[Hackathon] Ionization docs fixes (#5270)

* Fixed typos in generalized Ohm's law * Fixed Testing link * Removed minus sign from BTO equation (Zhang eq 8 is wrong) * Clarify comment about difference to published equation. Co-authored-by: Axel Huebl <[email protected]> --------- Co-authored-by: Johannes Van de Wetering <[email protected]> Co-authored-by: Roelof Groenewald <[email protected]> Co-authored-by: Axel Huebl <[email protected]>
ECP-WarpX · Sep 17, 2024 · f7dd6a9 · f7dd6a9
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diff --git a/Docs/source/theory/kinetic_fluid_hybrid_model.rst b/Docs/source/theory/kinetic_fluid_hybrid_model.rst
@@ -46,7 +46,7 @@ integrating over velocity), also called the generalized Ohm's law, is given by:
 
     .. math::
 
-        en_e\vec{E} = \frac{m}{e}\frac{\partial \vec{J}_e}{\partial t} + \frac{m}{e^2}\left( \vec{U}_e\cdot\nabla \right) \vec{J}_e - \nabla\cdot {\overleftrightarrow P}_e - \vec{J}_e\times\vec{B}+\vec{R}_e
+        en_e\vec{E} = \frac{m}{e}\frac{\partial \vec{J}_e}{\partial t} + \frac{m}{e}\left( \vec{U}_e\cdot\nabla \right) \vec{J}_e - \nabla\cdot {\overleftrightarrow P}_e - \vec{J}_e\times\vec{B}+\vec{R}_e
 
 where :math:`\vec{U}_e = \vec{J}_e/(en_e)` is the electron fluid velocity,
 :math:`{\overleftrightarrow P}_e` is the electron pressure tensor and
@@ -64,7 +64,7 @@ Plugging this back into the generalized Ohm' law gives:
 
         \left(en_e +\frac{m}{e\mu_0}\nabla\times\nabla\times\right)\vec{E} =&
         - \frac{m}{e}\left( \frac{\partial\vec{J}_{ext}}{\partial t} + \sum_{s\neq e}\frac{\partial\vec{J}_s}{\partial t} \right) \\
-        &+ \frac{m}{e^2}\left( \vec{U}_e\cdot\nabla \right) \vec{J}_e - \nabla\cdot {\overleftrightarrow P}_e - \vec{J}_e\times\vec{B}+\vec{R}_e.
+        &+ \frac{m}{e}\left( \vec{U}_e\cdot\nabla \right) \vec{J}_e - \nabla\cdot {\overleftrightarrow P}_e - \vec{J}_e\times\vec{B}+\vec{R}_e.
 
 If we now further assume electrons are inertialess (i.e. :math:`m=0`), the above equation simplifies to,
 

diff --git a/Docs/source/theory/multiphysics/ionization.rst b/Docs/source/theory/multiphysics/ionization.rst
@@ -56,18 +56,18 @@ where :math:`\mathrm{d}\tau` is the simulation timestep, which is divided by the
 Empirical Extension to Over-the-Barrier Regime for Hydrogen
 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 
-For hydrogen, WarpX offers the modified empirical ADK extension to the Over-the-Barrier (OTB) published in :cite:t:`mpion-zhang_empirical_2014` Eq. (8).
+For hydrogen, WarpX offers the modified empirical ADK extension to the Over-the-Barrier (OTB) published in :cite:t:`mpion-zhang_empirical_2014` Eq. (8) (note there is a typo in the paper and there should not be a minus sign in Eq. 8).
 
 .. math::
 
-    W_\mathrm{M} = \exp\left[ -\left( a_1 \frac{E^2}{E_\mathrm{b}} + a_2 \frac{E}{E_\mathrm{b}} + a_3  \right) \right] W_\mathrm{ADK}
+    W_\mathrm{M} = \exp\left[ a_1 \frac{E^2}{E_\mathrm{b}} + a_2 \frac{E}{E_\mathrm{b}} + a_3 \right] W_\mathrm{ADK}
 
 The parameters :math:`a_1` through :math:`a_3` are independent of :math:`E` and can be found in the same reference. :math:`E_\mathrm{b}` is the classical Barrier Suppresion Ionization (BSI) field strength :math:`E_\mathrm{b} = U_\mathrm{ion}^2 / (4 Z)` given here in atomic units (AU). For a detailed description of conversion between unit systems consider the book by :cite:t:`mpion-Mulser2010`.
 
 Testing
 ^^^^^^^
 
-* `Testing the field ionization module <../../../../Examples/Tests/field_ionization/README.rst>`_.
+* `Testing the field ionization module <../../../../en/latest/usage/examples/field_ionization/README.html>`_.
 
 .. bibliography::
     :keyprefix: mpion-