[707] | 1 | |
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| 2 | % ================================================================ |
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| 3 | % Invariant of the Equations |
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| 4 | % ================================================================ |
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[1223] | 5 | \chapter{Invariants of the Primitive Equations} |
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[707] | 6 | \label{Invariant} |
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| 7 | \minitoc |
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| 8 | |
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| 9 | The continuous equations of motion have many analytic properties. Many |
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| 10 | quantities (total mass, energy, enstrophy, etc.) are strictly conserved in |
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| 11 | the inviscid and unforced limit, while ocean physics conserve the total |
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| 12 | quantities on which they act (momentum, temperature, salinity) but dissipate |
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| 13 | their total variance (energy, enstrophy, etc.). Unfortunately, the finite |
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| 14 | difference form of these equations is not guaranteed to retain all these |
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| 15 | important properties. In constructing the finite differencing schemes, we |
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| 16 | wish to ensure that certain integral constraints will be maintained. In |
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| 17 | particular, it is desirable to construct the finite difference equations so |
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| 18 | that horizontal kinetic energy and/or potential enstrophy of horizontally |
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| 19 | non-divergent flow, and variance of temperature and salinity will be |
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| 20 | conserved in the absence of dissipative effects and forcing. \citet{Arakawa1966} |
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| 21 | has first pointed out the advantage of this approach. He showed that if |
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| 22 | integral constraints on energy are maintained, the computation will be free |
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| 23 | of the troublesome "non linear" instability originally pointed out by |
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| 24 | \citet{Phillips1959}. A consistent formulation of the energetic properties is |
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| 25 | also extremely important in carrying out long-term numerical simulations for |
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| 26 | an oceanographic model. Such a formulation avoids systematic errors that |
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| 27 | accumulate with time \citep{Bryan1997}. |
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| 28 | |
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| 29 | The general philosophy of OPA which has led to the discrete formulation |
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| 30 | presented in {\S}II.2 and II.3 is to choose second order non-diffusive |
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| 31 | scheme for advective terms for both dynamical and tracer equations. At this |
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| 32 | 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 |
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| 35 | corrected schemes. This last option was rejected because we prefer a |
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| 36 | complete handling of the model diffusion, i.e. of the model physics rather |
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| 37 | than letting the advective scheme produces its own implicit diffusion |
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| 38 | without controlling the space and time structure of this implicit diffusion. |
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| 39 | Note that in some very specific cases as passive tracer studies, the |
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| 40 | positivity of the advective scheme is required. In that case, and in that |
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| 41 | case only, the advective scheme used for passive tracer is a flux correction |
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| 42 | scheme \citep{Marti1992, Levy1996, Levy1998}. |
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| 43 | |
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| 44 | % ------------------------------------------------------------------------------------------------------------- |
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| 45 | % Conservation Properties on Ocean Dynamics |
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| 46 | % ------------------------------------------------------------------------------------------------------------- |
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| 47 | \section{Conservation Properties on Ocean Dynamics} |
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| 48 | \label{Invariant_dyn} |
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| 49 | |
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| 50 | The non linear term of the momentum equations has been split into a |
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| 51 | vorticity term, a gradient of horizontal kinetic energy and a vertical |
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| 52 | advection term. Three schemes are available for the former (see {\S}~II.2) |
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| 53 | according to the CPP variable defined (default option\textbf{ |
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| 54 | }or \textbf{key{\_}vorenergy } or \textbf{key{\_}vorcombined |
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| 55 | } defined). They differ in their conservative |
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| 56 | properties (energy or enstrophy conserving scheme). The two latter terms |
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| 57 | preserve the total kinetic energy: the large scale kinetic energy is also |
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| 58 | preserved in practice. The remaining non-diffusive terms of the momentum |
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| 59 | equation (namely the hydrostatic and surface pressure gradient terms) also |
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| 60 | preserve the total kinetic energy and have no effect on the vorticity of the |
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| 61 | flow. |
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| 62 | |
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| 63 | \textbf{* relative, planetary and total vorticity term:} |
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| 64 | |
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| 65 | Let us define as either the relative, planetary and total potential |
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| 66 | vorticity, i.e. , , and , respectively. The continuous formulation of the |
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| 67 | vorticity term satisfies following integral constraints: |
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| 68 | \begin{equation} \label{Eq_vor_vorticity} |
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| 69 | \int_D {{\textbf {k}}\cdot \frac{1}{e_3 }\nabla \times \left( {\varsigma |
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| 70 | \;{\rm {\bf k}}\times {\textbf {U}}_h } \right)\;dv} =0 |
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| 71 | \end{equation} |
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| 72 | |
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| 73 | \begin{equation} \label{Eq_vor_enstrophy} |
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| 74 | if\quad \chi =0\quad \quad \int\limits_D {\varsigma \;{\textbf{k}}\cdot |
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| 75 | \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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| 76 | =0 |
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| 77 | \end{equation} |
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| 78 | |
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| 79 | \begin{equation} \label{Eq_vor_energy} |
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| 80 | \int_D {{\textbf{U}}_h \times \left( {\varsigma \;{\textbf{k}}\times {\textbf{U}}_h } \right)\;dv} =0 |
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| 81 | \end{equation} |
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| 82 | where $dv = e_1\, e_2\, e_3\, di\, dj\, dk$ is the volume element. |
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[996] | 83 | (II.4.1a) means that $\varsigma $ is conserved. (II.4.1b) is obtained by an |
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| 84 | integration by part. It means that $\varsigma^2$ is conserved for a horizontally |
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| 85 | non-divergent flow. |
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[707] | 86 | (II.4.1c) is even satisfied locally since the vorticity term is orthogonal |
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| 87 | to the horizontal velocity. It means that the vorticity term has no |
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| 88 | contribution to the evolution of the total kinetic energy. (II.4.1a) is |
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| 89 | obviously always satisfied, but (II.4.1b) and (II.4.1c) cannot be satisfied |
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| 90 | simultaneously with a second order scheme. Using the symmetry or |
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| 91 | anti-symmetry properties of the operators (Eqs II.1.10 and 11), it can be |
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| 92 | shown that the scheme (II.2.11) satisfies (II.4.1b) but not (II.4.1c), while |
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| 93 | scheme (II.2.12) satisfies (II.4.1c) but not (II.4.1b) (see appendix C). |
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| 94 | Note that the enstrophy conserving scheme on total vorticity has been chosen |
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| 95 | as the standard discrete form of the vorticity term. |
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| 96 | |
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| 97 | \textbf{* Gradient of kinetic energy / vertical advection} |
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| 98 | |
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| 99 | In continuous formulation, the gradient of horizontal kinetic energy has no |
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| 100 | contribution to the evolution of the vorticity as the curl of a gradient is |
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| 101 | zero. This property is satisfied locally with the discrete form of both the |
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| 102 | gradient and the curl operator we have made (property (II.1.9)~). Another |
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| 103 | continuous property is that the change of horizontal kinetic energy due to |
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| 104 | vertical advection is exactly balanced by the change of horizontal kinetic |
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| 105 | energy due to the horizontal gradient of horizontal kinetic energy: |
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| 106 | |
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| 107 | \begin{equation} \label{Eq_keg_zad} |
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| 108 | \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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| 109 | {\textbf{U}}_h }{\partial k}\;dv} |
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| 110 | \end{equation} |
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| 111 | |
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| 112 | Using the discrete form given in {\S}II.2-a and the symmetry or |
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| 113 | anti-symmetry properties of the mean and difference operators, \eqref{Eq_keg_zad} is |
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| 114 | demonstrated in the Appendix C. The main point here is that satisfying |
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| 115 | \eqref{Eq_keg_zad} links the choice of the discrete forms of the vertical advection |
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| 116 | and of the horizontal gradient of horizontal kinetic energy. Choosing one |
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| 117 | imposes the other. The discrete form of the vertical advection given in |
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| 118 | {\S}II.2-a is a direct consequence of formulating the horizontal kinetic |
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| 119 | energy as $1/2 \left( \overline{u^2}^i + \overline{v^2}^j \right) $ in the gradient term. |
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| 120 | |
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| 121 | \textbf{* hydrostatic pressure gradient term} |
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| 122 | |
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| 123 | In continuous formulation, a pressure gradient has no contribution to the |
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| 124 | evolution of the vorticity as the curl of a gradient is zero. This |
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| 125 | properties is satisfied locally with the choice of discretization we have |
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| 126 | made (property (II.1.9)~). In addition, when the equation of state is linear |
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| 127 | (i.e. when an advective-diffusive equation for density can be derived from |
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| 128 | those of temperature and salinity) the change of horizontal kinetic energy |
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| 129 | due to the work of pressure forces is balanced by the change of potential |
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| 130 | energy due to buoyancy forces: |
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| 131 | |
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| 132 | \begin{equation} \label{Eq_hpg_pe} |
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| 133 | \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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| 134 | \end{equation} |
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| 135 | |
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| 136 | Using the discrete form given in {\S}~II.2-a and the symmetry or |
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| 137 | anti-symmetry properties of the mean and difference operators, (II.4.3) is |
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| 138 | demonstrated in the Appendix C. The main point here is that satisfying |
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| 139 | (II.4.3) strongly constraints the discrete expression of the depth of |
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| 140 | $T$-points and of the term added to the pressure gradient in $s-$coordinates: the |
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| 141 | depth of a $T$-point, $z_T$, is defined as the sum the vertical scale |
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| 142 | factors at $w$-points starting from the surface. |
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| 143 | |
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| 144 | \textbf{* surface pressure gradient term} |
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| 145 | |
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| 146 | In continuous formulation, the surface pressure gradient has no contribution |
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| 147 | to the evolution of vorticity. This properties is trivially satisfied |
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| 148 | locally as (II.2.3) (the equation verified by $\psi$ has been |
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| 149 | derived from the discrete formulation of the momentum equations, vertical |
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| 150 | sum and curl. Nevertheless, the $\psi$-equation is solved numerically by an |
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| 151 | iterative solver (see {\S}~III.5), thus the property is only satisfied with |
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| 152 | the accuracy required on the solver. In addition, with the rigid-lid |
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| 153 | approximation, the change of horizontal kinetic energy due to the work of |
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| 154 | surface pressure forces is exactly zero: |
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| 155 | \begin{equation} \label{Eq_spg} |
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| 156 | \int_D {-\frac{1}{\rho _o }\nabla _h } \left( {p_s } \right)\cdot {\textbf{U}}_h \;dv=0 |
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| 157 | \end{equation} |
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| 158 | |
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| 159 | (II.4.4) is satisfied in discrete form only if the discrete barotropic |
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| 160 | streamfunction time evolution equation is given by (II.2.3) (see appendix |
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| 161 | C). This shows that (II.2.3) is the only way to compute the streamfunction, |
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| 162 | otherwise there is no guarantee that the surface pressure force work |
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| 163 | vanishes. |
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| 164 | |
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| 165 | % ------------------------------------------------------------------------------------------------------------- |
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| 166 | % Conservation Properties on Ocean Thermodynamics |
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| 167 | % ------------------------------------------------------------------------------------------------------------- |
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| 168 | \section{Conservation Properties on Ocean Thermodynamics} |
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| 169 | \label{Invariant_tra} |
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| 170 | |
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| 171 | In continuous formulation, the advective terms of the tracer equations |
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| 172 | conserve the tracer content and the quadratic form of the tracer, i.e. |
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| 173 | \begin{equation} \label{Eq_tra_tra2} |
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| 174 | \int_D {\nabla .\left( {T\;{\textbf{U}}} \right)\;dv} =0 |
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| 175 | \;\text{and} |
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| 176 | \int_D {T\;\nabla .\left( {T\;{\textbf{U}}} \right)\;dv} =0 |
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| 177 | \end{equation} |
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| 178 | |
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| 179 | The numerical scheme used ({\S}II.2-b) (equations in flux form, second order |
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| 180 | centred finite differences) satisfies (II.4.5) (see appendix C). Note that |
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| 181 | in both continuous and discrete formulations, there is generally no strict |
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| 182 | conservation of mass, since the equation of state is non linear with respect |
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| 183 | to $T$ and $S$. In practice, the mass is conserved with a very good accuracy. |
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| 184 | |
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| 185 | % ------------------------------------------------------------------------------------------------------------- |
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| 186 | % Conservation Properties on Momentum Physics |
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| 187 | % ------------------------------------------------------------------------------------------------------------- |
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| 188 | \subsection{Conservation Properties on Momentum Physics} |
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| 189 | \label{Invariant_dyn_physics} |
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| 190 | |
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| 191 | \textbf{* lateral momentum diffusion term} |
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| 192 | |
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| 193 | The continuous formulation of the horizontal diffusion of momentum satisfies |
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| 194 | the following integral constraints~: |
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| 195 | \begin{equation} \label{Eq_dynldf_dyn} |
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| 196 | \int\limits_D {\frac{1}{e_3 }{\rm {\bf k}}\cdot \nabla \times \left[ {\nabla |
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| 197 | _h \left( {A^{lm}\;\chi } \right)-\nabla _h \times \left( {A^{lm}\;\zeta |
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| 198 | \;{\rm {\bf k}}} \right)} \right]\;dv} =0 |
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| 199 | \end{equation} |
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| 200 | |
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| 201 | \begin{equation} \label{Eq_dynldf_div} |
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| 202 | \int\limits_D {\nabla _h \cdot \left[ {\nabla _h \left( {A^{lm}\;\chi } |
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| 203 | \right)-\nabla _h \times \left( {A^{lm}\;\zeta \;{\rm {\bf k}}} \right)} |
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| 204 | \right]\;dv} =0 |
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| 205 | \end{equation} |
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| 206 | |
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| 207 | \begin{equation} \label{Eq_dynldf_curl} |
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| 208 | \int_D {{\rm {\bf U}}_h \cdot \left[ {\nabla _h \left( {A^{lm}\;\chi } |
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| 209 | \right)-\nabla _h \times \left( {A^{lm}\;\zeta \;{\rm {\bf k}}} \right)} |
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| 210 | \right]\;dv} \leqslant 0 |
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| 211 | \end{equation} |
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| 212 | |
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| 213 | \begin{equation} \label{Eq_dynldf_curl2} |
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| 214 | \mbox{if}\quad A^{lm}=cste\quad \quad \int_D {\zeta \;{\rm {\bf k}}\cdot |
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| 215 | \nabla \times \left[ {\nabla _h \left( {A^{lm}\;\chi } \right)-\nabla _h |
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| 216 | \times \left( {A^{lm}\;\zeta \;{\rm {\bf k}}} \right)} \right]\;dv} |
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| 217 | \leqslant 0 |
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| 218 | \end{equation} |
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| 219 | |
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| 220 | \begin{equation} \label{Eq_dynldf_div2} |
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| 221 | \mbox{if}\quad A^{lm}=cste\quad \quad \int_D {\chi \;\nabla _h \cdot \left[ |
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| 222 | {\nabla _h \left( {A^{lm}\;\chi } \right)-\nabla _h \times \left( |
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| 223 | {A^{lm}\;\zeta \;{\rm {\bf k}}} \right)} \right]\;dv} \leqslant 0 |
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| 224 | \end{equation} |
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| 225 | |
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| 226 | |
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| 227 | (II.4.6a) and (II.4.6b) means that the horizontal diffusion of momentum |
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| 228 | conserve both the potential vorticity and the divergence of the flow, while |
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| 229 | Eqs (II.4.6c) to (II.4.6e) mean that it dissipates the energy, the enstrophy |
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| 230 | and the square of the divergence. The two latter properties are only |
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| 231 | satisfied when the eddy coefficients are horizontally uniform. |
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| 232 | |
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| 233 | Using (II.1.8) and (II.1.9), and the symmetry or anti-symmetry properties of |
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| 234 | the mean and difference operators, it is shown that the discrete form of the |
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| 235 | lateral momentum diffusion given in {\S}II.2-c satisfies all the integral |
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| 236 | constraints (II.4.6) (see appendix C). In particular, when the eddy |
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| 237 | coefficients are horizontally uniform, a complete separation of vorticity |
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| 238 | and horizontal divergence fields is ensured, so that diffusion (dissipation) |
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| 239 | of vorticity (enstrophy) does not generate horizontal divergence (variance |
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| 240 | of the horizontal divergence) and \textit{vice versa}. When the vertical curl of the horizontal |
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| 241 | diffusion of momentum (discrete sense) is taken, the term associated to the |
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| 242 | horizontal gradient of the divergence is zero locally. When the horizontal |
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| 243 | divergence of the horizontal diffusion of momentum (discrete sense) is |
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| 244 | taken, the term associated to the vertical curl of the vorticity is zero |
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| 245 | locally. The resulting term conserves $\chi$ and dissipates |
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| 246 | $\chi^2$ when the |
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| 247 | eddy coefficient is horizontally uniform. |
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| 248 | |
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| 249 | \textbf{* vertical momentum diffusion term} |
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| 250 | |
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| 251 | As for the lateral momentum physics, the continuous form of the vertical |
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| 252 | diffusion of momentum satisfies following integral constraints~: |
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| 253 | |
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| 254 | conservation of momentum, dissipation of horizontal kinetic energy |
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| 255 | |
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| 256 | \begin{equation} \label{Eq_dynzdf_dyn} |
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| 257 | \begin{aligned} |
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| 258 | & \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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| 259 | & \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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| 260 | \end{aligned} |
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| 261 | \end{equation} |
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| 262 | conservation of vorticity, dissipation of enstrophy |
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| 263 | \begin{equation} \label{Eq_dynzdf_vor} |
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| 264 | \begin{aligned} |
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| 265 | & \int_D {\frac{1}{e_3 }{\rm {\bf k}}\cdot \nabla \times \left( {\frac{1}{e_3 |
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| 266 | }\frac{\partial }{\partial k}\left( {\frac{A^{vm}}{e_3 }\frac{\partial {\rm |
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| 267 | {\bf U}}_h }{\partial k}} \right)} \right)\;dv} =0 \\ |
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| 268 | & \int_D {\zeta \,{\rm {\bf k}}\cdot \nabla \times \left( {\frac{1}{e_3 |
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| 269 | }\frac{\partial }{\partial k}\left( {\frac{A^{vm}}{e_3 }\frac{\partial {\rm |
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| 270 | {\bf U}}_h }{\partial k}} \right)} \right)\;dv} \leq 0 \\ |
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| 271 | \end{aligned} |
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| 272 | \end{equation} |
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| 273 | conservation of horizontal divergence, dissipation of square of the |
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| 274 | horizontal divergence |
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| 275 | \begin{equation} \label{Eq_dynzdf_div} |
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| 276 | \begin{aligned} |
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| 277 | &\int_D {\nabla \cdot \left( {\frac{1}{e_3 }\frac{\partial }{\partial |
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| 278 | k}\left( {\frac{A^{vm}}{e_3 }\frac{\partial {\rm {\bf U}}_h }{\partial k}} |
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| 279 | \right)} \right)\;dv} =0 \\ |
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| 280 | & \int_D {\chi \;\nabla \cdot \left( {\frac{1}{e_3 }\frac{\partial }{\partial |
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| 281 | k}\left( {\frac{A^{vm}}{e_3 }\frac{\partial {\rm {\bf U}}_h }{\partial k}} |
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| 282 | \right)} \right)\;dv} \leq 0 \\ |
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| 283 | \end{aligned} |
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| 284 | \end{equation} |
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| 285 | |
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| 286 | In discrete form, all these properties are satisfied in $z$-coordinate (see |
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| 287 | Appendix C). In $s$-coordinates, only first order properties can be |
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| 288 | demonstrated, i.e. the vertical momentum physics conserve momentum, |
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| 289 | potential vorticity, and horizontal divergence. |
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| 290 | |
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| 291 | % ------------------------------------------------------------------------------------------------------------- |
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| 292 | % Conservation Properties on Tracer Physics |
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| 293 | % ------------------------------------------------------------------------------------------------------------- |
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| 294 | \subsection{Conservation Properties on Tracer Physics} |
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| 295 | \label{Invariant_tra_physics} |
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| 296 | |
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| 297 | The numerical schemes used for tracer subgridscale physics are written in |
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| 298 | such a way that the heat and salt contents are conserved (equations in flux |
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| 299 | form, second order centred finite differences). As a form flux is used to |
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| 300 | compute the temperature and salinity, the quadratic form of these quantities |
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| 301 | (i.e. their variance) globally tends to diminish. As for the advective term, |
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| 302 | there is generally no strict conservation of mass even if, in practice, the |
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| 303 | mass is conserved with a very good accuracy. |
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| 304 | |
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| 305 | \textbf{* lateral physics: }conservation of tracer, dissipation of tracer |
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| 306 | variance, i.e. |
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| 307 | |
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| 308 | \begin{equation} \label{Eq_traldf_t_t2} |
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| 309 | \begin{aligned} |
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| 310 | &\int_D \nabla\, \cdot\, \left( A^{lT} \,\Re \,\nabla \,T \right)\;dv = 0 \\ |
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| 311 | &\int_D \,T\, \nabla\, \cdot\, \left( A^{lT} \,\Re \,\nabla \,T \right)\;dv \leq 0 \\ |
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| 312 | \end{aligned} |
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| 313 | \end{equation} |
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| 314 | |
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| 315 | \textbf{* vertical physics: }conservation of tracer, dissipation of tracer |
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| 316 | variance, i.e. |
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| 317 | |
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| 318 | \begin{equation} \label{Eq_trazdf_t_t2} |
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| 319 | \begin{aligned} |
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| 320 | & \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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| 321 | & \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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| 322 | \end{aligned} |
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| 323 | \end{equation} |
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| 324 | |
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| 325 | Using the symmetry or anti-symmetry properties of the mean and difference |
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| 326 | operators, it is shown that the discrete form of tracer physics given in |
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| 327 | {\S}~II.2-c satisfies all the integral constraints (II.4.8) and (II.4.9) |
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| 328 | except the dissipation of the square of the tracer when non-geopotential |
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| 329 | diffusion is used (see appendix C). A discrete form of the lateral tracer |
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| 330 | physics can be derived which satisfies these last properties. Nevertheless, |
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| 331 | it requires a horizontal averaging of the vertical component of the lateral |
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| 332 | physics that prevents the use of implicit resolution in the vertical. It has |
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| 333 | not been implemented. |
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| 334 | |
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