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-rw-r--r--source/know/concept/magnetohydrodynamics/index.md122
1 files changed, 61 insertions, 61 deletions
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**:
$$\begin{aligned}
- \vb{E} + \vb{u} \cross \vb{B}
- = 0
+ \boxed{
+ \vb{E} + \vb{u} \cross \vb{B}
+ = 0
+ }
\end{aligned}$$