1 | |
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2 | % ================================================================ |
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3 | % Invariant of the Equations |
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4 | % ================================================================ |
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5 | \chapter{Annex --- Invariants of the Primitive Equations} |
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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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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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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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