Ohm's Law and Ambipolar Diffusion

Ohm's Lawand Ambipolar Diffusion Astrophysical Fluid Dynamics E. Battaner Department of Physics National Tsing Hua University G.T. Chen 2004/12/9

Outline • Basic Properties of Fluid Dynamics • Diffusion in Classical Fluid • Diffusion in Plasma Fluid • Ohm’s Law • Ambipolar Diffusion • Summary

Basic Properties • Boltzmann’s Eq. • Multicomponent fluids • r

Basic Properties • Mean velocity • Peculiar velocity where average then

Basic Properties • Multiply Boltzmann’s eq. by Gi (v) and integrate over velocity space

Basic Properties • This moment eq. is sometimes known as “transfer eq. “ , depending on the choice of Gi • Choosing Gi is a function which has the property of collisional invariance corresponding to the conservation of mass, momentum and energy Mass MomentumEnergy

Basic Properties • According to collisional invariance, the collision term • If Gi=1  continuity eq. • If Gi=mi  continuity eq. of mass • If Gi=(mv2)/2  energy balance eq.

Basic Properties • If Gi=p (momentum) define , of course Sum all species Combine with continuity eq. Eq. of motion

Diffusion in Classical Fluid • To solve the diffusion velocity <Vi>, we take two assumptions where coefficient depending on intermolecular potential for elastic collision model between two rigid sphere reduce mass Sum of the radii of both spheres

Diffusion in Classical Fluid • Define the momentum transfer collision frequency recall

Diffusion in Classical Fluid • For simplicity, take and neglect diffusion acceleration and force consider no viscosity and isothermal

Diffusion in Classical Fluid • Now , solve <Vi> =the friction velocity • For moderate rate of diffusion the procedure converges rapidly, so the friction velocity often can be ignored. where

Diffusion in Classical Fluid • Diffusion coefficient • If consider a binary mixture , let n1<<n2 and isothermal  Di=constant and Fi=0 (Fick’s 1st law) (Fick’s 2nd law) Combine with continuity eq.

Diffusion in Plasma Fluid • Consider plasma fluid, and there are three species ni ,ne ,and nn ( ne = ni ) • Continuity eq.s electrons: (1) ions: (2) neutrals: (3)

Diffusion in Plasma Fluid • (1)x me ,(2)xmi ,(3)xmi individually(mn ~ mi ), and sum all species • (2)-(1) define charge current density

Diffusion in Plasma Fluid • because diffusion fluxes of ions and electrons Ambipolar diffusion flux

Diffusion in Plasma Fluid • The diffusion of ions and electrons has decomposed into 2 kinds of fluxes, one associated with charge current density, and the other called ambipolar diffusion • No ambipolar diffusion will occur in fully ionized plasma

Diffusion in Plasma Fluid • Remember diffusion in classical fluid , we have 2 assumptions • Recall Aj in plasma fluid is from Boltzmann’s eq.

Ohm’s Law • Consider absence of ambipolar difusion • Calculate by using the above two methods Assume external force is not important and ,and T=constant

Ohm’s Law • is often negligible • Assume can be ignored • Define scalar electrical conductivity This is called Ohm’s Law

PS: Ohm’s Law • Consider simple application • Define electrical conductivity tensor

Ohm’s Law • The electrical conductivity tensor is also equal to • Hall charge current density Pederson conductivity where Hall conductivity

Ohm’s Law • For Hall conductivity: When is very high . It is unimportant When , it’s also negligible It becomes important when , it would cause the media being anisotropy • When is very low ,Hall term 0 ,Pederson term  then σ will reduce to isotropic electrical conductivity

Ohm’s Law • When B is very high  is very high the charge current density lies in the direction of , and only the projection of along the direction of is effective in producing it  the σ becomes highly anisotropic

Ohm’s Law • If choosing OX-axis = the direction of B field and it also can reproduce the above properties of the electrical conductivity tensor

Ambipolar Diffusion • Consider a weakly ionized medium with nn >>ne • Calculate the Ai and Ae individually (a) (b)

Ambipolar Diffusion • Consider B field=0  ambipolar diffusion in the absence of magnetic field • Assume j=0 , E=0 ,and solve (a) (b) but Di is not equal to De  the problem should appear in the hypothesis

Ambipolar Diffusion • We must consider the electric field being generated  binding electric field is allowed to existence take The ambipolar diffusion coefficient is twice the ion diffusion coefficient , but lower than De

Ambipolar Diffusion • The binding electric field In the absence of B field , ambipolar diffusion is isotropic

Summary • a • a • Ohm’s Law • Ambipolar Diffusion

Thank You

Introduction to Plasma Theory , Nicholson

E x B Drift

g x B Drift

Alfven Wave

Ohm's Law and Ambipolar Diffusion