1. Turbulent transport in a structured three-dimensional solar wind
Daikou Shiota
National Institute of Information and Communications Technology (NICT), Japan
Institute for Space-Earth Environmental Research (ISEE), Nagoya University, Japan
Collaboration with :
G. P. Zank, L. Adhikari, P. Hunana (University of Alabama in Huntsville)
D. Telloni (INAF—Astrophysical Observatory of Torino)
R. Bruno (INAF-IAPS Istituto di Astrofisica e Planetologia Spaziali)
Parker Solar Probe Solar Working Group Meeting
Joint with the Solar Obiter SWT
＠APL, Johns Hopkins University,

2. Outlines Introduction Global MHD simulation of solar wind
Scale-separated MHD equations
MHD model coupled with turbulence transport (Shiota , ApJ, 837, 75)
Future direction for PSP and SO observations
Summary

3. Turbulence in solar wind
The spatial scale of turbulence eddy
Turbulence in solar wind with a wide range of spectrum has been observed in many in situ measurements as fluctuations in interplanetary magnetic field (IMF) and velocity since the pioneering studies (Coleman 1968, Belcher & Davis 1971).
global
scale
Turbulence
Bruno & Carbone (2013)

4. Radial evolution of turbulence
0.3AU
0.7AU
0.9AU
As solar wind propagates distant from the Sun Intensity of turbulence ↓ ↑ expansion of solar wind Correlation length ↑ The spatial scale of turbulence eddy becomes larger
1.4AU
4.8AU
Bruno & Carbone (2013)

5. Solar wind structure and turbulence
Solar wind shows a large variation in speed that may reflect the difference in energy deposition (heating and acceleration), in which turbulence or waves may play significant roles. Difference in turbulence can be produced due to the radial and lateral inhomogeneity of global magnetic field and solar wind plasma distribution.
McComas+ 2008
Coupling between global solar wind and turbulence

6. Motivation of this study
Because of the close coupling of turbulence and solar wind, a comprehensive model describing not only turbulence but also the large-scale inhomogeneity of the solar wind and the interplanetary magnetic field (IMF) is necessary to understand the physics of solar wind and energetic particles.
However, it is difficult, if not impossible currently, to perform a direct MHD simulation that can capture both the full range of the fluctuations and the entire evolution during the propagation in the 3D heliosphere because it requires extremely huge amount of numerical resource.
We have been developing MHD simulation coupled with a theoretical model of transport of turbulence to capture the entire evolutions of solar wind and turbulence.

7. 3D MHD simulation of solar wind (SUSANOO-SW)
Numerical domain in
25 Rs ≦ r ≦ 425 Rs (~ 2 au)
Yinyang Grid
(202 × 68 × 192 × 2)
Inner boundary
solar wind map
rotating and updating
Planets are revolving
Heliographic inertial coordinate
Solar wind map on the ecliptic plane
g= 1.46
Colors:　velocity on ecliptic plane　
White surface: 　neutral sheet
(Shiota )

8. Coronal magnetic field and solar wind velocity
Photospheric magnetic field map
Potential field source surface (PFSS) model
Solar wind map at 25 solar radii
Wang-Sheeley-Arge
(WSA) 2000 formula
(Arge & Pizzo 2000)
+Helios Obs.
(Hayashi+ 2003)
(Shiota )

9. Time-varying inner boundary condition
A time series of photospheric magnetic field maps (one map per day)
⇒ A time series of solar wind maps for the inner
boundary condition of MHD simulation t t
(Shiota )

10. Solar wind in 2013~2014 blue: Earth, green: Jupiter, red: Mars,
orange: Venus, light blue: Mercury
(Shiota )

11. Solar wind in 2007 at Earth position
Velocity
Azimuth angle
of IMF
in situ measurement
MHD simulation
kinematic model
Day of Year
(Shiota )

12. Automated forecast system (SUSANOO)
Activity probability
Radiation belt flux time profile
Solar wind time profile
Paying hommage to ENLIL
Earth

13. Scale separation of MHD equations
global - fluctuation

14. Equation of fluctuation transport
Transports of oppositely propagating modes are considered
Elsässer variables (Elsässer 1950)
u, b: fluctuations in velocity and magnetic field
Scale-separated MHD equation
(Zhou & Matthaeus 1990, Marsh & Tu 1989,1990)
Nonlinear
dissipation
Source
term
Global structures of flow and magnetic field

15. Turbulence transport model (Zank + 2012)
Zank + (2012) proposed a nonlinear model describing the transport of MHD turbulence.
Consider three types of moment of the small scale fluctuation (unit: energy per unit mass)
Forward propagating mode
global
scale
Turbulence
Backward propagating mode
Residual energy

16. Turbulence transport model (Zank + 2012)
They also consider more three variables that describe the corresponding correlation lengths
global
scale
Turbulence l

17. Turbulence transport equations
(Zank , Shiota )
where
These equations are solved with the global MHD simulation of solar wind

18. 3D structure of IMF and bimodal solar wind
(Shiota )
In order to investigate the effect of inhomogeneity of back ground solar wind, we obtained a steady state of solar wind a simple configuration ‘tilted dipole’ to mimic a solar minimum heliosphere. Magnetic field: a tilted dipole magnetic field whose magnetic axis is inclined at 30 degrees from the rotational axis of the Sun Bimodal solar wind: Solar wind speed model same as in Shiota+ (2014) (SUSANOO-SW) Specific heat ratio g= 5/3 Numerical domain: 0.3AU < r < 6AU

19. Boundary condition and Source terms
(Shiota )
Spherically symmetric boundary condition
Spherically symmetric sources of turbulence

20. 2D distribution on the Equatorial plane
(Shiota )

21. Synthetic profile of turbulence intensity
(Shiota )
Turbulence intensities are computed along the trajectories of Helios 2 and Ulysses.
Outward propagating mode
Fast wind
Slow wind
Inward propagating mode
Helios 2
1976
Ulysses
1995 Apr
Jul
Intensity [(km/s)2]
(Adhikari )
The synthetic profiles show reasonable agreements with the observations.

22. Profiles of other turbulence variables
(Shiota )
Normalized Cross helicity
Normalized residual energy
Alfven ratio
Velocity fluctuation
Correlation length for outward prop.
Correlation length for inward prop.
Correlation length for residual energy
Magnetic field fluctuation

23. Discussion: advantage of our model
The current model can capture the distribution of turbulence caused due to the interaction of 3D inhomogeneous solar wind (all input are spherical symmetric).
Based on the numerical results, the synthetic profiles of the turbulence variables were calculated along the trajectories of Helios and Ulysses. The radial profiles of the distributions show reasonable agreement with the observed radial dependences.
The based MHD simulation of solar wind is executed everyday for use in space weather forecast The future completed simulation can be operated similarly and predict in situ observation of solar wind and turbulence. The simulation with a realistic condition of background solar wind enables us to validate the result with direct comparison with in situ observations.

24. Discussion: necessary improvements
In the current model, we specified turbulence variables on the inner boundary (0.3AU) obtained from Helios observations (Adhikari +2015) for fast wind. It remains unclear those values for slow wind and how those turbulence is generated and propagates in the region before 0.3 AU.
The result of the current model indicate the existence of turbulence sources in the interplanetary space to explain the observed turbulence intensities. The heating of MHD plasma as a result of the dissipation of turbulence not included in the current model More self-consistent couplings of these processes are important for the comprehensive understanding of role of turbulence.

25. Future direction for PSP and SO observation
Strictly speaking, the turbulent transport model used in this study is valid in high beta plasma (valid in the region distant from the Sun) because of the assumption of incompressibility.
Zank + (2017) developed a nearly incompressible (NI) MHD turbulence formalism for a coupled two-component 2D (advected) and slab (propagating) turbulence. The model is suitable for the beta ~1 and beta <1 condition as in solar wind and corona. (Dr. Peter Hunana will give a talk on the NI MHD turbulence in this afternoon).
In order to model the corona region (inner than the boundary of the current model), we have been developing a MHD model to applying the NI MHD turbulence transport.

26. Coronal magnetic field and solar wind velocity
Photospheric magnetic field map
Potential field source surface (PFSS) model
Solar wind map at 25 solar radii
Wang-Sheeley-Arge
(WSA) 2000 formula
(Arge & Pizzo 2000)
+Helios Obs.
(Hayashi+ 2003)
In our MHD model, solar wind speed is specified reflecting global magnetic field structure of the Sun.
Similarly, turbulence variables on the inner boundary would be specified reflecting global magnetic field structure of the Sun.
(Shiota )

27. Fine magnetic structure and injection of turbulence
Tsuneta Ito , Shiota
McComas+ 2008
Hinode revealed complex distribution of magnetic field on the photosphere. The 3D fine structure can vary reflecting the relation to global magnetic field.
It is required to develop a model of turbulence injection according to the relationship with the global magnetic structure.
The validation with PSP and SO observations are expected to improve the model and yield greater understanding of turbulence in the solar wind.

28. Summary
To investigate the interaction with background inhomogeneity and the turbulence sources, we have been developing a 3D MHD model that includes the transport and dissipation of turbulence using the theoretical model (Zank ).
We calculated a steady state of solar wind and turbulence assuming a simple bimodal solar wind configuration with a tilted dipole to mimic a solar minimum condition. We compared the profiles of the predicted turbulence distribution with in situ measurements made by the Helios and Ulysses spacecraft, finding that the synthetic profiles of the turbulence intensities show reasonable agreements with observations.
Inner boundary conditions of turbulence variables would be specified reflecting global magnetic structure and turbulence injection. The validation with PSP and SO observations are expected to improve the model and yield greater understanding of source and behavior of turbulence in the solar wind.