13. Two-Fluid Model
13. Two-Fluid Model¶
Learning Objectives¶
- Derive fluid equations from kinetic theory by taking velocity-space moments of the Vlasov equation
- Understand the closure problem and various closure approximations (isothermal, adiabatic, CGL)
- Derive the generalized Ohm's law from the electron momentum equation and analyze each term's physical significance
- Explain the Hall effect and its role in decoupling ions from the magnetic field at small scales
- Distinguish between particle drifts and fluid drifts, particularly the diamagnetic drift
- Apply two-fluid theory to understand wave phenomena beyond single-fluid MHD
1. From Vlasov to Fluid Equations¶
1.1 The Moment Hierarchy¶
The Vlasov equation describes the evolution of the distribution function $f_s(\mathbf{r}, \mathbf{v}, t)$ for species $s$:
$$\frac{\partial f_s}{\partial t} + \mathbf{v} \cdot \nabla f_s + \frac{q_s}{m_s}(\mathbf{E} + \mathbf{v} \times \mathbf{B}) \cdot \frac{\partial f_s}{\partial \mathbf{v}} = \left(\frac{\partial f_s}{\partial t}\right)_{\text{coll}}$$
While the Vlasov equation contains complete information about the plasma, it is a 6D partial differential equation that is computationally expensive to solve. For many applications, we don't need the full distribution function—we only care about macroscopic quantities like density, flow velocity, and pressure.
The method of moments reduces the dimensionality by integrating the Vlasov equation over velocity space with different weights. The $n$-th moment is obtained by multiplying the Vlasov equation by $v^n$ and integrating:
$$\int (\text{Vlasov equation}) \times (\text{weight function}) \, d^3v$$
This generates a hierarchy of fluid equations, where each equation involves the next higher-order moment.
1.2 Zeroth Moment: Continuity Equation¶
The zeroth moment (weight = 1) gives the continuity equation:
$$\int \frac{\partial f_s}{\partial t} d^3v + \int \mathbf{v} \cdot \nabla f_s d^3v + \int \frac{q_s}{m_s}(\mathbf{E} + \mathbf{v} \times \mathbf{B}) \cdot \frac{\partial f_s}{\partial \mathbf{v}} d^3v = 0$$
The number density is: $$n_s(\mathbf{r}, t) = \int f_s(\mathbf{r}, \mathbf{v}, t) d^3v$$
For the first term: $$\int \frac{\partial f_s}{\partial t} d^3v = \frac{\partial}{\partial t} \int f_s d^3v = \frac{\partial n_s}{\partial t}$$
For the second term, using the divergence theorem in velocity space: $$\int \mathbf{v} \cdot \nabla f_s d^3v = \nabla \cdot \int \mathbf{v} f_s d^3v = \nabla \cdot (n_s \mathbf{u}_s)$$
where the mean flow velocity is: $$\mathbf{u}_s = \frac{1}{n_s} \int \mathbf{v} f_s d^3v$$
For the third term, the Lorentz force term vanishes because: $$\int \frac{\partial f_s}{\partial \mathbf{v}} d^3v = [f_s]_{v=-\infty}^{v=+\infty} = 0$$
(assuming $f_s \to 0$ as $|\mathbf{v}| \to \infty$).
The result is the continuity equation:
$$\boxed{\frac{\partial n_s}{\partial t} + \nabla \cdot (n_s \mathbf{u}_s) = 0}$$
This is conservation of particles. In the presence of ionization/recombination, a source term would appear on the right-hand side.
1.3 First Moment: Momentum Equation¶
The first moment (weight = $m_s \mathbf{v}$) gives the momentum equation. We multiply the Vlasov equation by $m_s \mathbf{v}$ and integrate:
Define the momentum density: $$\mathbf{p}_s = m_s n_s \mathbf{u}_s = m_s \int \mathbf{v} f_s d^3v$$
The peculiar velocity (thermal velocity) is: $$\mathbf{w} = \mathbf{v} - \mathbf{u}_s$$
The pressure tensor is: $$\overleftrightarrow{P}_s = m_s \int \mathbf{w} \mathbf{w} f_s d^3v$$
After considerable algebra (using integration by parts and the divergence theorem), the momentum equation becomes:
$$\boxed{m_s n_s \frac{d \mathbf{u}_s}{dt} = q_s n_s (\mathbf{E} + \mathbf{u}_s \times \mathbf{B}) - \nabla \cdot \overleftrightarrow{P}_s + \mathbf{R}_s}$$
where $d/dt = \partial/\partial t + \mathbf{u}_s \cdot \nabla$ is the convective derivative, and $\mathbf{R}_s$ is the momentum transfer from collisions with other species.
This is Newton's second law for a fluid element: - LHS: mass × acceleration - RHS: Lorentz force + pressure gradient force + collision force
Key point: This equation introduces a new quantity, the pressure tensor $\overleftrightarrow{P}_s$, which is a second-order moment of the distribution function.
1.4 Second Moment: Energy Equation¶
The second moment (weight = $\frac{1}{2} m_s v^2$) gives the energy equation:
Define the thermal energy density: $$\mathcal{E}_s = \frac{1}{2} m_s \int w^2 f_s d^3v$$
For an isotropic pressure ($\overleftrightarrow{P}_s = p_s \overleftrightarrow{I}$), we have: $$p_s = \frac{1}{3} m_s \int w^2 f_s d^3v = \frac{2}{3} \mathcal{E}_s$$
The energy equation becomes:
$$\frac{\partial \mathcal{E}_s}{\partial t} + \nabla \cdot (\mathcal{E}_s \mathbf{u}_s) = -p_s \nabla \cdot \mathbf{u}_s - \nabla \cdot \mathbf{q}_s + Q_s$$
where: - $\mathbf{q}_s = \frac{1}{2} m_s \int w^2 \mathbf{w} f_s d^3v$ is the heat flux vector (third-order moment) - $Q_s$ is the collisional energy transfer
Using $p_s = \frac{2}{3} \mathcal{E}_s$, this can be rewritten as:
$$\frac{3}{2} \frac{d p_s}{dt} + \frac{5}{2} p_s \nabla \cdot \mathbf{u}_s = -\nabla \cdot \mathbf{q}_s + Q_s$$
The closure problem: The energy equation introduces the heat flux $\mathbf{q}_s$, a third-order moment. If we took the third moment, we'd get an equation involving a fourth-order moment, and so on. This infinite hierarchy must be closed by making an assumption about the highest-order moment.
1.5 The Closure Problem¶
Moment hierarchy:
0th moment: ∂n/∂t + ∇·(nu) = 0 (introduces u)
1st moment: mn(du/dt) = qn(E+u×B) - ∇·P + R (introduces P)
2nd moment: dp/dt = -p∇·u - ∇·q + Q (introduces q)
3rd moment: ... (introduces next moment)
...
Each equation introduces a new unknown from the next higher moment.
This is the CLOSURE PROBLEM.
We need to truncate the hierarchy by assuming a relationship between the highest moment and lower moments. Common closures:
1. Isothermal closure: Assume constant temperature $$p_s = n_s k_B T_s, \quad T_s = \text{const}$$
This is valid when heat conduction is very efficient, so temperature equilibrates instantly.
2. Adiabatic closure: Assume no heat flux ($\mathbf{q}_s = 0$) and adiabatic evolution $$\frac{d}{dt}\left( \frac{p_s}{n_s^\gamma} \right) = 0$$
where $\gamma$ is the adiabatic index ($\gamma = 5/3$ for a monatomic gas). This is valid for rapid processes where heat conduction is negligible.
3. CGL closure (Chew-Goldberger-Low): For collisionless magnetized plasmas, pressure is anisotropic: $$\overleftrightarrow{P}_s = p_{\perp s} \overleftrightarrow{I} + (p_{\parallel s} - p_{\perp s}) \hat{\mathbf{b}} \hat{\mathbf{b}}$$
with double adiabatic equations: $$\frac{d}{dt}\left( \frac{p_{\perp s}}{n_s B} \right) = 0, \quad \frac{d}{dt}\left( \frac{p_{\parallel s} B^2}{n_s^3} \right) = 0$$
We'll discuss CGL in Lesson 14.
1.6 Two-Fluid Equations Summary¶
For each species (electrons $e$, ions $i$), we have:
Continuity: $$\frac{\partial n_s}{\partial t} + \nabla \cdot (n_s \mathbf{u}_s) = 0$$
Momentum: $$m_s n_s \frac{d \mathbf{u}_s}{dt} = q_s n_s (\mathbf{E} + \mathbf{u}_s \times \mathbf{B}) - \nabla p_s + \mathbf{R}_s$$
(assuming isotropic pressure)
Energy (with adiabatic closure): $$\frac{d}{dt}\left( \frac{p_s}{n_s^\gamma} \right) = 0$$
These are coupled to Maxwell's equations: $$\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}, \quad \nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0 \epsilon_0 \frac{\partial \mathbf{E}}{\partial t}$$ $$\nabla \cdot \mathbf{E} = \frac{\rho}{\epsilon_0}, \quad \nabla \cdot \mathbf{B} = 0$$
where the charge and current densities are: $$\rho = \sum_s q_s n_s, \quad \mathbf{J} = \sum_s q_s n_s \mathbf{u}_s$$
The collision terms $\mathbf{R}_s$ couple the species. For electron-ion collisions: $$\mathbf{R}_e = -\mathbf{R}_i = -\frac{m_e n_e}{\tau_{ei}} (\mathbf{u}_e - \mathbf{u}_i)$$
where $\tau_{ei}$ is the electron-ion collision time.
2. Generalized Ohm's Law¶
2.1 Derivation from Electron Momentum Equation¶
One of the most important results from two-fluid theory is the generalized Ohm's law, which relates the electric field to the current. In ideal MHD, we have the simple form: $$\mathbf{E} + \mathbf{v} \times \mathbf{B} = 0$$
But this is a severe approximation. Let's derive the full form from the electron momentum equation.
Starting with: $$m_e n_e \frac{d \mathbf{u}_e}{dt} = -e n_e (\mathbf{E} + \mathbf{u}_e \times \mathbf{B}) - \nabla p_e + \mathbf{R}_e$$
The collision term can be written as: $$\mathbf{R}_e = -\frac{m_e n_e}{\tau_{ei}} (\mathbf{u}_e - \mathbf{u}_i) \approx -\frac{m_e n_e \mathbf{u}_e}{\tau_{ei}}$$
(assuming $u_e \gg u_i$ for current-carrying electrons).
Rearranging: $$\mathbf{E} + \mathbf{u}_e \times \mathbf{B} = \frac{m_e}{e \tau_{ei}} \mathbf{u}_e - \frac{1}{e n_e} \nabla p_e + \frac{m_e}{e n_e} \frac{d \mathbf{u}_e}{dt}$$
Now, express everything in terms of the current density $\mathbf{J}$ and the center-of-mass velocity $\mathbf{v}$.
Define: $$\mathbf{J} = -e n_e \mathbf{u}_e + e n_i \mathbf{u}_i \approx -e n_e (\mathbf{u}_e - \mathbf{u}_i)$$ $$\mathbf{v} = \frac{m_i n_i \mathbf{u}_i + m_e n_e \mathbf{u}_e}{m_i n_i + m_e n_e} \approx \mathbf{u}_i$$
(using $m_i \gg m_e$ and quasi-neutrality $n_e \approx n_i \equiv n$).
From the current definition: $$\mathbf{u}_e = \mathbf{u}_i - \frac{\mathbf{J}}{e n} \approx \mathbf{v} - \frac{\mathbf{J}}{e n}$$
Substituting into the rearranged electron equation:
$$\mathbf{E} + \left( \mathbf{v} - \frac{\mathbf{J}}{en} \right) \times \mathbf{B} = \frac{m_e}{e^2 n \tau_{ei}} \mathbf{J} - \frac{1}{en} \nabla p_e + \frac{m_e}{e^2 n} \frac{d}{dt}\left( -\frac{\mathbf{J}}{e n} \right)$$
Simplifying the cross product: $$\mathbf{u}_e \times \mathbf{B} = \mathbf{v} \times \mathbf{B} - \frac{\mathbf{J} \times \mathbf{B}}{en}$$
This gives the generalized Ohm's law:
$$\boxed{\mathbf{E} + \mathbf{v} \times \mathbf{B} = \eta \mathbf{J} + \frac{1}{en} \mathbf{J} \times \mathbf{B} - \frac{1}{en} \nabla p_e + \frac{m_e}{e^2 n^2} \frac{d \mathbf{J}}{dt}}$$
where the resistivity is: $$\eta = \frac{m_e}{e^2 n \tau_{ei}}$$
2.2 Physical Interpretation of Each Term¶
Let's identify each term on the RHS:
- Resistive term: $\eta \mathbf{J}$
- Ohmic dissipation due to electron-ion collisions
- Causes magnetic diffusion (resistive MHD)
-
$\eta \sim T_e^{-3/2}$ (decreases with temperature)
-
Hall term: $\frac{1}{en} \mathbf{J} \times \mathbf{B}$
- Decoupling of ions from magnetic field
- Important at scales $\sim$ ion skin depth $d_i = c/\omega_{pi}$
-
Enables fast magnetic reconnection
-
Electron pressure term: $-\frac{1}{en} \nabla p_e$
- Pressure gradient drives current even without E field
-
Important in steep gradient regions (e.g., current sheets)
-
Electron inertia term: $\frac{m_e}{e^2 n^2} \frac{d \mathbf{J}}{dt}$
- Important at electron skin depth $d_e = c/\omega_{pe}$
- Relevant for very fast phenomena (whistler waves, reconnection)
2.3 Scale Analysis: When Does Each Term Matter?¶
Let's perform an order-of-magnitude analysis to determine when each term is important.
Define characteristic scales: - Length: $L$ - Velocity: $V$ - Magnetic field: $B_0$ - Density: $n_0$ - Current: $J_0 \sim B_0/(\mu_0 L)$ (from Ampère's law)
Ideal MHD term (LHS): $$\mathbf{v} \times \mathbf{B} \sim V B_0$$
Resistive term: $$\eta \mathbf{J} \sim \eta \frac{B_0}{\mu_0 L}$$
Ratio: $$\frac{\eta J}{\mathbf{v} \times \mathbf{B}} \sim \frac{\eta}{\mu_0 V L} = \frac{1}{R_m}$$
where $R_m = \mu_0 V L / \eta$ is the magnetic Reynolds number. Resistivity is important when $R_m \lesssim 1$.
Hall term: $$\frac{\mathbf{J} \times \mathbf{B}}{en} \sim \frac{B_0^2}{\mu_0 e n_0 L}$$
Ratio: $$\frac{J \times B / en}{\mathbf{v} \times \mathbf{B}} \sim \frac{B_0}{\mu_0 e n_0 V L} = \frac{V_A}{V} \frac{d_i}{L}$$
where $d_i = c/\omega_{pi} = \sqrt{m_i / (\mu_0 e^2 n_0)}$ is the ion skin depth and $V_A = B_0/\sqrt{\mu_0 m_i n_0}$ is the Alfvén speed.
The Hall term is important when $L \lesssim d_i$ or when $V \lesssim V_A$ at ion scales.
Electron pressure term: $$\frac{\nabla p_e}{en} \sim \frac{k_B T_e}{eL}$$
Ratio: $$\frac{\nabla p_e / en}{\mathbf{v} \times \mathbf{B}} \sim \frac{k_B T_e}{e V B_0 L} = \frac{v_{te}^2}{V^2} \frac{\rho_e}{L}$$
where $v_{te} = \sqrt{k_B T_e / m_e}$ is the electron thermal speed and $\rho_e = v_{te}/\omega_{ce}$ is the electron gyroradius.
This term is important in regions of steep pressure gradients.
Electron inertia term: $$\frac{m_e}{e^2 n^2} \frac{dJ}{dt} \sim \frac{m_e}{e^2 n_0^2} \frac{B_0}{\mu_0 L} \frac{V}{L} = \frac{m_e V B_0}{\mu_0 e^2 n_0 L^2}$$
Ratio: $$\frac{m_e dJ/dt / (e^2 n^2)}{v \times B} \sim \frac{m_e}{\mu_0 e^2 n_0 L^2} = \frac{d_e^2}{L^2}$$
where $d_e = c/\omega_{pe}$ is the electron skin depth.
This term is important when $L \lesssim d_e$.
Summary:
Term Scale When important
---------------- ----------------- ------------------------
Resistive 1/R_m R_m ~ 1 (low T, small L)
Hall d_i/L L ~ d_i (ion scales)
Electron pressure β_e ρ_e/L Steep gradients
Electron inertia (d_e/L)^2 L ~ d_e (electron scales)
Typical ordering: d_e << ρ_e << d_i << L (MHD)
2.4 Limiting Cases¶
Ideal MHD ($R_m \to \infty$, $L \gg d_i$): $$\mathbf{E} + \mathbf{v} \times \mathbf{B} = 0$$
Magnetic field is frozen into the fluid.
Resistive MHD (keep resistive term, drop others): $$\mathbf{E} + \mathbf{v} \times \mathbf{B} = \eta \mathbf{J}$$
Allows magnetic reconnection, but slow (Sweet-Parker rate).
Hall MHD (keep Hall term, drop resistive/inertia): $$\mathbf{E} + \mathbf{v} \times \mathbf{B} = \frac{1}{en} \mathbf{J} \times \mathbf{B}$$
Enables fast reconnection (Petschek rate), whistler waves.
Electron MHD (keep Hall + inertia, drop resistive): $$\mathbf{E} + \mathbf{v} \times \mathbf{B} = \frac{1}{en} \mathbf{J} \times \mathbf{B} + \frac{m_e}{e^2 n^2} \frac{d \mathbf{J}}{dt}$$
Relevant at electron scales (e.g., reconnection diffusion region).
3. The Hall Effect¶
3.1 Physics of the Hall Term¶
The Hall term $\frac{1}{en} \mathbf{J} \times \mathbf{B}$ arises from the difference in electron and ion motions. When a current flows across a magnetic field, electrons and ions experience different Lorentz forces, creating a charge separation and thus an electric field perpendicular to both $\mathbf{J}$ and $\mathbf{B}$.
Consider a current $\mathbf{J} = J_x \hat{\mathbf{x}}$ in a magnetic field $\mathbf{B} = B_0 \hat{\mathbf{z}}$:
$$\mathbf{J} \times \mathbf{B} = J_x B_0 \hat{\mathbf{y}}$$
This creates an electric field: $$E_y = \frac{J_x B_0}{en}$$
This is the Hall electric field.
3.2 Hall Parameter¶
The Hall parameter quantifies the importance of the magnetic field:
$$\Omega_s \tau_s = \omega_{cs} \tau_{cs}$$
where $\omega_{cs} = q_s B / m_s$ is the cyclotron frequency and $\tau_{cs}$ is the collision time.
- When $\Omega_s \tau_s \ll 1$: collisions dominate, particle orbits are interrupted before completing a gyration → unmagnetized
- When $\Omega_s \tau_s \gg 1$: particles complete many gyrations between collisions → magnetized
For electrons in typical plasmas, $\Omega_e \tau_e \gg 1$ (strongly magnetized). For ions, $\Omega_i \tau_i$ can vary (weakly magnetized in collisional plasmas, strongly magnetized in hot fusion plasmas).
3.3 Decoupling of Ions from Magnetic Field¶
At scales larger than the ion skin depth $d_i$, both electrons and ions are frozen to the magnetic field (ideal MHD). But at scales $L \lesssim d_i$, the Hall term becomes important, and ions decouple from the magnetic field.
To see this, consider the ion and electron momentum equations:
Ions: $$m_i n \frac{d \mathbf{u}_i}{dt} = e n (\mathbf{E} + \mathbf{u}_i \times \mathbf{B}) - \nabla p_i$$
Electrons (from generalized Ohm's law, keeping only Hall term): $$\mathbf{E} + \mathbf{u}_e \times \mathbf{B} \approx \frac{1}{en} \mathbf{J} \times \mathbf{B}$$
Using $\mathbf{J} = en(\mathbf{u}_i - \mathbf{u}_e)$: $$\mathbf{E} + \mathbf{u}_e \times \mathbf{B} = \frac{1}{en} en (\mathbf{u}_i - \mathbf{u}_e) \times \mathbf{B}$$
Rearranging: $$\mathbf{E} + \mathbf{u}_e \times \mathbf{B} = (\mathbf{u}_i - \mathbf{u}_e) \times \mathbf{B}$$ $$\mathbf{E} + \mathbf{u}_i \times \mathbf{B} = 0$$
So electrons satisfy the frozen-in condition: $$\mathbf{E} + \mathbf{u}_e \times \mathbf{B} = 0$$
But ions do not! They experience an electric field: $$\mathbf{E} = -\mathbf{u}_i \times \mathbf{B} + (\mathbf{u}_i - \mathbf{u}_e) \times \mathbf{B} = \mathbf{u}_e \times \mathbf{B} \neq -\mathbf{u}_i \times \mathbf{B}$$
This means the magnetic field is frozen into the electron fluid, not the ion fluid, at scales $\sim d_i$.
3.4 Hall MHD Waves¶
Including the Hall term modifies MHD wave dispersion. The key change is the appearance of whistler waves at high frequencies.
The Hall MHD dispersion relation (in the low-frequency, small-amplitude limit) gives:
Alfvén/whistler branch: $$\omega = k_\parallel V_A \sqrt{1 + k^2 d_i^2}$$
- At $k d_i \ll 1$ (large scales): $\omega \approx k_\parallel V_A$ (Alfvén wave)
- At $k d_i \gg 1$ (small scales): $\omega \approx k_\parallel V_A k d_i = k \sqrt{k_\parallel V_A d_i}$ (whistler)
Whistler waves have: - Right-hand circular polarization (in ion frame) - Dispersive: phase velocity increases with $k$ - No ion motion: only electrons respond
We'll compute this dispersion relation in the Python code below.
4. Diamagnetic Drift¶
4.1 Particle vs. Fluid Drifts¶
In Lesson 3, we derived particle drifts from single-particle orbit theory:
$$\mathbf{v}_E = \frac{\mathbf{E} \times \mathbf{B}}{B^2}, \quad \mathbf{v}_{\nabla B} = \frac{m v_\perp^2}{2 q B^3} \mathbf{B} \times \nabla B, \quad \text{etc.}$$
These are drifts of individual particles.
In fluid theory, we have fluid drifts that emerge from pressure gradients and other collective effects. The most important is the diamagnetic drift.
4.2 Derivation of Diamagnetic Drift¶
Consider the momentum equation in a magnetized plasma with a pressure gradient perpendicular to $\mathbf{B}$:
$$m_s n_s \frac{d \mathbf{u}_s}{dt} = q_s n_s (\mathbf{E} + \mathbf{u}_s \times \mathbf{B}) - \nabla p_s$$
In equilibrium ($d\mathbf{u}_s/dt = 0$) with no electric field ($\mathbf{E} = 0$):
$$0 = q_s n_s \mathbf{u}_s \times \mathbf{B} - \nabla p_s$$
Taking the cross product with $\mathbf{B}$:
$$q_s n_s (\mathbf{u}_s \times \mathbf{B}) \times \mathbf{B} = -\nabla p_s \times \mathbf{B}$$
Using the vector identity $(\mathbf{A} \times \mathbf{B}) \times \mathbf{C} = \mathbf{B}(\mathbf{A} \cdot \mathbf{C}) - \mathbf{A}(\mathbf{B} \cdot \mathbf{C})$:
$$q_s n_s [\mathbf{B} (\mathbf{u}_s \cdot \mathbf{B}) - \mathbf{u}_s B^2] = -\nabla p_s \times \mathbf{B}$$
If the flow is perpendicular to $\mathbf{B}$ (i.e., $\mathbf{u}_s \cdot \mathbf{B} = 0$):
$$\mathbf{u}_s = \frac{\nabla p_s \times \mathbf{B}}{q_s n_s B^2} = -\frac{\mathbf{B} \times \nabla p_s}{q_s n_s B^2}$$
This is the diamagnetic drift velocity:
$$\boxed{\mathbf{v}_{*s} = -\frac{\mathbf{B} \times \nabla p_s}{q_s n_s B^2}}$$
For electrons ($q_e = -e$): $$\mathbf{v}_{*e} = \frac{\mathbf{B} \times \nabla p_e}{e n_e B^2}$$
For ions ($q_i = +e$): $$\mathbf{v}_{*i} = -\frac{\mathbf{B} \times \nabla p_i}{e n_i B^2}$$
4.3 Diamagnetic Current¶
The diamagnetic current is:
$$\mathbf{J}_* = \sum_s q_s n_s \mathbf{v}_{*s} = -\frac{\mathbf{B} \times \nabla p_e}{B^2} - \frac{\mathbf{B} \times \nabla p_i}{B^2} = \frac{\mathbf{B} \times \nabla p}{B^2}$$
where $p = p_e + p_i$ is the total pressure.
This can also be written as: $$\mathbf{J}_* = -\nabla p \times \frac{\mathbf{B}}{B^2}$$
Key point: The diamagnetic drift is not a particle drift! If you solve for individual particle orbits, you won't find this drift. It arises from the spatial variation of the distribution function due to pressure gradients.
To see this, note that the diamagnetic drift velocity depends on the gradient scale length $L_p = p / |\nabla p|$:
$$v_* \sim \frac{p}{q n B L_p} = \frac{k_B T}{q B L_p} \sim \frac{\rho}{L_p} v_{th}$$
where $\rho = v_{th}/\omega_c$ is the gyroradius.
In a collisionless plasma, particles on different gyro-orbits have different densities, creating a net drift when averaged over a distribution.
4.4 Physical Interpretation: Magnetization Current¶
The diamagnetic current can be understood as a magnetization current arising from the magnetic moments of gyrating particles.
The magnetization is: $$\mathbf{M} = -n \mu \frac{\mathbf{B}}{B}$$
where $\mu = m v_\perp^2 / (2B)$ is the magnetic moment.
The magnetization current is: $$\mathbf{J}_m = \nabla \times \mathbf{M}$$
For a pressure gradient perpendicular to $\mathbf{B}$, this gives: $$\mathbf{J}_m = \frac{\mathbf{B} \times \nabla p_\perp}{B^2}$$
which is exactly the diamagnetic current.
4.5 Example: Cylindrical Plasma Column¶
Consider a cylindrical plasma column with: - Axial magnetic field: $\mathbf{B} = B_0 \hat{\mathbf{z}}$ - Radial pressure profile: $p(r) = p_0 \left(1 - \frac{r^2}{a^2}\right)$
The pressure gradient is: $$\nabla p = \frac{dp}{dr} \hat{\mathbf{r}} = -\frac{2 p_0 r}{a^2} \hat{\mathbf{r}}$$
The diamagnetic current is: $$\mathbf{J}_* = \frac{\mathbf{B} \times \nabla p}{B^2} = \frac{B_0 \hat{\mathbf{z}} \times \left( -\frac{2 p_0 r}{a^2} \hat{\mathbf{r}} \right)}{B_0^2} = \frac{2 p_0 r}{B_0 a^2} \hat{\boldsymbol{\theta}}$$
This is an azimuthal current that opposes the applied field (diamagnetic).
The diamagnetic drift velocity for electrons is: $$\mathbf{v}_{*e} = \frac{\mathbf{B} \times \nabla p_e}{e n_e B^2} = \frac{2 k_B T_e r}{e B_0 a^2} \hat{\boldsymbol{\theta}}$$
Electrons drift in the $+\hat{\boldsymbol{\theta}}$ direction (counterclockwise when viewed from above).
Ions drift in the opposite direction: $$\mathbf{v}_{*i} = -\frac{2 k_B T_i r}{e B_0 a^2} \hat{\boldsymbol{\theta}}$$
The net current is the sum of electron and ion contributions.
5. Two-Fluid Waves¶
5.1 Kinetic Alfvén Wave¶
At scales approaching the ion gyroradius, the Alfvén wave is modified by kinetic effects. The kinetic Alfvén wave (KAW) has a dispersion relation:
$$\omega^2 = k_\parallel^2 V_A^2 \left( 1 + k_\perp^2 \rho_s^2 \right)$$
where $\rho_s = c_s / \omega_{ci}$ is the ion sound gyroradius (or hybrid gyroradius), with $c_s = \sqrt{k_B T_e / m_i}$ the ion sound speed.
Key features: - Finite $k_\perp$ increases the wave frequency - Electric field has a parallel component: $E_\parallel \neq 0$ - Electrons can be accelerated parallel to $\mathbf{B}$
The KAW is important in: - Auroral acceleration - Solar wind turbulence - Tokamak edge turbulence
5.2 Whistler Wave from Two-Fluid Perspective¶
In Lesson 10, we derived whistler waves from kinetic theory. Here's the two-fluid perspective:
Starting from Hall MHD (electron inertia neglected), the dispersion relation for electromagnetic waves is:
$$\omega = \frac{k_\parallel^2 V_A^2}{\omega_{ci}} \equiv k_\parallel V_A k_\parallel d_i$$
This is the whistler wave: - High-frequency ($\omega \ll \omega_{ce}$, but $\omega \gg \omega_{ci}$) - Right-hand polarized (electrons gyrate, ions stationary) - Phase velocity increases with $k$ (dispersive)
The whistler wave plays a key role in: - Magnetic reconnection (enables fast inflow) - Radiation belt dynamics (pitch-angle scattering of energetic electrons) - Solar corona heating
5.3 Ion-Cyclotron Wave¶
At frequencies near the ion cyclotron frequency, the ion-cyclotron wave (or ion Bernstein wave) appears:
$$\omega \approx \omega_{ci} + k_\parallel^2 V_A^2 / \omega_{ci}$$
Features: - Left-hand polarized (ions gyrate, electrons respond adiabatically) - Resonant absorption at $\omega = \omega_{ci}$ - Used for plasma heating (ICRF heating in tokamaks)
5.4 Two-Stream Instability¶
When two fluids have relative streaming velocity $u_0$, the system can be unstable. Consider ions at rest and electrons streaming with velocity $u_0$:
The dispersion relation becomes: $$\omega^2 - k^2 c_s^2 - \omega_{pe}^2 = 0, \quad \text{(ion acoustic)}$$ $$(\omega - k u_0)^2 - \omega_{pe}^2 = 0 \quad \text{(Langmuir shifted by Doppler)}$$
When these modes couple, we get the two-stream instability if $u_0 > v_{te}$ (electron thermal speed).
Growth rate: $$\gamma \sim \frac{\omega_{pe}}{3^{1/3}} \left( \frac{u_0}{v_{te}} \right)^{2/3}$$
This is a kinetic instability, but it can be captured in two-fluid theory with appropriate closure.
6. Python Code Examples¶
6.1 Two-Fluid vs. Single-Fluid Dispersion Relations¶
import numpy as np
import matplotlib.pyplot as plt
# Plasma parameters
m_i = 1.67e-27 # proton mass (kg)
m_e = 9.11e-31 # electron mass (kg)
e = 1.6e-19 # elementary charge (C)
c = 3e8 # speed of light (m/s)
mu_0 = 4e-7 * np.pi # permeability
n = 1e19 # density (m^-3)
B = 0.1 # magnetic field (T)
T_e = 10 # electron temperature (eV)
T_i = 10 # ion temperature (eV)
# Convert temperature to Joules
k_B = 1.38e-23
T_e_J = T_e * e
T_i_J = T_i * e
# Derived quantities
omega_pe = np.sqrt(n * e**2 / (m_e * 8.85e-12))
omega_pi = np.sqrt(n * e**2 / (m_i * 8.85e-12))
omega_ce = e * B / m_e
omega_ci = e * B / m_i
v_A = B / np.sqrt(mu_0 * n * m_i) # Alfvén speed
c_s = np.sqrt((T_e_J + T_i_J) / m_i) # ion sound speed
d_i = c / omega_pi # ion skin depth
d_e = c / omega_pe # electron skin depth
print("Plasma parameters:")
print(f" Alfvén speed V_A = {v_A:.2e} m/s = {v_A/c:.2e} c")
print(f" Ion sound speed c_s = {c_s:.2e} m/s")
print(f" Ion skin depth d_i = {d_i:.2e} m")
print(f" Electron skin depth d_e = {d_e:.2e} m")
print(f" Ion gyrofrequency ω_ci = {omega_ci:.2e} rad/s")
print(f" Electron gyrofrequency ω_ce = {omega_ce:.2e} rad/s")
print()
# Wavenumber range (parallel to B)
k_min = 1 / (100 * d_i)
k_max = 1 / (0.1 * d_i)
k = np.logspace(np.log10(k_min), np.log10(k_max), 500)
# MHD Alfvén wave (single-fluid)
omega_MHD = k * v_A
# Hall MHD Alfvén/whistler wave (two-fluid)
omega_Hall = k * v_A * np.sqrt(1 + (k * d_i)**2)
# Kinetic Alfvén wave (with finite k_perp)
k_perp = k / 2 # assume oblique propagation
rho_s = c_s / omega_ci # ion sound gyroradius
omega_KAW = k * v_A * np.sqrt(1 + (k_perp * rho_s)**2)
# Plot dispersion relations
plt.figure(figsize=(10, 6))
plt.loglog(k * d_i, omega_MHD / omega_ci, 'b-', label='MHD Alfvén', linewidth=2)
plt.loglog(k * d_i, omega_Hall / omega_ci, 'r--', label='Hall MHD (whistler)', linewidth=2)
plt.loglog(k * d_i, omega_KAW / omega_ci, 'g-.', label='Kinetic Alfvén', linewidth=2)
plt.axvline(1, color='k', linestyle=':', alpha=0.5, label='$k d_i = 1$')
plt.xlabel(r'$k d_i$ (normalized wavenumber)', fontsize=12)
plt.ylabel(r'$\omega / \omega_{ci}$ (normalized frequency)', fontsize=12)
plt.title('Two-Fluid Dispersion Relations: Alfvén to Whistler Transition', fontsize=14)
plt.legend(fontsize=11)
plt.grid(True, which='both', alpha=0.3)
plt.tight_layout()
plt.savefig('two_fluid_dispersion.png', dpi=150)
plt.show()
print("At k d_i = 1:")
idx = np.argmin(np.abs(k * d_i - 1))
print(f" MHD: ω/ω_ci = {omega_MHD[idx]/omega_ci:.2f}")
print(f" Hall MHD: ω/ω_ci = {omega_Hall[idx]/omega_ci:.2f}")
print(f" Kinetic Alfvén: ω/ω_ci = {omega_KAW[idx]/omega_ci:.2f}")
6.2 Generalized Ohm's Law: Relative Term Magnitudes¶
import numpy as np
import matplotlib.pyplot as plt
def ohm_law_terms(n, T_e, B, L, V, eta=None):
"""
Calculate relative magnitudes of generalized Ohm's law terms.
Parameters:
n: density (m^-3)
T_e: electron temperature (eV)
B: magnetic field (T)
L: length scale (m)
V: flow velocity (m/s)
eta: resistivity (Ω·m), if None calculate from Spitzer
"""
e = 1.6e-19
m_e = 9.11e-31
m_i = 1.67e-27
mu_0 = 4e-7 * np.pi
k_B = 1.38e-23
c = 3e8
# Spitzer resistivity (if not provided)
if eta is None:
T_e_eV = T_e
ln_Lambda = 15 # Coulomb logarithm (typical)
eta = 5.2e-5 * ln_Lambda * T_e_eV**(-3/2) # Ω·m
# Current density (from Ampere's law estimate)
J = B / (mu_0 * L)
# Characteristic electric field (ideal MHD)
E_ideal = V * B
# Generalized Ohm's law terms
E_resistive = eta * J
E_Hall = J * B / (e * n)
E_pressure = k_B * T_e * e / (e * L) # ∇p_e ~ nkT/L
omega_pe = np.sqrt(n * e**2 / (m_e * 8.85e-12))
d_e = c / omega_pe
E_inertia = (m_e / (e**2 * n**2)) * J * (V / L)
# Normalize to ideal MHD term
terms = {
'Ideal (v×B)': E_ideal,
'Resistive (ηJ)': E_resistive,
'Hall (J×B/ne)': E_Hall,
'Pressure (∇p_e/ne)': E_pressure,
'Inertia (m_e dJ/dt)': E_inertia
}
return {k: v/E_ideal for k, v in terms.items()}, eta
# Parameter scan: vary length scale
L_range = np.logspace(-3, 3, 100) # 1 mm to 1 km
n = 1e19
T_e = 10
B = 0.1
V = 1e5 # 100 km/s
terms_vs_L = {k: [] for k in ['Ideal (v×B)', 'Resistive (ηJ)',
'Hall (J×B/ne)', 'Pressure (∇p_e/ne)',
'Inertia (m_e dJ/dt)']}
for L in L_range:
terms, _ = ohm_law_terms(n, T_e, B, L, V)
for k, v in terms.items():
terms_vs_L[k].append(v)
# Plot
plt.figure(figsize=(10, 6))
for key, values in terms_vs_L.items():
if key != 'Ideal (v×B)':
plt.loglog(L_range, values, label=key, linewidth=2)
# Mark characteristic scales
d_e = 3e8 / np.sqrt(n * (1.6e-19)**2 / (9.11e-31 * 8.85e-12))
d_i = 3e8 / np.sqrt(n * (1.6e-19)**2 / (1.67e-27 * 8.85e-12))
plt.axvline(d_e, color='r', linestyle=':', alpha=0.7, label=f'$d_e$ = {d_e:.2e} m')
plt.axvline(d_i, color='b', linestyle=':', alpha=0.7, label=f'$d_i$ = {d_i:.2e} m')
plt.xlabel('Length scale L (m)', fontsize=12)
plt.ylabel('Relative magnitude (normalized to v×B)', fontsize=12)
plt.title('Generalized Ohm\'s Law: Term Magnitudes vs. Scale', fontsize=14)
plt.legend(fontsize=10)
plt.grid(True, which='both', alpha=0.3)
plt.tight_layout()
plt.savefig('ohm_law_terms.png', dpi=150)
plt.show()
# Print values at specific scales
print("\nRelative term magnitudes:")
print(f"\nAt L = {d_e:.2e} m (electron skin depth):")
terms, _ = ohm_law_terms(n, T_e, B, d_e, V)
for k, v in terms.items():
print(f" {k:25s}: {v:.2e}")
print(f"\nAt L = {d_i:.2e} m (ion skin depth):")
terms, _ = ohm_law_terms(n, T_e, B, d_i, V)
for k, v in terms.items():
print(f" {k:25s}: {v:.2e}")
print(f"\nAt L = 1 m (macroscopic scale):")
terms, _ = ohm_law_terms(n, T_e, B, 1.0, V)
for k, v in terms.items():
print(f" {k:25s}: {v:.2e}")
6.3 Diamagnetic Drift Visualization¶
import numpy as np
import matplotlib.pyplot as plt
from matplotlib.patches import Circle
def diamagnetic_drift_cylinder():
"""
Visualize diamagnetic drift in a cylindrical plasma column.
"""
# Plasma parameters
a = 0.1 # plasma radius (m)
B_0 = 1.0 # axial magnetic field (T)
p_0 = 1e5 # peak pressure (Pa)
T_e = 10 # electron temperature (eV)
T_i = 10 # ion temperature (eV)
n_0 = 1e19 # peak density (m^-3)
e = 1.6e-19
k_B = 1.38e-23
# Radial grid
r = np.linspace(0, a, 100)
# Pressure profile (parabolic)
p = p_0 * (1 - (r/a)**2)
p_e = p / 2
p_i = p / 2
n = n_0 * (1 - (r/a)**2)
# Pressure gradient
dp_dr = -2 * p_0 * r / a**2
dp_e_dr = dp_dr / 2
dp_i_dr = dp_dr / 2
# Diamagnetic drift velocities
v_star_e = -dp_e_dr / (e * n * B_0) # azimuthal (θ) direction
v_star_i = dp_i_dr / (e * n * B_0)
# Diamagnetic current density
J_theta = -dp_dr / B_0
# Plot profiles
fig, axes = plt.subplots(2, 2, figsize=(12, 10))
# Pressure profile
axes[0, 0].plot(r*100, p/1e3, 'b-', linewidth=2, label='Total')
axes[0, 0].plot(r*100, p_e/1e3, 'r--', linewidth=2, label='Electron')
axes[0, 0].plot(r*100, p_i/1e3, 'g--', linewidth=2, label='Ion')
axes[0, 0].set_xlabel('Radius (cm)', fontsize=11)
axes[0, 0].set_ylabel('Pressure (kPa)', fontsize=11)
axes[0, 0].set_title('Pressure Profile', fontsize=12)
axes[0, 0].legend()
axes[0, 0].grid(alpha=0.3)
# Diamagnetic drift velocities
axes[0, 1].plot(r*100, v_star_e/1e3, 'r-', linewidth=2, label='Electron')
axes[0, 1].plot(r*100, v_star_i/1e3, 'g-', linewidth=2, label='Ion')
axes[0, 1].axhline(0, color='k', linestyle='--', alpha=0.5)
axes[0, 1].set_xlabel('Radius (cm)', fontsize=11)
axes[0, 1].set_ylabel('Drift velocity (km/s)', fontsize=11)
axes[0, 1].set_title('Diamagnetic Drift Velocity (azimuthal)', fontsize=12)
axes[0, 1].legend()
axes[0, 1].grid(alpha=0.3)
# Diamagnetic current
axes[1, 0].plot(r*100, J_theta/1e3, 'b-', linewidth=2)
axes[1, 0].set_xlabel('Radius (cm)', fontsize=11)
axes[1, 0].set_ylabel('Current density (kA/m²)', fontsize=11)
axes[1, 0].set_title('Diamagnetic Current Density (azimuthal)', fontsize=12)
axes[1, 0].grid(alpha=0.3)
# 2D visualization: top view
ax = axes[1, 1]
theta = np.linspace(0, 2*np.pi, 50)
R, Theta = np.meshgrid(r, theta)
X = R * np.cos(Theta)
Y = R * np.sin(Theta)
# Pressure contour
P_grid = np.outer(np.ones_like(theta), p)
contour = ax.contourf(X*100, Y*100, P_grid/1e3, levels=20, cmap='hot')
plt.colorbar(contour, ax=ax, label='Pressure (kPa)')
# Velocity vectors (sample points)
n_arrows = 8
r_arrows = np.linspace(0.2*a, 0.9*a, 5)
theta_arrows = np.linspace(0, 2*np.pi, n_arrows, endpoint=False)
for ri in r_arrows:
for ti in theta_arrows:
xi = ri * np.cos(ti)
yi = ri * np.sin(ti)
# Diamagnetic drift is in theta direction
# In Cartesian: v_theta = -sin(θ) v_r_hat + cos(θ) v_θ_hat
idx = np.argmin(np.abs(r - ri))
v_mag = v_star_e[idx]
vx = -v_mag * np.sin(ti)
vy = v_mag * np.cos(ti)
ax.arrow(xi*100, yi*100, vx*1e-3, vy*1e-3,
head_width=0.5, head_length=0.3, fc='cyan', ec='cyan', alpha=0.8)
ax.set_xlabel('x (cm)', fontsize=11)
ax.set_ylabel('y (cm)', fontsize=11)
ax.set_title('Electron Diamagnetic Drift (top view)', fontsize=12)
ax.set_aspect('equal')
ax.add_patch(Circle((0, 0), a*100, fill=False, edgecolor='white', linewidth=2))
plt.tight_layout()
plt.savefig('diamagnetic_drift.png', dpi=150)
plt.show()
# Print values at r = a/2
idx = np.argmin(np.abs(r - a/2))
print(f"\nAt r = a/2 = {a/2*100:.1f} cm:")
print(f" Pressure: {p[idx]/1e3:.2f} kPa")
print(f" Electron drift: {v_star_e[idx]/1e3:.2f} km/s")
print(f" Ion drift: {v_star_i[idx]/1e3:.2f} km/s")
print(f" Current density: {J_theta[idx]/1e3:.2f} kA/m²")
print(f" Drift frequency: {v_star_e[idx]/(a/2):.2e} rad/s")
print(f" Compare to ω_ci = {e*B_0/(1.67e-27):.2e} rad/s")
diamagnetic_drift_cylinder()
6.4 Two-Fluid Closure Comparison¶
import numpy as np
import matplotlib.pyplot as plt
def compare_closures():
"""
Compare different closure models: isothermal vs. adiabatic.
Simulate compression of a plasma element.
"""
# Initial conditions
n_0 = 1e19 # m^-3
T_0 = 10 # eV
V_0 = 1.0 # m^3
gamma = 5/3 # adiabatic index
# Compression ratio
V = np.linspace(V_0, 0.1*V_0, 100)
n = n_0 * (V_0 / V) # density increases as volume decreases
# Isothermal: T = const
T_isothermal = np.ones_like(V) * T_0
p_isothermal = n * T_isothermal
# Adiabatic: p V^γ = const
p_adiabatic = n_0 * T_0 * (V_0 / V)**gamma
T_adiabatic = p_adiabatic / n
# Plot
fig, axes = plt.subplots(1, 2, figsize=(12, 5))
# Temperature vs. compression
axes[0].plot(V/V_0, T_isothermal, 'b-', linewidth=2, label='Isothermal')
axes[0].plot(V/V_0, T_adiabatic, 'r--', linewidth=2, label='Adiabatic (γ=5/3)')
axes[0].set_xlabel('V / V₀', fontsize=12)
axes[0].set_ylabel('Temperature (eV)', fontsize=12)
axes[0].set_title('Temperature Evolution under Compression', fontsize=13)
axes[0].legend(fontsize=11)
axes[0].grid(alpha=0.3)
# Pressure vs. density
axes[1].loglog(n/n_0, p_isothermal/(n_0*T_0), 'b-', linewidth=2, label='Isothermal (p ∝ n)')
axes[1].loglog(n/n_0, p_adiabatic/(n_0*T_0), 'r--', linewidth=2, label='Adiabatic (p ∝ n^γ)')
axes[1].set_xlabel('n / n₀', fontsize=12)
axes[1].set_ylabel('p / (n₀ T₀)', fontsize=12)
axes[1].set_title('Pressure vs. Density', fontsize=13)
axes[1].legend(fontsize=11)
axes[1].grid(alpha=0.3, which='both')
plt.tight_layout()
plt.savefig('closure_comparison.png', dpi=150)
plt.show()
# At 50% compression
idx = np.argmin(np.abs(V/V_0 - 0.5))
print("\nAt 50% compression (V = 0.5 V₀):")
print(f" Density: {n[idx]/n_0:.2f} n₀")
print(f" Isothermal:")
print(f" T = {T_isothermal[idx]:.2f} eV (unchanged)")
print(f" p = {p_isothermal[idx]/(n_0*T_0):.2f} (n₀ T₀)")
print(f" Adiabatic:")
print(f" T = {T_adiabatic[idx]:.2f} eV")
print(f" p = {p_adiabatic[idx]/(n_0*T_0):.2f} (n₀ T₀)")
print(f" Pressure ratio (adiabatic/isothermal): {p_adiabatic[idx]/p_isothermal[idx]:.2f}")
compare_closures()
Summary¶
In this lesson, we derived the two-fluid model by taking velocity-space moments of the Vlasov equation. Key points:
-
Moment hierarchy: Each moment equation introduces the next higher-order moment, leading to a closure problem.
-
Closure models: Isothermal, adiabatic, and CGL closures truncate the hierarchy with different physical assumptions.
-
Generalized Ohm's law: The full form includes resistive, Hall, electron pressure, and electron inertia terms. Each term becomes important at different length scales:
- Resistive: low $R_m$ (collisional plasmas)
- Hall: $L \sim d_i$ (ion skin depth)
- Electron pressure: steep gradients
-
Electron inertia: $L \sim d_e$ (electron skin depth)
-
Hall effect: At scales $\lesssim d_i$, ions decouple from the magnetic field while electrons remain frozen-in. This enables fast magnetic reconnection and whistler wave propagation.
-
Diamagnetic drift: A fluid drift arising from pressure gradients, not a single-particle drift. Creates a current $\mathbf{J}_* = \mathbf{B} \times \nabla p / B^2$.
-
Two-fluid waves: Hall MHD modifies Alfvén waves into whistler waves at small scales. Kinetic Alfvén waves include finite-$k_\perp$ effects.
The two-fluid model bridges the gap between single-particle kinetic theory and single-fluid MHD. It captures important physics at intermediate scales (ion gyroradius to ion skin depth) that are missed by ideal MHD but don't require the full complexity of kinetic theory.
Practice Problems¶
Problem 1: Moment Calculation¶
Starting from the Vlasov equation, derive the second moment (energy equation) explicitly. Show that the heat flux $\mathbf{q}_s = \frac{1}{2} m_s \int w^2 \mathbf{w} f_s d^3v$ appears. What physical process does the heat flux represent?
Problem 2: Hall MHD Dispersion¶
Derive the dispersion relation for whistler waves in Hall MHD: $$\omega = \frac{k_\parallel^2 V_A^2}{\omega_{ci}}$$ Start from the two-fluid equations with the Hall term, assume $\omega \ll \omega_{ce}$ and $\omega \gg \omega_{ci}$, and use the cold plasma approximation ($p = 0$).
Problem 3: Diamagnetic Current in a Tokamak¶
In a tokamak with major radius $R_0 = 3$ m and minor radius $a = 1$ m, the electron pressure profile is: $$p_e(r) = p_0 \left(1 - \frac{r^2}{a^2}\right)^2$$ with $p_0 = 5 \times 10^5$ Pa. The toroidal magnetic field is $B_\phi = 5$ T. Calculate: (a) The diamagnetic current density at $r = a/2$. (b) The total poloidal current from the diamagnetic effect (integrate $J_\theta$ over the cross-section). (c) Compare this to the bootstrap current (which has a similar profile).
Problem 4: Generalized Ohm's Law in a Current Sheet¶
In a magnetic reconnection current sheet, the length scale is $L = 10 d_i$, where $d_i = 100$ km is the ion skin depth. The plasma density is $n = 10^7$ m$^{-3}$ (solar wind), electron temperature $T_e = 100$ eV, and magnetic field $B = 10$ nT. Calculate the relative magnitudes of: (a) The ideal MHD term $\mathbf{v} \times \mathbf{B}$ (b) The Hall term $\mathbf{J} \times \mathbf{B} / (en)$ (c) The electron pressure term $\nabla p_e / (en)$ (d) The electron inertia term Which term(s) are important in this current sheet?
Problem 5: Two-Fluid Instability¶
Consider a two-fluid plasma with $T_e = T_i$ and a density gradient $\nabla n = -n_0 / L_n \hat{\mathbf{x}}$ in a magnetic field $\mathbf{B} = B_0 \hat{\mathbf{z}}$. (a) Calculate the electron and ion diamagnetic drift velocities. (b) The drift-wave instability occurs when the phase difference between density and potential perturbations causes wave growth. Using the continuity equation and quasi-neutrality, show that electrostatic drift waves have the dispersion relation: $$\omega = \frac{k_y k_B T_e}{e B_0 L_n}$$ where $k_y$ is the wavenumber perpendicular to both $\mathbf{B}$ and $\nabla n$. (c) For $L_n = 1$ cm, $T_e = 1$ eV, $B_0 = 0.1$ T, and $k_y = 100$ m$^{-1}$, calculate the drift wave frequency.
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