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NEMO/trunk/doc/latex/NEMO/subfiles/chap_model_basics_zstar.tex
r10146 r10354 16 16 \colorbox{yellow}{ to be updated } 17 17 18 In that case, the free surface equation is nonlinear, and the variations of 19 volume are fully taken into account. These coordinates systems is presented in 20 a report \citep{Levier2007} available on the \NEMO web site. 18 In that case, the free surface equation is nonlinear, and the variations of volume are fully taken into account. 19 These coordinates systems is presented in a report \citep{Levier2007} available on the \NEMO web site. 21 20 22 21 \colorbox{yellow}{ end of to be updated} … … 25 24 % from MOM4p1 documentation 26 25 27 To overcome problems with vanishing surface and/or bottom cells, we consider the 28 zstar coordinate 26 To overcome problems with vanishing surface and/or bottom cells, we consider the zstar coordinate 29 27 \begin{equation} \label{eq:PE_} 30 28 z^\star = H \left( \frac{z-\eta}{H+\eta} \right) 31 29 \end{equation} 32 30 33 This coordinate is closely related to the "eta" coordinate used in many atmospheric 34 models (see Black (1994) for a review of eta coordinate atmospheric models). It 35 was originally used in ocean models by Stacey et al. (1995) for studies of tides 36 next to shelves, and it has been recently promoted by Adcroft and Campin (2004) 37 for global climate modelling. 38 39 The surfaces of constant $z^\star$ are quasi-horizontal. Indeed, the $z^\star$ coordinate reduces to $z$ when $\eta$ is zero. In general, when noting the large differences between 40 undulations of the bottom topography versus undulations in the surface height, it 41 is clear that surfaces constant $z^\star$ are very similar to the depth surfaces. These properties greatly reduce difficulties of computing the horizontal pressure gradient relative to terrain following sigma models discussed in \autoref{subsec:PE_sco}. 42 Additionally, since $z^\star$ when $\eta = 0$, no flow is spontaneously generated in an 43 unforced ocean starting from rest, regardless the bottom topography. This behaviour is in contrast to the case with "s"-models, where pressure gradient errors in 44 the presence of nontrivial topographic variations can generate nontrivial spontaneous flow from a resting state, depending on the sophistication of the pressure 45 gradient solver. The quasi-horizontal nature of the coordinate surfaces also facilitates the implementation of neutral physics parameterizations in $z^\star$ models using 46 the same techniques as in $z$-models (see Chapters 13-16 of Griffies (2004) for a 47 discussion of neutral physics in $z$-models, as well as \autoref{sec:LDF_slp} 48 in this document for treatment in \NEMO). 49 50 The range over which $z^\star$ varies is time independent $-H \leq z^\star \leq 0$. Hence, all 51 cells remain nonvanishing, so long as the surface height maintains $\eta > ?H$. This 52 is a minor constraint relative to that encountered on the surface height when using 53 $s = z$ or $s = z - \eta$. 54 55 Because $z^\star$ has a time independent range, all grid cells have static increments 56 ds, and the sum of the ver tical increments yields the time independent ocean 57 depth %�k ds = H. 58 The $z^\star$ coordinate is therefore invisible to undulations of the 59 free surface, since it moves along with the free surface. This proper ty means that 60 no spurious ver tical transpor t is induced across surfaces of constant $z^\star$ by the 61 motion of external gravity waves. Such spurious transpor t can be a problem in 62 z-models, especially those with tidal forcing. Quite generally, the time independent 63 range for the $z^\star$ coordinate is a very convenient proper ty that allows for a nearly 64 arbitrary ver tical resolution even in the presence of large amplitude fluctuations of 31 This coordinate is closely related to the "eta" coordinate used in many atmospheric models 32 (see Black (1994) for a review of eta coordinate atmospheric models). 33 It was originally used in ocean models by Stacey et al. (1995) for studies of tides next to shelves, 34 and it has been recently promoted by Adcroft and Campin (2004) for global climate modelling. 35 36 The surfaces of constant $z^\star$ are quasi-horizontal. 37 Indeed, the $z^\star$ coordinate reduces to $z$ when $\eta$ is zero. 38 In general, when noting the large differences between undulations of the bottom topography versus undulations in 39 the surface height, it is clear that surfaces constant $z^\star$ are very similar to the depth surfaces. 40 These properties greatly reduce difficulties of computing the horizontal pressure gradient relative to 41 terrain following sigma models discussed in \autoref{subsec:PE_sco}. 42 Additionally, since $z^\star$ when $\eta = 0$, no flow is spontaneously generated in 43 an unforced ocean starting from rest, regardless the bottom topography. 44 This behaviour is in contrast to the case with "s"-models, where pressure gradient errors in the presence of 45 nontrivial topographic variations can generate nontrivial spontaneous flow from a resting state, 46 depending on the sophistication of the pressure gradient solver. 47 The quasi-horizontal nature of the coordinate surfaces also facilitates the implementation of 48 neutral physics parameterizations in $z^\star$ models using the same techniques as in $z$-models 49 (see Chapters 13-16 of Griffies (2004) for a discussion of neutral physics in $z$-models, 50 as well as \autoref{sec:LDF_slp} in this document for treatment in \NEMO). 51 52 The range over which $z^\star$ varies is time independent $-H \leq z^\star \leq 0$. 53 Hence, all cells remain nonvanishing, so long as the surface height maintains $\eta > ?H$. 54 This is a minor constraint relative to that encountered on the surface height when using $s = z$ or $s = z - \eta$. 55 56 Because $z^\star$ has a time independent range, all grid cells have static increments ds, 57 and the sum of the ver tical increments yields the time independent ocean depth %�k ds = H. 58 The $z^\star$ coordinate is therefore invisible to undulations of the free surface, 59 since it moves along with the free surface. 60 This proper ty means that no spurious ver tical transpor t is induced across surfaces of 61 constant $z^\star$ by the motion of external gravity waves. 62 Such spurious transpor t can be a problem in z-models, especially those with tidal forcing. 63 Quite generally, the time independent range for the $z^\star$ coordinate is a very convenient property that 64 allows for a nearly arbitrary vertical resolution even in the presence of large amplitude fluctuations of 65 65 the surface height, again so long as $\eta > -H$. 66 66 … … 81 81 \nlst{nam_dynspg} 82 82 %------------------------------------------------------------------------------------------------------------ 83 Options are defined through the \ngn{nam\_dynspg} namelist variables. 84 The surface pressure gradient term is related to the representation of the free surface (\autoref{sec:PE_hor_pg}). The main distinction is between the fixed volume case (linear free surface or rigid lid) and the variable volume case (nonlinear free surface, \key{vvl} is active). In the linear free surface case (\autoref{subsec:PE_free_surface}) and rigid lid (\autoref{PE_rigid_lid}), the vertical scale factors $e_{3}$ are fixed in time, while in the nonlinear case (\autoref{subsec:PE_free_surface}) they are time-dependent. With both linear and nonlinear free surface, external gravity waves are allowed in the equations, which imposes a very small time step when an explicit time stepping is used. Two methods are proposed to allow a longer time step for the three-dimensional equations: the filtered free surface, which is a modification of the continuous equations (see \autoref{eq:PE_flt}), and the split-explicit free surface described below. The extra term introduced in the filtered method is calculated implicitly, so that the update of the next velocities is done in module \mdl{dynspg\_flt} and not in \mdl{dynnxt}. 83 Options are defined through the \ngn{nam\_dynspg} namelist variables. 84 The surface pressure gradient term is related to the representation of the free surface (\autoref{sec:PE_hor_pg}). 85 The main distinction is between the fixed volume case (linear free surface or rigid lid) and 86 the variable volume case (nonlinear free surface, \key{vvl} is active). 87 In the linear free surface case (\autoref{subsec:PE_free_surface}) and rigid lid (\autoref{PE_rigid_lid}), 88 the vertical scale factors $e_{3}$ are fixed in time, 89 while in the nonlinear case (\autoref{subsec:PE_free_surface}) they are time-dependent. 90 With both linear and nonlinear free surface, external gravity waves are allowed in the equations, 91 which imposes a very small time step when an explicit time stepping is used. 92 Two methods are proposed to allow a longer time step for the three-dimensional equations: 93 the filtered free surface, which is a modification of the continuous equations (see \autoref{eq:PE_flt}), 94 and the split-explicit free surface described below. 95 The extra term introduced in the filtered method is calculated implicitly, 96 so that the update of the next velocities is done in module \mdl{dynspg\_flt} and not in \mdl{dynnxt}. 85 97 86 98 %------------------------------------------------------------- … … 90 102 \label{subsec:DYN_spg_exp} 91 103 92 In the explicit free surface formulation, the model time step is chosen small enough to describe the external gravity waves (typically a few ten seconds). The sea surface height is given by : 104 In the explicit free surface formulation, the model time step is chosen small enough to 105 describe the external gravity waves (typically a few ten seconds). 106 The sea surface height is given by: 93 107 \begin{equation} \label{eq:dynspg_ssh} 94 108 \frac{\partial \eta }{\partial t}\equiv \frac{\text{EMP}}{\rho _w }+\frac{1}{e_{1T} … … 97 111 \end{equation} 98 112 99 where EMP is the surface freshwater budget (evaporation minus precipitation, and minus river runoffs (if the later are introduced as a surface freshwater flux, see \autoref{chap:SBC}) expressed in $Kg.m^{-2}.s^{-1}$, and $\rho _w =1,000\,Kg.m^{-3}$ is the volumic mass of pure water. The sea-surface height is evaluated using a leapfrog scheme in combination with an Asselin time filter, i.e. the velocity appearing in (\autoref{eq:dynspg_ssh}) is centred in time (\textit{now} velocity). 100 101 The surface pressure gradient, also evaluated using a leap-frog scheme, is then simply given by : 113 where EMP is the surface freshwater budget (evaporation minus precipitation, and minus river runoffs 114 (if the later are introduced as a surface freshwater flux, see \autoref{chap:SBC}) expressed in $Kg.m^{-2}.s^{-1}$, 115 and $\rho _w =1,000\,Kg.m^{-3}$ is the volumic mass of pure water. 116 The sea-surface height is evaluated using a leapfrog scheme in combination with an Asselin time filter, 117 i.e. the velocity appearing in (\autoref{eq:dynspg_ssh}) is centred in time (\textit{now} velocity). 118 119 The surface pressure gradient, also evaluated using a leap-frog scheme, is then simply given by: 102 120 \begin{equation} \label{eq:dynspg_exp} 103 121 \left\{ \begin{aligned} … … 108 126 \end{equation} 109 127 110 Consistent with the linearization, a $\left. \rho \right|_{k=1} / \rho _o$ factor is omitted in (\autoref{eq:dynspg_exp}). 128 Consistent with the linearization, a $\left. \rho \right|_{k=1} / \rho _o$ factor is omitted in 129 (\autoref{eq:dynspg_exp}). 111 130 112 131 %------------------------------------------------------------- … … 119 138 \nlst{namdom} 120 139 %-------------------------------------------------------------------------------------------------------------- 121 The split-explicit free surface formulation used in OPA follows the one proposed by \citet{Griffies2004}. The general idea is to solve the free surface equation with a small time step, while the three dimensional prognostic variables are solved with a longer time step that is a multiple of \np{rdtbt} 122 in the \ngn{namdom} namelist. 123 (Figure III.3). 140 The split-explicit free surface formulation used in OPA follows the one proposed by \citet{Griffies2004}. 141 The general idea is to solve the free surface equation with a small time step, 142 while the three dimensional prognostic variables are solved with a longer time step that 143 is a multiple of \np{rdtbt} in the \ngn{namdom} namelist (Figure III.3). 124 144 125 145 %> > > > > > > > > > > > > > > > > > > > > > > > > > > > … … 127 147 \includegraphics[width=0.90\textwidth]{Fig_DYN_dynspg_ts} 128 148 \caption{ \protect\label{fig:DYN_dynspg_ts} 129 Schematic of the split-explicit time stepping scheme for the barotropic and baroclinic modes, 130 after \citet{Griffies2004}. Time increases to the right. Baroclinic time steps are denoted by 131 $t-\Delta t$, $t, t+\Delta t$, and $t+2\Delta t$. The curved line represents a leap-frog time step, 132 and the smaller barotropic time steps $N \Delta t=2\Delta t$ are denoted by the zig-zag line. 133 The vertically integrated forcing \textbf{M}(t) computed at baroclinic time step t represents 134 the interaction between the barotropic and baroclinic motions. While keeping the total depth, 135 tracer, and freshwater forcing fields fixed, a leap-frog integration carries the surface height 136 and vertically integrated velocity from t to $t+2 \Delta t$ using N barotropic time steps of length 137 $\Delta t$. Time averaging the barotropic fields over the N+1 time steps (endpoints included) 138 centers the vertically integrated velocity at the baroclinic timestep $t+\Delta t$. 139 A baroclinic leap-frog time step carries the surface height to $t+\Delta t$ using the convergence 140 of the time averaged vertically integrated velocity taken from baroclinic time step t. } 149 Schematic of the split-explicit time stepping scheme for the barotropic and baroclinic modes, 150 after \citet{Griffies2004}. 151 Time increases to the right. 152 Baroclinic time steps are denoted by $t-\Delta t$, $t, t+\Delta t$, and $t+2\Delta t$. 153 The curved line represents a leap-frog time step, 154 and the smaller barotropic time steps $N \Delta t=2\Delta t$ are denoted by the zig-zag line. 155 The vertically integrated forcing \textbf{M}(t) computed at 156 baroclinic time step t represents the interaction between the barotropic and baroclinic motions. 157 While keeping the total depth, tracer, and freshwater forcing fields fixed, 158 a leap-frog integration carries the surface height and vertically integrated velocity from 159 t to $t+2 \Delta t$ using N barotropic time steps of length $\Delta t$. 160 Time averaging the barotropic fields over the N+1 time steps (endpoints included) 161 centers the vertically integrated velocity at the baroclinic timestep $t+\Delta t$. 162 A baroclinic leap-frog time step carries the surface height to $t+\Delta t$ using the convergence of 163 the time averaged vertically integrated velocity taken from baroclinic time step t. } 141 164 \end{center} 142 165 \end{figure} 143 166 %> > > > > > > > > > > > > > > > > > > > > > > > > > > > 144 167 145 The split-explicit formulation has a damping effect on external gravity waves, which is weaker than the filtered free surface but still significant as shown by \citet{Levier2007} in the case of an analytical barotropic Kelvin wave. 168 The split-explicit formulation has a damping effect on external gravity waves, 169 which is weaker than the filtered free surface but still significant as shown by \citet{Levier2007} in 170 the case of an analytical barotropic Kelvin wave. 146 171 147 172 %from griffies book: ..... copy past ! … … 149 174 \textbf{title: Time stepping the barotropic system } 150 175 151 Assume knowledge of the full velocity and tracer fields at baroclinic time $\tau$. Hence,152 we can update the surface height and vertically integrated velocity with a leap-frog 153 scheme using the small barotropic time step $\Delta t$. We have 154 176 Assume knowledge of the full velocity and tracer fields at baroclinic time $\tau$. 177 Hence, we can update the surface height and vertically integrated velocity with a leap-frog scheme using 178 the small barotropic time step $\Delta t$. 179 We have 155 180 \begin{equation} \label{eq:DYN_spg_ts_eta} 156 181 \eta^{(b)}(\tau,t_{n+1}) - \eta^{(b)}(\tau,t_{n+1}) (\tau,t_{n-1}) … … 164 189 \ 165 190 166 In these equations, araised (b) denotes values of surface height and vertically integrated velocity updated with the barotropic time steps. The $\tau$ time label on $\eta^{(b)}$ 167 and $U^{(b)}$ denotes the baroclinic time at which the vertically integrated forcing $\textbf{M}(\tau)$ (note that this forcing includes the surface freshwater forcing), the tracer fields, the freshwater flux $\text{EMP}_w(\tau)$, and total depth of the ocean $H(\tau)$ are held for the duration of the barotropic time stepping over a single cycle. This is also the time 168 that sets the barotropic time steps via 191 In these equations, araised (b) denotes values of surface height and 192 vertically integrated velocity updated with the barotropic time steps. 193 The $\tau$ time label on $\eta^{(b)}$ and $U^{(b)}$ denotes the baroclinic time at which 194 the vertically integrated forcing $\textbf{M}(\tau)$ 195 (note that this forcing includes the surface freshwater forcing), the tracer fields, 196 the freshwater flux $\text{EMP}_w(\tau)$, and total depth of the ocean $H(\tau)$ are held for 197 the duration of the barotropic time stepping over a single cycle. 198 This is also the time that sets the barotropic time steps via 169 199 \begin{equation} \label{eq:DYN_spg_ts_t} 170 200 t_n=\tau+n\Delta t 171 201 \end{equation} 172 with $n$ an integer. The density scaled surface pressure is evaluated via 202 with $n$ an integer. 203 The density scaled surface pressure is evaluated via 173 204 \begin{equation} \label{eq:DYN_spg_ts_ps} 174 205 p_s^{(b)}(\tau,t_{n}) = \begin{cases} … … 189 220 \overline{\eta^{(b)}(\tau)} = \frac{1}{N+1} \sum\limits_{n=0}^N \eta^{(b)}(\tau-\Delta t,t_{n}) 190 221 \end{equation} 191 the time averaged surface height taken from the previous barotropic cycle. Likewise, 222 the time averaged surface height taken from the previous barotropic cycle. 223 Likewise, 192 224 \begin{equation} \label{eq:DYN_spg_ts_u} 193 225 \textbf{U}^{(b)}(\tau,t_{n=0}) = \overline{\textbf{U}^{(b)}(\tau)} \\ … … 200 232 = \frac{1}{N+1} \sum\limits_{n=0}^N\textbf{U}^{(b)}(\tau-\Delta t,t_{n}) 201 233 \end{equation} 202 the time averaged vertically integrated transport. Notably, there is no Robert-Asselin time filter used in the barotropic portion of the integration. 203 204 Upon reaching $t_{n=N} = \tau + 2\Delta \tau$ , the vertically integrated velocity is time averaged to produce the updated vertically integrated velocity at baroclinic time $\tau + \Delta \tau$ 234 the time averaged vertically integrated transport. 235 Notably, there is no Robert-Asselin time filter used in the barotropic portion of the integration. 236 237 Upon reaching $t_{n=N} = \tau + 2\Delta \tau$ , the vertically integrated velocity is time averaged to 238 produce the updated vertically integrated velocity at baroclinic time $\tau + \Delta \tau$ 205 239 \begin{equation} \label{eq:DYN_spg_ts_u} 206 240 \textbf{U}(\tau+\Delta t) = \overline{\textbf{U}^{(b)}(\tau+\Delta t)} 207 241 = \frac{1}{N+1} \sum\limits_{n=0}^N\textbf{U}^{(b)}(\tau,t_{n}) 208 242 \end{equation} 209 The surface height on the new baroclinic time step is then determined via a baroclinic leap-frog using the following form210 243 The surface height on the new baroclinic time step is then determined via 244 a baroclinic leap-frog using the following form 211 245 \begin{equation} \label{eq:DYN_spg_ts_ssh} 212 246 \eta(\tau+\Delta) - \eta^{F}(\tau-\Delta) = 2\Delta t \ \left[ - \nabla \cdot \textbf{U}(\tau) + \text{EMP}_w \right] 213 247 \end{equation} 214 248 215 The use of this "big-leap-frog" scheme for the surface height ensures compatibility between the mass/volume budgets and the tracer budgets. More discussion of this point is provided in Chapter 10 (see in particular Section 10.2). 249 The use of this "big-leap-frog" scheme for the surface height ensures compatibility between 250 the mass/volume budgets and the tracer budgets. 251 More discussion of this point is provided in Chapter 10 (see in particular Section 10.2). 216 252 217 In general, some form of time filter is needed to maintain integrity of the surface 218 height field due to the leap-frog splitting mode in equation \autoref{eq:DYN_spg_ts_ssh}. We 219 have tried various forms of such filtering, with the following method discussed in 220 Griffies et al. (2001) chosen due to its stability and reasonably good maintenance of 221 tracer conservation properties (see ??)253 In general, some form of time filter is needed to maintain integrity of the surface height field due to 254 the leap-frog splitting mode in equation \autoref{eq:DYN_spg_ts_ssh}. 255 We have tried various forms of such filtering, 256 with the following method discussed in Griffies et al. (2001) chosen due to its stability and 257 reasonably good maintenance of tracer conservation properties (see ??) 222 258 223 259 \begin{equation} \label{eq:DYN_spg_ts_sshf} … … 232 268 \end{equation} 233 269 234 which is useful since it isolates all the time filtering aspects into the term multiplied 235 by $\alpha$. This isolation allows for an easy check that tracer conservation is exact when 236 eliminating tracer and surface height time filtering (see ?? for more complete discussion). However, in the general case with a non-zero $\alpha$, the filter \autoref{eq:DYN_spg_ts_sshf} was found to be more conservative, and so is recommended. 270 which is useful since it isolates all the time filtering aspects into the term multiplied by $\alpha$. 271 This isolation allows for an easy check that tracer conservation is exact when eliminating tracer and 272 surface height time filtering (see ?? for more complete discussion). 273 However, in the general case with a non-zero $\alpha$, the filter \autoref{eq:DYN_spg_ts_sshf} was found to 274 be more conservative, and so is recommended. 237 275 238 276 … … 246 284 \label{subsec:DYN_spg_flt} 247 285 248 The filtered formulation follows the \citet{Roullet2000} implementation. The extra term introduced in the equations (see {\S}I.2.2) is solved implicitly. The elliptic solvers available in the code are 249 documented in \autoref{chap:MISC}. The amplitude of the extra term is given by the namelist variable \np{rnu}. The default value is 1, as recommended by \citet{Roullet2000} 286 The filtered formulation follows the \citet{Roullet2000} implementation. 287 The extra term introduced in the equations (see {\S}I.2.2) is solved implicitly. 288 The elliptic solvers available in the code are documented in \autoref{chap:MISC}. 289 The amplitude of the extra term is given by the namelist variable \np{rnu}. 290 The default value is 1, as recommended by \citet{Roullet2000} 250 291 251 292 \colorbox{red}{\np{rnu}\forcode{ = 1} to be suppressed from namelist !} … … 257 298 \label{subsec:DYN_spg_vvl} 258 299 259 In the non-linear free surface formulation, the variations of volume are fully taken into account. This option is presented in a report \citep{Levier2007} available on the NEMO web site. The three time-stepping methods (explicit, split-explicit and filtered) are the same as in \autoref{DYN_spg_linear} except that the ocean depth is now time-dependent. In particular, this means that in filtered case, the matrix to be inverted has to be recomputed at each time-step. 300 In the non-linear free surface formulation, the variations of volume are fully taken into account. 301 This option is presented in a report \citep{Levier2007} available on the NEMO web site. 302 The three time-stepping methods (explicit, split-explicit and filtered) are the same as in 303 \autoref{DYN_spg_linear} except that the ocean depth is now time-dependent. 304 In particular, this means that in filtered case, the matrix to be inverted has to be recomputed at each time-step. 260 305 261 306
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