self-similarity in classical fluids and mhd b.c. low high altitude observatory national center for...
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Self-Similarity in Classical Fluids and MHD
B.C. LowHigh Altitude Observatory
National Center for Atmospheric Research
Boulder, Colorado, USA
The National Center for Atmospheric Research is operated by the University Corporation for Atmospheric Researchunder sponsorship of the National Science Foundation. An Equal Opportunity/Affirmative Action Employer.
1-D Thermal Pulse
')0,'(4
)'(exp
1),(
;4
exp),(
4exp),(
0
2
2/1
2
0
20
2
2
dxxTt
xx
ttxT
t
xdTdxtxTE
t
x
t
TtxT
x
T
t
T
Reduction from pde to ode
4exp
02
1
)(1
),(&
2
2
2
2/12/1
d
d
d
d
ttxT
t
x
Spherically-Symmetric Flows
0v
0v1
pvv
v
-
22
pr
pt
rrrt
rrt
Sedov-Taylor Point Explosion
• Initial state:
• Similarity variable:
• The solution:
30
220 ~,~E MLTML
5
1
20
0
E
tr
)()1(25
8
)(1
1
)V(1)5(
4v
2
20
0
Pt
rp
D
t
r
A Realistic “Point” Explosion
• Energy released:
• Fireball radius:
• Warm medium:
• Length scales:
• Time scales:
• Similarity regime:
0E
0R
00 , p 3/1
0
0201 p
E,R
rr
2/13/50
2/3002
2/1
05001 /E,E/R ptt
2121 , tttrrr
Intermediate AsymptoticsBarenblatt & Zel’dovich (1972, Ann.Rev. Fluid Mech. 4, 285)
• Self-similar solutions are the intermediate asymptotic forms of a solution in a local space time region away from boundaries and its initial state.
• Self-similar solutions often turn out to be the “preferred” end state of a diversity of possible evolutions originating from very different initial states.
CME Kinetic Properties
• Typical CME mass ~
• Median Speed ~
• Range ~
• G-escape Speed ~
• Coronal Sound Speed ~
• Coronal Alfven Speed ~
g10515
skm /450
skm /550skm /120
skm /1000
skm /10250 3
Time-Dependent Ideal MHD
0)(pρdt
d
)Bv(t
B
0)v(ρt
ρ
r̂r
GMρpB)B(
4π
1
dt
vdρ
γ
2Sun
Darnell & Burkepile 2002
Homologous Expansion
4
3
422
43
1
~,3/4
;~~
~;~
~;~
~
Rpthenif
Rp
RBRB
RR
R
Time-Dependent Ideal MHD
0)(pρdt
d
)Bv(t
B
0)v(ρt
ρ
r̂r
GMρpB)B(
4π
1v.v
t
vρ
4/3
2Sun
Homologous Radial Flow: Zero-Force Flow
0
0
t
r
dt
d
t
r
dt
drt
t
r
vv.v
rv
Homologous Radial Flow: Mass Conservation
),,t
rD(
10
1
0v1
3
3
2
r2
2
tt
r
rrt
rrrt
t
r
rv
A Simple Class of Self-Similar MHD Solutions
0r̂GM
D*H)H*(4π
1
),,(H),,(H1
B
),,P(),,P(1
),,D(),,D(1
,
2Sun
2
2
2
4
4
4
3
3
3
P
rt
rtp
rt
t
r
t
rrv
A More Sophisticated Class of Self-Similar MHD Solutions
22
2
2Sun
2
4
3
d
0r̂GM
D*H)H*(4π
1
),,(H1
B
),,P(1
),,D(1
)(,
)(
dt
P
p
t
r
dt
d
t
rrv
A Non-Galilean Transformation
• A non-Galiliean transformation introduces inertial forces, e.g., the Coriolis force in a uniformly rotating frame of reference.
• The similarity space
introduces a pure radial force of magnitude
- the net force driving the flow.
,,)(t
r
Stability Problems in Similarity Space
• With no loss of generality:
• Time-dependent linear perturbations about a static state in space – a classic stability problem.
),,,)(
(},,,( tt
ttr
),,(
A More Sophisticated Class of Self-Similar MHD Solutions
22
2
2Sun
2
4
3
d
0r̂GM
D*H)H*(4π
1
),,(H1
B
),,P(1
),,D(1
)(,
)(
dt
P
p
t
r
dt
d
t
rrv
Velocity vs Distance
A More Sophisticated Class of Self-Similar MHD Solutions
22
2
2Sun
2
4
3
d
0r̂GM
D*H)H*(4π
1
),,(H1
B
),,P(1
),,D(1
)(,
)(
dt
P
p
t
r
dt
d
t
rrv
Squeezing a 3D Solution out of a 2D Solution
Unsqueezed Solution
Squeezed Solution
A Theoretical 3-Part CME
Gibson & Low (1998, 2000)
Summary
• Similarity solutions as long-time end-states.• Compatible scaling laws of the thermal
energy of a polytropic gas of 4/3 index with gravitational and magnetic energy.
• Observed broad range of CME speeds.• The 3-part structure of CMEs has its origin
in the initial state.• Do similarity flows occur in CMEs?
Movies from Gibson & Low
• Frot~1 shows a time sequence of the magnetic field of the CME making its way out surrounded by the global open magnetic field of the corona.
• Sigmoid shows the 3 D magnetic field of the CME from different perspective.
• Pb3 shows a time sequence of the CME in simulated Thomson-scattered light.