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NEMO/trunk/doc/latex/NEMO/subfiles/chap_model_basics.tex
r11151 r11335 32 32 \begin{enumerate} 33 33 \item 34 \textit{spherical earth approximation}: the geopotential surfaces are assumed to be spheres so that 35 gravity (local vertical) is parallel to the earth's radius 34 \textit{spherical Earth approximation}: the geopotential surfaces are assumed to be oblate spheriods 35 that follow the Earth's bulge; these spheroids are approximated by spheres with 36 gravity locally vertical (parallel to the Earth's radius) and independent of latitude 37 \citep[][section 2]{white.hoskins.ea_QJRMS05}. 36 38 \item 37 39 \textit{thin-shell approximation}: the ocean depth is neglected compared to the earth's radius … … 63 65 \nabla \cdot \vect U = 0 64 66 \end{equation} 67 \item 68 \textit{Neglect of additional Coriolis terms}: the Coriolis terms that vary with the cosine of latitude are neglected. 69 These terms may be non-negligible where the Brunt-Vaisala frequency $N$ is small, either in the deep ocean or 70 in the sub-mesoscale motions of the mixed layer, or near the equator \citep[][section 1]{white.hoskins.ea_QJRMS05}. 71 They can be consistently included as part of the ocean dynamics \citep[][section 3(d)]{white.hoskins.ea_QJRMS05} and are 72 retained in the MIT ocean model. 65 73 \end{enumerate} 66 74 67 75 Because the gravitational force is so dominant in the equations of large-scale motions, 68 it is useful to choose an orthogonal set of unit vectors $(i,j,k)$ linked to the earth such that76 it is useful to choose an orthogonal set of unit vectors $(i,j,k)$ linked to the Earth such that 69 77 $k$ is the local upward vector and $(i,j)$ are two vectors orthogonal to $k$, 70 78 \ie tangent to the geopotential surfaces. … … 107 115 an air-sea or ice-sea interface at its top. 108 116 These boundaries can be defined by two surfaces, $z = - H(i,j)$ and $z = \eta(i,j,k,t)$, 109 where $H$ is the depth of the ocean bottom and $\eta$ is the height of the sea surface. 110 Both $H$ and $\eta$ are usually referenced to a given surface, $z = 0$, chosen as a mean sea surface 117 where $H$ is the depth of the ocean bottom and $\eta$ is the height of the sea surface 118 (discretisation can introduce additional artificial ``side-wall'' boundaries). 119 Both $H$ and $\eta$ are referenced to a surface of constant geopotential (\ie a mean sea surface height) on which $z = 0$. 111 120 (\autoref{fig:ocean_bc}). 112 121 Through these two boundaries, the ocean can exchange fluxes of heat, fresh water, salt, and momentum with … … 210 219 The flow is barotropic and the surface moves up and down with gravity as the restoring force. 211 220 The phase speed of such waves is high (some hundreds of metres per second) so that 212 the time step would have to be very short if they were present in the model.221 the time step has to be very short when they are present in the model. 213 222 The latter strategy filters out these waves since the rigid lid approximation implies $\eta = 0$, 214 223 \ie the sea surface is the surface $z = 0$. … … 217 226 The rigid-lid hypothesis is an obsolescent feature in modern OGCMs. 218 227 It has been available until the release 3.1 of \NEMO, and it has been removed in release 3.2 and followings. 219 Only the free surface formulation is now described in th e this document (see the next sub-section).228 Only the free surface formulation is now described in this document (see the next sub-section). 220 229 221 230 % ------------------------------------------------------------------------------------------------------------- … … 237 246 Allowing the air-sea interface to move introduces the external gravity waves (EGWs) as 238 247 a class of solution of the primitive equations. 239 These waves are barotropic because of hydrostatic assumption,and their phase speed is quite high.248 These waves are barotropic (\ie nearly independent of depth) and their phase speed is quite high. 240 249 Their time scale is short with respect to the other processes described by the primitive equations. 241 250 … … 266 275 the implicit scheme \citep{dukowicz.smith_JGR94} or 267 276 the addition of a filtering force in the momentum equation \citep{roullet.madec_JGR00}. 268 With the present release, \NEMO offers the choice between277 With the present release, \NEMO offers the choice between 269 278 an explicit free surface (see \autoref{subsec:DYN_spg_exp}) or 270 279 a split-explicit scheme strongly inspired the one proposed by \citet{shchepetkin.mcwilliams_OM05} … … 338 347 the vertical scale factor is a single function of $k$ as $k$ is parallel to $z$. 339 348 The scalar and vector operators that appear in the primitive equations 340 (\autoref{eq:PE_dyn} to \autoref{eq:PE_eos}) can be written in the tensorial form,349 (\autoref{eq:PE_dyn} to \autoref{eq:PE_eos}) can then be written in the tensorial form, 341 350 invariant in any orthogonal horizontal curvilinear coordinate system transformation: 342 351 \begin{subequations} … … 384 393 \end{gather} 385 394 386 Using the fact that the horizontal scale factors $e_1$ and $e_2$ are independent of $k$ and that395 Using again the fact that the horizontal scale factors $e_1$ and $e_2$ are independent of $k$ and that 387 396 $e_3$ is a function of the single variable $k$, 388 397 $NLT$ the nonlinear term of \autoref{eq:PE_dyn} can be transformed as follows: … … 456 465 & &= &\nabla \cdot (\vect U \, u) - (\nabla \cdot \vect U) \ u 457 466 + \frac{1}{e_1 e_2} \lt( -v^2 \pd[e_2]{i} + u v \, \pd[e_1]{j} \rt) \\ 458 \intertext{as $\nabla \cdot {\vect U} \; = 0$ (incompressibility) it comes:}467 \intertext{as $\nabla \cdot {\vect U} \; = 0$ (incompressibility) it becomes:} 459 468 & &= &\, \nabla \cdot (\vect U \, u) + \frac{1}{e_1 e_2} \lt( v \; \pd[e_2]{i} - u \; \pd[e_1]{j} \rt) (-v) 460 469 \end{alignat*} … … 516 525 % \label{eq:PE_dyn_flux_v} 517 526 \pd[v]{t} = - \lt[ f + \frac{1}{e_1 \; e_2} \lt( v \pd[e_2]{i} - u \pd[e_1]{j} \rt) \rt] \, u \\ 518 +\frac{1}{e_1 \; e_2} \lt( \pd[(e_2 \, u \, v)]{i} + \pd[(e_1 \, v \, v)]{j} \rt) \\527 - \frac{1}{e_1 \; e_2} \lt( \pd[(e_2 \, u \, v)]{i} + \pd[(e_1 \, v \, v)]{j} \rt) \\ 519 528 - \frac{1}{e_3} \pd[(w \, v)]{k} - \frac{1}{e_2} \pd[]{j} \lt( \frac{p_s + p_h}{\rho_o} \rt) 520 529 + D_v^{\vect U} + F_v^{\vect U} … … 526 535 p_s = \rho \,g \, \eta 527 536 \] 528 with $\eta$ issolution of \autoref{eq:PE_ssh}.537 and $\eta$ is the solution of \autoref{eq:PE_ssh}. 529 538 530 539 The vertical velocity and the hydrostatic pressure are diagnosed from the following equations: … … 536 545 \] 537 546 where the divergence of the horizontal velocity, $\chi$ is given by \autoref{eq:PE_div_Uh}. 538 \item \textit{tracer equations}: 539 \[ 540 %\label{eq:S} 541 \pd[T]{t} = - \frac{1}{e_1 e_2} \lt[ \pd[(e_2 T \, u)]{i} + \pd[(e_1 T \, v)]{j} \rt] 547 548 \item 549 \textbf{tracer equations}: 550 \begin{equation} 551 \begin{split} 552 \pd[T]{t} = & - \frac{1}{e_1 e_2} \lt[ \pd[(e_2 T \, u)]{i} + \pd[(e_1 T \, v)]{j} \rt] 542 553 - \frac{1}{e_3} \pd[(T \, w)]{k} + D^T + F^T \\ 543 %\label{eq:T} 544 \pd[S]{t} = - \frac{1}{e_1 e_2} \lt[ \pd[(e_2 S \, u)]{i} + \pd[(e_1 S \, v)]{j} \rt] 545 - \frac{1}{e_3} \pd[(S \, w)]{k} + D^S + F^S 546 %\label{eq:rho} 547 \rho = \rho \big( T,S,z(k) \big) 548 \] 554 \pd[S]{t} = & - \frac{1}{e_1 e_2} \lt[ \pd[(e_2 S \, u)]{i} + \pd[(e_1 S \, v)]{j} \rt] 555 - \frac{1}{e_3} \pd[(S \, w)]{k} + D^S + F^S \\ 556 \rho = & \rho \big( T,S,z(k) \big) 557 \end{split} 558 \end{equation} 549 559 \end{itemize} 550 560 … … 575 585 follows the isopycnal surfaces, \eg an isopycnic coordinate. 576 586 577 In order to satisfy two or more constrain s one can even be tempted to mixed these coordinate systems, as in587 In order to satisfy two or more constraints one can even be tempted to mixed these coordinate systems, as in 578 588 HYCOM (mixture of $z$-coordinate at the surface, isopycnic coordinate in the ocean interior and $\sigma$ at 579 589 the ocean bottom) \citep{chassignet.smith.ea_JPO03} or … … 594 604 This so-called \textit{generalised vertical coordinate} \citep{kasahara_MWR74} is in fact 595 605 an Arbitrary Lagrangian--Eulerian (ALE) coordinate. 596 Indeed, choosing an expression for $s$ is an arbitrary choice thatdetermines606 Indeed, one has a great deal of freedom in the choice of expression for $s$. The choice determines 597 607 which part of the vertical velocity (defined from a fixed referential) will cross the levels (Eulerian part) and 598 608 which part will be used to move them (Lagrangian part). … … 601 611 Its most often used implementation is via an ALE algorithm, 602 612 in which a pure lagrangian step is followed by regridding and remapping steps, 603 the lat er step implicitly embedding the vertical advection613 the latter step implicitly embedding the vertical advection 604 614 \citep{hirt.amsden.ea_JCP74, chassignet.smith.ea_JPO03, white.adcroft.ea_JCP09}. 605 615 Here we follow the \citep{kasahara_MWR74} strategy: 606 a regridding step (an update of the vertical coordinate) followed by an eulerian step with616 a regridding step (an update of the vertical coordinate) followed by an Eulerian step with 607 617 an explicit computation of vertical advection relative to the moving s-surfaces. 608 618 609 619 %\gmcomment{ 610 620 %A key point here is that the $s$-coordinate depends on $(i,j)$ ==> horizontal pressure gradient... 611 the generalized vertical coordinates used in ocean modelling are not orthogonal,621 The generalized vertical coordinates used in ocean modelling are not orthogonal, 612 622 which contrasts with many other applications in mathematical physics. 613 623 Hence, it is useful to keep in mind the following properties that may seem odd on initial encounter. … … 615 625 The horizontal velocity in ocean models measures motions in the horizontal plane, 616 626 perpendicular to the local gravitational field. 617 That is, horizontal velocity is mathematically the same regardless the vertical coordinate, be it geopotential,627 That is, horizontal velocity is mathematically the same regardless of the vertical coordinate, be it geopotential, 618 628 isopycnal, pressure, or terrain following. 619 629 The key motivation for maintaining the same horizontal velocity component is that … … 660 670 \[ 661 671 % \label{eq:PE_sco_w} 662 \omega = w - e_3 \, \pd[s]{t}- \sigma_1 \, u - \sigma_2 \, v672 \omega = w - \, \lt. \pd[z]{t} \rt|_s - \sigma_1 \, u - \sigma_2 \, v 663 673 \] 664 674 … … 671 681 % \label{eq:PE_sco_u_vector} 672 682 \pd[u]{t} = + (\zeta + f) \, v - \frac{1}{2 \, e_1} \pd[]{i} (u^2 + v^2) - \frac{1}{e_3} \omega \pd[u]{k} \\ 673 - \frac{1}{e_1} \pd[]{i} \lt( \frac{p_s + p_h}{\rho_o} \rt) +g \frac{\rho}{\rho_o} \sigma_1683 - \frac{1}{e_1} \pd[]{i} \lt( \frac{p_s + p_h}{\rho_o} \rt) - g \frac{\rho}{\rho_o} \sigma_1 674 684 + D_u^{\vect U} + F_u^{\vect U} 675 685 \end{multline*} … … 677 687 % \label{eq:PE_sco_v_vector} 678 688 \pd[v]{t} = - (\zeta + f) \, u - \frac{1}{2 \, e_2} \pd[]{j}(u^2 + v^2) - \frac{1}{e_3} \omega \pd[v]{k} \\ 679 - \frac{1}{e_2} \pd[]{j} \lt( \frac{p_s + p_h}{\rho_o} \rt) +g \frac{\rho}{\rho_o} \sigma_2689 - \frac{1}{e_2} \pd[]{j} \lt( \frac{p_s + p_h}{\rho_o} \rt) - g \frac{\rho}{\rho_o} \sigma_2 680 690 + D_v^{\vect U} + F_v^{\vect U} 681 691 \end{multline*} … … 687 697 - \frac{1}{e_3} \pd[(\omega \, u)]{k} 688 698 - \frac{1}{e_1} \pd[]{i} \lt( \frac{p_s + p_h}{\rho_o} \rt) 689 +g \frac{\rho}{\rho_o} \sigma_1 + D_u^{\vect U} + F_u^{\vect U}699 - g \frac{\rho}{\rho_o} \sigma_1 + D_u^{\vect U} + F_u^{\vect U} 690 700 \end{multline*} 691 701 \begin{multline*} … … 695 705 - \frac{1}{e_3} \pd[(\omega \, v)]{k} 696 706 - \frac{1}{e_2} \pd[]{j} \lt( \frac{p_s + p_h}{\rho_o} \rt) 697 +g \frac{\rho}{\rho_o}\sigma_2 + D_v^{\vect U} + F_v^{\vect U}707 - g \frac{\rho}{\rho_o}\sigma_2 + D_v^{\vect U} + F_v^{\vect U} 698 708 \end{multline*} 699 709 where the relative vorticity, $\zeta$, the surface pressure gradient, … … 750 760 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 751 761 752 In th atcase, the free surface equation is nonlinear, and the variations of volume are fully taken into account.762 In this case, the free surface equation is nonlinear, and the variations of volume are fully taken into account. 753 763 These coordinates systems is presented in a report \citep{levier.treguier.ea_rpt07} available on the \NEMO web site. 754 764 … … 766 776 The position (\zstar) and vertical discretization (\zstar) are expressed as: 767 777 \[ 768 % \label{eq: z-star}778 % \label{eq:PE_z-star} 769 779 H + \zstar = (H + z) / r \quad \text{and} \quad \delta \zstar 770 780 = \delta z / r \quad \text{with} \quad r 771 = \frac{H + \eta}{H} 781 = \frac{H + \eta}{H} . 782 \] 783 Simple re-organisation of the above expressions gives 784 \[ 785 % \label{eq:PE_zstar_2} 786 \zstar = H \lt( \frac{z - \eta}{H + \eta} \rt) . 772 787 \] 773 788 Since the vertical displacement of the free surface is incorporated in the vertical coordinate \zstar, … … 776 791 Also the divergence of the flow field is no longer zero as shown by the continuity equation: 777 792 \[ 778 \pd[r]{t} = \nabla_{\zstar} \cdot \lt( r \; \vect U_h \rt) (r \; w *) = 0793 \pd[r]{t} = \nabla_{\zstar} \cdot \lt( r \; \vect U_h \rt) + \pd[r \; w^*]{\zstar} = 0 . 779 794 \] 780 781 % from MOM4p1 documentation 782 To overcome problems with vanishing surface and/or bottom cells, we consider the zstar coordinate 783 \[ 784 % \label{eq:PE_} 785 \zstar = H \lt( \frac{z - \eta}{H + \eta} \rt) 786 \] 787 788 This coordinate is closely related to the "eta" coordinate used in many atmospheric models 795 This \zstar coordinate is closely related to the "eta" coordinate used in many atmospheric models 789 796 (see Black (1994) for a review of eta coordinate atmospheric models). 790 797 It was originally used in ocean models by Stacey et al. (1995) for studies of tides next to shelves, … … 798 805 These properties greatly reduce difficulties of computing the horizontal pressure gradient relative to 799 806 terrain following sigma models discussed in \autoref{subsec:PE_sco}. 800 Additionally, since \zstarwhen $\eta = 0$,807 Additionally, since $\zstar = z$ when $\eta = 0$, 801 808 no flow is spontaneously generated in an unforced ocean starting from rest, regardless the bottom topography. 802 809 This behaviour is in contrast to the case with "s"-models, where pressure gradient errors in the presence of … … 804 811 depending on the sophistication of the pressure gradient solver. 805 812 The quasi -horizontal nature of the coordinate surfaces also facilitates the implementation of 806 neutral physics parameterizations in \zstar models using the same techniques as in $z$-models813 neutral physics parameterizations in \zstar models using the same techniques as in $z$-models 807 814 (see Chapters 13-16 of \cite{griffies_bk04}) for a discussion of neutral physics in $z$-models, 808 815 as well as \autoref{sec:LDF_slp} in this document for treatment in \NEMO). 809 816 810 The range over which \zstar varies is time independent $-H \leq \zstar \leq 0$.811 Hence, all cells remain nonvanishing, so long as the surface height maintains $\eta > ?H$.817 The range over which \zstar varies is time independent $-H \leq \zstar \leq 0$. 818 Hence, all cells remain nonvanishing, so long as the surface height maintains $\eta > -H$. 812 819 This is a minor constraint relative to that encountered on the surface height when using $s = z$ or $s = z - \eta$. 813 820 814 Because \zstar has a time independent range, all grid cells have static increments ds,815 and the sum of the ver 821 Because \zstar has a time independent range, all grid cells have static increments ds, 822 and the sum of the vertical increments yields the time independent ocean depth. %k ds = H. 816 823 The \zstar coordinate is therefore invisible to undulations of the free surface, 817 824 since it moves along with the free surface. 818 This proper ty means that no spurious vertical transport is induced across surfaces of constant \zstarby825 This property means that no spurious vertical transport is induced across surfaces of constant \zstar by 819 826 the motion of external gravity waves. 820 Such spurious transpor 821 Quite generally, the time independent range for the \zstar coordinate is a very convenient property that822 allows for a nearly arbitrary ver 827 Such spurious transport can be a problem in z-models, especially those with tidal forcing. 828 Quite generally, the time independent range for the \zstar coordinate is a very convenient property that 829 allows for a nearly arbitrary vertical resolution even in the presence of large amplitude fluctuations of 823 830 the surface height, again so long as $\eta > -H$. 824 831 %end MOM doc %%% … … 870 877 \begin{equation} 871 878 \label{eq:PE_p_sco} 872 \nabla p |_z = \nabla p |_s - \ pd[p]{s} \nabla z |_s879 \nabla p |_z = \nabla p |_s - \frac{1}{e_3} \pd[p]{s} \nabla z |_s 873 880 \end{equation} 874 881 875 882 The second term in \autoref{eq:PE_p_sco} depends on the tilt of the coordinate surface and 876 introducesa truncation error that is not present in a $z$-model.883 leads to a truncation error that is not present in a $z$-model. 877 884 In the special case of a $\sigma$-coordinate (i.e. a depth-normalised coordinate system $\sigma = z/H$), 878 885 \citet{haney_JPO91} and \citet{beckmann.haidvogel_JPO93} have given estimates of the magnitude of this truncation error. … … 887 894 However, the definition of the model domain vertical coordinate becomes then a non-trivial thing for 888 895 a realistic bottom topography: 889 a envelope topography is defined in $s$-coordinate on which a full or896 an envelope topography is defined in $s$-coordinate on which a full or 890 897 partial step bottom topography is then applied in order to adjust the model depth to the observed one 891 898 (see \autoref{sec:DOM_zgr}. … … 922 929 923 930 The \ztilde -coordinate has been developed by \citet{leclair.madec_OM11}. 924 It is available in \NEMO since the version 3.4 .931 It is available in \NEMO since the version 3.4 and is more robust in version 4.0 than previously. 925 932 Nevertheless, it is currently not robust enough to be used in all possible configurations. 926 933 Its use is therefore not recommended. … … 934 941 \label{sec:PE_zdf_ldf} 935 942 936 The primitive equations describe the behaviour of a geophysical fluid at space and time scales larger than943 The hydrostatic primitive equations describe the behaviour of a geophysical fluid at space and time scales larger than 937 944 a few kilometres in the horizontal, a few meters in the vertical and a few minutes. 938 945 They are usually solved at larger scales: the specified grid spacing and time step of the numerical model. … … 984 991 All the vertical physics is embedded in the specification of the eddy coefficients. 985 992 They can be assumed to be either constant, or function of the local fluid properties 986 (\eg Richardson number, Brunt-Vais\"{a}l\"{a} frequency ...),993 (\eg Richardson number, Brunt-Vais\"{a}l\"{a} frequency, distance from the boundary ...), 987 994 or computed from a turbulent closure model. 988 995 The choices available in \NEMO are discussed in \autoref{chap:ZDF}). … … 1016 1023 both horizontal and isoneutral operators have no effect on mean (\ie large scale) potential energy whereas 1017 1024 potential energy is a main source of turbulence (through baroclinic instabilities). 1018 \citet{gent.mcwilliams_JPO90} haveproposed a parameterisation of mesoscale eddy-induced turbulence which1025 \citet{gent.mcwilliams_JPO90} proposed a parameterisation of mesoscale eddy-induced turbulence which 1019 1026 associates an eddy-induced velocity to the isoneutral diffusion. 1020 1027 Its mean effect is to reduce the mean potential energy of the ocean. … … 1033 1040 Another approach is becoming more and more popular: 1034 1041 instead of specifying explicitly a sub-grid scale term in the momentum and tracer time evolution equations, 1035 one uses a advective scheme which is diffusive enough to maintain the model stability.1042 one uses an advective scheme which is diffusive enough to maintain the model stability. 1036 1043 It must be emphasised that then, all the sub-grid scale physics is included in the formulation of 1037 1044 the advection scheme. 1038 1045 1039 1046 All these parameterisations of subgrid scale physics have advantages and drawbacks. 1040 The reare not all available in \NEMO. For active tracers (temperature and salinity) the main ones are:1047 They are not all available in \NEMO. For active tracers (temperature and salinity) the main ones are: 1041 1048 Laplacian and bilaplacian operators acting along geopotential or iso-neutral surfaces, 1042 1049 \citet{gent.mcwilliams_JPO90} parameterisation, and various slightly diffusive advection schemes. … … 1141 1148 - \nabla_h \times \big( A^{lm} \, \zeta \; \vect k \big) \\ 1142 1149 &= \lt( \frac{1}{e_1} \pd[ \lt( A^{lm} \chi \rt) ]{i} \rt. 1143 - \frac{1}{e_2 e_3} \pd[ \lt( A^{lm} \; e_3 \zeta \rt) ]{j} 1150 - \frac{1}{e_2 e_3} \pd[ \lt( A^{lm} \; e_3 \zeta \rt) ]{j} , 1144 1151 \frac{1}{e_2} \pd[ \lt( A^{lm} \chi \rt) ]{j} 1145 1152 \lt. + \frac{1}{e_1 e_3} \pd[ \lt( A^{lm} \; e_3 \zeta \rt) ]{i} \rt) … … 1164 1171 a geographical coordinate system \citep{lengaigne.madec.ea_JGR03}. 1165 1172 1166 \subsubsection{ lateral bilaplacian momentum diffusive operator}1173 \subsubsection{Lateral bilaplacian momentum diffusive operator} 1167 1174 1168 1175 As for tracers, the bilaplacian order momentum diffusive operator is a re-entering Laplacian operator with
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