Warwick PX420 Solar MHD 2015-2016: Hydrostatic Pressure Balance
1
Magnetohydrostatic Equilibria:
Hydrostatic Pressure Balance
The magnetohydrostatic equilibrium condition is
0 = −∇p + j × B + ρg,
(1)
∇ · B = 0,
(2)
µj = ∇ × B,
(3)
coupled with
ρRT
,
µ̃
p=
(4)
and T satisfies an energy equation.
Before investigating any specific phenomena we need to consider the basic pressure balance when the
magnetic field does not exert any force.
Consider the simple case of a uniform vertical magnetic field. For simplicity we assume that the temperature is known.
Thus,
g = −gẑ.
B = B0 ẑ,
Hence, j = 0 and there is no Lorentz force.
In addition, the pressure is p = p(z) and (1) becomes
dp
g µ̃
p(z)
= −ρ(z)g = −
p(z) = −
,
dz
RT (z)
Λ(z)
where
Λ(z) =
RT (z)
,
µ̃g
(5)
(6)
is the pressure scale height.
Eq. (5) is a separable, first order ordinary differential equation so that
dp
1
=−
dz,
p
Λ(z)
and integrating gives
log p = −n(z) + log P (0),
where
Z
n(z) =
0
z
1
du,
Λ(u)
is the ‘integrated number’ of scale heights between the arbitrary level at which the pressure is p(0) and
the height z. Therefore,
p(z) = p(0) exp [−n(z)].
(7)
If the atmosphere is isothermal so that both T and Λ are constant, then (7) gives
p(z) = p(0) exp (−z/Λ),
ρ(z) = ρ(0) exp [−z/Λ],
(8)
Warwick PX420 Solar MHD 2015-2016: Hydrostatic Pressure Balance
2
so that the pressure decreases exponentially on a typical length scale given by the pressure scale height
Λ:
(Here, three curves are shown, corresponding to Λ = 1, Λ = 2 and Λ = 3. Mark the curves in the figure
with the appropriate value of Λ.)
Consider typical values of the pressure scale height. Taking the solar gravitational constant as g =
274 ms−2 and R = 8.3 × 103 J K−1 mol−1 then Λ takes the following values:
1. In the photosphere T = 6, 000 K and µ̃ = 1.3 so that
Λ=
8.3 × 103 × 6 × 103
RT
=
= 140 km.
µ̃g
1.3 × 274
2. In the corona T > 106 K and µ̃ = 0.6 giving
Λ=
RT
8.3 × 103 T
=
≈ 50.5T m.
µ̃g
0.5 × 274
Thus, the scale height can be estimated as
Λ/Mm ≈ 50T /MK.
E.g., the scale height of the corona observed by TRACE-171Å (the temperature is 1 MK) is 50 Mm.
(This figure is comparable to the size of a loop).
Warwick PX420 Solar MHD 2015-2016: Hydrostatic Pressure Balance
3
Warwick PX420 Solar MHD 2015-2016: Hydrostatic Pressure Balance
4
Example: Density stratification in a polar plume.
Polar plumes are cool, dense, linear, magnetically open structures that arise from predominantly
unipolar magnetic footpoints in the solar polar coronal holes.
The solid line shows the hydrostatic solution for T = 106 K.
3. In the Earth’s atmosphere T = 300 K, g = 9.81 ms−1 , µ̃ = 29 in air and so
Λ=
8.3 × 300
= 8.7 km.
29 × 9.81
Note that the height of Mount Everest is about 8.8 km. Thus, the air pressure at the summit of Everest
is about 1/e = 0.37 that of the air pressure at sea level.
Warwick PX420 Solar MHD 2015-2016: Hydrostatic Pressure Balance
5
Hydrostatic equilibrium at larger heights
(Image of the solar corona taken with a coronograph).
Determination of the density stratification on larger scale, e.g. in coronal holes, requires taking into
account the effects of the spherical geometry and the change of the gravitational acceleration with height,
GM
r2
(9)
2
B0 R
.
2
r
(10)
g(r) =
where r is the radial coordinate.
The magnetic field is assumed to be strictly radial,
B=
In the following we consider spherically symmetric isothermal (T = const) atmosphere.
Again, there is no the Lorenz force. The magnetostatic equation is similar to (5),
dp(r)
= −ρ(r)g(r),
dr
(11)
which, with the use of the state equation,
p=
can be rewritten as
ρRT
µ̃
R2 1
dρ(r)
= − 2 ρ(r).
(12)
dr
r Λ
Here, the scale height Λ was defined by substituting the value of the gravitational acceleration g(R ) =
2
GM /R
at the solar surface into Eq. (6).
Warwick PX420 Solar MHD 2015-2016: Hydrostatic Pressure Balance
6
ODE (18) is separable,
Z
dρ
=−
ρ
Z
with the solution
2
R
dr
Λr2
ρ(r) = C exp
2
R
Λr
(13)
.
Determining the constant C from the condition ρ(R ) = ρ0 , we obtain
R R
−1 .
ρ(r) = ρ0 exp
Λ
r
(14)
(15)
This solution coincides well with the observationally determined empirical dependence. E.g., the profile of plasma concentration in polar coronal holes determined with SPARTAN 201-01 (from Fisher &
Guhathakurta 1995):
An empirical model was constructed by Esser et al. (1999),
ne =
2.494 × 106
1.034 × 107
3.711 × 108
+
+
,
3.76
9.64
r
r
r16.86
(16)
which corresponds to the theoretical dependence reasonably well.
Notice that Eq. (21) gives infinite density at r → ∞, so can be applied at the heights below 5–6 R only.
At larger distances from the Sun steady flows of plasma (the solar wind) must be accounted for.