Migrate from 'jekyll-katex' to 'kramdown-math-sskatex'

1 files changed, 48 insertions, 48 deletions

diff --git a/source/know/concept/magnetohydrodynamics/index.md b/source/know/concept/magnetohydrodynamics/index.md
index 115ce04..bcc23f3 100644
--- a/source/know/concept/magnetohydrodynamics/index.md
+++ b/source/know/concept/magnetohydrodynamics/index.md
@@ -19,8 +19,8 @@ but the results are not specific to plasmas.
 
 In the two-fluid model, we described the plasma as two separate fluids,
 but in MHD we treat it as a single conductive fluid.
-The macroscopic pressure $p$
-and electric current density $\vb{J}$ are:
+The macroscopic pressure $$p$$
+and electric current density $$\vb{J}$$ are:
 
 $$\begin{aligned}
     p
@@ -30,8 +30,8 @@ $$\begin{aligned}
     = q_i n_i \vb{u}_i + q_e n_e \vb{u}_e
 \end{aligned}$$
 
-Meanwhile, the macroscopic mass density $\rho$
-and center-of-mass flow velocity $\vb{u}$
+Meanwhile, the macroscopic mass density $$\rho$$
+and center-of-mass flow velocity $$\vb{u}$$
 are as follows, although the ions dominate due to their large mass:
 
 $$\begin{aligned}
@@ -76,8 +76,8 @@ $$\begin{aligned}
 
 We will assume that electrons' inertia
 is negligible compared to the [Lorentz force](/know/concept/lorentz-force/).
-Let $\tau_\mathrm{char}$ be the characteristic timescale of the plasma's dynamics,
-i.e. nothing noticable happens in times shorter than $\tau_\mathrm{char}$,
+Let $$\tau_\mathrm{char}$$ be the characteristic timescale of the plasma's dynamics,
+i.e. nothing noticable happens in times shorter than $$\tau_\mathrm{char}$$,
 then this assumption can be written as:
 
 $$\begin{aligned}
@@ -89,10 +89,10 @@ $$\begin{aligned}
     \ll 1
 \end{aligned}$$
 
-Where we have recognized the cyclotron frequency $\omega_c$ (see Lorentz force article).
+Where we have recognized the cyclotron frequency $$\omega_c$$ (see Lorentz force article).
 In other words, our assumption is equivalent to
-the electron gyration period $2 \pi / \omega_{ce}$
-being small compared to the macroscopic dynamics' timescale $\tau_\mathrm{char}$.
+the electron gyration period $$2 \pi / \omega_{ce}$$
+being small compared to the macroscopic dynamics' timescale $$\tau_\mathrm{char}$$.
 By construction, we can thus ignore the left-hand side
 of the electron momentum equation, leaving:
 
@@ -105,7 +105,7 @@ $$\begin{aligned}
 \end{aligned}$$
 
 We add up these momentum equations,
-recognizing the pressure $p$ and current $\vb{J}$:
+recognizing the pressure $$p$$ and current $$\vb{J}$$:
 
 $$\begin{aligned}
     m_i n_i \frac{\mathrm{D} \vb{u}_i}{\mathrm{D} t}
@@ -115,7 +115,7 @@ $$\begin{aligned}
     &= (q_i n_i + q_e n_e) \vb{E} + \vb{J} \cross \vb{B} - \nabla p
 \end{aligned}$$
 
-Where we have used $f_{ie} m_i n_i = f_{ei} m_e n_e$
+Where we have used $$f_{ie} m_i n_i = f_{ei} m_e n_e$$
 because momentum is conserved by the underlying
 [Rutherford scattering](/know/concept/rutherford-scattering/) process,
 which is [elastic](/know/concept/elastic-collision/).
@@ -123,10 +123,10 @@ In other words, the momentum given by ions to electrons
 is equal to the momentum received by electrons from ions.
 
 Since the two-fluid model assumes that
-the [Debye length](/know/concept/debye-length/) $\lambda_D$
-is small compared to a "blob" $\dd{V}$ of the fluid,
-we can invoke quasi-neutrality $q_i n_i + q_e n_e = 0$.
-Using that $\rho \approx m_i n_i$ and $\vb{u} \approx \vb{u}_i$,
+the [Debye length](/know/concept/debye-length/) $$\lambda_D$$
+is small compared to a "blob" $$\dd{V}$$ of the fluid,
+we can invoke quasi-neutrality $$q_i n_i + q_e n_e = 0$$.
+Using that $$\rho \approx m_i n_i$$ and $$\vb{u} \approx \vb{u}_i$$,
 we thus arrive at the **momentum equation**:
 
 $$\begin{aligned}
@@ -147,8 +147,8 @@ $$\begin{aligned}
     = \frac{f_{ei} m_e}{q_e} (\vb{u}_e - \vb{u}_i)
 \end{aligned}$$
 
-Again using quasi-neutrality $q_i n_i = - q_e n_e$,
-the current density $\vb{J} = q_e n_e (\vb{u}_e \!-\! \vb{u}_i)$,
+Again using quasi-neutrality $$q_i n_i = - q_e n_e$$,
+the current density $$\vb{J} = q_e n_e (\vb{u}_e \!-\! \vb{u}_i)$$,
 so:
 
 $$\begin{aligned}
@@ -159,13 +159,13 @@ $$\begin{aligned}
     \equiv \frac{f_{ei} m_e}{n_e q_e^2}
 \end{aligned}$$
 
-Where $\eta$ is the electrical resistivity of the plasma,
+Where $$\eta$$ is the electrical resistivity of the plasma,
 see [Spitzer resistivity](/know/concept/spitzer-resistivity/)
 for more information, and a rough estimate of this quantity for a plasma.
 
-Now, using that $\vb{u} \approx \vb{u}_i$,
-we add $(\vb{u} \!-\! \vb{u}_i) \cross \vb{B} \approx 0$ to the equation,
-and insert $\vb{J}$ again:
+Now, using that $$\vb{u} \approx \vb{u}_i$$,
+we add $$(\vb{u} \!-\! \vb{u}_i) \cross \vb{B} \approx 0$$ to the equation,
+and insert $$\vb{J}$$ again:
 
 $$\begin{aligned}
     \eta \vb{J}
@@ -185,8 +185,8 @@ $$\begin{aligned}
 
 Where we have used Faraday's law.
 This is the **induction equation**,
-and is used to compute $\vb{B}$.
-The pressure term can be rewritten using the ideal gas law $p_e = k_B T_e n_e$:
+and is used to compute $$\vb{B}$$.
+The pressure term can be rewritten using the ideal gas law $$p_e = k_B T_e n_e$$:
 
 $$\begin{aligned}
     \nabla \cross \frac{\nabla p_e}{q_e n_e}
@@ -195,8 +195,8 @@ $$\begin{aligned}
 \end{aligned}$$
 
 The curl of a gradient is always zero,
-and we notice that $\nabla n_e / n_e = \nabla\! \ln(n_e)$.
-Then we use the vector identity $\nabla \cross (f \nabla g) = \nabla f \cross \nabla g$,
+and we notice that $$\nabla n_e / n_e = \nabla\! \ln(n_e)$$.
+Then we use the vector identity $$\nabla \cross (f \nabla g) = \nabla f \cross \nabla g$$,
 leading to:
 
 $$\begin{aligned}
@@ -206,10 +206,10 @@ $$\begin{aligned}
     = \frac{k_B}{q_e n_e} \big( \nabla T_e \cross \nabla n_e \big)
 \end{aligned}$$
 
-It is reasonable to assume that $\nabla T_e$ and $\nabla n_e$
+It is reasonable to assume that $$\nabla T_e$$ and $$\nabla n_e$$
 point in roughly the same direction,
 in which case the pressure term can be neglected.
-Consequently, $p_e$ has no effect on the dynamics of $\vb{B}$,
+Consequently, $$p_e$$ has no effect on the dynamics of $$\vb{B}$$,
 so we argue that it can be dropped from the original (non-curled) equation too, leaving:
 
 $$\begin{aligned}
@@ -220,7 +220,7 @@ $$\begin{aligned}
 \end{aligned}$$
 
 This is known as the **generalized Ohm's law**,
-since it contains the relation $\vb{E} = \eta \vb{J}$.
+since it contains the relation $$\vb{E} = \eta \vb{J}$$.
 
 Next, consider [Ampère's law](/know/concept/maxwells-equations/),
 where we would like to neglect the last term:
@@ -230,9 +230,9 @@ $$\begin{aligned}
     = \mu_0 \vb{J} + \frac{1}{c^2} \pdv{\vb{E}}{t}
 \end{aligned}$$
 
-From Faraday's law, we can obtain a scale estimate for $\vb{E}$.
-Recall that $\tau_\mathrm{char}$ is the characteristic timescale of the plasma,
-and let $\lambda_\mathrm{char} \gg \lambda_D$ be its characteristic lengthscale:
+From Faraday's law, we can obtain a scale estimate for $$\vb{E}$$.
+Recall that $$\tau_\mathrm{char}$$ is the characteristic timescale of the plasma,
+and let $$\lambda_\mathrm{char} \gg \lambda_D$$ be its characteristic lengthscale:
 
 $$\begin{aligned}
     \nabla \cross \vb{E}
@@ -244,8 +244,8 @@ $$\begin{aligned}
 
 From this, we find when we can neglect
 the last term in Ampère's law:
-the characteristic velocity $v_\mathrm{char}$
-must be tiny compared to $c$,
+the characteristic velocity $$v_\mathrm{char}$$
+must be tiny compared to $$c$$,
 i.e. the plasma must be non-relativistic:
 
 $$\begin{aligned}
@@ -284,7 +284,7 @@ $$\begin{aligned}
 
 The continuity equation allows us to rewrite
 the [material derivative](/know/concept/material-derivative/)
-$\mathrm{D} \rho / \mathrm{D} t$ as follows:
+$$\mathrm{D} \rho / \mathrm{D} t$$ as follows:
 
 $$\begin{aligned}
     \pdv{\rho}{t} + \nabla \cdot (\rho \vb{u})
@@ -294,7 +294,7 @@ $$\begin{aligned}
 \end{aligned}$$
 
 Inserting this into the equation of state
-leads us to a differential equation for $p$:
+leads us to a differential equation for $$p$$:
 
 $$\begin{aligned}
     0
@@ -307,8 +307,8 @@ $$\begin{aligned}
 
 This closes the set of 14 MHD equations for 14 unknowns.
 Originally, the two-fluid model had 16 of each,
-but we have merged $n_i$ and $n_e$ into $\rho$,
-and $p_i$ and $p_i$ into $p$.
+but we have merged $$n_i$$ and $$n_e$$ into $$\rho$$,
+and $$p_i$$ and $$p_i$$ into $$p$$.
 
 
 ## Ohm's law variants
@@ -323,7 +323,7 @@ $$\begin{aligned}
 
 However, most authors neglect some of its terms:
 this form is used for **Hall MHD**,
-where $\vb{J} \cross \vb{B}$ is called the *Hall term*.
+where $$\vb{J} \cross \vb{B}$$ is called the *Hall term*.
 This term can be dropped in any of the following cases:
 
 $$\begin{gathered}
@@ -342,11 +342,11 @@ $$\begin{gathered}
     \ll 1
 \end{gathered}$$
 
-Where we have used the MHD momentum equation with $\nabla p \approx 0$
-to obtain the scale estimate $\vb{J} \cross \vb{B} \sim \rho v_\mathrm{char} / \tau_\mathrm{char}$.
-In other words, if the ion gyration period is short $\tau_\mathrm{char} \gg \omega_{ci}$,
+Where we have used the MHD momentum equation with $$\nabla p \approx 0$$
+to obtain the scale estimate $$\vb{J} \cross \vb{B} \sim \rho v_\mathrm{char} / \tau_\mathrm{char}$$.
+In other words, if the ion gyration period is short $$\tau_\mathrm{char} \gg \omega_{ci}$$,
 and/or if the electron gyration period is long
-compared to the electron-ion collision period $\omega_{ce} \ll f_{ei}$,
+compared to the electron-ion collision period $$\omega_{ce} \ll f_{ei}$$,
 then we are left with this form of Ohm's law, used in **resistive MHD**:
 
 $$\begin{aligned}
@@ -354,10 +354,10 @@ $$\begin{aligned}
     = \eta \vb{J}
 \end{aligned}$$
 
-Finally, we can neglect the resisitive term $\eta \vb{J}$
+Finally, we can neglect the resisitive term $$\eta \vb{J}$$
 if the Lorentz force is much larger.
 We formalize this condition as follows,
-where we have used Ampère's law to find $\vb{J} \sim \vb{B} / \mu_0 \lambda_\mathrm{char}$:
+where we have used Ampère's law to find $$\vb{J} \sim \vb{B} / \mu_0 \lambda_\mathrm{char}$$:
 
 $$\begin{aligned}
     1
@@ -368,8 +368,8 @@ $$\begin{aligned}
     \gg 1
 \end{aligned}$$
 
-Where we have defined the **magnetic Reynolds number** $\mathrm{R_m}$ as follows,
-which is analogous to the fluid [Reynolds number](/know/concept/reynolds-number/) $\mathrm{Re}$:
+Where we have defined the **magnetic Reynolds number** $$\mathrm{R_m}$$ as follows,
+which is analogous to the fluid [Reynolds number](/know/concept/reynolds-number/) $$\mathrm{Re}$$:
 
 $$\begin{aligned}
     \boxed{
@@ -378,9 +378,9 @@ $$\begin{aligned}
     }
 \end{aligned}$$
 
-If $\mathrm{R_m} \ll 1$, the plasma is "electrically viscous",
+If $$\mathrm{R_m} \ll 1$$, the plasma is "electrically viscous",
 such that resistivity needs to be accounted for,
-whereas if $\mathrm{R_m} \gg 1$, the resistivity is negligible,
+whereas if $$\mathrm{R_m} \gg 1$$, the resistivity is negligible,
 in which case we have **ideal MHD**:
 
 $$\begin{aligned}