1 | \include{Preamble} |
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2 | |
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3 | \begin{document} |
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4 | |
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5 | \title{Draft description of NEMO wetting and drying scheme: 22 November 2016 } |
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6 | |
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7 | \author{ Hedong Liu, Jason Holt, Andrew Coward and Michael J. Bell } |
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8 | |
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9 | %------------------------------------------------------------------------ |
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10 | % End of temporary latex header (to be removed) |
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11 | %------------------------------------------------------------------------ |
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12 | |
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13 | % ================================================================ |
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14 | % Chapter Ocean Dynamics (DYN) |
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15 | % ================================================================ |
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16 | \chapter{Ocean Dynamics (DYN)} |
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17 | \label{DYN} |
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18 | \minitoc |
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19 | |
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20 | % add a figure for dynvor ens, ene latices |
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21 | |
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22 | $\ $\newline % force a new ligne |
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23 | |
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24 | % ================================================================ |
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25 | % Wetting and drying |
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26 | % ================================================================ |
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27 | \section{Wetting and drying } |
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28 | \label{DYN_wetdry} |
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29 | |
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30 | This is preliminary documentation for the wetting and drying code (WAD). The emphasis is |
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31 | on explaining the rationale for the code. The approach used by the WAD is similar to that |
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32 | developed for POM by \cite{Oey06} and that developed for ROMS by \cite{WarnerEtal13} but |
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33 | the WAD uses schemes that have not been published. |
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34 | |
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35 | The following terminology is used. The depth of the topography (positive downwards) |
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36 | at each $(i,j)$ point is the quantity stored in array $\mathrm{ht\_wd}$ in the NEMO code. |
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37 | The height of the free surface (positive upwards) will be denoted by $ \mathrm{ssh}$. Both |
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38 | quantities are measured relative to a reference sea level at z$=$0m. Given the sign |
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39 | conventions used, the water depth is the height of the free surface plus the depth of the |
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40 | topography (i.e. $\mathrm{ssh} + \mathrm{ht\_wd}$). |
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41 | |
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42 | \namdisplay{nam_wad} |
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43 | |
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44 | WAD is activated by setting $\mathrm{ln\_wd} = \mathrm{.true.}$. Currently, this option |
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45 | works with six test cases provided in the WAD\_TEST\_CASES configuration. These are all |
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46 | pure sigma coordinate configurations which define their domain, surface forcing and |
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47 | initial conditions via a set of 'usrdef' routines in MY\_SRC. Extending this option to |
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48 | more realistic domains will require the derivation and provision of a suitable |
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49 | $\mathrm{ht\_wd}$ field in addition to the normal information provided in the domcfg.nc |
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50 | file. The six test cases are described in section \S\ref{WAD_test_cases}. |
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51 | |
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52 | The WAD takes all points in the domain below a land elevation of $\mathrm{rn\_wdld}$ to be |
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53 | covered by water. Points where the water depth is less than $\mathrm{rn\_wdmin1}$ are to |
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54 | be interpreted as ``dry''. The WAD requires the topography specified with a model |
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55 | configuration to have negative depths at points where the land is higher than the |
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56 | topography's reference sea-level. The vertical grid in NEMO is computed relative to an |
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57 | initial state with zero sea surface height elevation. These reference metrics and depths |
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58 | (i.e. the $\mathrm{e3t\_0, ht\_0}$ etc. arrays) are unaltered by WAD. |
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59 | $\mathrm{rn\_wdmin1}$ is usually chosen to be of order $0.075$m but complex topographies |
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60 | with steep slopes may require larger values. The scheme also makes use of a second |
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61 | parameter, $\mathrm{rn\_wdmin2}$, which is intended to be much smaller than |
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62 | $\mathrm{rn\_wdmin1}$, of order $10^{-6}$m or smaller {\it (Q: What is the purpose of |
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63 | $\mathrm{rn\_wdmin2}$? Seems a non-zero value is required for the flux limiter iterations |
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64 | to converge)}. |
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65 | |
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66 | The WAD modifies the fluxes across the faces of cells that are either already ``dry'' |
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67 | or may become dry within the next time-step using an iterative method. The |
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68 | first sub-section below describes this scheme. It also briefly describes the simpler ROMS |
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69 | method that has not been implemented. |
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70 | |
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71 | The following sub-section describes how the surface pressure gradients are modified by the |
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72 | WAD. The next sub-section should describe how the WAD maintains consistency between the |
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73 | points that are ``wet'' on the barotropic sub-steps and those that are wet on the longer |
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74 | baroclinic time-step. This sub-section has not yet been written. The final sub-section |
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75 | should describe the test cases that have been used to assess the performance of the WAD. |
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76 | |
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77 | %----------------------------------------------------------------------------------------- |
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78 | % Flux limiters |
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79 | %----------------------------------------------------------------------------------------- |
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80 | \subsection [Flux limiters (\textit{wet\_dry})] |
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81 | {Flux limiters (\mdl{wet\_dry})} |
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82 | \label{DYN_wd_flux_limit} |
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83 | |
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84 | The flux limiter for the barotropic flow devised by Hedong Liu can be understood as follows: |
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85 | |
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86 | The continuity equation for the total water depth in a column |
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87 | \begin{equation} \label{dyn_wd_continuity} |
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88 | \frac{\partial h}{\partial t} + \mathbf{\nabla.}(h\mathbf{u}) = 0 . |
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89 | \end{equation} |
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90 | can be written in discrete form as |
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91 | |
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92 | \begin{align} \label{dyn_wd_continuity_2} |
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93 | \frac{e_1 e_2}{\Delta t} ( h_{i,j}(t_{n+1}) - h_{i,j}(t_e) ) |
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94 | &= - ( \mathrm{flxu}_{i+1,j} - \mathrm{flxu}_{i,j} + \mathrm{flxv}_{i,j+1} - \mathrm{flxv}_{i,j} ) \\ |
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95 | &= \mathrm{zzflx}_{i,j} . |
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96 | \end{align} |
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97 | |
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98 | In the above $h$ is the depth of the water in the column at point $(i,j)$, |
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99 | $\mathrm{flxu}_{i+1,j}$ is the flux out of the ``eastern'' face of the cell and |
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100 | $\mathrm{flxv}_{i,j+1}$ the flux out of the ``northern'' face of the cell; $t_{n+1}$ is |
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101 | the new timestep, $t_e$ is the old timestep (either $t_b$ or $t_n$) and $ \Delta t = |
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102 | t_{n+1} - t_e$; $e_1 e_2$ is the area of the tracer cells centred at $(i,j)$ and |
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103 | $\mathrm{zzflx}$ is the sum of the fluxes through all the faces. |
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104 | |
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105 | The flux limiter splits the flux $\mathrm{zzflx}$ into fluxes that are out of the cell |
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106 | (zzflxp) and fluxes that are into the cell (zzflxn). Clearly |
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107 | |
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108 | \begin{equation} \label{dyn_wd_zzflx_p_n_1} |
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109 | \mathrm{zzflx}_{i,j} = \mathrm{zzflxp}_{i,j} + \mathrm{zzflxn}_{i,j} . |
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110 | \end{equation} |
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111 | |
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112 | The flux limiter iteratively adjusts the fluxes $\mathrm{flxu}$ and $\mathrm{flxv}$ until |
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113 | none of the cells will ``dry out''. To be precise the fluxes are limited until none of the |
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114 | cells has water depth less than $\mathrm{rn\_wdmin1}$ on step $n+1$. |
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115 | |
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116 | Let the fluxes on the $m$th iteration step be denoted by $\mathrm{flxu}^{(m)}$ and |
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117 | $\mathrm{flxv}^{(m)}$. Then the adjustment is achieved by seeking a set of coefficients, |
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118 | $\mathrm{zcoef}_{i,j}^{(m)}$ such that: |
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119 | |
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120 | \begin{equation} \label{dyn_wd_continuity_coef} |
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121 | \begin{split} |
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122 | \mathrm{zzflxp}^{(m)}_{i,j} =& \mathrm{zcoef}_{i,j}^{(m)} \mathrm{zzflxp}^{(0)}_{i,j} \\ |
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123 | \mathrm{zzflxn}^{(m)}_{i,j} =& \mathrm{zcoef}_{i,j}^{(m)} \mathrm{zzflxn}^{(0)}_{i,j} |
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124 | \end{split} |
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125 | \end{equation} |
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126 | |
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127 | where the coefficients are $1.0$ generally but can vary between $0.0$ and $1.0$ around |
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128 | cells that would otherwise dry. |
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129 | |
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130 | The iteration is initialised by setting |
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131 | |
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132 | \begin{equation} \label{dyn_wd_zzflx_initial} |
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133 | \mathrm{zzflxp^{(0)}}_{i,j} = \mathrm{zzflxp}_{i,j} , \quad \mathrm{zzflxn^{(0)}}_{i,j} = \mathrm{zzflxn}_{i,j} . |
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134 | \end{equation} |
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135 | |
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136 | The fluxes out of cell $(i,j)$ are updated at the $m+1$th iteration if the depth of the |
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137 | cell on timestep $t_e$, namely $h_{i,j}(t_e)$, is less than the total flux out of the cell |
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138 | times the timestep divided by the cell area. Using (\ref{dyn_wd_continuity_2}) this |
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139 | condition is |
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140 | |
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141 | \begin{equation} \label{dyn_wd_continuity_if} |
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142 | h_{i,j}(t_e) - \mathrm{rn\_wdmin1} < \frac{\Delta t}{e_1 e_2} ( \mathrm{zzflxp}^{(m)}_{i,j} + \mathrm{zzflxn}^{(m)}_{i,j} ) . |
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143 | \end{equation} |
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144 | |
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145 | Rearranging (\ref{dyn_wd_continuity_if}) we can obtain an expression for the maximum |
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146 | outward flux that can be allowed and still maintain the minimum wet depth: |
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147 | |
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148 | \begin{equation} \label{dyn_wd_max_flux} |
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149 | \begin{split} |
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150 | \mathrm{zzflxp}^{(m+1)}_{i,j} = \Big[ (h_{i,j}(t_e) & - \mathrm{rn\_wdmin1} - \mathrm{rn\_wdmin2}) \frac{e_1 e_2}{\Delta t} \phantom{]} \\ |
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151 | \phantom{[} & - \mathrm{zzflxn}^{(m)}_{i,j} \Big] |
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152 | \end{split} |
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153 | \end{equation} |
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154 | |
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155 | Note a small tolerance ($\mathrm{rn\_wdmin2}$) has been introduced here {\it [Q: Why is |
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156 | this necessary/desirable?]}. Substituting from (\ref{dyn_wd_continuity_coef}) gives an |
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157 | expression for the coefficient needed to multiply the outward flux at this cell in order |
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158 | to avoid drying. |
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159 | |
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160 | \begin{equation} \label{dyn_wd_continuity_nxtcoef} |
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161 | \begin{split} |
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162 | \mathrm{zcoef}^{(m+1)}_{i,j} = \Big[ (h_{i,j}(t_e) & - \mathrm{rn\_wdmin1} - \mathrm{rn\_wdmin2}) \frac{e_1 e_2}{\Delta t} \phantom{]} \\ |
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163 | \phantom{[} & - \mathrm{zzflxn}^{(m)}_{i,j} \Big] \frac{1}{ \mathrm{zzflxp}^{(0)}_{i,j} } |
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164 | \end{split} |
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165 | \end{equation} |
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166 | |
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167 | Only the outward flux components are altered but, of course, outward fluxes from one cell |
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168 | are inward fluxes to adjacent cells and the balance in these cells may need subsequent |
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169 | adjustment; hence the iterative nature of this scheme. Note, for example, that the flux |
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170 | across the ``eastern'' face of the $(i,j)$th cell is only updated at the $m+1$th iteration |
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171 | if that flux at the $m$th iteration is out of the $(i,j)$th cell. If that is the case then |
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172 | the flux across that face is into the $(i+1,j)$ cell and that flux will not be updated by |
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173 | the calculation for the $(i+1,j)$th cell. In this sense the updates to the fluxes across |
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174 | the faces of the cells do not ``compete'' (they do not over-write each other) and one |
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175 | would expect the scheme to converge relatively quickly. The scheme is also flux based so |
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176 | conserves mass. |
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177 | |
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178 | The ROMS scheme to prevent drying out of a cell is somewhat simpler. It specifies that if |
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179 | a tracer cell is dry (the water depth is less than $\mathrm{rn\_wdmin1}$) on the backward |
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180 | timestep, $t_e$, then any outward flux through its cell faces should be set to zero. This |
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181 | scheme has a clear physical rationale. This scheme is equivalent to setting |
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182 | $\mathrm{zcoef}^{(m+1)}_{i,j}$ to $0.0$ whenever a cell is at risk of drying. One |
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183 | objection to the ROMS scheme is that it introduces a spurious step function in the flux |
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184 | out of a cell as the water depth in the cell passes through the ``critical'' value |
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185 | $\mathrm{rn\_wdmin1}$. |
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186 | |
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187 | %---------------------------------------------------------------------------------------- |
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188 | % Surface pressure gradients |
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189 | %---------------------------------------------------------------------------------------- |
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190 | \subsection [Modification of surface pressure gradients (\textit{dynhpg})] |
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191 | {Modification of surface pressure gradients (\mdl{dynhpg})} |
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192 | \label{DYN_wd_spg} |
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193 | |
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194 | At ``dry'' points the water depth is usually close to $\mathrm{rn\_wdmin1}$. If the |
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195 | topography is sloping at these points the sea-surface will have a similar slope and there |
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196 | will hence be very large horizontal pressure gradients at these points. The WAD modifies |
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197 | the magnitude but not the sign of the surface pressure gradients (zhpi and zhpj) at such |
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198 | points by mulitplying them by positive factors (zcpx and zcpy respectively) that lie |
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199 | between $0$ and $1$. |
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200 | |
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201 | We describe how the scheme works for the ``eastward'' pressure gradient, zhpi, calculated |
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202 | at the $(i,j)$th $u$-point. The scheme uses the ht\_wd depths and surface heights at the |
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203 | neighbouring $(i+1,j)$ and $(i,j)$ tracer points. zcpx is calculated using two logicals |
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204 | variables, $\mathrm{ll\_tmp1}$ and $\mathrm{ll\_tmp2}$ which are evaluated for each grid |
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205 | column. The three possible combinations are illustrated in figure \ref{Fig_WAD_dynhpg}. |
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206 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
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207 | \begin{figure}[!ht] \begin{center} |
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208 | \includegraphics[width=0.8\textwidth]{Fig_WAD_dynhpg} |
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209 | \caption{ \label{Fig_WAD_dynhpg} |
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210 | Illustrations of the three possible combinations of the logical variables controlling the |
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211 | limiting of the horizontal pressure gradient in wetting and drying regimes} |
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212 | \end{center}\end{figure} |
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213 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
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214 | |
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215 | The first logical, $\mathrm{ll\_tmp1}$, is set to true if and only if the water depth at |
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216 | both neighbouring points is greater than $\mathrm{rn\_wdmin1} + \mathrm{rn\_wdmin2}$ and |
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217 | the minimum height of the sea surface at the two points is greater than the maximum height |
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218 | of the topography at the two points: |
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219 | |
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220 | \begin{equation} \label{dyn_ll_tmp1} |
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221 | \begin{split} |
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222 | \mathrm{ll\_tmp1} = & \mathrm{MIN(sshn(ji,jj), sshn(ji+1,jj))} > \\ |
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223 | & \quad \mathrm{MAX(-ht\_wd(ji,jj), -ht\_wd(ji+1,jj))\ .and.} \\ |
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224 | & \mathrm{MAX(sshn(ji,jj) + ht\_wd(ji,jj),} \\ |
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225 | & \mathrm{\phantom{MAX(}sshn(ji+1,jj) + ht\_wd(ji+1,jj))} >\\ |
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226 | & \quad\quad\mathrm{rn\_wdmin1 + rn\_wdmin2 } |
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227 | \end{split} |
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228 | \end{equation} |
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229 | |
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230 | The second logical, $\mathrm{ll\_tmp2}$, is set to true if and only if the maximum height |
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231 | of the sea surface at the two points is greater than the maximum height of the topography |
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232 | at the two points plus $\mathrm{rn\_wdmin1} + \mathrm{rn\_wdmin2}$ |
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233 | |
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234 | \begin{equation} \label{dyn_ll_tmp2} |
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235 | \begin{split} |
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236 | \mathrm{ ll\_tmp2 } = & \mathrm{( ABS( sshn(ji,jj) - sshn(ji+1,jj) ) > 1.E-12 )\ .AND.}\\ |
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237 | & \mathrm{( MAX(sshn(ji,jj), sshn(ji+1,jj)) > } \\ |
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238 | & \mathrm{\phantom{(} MAX(-ht\_wd(ji,jj), -ht\_wd(ji+1,jj)) + rn\_wdmin1 + rn\_wdmin2}) . |
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239 | \end{split} |
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240 | \end{equation} |
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241 | |
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242 | If $\mathrm{ll\_tmp1}$ is true then the surface pressure gradient, zhpi at the $(i,j)$ |
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243 | point is unmodified. If both logicals are false zhpi is set to zero. |
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244 | |
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245 | If $\mathrm{ll\_tmp1}$ is true and $\mathrm{ll\_tmp2}$ is false then the surface pressure |
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246 | gradient is multiplied through by zcpx which is the absolute value of the difference in |
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247 | the water depths at the two points divided by the difference in the surface heights at the |
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248 | two points. Thus the sign of the sea surface height gradient is retained but the magnitude |
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249 | of the pressure force is determined by the difference in water depths rather than the |
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250 | difference in surface height between the two points. Note that dividing by the difference |
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251 | between the sea surface heights can be problematic if the heights approach parity. An |
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252 | additional condition is applied to $\mathrm{ ll\_tmp2 }$ to ensure it is .false. in such |
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253 | conditions. |
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254 | |
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255 | %---------------------------------------------------------------------------------------- |
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256 | % The WAD test cases |
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257 | %---------------------------------------------------------------------------------------- |
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258 | \subsection [The WAD test cases (\textit{usrdef\_zgr})] |
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259 | {The WAD test cases (\mdl{usrdef\_zgr})} |
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260 | \label{WAD_test_cases} |
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261 | |
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262 | This section contains details of the six test cases that can be run as part of the |
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263 | WAD\_TEST\_CASES configuration. All the test cases are shallow (less than 10m deep), |
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264 | closed basins with 4m high walls and some of topography that can wet and dry up to 2m |
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265 | above sea-level. The horizontal grid is uniform with a 1km resolution and measures 52km by |
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266 | 34km. These dimensions are determined by a combination of code in the \mdl{usrdef\_nam} |
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267 | module located in the WAD\_TEST\_CASES/MY\_SRC directory and setting read in from the |
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268 | namusr\_def namelist. There is no rotation or external forcing and motion is simply |
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269 | initiated by an initial ssh slope. |
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270 | |
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271 | \namdisplay{nam_wad_usr} |
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272 | |
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273 | The $\mathrm{nn\_wad\_test}$ parameter can takes values 1 to 6 and it is this parameter |
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274 | that determines which of the test cases will be run. All cases will run with the default |
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275 | settings but the simple linear slope cases (tests 1 and 5) can be run with lower values of |
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276 | $\mathrm{rn\_wdmin1}$ as will be illustarted below. |
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277 | |
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278 | \subsubsection [WAD test case 1 : A simple linear slope] |
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279 | {WAD test case 1 : A simple linear slope} |
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280 | \label{WAD_test_case1} |
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281 | |
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282 | The first test case is a simple linear slope (in the x-direction, uniform in y) with an |
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283 | adverse SSH gradient that, when released, creates a surge up the slope. The parameters are |
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284 | chosen such that the surge rises above sea-level before falling back and oscillating |
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285 | towards an equilibrium position. This case can be run with $\mathrm{rn\_wdmin1}$ values as |
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286 | low as 0.075m. |
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287 | |
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288 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
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289 | \begin{figure}[!ht] \begin{center} |
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290 | \includegraphics[width=0.8\textwidth]{Fig_WAD_TC1} |
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291 | \caption{ \label{Fig_WAD_TC1} |
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292 | The evolution of the sea surface height in WAD\_TEST\_CASE 1 from the initial state (t=0) |
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293 | over the first three hours of simulation. Note that in this time-frame the resultant surge |
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294 | reaches to nearly 2m above sea-level before retreating.} |
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295 | \end{center}\end{figure} |
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296 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
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297 | |
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298 | \subsubsection [WAD test case 2 : A parabolic channel ] |
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299 | {WAD test case 2 : A parabolic channel} |
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300 | \label{WAD_test_case2} |
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301 | |
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302 | The second and third test cases use a closed channel which is parabolic in x and uniform in y. |
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303 | Test case 2 uses a gentler initial SSH slope which nevertheless demonstrates the ability to |
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304 | wet and dry on both sides of the channel. This solution requires values of $\mathrm{rn\_wdmin1}$ greater than 0.25m ({\it Q.: A function of the maximum topographic slope?}) |
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305 | |
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306 | |
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307 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
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308 | \begin{figure}[!ht] \begin{center} |
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309 | \includegraphics[width=0.8\textwidth]{Fig_WAD_TC2} |
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310 | \caption{ \label{Fig_WAD_TC2} |
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311 | The evolution of the sea surface height in WAD\_TEST\_CASE 2 from the initial state (t=0) |
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312 | over the first three hours of simulation. Note that in this time-frame the resultant sloshing |
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313 | causes wetting and drying on both sides of the parabolic channel.} |
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314 | \end{center}\end{figure} |
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315 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
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316 | |
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317 | \subsubsection [WAD test case 3 : A parabolic channel (extreme slope) ] |
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318 | {WAD test case 3 : A parabolic channel (extreme slope)} |
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319 | \label{WAD_test_case3} |
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320 | |
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321 | Same again but with a steeper initial SSH slope. The solution is similar but more vigorous. |
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322 | |
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323 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
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324 | \begin{figure}[!ht] \begin{center} |
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325 | \includegraphics[width=0.8\textwidth]{Fig_WAD_TC3} |
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326 | \caption{ \label{Fig_WAD_TC3} |
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327 | The evolution of the sea surface height in WAD\_TEST\_CASE 3 from the initial state (t=0) |
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328 | over the first three hours of simulation. Note that in this time-frame the resultant sloshing |
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329 | causes wetting and drying on both sides of the parabolic channel.} |
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330 | \end{center}\end{figure} |
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331 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
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332 | |
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333 | \subsubsection [WAD test case 4 : A parabolic bowl ] |
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334 | {WAD test case 4 : A parabolic bowl} |
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335 | \label{WAD_test_case4} |
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336 | |
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337 | Test case 4 includes variation in the y-direction in the form of a parabolic bowl. The |
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338 | initial condition is now a raised bulge centred over the bowl. Figure \ref{Fig_WAD_TC4} |
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339 | shows a cross-section of the SSH in the X-direction but features can be seen to propagate |
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340 | in all directions and interfere when return paths cross. |
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341 | |
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342 | |
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343 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
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344 | \begin{figure}[!ht] \begin{center} |
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345 | \includegraphics[width=0.8\textwidth]{Fig_WAD_TC4} |
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346 | \caption{ \label{Fig_WAD_TC4} |
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347 | The evolution of the sea surface height in WAD\_TEST\_CASE 4 from the initial state (t=0) |
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348 | over the first three hours of simulation. Note that this test case is a parabolic bowl with |
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349 | variations occurring in the y-direction too (not shown here).} |
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350 | \end{center}\end{figure} |
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351 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
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352 | |
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353 | \subsubsection [WAD test case 5 : A double slope with shelf channel ] |
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354 | {WAD test case 5 : A double slope with shelf channel} |
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355 | \label{WAD_test_case5} |
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356 | |
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357 | Similar in nature to test case 1 but with a change in slope and a mid-depth shelf. |
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358 | |
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359 | |
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360 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
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361 | \begin{figure}[!ht] \begin{center} |
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362 | \includegraphics[width=0.8\textwidth]{Fig_WAD_TC5} |
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363 | \caption{ \label{Fig_WAD_TC5} |
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364 | The evolution of the sea surface height in WAD\_TEST\_CASE 5 from the initial state (t=0) |
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365 | over the first three hours of simulation. The surge resulting in this case wets to the full |
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366 | depth permitted (2.5m above sea-level) and is only halted by the 4m high side walls.} |
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367 | \end{center}\end{figure} |
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368 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
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369 | |
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370 | \subsubsection [WAD test case 6 : A parabolic channel with central bar ] |
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371 | {WAD test case 6 : A parabolic channel with central bar} |
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372 | \label{WAD_test_case6} |
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373 | |
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374 | Test cases 1 to 5 have all used uniform T and S conditions. The dashed line in each plot |
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375 | shows the surface salinity along the y=17 line which remains satisfactorily constant. Test |
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376 | case 6 introduces variation in salinity by taking a parabolic channel divided by a central |
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377 | bar (gaussian) and using two different salinity values in each half of the channel. This |
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378 | step change in salinity is initially enforced by the central bar but the bar is |
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379 | subsequently over-topped after the initial SSH gradient is released. The time series in |
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380 | this case shows the SSH evolution with the water coloured according to local salinity |
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381 | values. Encroachment of the high salinity (red) waters into the low salinity (blue) basin |
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382 | can clearly be seen. |
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383 | |
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384 | |
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385 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
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386 | \begin{figure}[!ht] \begin{center} |
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387 | \includegraphics[width=0.8\textwidth]{Fig_WAD_TC6} |
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388 | \caption{ \label{Fig_WAD_TC6} |
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389 | The evolution of the sea surface height in WAD\_TEST\_CASE 6 from the initial state (t=0) |
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390 | over the first three hours of simulation. Water is coloured according to local salinity |
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391 | values. Encroachment of the high salinity (red) waters into the low salinity (blue) basin |
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392 | can clearly be seen although the largest influx occurs early in the sequence between the |
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393 | frames shown.} |
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394 | \end{center}\end{figure} |
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395 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
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396 | |
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397 | |
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398 | % ================================================================ |
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399 | |
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400 | \bibliographystyle{wileyqj} |
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401 | \bibliography{references} |
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402 | |
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403 | \end{document} |
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