1 | % ================================================================ |
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2 | % Chapter 1 Ñ Model Basics |
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3 | % ================================================================ |
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4 | |
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5 | \chapter{Model basics} |
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6 | \label{PE} |
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7 | \minitoc |
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8 | |
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9 | |
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10 | % ================================================================ |
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11 | % Primitive Equations |
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12 | % ================================================================ |
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13 | \section{Primitive Equations} |
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14 | \label{PE_PE} |
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15 | |
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16 | % ------------------------------------------------------------------------------------------------------------- |
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17 | % Vector Invariant Formulation |
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18 | % ------------------------------------------------------------------------------------------------------------- |
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19 | |
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20 | \subsection{Vector Invariant Formulation} |
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21 | \label{PE_Vector} |
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22 | |
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23 | |
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24 | The ocean is a fluid that can be described to a good approximation by the primitive |
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25 | equations, $i.e.$ the Navier-Stokes equations along with a nonlinear equation of |
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26 | state which couples the two active tracers (temperature and salinity) to the fluid |
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27 | velocity, plus the following additional assumptions made from scale considerations: |
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28 | |
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29 | \textit{(1) spherical earth approximation: }the geopotential surfaces are assumed to |
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30 | be spheres so that gravity (local vertical) is parallel to the earth's radius |
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31 | |
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32 | \textit{(2) thin-shell approximation: }the ocean depth is neglected compared to the earth's radius |
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33 | |
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34 | \textit{(3) turbulent closure hypothesis: }the turbulent fluxes (which represent the effect |
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35 | of small scale processes on the large-scale) are expressed in terms of large-scale features |
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36 | |
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37 | \textit{(4) Boussinesq hypothesis:} density variations are neglected except in their |
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38 | contribution to the buoyancy force |
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39 | |
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40 | \textit{(5) Hydrostatic hypothesis: }the vertical momentum equation is reduced to a |
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41 | balance between the vertical pressure gradient and the buoyancy force (this removes |
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42 | convective processes from the initial Navier-Stokes equations and so convective processes |
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43 | must be parameterized instead) |
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44 | |
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45 | \textit{(6) Incompressibility hypothesis: }the three dimensional divergence of the velocity |
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46 | vector is assumed to be zero. |
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47 | |
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48 | Because the gravitational force is so dominant in the equations of large-scale motions, |
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49 | it is useful to choose an orthogonal set of unit vectors (\textbf{i},\textbf{j},\textbf{k}) linked |
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50 | to the earth such that \textbf{k} is the local upward vector and (\textbf{i},\textbf{j}) are two |
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51 | vectors orthogonal to \textbf{k}, $i.e.$ tangent to the geopotential surfaces. Let us define |
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52 | the following variables: \textbf{U} the vector velocity, $\textbf{U}=\textbf{U}_h + w\, \textbf{k}$ |
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53 | (the subscript $h$ denotes the local horizontal vector, $i.e.$ over the (\textbf{i},\textbf{j}) plane), |
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54 | $T$ the potential temperature, $S$ the salinity, \textit{$\rho $} the \textit{in situ} density. |
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55 | The vector invariant form of the primitive equations in the (\textbf{i},\textbf{j},\textbf{k}) |
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56 | vector system provides the following six equations (namely the momentum balance, the |
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57 | hydrostatic equilibrium, the incompressibility equation, the heat and salt conservation |
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58 | equations and an equation of state): |
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59 | \begin{subequations} \label{Eq_PE} |
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60 | \begin{equation} \label{Eq_PE_dyn} |
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61 | \frac{\partial {\rm {\bf U}}_h }{\partial t}= |
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62 | -\left[ {\left( {\nabla \times {\rm {\bf U}}} \right)\times {\rm {\bf U}} |
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63 | +\frac{1}{2}\nabla \left( {{\rm {\bf U}}^2} \right)} \right]_h |
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64 | -f\;{\rm {\bf k}}\times {\rm {\bf U}}_h |
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65 | -\frac{1}{\rho _o }\nabla _h p + {\rm {\bf D}}^{\rm {\bf U}} + {\rm {\bf F}}^{\rm {\bf U}} |
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66 | \end{equation} |
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67 | \begin{equation} \label{Eq_PE_hydrostatic} |
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68 | \frac{\partial p }{\partial z} = - \rho \ g |
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69 | \end{equation} |
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70 | \begin{equation} \label{Eq_PE_continuity} |
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71 | \nabla \cdot {\bf U}= 0 |
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72 | \end{equation} |
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73 | \begin{equation} \label{Eq_PE_tra_T} |
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74 | \frac{\partial T}{\partial t} = - \nabla \cdot \left( T \ \rm{\bf U} \right) + D^T + F^T |
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75 | \end{equation} |
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76 | \begin{equation} \label{Eq_PE_tra_S} |
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77 | \frac{\partial S}{\partial t} = - \nabla \cdot \left( S \ \rm{\bf U} \right) + D^S + F^S |
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78 | \end{equation} |
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79 | \begin{equation} \label{Eq_PE_eos} |
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80 | \rho = \rho \left( T,S,p \right) |
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81 | \end{equation} |
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82 | \end{subequations} |
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83 | where $\nabla$ is the generalised derivative vector operator in $(\bf i,\bf j, \bf k)$ directions, |
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84 | $t$ is the time, $z$ is the vertical coordinate, $\rho $ is the \textit{in situ} density given by |
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85 | the equation of state (\ref{Eq_PE_eos}), $\rho_o$ is a reference density, $p$ the pressure, |
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86 | $f=2 \bf \Omega \cdot \bf k$ is the Coriolis acceleration (where $\bf \Omega$ is the Earth's |
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87 | angular velocity vector), and $g$ is the gravitational acceleration. |
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88 | ${\rm {\bf D}}^{\rm {\bf U}}$, $D^T$ and $D^S$ are the parameterisations of small-scale |
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89 | physics for momentum, temperature and salinity, and ${\rm {\bf F}}^{\rm {\bf U}}$, $F^T$ |
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90 | and $F^S$ surface forcing terms. Their nature and formulation are discussed in |
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91 | \S\ref{PE_zdf_ldf} and page \S\ref{PE_boundary_condition}. |
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92 | |
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93 | . |
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94 | |
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95 | % ------------------------------------------------------------------------------------------------------------- |
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96 | % Boundary condition |
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97 | % ------------------------------------------------------------------------------------------------------------- |
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98 | \subsection{Boundary Conditions} |
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99 | \label{PE_boundary_condition} |
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100 | |
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101 | An ocean is bounded by complex coastlines, bottom topography at its base and an air-sea |
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102 | or ice-sea interface at its top. These boundaries can be defined by two surfaces, $z=-H(i,j)$ |
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103 | and $z=\eta(i,j,k,t)$, where $H$ is the depth of the ocean bottom and $\eta$ is the height |
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104 | of the sea surface. Both $H$ and $\eta$ are usually referenced to a given surface, $z=0$, |
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105 | chosen as a mean sea surface (Fig.~\ref{Fig_ocean_bc}). Through these two boundaries, |
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106 | the ocean can exchange fluxes of heat, fresh water, salt, and momentum with the solid earth, |
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107 | the continental margins, the sea ice and the atmosphere. However, some of these fluxes are |
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108 | so weak that even on climatic time scales of thousands of years they can be neglected. |
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109 | In the following, we briefly review the fluxes exchanged at the interfaces between the ocean |
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110 | and the other components of the earth system. |
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111 | |
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112 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
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113 | \begin{figure}[!ht] \label{Fig_ocean_bc} \begin{center} |
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114 | \includegraphics[width=0.90\textwidth]{./TexFiles/Figures/Fig_I_ocean_bc.pdf} |
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115 | \caption{The ocean is bounded by two surfaces, $z=-H(i,j)$ and $z=\eta(i,j,k,t)$, where $H$ |
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116 | is the depth of the sea floor and $\eta$ the height of the sea surface. Both $H$ and $\eta $ |
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117 | are referenced to $z=0$.} |
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118 | \end{center} \end{figure} |
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119 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
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120 | |
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121 | |
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122 | \begin{description} |
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123 | \item[Land - ocean interface:] the major flux between continental margins and the ocean is |
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124 | a mass exchange of fresh water through river runoff. Such an exchange modifies the sea |
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125 | surface salinity especially in the vicinity of major river mouths. It can be neglected for short |
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126 | range integrations but has to be taken into account for long term integrations as it influences |
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127 | the characteristics of water masses formed (especially at high latitudes). It is required in order |
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128 | to close the water cycle of the climate system. It is usually specified as a fresh water flux at |
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129 | the air-sea interface in the vicinity of river mouths. |
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130 | \item[Solid earth - ocean interface:] heat and salt fluxes through the sea floor are small, |
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131 | except in special areas of little extent. They are usually neglected in the model |
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132 | \footnote{In fact, it has been shown that the heat flux associated with the solid Earth cooling |
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133 | ($i.e.$the geothermal heating) is not negligible for the thermohaline circulation of the world |
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134 | ocean (see \ref{TRA_bbc}).}. |
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135 | The boundary condition is thus set to no flux of heat and salt across solid boundaries. |
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136 | For momentum, the situation is different. There is no flow across solid boundaries, |
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137 | $i.e.$ the velocity normal to the ocean bottom and coastlines is zero (in other words, |
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138 | the bottom velocity is parallel to solid boundaries). This kinematic boundary condition |
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139 | can be expressed as: |
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140 | \begin{equation} \label{Eq_PE_w_bbc} |
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141 | w = -{\rm {\bf U}}_h \cdot \nabla _h \left( H \right) |
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142 | \end{equation} |
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143 | In addition, the ocean exchanges momentum with the earth through frictional processes. |
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144 | Such momentum transfer occurs at small scales in a boundary layer. It must be parameterized |
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145 | in terms of turbulent fluxes using bottom and/or lateral boundary conditions. Its specification |
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146 | depends on the nature of the physical parameterisation used for ${\rm {\bf D}}^{\rm {\bf U}}$ |
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147 | in \eqref{Eq_PE_dyn}. It is discussed in \S\ref{PE_zdf}, page~\pageref{PE_zdf}.% and Chap. III.6 to 9. |
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148 | \item[Atmosphere - ocean interface:] the kinematic surface condition plus the mass flux |
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149 | of fresh water PE (the precipitation minus evaporation budget) leads to: |
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150 | \begin{equation} \label{Eq_PE_w_sbc} |
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151 | w = \frac{\partial \eta }{\partial t} |
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152 | + \left. {{\rm {\bf U}}_h } \right|_{z=\eta } \cdot \nabla _h \left( \eta \right) |
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153 | + P-E |
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154 | \end{equation} |
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155 | The dynamic boundary condition, neglecting the surface tension (which removes capillary |
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156 | waves from the system) leads to the continuity of pressure across the interface $z=\eta$. |
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157 | The atmosphere and ocean also exchange horizontal momentum (wind stress), and heat. |
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158 | \item[Sea ice - ocean interface:] the ocean and sea ice exchange heat, salt, fresh water |
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159 | and momentum. The sea surface temperature is constrained to be at the freezing point |
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160 | at the interface. Sea ice salinity is very low ($\sim4-6 \,psu$) compared to those of the |
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161 | ocean ($\sim34 \,psu$). The cycle of freezing/melting is associated with fresh water and |
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162 | salt fluxes that cannot be neglected. |
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163 | \end{description} |
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164 | |
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165 | |
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166 | % ================================================================ |
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167 | % The Horizontal Pressure Gradient |
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168 | % ================================================================ |
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169 | \section{The Horizontal Pressure Gradient } |
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170 | \label{PE_hor_pg} |
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171 | |
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172 | % ------------------------------------------------------------------------------------------------------------- |
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173 | % Pressure Formulation |
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174 | % ------------------------------------------------------------------------------------------------------------- |
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175 | \subsection{Pressure Formulation} |
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176 | \label{PE_p_formulation} |
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177 | |
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178 | The total pressure at a given depth $z$ is composed of a surface pressure $p_s$ at a |
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179 | reference geopotential surface ($z=0$) and a hydrostatic pressure $p_h$ such that: |
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180 | $p(i,j,k,t)=p_s(i,j,t)+p_h(i,j,k,t)$. The latter is computed by integrating (\ref{Eq_PE_hydrostatic}), |
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181 | assuming that pressure in decibars can be approximated by depth in meters in (\ref{Eq_PE_eos}). |
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182 | The hydrostatic pressure is then given by: |
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183 | \begin{equation} \label{Eq_PE_pressure} |
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184 | p_h \left( {i,j,z,t} \right) |
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185 | = \int_{\varsigma =z}^{\varsigma =0} {g\;\rho \left( {T,S,\varsigma} \right)\;d\varsigma } |
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186 | \end{equation} |
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187 | Two strategies can be considered for the surface pressure term: $(a)$ introduce of a |
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188 | new variable $\eta$, the free-surface elevation, for which a prognostic equation can be |
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189 | established and solved; $(b)$ assume that the ocean surface is a rigid lid, on which the |
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190 | pressure (or its horizontal gradient) can be diagnosed. When the former strategy is used, |
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191 | one solution of the free-surface elevation consists of the excitation of external gravity waves. |
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192 | The flow is barotropic and the surface moves up and down with gravity as the restoring force. |
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193 | The phase speed of such waves is high (some hundreds of metres per second) so that |
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194 | the time step would have to be very short if they were present in the model. The latter |
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195 | strategy filters out these waves since the rigid lid approximation implies $\eta=0$, $i.e.$ |
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196 | the sea surface is the surface $z=0$. This well known approximation increases the surface |
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197 | wave speed to infinity and modifies certain other longwave dynamics ($e.g.$ barotropic |
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198 | Rossby or planetary waves). In the present release of \NEMO, both strategies are still available. |
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199 | They are further described in the next two sub-sections. |
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200 | |
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201 | % ------------------------------------------------------------------------------------------------------------- |
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202 | % Free Surface Formulation |
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203 | % ------------------------------------------------------------------------------------------------------------- |
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204 | \subsection{Free Surface Formulation} |
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205 | \label{PE_free_surface} |
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206 | |
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207 | In the free surface formulation, a variable $\eta$, the sea-surface height, is introduced |
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208 | which describes the shape of the air-sea interface. This variable is solution of a |
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209 | prognostic equation which is established by forming the vertical average of the kinematic |
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210 | surface condition (\ref{Eq_PE_w_bbc}): |
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211 | \begin{equation} \label{Eq_PE_ssh} |
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212 | \frac{\partial \eta }{\partial t}=-D+P-E |
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213 | \quad \text{where} \ |
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214 | D=\nabla \cdot \left[ {\left( {H+\eta } \right) \; {\rm{\bf \overline{U}}}_h \,} \right] |
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215 | \end{equation} |
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216 | and using (\ref{Eq_PE_hydrostatic}) the surface pressure is given by: $p_s = \rho \, g \, \eta$. |
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217 | |
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218 | Allowing the air-sea interface to move introduces the external gravity waves (EGWs) |
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219 | as a class of solution of the primitive equations. These waves are barotropic because |
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220 | of hydrostatic assumption, and their phase speed is quite high. Their time scale is |
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221 | short with respect to the other processes described by the primitive equations. |
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222 | |
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223 | Three choices can be made regarding the implementation of the free surface in the model, |
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224 | depending on the physical processes of interest. |
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225 | |
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226 | $\bullet$ If one is interested in EGWs, in particular the tides and their interaction |
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227 | with the baroclinic structure of the ocean (internal waves) possibly in shallow seas, |
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228 | then a non linear free surface is the most appropriate. This means that no |
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229 | approximation is made in (\ref{Eq_PE_ssh}) and that the variation of the ocean |
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230 | volume is fully taken into account. Note that in order to study the fast time scales |
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231 | associated with EGWs it is necessary to minimize time filtering effects (use an |
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232 | explicit time scheme with very small time step, or a split-explicit scheme with |
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233 | reasonably small time step, see \S\ref{DYN_spg_exp} or \S\ref{DYN_spg_ts}. |
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234 | |
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235 | $\bullet$ If one is not interested in EGW but rather sees them as high frequency |
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236 | noise, it is possible to apply an explicit filter to slow down the fastest waves while |
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237 | not altering the slow barotropic Rossby waves. If further, an approximative conservation |
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238 | of heat and salt contents is sufficient for the problem solved, then it is |
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239 | sufficient to solve a linearized version of (\ref{Eq_PE_ssh}), which still allows |
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240 | to take into account freshwater fluxes applied at the ocean surface \citep{Roullet2000}. |
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241 | |
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242 | $\bullet$ For process studies not involving external waves nor surface freshwater |
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243 | fluxes, it is possible to use the rigid lid approximation see (next |
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244 | section). The ocean surface is then considered as a fixed surface, so that all |
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245 | external waves are removed from the system. |
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246 | |
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247 | The filtering of EGWs in models with a free surface is usually a matter of discretisation |
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248 | of the temporal derivatives, using the time splitting method \citep{Killworth1991, Zhang1992} |
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249 | or the implicit scheme \citep{Dukowicz1994}. In \NEMO, we use a slightly different approach |
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250 | developed by \citet{Roullet2000}: the damping of EGWs is ensured by introducing an |
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251 | additional force in the momentum equation. \eqref{Eq_PE_dyn} becomes: |
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252 | \begin{equation} \label{Eq_PE_flt} |
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253 | \frac{\partial {\rm {\bf U}}_h }{\partial t}= {\rm {\bf M}} |
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254 | - g \nabla \left( \tilde{\rho} \ \eta \right) |
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255 | - g \ T_c \nabla \left( \widetilde{\rho} \ \partial_t \eta \right) |
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256 | \end{equation} |
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257 | where $T_c$, is a parameter with dimensions of time which characterizes the force, |
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258 | $\widetilde{\rho} = \rho / \rho_o$ is the dimensionless density, and $\rm {\bf M}$ |
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259 | represents the collected contributions of the Coriolis, hydrostatic pressure gradient, |
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260 | non-linear and viscous terms in \eqref{Eq_PE_dyn}. |
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261 | |
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262 | The new force can be interpreted as a diffusion of vertically integrated volume flux divergence. |
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263 | The time evolution of $D$ is thus governed by a balance of two terms, $-g$ \textbf{A} $\eta$ |
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264 | and $g \, T_c \,$ \textbf{A} $D$, associated with a propagative regime and a diffusive regime |
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265 | in the temporal spectrum, respectively. In the diffusive regime, the EGWs no longer propagate, |
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266 | $i.e.$ they are stationary and damped. The diffusion regime applies to the modes shorter than |
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267 | $T_c$. For longer ones, the diffusion term vanishes. Hence, the temporally unresolved EGWs |
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268 | can be damped by choosing $T_c > \Delta t$. \citet{Roullet2000} demonstrate that |
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269 | (\ref{Eq_PE_flt}) can be integrated with a leap frog scheme except the additional term which |
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270 | has to be computed implicitly. This is not surprising since the use of a large time step has a |
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271 | necessarily numerical cost. Two gains arise in comparison with the previous formulations. |
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272 | Firstly, the damping of EGWs can be quantified through the magnitude of the additional term. |
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273 | Secondly, the numerical scheme does not need any tuning. Numerical stability is ensured as |
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274 | soon as $T_c > \Delta t$. |
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275 | |
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276 | When the variations of free surface elevation are small compared to the thickness of the first |
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277 | model layer, the free surface equation (\ref{Eq_PE_ssh}) can be linearized. As emphasized |
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278 | by \citet{Roullet2000} the linearization of (\ref{Eq_PE_ssh}) has consequences on the |
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279 | conservation of salt in the model. With the nonlinear free surface equation, the time evolution |
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280 | of the total salt content is |
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281 | \begin{equation} \label{Eq_PE_salt_content} |
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282 | \frac{\partial }{\partial t}\int\limits_{D\eta } {S\;dv} |
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283 | =\int\limits_S {S\;(-\frac{\partial \eta }{\partial t}-D+P-E)\;ds} |
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284 | \end{equation} |
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285 | where $S$ is the salinity, and the total salt is integrated over the whole ocean volume |
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286 | $D_\eta$ bounded by the time-dependent free surface. The right hand side (which is an |
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287 | integral over the free surface) vanishes when the nonlinear equation (\ref{Eq_PE_ssh}) |
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288 | is satisfied, so that the salt is perfectly conserved. When the free surface equation is |
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289 | linearized, \citet{Roullet2000} show that the total salt content integrated in the fixed |
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290 | volume $D$ (bounded by the surface $z=0$) is no longer conserved: |
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291 | \begin{equation} \label{Eq_PE_salt_content_linear} |
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292 | \frac{\partial }{\partial t}\int\limits_D {S\;dv} |
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293 | = - \int\limits_S {S\;\frac{\partial \eta }{\partial t}ds} |
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294 | \end{equation} |
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295 | |
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296 | The right hand side of (\ref{Eq_PE_salt_content_linear}) is small in equilibrium solutions |
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297 | \citep{Roullet2000}. It can be significant when the freshwater forcing is not balanced and |
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298 | the globally averaged free surface is drifting. An increase in sea surface height \textit{$\eta $} |
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299 | results in a decrease of the salinity in the fixed volume $D$. Even in that case though, |
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300 | the total salt integrated in the variable volume $D_{\eta}$ varies much less, since |
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301 | (\ref{Eq_PE_salt_content_linear}) can be rewritten as |
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302 | \begin{equation} \label{Eq_PE_salt_content_corrected} |
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303 | \frac{\partial }{\partial t}\int\limits_{D\eta } {S\;dv} |
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304 | =\frac{\partial}{\partial t} \left[ \;{\int\limits_D {S\;dv} +\int\limits_S {S\eta \;ds} } \right] |
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305 | =\int\limits_S {\eta \;\frac{\partial S}{\partial t}ds} |
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306 | \end{equation} |
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307 | |
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308 | Although the total salt content is not exactly conserved with the linearized free surface, |
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309 | its variations are driven by correlations of the time variation of surface salinity with the |
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310 | sea surface height, which is a negligible term. This situation contrasts with the case of |
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311 | the rigid lid approximation (following section) in which case freshwater forcing is |
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312 | represented by a virtual salt flux, leading to a spurious source of salt at the ocean |
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313 | surface \citep{Roullet2000}. |
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314 | |
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315 | % ------------------------------------------------------------------------------------------------------------- |
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316 | % Rigid-Lid Formulation |
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317 | % ------------------------------------------------------------------------------------------------------------- |
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318 | \subsection{Rigid-Lid formulation} |
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319 | \label{PE_rigid_lid} |
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320 | |
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321 | With the rigid lid approximation, we assume that the ocean surface ($z=0$) is a rigid lid |
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322 | on which a pressure $p_s$ is exerted. This implies that the vertical velocity at the surface |
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323 | is equal to zero. From the continuity equation \eqref{Eq_PE_continuity} and the kinematic |
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324 | condition at the bottom \eqref{Eq_PE_w_bbc} (no flux across the bottom), it can be shown |
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325 | that the vertically integrated flow $H{\rm {\bf \bar {U}}}_h$ is non-divergent (where the |
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326 | overbar indicates a vertical average over the whole water column, i.e. from $z=-H$, |
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327 | the ocean bottom, to $z=0$ , the rigid-lid). Thus, $\rm {\bf \bar {U}}_h$ can be derived |
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328 | from a volume transport streamfunction $\psi$: |
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329 | \begin{equation} \label{Eq_PE_u_psi} |
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330 | \overline{\vect{U}}_h =\frac{1}{H}\left( \vect{k} \times \nabla \psi \right) |
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331 | \end{equation} |
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332 | |
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333 | As $p_s$ does not depend on depth, its horizontal gradient is obtained by forming the |
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334 | vertical average of \eqref{Eq_PE_dyn} and using \eqref{Eq_PE_u_psi}: |
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335 | |
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336 | \begin{equation} \label{Eq_PE_u_barotrope} |
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337 | \frac{1}{\rho _o }\nabla _h p_s |
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338 | =\overline{\vect{M}} -\frac{\partial \overline{\vect{U}} _h }{\partial t} |
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339 | =\overline{\vect{M}} |
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340 | -\frac{1}{H} \left[ \vect{k} \times \nabla \left( \frac{\partial \psi}{\partial t} \right) \right] |
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341 | \end{equation} |
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342 | |
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343 | Here ${\rm {\bf M}} = \left( M_u,M_v \right)$ represents the collected contributions of the |
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344 | Coriolis, hydrostatic pressure gradient, nonlinear and viscous terms in \eqref{Eq_PE_dyn}. |
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345 | The time derivative of $\psi $ is the solution of an elliptic equation which is obtained from |
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346 | the vertical component of the curl of (\ref{Eq_PE_u_barotrope}): |
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347 | \begin{equation} \label{Eq_PE_psi} |
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348 | \left[ {\nabla \times \left[ {\frac{1}{H} \vect{\bf k} |
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349 | \times \nabla \left( {\frac{\partial \psi }{\partial t}} \right)} \right]} \; \right]_z |
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350 | =\left[ {\nabla \times \overline{\vect{M}} } \right]_z |
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351 | \end{equation} |
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352 | |
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353 | Using the proper boundary conditions, \eqref{Eq_PE_psi} can be solved to find $\partial_t \psi$ |
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354 | and thus using \eqref{Eq_PE_u_barotrope} the horizontal surface pressure gradient. |
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355 | It should be noted that $p_s$ can be computed by taking the divergence of |
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356 | \eqref{Eq_PE_u_barotrope} and solving the resulting elliptic equation. Thus the surface |
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357 | pressure is a diagnostic quantity that can be recovered for analysis purposes. |
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358 | |
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359 | A difficulty lies in the determination of the boundary condition on $\partial_t \psi$. |
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360 | The boundary condition on velocity is that there is no flow normal to a solid wall, |
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361 | $i.e.$ the coastlines are streamlines. Therefore \eqref{Eq_PE_psi} is solved with |
---|
362 | the following Dirichlet boundary condition: $\partial_t \psi$ is constant along each |
---|
363 | coastline of the same continent or of the same island. When all the coastlines are |
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364 | connected (there are no islands), the constant value of $\partial_t \psi$ along the |
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365 | coast can be arbitrarily chosen to be zero. When islands are present in the domain, |
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366 | the value of the barotropic streamfunction will generally be different for each island |
---|
367 | and for the continent, and will vary with respect to time. So the boundary condition is: |
---|
368 | $\psi=0$ along the continent and $\psi=\mu_n$ along island $n$ ($1 \leq n \leq Q$), |
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369 | where $Q$ is the number of islands present in the domain and $\mu_n$ is a time |
---|
370 | dependent variable. A time evolution equation of the unknown $\mu_n$ can be found |
---|
371 | by evaluating the circulation of the time derivative of the vertical average (barotropic) |
---|
372 | velocity field along a closed contour around each island. Since the circulation of a |
---|
373 | gradient field along a closed contour is zero, from \eqref{Eq_PE_u_barotrope} we have: |
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374 | \begin{equation} \label{Eq_PE_isl_circulation} |
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375 | \oint_n {\frac{1}{H}\left[ {{\rm {\bf k}}\times \nabla \left( |
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376 | {\frac{\partial \psi }{\partial t}} \right)} \right] \cdot {\rm {\bf d}}\ell } |
---|
377 | = \oint_n {\overline {\rm {\bf M}} \cdot {\rm {\bf d}}\ell } |
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378 | \qquad 1 \leq n \leq Q |
---|
379 | \end{equation} |
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380 | |
---|
381 | Since (\ref{Eq_PE_psi}) is linear, its solution \textit{$\psi $} can be decomposed |
---|
382 | as follows: |
---|
383 | \begin{equation} \label{Eq_PE_psi_isl} |
---|
384 | \psi =\psi _o +\sum\limits_{n=1}^{n=Q} {\mu _n \psi _n } |
---|
385 | \end{equation} |
---|
386 | where $\psi _o$ is the solution of \eqref{Eq_PE_psi} with $\psi _o=0$ long all |
---|
387 | the coastlines, and where $\psi _n$ is the solution of \eqref{Eq_PE_psi} with |
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388 | the right-hand side equal to $0$, and with $\psi _n =1$ long the island $n$, |
---|
389 | $\psi _n =0$ along the other boundaries. The function $\psi _n$ is thus |
---|
390 | independent of time. Introducing \eqref{Eq_PE_psi_isl} into |
---|
391 | \eqref{Eq_PE_isl_circulation} yields: |
---|
392 | \begin{multline} \label{Eq_PE_psi_isl_circulation} |
---|
393 | \left[ {\oint_n {\frac{1}{H} \left[ {{\rm {\bf k}}\times \nabla \psi _m } \right]\cdot {\rm {\bf d}}\ell } } \right]_{1\leq m\leqslant Q \atop 1\leq n\leqslant Q } |
---|
394 | \left( {\frac{\partial \mu _n }{\partial t}} |
---|
395 | \right)_{1\leqslant n\leqslant Q} \\ |
---|
396 | =\left( {\oint_n {\left[ {\overline {\rm |
---|
397 | {\bf M}} -\frac{1}{H}\left[ {{\rm {\bf k}}\times \nabla \left( |
---|
398 | {\frac{\partial \psi _o }{\partial t}} \right)} \right]} \right]\cdot {\rm |
---|
399 | {\bf d}}\ell } } \right)_{1\leqslant n\leqslant Q} |
---|
400 | \end{multline} |
---|
401 | which can be rewritten as: |
---|
402 | \begin{equation} \label{Eq_PE_psi_isl_matrix} |
---|
403 | {\rm {\bf A}}\;\left( {\frac{\partial \mu _n }{\partial t}} |
---|
404 | \right)_{1\leqslant n\leqslant Q} ={\rm {\bf B}} |
---|
405 | \end{equation} |
---|
406 | where \textbf{A} is a $Q \times Q$ matrix and \textbf{B} is a time dependent vector. |
---|
407 | As \textbf{A} is independent of time, it can be calculated and inverted once. The time |
---|
408 | derivative of the streamfunction when islands are present is thus given by: |
---|
409 | \begin{equation} \label{Eq_PE_psi_isl_dt} |
---|
410 | \frac{\partial \psi }{\partial t}=\frac{\partial \psi _o }{\partial |
---|
411 | t}+\sum\limits_{n=1}^{n=Q} {{\rm {\bf A}}^{-1}{\rm {\bf B}}\;\psi _n } |
---|
412 | \end{equation} |
---|
413 | |
---|
414 | |
---|
415 | |
---|
416 | % ================================================================ |
---|
417 | % Curvilinear z-coordinate System |
---|
418 | % ================================================================ |
---|
419 | \section{Curvilinear \textit{z-}coordinate System} |
---|
420 | \label{PE_zco} |
---|
421 | |
---|
422 | |
---|
423 | % ------------------------------------------------------------------------------------------------------------- |
---|
424 | % Tensorial Formalism |
---|
425 | % ------------------------------------------------------------------------------------------------------------- |
---|
426 | \subsection{Tensorial Formalism} |
---|
427 | \label{PE_tensorial} |
---|
428 | |
---|
429 | In many ocean circulation problems, the flow field has regions of enhanced dynamics |
---|
430 | ($i.e.$ surface layers, western boundary currents, equatorial currents, or ocean fronts). |
---|
431 | The representation of such dynamical processes can be improved by specifically increasing |
---|
432 | the model resolution in these regions. As well, it may be convenient to use a lateral |
---|
433 | boundary-following coordinate system to better represent coastal dynamics. Moreover, |
---|
434 | the common geographical coordinate system has a singular point at the North Pole that |
---|
435 | cannot be easily treated in a global model without filtering. A solution consists of introducing |
---|
436 | an appropriate coordinate transformation that shifts the singular point onto land |
---|
437 | \citep{MadecImb1996, Murray1996}. As a consequence, it is important to solve the primitive |
---|
438 | equations in various curvilinear coordinate systems. An efficient way of introducing an |
---|
439 | appropriate coordinate transform can be found when using a tensorial formalism. |
---|
440 | This formalism is suited to any multidimensional curvilinear coordinate system. Ocean |
---|
441 | modellers mainly use three-dimensional orthogonal grids on the sphere (spherical earth |
---|
442 | approximation), with preservation of the local vertical. Here we give the simplified equations |
---|
443 | for this particular case. The general case is detailed by \citet{Eiseman1980} in their survey |
---|
444 | of the conservation laws of fluid dynamics. |
---|
445 | |
---|
446 | Let (\textbf{i},\textbf{j},\textbf{k}) be a set of orthogonal curvilinear coordinates on the sphere |
---|
447 | associated with the positively oriented orthogonal set of unit vectors (\textbf{i},\textbf{j},\textbf{k}) |
---|
448 | linked to the earth such that \textbf{k} is the local upward vector and (\textbf{i},\textbf{j}) are |
---|
449 | two vectors orthogonal to \textbf{k}, $i.e.$ along geopotential surfaces (Fig.\ref{Fig_referential}). |
---|
450 | Let $(\lambda,\varphi,z)$ be the geographical coordinate system in which a position is defined |
---|
451 | by the latitude $\varphi(i,j)$, the longitude $\lambda(i,j)$ and the distance from the centre of |
---|
452 | the earth $a+z(k)$ where $a$ is the earth's radius and $z$ the altitude above a reference sea |
---|
453 | level (Fig.\ref{Fig_referential}). The local deformation of the curvilinear coordinate system is |
---|
454 | given by $e_1$, $e_2$ and $e_3$, the three scale factors: |
---|
455 | \begin{equation} \label{Eq_scale_factors} |
---|
456 | \begin{aligned} |
---|
457 | e_1 &=\left( {a+z} \right)\;\left[ {\left( {\frac{\partial \lambda |
---|
458 | }{\partial i}\cos \varphi } \right)^2+\left( {\frac{\partial \varphi |
---|
459 | }{\partial i}} \right)^2} \right]^{1/2} \\ |
---|
460 | e_2 &=\left( {a+z} \right)\;\left[ {\left( {\frac{\partial \lambda |
---|
461 | }{\partial j}\cos \varphi } \right)^2+\left( {\frac{\partial \varphi |
---|
462 | }{\partial j}} \right)^2} \right]^{1/2} \\ |
---|
463 | e_3 &=\left( {\frac{\partial z}{\partial k}} \right) \\ |
---|
464 | \end{aligned} |
---|
465 | \end{equation} |
---|
466 | |
---|
467 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
---|
468 | \begin{figure}[!tb] \label{Fig_referential} \begin{center} |
---|
469 | \includegraphics[width=0.60\textwidth]{./TexFiles/Figures/Fig_I_earth_referential.pdf} |
---|
470 | \caption{the geographical coordinate system $(\lambda,\varphi,z)$ and the curvilinear |
---|
471 | coordinate system (\textbf{i},\textbf{j},\textbf{k}). } |
---|
472 | \end{center} \end{figure} |
---|
473 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
---|
474 | |
---|
475 | Since the ocean depth is far smaller than the earth's radius, $a+z$, can be replaced by |
---|
476 | $a$ in (\ref{Eq_scale_factors}) (thin-shell approximation). The resulting horizontal scale |
---|
477 | factors $e_1$, $e_2$ are independent of $k$ while the vertical scale factor is a single |
---|
478 | function of $k$ as \textbf{k} is parallel to \textbf{z}. The scalar and vector operators that |
---|
479 | appear in the primitive equations (Eqs. \eqref{Eq_PE_dyn} to \eqref{Eq_PE_eos}) can |
---|
480 | be written in the tensorial form, invariant in any orthogonal horizontal curvilinear coordinate |
---|
481 | system transformation: |
---|
482 | \begin{subequations} \label{Eq_PE_discrete_operators} |
---|
483 | \begin{equation} \label{Eq_PE_grad} |
---|
484 | \nabla q=\frac{1}{e_1 }\frac{\partial q}{\partial i}\;{\rm {\bf |
---|
485 | i}}+\frac{1}{e_2 }\frac{\partial q}{\partial j}\;{\rm {\bf j}}+\frac{1}{e_3 |
---|
486 | }\frac{\partial q}{\partial k}\;{\rm {\bf k}} \\ |
---|
487 | \end{equation} |
---|
488 | \begin{equation} \label{Eq_PE_div} |
---|
489 | \nabla \cdot {\rm {\bf A}} |
---|
490 | = \frac{1}{e_1 \; e_2} \left[ |
---|
491 | \frac{\partial \left(e_2 \; a_1\right)}{\partial i } |
---|
492 | +\frac{\partial \left(e_1 \; a_2\right)}{\partial j } \right] |
---|
493 | + \frac{1}{e_3} \left[ \frac{\partial a_3}{\partial k } \right] |
---|
494 | \end{equation} |
---|
495 | \begin{equation} \label{Eq_PE_curl} |
---|
496 | \begin{split} |
---|
497 | \nabla \times \vect{A} = |
---|
498 | \left[ {\frac{1}{e_2 }\frac{\partial a_3}{\partial j} |
---|
499 | -\frac{1}{e_3 }\frac{\partial a_2 }{\partial k}} \right] \; \vect{i} |
---|
500 | &+\left[ {\frac{1}{e_3 }\frac{\partial a_1 }{\partial k} |
---|
501 | -\frac{1}{e_1 }\frac{\partial a_3 }{\partial i}} \right] \; \vect{j} \\ |
---|
502 | &+\frac{1}{e_1 e_2 } \left[ {\frac{\partial \left( {e_2 a_2 } \right)}{\partial i} |
---|
503 | -\frac{\partial \left( {e_1 a_1 } \right)}{\partial j}} \right] \; \vect{k} |
---|
504 | \end{split} |
---|
505 | \end{equation} |
---|
506 | \begin{equation} \label{Eq_PE_lap} |
---|
507 | \Delta q = \nabla \cdot \left( \nabla q \right) |
---|
508 | \end{equation} |
---|
509 | \begin{equation} \label{Eq_PE_lap_vector} |
---|
510 | \Delta {\rm {\bf A}} = |
---|
511 | \nabla \left( \nabla \cdot {\rm {\bf A}} \right) |
---|
512 | - \nabla \times \left( \nabla \times {\rm {\bf A}} \right) |
---|
513 | \end{equation} |
---|
514 | \end{subequations} |
---|
515 | where $q$ is a scalar quantity and ${\rm {\bf A}}=(a_1,a_2,a_3)$ a vector in the $(i,j,k)$ coordinate system. |
---|
516 | |
---|
517 | % ------------------------------------------------------------------------------------------------------------- |
---|
518 | % Continuous Model Equations |
---|
519 | % ------------------------------------------------------------------------------------------------------------- |
---|
520 | \subsection{Continuous Model Equations} |
---|
521 | \label{PE_zco_Eq} |
---|
522 | |
---|
523 | In order to express the Primitive Equations in tensorial formalism, it is necessary to compute |
---|
524 | the horizontal component of the non-linear and viscous terms of the equation using |
---|
525 | \eqref{Eq_PE_grad}) to \eqref{Eq_PE_lap_vector}. |
---|
526 | Let us set $\vect U=(u,v,w)={\vect{U}}_h +w\;\vect{k}$, the velocity in the $(i,j,k)$ coordinate |
---|
527 | system and define the relative vorticity $\zeta$ and the divergence of the horizontal velocity |
---|
528 | field $\chi$, by: |
---|
529 | \begin{equation} \label{Eq_PE_curl_Uh} |
---|
530 | \zeta =\frac{1}{e_1 e_2 }\left[ {\frac{\partial \left( {e_2 \,v} |
---|
531 | \right)}{\partial i}-\frac{\partial \left( {e_1 \,u} \right)}{\partial j}} |
---|
532 | \right] |
---|
533 | \end{equation} |
---|
534 | \begin{equation} \label{Eq_PE_div_Uh} |
---|
535 | \chi =\frac{1}{e_1 e_2 }\left[ {\frac{\partial \left( {e_2 \,u} |
---|
536 | \right)}{\partial i}+\frac{\partial \left( {e_1 \,v} \right)}{\partial j}} |
---|
537 | \right] |
---|
538 | \end{equation} |
---|
539 | |
---|
540 | Using the fact that the horizontal scale factors $e_1$ and $e_2$ are independent of $k$ |
---|
541 | and that $e_3$ is a function of the single variable $k$, the nonlinear term of |
---|
542 | \eqref{Eq_PE_dyn} can be transformed as follows: |
---|
543 | \begin{flalign*} |
---|
544 | &\left[ {\left( { \nabla \times {\rm {\bf U}} } \right) \times {\rm {\bf U}} |
---|
545 | +\frac{1}{2} \nabla \left( {{\rm {\bf U}}^2} \right)} \right]_h & |
---|
546 | \end{flalign*} |
---|
547 | \begin{flalign*} |
---|
548 | &\qquad=\left( {{\begin{array}{*{20}c} |
---|
549 | {\left[ { \frac{1}{e_3} \frac{\partial u }{\partial k} |
---|
550 | -\frac{1}{e_1} \frac{\partial w }{\partial i} } \right] w - \zeta \; v } \\ |
---|
551 | {\zeta \; u - \left[ { \frac{1}{e_2} \frac{\partial w}{\partial j} |
---|
552 | -\frac{1}{e_3} \frac{\partial v}{\partial k} } \right] \ w} \\ |
---|
553 | \end{array} }} \right) |
---|
554 | +\frac{1}{2} \left( {{\begin{array}{*{20}c} |
---|
555 | { \frac{1}{e_1} \frac{\partial \left( u^2+v^2+w^2 \right)}{\partial i}} \hfill \\ |
---|
556 | { \frac{1}{e_2} \frac{\partial \left( u^2+v^2+w^2 \right)}{\partial j}} \hfill \\ |
---|
557 | \end{array} }} \right) & |
---|
558 | \end{flalign*} |
---|
559 | \begin{flalign*} |
---|
560 | & \qquad =\left( {{ \begin{array}{*{20}c} |
---|
561 | {-\zeta \; v} \hfill \\ |
---|
562 | { \zeta \; u} \hfill \\ |
---|
563 | \end{array} }} \right) |
---|
564 | +\frac{1}{2}\left( {{ \begin{array}{*{20}c} |
---|
565 | {\frac{1}{e_1 }\frac{\partial \left( {u^2+v^2} \right)}{\partial i}} \hfill \\ |
---|
566 | {\frac{1}{e_2 }\frac{\partial \left( {u^2+v^2} \right)}{\partial j}} \hfill \\ |
---|
567 | \end{array} }} \right) |
---|
568 | +\frac{1}{e_3 }\left( {{ \begin{array}{*{20}c} |
---|
569 | { w \; \frac{\partial u}{\partial k}} \\ |
---|
570 | { w \; \frac{\partial v}{\partial k}} \\ |
---|
571 | \end{array} }} \right) |
---|
572 | -\left( {{ \begin{array}{*{20}c} |
---|
573 | {\frac{w}{e_1}\frac{\partial w}{\partial i} |
---|
574 | -\frac{1}{2e_1}\frac{\partial w^2}{\partial i}} \hfill \\ |
---|
575 | {\frac{w}{e_2}\frac{\partial w}{\partial j} |
---|
576 | -\frac{1}{2e_2}\frac{\partial w^2}{\partial j}} \hfill \\ |
---|
577 | \end{array} }} \right) & |
---|
578 | \end{flalign*} |
---|
579 | |
---|
580 | The last term of the right hand side is obviously zero, and thus the nonlinear term of |
---|
581 | \eqref{Eq_PE_dyn} is written in the $(i,j,k)$ coordinate system: |
---|
582 | \begin{equation} \label{Eq_PE_vector_form} |
---|
583 | \left[ {\left( { \nabla \times {\rm {\bf U}} } \right) \times {\rm {\bf U}} |
---|
584 | +\frac{1}{2} \nabla \left( {{\rm {\bf U}}^2} \right)} \right]_h |
---|
585 | =\zeta |
---|
586 | \;{\rm {\bf k}}\times {\rm {\bf U}}_h +\frac{1}{2}\nabla _h \left( {{\rm |
---|
587 | {\bf U}}_h^2 } \right)+\frac{1}{e_3 }w\frac{\partial {\rm {\bf U}}_h |
---|
588 | }{\partial k} |
---|
589 | \end{equation} |
---|
590 | |
---|
591 | This is the so-called \textit{vector invariant form} of the momentum advection term. |
---|
592 | For some purposes, it can be advantageous to write this term in the so-called flux form, |
---|
593 | $i.e.$ to write it as the divergence of fluxes. For example, the first component of |
---|
594 | \eqref{Eq_PE_vector_form} (the $i$-component) is transformed as follows: |
---|
595 | \begin{flalign*} |
---|
596 | &{ \begin{array}{*{20}l} |
---|
597 | \left[ {\left( {\nabla \times \vect{U}} \right)\times \vect{U} |
---|
598 | +\frac{1}{2}\nabla \left( {\vect{U}}^2 \right)} \right]_i % \\ |
---|
599 | %\\ |
---|
600 | = - \zeta \;v |
---|
601 | + \frac{1}{2\;e_1 } \frac{\partial \left( {u^2+v^2} \right)}{\partial i} |
---|
602 | + \frac{1}{e_3}w \ \frac{\partial u}{\partial k} \\ |
---|
603 | \\ |
---|
604 | \qquad =\frac{1}{e_1 \; e_2} \left( -v\frac{\partial \left( {e_2 \,v} \right)}{\partial i} |
---|
605 | +v\frac{\partial \left( {e_1 \,u} \right)}{\partial j} \right) |
---|
606 | +\frac{1}{e_1 e_2 }\left( +e_2 \; u\frac{\partial u}{\partial i} |
---|
607 | +e_2 \; v\frac{\partial v}{\partial i} \right) |
---|
608 | +\frac{1}{e_3} \left( w\;\frac{\partial u}{\partial k} \right) \\ |
---|
609 | \end{array} } & |
---|
610 | \end{flalign*} |
---|
611 | \begin{flalign*} |
---|
612 | &{ \begin{array}{*{20}l} |
---|
613 | \qquad =\frac{1}{e_1 \; e_2} \left\{ |
---|
614 | -\left( v^2 \frac{\partial e_2 }{\partial i} |
---|
615 | +e_2 \,v \frac{\partial v }{\partial i} \right) |
---|
616 | +\left( \frac{\partial \left( {e_1 \,u\,v} \right)}{\partial j} |
---|
617 | -e_1 \,u \frac{\partial v }{\partial j} \right) \right. |
---|
618 | \\ \left. \qquad \qquad \quad |
---|
619 | +\left( \frac{\partial \left( {e_2 u\,u} \right)}{\partial i} |
---|
620 | -u \frac{\partial \left( {e_2 u} \right)}{\partial i} \right) |
---|
621 | +e_2 v \frac{\partial v }{\partial i} |
---|
622 | \right\} |
---|
623 | +\frac{1}{e_3} \left( |
---|
624 | \frac{\partial \left( {w\,u} \right) }{\partial k} |
---|
625 | -u \frac{\partial w }{\partial k} \right) \\ |
---|
626 | \end{array} } & |
---|
627 | \end{flalign*} |
---|
628 | \begin{flalign*} |
---|
629 | &{ \begin{array}{*{20}l} |
---|
630 | \qquad =\frac{1}{e_1 \; e_2} \left( |
---|
631 | \frac{\partial \left( {e_2 \,u\,u} \right)}{\partial i} |
---|
632 | + \frac{\partial \left( {e_1 \,u\,v} \right)}{\partial j} \right) |
---|
633 | +\frac{1}{e_3 } \frac{\partial \left( {w\,u } \right)}{\partial k} |
---|
634 | \\ \qquad \qquad \quad |
---|
635 | +\frac{1}{e_1 e_2 } \left( |
---|
636 | -u \left( \frac{\partial \left( {e_1 v } \right)}{\partial j} |
---|
637 | -v\,\frac{\partial e_1 }{\partial j} \right) |
---|
638 | -u \frac{\partial \left( {e_2 u } \right)}{\partial i} |
---|
639 | \right) |
---|
640 | -\frac{1}{e_3 } \frac{\partial w}{\partial k} u |
---|
641 | +\frac{1}{e_1 e_2 }\left( -v^2\frac{\partial e_2 }{\partial i} \right) |
---|
642 | \end{array} } & |
---|
643 | \end{flalign*} |
---|
644 | \begin{flalign*} |
---|
645 | &{ \begin{array}{*{20}l} |
---|
646 | \qquad = \nabla \cdot \left( {{\rm {\bf U}}\,u} \right) |
---|
647 | - \left( \nabla \cdot {\rm {\bf U}} \right) \ u |
---|
648 | +\frac{1}{e_1 e_2 }\left( |
---|
649 | -v^2 \frac{\partial e_2 }{\partial i} |
---|
650 | +uv \, \frac{\partial e_1 }{\partial j} \right) \\ |
---|
651 | \end{array} } & |
---|
652 | \end{flalign*} |
---|
653 | as $\nabla \cdot {\rm {\bf U}}\;=0$ (incompressibility) it comes: |
---|
654 | \begin{flalign*} |
---|
655 | &{ \begin{array}{*{20}l} |
---|
656 | \qquad = \nabla \cdot \left( {{\rm {\bf U}}\,u} \right) |
---|
657 | + \frac{1}{e_1 e_2 } \left( v \; \frac{\partial e_2}{\partial i} |
---|
658 | -u \; \frac{\partial e_1}{\partial j} \right) \left( -v \right) |
---|
659 | \end{array} } & |
---|
660 | \end{flalign*} |
---|
661 | |
---|
662 | The flux form of the momentum advection term is therefore given by: |
---|
663 | \begin{multline} \label{Eq_PE_flux_form} |
---|
664 | \left[ |
---|
665 | \left( {\nabla \times {\rm {\bf U}}} \right) \times {\rm {\bf U}} |
---|
666 | +\frac{1}{2} \nabla \left( {{\rm {\bf U}}^2} \right) |
---|
667 | \right]_h |
---|
668 | \\ |
---|
669 | = \nabla \cdot \left( {{\begin{array}{*{20}c} {\rm {\bf U}} \, u \hfill \\ |
---|
670 | {\rm {\bf U}} \, v \hfill \\ |
---|
671 | \end{array} }} |
---|
672 | \right) |
---|
673 | +\frac{1}{e_1 e_2 } \left( |
---|
674 | v\frac{\partial e_2}{\partial i} |
---|
675 | -u\frac{\partial e_1}{\partial j} |
---|
676 | \right) {\rm {\bf k}} \times {\rm {\bf U}}_h |
---|
677 | \end{multline} |
---|
678 | |
---|
679 | The flux form has two terms, the first one is expressed as the divergence of momentum |
---|
680 | fluxes (hence the flux form name given to this formulation) and the second one is due to |
---|
681 | the curvilinear nature of the coordinate system used. The latter is called the \emph{metric} |
---|
682 | term and can be viewed as a modification of the Coriolis parameter: |
---|
683 | \begin{equation} \label{Eq_PE_cor+metric} |
---|
684 | f \to f + \frac{1}{e_1 \; e_2} \left( v \frac{\partial e_2}{\partial i} |
---|
685 | -u \frac{\partial e_1}{\partial j} \right) |
---|
686 | \end{equation} |
---|
687 | |
---|
688 | Note that in the case of geographical coordinate, $i.e.$ when $(i,j) \to (\lambda ,\varphi )$ |
---|
689 | and $(e_1 ,e_2) \to (a \,\cos \varphi ,a)$, we recover the commonly used modification of |
---|
690 | the Coriolis parameter $f \to f+(u/a) \tan \varphi$. |
---|
691 | |
---|
692 | To sum up, the equations solved by the ocean model can be written in the following tensorial formalism: |
---|
693 | |
---|
694 | \vspace{+10pt} |
---|
695 | $\bullet$ \textit{momentum equations} : |
---|
696 | |
---|
697 | vector invariant form : |
---|
698 | \begin{subequations} \label{Eq_PE_dyn_vect} |
---|
699 | \begin{multline} \label{Eq_PE_dyn_vect_u} |
---|
700 | \frac{\partial u}{\partial t}= |
---|
701 | + \left( {\zeta +f} \right)\,v |
---|
702 | - \frac{1}{2\,e_1} \frac{\partial}{\partial i} \left( u^2+v^2 \right) |
---|
703 | - \frac{1}{e_3} w \frac{\partial u}{\partial k} \\ |
---|
704 | - \frac{1}{e_1} \frac{\partial}{\partial i} \left( \frac{p_s+p_h }{\rho _o} \right) |
---|
705 | + D_u^{\vect{U}} + F_u^{\vect{U}} |
---|
706 | \end{multline} |
---|
707 | \begin{multline} \label{Eq_PE_dyn_vect_v} |
---|
708 | \frac{\partial v}{\partial t}= |
---|
709 | - \left( {\zeta +f} \right)\,u |
---|
710 | - \frac{1}{2\,e_2 }\frac{\partial }{\partial j}\left( u^2+v^2 \right) - \frac{1}{e_3 }w\frac{\partial v}{\partial k} \\ |
---|
711 | - \frac{1}{e_2 }\frac{\partial }{\partial j}\left( \frac{p_s+p_h }{\rho _o} \right) |
---|
712 | + D_v^{\vect{U}} + F_v^{\vect{U}} |
---|
713 | \end{multline} |
---|
714 | \end{subequations} |
---|
715 | |
---|
716 | flux form: |
---|
717 | \begin{subequations} \label{Eq_PE_dyn_flux} |
---|
718 | \begin{multline} \label{Eq_PE_dyn_flux_u} |
---|
719 | \frac{\partial u}{\partial t}= |
---|
720 | + \left( { f + \frac{1}{e_1 \; e_2} |
---|
721 | \left( v \frac{\partial e_2}{\partial i} |
---|
722 | -u \frac{\partial e_1}{\partial j} \right)} \right) \, v \\ |
---|
723 | - \frac{1}{e_1 \; e_2} \left( |
---|
724 | \frac{\partial \left( {e_2 \,u\,u} \right)}{\partial i} |
---|
725 | + \frac{\partial \left( {e_1 \,v\,u} \right)}{\partial j} \right) |
---|
726 | - \frac{1}{e_3 }\frac{\partial \left( { w\,u} \right)}{\partial k} \\ |
---|
727 | - \frac{1}{e_1 }\frac{\partial}{\partial i}\left( \frac{p_s+p_h }{\rho _o} \right) |
---|
728 | + D_u^{\vect{U}} + F_u^{\vect{U}} |
---|
729 | \end{multline} |
---|
730 | \begin{multline} \label{Eq_PE_dyn_flux_v} |
---|
731 | \frac{\partial v}{\partial t}= |
---|
732 | - \left( { f + \frac{1}{e_1 \; e_2} |
---|
733 | \left( v \frac{\partial e_2}{\partial i} |
---|
734 | -u \frac{\partial e_1}{\partial j} \right)} \right) \, u \\ |
---|
735 | \frac{1}{e_1 \; e_2} \left( |
---|
736 | \frac{\partial \left( {e_2 \,u\,v} \right)}{\partial i} |
---|
737 | + \frac{\partial \left( {e_1 \,v\,v} \right)}{\partial j} \right) |
---|
738 | - \frac{1}{e_3 } \frac{\partial \left( { w\,v} \right)}{\partial k} \\ |
---|
739 | - \frac{1}{e_2 }\frac{\partial }{\partial j}\left( \frac{p_s+p_h }{\rho _o} \right) |
---|
740 | + D_v^{\vect{U}} + F_v^{\vect{U}} |
---|
741 | \end{multline} |
---|
742 | \end{subequations} |
---|
743 | where $\zeta$ is given by \eqref{Eq_PE_curl_Uh} and the surface pressure gradient formulation |
---|
744 | depends on the one of the free surface: |
---|
745 | |
---|
746 | $*$ free surface formulation |
---|
747 | \begin{equation}\label{Eq_PE_dyn_sco} |
---|
748 | \frac{1}{\rho _o }\nabla _h p_s =\left( {{\begin{array}{*{20}c} |
---|
749 | {\frac{g}{\;e_1 }\frac{\partial \eta }{\partial i}} \hfill \\ |
---|
750 | {\frac{g}{\;e_2 }\frac{\partial \eta }{\partial j}} \hfill \\ |
---|
751 | \end{array} }} \right) |
---|
752 | \qquad \text{where $\eta$ is solution of \eqref{Eq_PE_ssh} } |
---|
753 | \end{equation} |
---|
754 | |
---|
755 | $*$ rigid-lid approximation |
---|
756 | \begin{equation}\label{Eq_PE_dyn_zco} |
---|
757 | \frac{1}{\rho _o }\nabla _h p_s =\left( {{\begin{array}{*{20}c} |
---|
758 | {\overline M _u +\frac{1}{H\;e_2 }\frac{\partial }{\partial j}\left( |
---|
759 | {\frac{\partial \psi }{\partial t}} \right)} \\ |
---|
760 | {\overline M _v -\frac{1}{H\;e_1 }\frac{\partial }{\partial i}\left( |
---|
761 | {\frac{\partial \psi }{\partial t}} \right)} \\ |
---|
762 | \end{array} }} \right) |
---|
763 | \end{equation} |
---|
764 | where ${\vect{M}}= \left( M_u,M_v \right)$ represents the collected contributions of nonlinear, |
---|
765 | viscosity and hydrostatic pressure gradient terms in \eqref{Eq_PE_dyn_vect} and the overbar |
---|
766 | indicates a vertical average over the whole water column ($i.e.$ from $z=-H$, the ocean bottom, |
---|
767 | to $z=0$, the rigid-lid), and where the time derivative of $\psi$ is the solution of an elliptic equation: |
---|
768 | \begin{multline} \label{Eq_psi_total} |
---|
769 | \frac{\partial }{\partial i}\left[ {\frac{e_2 }{H\,e_1}\frac{\partial}{\partial i} |
---|
770 | \left( {\frac{\partial \psi }{\partial t}} \right)} \right] |
---|
771 | +\frac{\partial }{\partial j}\left[ {\frac{e_1 }{H\,e_2}\frac{\partial }{\partial j} |
---|
772 | \left( {\frac{\partial \psi }{\partial t}} \right)} \right] |
---|
773 | = \\ |
---|
774 | \frac{\partial }{\partial i}\left( {e_2 \overline M _v } \right) |
---|
775 | - \frac{\partial }{\partial j}\left( {e_1 \overline M _u } \right) |
---|
776 | \end{multline} |
---|
777 | |
---|
778 | The vertical velocity and the hydrostatic pressure are diagnosed from the following equations: |
---|
779 | \begin{equation} \label{Eq_w_diag} |
---|
780 | \frac{\partial w}{\partial k}=-\chi \;e_3 |
---|
781 | \end{equation} |
---|
782 | \begin{equation} \label{Eq_hp_diag} |
---|
783 | \frac{\partial p_h }{\partial k}=-\rho \;g\;e_3 |
---|
784 | \end{equation} |
---|
785 | |
---|
786 | where the divergence of the horizontal velocity, $\chi$ is given by \eqref{Eq_PE_div_Uh}. |
---|
787 | |
---|
788 | \vspace{+10pt} |
---|
789 | $\bullet$ \textit{tracer equations} : |
---|
790 | \begin{equation} \label{Eq_S} |
---|
791 | \frac{\partial T}{\partial t} = |
---|
792 | -\frac{1}{e_1 e_2 }\left[ { \frac{\partial \left( {e_2 T\,u} \right)}{\partial i} |
---|
793 | +\frac{\partial \left( {e_1 T\,v} \right)}{\partial j}} \right] |
---|
794 | -\frac{1}{e_3 }\frac{\partial \left( {T\,w} \right)}{\partial k} + D^T + F^T |
---|
795 | \end{equation} |
---|
796 | \begin{equation} \label{Eq_T} |
---|
797 | \frac{\partial S}{\partial t} = |
---|
798 | -\frac{1}{e_1 e_2 }\left[ {\frac{\partial \left( {e_2 S\,u} \right)}{\partial i} |
---|
799 | +\frac{\partial \left( {e_1 S\,v} \right)}{\partial j}} \right] |
---|
800 | -\frac{1}{e_3 }\frac{\partial \left( {S\,w} \right)}{\partial k} + D^S + F^S |
---|
801 | \end{equation} |
---|
802 | \begin{equation} \label{Eq_rho} |
---|
803 | \rho =\rho \left( {T,S,z(k)} \right) |
---|
804 | \end{equation} |
---|
805 | |
---|
806 | The expression of \textbf{D}$^{U}$, $D^{S}$ and $D^{T}$ depends on the subgrid scale |
---|
807 | parameterisation used. It will be defined in \S\ref{PE_zdf}. The nature and formulation of |
---|
808 | ${\rm {\bf F}}^{\rm {\bf U}}$, $F^T$ and $F^S$, the surface forcing terms, are discussed |
---|
809 | in Chapter~\ref{SBC}. |
---|
810 | |
---|
811 | \newpage |
---|
812 | % ================================================================ |
---|
813 | % Curvilinear z*-coordinate System |
---|
814 | % ================================================================ |
---|
815 | \section{Curvilinear \textit{z*}-coordinate System} |
---|
816 | |
---|
817 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
---|
818 | \begin{figure}[!b] \label{Fig_z_zstar} \begin{center} |
---|
819 | \includegraphics[width=1.0\textwidth]{./TexFiles/Figures/Fig_z_zstar.pdf} |
---|
820 | \caption{(a) $z$-coordinate in linear free-surface case ; (b) $z-$coordinate in non-linear |
---|
821 | free surface case (c) re-scaled height coordinate (become popular as the \textit{z*-}coordinate |
---|
822 | \citep{Adcroft_Campin_OM04} ).} |
---|
823 | \end{center} \end{figure} |
---|
824 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
---|
825 | |
---|
826 | |
---|
827 | In that case, the free surface equation is nonlinear, and the variations of volume are fully |
---|
828 | taken into account. These coordinates systems is presented in a report \citep{Levier2007} |
---|
829 | available on the \NEMO web site. |
---|
830 | |
---|
831 | \gmcomment{ |
---|
832 | The \textit{z*} coordinate approach is an unapproximated, non-linear free surface implementation |
---|
833 | which allows one to deal with large amplitude free-surface |
---|
834 | variations relative to the vertical resolution \citep{Adcroft_Campin_OM04}. In |
---|
835 | the \textit{z*} formulation, the variation of the column thickness due to sea-surface |
---|
836 | undulations is not concentrated in the surface level, as in the $z$-coordinate formulation, |
---|
837 | but is equally distributed over the full water column. Thus vertical |
---|
838 | levels naturally follow sea-surface variations, with a linear attenuation with |
---|
839 | depth, as illustrated by figure fig.1c . Note that with a flat bottom, such as in |
---|
840 | fig.1c, the bottom-following $z$ coordinate and \textit{z*} are equivalent. |
---|
841 | The definition and modified oceanic equations for the rescaled vertical coordinate |
---|
842 | \textit{z*}, including the treatment of fresh-water flux at the surface, are |
---|
843 | detailed in Adcroft and Campin (2004). The major points are summarized |
---|
844 | here. The position ( \textit{z*}) and vertical discretization (\textit{z*}) are expressed as: |
---|
845 | |
---|
846 | $H + \textit{z*} = (H + z) / r$ and $\delta \textit{z*} = \delta z / r$ with $r = \frac{H+\eta} {H}$ |
---|
847 | |
---|
848 | Since the vertical displacement of the free surface is incorporated in the vertical |
---|
849 | coordinate \textit{z*}, the upper and lower boundaries are at fixed \textit{z*} position, |
---|
850 | $\textit{z*} = 0$ and $\textit{z*} = ?H$ respectively. Also the divergence of the flow field |
---|
851 | is no longer zero as shown by the continuity equation: |
---|
852 | |
---|
853 | $\frac{\partial r}{\partial t} = \nabla_{\textit{z*}} \cdot \left( r \; \rm{\bf U}_h \right) |
---|
854 | \left( r \; w\textit{*} \right) = 0 $ |
---|
855 | |
---|
856 | } |
---|
857 | |
---|
858 | |
---|
859 | \newpage |
---|
860 | % ================================================================ |
---|
861 | % Curvilinear s-coordinate System |
---|
862 | % ================================================================ |
---|
863 | \section{Curvilinear \textit{s}-coordinate System} |
---|
864 | \label{PE_sco} |
---|
865 | |
---|
866 | % ------------------------------------------------------------------------------------------------------------- |
---|
867 | % Introduction |
---|
868 | % ------------------------------------------------------------------------------------------------------------- |
---|
869 | \subsection{Introduction} |
---|
870 | |
---|
871 | Several important aspects of the ocean circulation are influenced by bottom topography. |
---|
872 | Of course, the most important is that bottom topography determines deep ocean sub-basins, |
---|
873 | barriers, sills and channels that strongly constrain the path of water masses, but more subtle |
---|
874 | effects exist. For example, the topographic $\beta$-effect is usually larger than the planetary |
---|
875 | one along continental slopes. Topographic Rossby waves can be excited and can interact |
---|
876 | with the mean current. In the $z-$coordinate system presented in the previous section |
---|
877 | (\S\ref{PE_zco}), $z-$surfaces are geopotential surfaces. The bottom topography is |
---|
878 | discretised by steps. This often leads to a misrepresentation of a gradually sloping bottom |
---|
879 | and to large localized depth gradients associated with large localized vertical velocities. |
---|
880 | The response to such a velocity field often leads to numerical dispersion effects. |
---|
881 | One solution to strongly reduce this error is to use a partial step representation of bottom |
---|
882 | topography instead of a full step one \cite{Pacanowski_Gnanadesikan_MWR98}. |
---|
883 | Another solution is to introduce a terrain-following coordinate system (hereafter $s-$coordinate) |
---|
884 | |
---|
885 | The $s$-coordinate avoids the discretisation error in the depth field since the layers of |
---|
886 | computation are gradually adjusted with depth to the ocean bottom. Relatively small |
---|
887 | topographic features as well as gentle, large-scale slopes of the sea floor in the deep |
---|
888 | ocean, which would be ignored in typical $z$-model applications with the largest grid |
---|
889 | spacing at greatest depths, can easily be represented (with relatively low vertical resolution). |
---|
890 | A terrain-following model (hereafter $s-$model) also facilitates the modelling of the |
---|
891 | boundary layer flows over a large depth range, which in the framework of the $z$-model |
---|
892 | would require high vertical resolution over the whole depth range. Moreover, with a |
---|
893 | $s$-coordinate it is possible, at least in principle, to have the bottom and the sea surface |
---|
894 | as the only boundaries of the domain (nomore lateral boundary condition to specify). |
---|
895 | Nevertheless, a $s$-coordinate also has its drawbacks. Perfectly adapted to a |
---|
896 | homogeneous ocean, it has strong limitations as soon as stratification is introduced. |
---|
897 | The main two problems come from the truncation error in the horizontal pressure |
---|
898 | gradient and a possibly increased diapycnal diffusion. The horizontal pressure force |
---|
899 | in $s$-coordinate consists of two terms (see Appendix~\ref{Apdx_A}), |
---|
900 | |
---|
901 | \begin{equation} \label{Eq_PE_p_sco} |
---|
902 | \left. {\nabla p} \right|_z =\left. {\nabla p} \right|_s -\frac{\partial |
---|
903 | p}{\partial s}\left. {\nabla z} \right|_s |
---|
904 | \end{equation} |
---|
905 | |
---|
906 | The second term in \eqref{Eq_PE_p_sco} depends on the tilt of the coordinate surface |
---|
907 | and introduces a truncation error that is not present in a $z$-model. In the special case |
---|
908 | of a $\sigma-$coordinate (i.e. a depth-normalised coordinate system $\sigma = z/H$), |
---|
909 | \citet{Haney1991} and \citet{Beckmann1993} have given estimates of the magnitude |
---|
910 | of this truncation error. It depends on topographic slope, stratification, horizontal and |
---|
911 | vertical resolution, the equation of state, and the finite difference scheme. This error |
---|
912 | limits the possible topographic slopes that a model can handle at a given horizontal |
---|
913 | and vertical resolution. This is a severe restriction for large-scale applications using |
---|
914 | realistic bottom topography. The large-scale slopes require high horizontal resolution, |
---|
915 | and the computational cost becomes prohibitive. This problem can be at least partially |
---|
916 | overcome by mixing $s$-coordinate and step-like representation of bottom topography \citep{Gerdes1993a,Gerdes1993b,Madec1996}. However, the definition of the model |
---|
917 | domain vertical coordinate becomes then a non-trivial thing for a realistic bottom |
---|
918 | topography: a envelope topography is defined in $s$-coordinate on which a full or |
---|
919 | partial step bottom topography is then applied in order to adjust the model depth to |
---|
920 | the observed one (see \S\ref{DOM_zgr}. |
---|
921 | |
---|
922 | For numerical reasons a minimum of diffusion is required along the coordinate surfaces |
---|
923 | of any finite difference model. It causes spurious diapycnal mixing when coordinate |
---|
924 | surfaces do not coincide with isoneutral surfaces. This is the case for a $z$-model as |
---|
925 | well as for a $s$-model. However, density varies more strongly on $s-$surfaces than |
---|
926 | on horizontal surfaces in regions of large topographic slopes, implying larger diapycnal |
---|
927 | diffusion in a $s$-model than in a $z$-model. Whereas such a diapycnal diffusion in a |
---|
928 | $z$-model tends to weaken horizontal density (pressure) gradients and thus the horizontal |
---|
929 | circulation, it usually reinforces these gradients in a $s$-model, creating spurious circulation. |
---|
930 | For example, imagine an isolated bump of topography in an ocean at rest with a horizontally |
---|
931 | uniform stratification. Spurious diffusion along $s$-surfaces will induce a bump of isoneutral |
---|
932 | surfaces over the topography, and thus will generate there a baroclinic eddy. In contrast, |
---|
933 | the ocean will stay at rest in a $z$-model. As for the truncation error, the problem can be reduced by introducing the terrain-following coordinate below the strongly stratified portion of the water column |
---|
934 | ($i.e.$ the main thermocline) \citep{Madec1996}. An alternate solution consists of rotating |
---|
935 | the lateral diffusive tensor to geopotential or to isoneutral surfaces (see \S\ref{PE_ldf}. |
---|
936 | Unfortunately, the slope of isoneutral surfaces relative to the $s$-surfaces can very large, |
---|
937 | strongly exceeding the stability limit of such a operator when it is discretized (see Chapter~\ref{LDF}). |
---|
938 | |
---|
939 | The $s-$coordinates introduced here \citep{Lott1990,Madec1996} differ mainly in two |
---|
940 | aspects from similar models: it allows a representation of bottom topography with mixed |
---|
941 | full or partial step-like/terrain following topography ; It also offers a completely general |
---|
942 | transformation, $s=s(i,j,z)$ for the vertical coordinate. |
---|
943 | |
---|
944 | % ------------------------------------------------------------------------------------------------------------- |
---|
945 | % The s-coordinate Formulation |
---|
946 | % ------------------------------------------------------------------------------------------------------------- |
---|
947 | \subsection{The \textit{s-}coordinate Formulation} |
---|
948 | |
---|
949 | Starting from the set of equations established in \S\ref{PE_zco} for the special case $k=z$ |
---|
950 | and thus $e_3=1$, we introduce an arbitrary vertical coordinate $s=s(i,j,k,t)$, which includes |
---|
951 | $z$-, \textit{z*}- and $\sigma-$coordinates as special cases ($s=z$, $s=\textit{z*}$, and |
---|
952 | $s=\sigma=z/H$ or $=z/\left(H+\eta \right)$, resp.). A formal derivation of the transformed |
---|
953 | equations is given in Appendix~\ref{Apdx_A}. Let us define the vertical scale factor by |
---|
954 | $e_3=\partial_s z$ ($e_3$ is now a function of $(i,j,k,t)$ ), and the slopes in the |
---|
955 | (\textbf{i},\textbf{j}) directions between $s-$ and $z-$surfaces by : |
---|
956 | \begin{equation} \label{Eq_PE_sco_slope} |
---|
957 | \sigma _1 =\frac{1}{e_1 }\;\left. {\frac{\partial z}{\partial i}} \right|_s |
---|
958 | \quad \text{, and } \quad |
---|
959 | \sigma _2 =\frac{1}{e_2 }\;\left. {\frac{\partial z}{\partial j}} \right|_s |
---|
960 | \end{equation} |
---|
961 | We also introduce $\omega $, a dia-surface velocity component, defined as the velocity |
---|
962 | relative to the moving $s$-surfaces and normal to them: |
---|
963 | \begin{equation} \label{Eq_PE_sco_w} |
---|
964 | \omega = w - e_3 \, \frac{\partial s}{\partial t} - \sigma _1 \,u - \sigma _2 \,v \\ |
---|
965 | \end{equation} |
---|
966 | |
---|
967 | The equations solved by the ocean model \eqref{Eq_PE} in $s-$coordinate can be written as follows: |
---|
968 | |
---|
969 | \vspace{0.5cm} |
---|
970 | * momentum equation: |
---|
971 | \begin{multline} \label{Eq_PE_sco_u} |
---|
972 | \frac{1}{e_3} \frac{\partial \left( e_3\,u \right) }{\partial t}= |
---|
973 | + \left( {\zeta +f} \right)\,v |
---|
974 | - \frac{1}{2\,e_1} \frac{\partial}{\partial i} \left( u^2+v^2 \right) |
---|
975 | - \frac{1}{e_3} \omega \frac{\partial u}{\partial k} \\ |
---|
976 | - \frac{1}{e_1} \frac{\partial}{\partial i} \left( \frac{p_s + p_h}{\rho _o} \right) |
---|
977 | + g\frac{\rho }{\rho _o}\sigma _1 |
---|
978 | + D_u^{\vect{U}} + F_u^{\vect{U}} \quad |
---|
979 | \end{multline} |
---|
980 | \begin{multline} \label{Eq_PE_sco_v} |
---|
981 | \frac{1}{e_3} \frac{\partial \left( e_3\,v \right) }{\partial t}= |
---|
982 | - \left( {\zeta +f} \right)\,u |
---|
983 | - \frac{1}{2\,e_2 }\frac{\partial }{\partial j}\left( u^2+v^2 \right) |
---|
984 | - \frac{1}{e_3 } \omega \frac{\partial v}{\partial k} \\ |
---|
985 | - \frac{1}{e_2 }\frac{\partial }{\partial j}\left( \frac{p_s+p_h }{\rho _o} \right) |
---|
986 | + g\frac{\rho }{\rho _o }\sigma _2 |
---|
987 | + D_v^{\vect{U}} + F_v^{\vect{U}} \quad |
---|
988 | \end{multline} |
---|
989 | where the relative vorticity, \textit{$\zeta $}, the surface pressure gradient, and the hydrostatic |
---|
990 | pressure have the same expressions as in $z$-coordinates although they do not represent |
---|
991 | exactly the same quantities. $\omega$ is provided by the continuity equation |
---|
992 | (see Appendix~\ref{Apdx_A}): |
---|
993 | |
---|
994 | \begin{equation} \label{Eq_PE_sco_continuity} |
---|
995 | \frac{\partial e_3}{\partial t} + e_3 \; \chi + \frac{\partial \omega }{\partial s} = 0 |
---|
996 | \qquad \text{with }\;\; |
---|
997 | \chi =\frac{1}{e_1 e_2 e_3 }\left[ {\frac{\partial \left( {e_2 e_3 \,u} |
---|
998 | \right)}{\partial i}+\frac{\partial \left( {e_1 e_3 \,v} \right)}{\partial |
---|
999 | j}} \right] |
---|
1000 | \end{equation} |
---|
1001 | |
---|
1002 | \vspace{0.5cm} |
---|
1003 | * tracer equations: |
---|
1004 | \begin{multline} \label{Eq_PE_sco_t} |
---|
1005 | \frac{1}{e_3} \frac{\partial \left( e_3\,T \right) }{\partial t}= |
---|
1006 | -\frac{1}{e_1 e_2 e_3 }\left[ {\frac{\partial \left( {e_2 e_3\,u\,T} \right)}{\partial i} |
---|
1007 | +\frac{\partial \left( {e_1 e_3\,v\,T} \right)}{\partial j}} \right] \\ |
---|
1008 | -\frac{1}{e_3 }\frac{\partial \left( {T\,\omega } \right)}{\partial k} + D^T + F^S \qquad |
---|
1009 | \end{multline} |
---|
1010 | |
---|
1011 | \begin{multline} \label{Eq_PE_sco_s} |
---|
1012 | \frac{1}{e_3} \frac{\partial \left( e_3\,S \right) }{\partial t}= |
---|
1013 | -\frac{1}{e_1 e_2 e_3 }\left[ {\frac{\partial \left( {e_2 e_3\,u\,S} \right)}{\partial i} |
---|
1014 | +\frac{\partial \left( {e_1 e_3\,v\,S} \right)}{\partial j}} \right] \\ |
---|
1015 | -\frac{1}{e_3 }\frac{\partial \left( {S\,\omega } \right)}{\partial k} + D^S + F^S \qquad |
---|
1016 | \end{multline} |
---|
1017 | |
---|
1018 | The equation of state has the same expression as in $z$-coordinate, and similar expressions |
---|
1019 | are used for mixing and forcing terms. |
---|
1020 | |
---|
1021 | \gmcomment{ |
---|
1022 | \colorbox{yellow}{ to be updated $= = >$} |
---|
1023 | Add a few works on z and zps and s and underlies the differences between all of them |
---|
1024 | \colorbox{yellow}{ $< = =$ end update} } |
---|
1025 | |
---|
1026 | \newpage |
---|
1027 | % ================================================================ |
---|
1028 | % Subgrid Scale Physics |
---|
1029 | % ================================================================ |
---|
1030 | \section{Subgrid Scale Physics} |
---|
1031 | \label{PE_zdf_ldf} |
---|
1032 | |
---|
1033 | The primitive equations describe the behaviour of a geophysical fluid at |
---|
1034 | space and time scales larger than a few kilometres in the horizontal, a few |
---|
1035 | meters in the vertical and a few minutes. They are usually solved at larger |
---|
1036 | scales: the specified grid spacing and time step of the numerical model. The |
---|
1037 | effects of smaller scale motions (coming from the advective terms in the |
---|
1038 | Navier-Stokes equations) must be represented entirely in terms of |
---|
1039 | large-scale patterns to close the equations. These effects appear in the |
---|
1040 | equations as the divergence of turbulent fluxes ($i.e.$ fluxes associated with |
---|
1041 | the mean correlation of small scale perturbations). Assuming a turbulent |
---|
1042 | closure hypothesis is equivalent to choose a formulation for these fluxes. |
---|
1043 | It is usually called the subgrid scale physics. It must be emphasized that |
---|
1044 | this is the weakest part of the primitive equations, but also one of the |
---|
1045 | most important for long-term simulations as small scale processes \textit{in fine} |
---|
1046 | balance the surface input of kinetic energy and heat. |
---|
1047 | |
---|
1048 | The control exerted by gravity on the flow induces a strong anisotropy |
---|
1049 | between the lateral and vertical motions. Therefore subgrid-scale physics |
---|
1050 | \textbf{D}$^{\vect{U}}$, $D^{S}$ and $D^{T}$ in \eqref{Eq_PE_dyn}, |
---|
1051 | \eqref{Eq_PE_tra_T} and \eqref{Eq_PE_tra_S} are divided into a lateral part |
---|
1052 | \textbf{D}$^{l \vect{U}}$, $D^{lS}$ and $D^{lT}$ and a vertical part |
---|
1053 | \textbf{D}$^{vU}$, $D^{vS}$ and $D^{vT}$. The formulation of these terms |
---|
1054 | and their underlying physics are briefly discussed in the next two subsections. |
---|
1055 | |
---|
1056 | % ------------------------------------------------------------------------------------------------------------- |
---|
1057 | % Vertical Subgrid Scale Physics |
---|
1058 | % ------------------------------------------------------------------------------------------------------------- |
---|
1059 | \subsection{Vertical Subgrid Scale Physics} |
---|
1060 | \label{PE_zdf} |
---|
1061 | |
---|
1062 | The model resolution is always larger than the scale at which the major |
---|
1063 | sources of vertical turbulence occur (shear instability, internal wave |
---|
1064 | breaking...). Turbulent motions are thus never explicitly solved, even |
---|
1065 | partially, but always parameterized. The vertical turbulent fluxes are |
---|
1066 | assumed to depend linearly on the gradients of large-scale quantities (for |
---|
1067 | example, the turbulent heat flux is given by $\overline{T'w'}=-A^{vT} \partial_z \overline T$, |
---|
1068 | where $A^{vT}$ is an eddy coefficient). This formulation is |
---|
1069 | analogous to that of molecular diffusion and dissipation. This is quite |
---|
1070 | clearly a necessary compromise: considering only the molecular viscosity |
---|
1071 | acting on large scale severely underestimates the role of turbulent |
---|
1072 | diffusion and dissipation, while an accurate consideration of the details of |
---|
1073 | turbulent motions is simply impractical. The resulting vertical momentum and |
---|
1074 | tracer diffusive operators are of second order: |
---|
1075 | \begin{equation} \label{Eq_PE_zdf} |
---|
1076 | \begin{split} |
---|
1077 | {\vect{D}}^{v \vect{U}} &=\frac{\partial }{\partial z}\left( {A^{vm}\frac{\partial {\vect{U}}_h }{\partial z}} \right) \ , \\ |
---|
1078 | D^{vT} &= \frac{\partial }{\partial z}\left( {A^{vT}\frac{\partial T}{\partial z}} \right) \ , |
---|
1079 | \quad |
---|
1080 | D^{vS}=\frac{\partial }{\partial z}\left( {A^{vT}\frac{\partial S}{\partial z}} \right) |
---|
1081 | \end{split} |
---|
1082 | \end{equation} |
---|
1083 | where $A^{vm}$ and $A^{vT}$ are the vertical eddy viscosity and diffusivity coefficients, |
---|
1084 | respectively. At the sea surface and at the bottom, turbulent fluxes of momentum, heat |
---|
1085 | and salt must be specified (see Chap.~\ref{SBC} and \ref{ZDF} and \S\ref{TRA_bbl}). |
---|
1086 | All the vertical physics is embedded in the specification of the eddy coefficients. |
---|
1087 | They can be assumed to be either constant, or function of the local fluid properties |
---|
1088 | ($e.g.$ Richardson number, Brunt-Vais\"{a}l\"{a} frequency...), or computed from a |
---|
1089 | turbulent closure model. The choices available in \NEMO are discussed in \S\ref{ZDF}). |
---|
1090 | |
---|
1091 | % ------------------------------------------------------------------------------------------------------------- |
---|
1092 | % Lateral Diffusive and Viscous Operators Formulation |
---|
1093 | % ------------------------------------------------------------------------------------------------------------- |
---|
1094 | \subsection{Lateral Diffusive and Viscous Operators Formulation} |
---|
1095 | \label{PE_ldf} |
---|
1096 | |
---|
1097 | Lateral turbulence can be roughly divided into a mesoscale turbulence |
---|
1098 | associated with eddies (which can be solved explicitly if the resolution is |
---|
1099 | sufficient since their underlying physics are included in the primitive |
---|
1100 | equations), and a sub mesoscale turbulence which is never explicitly solved |
---|
1101 | even partially, but always parameterized. The formulation of lateral eddy |
---|
1102 | fluxes depends on whether the mesoscale is below or above the grid-spacing |
---|
1103 | ($i.e.$ the model is eddy-resolving or not). |
---|
1104 | |
---|
1105 | In non-eddy-resolving configurations, the closure is similar to that used |
---|
1106 | for the vertical physics. The lateral turbulent fluxes are assumed to depend |
---|
1107 | linearly on the lateral gradients of large-scale quantities. The resulting |
---|
1108 | lateral diffusive and dissipative operators are of second order. |
---|
1109 | Observations show that lateral mixing induced by mesoscale turbulence tends |
---|
1110 | to be along isopycnal surfaces (or more precisely neutral surfaces \cite{McDougall1987}) |
---|
1111 | rather than across them. |
---|
1112 | As the slope of neutral surfaces is small in the ocean, a common |
---|
1113 | approximation is to assume that the `lateral' direction is the horizontal, |
---|
1114 | $i.e.$ the lateral mixing is performed along geopotential surfaces. This leads |
---|
1115 | to a geopotential second order operator for lateral subgrid scale physics. |
---|
1116 | This assumption can be relaxed: the eddy-induced turbulent fluxes can be |
---|
1117 | better approached by assuming that they depend linearly on the gradients of |
---|
1118 | large-scale quantities computed along neutral surfaces. In such a case, |
---|
1119 | the diffusive operator is an isoneutral second order operator and it has |
---|
1120 | components in the three space directions. However, both horizontal and |
---|
1121 | isoneutral operators have no effect on mean ($i.e.$ large scale) potential |
---|
1122 | energy whereas potential energy is a main source of turbulence (through |
---|
1123 | baroclinic instabilities). \citet{Gent1990} have proposed a |
---|
1124 | parameterisation of mesoscale eddy-induced turbulence which associates an |
---|
1125 | eddy-induced velocity to the isoneutral diffusion. Its mean effect is to |
---|
1126 | reduce the mean potential energy of the ocean. This leads to a formulation |
---|
1127 | of lateral subgrid-scale physics made up of an isoneutral second order |
---|
1128 | operator and an eddy induced advective part. In all these lateral diffusive |
---|
1129 | formulations, the specification of the lateral eddy coefficients remains the |
---|
1130 | problematic point as there is no really satisfactory formulation of these |
---|
1131 | coefficients as a function of large-scale features. |
---|
1132 | |
---|
1133 | In eddy-resolving configurations, a second order operator can be used, but |
---|
1134 | usually a more scale selective one (biharmonic operator) is preferred as the |
---|
1135 | grid-spacing is usually not small enough compared to the scale of the |
---|
1136 | eddies. The role devoted to the subgrid-scale physics is to dissipate the |
---|
1137 | energy that cascades toward the grid scale and thus ensures the stability of |
---|
1138 | the model while not interfering with the solved mesoscale activity. Another approach |
---|
1139 | is becoming more and more popular: instead of specifying explicitly a sub-grid scale |
---|
1140 | term in the momentum and tracer time evolution equations, one uses a advective |
---|
1141 | scheme which is diffusive enough to maintain the model stability. It must be emphasised |
---|
1142 | that then, all the sub-grid scale physics is in this case include in the formulation of the |
---|
1143 | advection scheme. |
---|
1144 | |
---|
1145 | All these parameterisations of subgrid scale physics present advantages and |
---|
1146 | drawbacks. There are not all available in \NEMO. In the $z$-coordinate |
---|
1147 | formulation, five options are offered for active tracers (temperature and |
---|
1148 | salinity): second order geopotential operator, second order isoneutral |
---|
1149 | operator, \citet{Gent1990} parameterisation, fourth order |
---|
1150 | geopotential operator, and various slightly diffusive advection schemes. |
---|
1151 | The same options are available for momentum, except |
---|
1152 | \citet{Gent1990} parameterisation which only involves tracers. In the |
---|
1153 | $s$-coordinate formulation, additional options are offered for tracers: second |
---|
1154 | order operator acting along $s-$surfaces, and for momentum: fourth order |
---|
1155 | operator acting along $s-$surfaces (see \S\ref{LDF}). |
---|
1156 | |
---|
1157 | \subsubsection{lateral second order tracer diffusive operator} |
---|
1158 | |
---|
1159 | The lateral second order tracer diffusive operator is defined by (see Appendix~\ref{Apdx_B}): |
---|
1160 | \begin{equation} \label{Eq_PE_iso_tensor} |
---|
1161 | D^{lT}=\nabla {\rm {\bf .}}\left( {A^{lT}\;\Re \;\nabla T} \right) \qquad |
---|
1162 | \mbox{with}\quad \;\;\Re =\left( {{\begin{array}{*{20}c} |
---|
1163 | 1 \hfill & 0 \hfill & {-r_1 } \hfill \\ |
---|
1164 | 0 \hfill & 1 \hfill & {-r_2 } \hfill \\ |
---|
1165 | {-r_1 } \hfill & {-r_2 } \hfill & {r_1 ^2+r_2 ^2} \hfill \\ |
---|
1166 | \end{array} }} \right) |
---|
1167 | \end{equation} |
---|
1168 | where $r_1 \;\mbox{and}\;r_2 $ are the slopes between the surface along |
---|
1169 | which the diffusive operator acts and the model level ($e. g.$ $z$- or |
---|
1170 | $s$-surfaces). Note that the formulation \eqref{Eq_PE_iso_tensor} is exact for the |
---|
1171 | rotation between geopotential and $s$-surfaces, while it is only an approximation |
---|
1172 | for the rotation between isoneutral and $z$- or $s$-surfaces. Indeed, in the latter |
---|
1173 | case, two assumptions are made to simplify \eqref{Eq_PE_iso_tensor} \citep{Cox1987}. |
---|
1174 | First, the horizontal contribution of the dianeutral mixing is neglected since the ratio |
---|
1175 | between iso and dia-neutral diffusive coefficients is known to be several orders of |
---|
1176 | magnitude smaller than unity. Second, the two isoneutral directions of diffusion are |
---|
1177 | assumed to be independent since the slopes are generally less than $10^{-2}$ in the |
---|
1178 | ocean (see Appendix~\ref{Apdx_B}). |
---|
1179 | |
---|
1180 | For \textit{geopotential} diffusion, $r_1$ and $r_2 $ are the slopes between the |
---|
1181 | geopotential and computational surfaces: in $z$-coordinates they are zero |
---|
1182 | ($r_1 = r_2 = 0$) while in $s$-coordinate (including $\textit{z*}$ case) they are |
---|
1183 | equal to $\sigma _1$ and $\sigma _2$, respectively (see \eqref{Eq_PE_sco_slope} ). |
---|
1184 | |
---|
1185 | For \textit{isoneutral} diffusion $r_1$ and $r_2$ are the slopes between the isoneutral |
---|
1186 | and computational surfaces. Therefore, they have a same expression in $z$- and $s$-coordinates: |
---|
1187 | \begin{equation} \label{Eq_PE_iso_slopes} |
---|
1188 | r_1 =\frac{e_3 }{e_1 } \left( {\frac{\partial \rho }{\partial i}} \right) |
---|
1189 | \left( {\frac{\partial \rho }{\partial k}} \right)^{-1} \ , \quad |
---|
1190 | r_1 =\frac{e_3 }{e_1 } \left( {\frac{\partial \rho }{\partial i}} \right) |
---|
1191 | \left( {\frac{\partial \rho }{\partial k}} \right)^{-1} |
---|
1192 | \end{equation} |
---|
1193 | |
---|
1194 | When the \textit{Eddy Induced Velocity} parametrisation (eiv) \citep{Gent1990} is used, |
---|
1195 | an additional tracer advection is introduced in combination with the isoneutral diffusion of tracers: |
---|
1196 | \begin{equation} \label{Eq_PE_iso+eiv} |
---|
1197 | D^{lT}=\nabla \cdot \left( {A^{lT}\;\Re \;\nabla T} \right) |
---|
1198 | +\nabla \cdot \left( {{\vect{U}}^\ast \,T} \right) |
---|
1199 | \end{equation} |
---|
1200 | where ${\vect{U}}^\ast =\left( {u^\ast ,v^\ast ,w^\ast } \right)$ is a non-divergent, |
---|
1201 | eddy-induced transport velocity. This velocity field is defined by: |
---|
1202 | \begin{equation} \label{Eq_PE_eiv} |
---|
1203 | \begin{split} |
---|
1204 | u^\ast &= +\frac{1}{e_3 }\frac{\partial }{\partial k}\left[ {A^{eiv}\;\tilde{r}_1 } \right] \\ |
---|
1205 | v^\ast &= +\frac{1}{e_3 }\frac{\partial }{\partial k}\left[ {A^{eiv}\;\tilde{r}_2 } \right] \\ |
---|
1206 | w^\ast &= -\frac{1}{e_1 e_2 }\left[ |
---|
1207 | \frac{\partial }{\partial i}\left( {A^{eiv}\;e_2\,\tilde{r}_1 } \right) |
---|
1208 | +\frac{\partial }{\partial j}\left( {A^{eiv}\;e_1\,\tilde{r}_2 } \right) \right] |
---|
1209 | \end{split} |
---|
1210 | \end{equation} |
---|
1211 | where $A^{eiv}$ is the eddy induced velocity coefficient (or equivalently the isoneutral |
---|
1212 | thickness diffusivity coefficient), and $\tilde{r}_1$ and $\tilde{r}_2$ are the slopes |
---|
1213 | between isoneutral and \emph{geopotential} surfaces and thus depends on the coordinate |
---|
1214 | considered: |
---|
1215 | \begin{align} \label{Eq_PE_slopes_eiv} |
---|
1216 | \tilde{r}_n = \begin{cases} |
---|
1217 | r_n & \text{in $z$-coordinate} \\ |
---|
1218 | r_n + \sigma_n & \text{in \textit{z*} and $s$-coordinates} |
---|
1219 | \end{cases} |
---|
1220 | \quad \text{where } n=1,2 |
---|
1221 | \end{align} |
---|
1222 | |
---|
1223 | The normal component of the eddy induced velocity is zero at all the boundaries. |
---|
1224 | This can be achieved in a model by tapering either the eddy coefficient or the slopes |
---|
1225 | to zero in the vicinity of the boundaries. The latter strategy is used in \NEMO (cf. Chap.~\ref{LDF}). |
---|
1226 | |
---|
1227 | \subsubsection{lateral fourth order tracer diffusive operator} |
---|
1228 | |
---|
1229 | The lateral fourth order tracer diffusive operator is defined by: |
---|
1230 | \begin{equation} \label{Eq_PE_bilapT} |
---|
1231 | D^{lT}=\Delta \left( {A^{lT}\;\Delta T} \right) |
---|
1232 | \qquad \text{where} \ D^{lT}=\Delta \left( {A^{lT}\;\Delta T} \right) |
---|
1233 | \end{equation} |
---|
1234 | |
---|
1235 | It is the second order operator given by \eqref{Eq_PE_iso_tensor} applied twice with |
---|
1236 | the eddy diffusion coefficient correctly placed. |
---|
1237 | |
---|
1238 | |
---|
1239 | \subsubsection{lateral second order momentum diffusive operator} |
---|
1240 | |
---|
1241 | The second order momentum diffusive operator along $z$- or $s$-surfaces is found by |
---|
1242 | applying \eqref{Eq_PE_lap_vector} to the horizontal velocity vector (see Appendix~\ref{Apdx_B}): |
---|
1243 | \begin{equation} \label{Eq_PE_lapU} |
---|
1244 | \begin{split} |
---|
1245 | {\rm {\bf D}}^{l{\rm {\bf U}}} |
---|
1246 | &= \quad \ \nabla _h \left( {A^{lm}\chi } \right) |
---|
1247 | \ - \ \nabla _h \times \left( {A^{lm}\,\zeta \;{\rm {\bf k}}} \right) \\ |
---|
1248 | &= \left( \begin{aligned} |
---|
1249 | \frac{1}{e_1 } \frac{\partial \left( A^{lm} \chi \right)}{\partial i} |
---|
1250 | &-\frac{1}{e_2 e_3}\frac{\partial \left( {A^{lm} \;e_3 \zeta} \right)}{\partial j} \\ |
---|
1251 | \frac{1}{e_2 }\frac{\partial \left( {A^{lm} \chi } \right)}{\partial j} |
---|
1252 | &+\frac{1}{e_1 e_3}\frac{\partial \left( {A^{lm} \;e_3 \zeta} \right)}{\partial i} |
---|
1253 | \end{aligned} \right) |
---|
1254 | \end{split} |
---|
1255 | \end{equation} |
---|
1256 | |
---|
1257 | Such a formulation ensures a complete separation between the vorticity and |
---|
1258 | horizontal divergence fields (see Appendix~\ref{Apdx_C}). Unfortunately, it is not |
---|
1259 | available for geopotential diffusion in $s-$coordinates and for isoneutral |
---|
1260 | diffusion in both $z$- and $s$-coordinates ($i.e.$ when a rotation is required). |
---|
1261 | In these two cases, the $u$ and $v-$fields are considered as independent scalar |
---|
1262 | fields, so that the diffusive operator is given by: |
---|
1263 | \begin{equation} \label{Eq_PE_lapU_iso} |
---|
1264 | \begin{split} |
---|
1265 | D_u^{l{\rm {\bf U}}} &= \nabla .\left( {\Re \;\nabla u} \right) \\ |
---|
1266 | D_v^{l{\rm {\bf U}}} &= \nabla .\left( {\Re \;\nabla v} \right) |
---|
1267 | \end{split} |
---|
1268 | \end{equation} |
---|
1269 | where $\Re$ is given by \eqref{Eq_PE_iso_tensor}. It is the same expression as |
---|
1270 | those used for diffusive operator on tracers. It must be emphasised that such a |
---|
1271 | formulation is only exact in a Cartesian coordinate system, $i.e.$ on a $f-$ or |
---|
1272 | $\beta-$plane, not on the sphere. It is also a very good approximation in vicinity |
---|
1273 | of the Equator in a geographical coordinate system \citep{Lengaigne_al_JGR03}. |
---|
1274 | |
---|
1275 | \subsubsection{lateral fourth order momentum diffusive operator} |
---|
1276 | |
---|
1277 | As for tracers, the fourth order momentum diffusive operator along $z$ or $s$-surfaces |
---|
1278 | is a re-entering second order operator \eqref{Eq_PE_lapU} or \eqref{Eq_PE_lapU} |
---|
1279 | with the eddy viscosity coefficient correctly placed: |
---|
1280 | |
---|
1281 | geopotential diffusion in $z$-coordinate: |
---|
1282 | \begin{equation} \label{Eq_PE_bilapU} |
---|
1283 | \begin{split} |
---|
1284 | {\rm {\bf D}}^{l{\rm {\bf U}}} &=\nabla _h \left\{ {\;\nabla _h {\rm {\bf |
---|
1285 | .}}\left[ {A^{lm}\,\nabla _h \left( \chi \right)} \right]\;} |
---|
1286 | \right\}\; \\ |
---|
1287 | &+\nabla _h \times \left\{ {\;{\rm {\bf k}}\cdot \nabla \times |
---|
1288 | \left[ {A^{lm}\,\nabla _h \times \left( {\zeta \;{\rm {\bf k}}} \right)} |
---|
1289 | \right]\;} \right\} |
---|
1290 | \end{split} |
---|
1291 | \end{equation} |
---|
1292 | |
---|
1293 | \gmcomment{ change the position of the coefficient, both here and in the code} |
---|
1294 | |
---|
1295 | geopotential diffusion in $s$-coordinate: |
---|
1296 | \begin{equation} \label{Eq_bilapU_iso} |
---|
1297 | \left\{ \begin{aligned} |
---|
1298 | D_u^{l{\rm {\bf U}}} =\Delta \left( {A^{lm}\;\Delta u} \right) \\ |
---|
1299 | D_v^{l{\rm {\bf U}}} =\Delta \left( {A^{lm}\;\Delta v} \right) |
---|
1300 | \end{aligned} \right. |
---|
1301 | \quad \text{where} \quad |
---|
1302 | \Delta \left( \bullet \right) = \nabla \cdot \left( \Re \nabla(\bullet) \right) |
---|
1303 | \end{equation} |
---|
1304 | |
---|