Warwick PX420 Solar MHD 2008-2009: MHD Modes of a Magnetic Interface
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MHD Modes of a Magnetic Interface
1. Equilibrium
Consider a sharp interface between two uniform plasmas.
x
B0
T0 , p0 , ρ
0
z
0
Te , p , ρe
e
Be
The equilibrium parameters of the upper plasma are the density ρ0 , gas pressure p0 , and
it is penetrated by the magnetic field B = B0 ez . The lower plasma has the density ρe ,
gas pressure pe and the magnetic field B = Be ez . So,
B0 (x) =
(
B0 , x > 0,
Be , x < 0,
T0 (x) =
(
T0 , x > 0,
Te , x < 0,
ρ0 (x) =
(
ρ0 , x > 0,
ρe , x < 0,
p0 (x) =
(
p0 , x > 0,
pe , x < 0.
(1)
In the upper and the lower media, the sound speeds are Cs0 and Cse , Alfvén speeds CA0
and CAe , respectively.
Express the characteristic speeds through the other parameters of the equilibrium.
Stationary physical values are uniform everywhere except the jump at the interface x = 0,
where the total pressure balance condition is fulfilled.
The total pressure balance condition is
B02
Be2
p0 +
= pe +
.
2µ
2µ
(2)
Warwick PX420 Solar MHD 2008-2009: MHD Modes of a Magnetic Interface
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Re-write this equation though the characteristic speeds:.
In the following we restrict our attention to the case when the upper plasma is cold
plasma, so p0 = 0 and the low plasma is unmagnetized (field-free), Be = 0.
2. Equations for perturbations
Consider linear perturbations propagating along the direction of the magnetic field, k ez
2.1 Upper medium
The perturbations of the equilibrium physical values are governed by the MHD equations.
We need two of them:
∂B
= ∇ × (V × B),
∂t
(3)
dV
1
= − B × (∇ × B).
dt
µ
(4)
and
ρ
Linearizing the equations, we get
∂ B̃
= ∇ × (Ṽ × B0 ),
∂t
(5)
and
ρ0
∂ Ṽ
1
= − B0 × (∇ × B̃).
∂t
µ
(6)
Considering harmonic waves,
∝ exp(iωt − ikz),
(7)
and projecting the equations (or determining their components with the use of the components of the vectors Ṽ = (Vx , Vy , Vz ) and B̃ = (Bx , By , Bz )), we get
iωBx = −ikB0 Vx ,
(8)
∂Vx
,
∂x
(9)
iωBz = −B0
Warwick PX420 Solar MHD 2008-2009: MHD Modes of a Magnetic Interface
Ã
!
∂Bz
B0
ikBx +
.
iρ0 ωVx = −
µ
∂x
3
(10)
Equations (8)-(10) can be combined into a single equation,
∂ 2 Vx
ω2
2
− k − 2
∂x2
CA0
Ã
!
Vx = 0.
(11)
Perturbations of the magnetic field can be obtained with the use of (8) and (9),
kB0
Vx ,
ω
(12)
iB0 ∂Vx
.
ω ∂x
(13)
Bx = −
Bz =
2.2 Lower medium
The perturbations of the equilibrium physical values in the field-free medium are governed
by the linearized hydrodynamic equations:
∂ ρ̃
+ ∇(ρe V) = 0,
∂t
ρe
∂V
= −∇p̃,
∂t
(14)
(15)
and the isothermal form of the energy equation,
2
p̃ = Cse
ρ̃.
(16)
These equations can be rewritten as three scalar equations,
iω ρ̃ + ρe
∂Vx
− ikρe Vz = 0,
∂x
2
iωρe Vx = −Cse
∂ ρ̃
,
∂x
2
iωρe Vz = ikCse
ρ̃.
(17)
(18)
(19)
They can be combined into the second order ODE,
∂ 2 Vx
ω2
2
−
k
−
2
∂x2
Cse
Ã
!
Vx = 0.
(20)
Warwick PX420 Solar MHD 2008-2009: MHD Modes of a Magnetic Interface
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Compare this equation with (11). When these equations have oscillitory solutions, and
when evanescent?
Perturbations of other physical values are
ρ̃ =
iρe ω
∂Vx
,
2 k 2 ∂x
ω 2 − Cse
∂Vz
= −ikVx .
∂x
(21)
(22)
3. Boundary conditions
Solutions of the MHD equations in the upper and the lower media have to be matched
by the boundary conditions: the continuity of the perturbations of total pressure,
(0)
(e)
pT (x = 0) = pT (x = 0),
(23)
Vx(0) (x = 0) = Vx(e) (x = 0),
(24)
the normal velocity continuity,
where the indices (0) and (e) correspond to the upper and the lower media, respectively
and the condition of the mode localization,
Vx(0) (x → +∞) → 0,
Vx(e) (x → −∞) → 0.
(25)
In the upper medium, the thermal pressure is zero, and the perturbation of the total
pressure can be expressed through Vx :
Bx2 + By2 + (B0 + Bz )2
B0 Bz
pT =
≈
.
2µ
µ
(26)
Using Eq. (9), we obtain
pT =
iB 2 ∂Vx
B0 Bz
= 0
.
µ
µω ∂x
(27)
(The LHS should be supplemented with the index (0).)
In the lower, field-free medium, the perturbation of the magnetic pressure is zero, and
the total pressure perturbation coincides with the perturbation of the thermal pressure.
Thus, using (21), we obtain
2
pT = p̃ = Cse
ρ̃ =
2
iCse
ρe ω ∂Vx
,
2
2 k 2 ∂x
ω − Cse
(The LHS should be supplemented with the index (e).)
(28)
Warwick PX420 Solar MHD 2008-2009: MHD Modes of a Magnetic Interface
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4. Derivation of dispersion relations
Equation (11) has the solution
Vx(0) = A1 exp(m0 x) + A2 exp(−m0 x),
(29)
where A1 and A2 are arbitrary constants, and
m0 =
s
k2 −
ω2
.
2
CA0
(30)
Similarly, equation (20) has the solution
Vx(e) = B1 exp(me x) + B2 exp(−me x),
(31)
where B1 and B2 are arbitrary constants, and
v
u
u
ω2
me = tk 2 − 2 .
Cse
(32)
To satisfy the boundary conditions at infinity, we must choose A1 = 0 and B2 = 0.
Then, condition (24) with the use of solutions (29) and (31) becomes
Vx(0) (0) = Vx(e) (0) ⇒ A2 exp(−m0 0) = B1 exp(me 0) ⇒ A2 = B1 .
(33)
And, finally, condition (23) with (27) and (28) gives:
(0)
pT (0)
=
(e)
pT (0)
2
iCse
ρe ω ∂Vx(e)
iB02 ∂Vx(0)
(0) = 2
(0).
⇒
2 k 2 ∂x
µω ∂x
ω − Cse
(34)
Calculating the derivatives at x = 0,
∂Vx(0)
(0) = −m0 A2 exp(−m0 0) = −m0 A2 ,
∂x
(35)
∂Vx(e)
(0) = me B1 exp(me 0) = me B1 ,
∂x
(36)
and using that A2 = B1 , we rewrite equation (34) as
−
iC 2 ρe ω
iB02
m0 = 2 se 2 2 me ,
µω
ω − Cse k
(37)
Warwick PX420 Solar MHD 2008-2009: MHD Modes of a Magnetic Interface
or
2
−ρ0 CA0
m0 =
2
Cse
ρe ω 2
m
2 k2 e
ω 2 − Cse
6
(38)
This is the dispersion relations of the magnetic interface modes.
It is convenient to rewrite it introducing the phase speed, a = ω/k:
2
ρ0 CA0
−
2
ρe Cse
s
v
u
2
u
a2
a2
t1 − a
1− 2 = 2
2
2
CA0
a − Cse
Cse
(39)
The dispersion relation is a transcendental algebraic equation.
5. Analysis of dispersion relation
Obviously, the phase speed is less than both the characteristic speeds in the upper and
the lower media, CA0 and Cse . Thus, interface modes propagate slower the the waves in
both the upper and the lower media.
Also, dispersion relation (39) does not contain either the wave number k nor the frequency
ω. Thus, the phase speed a is independent of the frequency and the wave number.
Consequently, the modes are dispersionless.
In a more general case, when the upper (magnetized) plasma is not cold, p0 > 0 ⇒ Cs0 >
0, dispersion relation has the same form, with the only change,
m20 =
2
2
(k 2 Cs0
− ω 2 )(k 2 CA0
− ω2)
,
2
2
+ CA0
)(k 2 CT2 0 − ω 2 )
(Cs0
where
CT 0 =
is so-called tube or cusp speed.
Cs0 CA0
,
2 1/2
+ CA0
)
2
(Cs0
(40)
(41)