Appendix A
Plasma Parameter and Mathematical Tools
A.1
Tables of Plasma Parameters
The first table gives some fundamental constants:
Property
Speed of light
Boltzmann constant
Electron mass
Proton mass
Elementary charge
Dielectric constant
Permeability of free space
Symbol
c
k
me
mi
e
0
µ0
SI
3×108 m s−1
1.38×10−23 J K−1
9.1 ×10−31 kg
1836 me
1.6×10−19 C
8.85×10−12 F m−1
4π×10−7 H m−1
cgs
3×1010 m s−1
1.38 erg K−1
9.1 ×10−28 g
1836 me
4.8×10−10 statcoulomb
-
Note that 0 µ0 = 1/c2 . The following table presents several basic plasma frequencies, length scales and
velocities:
Property
Symbol
Plasma frequency
ωpe
Electron gyro frequency
Coulomb collision frequency
ωge
νei
Debye length
Skin depth (electron inertia)
Electron Gyroradius
λD
λe
rge
Electron Thermal velocity
vthe
Alfven speed
vA
Number of particles in a Debye sphere
The corresponding ion properties
are ωpi =
p
p
mi /me rge ; and vthi = mi /me vthe .
ND =
1
g
SI
1/2
2
ne e
0 me
eB
me
ωpe
ln Λ
4πnλ3D
0 kT 1/2
ne2
c
ωpe
vthe
ωpe1/2
kT
me
B
(µ0 nmi )1/2
4π
nλ3D
3
cgs
1/2
2
4πne e
me
eB
me c
ωpe
ln Λ
4πnλ3D
1/2
kT
4πne2
c
ωpe
vthe
ωpe1/2
kT
me
B
(4πnmi )1/2
4π
nλ3D
3
p
me /mi ωpe ; ωgi = (me /mi )ωge ; νie = (me /mi )νei ; rgi =
183
APPENDIX A. PLASMA PARAMETER AND MATHEMATICAL TOOLS
184
The following list provides numerical values for the various plasma parameters in a convenient form.
Because it is more common in the field of space physics everything is measured in cgs units in this table,
i.e., n in cm−3 , B in Gauss, T in eV, and ln Λ ≈ 20. Note that 1 T = 104 Gauss and 1 nT =10−5 Gauss.
Note also 1eV = 1.16 × 104 K.
ωpe
ωge
ωgi
νei
=
=
=
=
5.64 × 104 n1/2 [rad/sec]
1.76 × 107 B [rad/sec]
9.58 × 103 B [rad/sec]
1.1 × 10−5 ln(Λ)nTe3/2 sec−1
2
−1/2
rge = 2.38 × 100 Te1/2 B −1 [cm]
1/2
rgi = 1.02 × 102 Ti B −1 [cm]
vthe = 4.19 × 107 Te1/2 [cm/sec]
1/2
vthi = 9.79 × 105 Ti [cm/sec]
vA = 2.18 × 1011 Bn−1/2 [cm/sec]
ND = 1.72 × 109 Te3/2 n−1/2
Te1/2
[cm]
λDe = 7.43 × 10 n
5 −1/2
λe = 5.31 × 10 n
[cm]
7 −1/2
λi = 2.28 × 10 n
[cm]
The last table presents an overview of typical plasma properties in the inner and outer magnetosphere/tail,
the solar chromosphere and corona, warm interstallar gas, and a tokamak. Lei is the mean free path and
ND is the number of particles in a Debye cube.
Inner
M’Sphere
−3
n [cm ]
102
B [nT]
104
Te [eV]
103
Ti [eV]
103
ωpe
5.6× 105
ωge
1.8× 106
ωgi
9.6× 102
νei
7.0× 10−7
Lei [m]
1.9 × 1013
λDe [m]
2.3× 101
λe [m]
5.3× 102
λi [m]
2.3× 104
rge [m]
7.5× 100
rgi [m]
3.2× 102
vthe [m/sec] 1.3× 107
vthi [m/sec] 3.1× 105
vA [m/sec] 2.18× 107
ND
5.4× 1012
Outer
M’sphere
1
40
500
2×103
5.7× 104
7.0× 103
3.6× 100
2.0× 10−8
4.7 × 1014
1.7× 102
5.3× 103
2.3× 105
1.3× 103
1.1× 105
9.4× 106
4.4× 105
8.7× 105
1.9× 1013
Chromosphere
1012
0.1
0.5
0.5
1.8 × 1010
1.8 × 1010
9.6 × 106
2.3 × 107
1.7 × 10−2
1.5 × 10−5
1.7× 10−2
0.72
2.2 × 10−5
9.6 × 10−4
3.9 × 105
9.1 × 103
6.9 × 106
3.6 × 102
Solar
Corona
3 · 108
107
100
100
9.8 × 108
1.8 × 109
9.6 × 105
6.9
8.0 × 105
4.0 × 10−3
0.31
13
3.1 × 10−3
0.13
5.5 × 106
1.3 × 105
1.3 × 107
1.9 × 107
Interstellar
Gas
1
1
0.8
0.8
5.6 × 104
1.8 × 102
9.6 × 10−2
3.7 × 10−4
1.3 × 109
6.2
5.3 × 103
2.3 × 105
2.8 × 103
1.2 × 105
4.9 × 105
1.1 × 104
2.2 × 104
2.4 × 108
Fusion
Device
1014
2 · 109
103
103
5.6 × 1011
3.5 × 1011
1.9 × 108
6.0 × 1011
2.8 × 10−5
2.2 × 10−5
5.3 × 10−4
2.3 × 10−3
4.9 × 10−5
2.2 × 10−2
1.7 × 107
4.1 × 105
4.4 × 106
1.0 × 106
Most of these parameters can easily vary by an order of magnitude, however the table demonstrates that
plasma conditions can vary over a wide range of values depending on the system.
APPENDIX A. PLASMA PARAMETER AND MATHEMATICAL TOOLS
A.2
185
Entropy and Adiabatic Convection
In an ordinary gas or fluid the change of heat due to pressure or temperature changes is
∆Q = CV dT + pdV
Assuming an ideal gas (with 3 degrees of freedom) with p = nkB T = N kB T /V :
T = pV /R
with R = N kB and the specific heat and coefficients of specific heat
3
CV = R
2
3
cV =
2
5
Cp = R
2
5
cp =
2
and the ratio γ = cp /cV = 5/3.
Thus the change in heat becomes
∆Q = cV V dp + cp pdV
dV
dp
+γ
= cV pV
p
V
dV
dp
+γ
∆Q/T = cv R
p
V
Entropy changes are defined as dS = ∆Q/T or
S − S0 = CV ln (pV γ )
For adiabatic changes of the state of a system we have dS = 0 which can also be expressed as
d (pρ−γ )
∂
= pρ−γ + u · ∇ pρ−γ = 0
dt
∂t
(A.1)
with the specific volume V ∝ 1/ρ. This equation implies that the entropy of a plasma element does not
change (moving with this element) in the absence of shocks or ohmic heating.
A.3
Complex Variables, Residue Theorem, and Integration Contours
For the evaluation of Landau damping and dispersion relations of plasma waves it is helpful to introduce
some background in complex variables. In the following we assume the complex variable z = x + iy
with x and y being real variables. We also assume f (z) to be a complex function.
APPENDIX A. PLASMA PARAMETER AND MATHEMATICAL TOOLS
186
Laurent series expansion
If f (z) is analytic in a circular ring with r1 < |z − a| ≤ r2 there is a unique power law expansion within
this ring called a Laurent series
f (z) =
∞
X
k
ak (z − a) +
k=0
∞
X
bk (z − a)k
(A.2)
k=1
with the coefficients
1
ak =
2πi
I
γ
f (ζ) dζ
,
(ζ − a)k+1
1
bk =
2πi
I
(ζ − a)k−1 f (ζ) dζ
γ
in the expansion (A.2) the first part of the series is called the regular and the second part is called the
principal part of the series
One can combine the two series in a single expansion
f (z) =
∞
X
k=−∞
k
ck (z − a)
with
1
ck =
2πi
I
γ
f (ζ) dζ
(ζ − a)k+1
where γ represents an arbitrary closed curve in the counter clockwise (positive) direction as illustrated in
Figure A.1.
Figure A.1: The contour integral is the counterclockwise (mathematical positive) integral around the
singularity.
Residue
If an analytic function f (z) has an isolated singularity at z = a the coefficient c−1 of the power (z − a)−1
of the Laurent series is called the residue. For a 6= ∞ the residue is given by
APPENDIX A. PLASMA PARAMETER AND MATHEMATICAL TOOLS
1
Res [f (z) , a] =
2πi
187
I
f (ζ) dζ
(A.3)
γ
If f (z) has a pole of order m at z = a the Laurent series is
f (z) =
∞
X
ck (z − a)k
k=−m
and the residue can be computed from
c−1
Example: f (z) =
exp(iz)
z 2 +a2
1
dm−1
= Res [f (z) , a] =
lim m−1 [(z − a)m f (z)]
z→a
(m − 1)!
dz
has two poles at z = ±ia
exp (iz)
exp (−a)
exp (−a)
Res [f (z) , +ia] = lim (z − ia) 2
=
=
lim
z→ia
z→ia z + ia
z + a2
2ia
Residue Theorem
If a function f (z) is analytic in a domain D except at a finite number of points a1 , ..., ak and γ is a
piecewise smooth curve enclosing all singularities in D then
1
2πi
I
f (ζ) dζ =
γ
k
X
Res [f (z) , ai ]
(A.4)
i=1
Application of the residue theorem: Consider the integral
Im (z) > 0 except at z = i.
R∞
dx
.
−∞ 1+x2
The integrand is analytic for
1
1
1
1
Res
, +i = lim (z − i)
=
= lim
2
2
z→i
z→i z + i
1+z
1+z
2i
such that
Z
∞
−∞
dx
1
= 2πi = π
2
1+x
2i
Fourier and Laplace Transforms
In many cases Fourier and Laplace transformations are helpful to determine the solution of differential
equations. Here we use the following conventions for Fourier and its inverse transform in one dimension
APPENDIX A. PLASMA PARAMETER AND MATHEMATICAL TOOLS
f˜ (k) = (2π)
−1/2
Z
188
∞
dxf (x) exp (−ikx)
(A.5)
dkf (k) exp (ikx)
(A.6)
dtf (t) exp (−pt)
Z σ+i∞
1
f (t) =
dpF (p) exp (pt)
2πi σ−i∞
(A.7)
f (x) = (2π)−1/2
Z−∞
∞
−∞
and for the Laplace transform
Z
∞
F (p) =
0
(A.8)
If f (t) is exponentially ∼ exp (γt) growing in time with a growth rate (real part of γ) γr > 0 the Laplace
transformation requires that p is chosen larger than γ for convergence.
Figure A.2: Integral for the backtransformation of the Laplace transform
For the back transformation of the Laplace transform it must be noted that the integration is parallel to the
imaginary axis with σ larger than the real part of all singularities of F (p) as illustrated in Figure A.2. Note
that some formulations of the Laplace transform use −iω instead of p such that the back transformation
path is parallel to the real axis.
Consider the following example of a simple differential equation with a real value of γ > 0:
df
= γf
dt
with f (0) = f0 . The Laplace transform of the equation is optained by multiplication with exp (−pt) and
integration from 0 to ∞. This yields for the left side
APPENDIX A. PLASMA PARAMETER AND MATHEMATICAL TOOLS
Z
0
∞
df (t)
dt
exp (−pt) = f (t) exp (−pt)|∞
0 +p
dt
= −f (t = 0) + pF (p)
Z
189
∞
dtf (t) exp (−pt)
0
provided that p is sufficiently large. Thus the Laplace transform of the equation is
−f (t = 0) + pF (p) = γF (p)
or
f (t = 0)
p−γ
F (p) =
(A.9)
The inverse transform of F (p) is
1
f (t) =
2πi
Z
σ+i∞
dp
σ−i∞
f (t = 0)
exp (pt)
p−γ
where σ > γ and the Laplace transform assumed psufficiently large and positive! However, the Laplace
transform (A.9) can be continued analytically into the entire complex plane and to evaluete the integral
we consider the path cc1 cc0 c2 c3 c02 c0 c01 as illustrated in Figure A.3.
Figure A.3: Illustration of the deformation of the integration path for the inverse Laplace transform.
This integral is a closed integral that excludes the singularity such that theRvalue
R of the closed integral is
0
0. The contributions from the path’s c1 and c1 cancel each other such that c − Γ = 0 or
Z
Z
=
c
Γ
However to evaluate the integral over Γ we can close this through a semi circle with infinite radius in
the negative half of the real p plane where this semicircle has no contripution when the radius goes to
infinity. The new closed integral ahs one sigularity at p = γ. The residue theorem then yields
APPENDIX A. PLASMA PARAMETER AND MATHEMATICAL TOOLS
190
I
1
f (t = 0)
f (t) =
dp
exp (pt)
2πi Γ
p−γ
f (t = 0)
2πi
Res
exp (pt) , γ
=
2πi
p−γ
= f0 exp (γt)
Analytic Continuation
In the case of (A.9) the analytic continuation into the entire complex plane is no issue. However, let us
consider the function
Z
∞
f (w) =
−∞
dz
(z 2
+
a2 ) (z
− w)
defined for wi = Imw > 0 and a real and > 0.
If we close the contour in the lower half plane it encloses only one singularity at z = −ia with the result
1
, z = −ia
f (w) = −2πiRes
(z 2 + a2 ) (z − w)
1
1
= −2πi
−2ia −w − ia
π 1
= −
a w + ia
The same result is obtained if the integral is computed through a semi-circle in the upper half of the
complex plane. However, when wi < 0 the contour closed over the lower half contains 2 singularities
and leads to
f (w) = −
π 1
2πi
− 2
a w + ia w + a2
such that the function f (w) is discontinuous. A proper definition of an analytic function f (w) requires
to subtract this contribution such that for negative wi the proper analytic continuation is
Z
∞
f (w) =
−∞
dz
(z 2
+
a2 ) (z
− w)
+
2πi
+ a2
w2
Another and consistent way to achieve this is to deform the contour as illustrated in Figure A.4. For a
positive imagnary part of w the value of the integral along the path c can be obtained by closing the path
in the lower hlaf of the complex plane. If the imagninary part of w is negative we need to evaluate the
integral through the deformed contour Γ such that
APPENDIX A. PLASMA PARAMETER AND MATHEMATICAL TOOLS
191
Figure A.4: Deformation of the integration contour for the analytic continuation of f (w).
Z
dz
dz
1
f (w) =
=
+ 2πiRes
,z = w
2
2
2
2
(z 2 + a2 ) (z − w)
Γ (z + a ) (z − w)
c (z + a ) (z − w)
Z
dz
1
=
+ 2πi 2
2
2
(w + a2 )
c (z + a ) (z − w)
Z
Plemelj Formula
The prior example omitted the case where the imaginary part of w is wi = 0. Let us consider the case
where a singularity approaches the real axis in the form
∞
Z
dx
I = lim
δ→0
−∞
f (x)
x − a ± iδ
with δ > 0.
Figure A.5: Illustration of the Plemlj formula for a singularity on the real axis.
In this case the proper evaluation consits of the principal part of the integral plus one half of the residual
at the point z = a which can be illustrated through a contour with a semi circle around the singularity as
illustrated in which leads to
Z
∞
I=P
dx
−∞
f (x)
+ πif (a)
x−a
with
Z
∞
f (x)
P
dx
= lim
x − a ε→0
−∞
Z
a−ε
−∞
f (x)
dx
+
x−a
Z
∞
f (x)
dx
x−a
a+ε
APPENDIX A. PLASMA PARAMETER AND MATHEMATICAL TOOLS
A.4
192
Plasma Dispersion Function
Z(ζ) =
Z
1
π 1/2
∞
−∞
exp (−z 2 ) dz
z−ζ
Differential equation:
dZ(ζ)
= −2 [1 + ζZ(ζ)]
dζ
Useful relations
Z(−ζ) = −Z(ζ) + 2π 1/2 i exp −ζ 2 for Imζ > 0
e
Z(ζ)
= Z(ζ) − 2π 1/2 exp −ζ 2 for Imζ < 0
e
where Z(ζ)
is the analytic continuation of Z(ζ). The complex conjugate of the plasma dispersion function is
[Z(ζ)]∗ = Z(ζ ∗ ) − 2π 1/2 exp −ζ 2
Expansion for ζ < 1:
Z (ζ) = iπ
1/2
= π 1/2
exp −ζ
∞
X
n=0
2
2ζ 2 4ζ 4
− 2ζ 1 −
+
− ...
3
15
in+1 ζ n
Γ(1 + n/2)
Expansion for ζ 1:
1
Z (ζ) = −
ζ
3
1
1 + 2 + 4 + ... + σπ 1/2 i exp −ζ 2
2ζ
4ζ
with

 0,
1,
σ=

2,
for Imζ < 0
for Imζ = 0
for Imζ > 0
The plasma dispersion function is related to the error function:
erf(z) =
2i
1 + 1/2
π
Z(ζ) = iπ
1/2
erf(ζ)
Z
0
z
2
e dt e−z
t2
APPENDIX A. PLASMA PARAMETER AND MATHEMATICAL TOOLS
A.5
Bessel Functions and Modified Bessel Functions
Bessel’s differential equation
x2 y 00 + xy 0 + x2 − n2 y = 0
Solutions are the Bessel functions of the first kind
Jn (x) =
∞
X
v=0
x n+2v
(−1)v
v!Γ (n + v + 1) 2
Modified Bessel functions of the first kind
In (x) = i−n Jn (x) =
∞
X
v=0
x n+2v
1
v!Γ (n + v + 1) 2
are the solution of the differential equation
x2 y 00 + xy 0 − x2 − n2 y = 0
Figure A.6: Bessel functions Jn and modified Bessel functions In of the first kind.
Recurrence relations
2n
Jn (x) = Jn−1 (x) + Jn+1 (x)
x
d n
(x Jn (x)) = xn Jn−1 (x)
dx
d
x−n Jn (x) = −x−n Jn+1 (x)
dx
and for the modified Bessel functions
193
APPENDIX A. PLASMA PARAMETER AND MATHEMATICAL TOOLS
194
2n
In (x) = In−1 (x) − In+1 (x)
x
Asymptotic expansion for x 1
1/2
2
[cos (x − πn/2 − π/4)] + O(1/x)
Jn (x) =
πx
exp (x)
In (x) =
[1 + O(1/x)]
(2πx)1/2
Weber Integrals
The following Weber integrals are used to compute the magnetized plasma response function
Z
∞
2 2
=
xdxJl (px) Jl (rx) exp −q 2 x2
=
xdxJ0 (px) exp −q x
0
Z
0
∞
p2
2q
exp − 4
4q
2
−p + r2
pr
2 −1
2q
exp
Il
4q 4
2q 2
2 −1
(A.10)
Appendix B
Appendix
B.1
Derivation of the coefficients in the potential energy for the Z
pinch
U2
1
=
2
Z γp |∇ · ξ|2 + (ξ · ∇p) ∇ · ξ ∗ +
V
1
|B1 |2
µ0
−B1 · (ξ × j)] dx
with
∇·ξ =
B1 = ∇ × (ξ × B) =
=
=
ξ × (∇ × B) =
1 ∂
1 ∂ξθ ∂ξz
(rξr ) +
+
r ∂r
r ∂θ
∂z
1 ∂Gz ∂Gθ
∂Gr ∂Gz
1 ∂rGθ ∂Gr
−
−
−
er +
eθ +
ez
r ∂θ
∂z
∂z
∂r
r
∂r
∂θ
1 ∂ξr B
∂ξz B ∂Bξr
1 ∂ξz B
er + −
−
eθ +
ez
r ∂θ
∂z
∂r
r ∂θ
1 ∂
∂ξz
1 ∂ξz
1 ∂ξr
er −
(Bξr ) +
eθ +
ez
B
r ∂θ
B ∂r
∂z
r ∂θ
1 ∂
(rB) (ξθ er − ξr eθ )
r ∂r
and assuming no azimuthat dependence
195
APPENDIX B. APPENDIX
196
dp
B d
= −
(rB)
dr
µ0 r dr
1 ∂
∂ξz
ξr ∂ξr ∂ξz
∇·ξ =
(rξr ) +
=
+
+
r ∂r
∂z
r
∂r
∂z
1 ∂
∂ξz
ξr ∂B ∂ξr ∂ξz
(Bξr ) +
=
+
+
B ∂r
∂z
B ∂r
∂r ∂z
1 ∂B 1
−
+∇·ξ
= ξr
B ∂r
r
∂ ln B
ξr
=
−1
+∇·ξ
∂ ln r
r
B2
∂ ln p ξr
γβ (∇ · ξ)2 + β
∇·ξ
∂ ln r r
V 2µ0
2 !
d ln B
ξr
+2
−1
+∇·ξ
d ln r
r
2
d ln B
ξr
ξr
d ln p
− 1 2 + ∇ · ξ dx
+β
d ln r
d ln r
r
r
Z
2 1
B
=
(γβ + 2) (∇ · ξ)2
2 V 2µ0
d ln B
d ln p ξr
+2 2
−1 +β
∇·ξ
d ln r
d ln r r
2 d ln B
d ln p
d ln B
ξr
+
−1 β
+2
−1
dx
d ln r
d ln r
d ln r
r2
1
U =
2
Z
Collecting the coefficients for (∇ · ξ) and ξr /r terms leads to:
1
U=
2
Z ξr2
ξr
2
a11 (∇ · ξ) + 2a12 ∇ · ξ + a22 2 dx
r
r
V
where the coefficients are
a11 = γβ + 2
a12
a22
2µ0 p
β=
2
B
d ln B β d ln p
= 2
+
−1
d ln r
2 d ln r
d ln B
=
− 1 a12
d ln r
and stability requires
a11 a22 − a212 > 0
APPENDIX B. APPENDIX
197
Equilibrium condition
d ln B
β d ln p
=−
−1
d ln r
2 d ln r
such that a12 = −4
a11 a22 /a12 − a12
d ln B
d ln B β d ln p
= (γβ + 2)
−1 −2
+
−1
d ln r
d ln r
2 d ln r
β d ln p
= (γβ + 2) −
−2 +4
2 d ln r
β d ln p
= − (γβ + 2)
− 2γβ
2 d ln r
such that
Therefore the condition for stability (with a12 being negative) is
− (γβ + 2)
β d ln p
− 2γβ < 0
2 d ln r
or
d ln p
4γ
−
<
d ln r
γβ + 2
with
µ0 j02 2
r + p0
4
µ0 j0
B =
r
2
2µ0 p
−µ0 j02 r2 + 4p0 8µ0
2µ0 p0
β =
=
=
−2
2 2 2
2
B
4
µ0 j0 r
B2
d ln p
r µ0 j02
2B 2
4
−
=
r=
=
d ln r
p 2
µ0 p
β
p (r) = −
or
γβ + 2
β
< γ
or
γβ + 2 < γβ
This relation cannot be satisfied and demonstrates that the configuration is always unstable for constant
current density.
APPENDIX B. APPENDIX
B.2
198
X-mode dispersion relation
X(extraordinary)-mode: If the electric field is in the the x, y plane the electric field generates a velocity
u in the x, y plane and the u × B0 term generates a component perpendicular to the original velocity
and perpendicular to B0 such that this velocity is also entirely in the x, y plane. Thus we do not need to
consider the z component of the momentum equation. This illustrates that the ordinary and extraordinary
modes separate. In this case with E1 = Ex ex + Ey ey the components of the linear equations are
z
X-mode
B , B
0
1
y
e
u
k
1
E
x
−iωme ux = −eEx − euy B0
−iωme uy = −eEy + eux B0
ikEy = iωB1
iω
0 = −µ0 en0 ux − 2 Ex
c
iω
−ikB1 = −µ0 en0 uy − 2 Ey
c
These are 5 equations for 5 unknowns. Using the last three equations we express the velocities in terms
of the electric field
iω
Ex
µ0 en0 c2
iω
ik
iω
c2 k 2
= −
Ey +
B1 = −
1 − 2 Ey
µ0 en0 c2
µ0 en0
µ0 en0 c2
ω
ux = −
uy
Substitution in the first two equations
−iωme ux + eEx + eB0 uy = 0
−eB0 ux − iωme uy + eEy = 0
yields
ω 2 me
ωB0
c2 k 2
−
+ e Ex − i
1 − 2 Ey = 0
µ0 en0 c2
µ 0 c2 n 0
ω
2
2 2
ω me
ck
ωB0
i
Ex −
1 − 2 Ey + eEy = 0
2
2
µ0 c n0
µ0 en0 c
ω
APPENDIX B. APPENDIX
which is easily modified to
199
ω2
c2 k 2
ωωge
− 2 + 1 eEx − i 2
1 − 2 eEy = 0
ωpe
ωpe
ω
2 2 2
ck
ωωge
ω
i 2 eEx − 2 1 − 2 − 1 eEy = 0
ωpe
ωpe
ω
For nontrivial solutions the determinant must vanish such that the dispersion relation is
ω2
1− 2
ωpe
2
2 2 2
ω 2 ωge
ck
ωge
ω2
c2 k 2
1− 2 + 2 − 4 +
=0
4
ωpe
ωpe
ωpe
ωpe
With some manipulation we can rewrite the dispersion relation as
n2 =
2
2
ωpe
ω 2 − ωpe
k 2 c2
=
1
−
2
ω2
ω 2 ω 2 − ωuh
2
2
2
for the extraordinary or X-mode. Here n is the index of refraction
+ ωge
with ωuh
= ωpe
B.3
Dispersion relation for electromagnetic wave along B0
To discuss electromagnetic waves along B0 we use the same set of linear equations as in the case of
waves perpendicular to B0 which for the plane wave solutions assume the form
z
iωme n0 u = −en0 E − en0 u × B0
ik × E = iωB
iω
ik × B = −µ0 en0 u − 2 E
c
B , k
0
y
and E = E1 , B = B1 , u = ue1 , and B0 = B ez .
For k = k ez a consistent solution can be found by
assuming that all perturbations E, B, and u are in
the x y plane. Writing out the components of the
linear equations
−iωme ux
−iωme uy
−ikEy
ikEx
−ikBy
ikBx
B
e
u
x
−eEx − euy B0
−eEy + eux B0
iωBx
iωBy
iω
= −µ0 en0 ux − 2 Ex
c
iω
= −µ0 en0 uy − 2 Ey
c
=
=
=
=
1
E
1
APPENDIX B. APPENDIX
200
Using the induction equation we can substitute Bx and By in the last two equations.
iω
k2
Ex = −µ0 en0 ux − 2 Ex
ω
c
2
k
iω
−i Ey = −µ0 en0 uy − 2 Ey
ω
c
−i
with the result
ux
uy
iω
1−
= − 2
c µ0 en0
iω
= − 2
1−
c µ0 en0
c2 k 2
Ex
ω2
c2 k 2
Ey
ω2
Substitution into the momentum (first two) equations
me ω 2
c2 k 2
c2 k 2
iωB0
e− 2
1− 2
1 − 2 Ey = 0
Ex − i 2
c µ0 en0
ω
c µ0 n0
ω
2 2
2 2
2
ck
ck
me ω
ωB0
1 − 2 Ex + e − 2
1− 2
Ey = 0
−i 2
c µ0 n0
ω
c µ0 en0
ω
or in matrix form


1−
ω2
2
ωpe
+
ge
1−
−i ωω
2
ω
pe
c2 k2
2
ωpe
c2 k 2
ω2
ge
−i ωω
2
ωpe
1−
1−
ω2
2
ωpe
+
c2 k2
ω2
c2 k 2
2
ωpe


Ex
Ey
Setting the determinant to zero yields
2
2
2 ωge
ω2
c2 k 2
c2 k 2
1− 2 + 2
= 4 1− 2
ωpe
ωpe
ωpe
ω
or by taking the square root
ω
1− 2
ωpe
or
c2 k 2
c2 k 2
ωge
ω−
=± 2 ω−
ω
ωpe
ω
1
c2 k 2
1 = 2 (ω ± ωge ) ω −
ωpe
ω
=0
APPENDIX B. APPENDIX
B.4
201
Magnetohydrodynamic Waves
In the following we want to examine typical waves in the single fluid plasma. The treatment is similar
to for instance sound waves but clearly one expects that the magnetofluid has more degrees of freedom
and wave propagation should depend on the orientation of the magnetic field, i.e. it is not isotropic as in
the case of sound waves. Note that in the cause of waves in gravitationally stratified atmosphere wave
propagation is also anisotropic and gravity allows for a new wave mode the so-called gravity wave.
In the case of MHD we start from the full set of linearized ideal MHD equations.
∂ρ
∂t
∂ρu
∂t
∂B
∂t
∂p
∂t
= −∇ · ρu
= −∇ · (ρuu) − ∇p +
1
(∇ × B) × B
µ0
= ∇ × (u × B)
= −u · ∇p − γp∇ · u
where we have already combined the induction equation and Ohm’s law. Now we linearize the equation
as in the previous section, however, we do not introduce a small displace but rather keep u1 as a variable.
Specifically the MHD variables are expressed as
ρ
u
B
p
=
=
=
=
ρ0 + ρ1
u1
B0 + B1
p0 + p 1
For simplicity we assume all equilibrium quantities to be constant, i.e., we do not consider and inhomogeneous plasma. Substitution of the perturbations into the MHD equations yields similar to the small
displacement treatment
∂ρ1
∂t
∂u1
ρ0
∂t
∂B1
∂t
∂p1
∂t
= −ρ0 ∇ · u1
= −∇p1 +
1
(∇ × B1 ) × B0
µ0
= ∇ × (u1 × B0 )
= −γp0 ∇ · u1
Taking the second derivative in time of the momentum equation yields
APPENDIX B. APPENDIX
202
1
∂ 2 u1
∂p1
+
ρ0 2 = −∇
∂t
∂t
µ0
∂B1
∇×
× B0
∂t
B2
= γp0 ∇ (∇ · u1 ) + 0 (∇ × (∇ × (u1 × eB ))) × eB
µ0
where we used B0 = B0 eB . Now dividing by ρ0 and with the definitions for the speed of sound cs and
the Alfven speed vA
γp0
ρ
B02
=
µ0 ρ0
c2s =
vA2
we obtain
∂ 2 u1
= c2s ∇ (∇ · u1 ) + vA2 (∇ × (∇ × (u1 × eB ))) × eB
∂t2
We will now try to find the solution as plane waves by assuming
b 1 exp (ik · x − iωt)
u1 = u
which leads us to
ω 2 u1 = c2s k (k · u1 ) − vA2 eB × (k × (k × (u1 × eB )))
We now choose a coordinate system where the z axis points along the equilibrium (constant) magnetic
field and the k vector is in the x, z plane and given by
k = k⊥ ex + kk ez
This leads to the following three components for the equation:
2
u1x − c2s kk k⊥ u1z = 0
ω 2 − vA2 kk2 − c2s + vA2 k⊥
ω 2 − vA2 kk2 u1y = 0
ω 2 − c2s kk2 u1z − c2s kk k⊥ u1x = 0
or in matrix form W · u1 = 0


2
ω 2 − vA2 kk2 − (c2s + vA2 ) k⊥
0
−c2s kk k⊥
u1x

  u1y  = 0
0
ω 2 − vA2 kk2
0
2
2
2 2
u1z
−cs kk k⊥
0
ω − cs kk

APPENDIX B. APPENDIX
203
Exercise: Derive the above matrix.
These equations are linear independent and provide nontrivial solutions if the determinant is equal to
zero.
a) Alfvén wave:
The first nontrivial solution is determined by the y component of the above equations. This component
separates from the other two because u1y does not appear in the first and third equation. In the determinant
ω 2 − vA2 kk2 appears a a common factor. The solution is
ω 2 = vA2 kk2 =
kk2 B02
µ0 ρ 0
or
ω=±
k · B0
(µ0 ρ0 )1/2
= ±kvA cos θ
The group velocity vg = dω/dk is
vg = ±
B0
(µ0 ρ0 )1/2
= ±vA eB
which is independent of k and always along the magnetic field. The phase velocity is
vph =
ω
= ±vA cos θ
k
where theta is the angle between k and B0 . This wave is the Alf\ven wave.
A common representation of phase and group velocities is the Clemmov-Mullaly-Allis diagram which
represents phase and group velocities in a polar plot
where the radius vector represents the magnitude of
the velocity and the angle is the propagation angle theta. Note that the direction of the group velocity for Alfv\’en waves (which is the direction in
which energy and momentum is carried) is always
along the magnetic field! Note also that the case for
θ = 90◦ is singular in that wave does not propagate
and the points with θ = 90◦ should be excluded from
the group velocity plot.
0
z, B
ω/k
θ
A
v
Further properties of Alfvén waves:
• Velocity perturbation u1 ⊥B0
• Wave is incompressible: ∇ · u1 = ik · u1 = 0 or u1 ⊥k
• Magnetic field perturbation: ωB1 = −k×(u1 × B0 ) = −kk u1 , i.e., the magnetic field perturbation
is aligned with the velocity perturbation (and both are perpendicular to k and B0 .
APPENDIX B. APPENDIX
204
• Current density j1 ∼ k × B0 , i.e., the current perturbation is perpendicular to k and B0 .
• Alfvén wave also exist without steepening with nonlinear amplitudes.
Considering a localized perturbation for B1 along the k vector propagation of an Alfven wave and the
deformation of the magnetic field is presented in the following figure.
B0
Localized
z
vg
Perturbation
k
along k
z
B
vg
u B
1
1
x
y
cut in y, z plane
The Alfvén wave is of large importance in many plasma systems. It is very effective in carrying energy
and momentum along the magnetic field. Among many space plasma applications the Alfvén wave is
particulalry important for the coupling between the magnetosphere and the ionosphere. Here perturbations (convection) from the magnetosphere are carried into the ionosphere and depending on ionospheric
conditions cause convection in the high latitude ionosphere.
A coomon diagnostic to identify Alfvén waves in space physics is the use of the so-called the Walen
relation.
B
∆u = ±∆vA = ±∆ √
µ0 ρ
This relation is valid for linear and nonlinear Alfvén waves, and can also be used to identify the corresponding (nonlinear) discontinuities.
b) Fast and slow mode:
The other nontrivial solution is given by det W = 0 or
2 2
2
=0
ω − c2s kk2 − c4s kk2 k⊥
ω 2 − vA2 kk2 − c2s + vA2 k⊥
This is the dispersion relation for 4 additional solutions which are given by the roots of the equation.
Since the equation is double quadratic, i.e., depends only on ω 2 there are two different solutions for ω 2 .
Abbreviating c2f = c2s + vA2 (where the f stands for fast mode - the reason for this will be clear in a
moment) one can re-write the equation as
ω 4 − c2f k 2 ω 2 + vA2 c2s kk2 k 2 = 0
with the solutions
(B.1)
APPENDIX B. APPENDIX
205

# 
"
2 1/2 

k
k
k
=
c2f ± c4f − 4vA2 c2s 2

2 
k
h
i1/2 k2
2
2
2
2 2
2 2
2
vA + cs ± vA + cs − 4vA cs cos θ
=
2
2
ω2
Here the solution with the + sign is called the fast mode and the solution with the − sign the slow mode.
Fast wave:
This is the MHD wave with the fastest propagation speed, i.e., information in an MHD plasma cannot
travel faster thant this velocity.
• Phase speed along the magnetic field: ω 2 /k 2 = max [vA2 , c2s ]
• Phase speed perpendicular to the magnetic field ω 2 /k 2 = c2f ≡ vA2 + c2s
• Limit for c2s vA2 : ω 2 − vA2 k 2 = 0 => the fast mode becomes the so-called compressional
Afvén wave. Note that the compressional Alfven waves has a group velocity equal to its phase
velocity (different from the regular or shear Alfven wave).
• Limit for vA2 c2s : ω 2 − c2s k 2 = 0 =>
the fast mode becomes a sound wave.
The most prominant example of a fast wave in space physics is the Earth’s bow shock. This shock forms
as a result of the solar wind velocity which is faster than the fast mode speed. The fast wave is also
important in terms of transporting energy perpendicular to the magnetic field. The slow mode group
velocity is as for the Alfven wave along the magnetic feild and it is comparably small. Thus the fast wave
is the only MHD wave able to carry energy perpendicular to the magnetic field.
v >c
A
s
v >c
0
z, B
A
Group
0
z, B
s
v
s
c
fast
Velocities
A
A
v
s
c
fast
mode
mode
slow
mode
f
f
c
c
slow
mode
Slow wave:
The slow wave is the third of the basic MHD waves. The phase velocity is always smaller than or equal
to min [vA2 , c2s ]. Basic properties:
Phase speed along the magnetic field: ω 2 /k 2 = min [vA2 , c2s ].
Phase speed perpendicular to the magnetic field ω 2 /k 2 = 0.
APPENDIX B. APPENDIX
206
In the limits of c2s vA2 and vA2 c2s the slow wave disappears.
Applications of the slow wave are not as eye catching than those for the other MHD wave but the wave is
important to explain plasma structure in the region between the bow shock and the magnetosphere, and
well know for applications such as magnetic reconnection.
Concluding remarks:
There is an additional wave which in MHD which is the sound wave. This is obtained by using k⊥ = 0
and u⊥ = 0 such that the velocity perturbation is along the magnetic field and thus the magnetic field
perturbation is 0. However, it is singular in that it exists only for propagation exactly along the magnetic
field, i.e. the velocity is exactly parallel to B0 and the k vector is also exactly parallel to B0 . In fact the
sound wave can be either part of the slow wave or part of the fast wave solution depending on whether
c2s < vA2 or c2s > vA2 .
Exercise: Derive the dispersion relations for k parallel to B0 . What are the three wave solutions.
It should also be remarked that similar to a sonic shock wave the MHD waves are associated with respective shocks or discontinuities, I.e., there is a fast shock for the transition from plasma flow that is
super-fast (faster than the fast mode speed) to sub-fast (slower than the fast mode speed), a slow shock,
and an intermediate shock corresponding the Alfvén wave.