an mhd model with wave turbulence driven heating and solar wind acceleration
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8/8/2019 An MHD Model with Wave Turbulence Driven Heating and Solar Wind Acceleration
http://slidepdf.com/reader/full/an-mhd-model-with-wave-turbulence-driven-heating-and-solar-wind-acceleration 1/17
An MHD Model with Wave Turbulence Driven Heating and
Solar Wind Acceleration
Roberto Lionello1
Jon A. Linker1
Zoran Mikic1
Pete Riley1
Marco Velli2
[email protected], [email protected], [email protected],
[email protected], [email protected]
1 2
AAS/SPD 2010 Meeting Miami, FL – p.
8/8/2019 An MHD Model with Wave Turbulence Driven Heating and Solar Wind Acceleration
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Summary
• The mechanisms responsible for heating the Sun’s corona and accelerating the
solar wind are still being actively investigated.
• It is largely accepted that photospheric motions provide the energy source and
that the magnetic field must play a key role in the process.• Three-dimensional MHD models have traditionally used an empirical prescription
for coronal heating (e.g., Lionello et al. 2009), together with WKB Alfvén wave
acceleration of the solar wind.
•
In wave turbulence driven models (e.g., Cranmer et al. 2007; Cranmer 2010)heating and solar wind acceleration by Alfvén waves are included
self-consistently.
• We demonstrate the initial implementation of this idea in an MHD model based
on turbulent cascade heating in the closed-field regions (Rappazzo et al. 2007,
2008), and Alfvén wave turbulent dissipation in open field regions (Verdini & Velli2007, 2010).
AAS/SPD 2010 Meeting Miami, FL – p.
8/8/2019 An MHD Model with Wave Turbulence Driven Heating and Solar Wind Acceleration
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The Thermodynamic MHD Model
∇×A = B,
∂ A
∂t = v ×B−c2η
4π ∇×B,
∂ρ
∂t+∇·(ρv) = 0,
1
γ − 1„∂T
∂t+ v
· ∇T « =
−T ∇ ·
v
−mp
2kρ(∇ · q + nenpQ(T ) −H ch),
ρ„∂ v
∂t
+ v
·∇v« =
∇×B×B
4π −∇ p
−∇ pw + ρg +
∇ ·(νρ
∇v),
γ = 5/3,
q =
(−κ0T 5/2bb · ∇T if R⊙ ≤ r 10R⊙αnekT v if r 10R⊙
,
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8/8/2019 An MHD Model with Wave Turbulence Driven Heating and Solar Wind Acceleration
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Wave Propagation and Dissipation
• Alfvén and acoustic waves are propagated into the corona by specifying a wave
flux at the coronal base.
• These waves interact with the plasma and dissipate in open and closed field
regions, accelerating and heating the solar wind.• For convenience we split the wave energy density ǫ = δB2/4π into two fields,
ǫr and ǫb:
∂ǫr,b
∂t+∇ ·Fr,b =
1
2v · ∇ǫr,b −
Cαǫ3/2r,b
λ⊥√
ρ−D (ǫrǫb)n
Fr,b = „3
2v ± vAb« ǫr,b
• The total energy density is given by ǫ = ǫr + ǫb
• The wave pressure is pw = 1
2ǫ
• The wave heating is H ch =P
r,b Cαǫ
3/2
r,bλ⊥√ ρ + D
`ǫr,bǫb,r
´n
AAS/SPD 2010 Meeting Miami, FL – p.
8/8/2019 An MHD Model with Wave Turbulence Driven Heating and Solar Wind Acceleration
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Wave Propagation and Dissipation: Open Field
• How do we get the Open Field term? We assume that Alfvén waves are injected
from the solar surface, reflected, and dissipated through Kolmogorov turbulence,
heating thus the solar wind.
z+
z+
z−
Positive open
field line
z−
z+
z+
field line
Negative open
Reflection
Dissipation
Injection
• Why Kolmogorov? Because turbulence develops orthogonally to the mean field
and the Kraichnan/Alfvén effect of crossing eddies is not important.
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8/8/2019 An MHD Model with Wave Turbulence Driven Heating and Solar Wind Acceleration
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8/8/2019 An MHD Model with Wave Turbulence Driven Heating and Solar Wind Acceleration
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Wave Propagation and Dissipation: Open Field (3)
• so that the reflection coefficient is
α(r) =|z−|
|z+
|=
v + vA
vA + vAc
vAc − vA
v − vA
• λ⊥ is the outer scale of the turbulence, and it expands with the flux tube
dimension:
λ⊥ = λ0s B(0)
B(r) .
• The relation between ǫ and z± is
ǫ = ρ |z+
|2 +
|z−|2
4 ,
• And putting all together, we obtain:
Heating ∼ αǫ3/2
r,b
λ⊥√
ρ
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8/8/2019 An MHD Model with Wave Turbulence Driven Heating and Solar Wind Acceleration
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Wave Propagation and Dissipation: Closed Field
• The nonlinear phenomenology in the closed-field region is based on the results of
Rappazzo et al. (2007, 2008).
• The incoming Poynting flux from the solar surface is dissipated by turbulence
inside the volume.• We impose that the energy density at the red spot (blue), ǫ0, is increased by the
reflected contribution that propagates from blue spot (red):
Dissipation
Reflection
Reflection
Loop
Injection
Boundaryconditions :(
ǫr(0) = ǫ0 + ǫb(0),ǫb(L) = ǫ0 + ǫr(L).
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8/8/2019 An MHD Model with Wave Turbulence Driven Heating and Solar Wind Acceleration
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Wave Propagation and Dissipation: Closed Field (2)
• The dissipated turbulent power density is:
D =δB2⊥
8π
1
τ NL=
(ǫrǫb)1/2
τ NL.
• Rappazzo et al. estimate the non-linear dissipation time τ NL as
τ NL ≃λ⊥
√4πρ
δB⊥„λ
⊥
√4πρ
δB⊥
B√4πρL
«β−1
AAS/SPD 2010 Meeting Miami, FL – p.
8/8/2019 An MHD Model with Wave Turbulence Driven Heating and Solar Wind Acceleration
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Wave Propagation and Dissipation: Closed Field (3)
• This gives the following dissipative term for the the closed-field region:
D (ǫrǫb)
n
=
(ǫrǫb)2+β4 Lβ−1
λβ⊥ρ 12Bβ−1
• Different values of β are associated with different regimes of turbulence:
• β = 1 −→ Kolmogorov.
• β = 2 −→ Kraichnan.• β →∞−→ Weak.
AAS/SPD 2010 Meeting Miami, FL – p.
8/8/2019 An MHD Model with Wave Turbulence Driven Heating and Solar Wind Acceleration
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Application to 1D Problems
• We tested in a 1D configuration the turbulence dissipation heating mechanism we
have described:• A 1D wind solution from the Sun to 1 A.U.•
A loop 177 Mm long.• Boundary conditions at R⊙:
• T = 20, 000 K
• ne = 2 × 1012 cm−3.
•
pw ≃ 0.4 dyn/cm2
• Following Verdini et al. 2010, we add a small compressive heating term in the
lower corona:
H ρ
= 3 × 1010 exp0@−
r
R⊙ −1.3
0.25
!21A cm2s−3
• In either case the configuration evolves until it reaches a steady state.
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8/8/2019 An MHD Model with Wave Turbulence Driven Heating and Solar Wind Acceleration
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Example of 1D Wind Solution
0
100
200
300
400
500
600
700
800
50 100 150 200
k m / s
r / RS
Speed
1
100
10000
1e+06
1e+08
1e+10
1e+12
1e+14
50 100 150 200
c m - 3
r / RS
Density
0
100000
200000
300000
400000
500000
600000
700000
800000
900000
1e+06
50 100 150 200
K
r / RS
Temperature
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
1 10 100
3 × 1
0 1 0
c m
2 s
- 3
r / RS
Heating/density
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8/8/2019 An MHD Model with Wave Turbulence Driven Heating and Solar Wind Acceleration
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Example of 1D Loop Solution
-0.8
-0.7
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0
0 20 40 60 80 100 120 140 160 180
k m / s
Mm
Speed
1e+08
1e+09
1e+10
1e+11
1e+12
1e+13
0 20 40 60 80 100 120 140 160 180
c m - 3
Mm
Density
0
200000
400000
600000
800000
1e+06
1.2e+06
1.4e+06
0 20 40 60 80 100 120 140 160 180
K
Mm
Temperature
0
0.0001
0.0002
0.0003
0.0004
0.0005
0 20 40 60 80 100 120 140 160 180
3 × 1
0 1 0
c m
2 s
- 3
Mm
Heating/density
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8/8/2019 An MHD Model with Wave Turbulence Driven Heating and Solar Wind Acceleration
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2D Streamer and Solar Wind Solution
• We have begun testing the Alfvén wave heating mechanism with our 3D MHD
code.
• We specify a dipole of amplitude 1.5 G at r = R⊙.
• At the solar surface, we impose the same boundary conditions used in the 1Dexamples.
• Initial plasma, temperature, density, and velocity were obtained from a 1D solar
wind solution calculated previously.
•
Thermal conductivity κ and radiation loss function Q are modified to broaden thegradient in the transition region.
• Nonuniform grid in r × θ of 301× 401 points. Finest radial grid resolution at
r = R⊙ was 321 km; angular resolution was uniform.
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8/8/2019 An MHD Model with Wave Turbulence Driven Heating and Solar Wind Acceleration
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2D Streamer
0 1
MK
Temperature Magnetic Flux
1R
Sun Sun
1R
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Wind in the Heliosphere
0
1
2
3
4
5
6
7
0 0.5 1 1.5 2 2.5 3
c m - 3
θ
ρ at 215 RS
300
350
400
450
500
550
600
650
700
0 0.5 1 1.5 2 2.5 3
k m / s
θ
V at 215 RS
km/s
0 200 400 600
1 A.U.Sun
Speed vs. Latitude at 1 A.U.
Density vs. Latitude at 1 A.U.
Wind Speed
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8/8/2019 An MHD Model with Wave Turbulence Driven Heating and Solar Wind Acceleration
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Conclusions
• We have incorporated into our 3D MHD code:
• A self-consistent heating and acceleration mechanism for the solar wind
based on turbulence dissipation.
•
A heating mechanism based on non-linear cascade dissipation for theclosed field regions.
• The model has been tested first in 1D wind and loop simulations.
• We have performed a first 2D streamer/solar wind simulation with the main code.
• First results are encouraging.• Further testing is necessary.
Worked performed thanks to NASA Solar and Heliospheric Physics program and
Heliophysics Theory Program.
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