8.3 Resistivity and thermal conductivity in plasmas

Plasma resistivity and thermal conductivity play crucial roles in understanding how plasmas behave. These properties affect how electric currents flow and heat moves through the plasma, impacting everything from fusion experiments to space weather.

Collisions between particles in the plasma are key to these processes. They determine how easily electrons can move and transfer energy, influencing the plasma's electrical and thermal properties. Understanding these interactions is essential for predicting plasma behavior in various settings.

Plasma Resistivity

Spitzer Resistivity and Electron-Ion Collisions

• Spitzer resistivity quantifies plasma's resistance to electric current flow
• Arises from electron-ion collisions impeding electron movement
• Depends on plasma temperature and density
• Calculated using the formula $\eta = \frac{m_e}{n_e e^2 \tau_{ei}}$
• Electron-ion collision frequency $\nu_{ei} = \frac{1}{\tau_{ei}}$ determines resistivity
• Collision frequency increases with ion density and decreases with electron temperature

Plasma Resistivity and Coulomb Logarithm

• Plasma resistivity varies inversely with temperature $\eta \propto T^{-3/2}$
• Higher temperatures lead to lower resistivity due to reduced collision frequency
• Coulomb logarithm accounts for long-range nature of Coulomb interactions
• Defined as $\ln \Lambda = \ln \left(\frac{\lambda_D}{b_0}\right)$
• Debye length (λD) and impact parameter (b0) determine Coulomb logarithm value
• Typical values range from 10 to 20 for most laboratory and space plasmas

Electron Mean Free Path in Plasmas

• Electron mean free path represents average distance traveled between collisions
• Calculated using $\lambda_{mfp} = \frac{v_{th,e}}{\nu_{ei}}$
• Thermal velocity of electrons (vth,e) influences mean free path
• Longer mean free paths indicate less frequent collisions and lower resistivity
• In fully ionized plasmas, mean free path increases with temperature $\lambda_{mfp} \propto T^2$
• Affects various plasma phenomena (heat transport, current flow, wave propagation)

Joule Heating

Plasma Heating Through Resistivity

• Joule heating describes energy dissipation due to electrical resistance
• Occurs when electric current flows through a resistive medium (plasma)
• Power dissipated per unit volume given by $P = \eta J^2$
• Plasma resistivity (η) and current density (J) determine heating rate
• Contributes to plasma temperature increase and energy balance
• Significant in laboratory plasmas (tokamaks, z-pinches) and astrophysical contexts (solar corona)

Electron-Ion Collisions and Energy Transfer

• Electron-ion collisions facilitate energy transfer from electrons to ions
• Collision frequency affects rate of energy exchange between species
• Equipartition time measures duration for electron and ion temperatures to equalize
• Calculated using $\tau_{eq} = \frac{3m_e m_i}{8(2\pi)^{1/2}n_e e^4 \ln \Lambda} \left(\frac{kT_e}{m_e} + \frac{kT_i}{m_i}\right)^{3/2}$
• Longer equipartition times in high-temperature, low-density plasmas
• Influences plasma heating efficiency and temperature distribution

Thermal Conductivity

Heat Flux and Temperature Gradients

• Thermal conductivity measures plasma's ability to conduct heat
• Heat flux (q) relates to temperature gradient through Fourier's law $q = -\kappa \nabla T$
• Thermal conductivity (κ) depends on plasma parameters (density, temperature, magnetic field)
• Temperature gradients drive heat transport in plasmas
• Steeper gradients lead to higher heat fluxes
• Affects plasma confinement and energy balance in fusion devices

Electron Thermal Conductivity

• Electrons primarily responsible for heat conduction in plasmas
• Electron thermal conductivity given by $\kappa_e = \frac{n_e k_B T_e \tau_{ei}}{m_e}$
• Depends on electron density (ne), temperature (Te), and collision time (τei)
• Increases with temperature $\kappa_e \propto T_e^{5/2}$
• Anisotropic in magnetized plasmas (different parallel and perpendicular to magnetic field)
• Influences plasma cooling rates and heat transport in various plasma systems

Mean Free Path and Heat Transport

• Electron mean free path affects thermal conductivity
• Longer mean free paths lead to higher thermal conductivity
• In collisionless plasmas, heat transport limited by plasma instabilities
• Kinetic effects become important when temperature gradient scale length approaches mean free path
• Non-local heat transport occurs in steep temperature gradients
• Affects plasma edge behavior in fusion devices and astrophysical plasma phenomena