Merge pull request #189 from bendudson/more-docs

Adding documentation

Author: bendudson

docs/sphinx/tests.rst

@@ -139,3 +139,67 @@ stationary at the initial jump location.  Left state
 :math:`\left(\rho_L, u_L, p_L\right) = \left(1, -19.59745,
 1000.0\right)` Right state :math:`\left(\rho_R, u_R, p_R\right) =
 \left(1, -19.59745, 0.01\right)`.  Result at time :math:`t = 0.03`.
+
+Drift wave
+----------
+
+``tests/integrated/drift-wave``
+
+This calculates the growth rate and frequency of a resistive drift
+wave with finite electron mass. 
+
+The equations solved are:
+
+.. math::
+
+   \begin{aligned}
+   \frac{\partial n_i}{\partial t} =& -\nabla\cdot\left(n_i\mathbf{v}_{E\times B}\right) \\
+   n_e =& n_i \\
+   \frac{\partial}{\partial t}\nabla\cdot\left(\frac{n_0 m_i}{B^2}\nabla_\perp\phi\right) =& \nabla_{||}J_{||} = -\nabla_{||}\left(en_ev_{||e}\right) \\
+   \frac{\partial}{\partial t}\left(m_en_ev_{||e}\right) =& -\nabla\cdot\left(m_en_ev_{||e} \mathbf{b}v_{||e}\right) + en_e\partial_{||}\phi - \partial_{||}p_e - 0.51\nu_{ei}n_im_ev_{||e}
+   \end{aligned}
+
+Linearising around a stationary background with constant density :math:`n_0` and temperature :math:`T_0`,
+using :math:`\frac{\partial}{\partial t}\rightarrow -i\omega` gives:
+
+.. math::
+
+   \begin{aligned}
+   \tilde{n} =& \frac{k_\perp}{\omega}\frac{n_0}{BL_n}\tilde{\phi} \\
+   \tilde{\phi} =& -\frac{k_{||}}{\omega k_\perp^2}\frac{eB^2}{m_i}\tilde{v_{||e}} \\
+   \omega m_e \tilde{v_{||e}} =& -ek_{||}\tilde{\phi} + ek_{||}\frac{T_o}{n_0}\tilde{n} - i0.51\nu_{ei}m_e\tilde{v_{||e}}
+   \end{aligned}
+
+
+where the radial density length scale coming from the radial
+:math:`E\times B` advection of density is defined as
+
+.. math::
+
+   \frac{1}{L_n} \equiv \frac{1}{n_0}\frac{\partial n_0}{\partial r}
+
+Substituting and rearranging gives:
+
+.. math::
+
+   i\left(\frac{\omega}{\omega*}\right)^3 \frac{\omega_*}{0.51\nu_{ei}} = \left(\frac{\omega}{\omega_*} - 1\right)\frac{i\sigma_{||}}{\omega_*} + \left(\frac{\omega}{\omega*}\right)^2
+
+or
+
+.. math::
+
+   \frac{\omega_*}{0.51\nu_{ei}}\left(\frac{\omega}{\omega_*}\right)^3 + i\left(\frac{\omega}{\omega_*}\right)^2 - \frac{\sigma_{||}}{\omega_*}\left(\frac{\omega}{\omega_*}\right) + \frac{\sigma_{||}}{\omega_*} = 0
+
+where
+
+.. math::
+
+   \begin{aligned}
+   \omega_* =& \frac{k_\perp T_0}{BL_n} \\
+   \sigma_{||} =& \frac{k_{||}^2}{k_\perp^2}\frac{\Omega_i\Omega_e}{0.51\nu_{ei}} \\
+   \Omega_s =& eB / m_s
+   \end{aligned}
+
+This is a cubic dispersion relation, so we find the three roots (using
+NumPy), and choose the root with the most positive growth rate
+(imaginary component of :math:`\omega`).