build based on 6feaf74

previews/PR490/model_docs/lateral/kinwave/index.html

@@ -12,7 +12,7 @@
     e^{-f\subtext{z}{exp}}(z_t-\subtext{z}{exp}) & \text{if $z_i < \subtext{z}{exp}$}\\
     \\
     e^{-f\subtext{z}{exp}}(z_t - z_i) & \text{if $z_i \ge \subtext{z}{exp}$},
-    \end{cases}\]</p><p>&lt;!– potentially confusing reuse of symbol beta –&gt; where <span>$\beta$</span> is element slope angle, <span>$\SIb{Q}{m^3 d^{-1}}$</span> is subsurface flow, <span>$\SIb{K_0}{m d^{-1}}$</span> is the saturated hydraulic conductivity at the soil surface, <span>$\SIb{z_i}{m}$</span> is the water table depth, <span>$\SIb{z_{t}}{m}$</span> is the total soil depth, <span>$\SIb{f}{m^{-1}}$</span> is a scaling parameter that controls the decrease of <span>$K_0$</span> with depth and <span>$\SIb{\subtext{z}{exp}}{m}$</span> is the depth from soil surface for which the exponential decline of <span>$K_0$</span> is valid. For the <code>exponential</code> profile, <span>$\subtext{z}{exp}$</span> is equal to <span>$z_t$</span>.</p><p>Combining with the following continuity equation:</p><p class="math-container">\[    (\theta_s-\theta_r)w\frac{\partial h}{\partial t} = -\frac{\partial Q}{\partial x} + wr\]</p><p>where <span>$\SIb{h}{m}$</span> is the water table height, <span>$\SIb{x}{m}$</span> is the distance downslope, and <span>$\SIb{r}{m d^{-1}}$</span> is the net input rate to the saturated store. Substituting for <span>$h (\frac{\partial Q}{\partial h})$</span>, gives:</p><p class="math-container">\[  \frac{\partial Q}{\partial t} = -c\frac{\partial Q}{\partial x} + cwr\]</p><p>where celerity <span>$c$</span> is calculated as follows:</p><p class="math-container">\[    c = \frac{K_0 \tan(\beta)}{\theta_s-\theta_r}\begin{cases}
+    \end{cases}\]</p><p>where <span>$\beta$</span> is element slope angle, <span>$\SIb{Q}{m^3 d^{-1}}$</span> is subsurface flow, <span>$\SIb{K_0}{m d^{-1}}$</span> is the saturated hydraulic conductivity at the soil surface, <span>$\SIb{z_i}{m}$</span> is the water table depth, <span>$\SIb{z_{t}}{m}$</span> is the total soil depth, <span>$\SIb{f}{m^{-1}}$</span> is a scaling parameter that controls the decrease of <span>$K_0$</span> with depth and <span>$\SIb{\subtext{z}{exp}}{m}$</span> is the depth from soil surface for which the exponential decline of <span>$K_0$</span> is valid. For the <code>exponential</code> profile, <span>$\subtext{z}{exp}$</span> is equal to <span>$z_t$</span>.</p><p>Combining with the following continuity equation:</p><p class="math-container">\[    (\theta_s-\theta_r)w\frac{\partial h}{\partial t} = -\frac{\partial Q}{\partial x} + wr\]</p><p>where <span>$\SIb{h}{m}$</span> is the water table height, <span>$\SIb{x}{m}$</span> is the distance downslope, and <span>$\SIb{r}{m d^{-1}}$</span> is the net input rate to the saturated store. Substituting for <span>$h (\frac{\partial Q}{\partial h})$</span>, gives:</p><p class="math-container">\[  \frac{\partial Q}{\partial t} = -c\frac{\partial Q}{\partial x} + cwr\]</p><p>where celerity <span>$c$</span> is calculated as follows:</p><p class="math-container">\[    c = \frac{K_0 \tan(\beta)}{\theta_s-\theta_r}\begin{cases}
     e^{-fz_i}
     + e^{-f\subtext{z}{exp}} &amp; \text{if $z_i &lt; \subtext{z}{exp}$}\\
     \\
@@ -24,4 +24,4 @@
 pits = &quot;wflow_pits&quot;
 
 [model]
-pits = true</code></pre><h2 id="Limitations"><a class="docs-heading-anchor" href="#Limitations">Limitations</a><a id="Limitations-1"></a><a class="docs-heading-anchor-permalink" href="#Limitations" title="Permalink"></a></h2><p>The kinematic wave approach for channel, overland and lateral subsurface flow, assumes that the topography controls water flow mostly. This assumption holds for steep terrain, but in less steep terrain the hydraulic gradient is likely not equal to the surface slope (subsurface flow), or pressure differences and inertial momentum cannot be neglected (channel and overland flow). In addition, while the kinematic wave equations are solved with a nonlinear scheme using Newton&#39;s method (Chow, 1988), other model equations are solved through a simple explicit scheme. In summary the following limitations apply:</p><ul><li><p>Channel flow, and to a lesser degree overland flow, may be unrealistic in terrain that is not steep, and where pressure forces and inertial momentum are important.</p></li><li><p>The lateral movement of subsurface flow may be very wrong in terrain that is not steep.</p></li></ul><h2 id="External-inflows"><a class="docs-heading-anchor" href="#External-inflows">External inflows</a><a id="External-inflows-1"></a><a class="docs-heading-anchor-permalink" href="#External-inflows" title="Permalink"></a></h2><p>External inflows, for example water supply or abstractions, can be added to the kinematic wave via the <code>inflow</code> variable. For this, the user can supply a 2D map of the inflow, as a cyclic parameter or as part of forcing (see also <a href="../../../user_guide/step2_settings_file/#Input-section">Input section</a>). These inflows are added or abstracted from the upstream inflow <code>qin</code> before running the kinematic wave to solve the impact on resulting <code>q</code>. In case of a negative inflow (abstractions), a minimum of zero is applied to the upstream flow <code>qin</code>.</p><h2 id="References"><a class="docs-heading-anchor" href="#References">References</a><a id="References-1"></a><a class="docs-heading-anchor-permalink" href="#References" title="Permalink"></a></h2><ul><li>Chow, V., Maidment, D. and Mays, L., 1988, Applied Hydrology. McGraw-Hill Book Company, New York.</li></ul></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../gwf/">« Groundwater flow</a><a class="docs-footer-nextpage" href="../local-inertial/">Local inertial »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 0.27.25 on <span class="colophon-date" title="Monday 28 October 2024 07:14">Monday 28 October 2024</span>. Using Julia version 1.11.0.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
+pits = true</code></pre><h2 id="Limitations"><a class="docs-heading-anchor" href="#Limitations">Limitations</a><a id="Limitations-1"></a><a class="docs-heading-anchor-permalink" href="#Limitations" title="Permalink"></a></h2><p>The kinematic wave approach for channel, overland and lateral subsurface flow, assumes that the topography controls water flow mostly. This assumption holds for steep terrain, but in less steep terrain the hydraulic gradient is likely not equal to the surface slope (subsurface flow), or pressure differences and inertial momentum cannot be neglected (channel and overland flow). In addition, while the kinematic wave equations are solved with a nonlinear scheme using Newton&#39;s method (Chow, 1988), other model equations are solved through a simple explicit scheme. In summary the following limitations apply:</p><ul><li><p>Channel flow, and to a lesser degree overland flow, may be unrealistic in terrain that is not steep, and where pressure forces and inertial momentum are important.</p></li><li><p>The lateral movement of subsurface flow may be very wrong in terrain that is not steep.</p></li></ul><h2 id="External-inflows"><a class="docs-heading-anchor" href="#External-inflows">External inflows</a><a id="External-inflows-1"></a><a class="docs-heading-anchor-permalink" href="#External-inflows" title="Permalink"></a></h2><p>External inflows, for example water supply or abstractions, can be added to the kinematic wave via the <code>inflow</code> variable. For this, the user can supply a 2D map of the inflow, as a cyclic parameter or as part of forcing (see also <a href="../../../user_guide/step2_settings_file/#Input-section">Input section</a>). These inflows are added or abstracted from the upstream inflow <code>qin</code> before running the kinematic wave to solve the impact on resulting <code>q</code>. In case of a negative inflow (abstractions), a minimum of zero is applied to the upstream flow <code>qin</code>.</p><h2 id="References"><a class="docs-heading-anchor" href="#References">References</a><a id="References-1"></a><a class="docs-heading-anchor-permalink" href="#References" title="Permalink"></a></h2><ul><li>Chow, V., Maidment, D. and Mays, L., 1988, Applied Hydrology. McGraw-Hill Book Company, New York.</li></ul></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../gwf/">« Groundwater flow</a><a class="docs-footer-nextpage" href="../local-inertial/">Local inertial »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 0.27.25 on <span class="colophon-date" title="Monday 28 October 2024 13:47">Monday 28 October 2024</span>. Using Julia version 1.11.0.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>