------------------------------------- |
---|
190 | \subsection{Conservation Properties on Momentum Physics} |
---|
191 | \label{Invariant_dyn_physics} |
---|
192 | |
---|
193 | \textbf{* lateral momentum diffusion term} |
---|
194 | |
---|
195 | The continuous formulation of the horizontal diffusion of momentum satisfies |
---|
196 | the following integral constraints~: |
---|
197 | \begin{equation} \label{Eq_dynldf_dyn} |
---|
198 | \int\limits_D {\frac{1}{e_3 }{\rm {\bf k}}\cdot \nabla \times \left[ {\nabla |
---|
199 | _h \left( {A^{lm}\;\chi } \right)-\nabla _h \times \left( {A^{lm}\;\zeta |
---|
200 | \;{\rm {\bf k}}} \right)} \right]\;dv} =0 |
---|
201 | \end{equation} |
---|
202 | |
---|
203 | \begin{equation} \label{Eq_dynldf_div} |
---|
204 | \int\limits_D {\nabla _h \cdot \left[ {\nabla _h \left( {A^{lm}\;\chi } |
---|
205 | \right)-\nabla _h \times \left( {A^{lm}\;\zeta \;{\rm {\bf k}}} \right)} |
---|
206 | \right]\;dv} =0 |
---|
207 | \end{equation} |
---|
208 | |
---|
209 | \begin{equation} \label{Eq_dynldf_curl} |
---|
210 | \int_D {{\rm {\bf U}}_h \cdot \left[ {\nabla _h \left( {A^{lm}\;\chi } |
---|
211 | \right)-\nabla _h \times \left( {A^{lm}\;\zeta \;{\rm {\bf k}}} \right)} |
---|
212 | \right]\;dv} \leqslant 0 |
---|
213 | \end{equation} |
---|
214 | |
---|
215 | \begin{equation} \label{Eq_dynldf_curl2} |
---|
216 | \mbox{if}\quad A^{lm}=cste\quad \quad \int_D {\zeta \;{\rm {\bf k}}\cdot |
---|
217 | \nabla \times \left[ {\nabla _h \left( {A^{lm}\;\chi } \right)-\nabla _h |
---|
218 | \times \left( {A^{lm}\;\zeta \;{\rm {\bf k}}} \right)} \right]\;dv} |
---|
219 | \leqslant 0 |
---|
220 | \end{equation} |
---|
221 | |
---|
222 | \begin{equation} \label{Eq_dynldf_div2} |
---|
223 | \mbox{if}\quad A^{lm}=cste\quad \quad \int_D {\chi \;\nabla _h \cdot \left[ |
---|
224 | {\nabla _h \left( {A^{lm}\;\chi } \right)-\nabla _h \times \left( |
---|
225 | {A^{lm}\;\zeta \;{\rm {\bf k}}} \right)} \right]\;dv} \leqslant 0 |
---|
226 | \end{equation} |
---|
227 | |
---|
228 | |
---|
229 | (II.4.6a) and (II.4.6b) means that the horizontal diffusion of momentum |
---|
230 | conserve both the potential vorticity and the divergence of the flow, while |
---|
231 | Eqs (II.4.6c) to (II.4.6e) mean that it dissipates the energy, the enstrophy |
---|
232 | and the square of the divergence. The two latter properties are only |
---|
233 | satisfied when the eddy coefficients are horizontally uniform. |
---|
234 | |
---|
235 | Using (II.1.8) and (II.1.9), and the symmetry or anti-symmetry properties of |
---|
236 | the mean and difference operators, it is shown that the discrete form of the |
---|
237 | lateral momentum diffusion given in {\S}II.2-c satisfies all the integral |
---|
238 | constraints (II.4.6) (see appendix C). In particular, when the eddy |
---|
239 | coefficients are horizontally uniform, a complete separation of vorticity |
---|
240 | and horizontal divergence fields is ensured, so that diffusion (dissipation) |
---|
241 | of vorticity (enstrophy) does not generate horizontal divergence (variance |
---|
242 | of the horizontal divergence) and \textit{vice versa}. When the vertical curl of the horizontal |
---|
243 | diffusion of momentum (discrete sense) is taken, the term associated to the |
---|
244 | horizontal gradient of the divergence is zero locally. When the horizontal |
---|
245 | divergence of the horizontal diffusion of momentum (discrete sense) is |
---|
246 | taken, the term associated to the vertical curl of the vorticity is zero |
---|
247 | locally. The resulting term conserves $\chi$ and dissipates |
---|
248 | $\chi^2$ when the |
---|
249 | eddy coefficient is horizontally uniform. |
---|
250 | |
---|
251 | \textbf{* vertical momentum diffusion term} |
---|
252 | |
---|
253 | As for the lateral momentum physics, the continuous form of the vertical |
---|
254 | diffusion of momentum satisfies following integral constraints~: |
---|
255 | |
---|
256 | conservation of momentum, dissipation of horizontal kinetic energy |
---|
257 | |
---|
258 | \begin{equation} \label{Eq_dynzdf_dyn} |
---|
259 | \begin{aligned} |
---|
260 | & \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}} \\ |
---|
261 | & \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 \\ |
---|
262 | \end{aligned} |
---|
263 | \end{equation} |
---|
264 | conservation of vorticity, dissipation of enstrophy |
---|
265 | \begin{equation} \label{Eq_dynzdf_vor} |
---|
266 | \begin{aligned} |
---|
267 | & \int_D {\frac{1}{e_3 }{\rm {\bf k}}\cdot \nabla \times \left( {\frac{1}{e_3 |
---|
268 | }\frac{\partial }{\partial k}\left( {\frac{A^{vm}}{e_3 }\frac{\partial {\rm |
---|
269 | {\bf U}}_h }{\partial k}} \right)} \right)\;dv} =0 \\ |
---|
270 | & \int_D {\zeta \,{\rm {\bf k}}\cdot \nabla \times \left( {\frac{1}{e_3 |
---|
271 | }\frac{\partial }{\partial k}\left( {\frac{A^{vm}}{e_3 }\frac{\partial {\rm |
---|
272 | {\bf U}}_h }{\partial k}} \right)} \right)\;dv} \leq 0 \\ |
---|
273 | \end{aligned} |
---|
274 | \end{equation} |
---|
275 | conservation of horizontal divergence, dissipation of square of the |
---|
276 | horizontal divergence |
---|
277 | \begin{equation} \label{Eq_dynzdf_div} |
---|
278 | \begin{aligned} |
---|
279 | &\int_D {\nabla \cdot \left( {\frac{1}{e_3 }\frac{\partial }{\partial |
---|
280 | k}\left( {\frac{A^{vm}}{e_3 }\frac{\partial {\rm {\bf U}}_h }{\partial k}} |
---|
281 | \right)} \right)\;dv} =0 \\ |
---|
282 | & \int_D {\chi \;\nabla \cdot \left( {\frac{1}{e_3 }\frac{\partial }{\partial |
---|
283 | k}\left( {\frac{A^{vm}}{e_3 }\frac{\partial {\rm {\bf U}}_h }{\partial k}} |
---|
284 | \right)} \right)\;dv} \leq 0 \\ |
---|
285 | \end{aligned} |
---|
286 | \end{equation} |
---|
287 | |
---|
288 | In discrete form, all these properties are satisfied in $z$-coordinate (see |
---|
289 | Appendix C). In $s$-coordinates, only first order properties can be |
---|
290 | demonstrated, i.e. the vertical momentum physics conserve momentum, |
---|
291 | potential vorticity, and horizontal divergence. |
---|
292 | |
---|
293 | % ------------------------------------------------------------------------------------------------------------- |
---|
294 | % Conservation Properties on Tracer Physics |
---|
295 | % ------------------------------------------------------------------------------------------------------------- |
---|
296 | \subsection{Conservation Properties on Tracer Physics} |
---|
297 | \label{Invariant_tra_physics} |
---|
298 | |
---|
299 | The numerical schemes used for tracer subgridscale physics are written in |
---|
300 | such a way that the heat and salt contents are conserved (equations in flux |
---|
301 | form, second order centred finite differences). As a form flux is used to |
---|
302 | compute the temperature and salinity, the quadratic form of these quantities |
---|
303 | (i.e. their variance) globally tends to diminish. As for the advective term, |
---|
304 | there is generally no strict conservation of mass even if, in practice, the |
---|
305 | mass is conserved with a very good accuracy. |
---|
306 | |
---|
307 | \textbf{* lateral physics: }conservation of tracer, dissipation of tracer |
---|
308 | variance, i.e. |
---|
309 | |
---|
310 | \begin{equation} \label{Eq_traldf_t_t2} |
---|
311 | \begin{aligned} |
---|
312 | &\int_D \nabla\, \cdot\, \left( A^{lT} \,\Re \,\nabla \,T \right)\;dv = 0 \\ |
---|
313 | &\int_D \,T\, \nabla\, \cdot\, \left( A^{lT} \,\Re \,\nabla \,T \right)\;dv \leq 0 \\ |
---|
314 | \end{aligned} |
---|
315 | \end{equation} |
---|
316 | |
---|
317 | \textbf{* vertical physics: }conservation of tracer, dissipation of tracer |
---|
318 | variance, i.e. |
---|
319 | |
---|
320 | \begin{equation} \label{Eq_trazdf_t_t2} |
---|
321 | \begin{aligned} |
---|
322 | & \int_D \frac{1}{e_3 } \frac{\partial }{\partial k}\left( \frac{A^{vT}}{e_3 } \frac{\partial T}{\partial k} \right)\;dv = 0 \\ |
---|
323 | & \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 \\ |
---|
324 | \end{aligned} |
---|
325 | \end{equation} |
---|
326 | |
---|
327 | Using the symmetry or anti-symmetry properties of the mean and difference |
---|
328 | operators, it is shown that the discrete form of tracer physics given in |
---|
329 | {\S}~II.2-c satisfies all the integral constraints (II.4.8) and (II.4.9) |
---|
330 | except the dissipation of the square of the tracer when non-geopotential |
---|
331 | diffusion is used (see appendix C). A discrete form of the lateral tracer |
---|
332 | physics can be derived which satisfies these last properties. Nevertheless, |
---|
333 | it requires a horizontal averaging of the vertical component of the lateral |
---|
334 | physics that prevents the use of implicit resolution in the vertical. It has |
---|
335 | not been implemented. |
---|
336 | |
---|
337 | \end{document} |
---|