An MHD Model with Wave Turbulence Driven Heating and Solar Wind Acceleration

8/8/2019 An MHD Model with Wave Turbulence Driven Heating and Solar Wind Acceleration

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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.



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).




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⊙

,




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




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.




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




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).




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




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.




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.




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




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




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.




2D Streamer

0 1

MK

Temperature Magnetic Flux

1R

Sun Sun

1R




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




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.


An MHD Model with Wave Turbulence Driven Heating and Solar Wind Acceleration

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