mhd waves and turbulence in the expanding solar wind · mhd waves and turbulence in the expanding...

MHD waves and turbulence in the expanding solar wind

Chen Shi1, Anna Tenerani2, Marco Velli1, Victor Réville1

1, Earth, Planetary, and Space Sciences , UCLA

2, Department of Physics, UT Austin

SHINE Conference 2019, August 4th

Outline

Structure of the solar wind

Basics of turbulence: Kolmogorov’s theory

Solar wind turbulenceObservations

Equation & Kraichnan’s theory

Methods for study of solar wind turbulence

Expanding-Box-Model simulations

Conclusion

SHINE 2019 STUDENT DAY Solar Wind Turbulence 2

Solar Wind

Model of the wind far from the sun:Velocity is almost radial: 𝑼 = 𝑈 𝒆𝒓

Already accelerated within 0.2-0.3 AU: 𝑈 = 𝐶𝑜𝑛𝑠𝑡 beyond.

Spiral magnetic field: 𝑩 = 𝐵𝑟 𝒆𝒓 + 𝐵𝜙 𝒆𝝓

Expansion effect:

• 𝑛 ∝ 𝑟−2

• 𝑇 ∝ 𝑟−𝑎 with 𝑎 ≤ 1

• 𝐵𝑟 ∝ 𝑟−2

• 𝐵𝜙 ∝ 𝑟−1

Alfvén point 𝑟𝑐:

• 𝑈 𝑟𝑐 =𝐵𝑟

𝜇0𝜌(𝑟𝑐)

SHINE 2019 STUDENT DAY Solar Wind Turbulence

Alfvén point

3

Solar Wind

Fast windOrigin: coronal holes (high latitude)

Fast speed: > 600 km/s

Low density

High temperature

Slow windOrigin: streamer belt (low latitude)

Slow speed: 300 − 500 km/s

High density

Low temperature


Solar Wind

Stream interaction


Illustration of the corotating-interaction region

slow wind

fast wind

compression region

5

Basics on turbulence

The most important concepts in turbulence studyNonlinear interaction:

• Leads to energy transfers between scales

• ℱ𝒌 𝑨 ⋅ 𝛁𝑩 = 𝑑𝒑 𝑖 𝒌 − 𝒑 ⋅ 𝑨𝒑 𝑩𝒌−𝒑

Scales:

• Energy-containing scales 𝑙𝑒: Energy is injected only at these scales

• Energy-dissipation scales 𝑙𝑑: Energy is dissipated only at these scales

• Inertial scales 𝑙: Energy is neither injected nor dissipated but flows through scales (cascade)

• 𝑙𝑑 ≫ 𝑙 ≫ 𝑙𝑒 Energy transfer (dissipation) rate:

• 𝜀: Energy dissipation/transfer rate per unit mass

• Dimension: 𝑣2/𝜏

• Constant throughout the inertial range


Basics on turbulence

Energy (power) spectrum densityDefinition: 𝐸 𝑘 = 𝑣 𝑘 2𝑑𝑘, where 𝑣 𝑘 = ℱ 𝑣 𝑥 =

1

2𝜋 𝑑𝑥 𝑣(𝑥)𝑒−𝑖𝑘𝑥

Estimate of the perturbation amplitude on scale 𝑘: 𝑣 𝑘 ∼ 𝑘𝐸 𝑘

Scaling of 𝐸 𝑘 in inertial range is determined by the nonlinear interactions

Example: non(weakly)-magnetized fluid

𝜀 ∼𝑣2 𝑘

𝜏𝑁𝐿(𝑘)

From N-S equation: 𝜕𝑣

𝜕𝑡∼ 𝑣

𝜕𝑣

𝜕𝑥→

𝑣

𝜏𝑁𝐿∼ 𝑘𝑣2 → 𝜏𝑁𝐿 𝑘 ∼

1

𝑘𝑣 𝑘

𝜀 ∼ 𝑘𝑣3 → 𝐸 𝑘 ∼ 𝜀2

3𝑘−5

3


Observations

Early observations


Coleman, 19681

Russell, 19722

Feature of Turbulence

𝑓−1.2

Belcher and Davis, 19713

Outward Alfvén waves

1: Coleman, 1968, ApJ, 153, 3712: Russell, 1972, Comments on the measurement of power spectra of the interplanetary magnetic field3: Belcher and Davis, 1971, JGR, 76, 16

8

Modern views

The solar wind turbulence is most-of-the-time Alfvénic: Nearly-constant density and |𝑩| (incompressible) Dominated by outward-propagating Alfvén waves Inward-propagating Alfvén waves are also present but weaker

Nonlinear interaction between outward/inward waves is needed

Convective structures also exist in the solar wind, especially the slow wind

Questions:Where and how are the outward/inward waves generated?

• Outward waves may have solar origin

• Inward waves cannot have solar origin but may be generated by sheared streams, instabilities, etc.

Why the dominance of the outward waves decays with radial distance? How is the imbalance between magnetic and kinetic energies produced? The compressible MHD turbulences? Anisotropy of the turbulence?


Basic equation

Scale-separated, incompressible, MHD equation𝜌 = 𝜌0, 𝒖 = 𝒖𝟎 + 𝒖𝟏, 𝒃 = 𝒃𝟎 + 𝒃𝟏

Elsässer variables: 𝒛± = 𝒖𝟏 ±𝒃𝟏

4𝜋𝜌

(momentum equation) ± (induction equation/ 4𝜋𝜌)


𝜕𝒛±

𝜕𝑡+ 𝒖𝟎 ∓ 𝑽𝑨𝟎 ⋅ 𝛻𝒛± + 𝒛∓ ⋅ 𝛻 𝒖𝟎 ± 𝑽𝑨𝟎 ∓ 𝒛± ⋅ 𝛻 ln 𝜌 𝑽𝑨𝟎 ∓ 𝒖𝟎 ± 𝑽𝑨𝟎 ⋅ 𝛻 ln 𝜌

𝒛+ − 𝒛−

2= 𝑵𝑳±

𝑵𝑳± ∼ 𝒛∓ ⋅ 𝛻𝒛±

Propagationinhomogeneous background: linear evolution and couplingNonlinear interaction

10

Kraichnan Spectrum1: 𝐸 𝑘 ∼ 𝑘−3

2

Alfvénic turbulence with strong background field 𝑩𝟎

𝑩𝟎 weakens the nonlinear interaction 𝜏𝐴 = 𝑙/𝑉𝐴0

𝜀 ∝ 𝜏𝐴Dimensional analysis

• Write 𝜀 ∼ 𝜏𝐴𝑧𝛼𝑘𝛽 with 𝛼, 𝛽 to be determined

• 𝜀 has dimension 𝑣3𝑘

• 𝛼 = 4, 𝛽 = 2

• Substitute 𝑧 = 𝑘𝐸(𝑘)

• 𝐸 𝑘 ∼ 𝜀𝑉𝐴0

1

2𝑘−3

2


𝑩𝟎

𝑙 = 1/𝑘

𝑉𝐴0 𝑉𝐴0

1: Kraichnan, 1965, Physics of Fluids, 8, 1385

11

Two important parameters

Normalized cross helicity

𝜎𝑐 =𝒛− 2− 𝒛+

2

𝒛− 2+ 𝒛+ 2 = −2𝒖⋅𝒗𝑨

𝒖 2+ 𝒗𝑨2

Relative abundance of the energies of outward/inward Alfvén waves

Normalized residual energy

𝜎𝑟 =𝒖 2− 𝒗𝑨

2

𝒖 2+ 𝒗𝑨2 =

2𝒛+⋅𝒛−

𝒛+ 2+ 𝒛− 2

Relative abundance of the kinetic/magnetic energies


𝑧− = 𝑢1 −𝑏1

4𝜋𝜌: outward wave

𝑧+ = 𝑢1 +𝑏1

4𝜋𝜌: inward wave

Methods to study solar wind turbulence

In-situ measurements

Model1,2

derive equations for the averaged second-order moments of 𝒛±,

𝒛± 2& 𝒛+ ⋅ 𝒛− , which evolve on large (stream) scales

Direct Numerical Simulation (DNS)


1: Zank et al., 2012, ApJ, 745:352: Tu et al., 1984, JGR 89:A11

13

Direct Numerical Simulation

Solve the full MHD equation (or two fluid, kinetic, etc.)

Solar wind is spherically expanding and is huge

Impossible to simulate the whole heliosphere

Expanding Box Model (EBM)A rectangular box moving with the radial mean flow

Angular size of the box is constant

Neglect curvatures and assume cartesian coordinates locally


Tenerani and Velli, 20171

𝑼 = 𝑼𝟎 + 𝒖 = 𝑈𝑟 𝒆𝒓 + 𝒖𝑈𝑥 ≈ 𝑢𝑥 + 𝑈0

𝑈𝑦 ≈ 𝑢𝑦 + 𝑈0

𝑦

𝑅(𝑡)

𝑈𝑧 ≈ 𝑢𝑧 + 𝑈0

𝑧

𝑅(𝑡)

1: Tenerani and Velli, 2017, ApJ, 843(1), 26

14

Expanding Box Model

Translation and normalization: 𝑥′ = 𝑥 − 𝑅 𝑡 , 𝑦′ =𝑅0

𝑅 𝑡𝑦, 𝑧′ =

𝑅0

𝑅(𝑡)𝑧


𝜕𝜌

𝜕𝑡+ 𝛻 ⋅ 𝜌𝒖 = −2

𝜌

𝜏𝜕𝑝

𝜕𝑡+ 𝒖 ⋅ 𝛻𝑝 + 𝜅𝑝 𝛻 ⋅ 𝒖 = −2𝜅

𝑝

𝜏

𝜌𝜕𝒖

𝜕𝑡+ 𝒖 ⋅ 𝛻𝒖 + 𝛻 𝑝 +

1

2𝐵2 − 𝑩 ⋅ 𝛻𝑩 = −

𝜌

𝜏

0 0 00 1 00 0 1

𝒖

𝜕𝑩

𝜕𝑡+ 𝒖 ⋅ 𝛻𝑩 − 𝑩 ⋅ 𝛻𝒖 + 𝛻 ⋅ 𝒖 𝑩 = −

1

𝜏

2 0 00 1 00 0 1

𝑩

𝛁 =1

𝜕𝑥′,𝑅0

𝑅 𝑡

𝜕

𝜕𝑦′,𝑅0

𝑅 𝑡

𝜕

𝜕𝑧′, 𝜏 =

𝑅 𝑡

𝑈𝑟

𝜌 ∝ 𝑅−2

𝑝 ∝ 𝑅−2𝜅

𝑢𝜃,𝜙 ∝ 𝑅−1

𝐵𝑟 ∝ 𝑅−2

𝐵𝜃,𝜙 ∝ 𝑅−1

Periodic boundary conditions

15

Expanding Box Model

Corotating coordinates

Able to simulate the compression region: Initial fast-slow stream structure (see the 1D run):

𝑩𝟎 = 𝐵0 𝑒𝑥′ , 𝒖𝟎 = 𝑢0 𝑦′ 𝑒𝑥, 𝑇0 𝑦′ , 𝜌0 𝑦′

Parameters𝑅0 = 30𝑅𝑠, 𝐿𝑥 = 10𝑅𝑠, 𝐿𝑦 = 𝜋𝑅0: half circle

initial spiral angle 𝛼 = 8.1°, uniform 𝐵 = 250 𝑛𝑇

𝑝 = 5 𝑛𝑃𝑎, 𝜅 = 1.5

𝑈𝑟 = 464 𝑘𝑚/𝑠

Fast wind: 700 𝑘𝑚/𝑠, 140 𝑐𝑚−3

Slow wind: 340 𝑘𝑚/𝑠, 360 𝑐𝑚−3


Test: 1D simulation

No waves are added


Longitude

17

Initial condition: Alfvénic wave band

Circularly-polarized Alfvén waves

Initial energy spectrum: 𝑘𝑥′−1

𝑁𝑚𝑎𝑥 = 16

𝑟𝑖𝑜: in/out amplitude ratio


Outward-dominant 𝑟𝑖𝑜 = 0.2

Phase-mixing & depletion of wave energy


𝑍-component of the outward Elsässer variable 𝑧𝑜𝑢𝑡,𝑧

Lon

gitu

de

19



longitude longitude

𝜎𝑐 =𝑧− 2 − 𝑧+ 2

𝑧− 2 + 𝑧+ 2𝜎𝑟 =

𝑢 2 − 𝑣𝐴2

𝑢 2 + 𝑣𝐴2

𝐸𝑇 =1

2( 𝑧− 2 + 𝑧+ 2)

f c ss r

Total energy Density fluctuation

outward

inward

Pure kinetic

Pure magnetic



Power spectra of 𝒛 along 𝑦′ corrected by 𝑘𝑦′5/3

Power spectra of 𝒛 along 𝑥′ corrected by 𝑘𝑥′5/3

21

Radial evolution of Elsässer energies


Outward-dominant Outward-dominant𝛼 = 0

Balanced Inward-dominant

Radial evolution of 𝐸𝑜𝑢𝑡 and 𝐸𝑖𝑛 corrected by 𝑅/𝑅0

22

Conclusion of the simulations

Shear between fast and slow streams: depletes wave energy and decreases cross helicity

Compression between streams: facilitates the process

Kolmogorov-like spectrum is easier to form in the longitudinal direction (quasi-perpendicular) than in the parallel direction

The radial evolution of the Alfvén wave energy is highly dependent on the regions in the stream structure.


SEE MY POSTER FOR MORE DETAILS

23

SummaryThe solar wind turbulence is most-of-the-time Alfvénic:

Nearly-constant density and |𝑩| Dominated by outward-propagating Alfvén waves Inward-propagating Alfvén waves are also present but weaker

Nonlinear interaction between outward/inward waves is needed

Convective structures also exist in the solar wind, especially the slow wind

Questions:Where and how are the outward/inward waves generated?

• Outward waves may have solar origin

• Inward waves cannot have solar origin but may be generated by sheared streams, instabilities, etc.

Why the dominance of the outward waves decays with radial distance? How is the imbalance between magnetic and kinetic energies produced? The compressible MHD turbulences? Anisotropy of the turbulence?


Future reading

Turbulence: the legacy of AN Kolmogorov, Uriel Frisch, 1995, Cambridge university press

MHD structures, waves and turbulence in the solar wind: Observations and theories, Tu, C. Y. & Marsch, E., 1995, Space Science Reviews, 73(1-2), 1-210

The solar wind as a turbulence laboratory, Bruno, R. & Carbone, V., 2005, Springer



THANK YOU

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mhd waves and turbulence in the expanding solar wind · mhd waves and turbulence in the expanding...

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