Transcript Waves Launched by Diffusion in a Model of Magnetic

CORONAL HEATING
(Space Climate School, Saariselka, March, 2009)
Eric Priest (St Andrews)
Sept. 13, 2007
1. Introduction - The Corona (Eclipse)
Sept. 13, 2007
Skylab -- X-ray telescope
Sept. 13, 2007
Coronal holes -- loops -- X-ray bright points
Yohkoh
(5 arcsec)
A dynamic
magnetic
world
- subtle
interactions
B & plasma
Sept. 13, 2007
Hinode
(1 arcsec)
Stunning detail
on structure
& dynamics
(see Tsuneta)
Sept. 13, 2007
How is corona heated?
Waves or reconnection? - Space Obsns:
 Low-freq. waves in loops [TRACE]-too weak to heat
 High-freq. waves [UVCS] -- ?? heat outer corona
 Hinode -Chromospheric
Spicules
swaying
(straw, prairy)
Hansteen, Suematsu
--?? Solar wind/
coronal heating
Sept. 13, 2007
2. Reconnection - most likely in low corona
Quiet Sun:
[XRT on Hinode,
Tsuneta, golub]
Many brightenings
X-ray bright points above emerging and/or
cancelling fields
in photosphere
Sept. 13, 2007
[30-sec
cadence, 12-hour duration]
Hinode XRT - active region
(Schmeltz et al, 2009)
Observations inside white region
Differential emission measure
Sept. 13, 2007
Normal active region emission at 3 MK
Plus peak at 20 MK (?nanoflares)
Parker’s classical Nanoflare Model
by braiding (1972)
Initial B uniform / motions braiding
Sept. 13, 2007
Numerical Experiment (Galsgaard)
Braiding --> Current sheets grow --> turb. recon.
Sept. 13, 2007
3. Coronal Tectonics Model
(development of Parker’s model)
3.1 Effect “Magnetic Carpet”
Sept. 13, 2007
Magnetic
sources in surface are concentrated
Flux Sources Highly Dynamic
Magnetogram movie (white +ve , black -ve)


Flux emerges ... cancels
Reprocessed very quickly (14 hrs !!!)
Sept. 13, 2007
Many Sources-->
Corona has
Complex
Topology
In 2D -- Separatrix
curves
In 3D -- Separatrix
surfaces
In 2D, reconnection at X
In 3D, reconnection at
separator
In complex fields
we form the
SKELETON-set separatrices
3.3 “Simple” binary interaction of 2 photospheric
sources (Haynes et al)
- and + sources
in overlying B.
Separatrix
surfaces.
Move sources
& watch
Interaction
flux tube
joining sources
Separator
Cross-sections of Separatrix Surfaces
2 separators
5 separators
Separatrix surfaces (positive, negative) & Separators ( )
Number of separators: X

Life of Magnetic Flux
in Surface
(a) 50%? flux in Quiet Sun
emerges as ephemeral regions
[1 per 8 hrs per supergran, 3 x 1019 Mx]
 (b) Each pole migrates to
boundary (4 hours), fragments --> 10
"network elements" (3x1018 Mx)

(c) -- move along boundary (0.1
km/s)
-cancel
Sept. 13, 2007
From observed
magnetograms construct coronal
field lines
- each source
connects to 8 others
Time for all field lines
to reconnect
only 1.5 hours
(Close et al)
 much more tectonics
heating low down
where
Sept. 13,field
2007 is more
complex than higher up
Coronal Tectonics Model
(updated version of Parker nanoflare/topological dissipation)


(Priest, Heyvaerts & Title)
Each "Loop" --> surface in many sources
Flux from each
source
separated by
separatrix surfaces

As sources move
--> J sheets on separatrices & separators
--> Reconnect --> Heat

Corona filled w. myriads of J sheets,
Sept. 13, 2007
heating impulsively
Fundamental Flux Units
not Network Elements
 Intense tubes (B -- 1200 G, 100 km, 3 x 1017 Mx)


Each network element -- 10 intense tubes
Single ephemeral
region (XBP) --
100 sources
800 seprs, 1600 sepces

Each TRACE
Loop --
10 finer loops
rs, 160 sepces
80 sep
Sept. 13,
2007
TRACE Loop
Reaches to
surface in
many
footpoints.
Separatrices
form web in
corona
Sept. 13, 2007
Corona - Myriads Different Loops
Each flux element --> many neighbours
Sept. 13, 2007
But in practice each source has 8 connections
Results

Heating uniform along separatrix
Elementary (sub-telc) tube heated uniformly

But 95% photc. flux closes low down in carpet
-- remaining 5% forms large-scale connections
 --> Carpet heated more than large-scale corona

So unresolved observations of coronal loops
--> Enhanced heat near feet in carpet
--> Upper parts large-scale loops heated uniformly
& less strongly
Sept. 13, 2007
4. If reconnection heats corona
at many sheets,
1. How does energy spread out ?
-- conduction along B
-- reconnection jets
-- waves across B
2. If reconnection time-dependent,
how much energy liberated locally/globally?
Simple model problem
Sept. 13, 2007
[Longcope & Priest]
Magnetic field of Current Sheet in X
By  iBx  B' w 2  2  d/dw
At large r, B = B0 + B1
I0
B1 
(line current),
2 r
Lots of energy far from CS
B0  B' [y xˆ + x yˆ ],
Sept. 13, 2007
A 
B'
4
Suppose sheet reconnects
Current (I) dissipates
Local process but has
global consequences:
Decrease I --> B must change at large distances
Sept. 13, 2007
How ??
Model for effect of reconnection
Linearize about X-point B0 :
B1
   (v1  B0 )   2B1 ,
t
v1
0
 j1  B0 .
B0  B'[yxˆ  xyˆ ]
t
Assume B1 @ t=0 is due to current sheet
 is “turned on”
Sept. 13, 2007
current diffuses
i.e. reconnection

Combine equations:
Put
rB1  I(r,t) = twice current enclosed in r
2 

I
 1I
2   I 
  A r r  r 

2
t
r  r 
r r rt 
2
wave
diffusion
I (r)
Expect:
r
Sept. 13, 2007
I0

r
2 

I
 1I
2   I 
  A r r  r 

2
t
r  r 
r r rt 
2
wave
diffusion
R  ln(r /
(i) Large r (wave) limit: when
I(R,t)=I0-F(t-R)
I0

) >>1



A
R
(ii) Small r (diffusive) limit:
 r 2 
I(r,t)  I0  I0 exp

 4t 
NB
I0
2 t
1 I
j
--> 0 at origin as t increases
r r
Sept. 13, 2007
r
Numerical
Solution
I(r)
Locationwhere I  I
2
3 0
Wave solution R   A t
R
Transition: 
diffusive to
wave solution
Sept. 13, 2007

1
R

Diffusive solution
2 lnt
t
Sheath of Current propagates out
In wake of sheath a flow, assocd with
EV
increasing t

EV (r,t) zˆ = v1  B0
But flow near
X does not
disappear
-- it slowly
increases !
Sept. 13, 2007
Resolving the Paradox - 3rd regime
At large t
(i.e., t  1/ A )
2 

 I
  I 
 1 I
2
  A r r  r 

2
t
r  r 
r r rt 

2
Advection
=
diffusion
  A2 r 2 t 
I0 
I(r,t) 
1 exp

2 A t 
  
1 I I0 A
 rt
j

exp(
)
r r


2 2
A
Sept. 13, 2007
Peak in j remains at X
and
produces a steady E
(indep of  )
5. Summary
Coronal tectonics -- updated version of Parker braiding
Response to enhanced  in current sheet (CS)
during coronal tectonics:
(i) Diffusion spreads CS out
(ii) Wave carries current out at vA - as sheath
(iii) Peak in j at X remains --> steady E
independent of  i.e. fast
• Most magnetic energy is converted into
kinetic energy in wave -- may later dissipate.

• Coronal
heating -- reconnection + wave
Sept. 13, 2007
Sept. 13, 2007