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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} |