1 | \documentclass[../main/NEMO_manual]{subfiles} |
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2 | |
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3 | \begin{document} |
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4 | |
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5 | % ================================================================ |
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6 | % Invariant of the Equations |
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7 | % ================================================================ |
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8 | \chapter{Invariants of the Primitive Equations} |
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9 | \label{chap:Invariant} |
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10 | \minitoc |
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11 | |
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12 | The continuous equations of motion have many analytic properties. |
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13 | Many quantities (total mass, energy, enstrophy, etc.) are strictly conserved in the inviscid and unforced limit, |
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14 | while ocean physics conserve the total quantities on which they act (momentum, temperature, salinity) but |
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15 | dissipate their total variance (energy, enstrophy, etc.). |
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16 | Unfortunately, the finite difference form of these equations is not guaranteed to |
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17 | retain all these important properties. |
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18 | In constructing the finite differencing schemes, we wish to ensure that |
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19 | certain integral constraints will be maintained. |
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20 | In particular, it is desirable to construct the finite difference equations so that |
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21 | horizontal kinetic energy and/or potential enstrophy of horizontally non-divergent flow, |
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22 | and variance of temperature and salinity will be conserved in the absence of dissipative effects and forcing. |
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23 | \citet{arakawa_JCP66} has first pointed out the advantage of this approach. |
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24 | He showed that if integral constraints on energy are maintained, |
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25 | the computation will be free of the troublesome "non linear" instability originally pointed out by |
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26 | \citet{phillips_TAMS59}. |
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27 | A consistent formulation of the energetic properties is also extremely important in carrying out |
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28 | long-term numerical simulations for an oceanographic model. |
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29 | Such a formulation avoids systematic errors that accumulate with time \citep{bryan_JCP97}. |
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30 | |
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31 | 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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32 | choose second order non-diffusive scheme for advective terms for both dynamical and tracer equations. |
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33 | At this level of complexity, the resulting schemes are dispersive schemes. |
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34 | Therefore, they require the addition of a diffusive operator to be stable. |
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35 | The alternative is to use diffusive schemes such as upstream or flux corrected schemes. |
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36 | This last option was rejected because we prefer a complete handling of the model diffusion, |
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37 | \ie of the model physics rather than letting the advective scheme produces its own implicit diffusion without |
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38 | 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 positivity of the advective scheme is required. |
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40 | In that case, and in that case only, the advective scheme used for passive tracer is a flux correction scheme |
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41 | \citep{Marti1992?, Levy1996?, Levy1998?}. |
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42 | |
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43 | % ------------------------------------------------------------------------------------------------------------- |
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44 | % Conservation Properties on Ocean Dynamics |
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45 | % ------------------------------------------------------------------------------------------------------------- |
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46 | \section{Conservation properties on ocean dynamics} |
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47 | \label{sec:Invariant_dyn} |
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48 | |
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49 | The non linear term of the momentum equations has been split into a vorticity term, |
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50 | a gradient of horizontal kinetic energy and a vertical advection term. |
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51 | Three schemes are available for the former (see {\S}~II.2) according to the CPP variable defined |
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52 | (default option\textbf{?}or \textbf{key{\_}vorenergy} or \textbf{key{\_}vorcombined} defined). |
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53 | They differ in their conservative properties (energy or enstrophy conserving scheme). |
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54 | The two latter terms preserve the total kinetic energy: |
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55 | the large scale kinetic energy is also preserved in practice. |
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56 | The remaining non-diffusive terms of the momentum equation |
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57 | (namely the hydrostatic and surface pressure gradient terms) also preserve the total kinetic energy and |
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58 | have no effect on the vorticity of the flow. |
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59 | |
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60 | \textbf{* relative, planetary and total vorticity term:} |
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61 | |
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62 | Let us define as either the relative, planetary and total potential vorticity, \ie, ?, and ?, respectively. |
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63 | The continuous formulation of the vorticity term satisfies following integral constraints: |
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64 | \[ |
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65 | % \label{eq:vor_vorticity} |
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66 | \int_D {{\textbf {k}}\cdot \frac{1}{e_3 }\nabla \times \left( {\varsigma |
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67 | \;{\rm {\bf k}}\times {\textbf {U}}_h } \right)\;dv} =0 |
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68 | \] |
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69 | |
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70 | \[ |
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71 | % \label{eq:vor_enstrophy} |
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72 | if\quad \chi =0\quad \quad \int\limits_D {\varsigma \;{\textbf{k}}\cdot |
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73 | \frac{1}{e_3 }\nabla \times \left( {\varsigma {\textbf{k}}\times {\textbf{U}}_h } \right)\;dv} =-\int\limits_D {\frac{1}{2}\varsigma ^2\,\chi \;dv} |
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74 | =0 |
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75 | \] |
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76 | |
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77 | \[ |
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78 | % \label{eq:vor_energy} |
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79 | \int_D {{\textbf{U}}_h \times \left( {\varsigma \;{\textbf{k}}\times {\textbf{U}}_h } \right)\;dv} =0 |
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80 | \] |
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81 | where $dv = e_1\, e_2\, e_3\, di\, dj\, dk$ is the volume element. |
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82 | (II.4.1a) means that $\varsigma $ is conserved. (II.4.1b) is obtained by an integration by part. |
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83 | It means that $\varsigma^2$ is conserved for a horizontally non-divergent flow. |
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84 | (II.4.1c) is even satisfied locally since the vorticity term is orthogonal to the horizontal velocity. |
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85 | It means that the vorticity term has no contribution to the evolution of the total kinetic energy. |
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86 | (II.4.1a) is obviously always satisfied, but (II.4.1b) and (II.4.1c) cannot be satisfied simultaneously with |
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87 | a second order scheme. |
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88 | Using the symmetry or anti-symmetry properties of the operators (Eqs II.1.10 and 11), |
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89 | it can be shown that the scheme (II.2.11) satisfies (II.4.1b) but not (II.4.1c), |
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90 | while scheme (II.2.12) satisfies (II.4.1c) but not (II.4.1b) (see appendix C). |
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91 | Note that the enstrophy conserving scheme on total vorticity has been chosen as the standard discrete form of |
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92 | the vorticity term. |
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93 | |
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94 | \textbf{* Gradient of kinetic energy / vertical advection} |
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95 | |
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96 | In continuous formulation, the gradient of horizontal kinetic energy has no contribution to the evolution of |
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97 | the vorticity as the curl of a gradient is zero. |
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98 | This property is satisfied locally with the discrete form of both the gradient and the curl operator we have made |
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99 | (property (II.1.9)~). |
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100 | Another continuous property is that the change of horizontal kinetic energy due to |
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101 | vertical advection is exactly balanced by the change of horizontal kinetic energy due to |
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102 | the horizontal gradient of horizontal kinetic energy: |
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103 | |
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104 | \begin{equation} \label{eq:keg_zad} |
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105 | \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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106 | {\textbf{U}}_h }{\partial k}\;dv} |
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107 | \end{equation} |
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108 | |
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109 | Using the discrete form given in {\S}II.2-a and the symmetry or anti-symmetry properties of |
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110 | the mean and difference operators, \autoref{eq:keg_zad} is demonstrated in the Appendix C. |
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111 | The main point here is that satisfying \autoref{eq:keg_zad} links the choice of the discrete forms of |
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112 | the vertical advection and of the horizontal gradient of horizontal kinetic energy. |
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113 | Choosing one imposes the other. |
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114 | The discrete form of the vertical advection given in {\S}II.2-a is a direct consequence of |
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115 | formulating the horizontal kinetic energy as $1/2 \left( \overline{u^2}^i + \overline{v^2}^j \right) $ in |
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116 | the gradient term. |
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117 | |
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118 | \textbf{* hydrostatic pressure gradient term} |
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119 | |
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120 | In continuous formulation, a pressure gradient has no contribution to the evolution of the vorticity as |
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121 | the curl of a gradient is zero. |
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122 | This properties is satisfied locally with the choice of discretization we have made (property (II.1.9)~). |
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123 | In addition, when the equation of state is linear |
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124 | (\ie when an advective-diffusive equation for density can be derived from those of temperature and salinity) |
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125 | the change of horizontal kinetic energy due to the work of pressure forces is balanced by the change of |
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126 | potential energy due to buoyancy forces: |
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127 | |
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128 | \[ |
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129 | % \label{eq:hpg_pe} |
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130 | \int_D {-\frac{1}{\rho_o }\left. {\nabla p^h} \right|_z \cdot {\textbf {U}}_h \;dv} \;=\;\int_D {\nabla .\left( {\rho \,{\textbf{U}}} \right)\;g\;z\;\;dv} |
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131 | \] |
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132 | |
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133 | Using the discrete form given in {\S}~II.2-a and the symmetry or anti-symmetry properties of |
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134 | the mean and difference operators, (II.4.3) is demonstrated in the Appendix C. |
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135 | The main point here is that satisfying (II.4.3) strongly constraints the discrete expression of the depth of |
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136 | $T$-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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137 | is defined as the sum the vertical scale factors at $w$-points starting from the surface. |
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138 | |
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139 | \textbf{* surface pressure gradient term} |
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140 | |
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141 | In continuous formulation, the surface pressure gradient has no contribution to the evolution of vorticity. |
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142 | This properties is trivially satisfied locally as (II.2.3) |
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143 | (the equation verified by $\psi$ has been derived from the discrete formulation of the momentum equations, |
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144 | vertical sum and curl). |
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145 | Nevertheless, the $\psi$-equation is solved numerically by an iterative solver (see {\S}~III.5), |
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146 | thus the property is only satisfied with the accuracy required on the solver. |
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147 | In addition, with the rigid-lid approximation, the change of horizontal kinetic energy due to the work of |
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148 | surface pressure forces is exactly zero: |
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149 | \[ |
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150 | % \label{eq:spg} |
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151 | \int_D {-\frac{1}{\rho_o }\nabla _h } \left( {p_s } \right)\cdot {\textbf{U}}_h \;dv=0 |
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152 | \] |
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153 | |
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154 | (II.4.4) is satisfied in discrete form only if |
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155 | the discrete barotropic streamfunction time evolution equation is given by (II.2.3) (see appendix C). |
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156 | This shows that (II.2.3) is the only way to compute the streamfunction, |
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157 | otherwise there is no guarantee that the surface pressure force work vanishes. |
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158 | |
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159 | % ------------------------------------------------------------------------------------------------------------- |
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160 | % Conservation Properties on Ocean Thermodynamics |
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161 | % ------------------------------------------------------------------------------------------------------------- |
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162 | \section{Conservation properties on ocean thermodynamics} |
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163 | \label{sec:Invariant_tra} |
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164 | |
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165 | In continuous formulation, the advective terms of the tracer equations conserve the tracer content and |
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166 | the quadratic form of the tracer, \ie |
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167 | \[ |
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168 | % \label{eq:tra_tra2} |
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169 | \int_D {\nabla .\left( {T\;{\textbf{U}}} \right)\;dv} =0 |
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170 | \;\text{and} |
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171 | \int_D {T\;\nabla .\left( {T\;{\textbf{U}}} \right)\;dv} =0 |
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172 | \] |
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173 | |
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174 | The numerical scheme used ({\S}II.2-b) (equations in flux form, second order centred finite differences) satisfies |
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175 | (II.4.5) (see appendix C). |
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176 | Note that in both continuous and discrete formulations, there is generally no strict conservation of mass, |
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177 | since the equation of state is non linear with respect to $T$ and $S$. |
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178 | In practice, the mass is conserved with a very good accuracy. |
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179 | |
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180 | % ------------------------------------------------------------------------------------------------------------- |
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181 | % Conservation Properties on Momentum Physics |
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182 | % ------------------------------------------------------------------------------------------------------------- |
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183 | \subsection{Conservation properties on momentum physics} |
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184 | \label{subsec:Invariant_dyn_physics} |
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185 | |
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186 | \textbf{* lateral momentum diffusion term} |
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187 | |
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188 | The continuous formulation of the horizontal diffusion of momentum satisfies the following integral constraints~: |
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189 | \[ |
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190 | % \label{eq:dynldf_dyn} |
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191 | \int\limits_D {\frac{1}{e_3 }{\rm {\bf k}}\cdot \nabla \times \left[ {\nabla |
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192 | _h \left( {A^{lm}\;\chi } \right)-\nabla _h \times \left( {A^{lm}\;\zeta |
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193 | \;{\rm {\bf k}}} \right)} \right]\;dv} =0 |
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194 | \] |
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195 | |
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196 | \[ |
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197 | % \label{eq:dynldf_div} |
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198 | \int\limits_D {\nabla _h \cdot \left[ {\nabla _h \left( {A^{lm}\;\chi } |
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199 | \right)-\nabla _h \times \left( {A^{lm}\;\zeta \;{\rm {\bf k}}} \right)} |
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200 | \right]\;dv} =0 |
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201 | \] |
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202 | |
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203 | \[ |
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204 | % \label{eq:dynldf_curl} |
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205 | \int_D {{\rm {\bf U}}_h \cdot \left[ {\nabla _h \left( {A^{lm}\;\chi } |
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206 | \right)-\nabla _h \times \left( {A^{lm}\;\zeta \;{\rm {\bf k}}} \right)} |
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207 | \right]\;dv} \leqslant 0 |
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208 | \] |
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209 | |
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210 | \[ |
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211 | % \label{eq:dynldf_curl2} |
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212 | \mbox{if}\quad A^{lm}=cste\quad \quad \int_D {\zeta \;{\rm {\bf k}}\cdot |
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213 | \nabla \times \left[ {\nabla _h \left( {A^{lm}\;\chi } \right)-\nabla _h |
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214 | \times \left( {A^{lm}\;\zeta \;{\rm {\bf k}}} \right)} \right]\;dv} |
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215 | \leqslant 0 |
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216 | \] |
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217 | |
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218 | \[ |
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219 | % \label{eq:dynldf_div2} |
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220 | \mbox{if}\quad A^{lm}=cste\quad \quad \int_D {\chi \;\nabla _h \cdot \left[ |
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221 | {\nabla _h \left( {A^{lm}\;\chi } \right)-\nabla _h \times \left( |
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222 | {A^{lm}\;\zeta \;{\rm {\bf k}}} \right)} \right]\;dv} \leqslant 0 |
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223 | \] |
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224 | |
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225 | |
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226 | (II.4.6a) and (II.4.6b) means that the horizontal diffusion of momentum conserve both the potential vorticity and |
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227 | 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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228 | the square of the divergence. |
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229 | The two latter properties are only satisfied when the eddy coefficients are horizontally uniform. |
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230 | |
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231 | 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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232 | it is shown that the discrete form of the lateral momentum diffusion given in |
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233 | {\S}II.2-c satisfies all the integral constraints (II.4.6) (see appendix C). |
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234 | In particular, when the eddy coefficients are horizontally uniform, |
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235 | a complete separation of vorticity and horizontal divergence fields is ensured, |
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236 | so that diffusion (dissipation) of vorticity (enstrophy) does not generate horizontal divergence |
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237 | (variance of the horizontal divergence) and \textit{vice versa}. |
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238 | When the vertical curl of the horizontal diffusion of momentum (discrete sense) is taken, |
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239 | the term associated to the horizontal gradient of the divergence is zero locally. |
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240 | When the horizontal divergence of the horizontal diffusion of momentum (discrete sense) is taken, |
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241 | the term associated to the vertical curl of the vorticity is zero locally. |
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242 | The resulting term conserves $\chi$ and dissipates $\chi^2$ when the eddy coefficient is horizontally uniform. |
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243 | |
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244 | \textbf{* vertical momentum diffusion term} |
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245 | |
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246 | As for the lateral momentum physics, the continuous form of the vertical diffusion of |
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247 | momentum satisfies following integral constraints~: |
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248 | |
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249 | conservation of momentum, dissipation of horizontal kinetic energy |
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250 | |
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251 | \[ |
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252 | % \label{eq:dynzdf_dyn} |
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253 | \begin{aligned} |
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254 | & \int_D {\frac{1}{e_3 }} \frac{\partial }{\partial k}\left( \frac{A^{vm}}{e_3 }\frac{\partial {\textbf{U}}_h }{\partial k} \right) \;dv = \overrightarrow{\textbf{0}} \\ |
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255 | & \int_D \textbf{U}_h \cdot \frac{1}{e_3} \frac{\partial}{\partial k} \left( {\frac{A^{vm}}{e_3 }}{\frac{\partial \textbf{U}_h }{\partial k}} \right) \;dv \leq 0 \\ |
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256 | \end{aligned} |
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257 | \] |
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258 | conservation of vorticity, dissipation of enstrophy |
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259 | \[ |
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260 | % \label{eq:dynzdf_vor} |
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261 | \begin{aligned} |
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262 | & \int_D {\frac{1}{e_3 }{\rm {\bf k}}\cdot \nabla \times \left( {\frac{1}{e_3 |
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263 | }\frac{\partial }{\partial k}\left( {\frac{A^{vm}}{e_3 }\frac{\partial {\rm |
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264 | {\bf U}}_h }{\partial k}} \right)} \right)\;dv} =0 \\ |
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265 | & \int_D {\zeta \,{\rm {\bf k}}\cdot \nabla \times \left( {\frac{1}{e_3 |
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266 | }\frac{\partial }{\partial k}\left( {\frac{A^{vm}}{e_3 }\frac{\partial {\rm |
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267 | {\bf U}}_h }{\partial k}} \right)} \right)\;dv} \leq 0 \\ |
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268 | \end{aligned} |
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269 | \] |
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270 | conservation of horizontal divergence, dissipation of square of the horizontal divergence |
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271 | \[ |
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272 | % \label{eq:dynzdf_div} |
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273 | \begin{aligned} |
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274 | &\int_D {\nabla \cdot \left( {\frac{1}{e_3 }\frac{\partial }{\partial |
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275 | k}\left( {\frac{A^{vm}}{e_3 }\frac{\partial {\rm {\bf U}}_h }{\partial k}} |
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276 | \right)} \right)\;dv} =0 \\ |
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277 | & \int_D {\chi \;\nabla \cdot \left( {\frac{1}{e_3 }\frac{\partial }{\partial |
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278 | k}\left( {\frac{A^{vm}}{e_3 }\frac{\partial {\rm {\bf U}}_h }{\partial k}} |
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279 | \right)} \right)\;dv} \leq 0 \\ |
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280 | \end{aligned} |
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281 | \] |
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282 | |
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283 | In discrete form, all these properties are satisfied in $z$-coordinate (see Appendix C). |
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284 | In $s$-coordinates, only first order properties can be demonstrated, |
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285 | \ie the vertical momentum physics conserve momentum, potential vorticity, and horizontal divergence. |
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286 | |
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287 | % ------------------------------------------------------------------------------------------------------------- |
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288 | % Conservation Properties on Tracer Physics |
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289 | % ------------------------------------------------------------------------------------------------------------- |
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290 | \subsection{Conservation properties on tracer physics} |
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291 | \label{subsec:Invariant_tra_physics} |
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292 | |
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293 | The numerical schemes used for tracer subgridscale physics are written in such a way that |
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294 | the heat and salt contents are conserved (equations in flux form, second order centred finite differences). |
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295 | As a form flux is used to compute the temperature and salinity, |
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296 | the quadratic form of these quantities (\ie their variance) globally tends to diminish. |
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297 | As for the advective term, there is generally no strict conservation of mass even if, |
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298 | in practice, the mass is conserved with a very good accuracy. |
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299 | |
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300 | \textbf{* lateral physics: }conservation of tracer, dissipation of tracer |
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301 | variance, i.e. |
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302 | |
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303 | \[ |
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304 | % \label{eq:traldf_t_t2} |
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305 | \begin{aligned} |
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306 | &\int_D \nabla\, \cdot\, \left( A^{lT} \,\Re \,\nabla \,T \right)\;dv = 0 \\ |
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307 | &\int_D \,T\, \nabla\, \cdot\, \left( A^{lT} \,\Re \,\nabla \,T \right)\;dv \leq 0 \\ |
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308 | \end{aligned} |
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309 | \] |
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310 | |
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311 | \textbf{* vertical physics: }conservation of tracer, dissipation of tracer variance, \ie |
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312 | |
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313 | \[ |
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314 | % \label{eq:trazdf_t_t2} |
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315 | \begin{aligned} |
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316 | & \int_D \frac{1}{e_3 } \frac{\partial }{\partial k}\left( \frac{A^{vT}}{e_3 } \frac{\partial T}{\partial k} \right)\;dv = 0 \\ |
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317 | & \int_D \,T \frac{1}{e_3 } \frac{\partial }{\partial k}\left( \frac{A^{vT}}{e_3 } \frac{\partial T}{\partial k} \right)\;dv \leq 0 \\ |
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318 | \end{aligned} |
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319 | \] |
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320 | |
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321 | Using the symmetry or anti-symmetry properties of the mean and difference operators, |
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322 | 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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323 | (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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324 | (see appendix C). |
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325 | A discrete form of the lateral tracer physics can be derived which satisfies these last properties. |
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326 | Nevertheless, it requires a horizontal averaging of the vertical component of the lateral physics that |
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327 | prevents the use of implicit resolution in the vertical. |
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328 | It has not been implemented. |
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329 | |
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330 | \biblio |
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331 | |
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332 | \pindex |
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333 | |
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334 | \end{document} |
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