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.