From 075683cdf4588fe16f41d9f7b46b9720b42b2553 Mon Sep 17 00:00:00 2001
From: Prefetch
Date: Wed, 17 Jul 2024 10:01:43 +0200
Subject: Improve knowledge base

---
 source/know/concept/magnetohydrodynamics/index.md | 122 +++++++++++-----------
 1 file changed, 61 insertions(+), 61 deletions(-)

(limited to 'source/know/concept/magnetohydrodynamics')

diff --git a/source/know/concept/magnetohydrodynamics/index.md b/source/know/concept/magnetohydrodynamics/index.md
index bcc23f3..4431dfa 100644
--- a/source/know/concept/magnetohydrodynamics/index.md
+++ b/source/know/concept/magnetohydrodynamics/index.md
@@ -24,24 +24,23 @@ and electric current density $$\vb{J}$$ are:
 
 $$\begin{aligned}
     p
-    = p_i + p_e
-    \qquad \quad
+    &= p_i + p_e
+    \\
     \vb{J}
-    = q_i n_i \vb{u}_i + q_e n_e \vb{u}_e
+    &= q_i n_i \vb{u}_i + q_e n_e \vb{u}_e
 \end{aligned}$$
 
 Meanwhile, the macroscopic mass density $$\rho$$
-and center-of-mass flow velocity $$\vb{u}$$
-are as follows, although the ions dominate due to their large mass:
+and center-of-mass flow velocity $$\vb{u}$$ are as follows,
+although the ions dominate both due to their large mass,
+so $$\rho \approx m_i n_i$$ and $$\vb{u} \approx \vb{u}_i$$:
 
 $$\begin{aligned}
     \rho
-    = m_i n_i + m_e n_e
-    \approx m_i n_i
-    \qquad \quad
+    &= m_i n_i + m_e n_e
+    \\
     \vb{u}
-    = \frac{1}{\rho} \Big( m_i n_i \vb{u}_i + m_e n_e \vb{u}_e \Big)
-    \approx \vb{u}_i
+    &= \frac{1}{\rho} \Big( m_i n_i \vb{u}_i + m_e n_e \vb{u}_e \Big)
 \end{aligned}$$
 
 With these quantities in mind,
@@ -75,9 +74,9 @@ $$\begin{aligned}
 \end{aligned}$$
 
 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 $$\tau_\mathrm{char}$$,
+is negligible compared to the Lorentz force.
+Let $$\tau_\mathrm{char}$$ be the characteristic timescale of the plasma's dynamics
+(i.e. nothing notable happens in times shorter than $$\tau_\mathrm{char}$$),
 then this assumption can be written as:
 
 $$\begin{aligned}
@@ -86,15 +85,14 @@ $$\begin{aligned}
     \sim \frac{m_e n_e |\vb{u}_e| / \tau_\mathrm{char}}{q_e n_e |\vb{u}_e| |\vb{B}|}
     = \frac{m_e}{q_e |\vb{B}| \tau_\mathrm{char}}
     = \frac{1}{\omega_{ce} \tau_\mathrm{char}}
-    \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](/know/concept/lorentz-force/)).
 In other words, our assumption is equivalent to
 the electron gyration period $$2 \pi / \omega_{ce}$$
-being small compared to the macroscopic dynamics' timescale $$\tau_\mathrm{char}$$.
-By construction, we can thus ignore the left-hand side
-of the electron momentum equation, leaving:
+being small compared to the macroscopic timescale $$\tau_\mathrm{char}$$.
+We can thus ignore the left-hand side of the electron momentum equation, leaving:
 
 $$\begin{aligned}
     m_i n_i \frac{\mathrm{D} \vb{u}_i}{\mathrm{D} t}
@@ -138,8 +136,8 @@ $$\begin{aligned}
 
 However, we found this by combining two equations into one,
 so some information was implicitly lost;
-we need a second momentum equation.
-Therefore, we return to the electrons' momentum equation,
+we need a second one to keep our system of equations complete.
+Therefore we return to the electrons' momentum equation,
 after a bit of rearranging:
 
 $$\begin{aligned}
@@ -154,14 +152,14 @@ so:
 $$\begin{aligned}
     \vb{E} + \vb{u}_e \cross \vb{B} - \frac{\nabla p_e}{q_e n_e}
     = \eta \vb{J}
-    \qquad \quad
+    \qquad \qquad
     \eta
     \equiv \frac{f_{ei} m_e}{n_e q_e^2}
 \end{aligned}$$
 
 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.
+for more information and a rough estimate of its value in 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,
@@ -183,34 +181,37 @@ $$\begin{aligned}
     - \nabla \cross \frac{\nabla p_e}{q_e n_e}
 \end{aligned}$$
 
-Where we have used Faraday's law.
+Where we have used [Faraday's law](/know/concept/maxwells-equations/).
 This is the **induction equation**,
 and is used to compute $$\vb{B}$$.
 The pressure term can be rewritten using the ideal gas law $$p_e = k_B T_e n_e$$:
 
 $$\begin{aligned}
     \nabla \cross \frac{\nabla p_e}{q_e n_e}
-    = \frac{k_B}{q_e} \nabla \cross \frac{\nabla (n_e T_e)}{n_e}
-    = \frac{k_B}{q_e} \nabla \cross \Big( \nabla T_e + T_e \frac{\nabla n_e}{n_e} \Big)
+    &= \frac{k_B}{q_e} \nabla \cross \frac{\nabla (n_e T_e)}{n_e}
+    \\
+    &= \frac{k_B}{q_e} \nabla \cross \Big( \nabla T_e + T_e \frac{\nabla n_e}{n_e} \Big)
 \end{aligned}$$
 
 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 $$\nabla \cross (f \nabla g) = \nabla f \cross \nabla g$$,
-leading to:
+Then we use the vector identity $$\nabla \cross (f \nabla g) = \nabla f \cross \nabla g$$ to get:
 
 $$\begin{aligned}
     \nabla \cross \frac{\nabla p_e}{q_e n_e}
-    = \frac{k_B}{q_e} \nabla \cross \big( T_e \: \nabla\! \ln(n_e) \big)
-    = \frac{k_B}{q_e} \big( \nabla T_e \cross \nabla\! \ln(n_e) \big)
-    = \frac{k_B}{q_e n_e} \big( \nabla T_e \cross \nabla n_e \big)
+    &= \frac{k_B}{q_e} \nabla \cross \big( T_e \: \nabla\! \ln(n_e) \big)
+    \\
+    &= \frac{k_B}{q_e} \big( \nabla T_e \cross \nabla\! \ln(n_e) \big)
+    \\
+    &= \frac{k_B}{q_e n_e} \big( \nabla T_e \cross \nabla n_e \big)
 \end{aligned}$$
 
 It is reasonable to assume that $$\nabla T_e$$ and $$\nabla n_e$$
 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 $$\vb{B}$$,
-so we argue that it can be dropped from the original (non-curled) equation too, leaving:
+so we argue that it can also be dropped
+from the original equation (before taking the curl):
 
 $$\begin{aligned}
     \boxed{
@@ -232,20 +233,18 @@ $$\begin{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:
+and let $$\lambda_\mathrm{char} \gg \lambda_D$$ be its characteristic length scale:
 
 $$\begin{aligned}
     \nabla \cross \vb{E}
     = - \pdv{\vb{B}}{t}
-    \quad \implies \quad
+    \qquad \implies \qquad
     |\vb{E}|
     \sim \frac{\lambda_\mathrm{char}}{\tau_\mathrm{char}} |\vb{B}|
 \end{aligned}$$
 
-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 $$c$$,
+From this, we find that we can neglect the last term in Ampère's law
+as long as the characteristic velocity $$v_\mathrm{char}$$ is tiny compared to $$c$$,
 i.e. the plasma must be non-relativistic:
 
 $$\begin{aligned}
@@ -254,7 +253,6 @@ $$\begin{aligned}
     \sim \frac{|\vb{E}| / \tau_\mathrm{char}}{|\vb{B}| c^2 / \lambda_\mathrm{char}}
     \sim \frac{|\vb{B}| \lambda_\mathrm{char}^2 / \tau_\mathrm{char}^2}{|\vb{B}| c^2}
     = \frac{v_\mathrm{char}^2}{c^2}
-    \ll 1
 \end{aligned}$$
 
 We thus have the following reduced form of Ampère's law,
@@ -265,7 +263,7 @@ $$\begin{aligned}
         \nabla \cross \vb{B}
         = \mu_0 \vb{J}
     }
-    \qquad \quad
+    \qquad \qquad
     \boxed{
         \nabla \cross \vb{E}
         = - \pdv{\vb{B}}{t}
@@ -287,10 +285,12 @@ the [material derivative](/know/concept/material-derivative/)
 $$\mathrm{D} \rho / \mathrm{D} t$$ as follows:
 
 $$\begin{aligned}
-    \pdv{\rho}{t} + \nabla \cdot (\rho \vb{u})
-    = \pdv{\rho}{t} + \rho \nabla \cdot \vb{u} + \vb{u} \cdot \nabla \rho
-    = \rho \nabla \cdot \vb{u} + \frac{\mathrm{D} \rho}{\mathrm{D} t}
-    = 0
+    0
+    &= \pdv{\rho}{t} + \nabla \cdot (\rho \vb{u})
+    \\
+    &= \pdv{\rho}{t} + \rho \nabla \cdot \vb{u} + \vb{u} \cdot \nabla \rho
+    \\
+    &= \rho \nabla \cdot \vb{u} + \frac{\mathrm{D} \rho}{\mathrm{D} t}
 \end{aligned}$$
 
 Inserting this into the equation of state
@@ -311,6 +311,7 @@ but we have merged $$n_i$$ and $$n_e$$ into $$\rho$$,
 and $$p_i$$ and $$p_i$$ into $$p$$.
 
 
+
 ## Ohm's law variants
 
 It is worth discussing the generalized Ohm's law in more detail.
@@ -321,29 +322,27 @@ $$\begin{aligned}
     = \eta \vb{J}
 \end{aligned}$$
 
-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*.
-This term can be dropped in any of the following cases:
+However, most authors neglect some terms:
+the full form is used for **Hall MHD**,
+where $$\vb{J} \cross \vb{B}$$ is called the **Hall term**.
+It can be dropped in any of the following cases:
 
-$$\begin{gathered}
+$$\begin{aligned}
     1
-    \gg \frac{\big| \vb{J} \cross \vb{B} / q_e n_e \big|}{\big| \vb{u} \cross \vb{B} \big|}
+    &\gg \frac{\big| \vb{J} \cross \vb{B} / q_e n_e \big|}{\big| \vb{u} \cross \vb{B} \big|}
     \sim \frac{\rho v_\mathrm{char} / \tau_\mathrm{char}}{v_\mathrm{char} |\vb{B}| q_i n_i}
     \approx \frac{m_i n_i}{|\vb{B}| q_i n_i \tau_\mathrm{char}}
     = \frac{1}{\omega_{ci} \tau_\mathrm{char}}
-    \ll 1
     \\
     1
-    \gg \frac{\big| \vb{J} \cross \vb{B} / q_e n_e \big|}{\big| \eta \vb{J} \big|}
+    &\gg \frac{\big| \vb{J} \cross \vb{B} / q_e n_e \big|}{\big| \eta \vb{J} \big|}
     \sim \frac{|\vb{J}| |\vb{B}| q_e^2 n_e}{f_{ei} m_e |\vb{J}| q_e n_e}
     = \frac{|\vb{B}| q_e}{f_{ei} m_e}
     = \frac{\omega_{ce}}{f_{ei}}
-    \ll 1
-\end{gathered}$$
+\end{aligned}$$
 
 Where we have used the MHD momentum equation with $$\nabla p \approx 0$$
-to obtain the scale estimate $$\vb{J} \cross \vb{B} \sim \rho v_\mathrm{char} / \tau_\mathrm{char}$$.
+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 $$\omega_{ce} \ll f_{ei}$$,
@@ -354,18 +353,17 @@ $$\begin{aligned}
     = \eta \vb{J}
 \end{aligned}$$
 
-Finally, we can neglect the resisitive term $$\eta \vb{J}$$
+Finally, we can neglect the resistive 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 $$|\vb{J}| \sim |\vb{B}| / \mu_0 \lambda_\mathrm{char}$$:
 
 $$\begin{aligned}
     1
     \ll \frac{\big| \vb{u} \cross \vb{B} \big|}{\big| \eta \vb{J} \big|}
-    \sim \frac{v_\mathrm{char} |\vb{B}|}{\eta \vb{J}}
+    \sim \frac{v_\mathrm{char} |\vb{B}|}{\eta |\vb{J}|}
     \sim \frac{v_\mathrm{char} |\vb{B}|}{\eta |\vb{B}| / \mu_0 \lambda_\mathrm{char}}
     = \mathrm{R_m}
-    \gg 1
 \end{aligned}$$
 
 Where we have defined the **magnetic Reynolds number** $$\mathrm{R_m}$$ as follows,
@@ -379,13 +377,15 @@ $$\begin{aligned}
 \end{aligned}$$
 
 If $$\mathrm{R_m} \ll 1$$, the plasma is "electrically viscous",
-such that resistivity needs to be accounted for,
+meaning resistivity needs to be accounted for,
 whereas if $$\mathrm{R_m} \gg 1$$, the resistivity is negligible,
 in which case we have **ideal MHD**:
 
 $$\begin{aligned}
-    \vb{E} + \vb{u} \cross \vb{B}
-    = 0
+    \boxed{
+        \vb{E} + \vb{u} \cross \vb{B}
+        = 0
+    }
 \end{aligned}$$
 
 
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