[9389] | 1 | \documentclass[../tex_main/NEMO_manual]{subfiles} |
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[6997] | 2 | \begin{document} |
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[707] | 3 | |
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| 4 | % ================================================================ |
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| 5 | % Invariant of the Equations |
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| 6 | % ================================================================ |
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[1223] | 7 | \chapter{Invariants of the Primitive Equations} |
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[9407] | 8 | \label{chap:Invariant} |
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[707] | 9 | \minitoc |
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| 10 | |
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[10354] | 11 | The continuous equations of motion have many analytic properties. |
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| 12 | Many quantities (total mass, energy, enstrophy, etc.) are strictly conserved in the inviscid and unforced limit, |
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| 13 | while ocean physics conserve the total quantities on which they act (momentum, temperature, salinity) but |
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| 14 | dissipate their total variance (energy, enstrophy, etc.). |
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| 15 | Unfortunately, the finite difference form of these equations is not guaranteed to |
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| 16 | retain all these important properties. |
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| 17 | In constructing the finite differencing schemes, we wish to ensure that |
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| 18 | certain integral constraints will be maintained. |
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| 19 | In particular, it is desirable to construct the finite difference equations so that |
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| 20 | horizontal kinetic energy and/or potential enstrophy of horizontally non-divergent flow, |
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| 21 | and variance of temperature and salinity will be conserved in the absence of dissipative effects and forcing. |
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| 22 | \citet{Arakawa1966} has first pointed out the advantage of this approach. |
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| 23 | He showed that if integral constraints on energy are maintained, |
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| 24 | the computation will be free of the troublesome "non linear" instability originally pointed out by |
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| 25 | \citet{Phillips1959}. |
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| 26 | A consistent formulation of the energetic properties is also extremely important in carrying out |
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| 27 | long-term numerical simulations for an oceanographic model. |
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| 28 | Such a formulation avoids systematic errors that accumulate with time \citep{Bryan1997}. |
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[707] | 29 | |
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[10354] | 30 | The general philosophy of OPA which has led to the discrete formulation presented in {\S}II.2 and II.3 is to |
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| 31 | choose second order non-diffusive scheme for advective terms for both dynamical and tracer equations. |
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| 32 | At this level of complexity, the resulting schemes are dispersive schemes. |
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| 33 | Therefore, they require the addition of a diffusive operator to be stable. |
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| 34 | The alternative is to use diffusive schemes such as upstream or flux corrected schemes. |
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| 35 | This last option was rejected because we prefer a complete handling of the model diffusion, |
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| 36 | i.e. of the model physics rather than letting the advective scheme produces its own implicit diffusion without |
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| 37 | controlling the space and time structure of this implicit diffusion. |
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| 38 | Note that in some very specific cases as passive tracer studies, the positivity of the advective scheme is required. |
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| 39 | In that case, and in that case only, the advective scheme used for passive tracer is a flux correction scheme |
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| 40 | \citep{Marti1992, Levy1996, Levy1998}. |
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[707] | 41 | |
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| 42 | % ------------------------------------------------------------------------------------------------------------- |
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| 43 | % Conservation Properties on Ocean Dynamics |
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| 44 | % ------------------------------------------------------------------------------------------------------------- |
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[9393] | 45 | \section{Conservation properties on ocean dynamics} |
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[9407] | 46 | \label{sec:Invariant_dyn} |
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[707] | 47 | |
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[10354] | 48 | The non linear term of the momentum equations has been split into a vorticity term, |
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| 49 | a gradient of horizontal kinetic energy and a vertical advection term. |
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| 50 | Three schemes are available for the former (see {\S}~II.2) according to the CPP variable defined |
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| 51 | (default option\textbf{?}or \textbf{key{\_}vorenergy} or \textbf{key{\_}vorcombined} defined). |
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| 52 | They differ in their conservative properties (energy or enstrophy conserving scheme). |
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| 53 | The two latter terms preserve the total kinetic energy: |
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| 54 | the large scale kinetic energy is also preserved in practice. |
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| 55 | The remaining non-diffusive terms of the momentum equation |
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| 56 | (namely the hydrostatic and surface pressure gradient terms) also preserve the total kinetic energy and |
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| 57 | have no effect on the vorticity of the flow. |
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[707] | 58 | |
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| 59 | \textbf{* relative, planetary and total vorticity term:} |
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| 60 | |
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[10354] | 61 | Let us define as either the relative, planetary and total potential vorticity, i.e. ?, ?, and ?, respectively. |
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| 62 | The continuous formulation of the vorticity term satisfies following integral constraints: |
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[9407] | 63 | \begin{equation} \label{eq:vor_vorticity} |
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[707] | 64 | \int_D {{\textbf {k}}\cdot \frac{1}{e_3 }\nabla \times \left( {\varsigma |
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| 65 | \;{\rm {\bf k}}\times {\textbf {U}}_h } \right)\;dv} =0 |
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| 66 | \end{equation} |
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| 67 | |
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[9407] | 68 | \begin{equation} \label{eq:vor_enstrophy} |
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[707] | 69 | if\quad \chi =0\quad \quad \int\limits_D {\varsigma \;{\textbf{k}}\cdot |
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| 70 | \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} |
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| 71 | =0 |
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| 72 | \end{equation} |
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| 73 | |
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[9407] | 74 | \begin{equation} \label{eq:vor_energy} |
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[707] | 75 | \int_D {{\textbf{U}}_h \times \left( {\varsigma \;{\textbf{k}}\times {\textbf{U}}_h } \right)\;dv} =0 |
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| 76 | \end{equation} |
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[10354] | 77 | where $dv = e_1\, e_2\, e_3\, di\, dj\, dk$ is the volume element. |
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| 78 | (II.4.1a) means that $\varsigma $ is conserved. (II.4.1b) is obtained by an integration by part. |
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| 79 | It means that $\varsigma^2$ is conserved for a horizontally non-divergent flow. |
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| 80 | (II.4.1c) is even satisfied locally since the vorticity term is orthogonal to the horizontal velocity. |
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| 81 | It means that the vorticity term has no contribution to the evolution of the total kinetic energy. |
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| 82 | (II.4.1a) is obviously always satisfied, but (II.4.1b) and (II.4.1c) cannot be satisfied simultaneously with |
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| 83 | a second order scheme. |
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| 84 | Using the symmetry or anti-symmetry properties of the operators (Eqs II.1.10 and 11), |
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| 85 | it can be shown that the scheme (II.2.11) satisfies (II.4.1b) but not (II.4.1c), |
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| 86 | while scheme (II.2.12) satisfies (II.4.1c) but not (II.4.1b) (see appendix C). |
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| 87 | Note that the enstrophy conserving scheme on total vorticity has been chosen as the standard discrete form of |
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| 88 | the vorticity term. |
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[707] | 89 | |
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| 90 | \textbf{* Gradient of kinetic energy / vertical advection} |
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| 91 | |
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[10354] | 92 | In continuous formulation, the gradient of horizontal kinetic energy has no contribution to the evolution of |
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| 93 | the vorticity as the curl of a gradient is zero. |
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| 94 | This property is satisfied locally with the discrete form of both the gradient and the curl operator we have made |
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| 95 | (property (II.1.9)~). |
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| 96 | Another continuous property is that the change of horizontal kinetic energy due to |
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| 97 | vertical advection is exactly balanced by the change of horizontal kinetic energy due to |
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| 98 | the horizontal gradient of horizontal kinetic energy: |
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[707] | 99 | |
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[9407] | 100 | \begin{equation} \label{eq:keg_zad} |
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[707] | 101 | \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 |
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| 102 | {\textbf{U}}_h }{\partial k}\;dv} |
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| 103 | \end{equation} |
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| 104 | |
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[10354] | 105 | Using the discrete form given in {\S}II.2-a and the symmetry or anti-symmetry properties of |
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| 106 | the mean and difference operators, \autoref{eq:keg_zad} is demonstrated in the Appendix C. |
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| 107 | The main point here is that satisfying \autoref{eq:keg_zad} links the choice of the discrete forms of |
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| 108 | the vertical advection and of the horizontal gradient of horizontal kinetic energy. |
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| 109 | Choosing one imposes the other. |
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| 110 | The discrete form of the vertical advection given in {\S}II.2-a is a direct consequence of |
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| 111 | formulating the horizontal kinetic energy as $1/2 \left( \overline{u^2}^i + \overline{v^2}^j \right) $ in |
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| 112 | the gradient term. |
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[707] | 113 | |
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| 114 | \textbf{* hydrostatic pressure gradient term} |
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| 115 | |
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[10354] | 116 | In continuous formulation, a pressure gradient has no contribution to the evolution of the vorticity as |
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| 117 | the curl of a gradient is zero. |
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| 118 | This properties is satisfied locally with the choice of discretization we have made (property (II.1.9)~). |
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| 119 | In addition, when the equation of state is linear |
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| 120 | (i.e. when an advective-diffusive equation for density can be derived from those of temperature and salinity) |
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| 121 | the change of horizontal kinetic energy due to the work of pressure forces is balanced by the change of |
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| 122 | potential energy due to buoyancy forces: |
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[707] | 123 | |
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[9407] | 124 | \begin{equation} \label{eq:hpg_pe} |
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[707] | 125 | \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} |
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| 126 | \end{equation} |
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| 127 | |
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[10354] | 128 | Using the discrete form given in {\S}~II.2-a and the symmetry or anti-symmetry properties of |
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| 129 | the mean and difference operators, (II.4.3) is demonstrated in the Appendix C. |
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| 130 | The main point here is that satisfying (II.4.3) strongly constraints the discrete expression of the depth of |
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| 131 | $T$-points and of the term added to the pressure gradient in $s-$coordinates: the depth of a $T$-point, $z_T$, |
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| 132 | is defined as the sum the vertical scale factors at $w$-points starting from the surface. |
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[707] | 133 | |
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| 134 | \textbf{* surface pressure gradient term} |
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| 135 | |
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[10354] | 136 | In continuous formulation, the surface pressure gradient has no contribution to the evolution of vorticity. |
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| 137 | This properties is trivially satisfied locally as (II.2.3) |
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| 138 | (the equation verified by $\psi$ has been derived from the discrete formulation of the momentum equations, |
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| 139 | vertical sum and curl). |
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| 140 | Nevertheless, the $\psi$-equation is solved numerically by an iterative solver (see {\S}~III.5), |
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| 141 | thus the property is only satisfied with the accuracy required on the solver. |
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| 142 | In addition, with the rigid-lid approximation, the change of horizontal kinetic energy due to the work of |
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[707] | 143 | surface pressure forces is exactly zero: |
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[9407] | 144 | \begin{equation} \label{eq:spg} |
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[707] | 145 | \int_D {-\frac{1}{\rho _o }\nabla _h } \left( {p_s } \right)\cdot {\textbf{U}}_h \;dv=0 |
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| 146 | \end{equation} |
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| 147 | |
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[10354] | 148 | (II.4.4) is satisfied in discrete form only if |
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| 149 | the discrete barotropic streamfunction time evolution equation is given by (II.2.3) (see appendix C). |
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| 150 | This shows that (II.2.3) is the only way to compute the streamfunction, |
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| 151 | otherwise there is no guarantee that the surface pressure force work vanishes. |
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[707] | 152 | |
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| 153 | % ------------------------------------------------------------------------------------------------------------- |
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| 154 | % Conservation Properties on Ocean Thermodynamics |
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| 155 | % ------------------------------------------------------------------------------------------------------------- |
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[9393] | 156 | \section{Conservation properties on ocean thermodynamics} |
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[9407] | 157 | \label{sec:Invariant_tra} |
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[707] | 158 | |
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[10354] | 159 | In continuous formulation, the advective terms of the tracer equations conserve the tracer content and |
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| 160 | the quadratic form of the tracer, $i.e.$ |
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[9407] | 161 | \begin{equation} \label{eq:tra_tra2} |
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[707] | 162 | \int_D {\nabla .\left( {T\;{\textbf{U}}} \right)\;dv} =0 |
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| 163 | \;\text{and} |
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| 164 | \int_D {T\;\nabla .\left( {T\;{\textbf{U}}} \right)\;dv} =0 |
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| 165 | \end{equation} |
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| 166 | |
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[10354] | 167 | The numerical scheme used ({\S}II.2-b) (equations in flux form, second order centred finite differences) satisfies |
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| 168 | (II.4.5) (see appendix C). |
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| 169 | Note that in both continuous and discrete formulations, there is generally no strict conservation of mass, |
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| 170 | since the equation of state is non linear with respect to $T$ and $S$. |
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| 171 | In practice, the mass is conserved with a very good accuracy. |
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[707] | 172 | |
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| 173 | % ------------------------------------------------------------------------------------------------------------- |
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| 174 | % Conservation Properties on Momentum Physics |
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| 175 | % ------------------------------------------------------------------------------------------------------------- |
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[9393] | 176 | \subsection{Conservation properties on momentum physics} |
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[9407] | 177 | \label{subsec:Invariant_dyn_physics} |
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[707] | 178 | |
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| 179 | \textbf{* lateral momentum diffusion term} |
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| 180 | |
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[10354] | 181 | The continuous formulation of the horizontal diffusion of momentum satisfies the following integral constraints~: |
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[9407] | 182 | \begin{equation} \label{eq:dynldf_dyn} |
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[707] | 183 | \int\limits_D {\frac{1}{e_3 }{\rm {\bf k}}\cdot \nabla \times \left[ {\nabla |
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| 184 | _h \left( {A^{lm}\;\chi } \right)-\nabla _h \times \left( {A^{lm}\;\zeta |
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| 185 | \;{\rm {\bf k}}} \right)} \right]\;dv} =0 |
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| 186 | \end{equation} |
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| 187 | |
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[9407] | 188 | \begin{equation} \label{eq:dynldf_div} |
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[707] | 189 | \int\limits_D {\nabla _h \cdot \left[ {\nabla _h \left( {A^{lm}\;\chi } |
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| 190 | \right)-\nabla _h \times \left( {A^{lm}\;\zeta \;{\rm {\bf k}}} \right)} |
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| 191 | \right]\;dv} =0 |
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| 192 | \end{equation} |
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| 193 | |
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[9407] | 194 | \begin{equation} \label{eq:dynldf_curl} |
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[707] | 195 | \int_D {{\rm {\bf U}}_h \cdot \left[ {\nabla _h \left( {A^{lm}\;\chi } |
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| 196 | \right)-\nabla _h \times \left( {A^{lm}\;\zeta \;{\rm {\bf k}}} \right)} |
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| 197 | \right]\;dv} \leqslant 0 |
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| 198 | \end{equation} |
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| 199 | |
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[9407] | 200 | \begin{equation} \label{eq:dynldf_curl2} |
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[707] | 201 | \mbox{if}\quad A^{lm}=cste\quad \quad \int_D {\zeta \;{\rm {\bf k}}\cdot |
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| 202 | \nabla \times \left[ {\nabla _h \left( {A^{lm}\;\chi } \right)-\nabla _h |
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| 203 | \times \left( {A^{lm}\;\zeta \;{\rm {\bf k}}} \right)} \right]\;dv} |
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| 204 | \leqslant 0 |
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| 205 | \end{equation} |
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| 206 | |
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[9407] | 207 | \begin{equation} \label{eq:dynldf_div2} |
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[707] | 208 | \mbox{if}\quad A^{lm}=cste\quad \quad \int_D {\chi \;\nabla _h \cdot \left[ |
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| 209 | {\nabla _h \left( {A^{lm}\;\chi } \right)-\nabla _h \times \left( |
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| 210 | {A^{lm}\;\zeta \;{\rm {\bf k}}} \right)} \right]\;dv} \leqslant 0 |
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| 211 | \end{equation} |
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| 212 | |
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| 213 | |
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[10354] | 214 | (II.4.6a) and (II.4.6b) means that the horizontal diffusion of momentum conserve both the potential vorticity and |
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| 215 | the divergence of the flow, while Eqs (II.4.6c) to (II.4.6e) mean that it dissipates the energy, the enstrophy and |
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| 216 | the square of the divergence. |
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| 217 | The two latter properties are only satisfied when the eddy coefficients are horizontally uniform. |
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[707] | 218 | |
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[10354] | 219 | Using (II.1.8) and (II.1.9), and the symmetry or anti-symmetry properties of the mean and difference operators, |
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| 220 | it is shown that the discrete form of the lateral momentum diffusion given in |
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| 221 | {\S}II.2-c satisfies all the integral constraints (II.4.6) (see appendix C). |
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| 222 | In particular, when the eddy coefficients are horizontally uniform, |
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| 223 | a complete separation of vorticity and horizontal divergence fields is ensured, |
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| 224 | so that diffusion (dissipation) of vorticity (enstrophy) does not generate horizontal divergence |
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| 225 | (variance of the horizontal divergence) and \textit{vice versa}. |
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| 226 | When the vertical curl of the horizontal diffusion of momentum (discrete sense) is taken, |
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| 227 | the term associated to the horizontal gradient of the divergence is zero locally. |
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| 228 | When the horizontal divergence of the horizontal diffusion of momentum (discrete sense) is taken, |
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| 229 | the term associated to the vertical curl of the vorticity is zero locally. |
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| 230 | The resulting term conserves $\chi$ and dissipates $\chi^2$ when the eddy coefficient is horizontally uniform. |
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[707] | 231 | |
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| 232 | \textbf{* vertical momentum diffusion term} |
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| 233 | |
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[10354] | 234 | As for the lateral momentum physics, the continuous form of the vertical diffusion of |
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| 235 | momentum satisfies following integral constraints~: |
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[707] | 236 | |
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| 237 | conservation of momentum, dissipation of horizontal kinetic energy |
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| 238 | |
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[9407] | 239 | \begin{equation} \label{eq:dynzdf_dyn} |
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[707] | 240 | \begin{aligned} |
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| 241 | & \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}} \\ |
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| 242 | & \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 \\ |
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| 243 | \end{aligned} |
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| 244 | \end{equation} |
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| 245 | conservation of vorticity, dissipation of enstrophy |
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[9407] | 246 | \begin{equation} \label{eq:dynzdf_vor} |
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[707] | 247 | \begin{aligned} |
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| 248 | & \int_D {\frac{1}{e_3 }{\rm {\bf k}}\cdot \nabla \times \left( {\frac{1}{e_3 |
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| 249 | }\frac{\partial }{\partial k}\left( {\frac{A^{vm}}{e_3 }\frac{\partial {\rm |
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| 250 | {\bf U}}_h }{\partial k}} \right)} \right)\;dv} =0 \\ |
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| 251 | & \int_D {\zeta \,{\rm {\bf k}}\cdot \nabla \times \left( {\frac{1}{e_3 |
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| 252 | }\frac{\partial }{\partial k}\left( {\frac{A^{vm}}{e_3 }\frac{\partial {\rm |
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| 253 | {\bf U}}_h }{\partial k}} \right)} \right)\;dv} \leq 0 \\ |
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| 254 | \end{aligned} |
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| 255 | \end{equation} |
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[10354] | 256 | conservation of horizontal divergence, dissipation of square of the horizontal divergence |
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[9407] | 257 | \begin{equation} \label{eq:dynzdf_div} |
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[707] | 258 | \begin{aligned} |
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| 259 | &\int_D {\nabla \cdot \left( {\frac{1}{e_3 }\frac{\partial }{\partial |
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| 260 | k}\left( {\frac{A^{vm}}{e_3 }\frac{\partial {\rm {\bf U}}_h }{\partial k}} |
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| 261 | \right)} \right)\;dv} =0 \\ |
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| 262 | & \int_D {\chi \;\nabla \cdot \left( {\frac{1}{e_3 }\frac{\partial }{\partial |
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| 263 | k}\left( {\frac{A^{vm}}{e_3 }\frac{\partial {\rm {\bf U}}_h }{\partial k}} |
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| 264 | \right)} \right)\;dv} \leq 0 \\ |
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| 265 | \end{aligned} |
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| 266 | \end{equation} |
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| 267 | |
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[10354] | 268 | In discrete form, all these properties are satisfied in $z$-coordinate (see Appendix C). |
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| 269 | In $s$-coordinates, only first order properties can be demonstrated, |
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| 270 | $i.e.$ the vertical momentum physics conserve momentum, potential vorticity, and horizontal divergence. |
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[707] | 271 | |
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| 272 | % ------------------------------------------------------------------------------------------------------------- |
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| 273 | % Conservation Properties on Tracer Physics |
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| 274 | % ------------------------------------------------------------------------------------------------------------- |
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[9393] | 275 | \subsection{Conservation properties on tracer physics} |
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[9407] | 276 | \label{subsec:Invariant_tra_physics} |
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[707] | 277 | |
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[10354] | 278 | The numerical schemes used for tracer subgridscale physics are written in such a way that |
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| 279 | the heat and salt contents are conserved (equations in flux form, second order centred finite differences). |
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| 280 | As a form flux is used to compute the temperature and salinity, |
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| 281 | the quadratic form of these quantities (i.e. their variance) globally tends to diminish. |
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| 282 | As for the advective term, there is generally no strict conservation of mass even if, |
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| 283 | in practice, the mass is conserved with a very good accuracy. |
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[707] | 284 | |
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| 285 | \textbf{* lateral physics: }conservation of tracer, dissipation of tracer |
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| 286 | variance, i.e. |
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| 287 | |
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[9407] | 288 | \begin{equation} \label{eq:traldf_t_t2} |
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[707] | 289 | \begin{aligned} |
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| 290 | &\int_D \nabla\, \cdot\, \left( A^{lT} \,\Re \,\nabla \,T \right)\;dv = 0 \\ |
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| 291 | &\int_D \,T\, \nabla\, \cdot\, \left( A^{lT} \,\Re \,\nabla \,T \right)\;dv \leq 0 \\ |
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| 292 | \end{aligned} |
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| 293 | \end{equation} |
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| 294 | |
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[10354] | 295 | \textbf{* vertical physics: }conservation of tracer, dissipation of tracer variance, $i.e.$ |
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[707] | 296 | |
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[9407] | 297 | \begin{equation} \label{eq:trazdf_t_t2} |
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[707] | 298 | \begin{aligned} |
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| 299 | & \int_D \frac{1}{e_3 } \frac{\partial }{\partial k}\left( \frac{A^{vT}}{e_3 } \frac{\partial T}{\partial k} \right)\;dv = 0 \\ |
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| 300 | & \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 \\ |
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| 301 | \end{aligned} |
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| 302 | \end{equation} |
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| 303 | |
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[10354] | 304 | Using the symmetry or anti-symmetry properties of the mean and difference operators, |
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| 305 | it is shown that the discrete form of tracer physics given in {\S}~II.2-c satisfies all the integral constraints |
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| 306 | (II.4.8) and (II.4.9) except the dissipation of the square of the tracer when non-geopotential diffusion is used |
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| 307 | (see appendix C). |
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| 308 | A discrete form of the lateral tracer physics can be derived which satisfies these last properties. |
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| 309 | Nevertheless, it requires a horizontal averaging of the vertical component of the lateral physics that |
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| 310 | prevents the use of implicit resolution in the vertical. |
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| 311 | It has not been implemented. |
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[707] | 312 | |
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[6997] | 313 | \end{document} |
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