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\end{align}\]</p><h3 id="Evolving-n"><a class="docs-heading-anchor" href="#Evolving-n">Evolving <span>$n$</span></a><a id="Evolving-n-1"></a><a class="docs-heading-anchor-permalink" href="#Evolving-n" title="Permalink"></a></h3><p>When evolving only <span>$n$</span> separately, we only need one constraint and the correction is just a re-scaling. This corresponds to <span>$B=0,C=0$</span> so that</p><p class="math-container">\[\begin{align}
1 &amp;= AI_0 \\
A &amp;= \frac{1}{I_0}
\end{align}\]</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../">« Home</a><a class="docs-footer-nextpage" href="../debugging-hints/">Debugging »</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 30 September 2024 14:01">Monday 30 September 2024</span>. Using Julia version 1.10.5.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
\end{align}\]</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../">« Home</a><a class="docs-footer-nextpage" href="../debugging-hints/">Debugging »</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 30 September 2024 16:57">Monday 30 September 2024</span>. Using Julia version 1.10.5.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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\end{align}\]</p><p>or in 1V simulations</p><p class="math-container">\[\begin{align}
S_i &amp;= A_i(r,z) \left[ \frac{1}{sqrt{\pi} \sqrt{2 T_{\mathrm{source},i} / m_i}} \exp\left( -\frac{v_\perp^2}{T_{\mathrm{source},i}} - f_i(v_\parallel) \right) \right] \\
S_n &amp;= A_n(r,z) \left[ \frac{1}{sqrt{\pi} \sqrt{2 T_{\mathrm{source},n} / m_n}} \exp\left( -\frac{v_z^2}{T_{\mathrm{source},n}} - f_n(v_z) \right) \right]
\end{align}\]</p><p>Note that this source does not give a fixed power input (although that might be a nice feature to have), it just swaps plasma/neutral particles at a constant rate.</p><h2 id="API"><a class="docs-heading-anchor" href="#API">API</a><a id="API-1"></a><a class="docs-heading-anchor-permalink" href="#API" title="Permalink"></a></h2><p>See <a href="../zz_external_sources/#external_sources">external_sources</a>.</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../developing/">« Developing</a><a class="docs-footer-nextpage" href="../geometry/">Magnetic Geometry »</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 30 September 2024 14:01">Monday 30 September 2024</span>. Using Julia version 1.10.5.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
\end{align}\]</p><p>Note that this source does not give a fixed power input (although that might be a nice feature to have), it just swaps plasma/neutral particles at a constant rate.</p><h2 id="API"><a class="docs-heading-anchor" href="#API">API</a><a id="API-1"></a><a class="docs-heading-anchor-permalink" href="#API" title="Permalink"></a></h2><p>See <a href="../zz_external_sources/#external_sources">external_sources</a>.</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../developing/">« Developing</a><a class="docs-footer-nextpage" href="../geometry/">Magnetic Geometry »</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 30 September 2024 16:57">Monday 30 September 2024</span>. Using Julia version 1.10.5.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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= J \frac{E_z B_\zeta}{B^2}.
\end{equation}\]</p><p>Due to the axisymmetry of the system, the differential operator <span>$\mathbf{b} \cdot \nabla (\cdot) = b_z \partial {(\cdot)}{\partial z}$</span>, and the convective derivative</p><p class="math-container">\[\begin{equation}
\frac{d B}{d t} = \frac{d z}{d t} \frac{\partial B}{ \partial z} + \frac{dr}{dt}\frac{\partial B}{\partial r}.
\end{equation}\]</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../external_sources_notes/">« External sources</a><a class="docs-footer-nextpage" href="../getting_started/">Getting started »</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 30 September 2024 14:01">Monday 30 September 2024</span>. Using Julia version 1.10.5.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
\end{equation}\]</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../external_sources_notes/">« External sources</a><a class="docs-footer-nextpage" href="../getting_started/">Getting started »</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 30 September 2024 16:57">Monday 30 September 2024</span>. Using Julia version 1.10.5.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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