ON THE INTERPLAY BETWEEN MICROKINETICS

AND

TURBULENCE IN SPACE PLASMAS

Ramiz A. Qudsi

Dept. of Phys. and Astronomy, University of Delaware,  DE

21 June, 2021

ahmadr@udel.edu

Advisor: Bennett A. Maruca

In this talk

  • Plasma and why it is important to study.
  • Different kinds of plasmas
  • How we study them
  • Instabilities in a plasma
  • Intermittency in plasmas
    • Origin
    • Measuring it
    • Consequence
  • Interplay between linear and nonlinear process
  • Magnetic field topology reconstruction
  • Conclusion

https://slides.com/qudsi/thesis/

Plasma

  • Ionosphere
  • Terrestrial Magnetosheath
  • Solar wind
    • At 1 au
    • Inner Heliosphere (0.2 au)
    • Outer Heliosphere
  • Interstellar Medium (ISM)
  • Intergalactic Medium (IGM)

99.9% of the observable universe is in plasma state

Simulations

  • PIC
  • MHD
  • Hybrid

Experiment/Lab Plasmas

Space Plasmas

https://news.engin.umich.edu/2018/08/the-end-of-the-mission/

  • It is the fourth state of matter.
  • Consists of charged particles and is generally neutral.

https://en.wikipedia.org/wiki/Magnetosphere

Interaction between Solar Wind and Earth's Magnetic Field

1) Bow shock.

 

2) Magnetosheath.

 

3) Magnetopause.

 

4) Magnetosphere.

 

5) Northern tail lobe.

 

6) Southern tail lobe.

 

7) Plasmasphere.

Typical Values

0.2 au 1 au Magnetosheath
Magnetic Field     70 5 20
Ion-density     150 5 30
Ion-speed     400 450 250
Ion-temperature            1 3 2.5

$$\rm{(cm^{-3})}$$

$$(\rm{nT})$$

$$\rm{(km/s)}$$

$$\rm{(10^6K)}$$

Studying Plasma

Vlasov Equation

\frac{\partial \mathnormal{f}_{j}}{\partial t} + \vec{v} \cdot \nabla_{\vec{x}}\mathnormal{f}_{j} + \frac{q}{m}\left(\vec{E} + \vec{v} \times \vec{B}\right) \cdot \nabla_{\vec{v}} \mathnormal{f}_{j} = 0
\frac{d \vec{P}}{d t} = m \frac{d^2 \vec{x}}{d t^2}

Equation of Motion

\nabla \cdot \vec{E} = \frac{\rho}{\epsilon_0} \\ \nabla \cdot \vec{B} = 0 \\ \nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t} \\ \nabla \times \vec{B} = \mu_0 \vec{J} + \frac{1}{c^2} \frac{\partial \vec{E}}{\partial t}

Maxwell's Equation

\mathcal{F}_n(\mathbf{x},\mathbf{v},t) = \sum_{i=1}^{n}\delta \left(\mathbf{x}-\mathbf{x}_i(t)\right)\delta \left(\mathbf{v}-\mathbf{v}_i(t)\right)
\frac{d}{dt}\left(\mathcal{F}_n(\mathbf{x},\mathbf{v},t)\right) = 0
m_i \frac{d\mathbf{v}_i}{dt}= q_i\left(\mathbf{E}_\mu + \mathbf{v}_i \times \mathbf{B}_\mu\right)
\mathnormal{f}_j(\mathbf{x}, \mathbf{v}, t) = \mathnormal{f}_j^0(\mathbf{x}, \mathbf{v}) + \mathnormal{f}_j^1(\mathbf{x}, \mathbf{v}, t)\\ \hspace{9em} = \mathnormal{f}_j^0(\mathbf{x}, \mathbf{v}) + \mathnormal{f}_j^1(\mathbf{k}, \omega, \mathbf{v})~e^{(i(\mathbf{k}\cdot \mathbf{x} -\omega t))}
\mathbf{B}_{\mu} = \left<\mathbf{B}_{\mu}\right> + \delta\mathbf{E}_{\mu} \\ = \mathbf{B}_{\mu} + \delta\mathbf{B}_{\mu}
\mathbf{E}_{\mu} = \left<\mathbf{E}_{\mu}\right> + \delta\mathbf{E}_{\mu} \\ = \mathbf{E} + \delta\mathbf{E}_{\mu}
\frac{\partial \mathnormal{f}}{\partial t} + \mathbf{v} \cdot \nabla_{\mathbf{x}}\mathnormal{f} + \frac{q}{m}\left(\mathbf{E} + \mathbf{v} \times \mathbf{B}\right) \cdot \nabla_{\mathbf{v}} \mathnormal{f} = \left(\frac{\partial \mathnormal{f}}{\partial t}\right)_c
\frac{\partial \mathnormal{f}}{\partial t} + \mathbf{v} \cdot \nabla_{\mathbf{x}}\mathnormal{f} + \frac{q}{m}\left(\mathbf{E} + \mathbf{v} \times \mathbf{B}\right) \cdot \nabla_{\mathbf{v}} \mathnormal{f} = \frac{q}{m}\left<\left(\delta\mathbf{E} + \mathbf{v} \times \delta \mathbf{B}\right) \cdot \nabla_{\mathbf{v}} \mathcal{F}\right>
\mathcal{F} = \left<\mathcal{F}\right> + \delta\mathcal{F} \\ = \mathnormal{f} + \delta\mathcal{F}
m_i \frac{d\mathbf{v}_i}{dt}= q_i\left(\mathbf{E}_\mu + \mathbf{v}_i \times \mathbf{B}_\mu\right)
\frac{\partial \mathcal{F}}{\partial t} + \mathbf{v} \cdot \nabla_{\mathbf{x}} \mathcal{F} + \frac{q}{m}\left(\mathbf{E}_\mu + \mathbf{v}_i \times \mathbf{B}_\mu\right) \cdot \nabla_{\mathbf{v}} \mathcal{F} = 0
\frac{\partial \mathcal{F}}{\partial t} + \frac{\partial \mathbf{x}}{\partial t} \cdot \frac{\partial \mathcal{F}}{\partial \mathbf{x}} + \frac{\partial \mathbf{v}}{\partial t} \cdot \frac{\partial \mathcal{F}}{\partial \mathbf{v}} = 0
\mathrm{det}(\mathbf{D}(\mathbf{k},\omega)) = 0
\mathbf{D}(\mathbf{k},\omega) = \left(\omega^2 - c^2\,k^2\right)\mathbf{I} + c^2\,\mathbf{k}\,\mathbf{k} + c^2\,k^2\,\sum_j\,\mathbf{S}_j(\mathbf{k},\omega)
\mathbf{D}(\mathbf{k},\omega)\cdot \mathbf{E}^1(\mathbf{k},\omega) = 0
\mathbf{\Gamma}_j^1(\mathbf{k}, \omega) = - \frac{i\,\epsilon_0\,k^2\,c^2}{q_j\,\omega}\,\mathbf{S}_j(\mathbf{k},\omega)\cdot \mathbf{E}^1(\mathbf{k},\omega)
\mathbf{J}^1(\mathbf{k}, \omega) = \sum_j q_j\,\mathbf{\Gamma}_j^1(\mathbf{k}, \omega)
\Gamma_j^1(\mathbf{k}, \omega) = \int_{-\infty}^{\infty}d^3 v\,\mathbf{v}\,\mathnormal{f}_j^1(\mathbf{k}, \omega, \mathbf{v})
\mathbf{E}(\mathbf{x}, t) = \mathbf{E}^0(\mathbf{x}) + \mathbf{E}^1(\mathbf{x}, t)\\ \hspace{8em}= \mathbf{E}^0(\mathbf{x}) + \mathbf{E}^1(\mathbf{k},\omega)~e^{(i(\mathbf{k}\cdot \mathbf{x} -\omega t))}
\mathbf{B}(\mathbf{x}, t) = \mathbf{B}^0(\mathbf{x}) + \mathbf{B}^1(\mathbf{x}, t)\\ \hspace{8em} = \mathbf{B}^0(\mathbf{x}) + \mathbf{B}^1(\mathbf{k},\omega)~e^{(i(\mathbf{k}\cdot \mathbf{x} -\omega t))}

Dispersion Relation

   is the distribution function of plasma for species j

$$f_j$$

Linear Dispersion Equation

Vlasov Equation

\mathnormal{f}_j(\vec{x}, \vec{v}, t) = \mathnormal{f}_j^0(\vec{x}, \vec{v}) + \mathnormal{f}_j^1(\vec{x}, \vec{v}, t)\\ = \mathnormal{f}_j^0(\vec{x}, \vec{v}) + \mathnormal{f}_j^1(\vec{k}, \omega, \vec{v})~e^{i(\vec{k}\cdot \vec{x} -\omega t)}

Linearization

real
\gamma_{\max}
\omega_{\rm r} + i \gamma

Maximum value of growth rate of a given mode for all k and directions

\overbrace{\mathnormal{f}_j^1(\vec{k}, \omega, \vec{v})~e^{i(\vec{k}\cdot \vec{x} -\omega_{\rm r} t)}~e^{\gamma t}}
\underbrace{\hspace{8.5em}}
instability growth rate
\left(\gamma > 0 \right)
imaginary
(Marsch, JGRL-1982)

VDF: Probability distribution function of phase space density

$$\hat{B}$$

Temperature Anisotropy:

Ratio of perpendicular and parallel temperatures

$$R_j = \frac{T_{\perp j}}{T_{\parallel j}}$$

Beta:

Ratio of thermal and magnetic pressure

$$\beta_{\parallel j} \equiv \frac{n_j\,k_{\rm B}\,T_{\parallel j}}{B^2\,/\,(2\,\mu_0)}$$

Solar Wind, 1 au

Magnetosheath

(Maruca, ApJ-2018)

Solar Wind, 1 au

Magnetosheath

(Hellinger, GRL-2006)
(Maruca, ApJ-2018)

Solar Wind, 1 au

Magnetosheath

(Qudsi, In-prep)
(Huang, ApJS-2020)

3-D PIC simulation

Solar Wind, 0.2-au

(Qudsi, ApJ-2020)

2.5-D PIC simulation

3-D PIC simulation

(Qudsi, ApJ-2020)

MMS Observation

(Qudsi, ApJ-2020)

Intermittency comparison between spacecraft observation and simulation

MMS

Wind

Measuring Intermittency

Intermittency: Burstiness

Distribution is not uniform and has localized structures

Measuring intermittency

\mathcal{I}(t, \tau) = \frac{|\Delta \mathbf{B}(t, \tau)|}{\sqrt{\langle |\Delta \mathbf{B}(t, \tau)|^2 \rangle_\Delta}}
\Delta \mathbf{B}(t, \tau) = \mathbf{B}(t+\tau) - \mathbf{B}(t)
(Greco, GRL-2008)
\ell = v \cdot \tau

Lag in distance

\mathcal{I} > 2.4 \implies

Non-Gaussianity

\tau

: Time lag

(Osman, PRL-2012)
\tau \ll \tau_{\rm correlation}\\ \Delta \gg \tau_{\rm correlation}

What value of    and     one should choose?

\tau \hspace{1.6em} \Delta

PVI

(Assuming Taylor hypothesis)

PSP : Encounter 1 (second half)

(Qudsi, ApJS-2020)
\widetilde{T}_p( \delta t, \theta_1, \theta_2) = \langle T_p(t_{\mathcal{I}}+\delta t)| \theta_1 \leq \mathcal{I}(t_\mathcal{I}) < \theta_2 \rangle

Conditional Temperature Averages

(Qudsi, ApJS-2020)

Non-linear Processes

Turbulence

  • microinstability processes
  • strongly nonlinear intermittent processes

Distortion of VDF

Microinstabilities

Intermittency

Turbulence

We estimate it from the spectral amplitude near the ion-inertial scale

Non-linear time scale

$$(\omega_{nl})$$

$$\omega_{\rm nl} \left(\vec{r}\right) \sim { \delta b_{\ell}}/{\ell}$$

 

$$\delta b_{\ell} = \left \lvert\hat{\boldsymbol{\ell}} {\cdot} \left[\vec{b} \left(\vec{r} + \vec{\ell}\right) - \vec{b} \left(\vec{r}\right)\right]\right\lvert$$

 

$$ b = \left\lvert \vec{B}/\vec{V_A}\right\lvert, \ell \sim 1/k_{\max}$$

\Gamma_{\max} = \max( \gamma_{\rm \max,\,cyclotron}, \gamma_{\rm \max,\,mirror}, \\ \hspace{6.2em} \gamma_{\rm \max,\,\parallel firehose}, \gamma_{\rm \max,\,\nparallel firehose})

Comparison between      and       :

$$\omega_{\rm nl}$$

$$\Gamma_{\max}$$

Comparison between      and

$$\omega_{\rm nl} \hspace{2em} \Gamma_{\max}$$

(Qudsi2021a, in prep)

MMS

Wind

Comparison between      and

$$\omega_{\rm nl} \hspace{2em} \Gamma_{\max}$$

(Bandyopadhyay, PRL-2021,                            
under review)

Comparison between      and

$$\omega_{\rm nl} \hspace{2em} \Gamma_{\max}$$

(Qudsi2021a, in prep)

Comparison between      and

$$\omega_{\rm nl} \hspace{2em} \Gamma_{\max}$$

(Qudsi2021a, in prep)

Solar Wind, 1 au

(Maruca, ApJ-2018)

Solar Wind, 1 au

Magnetosheath

For any given system how do we figure out which one is most relevant?

A lot better understanding of turbulence cascade in plasmas

Complete 3D structure of interplanetary magnetic field

Comparison between      and

$$\omega_{\rm nl} \hspace{2em} \Gamma_{\max}$$

(Qudsi2021a, in prep)

Magnetic Field Reconstruction

Gaussian Process Regression

It is a probabilistic data imputation method

$$m(\mathbf{x}) = \mathbb{E}[f(\mathbf{x})]$$

Mean function

$$k(\mathbf{x}, \mathbf{x'}) = \mathbb{E}[(f(\mathbf{x}) - m(\mathbf{x}))(f(\mathbf{x'}) - m(\mathbf{x'}))]$$

Covariance function

$$f(\mathbf{x}) \sim \mathcal{GP}\left(m(\mathbf{x}), k(\mathbf{x}, \mathbf{x'})\right)$$

Gaussian Processes

Constant

Linear

RBF

Matern

Kernels

Reconstructed Magnetic Field

(Maruca, Frontiers-2021)

Reconstructed Magnetic Field

Conclusion:

  • Linear instabilities are distributed intermittently as are coherent structures for all the cases.
  • Linear instabilities as well as the heating rates in plasmas are amplified by the presence of intermittency.
  • For all cases studied, non-linear processes are faster than the linear time scale, though in phase space their distribution is a little more complicated.
  • Though we showed an interplay between the two processes, a better understanding of type of turbulence/cascade is essential to conclusively predict denouement of the competition between the two.
  • Knowledge of full 3D structure of interplanetary magnetic field will help with this.
  • We showed that we need at least 24 spacecraft to reconstruct magnetic field with sufficient accuracy.

Acknowledgements

Questions?

https://slides.com/qudsi/thesis/

https://xkcd.com/1403/

Thank You! :)