Download Derivation of EMHD Equations

Derivation of EMHD Equations
Electron-magnetohydrodynamics (EMHD) deals with the phenomena occurring
on fast time scales so that the ions being heavier are not able to respond and
remain stationary. On such time scales ions provide just charge neutralizing
background and the dynamics is completely goverened by electrons. Therefore, equations of EMHD consist of just electron-fluid equations and Maxwell
equations. We write below these equations.
Electron-fluid equations:
∂~ve
me n e
=
+ (~ve .∇)~ve
∂t
∂ne
+ ∇.(ne~ve ) =
∂t
pe
=
nγe
~−
−ene E
ene
~ − ∇pe − νei ne me~ve (1)
(~ve × B)
c
0
(2)
constant
(3)
= −4πne e
(4)
= 0
(5)
Maxwell equations:
~
∇.E
~
∇.B
~
∇×E
~
∇×B
~
1 ∂B
= −
c ∂t
~
4π
1 ∂E
= − ne e~ve +
c
c ∂t
(6)
(7)
The symbols used in above equations have their usual meanings. In writting the
above equations we have considerd ions to be at rest (vi = 0) and therefore the
current J~ in the Maxwell equation (7) has been written as (−ne e~ve ). In EMHD
very high frequencies are also ignored by neglecting the displacement current
in the Maxwell equations. Now we shall see under what conditions this can be
done. Comparing the magnitudes of the displacement and conduction current
terms in equation (7),
~
| 1c ∂∂tE |
4π
| c ne e~ve |
=
ω e E
,
2 m v
ωpe
e e
(8)
where ωpe is the electron plasma frequency and ∂/∂t ∼ ω. Estimating the ratio
E/ve from the comparision of the first and second terms on the RHS of the
equation (1), we get,
E
ve
∼
1
B
c
Using this estimation of E/ve we obtain from equation (8)
~
| 1c ∂∂tE |
ve |
| 4π
c ne e~
ω eB
2 m c
ωpe
e
ωωce
2
ωpe
=
=
where ωce = eB/me c is the electron larmor gyration frequency. Therefore,
the displacement current in the Maxwell equation can be neglected if ω ≪
2
ωpe
/ωce . However, if we compare the first term of LHS and first term of
RHS in equation (1) to estimate E/ve , we get another condition for neglecting
displacement current. Such a comparision gives,
E
ve
me ω
,
e
∼
which when combined with equation (8) gives,
~
| 1c ∂∂tE |
4π
| c ne e~ve |
=
ω2
,
2
ωpe
showing ω ≪ ωpe as another condition for neglecting displacement current.
Therefore the condition of neglecting displacement current in EMHD can be
2
stated as ω ≪ min(ωpe , ωpe
/ωce ). The consequence of neglecting displacement
current is ∇.J~ = −e∇.(ne~ve ) = 0, which from equation (2) implies that the
density fluctuations can also be ignored in EMHD. Thus the equation (2) can
be avoided.
Now we attempt to obtain a compact form of the EMHD equations by eliminating variables. For this purpose we take the curl of equation (1), use Faraday
law (6) to eliminate electric field and use the vector identity,
~ B)
~
∇(A.
=
~ × (∇ × B)
~ +B
~ × (∇ × A)
~ + (A.∇)
~
~ + (B.∇)
~
~
A
B
A
to write (~ve .∇)~ve = ∇(ve2 /2)−~ve × (∇×~ve ). The resulting equation is following.
∂
(∇ × P~e )
∂t
= ∇ × [~ve × (∇ × P~e )] − νei me ∇ × ~ve
(9)
~ (A
~ being the magnetic vector potential) is the generalized
Here P~e = me~ve −eA/c
momentum of an electron and hence ∇ × P~e has the interpretaion of generalized
vorticity. In the equation (9), the curl operation has eliminated the pressure
~ and ~ve ,
term for constant temperatures. Since equation (9) contains only B
~
only one more equation relating ~ve to B is needed. Such an equation is Ampere
law (7) with displacement current ignored and can be written as,
~ve
= −
c
~
∇×B
4πne e
2
(10)
For uniform electron density, the generalized vorticity, with the help of equation
(10), can be expressed in terms of magnetic field alone as,
∇ × P~e
=
e 2 2~ ~
(d ∇ B − B),
c e
where de = c/ωpe is the electron skin depth. The coupled system of equations
(9) and (10) describes the physics of Electron-magnetohydrodynamics. Now
we normalize length by electron skin depth de , magnetic field by B00 , time
by inverse of the gyro-frequency corresponding to the magnetic field B00 and
write below the final normalized version of EMHD equations (9) and (10). In
our normalization velocity is normalized de ωce , which has the interpretation of
electron Alfven velocity.
∂
~ − B)
~
(∇2 B
∂t
~ve
~ − B)]
~ − νei ∇2 B
~
= ∇ × [~ve × (∇2 B
(11)
~
= −∇ × B
(12)
The normalized equations (11) and (12) are the equations which we have used
in our studies in this thesis.
3