Michael Rowan
Adviser: Ramesh Narayan; Co-adviser: Lorenzo Sironi
8-26-2019
Sgr A$^{*}$ (black hole at the center of our galaxy) has Schwarzschild radius $R_{\rm G} \sim 10^{12}\,{\rm cm}$
For the corona of Sgr A$^{*}$, plasma scale is: $c/\omega_{\rm pe} \sim 1\,{\rm cm}$
This is a challenge for numerical (MHD) simulations of black hole environments, which resolve only the macroscopic scales
Plasma physics controls energy dissipation at small scales:
Can turbulence or other instabilities induce reconnection?
Reconnection is well-studied in non-relativistic
and relativistic systems
BH coronae expected to be inbetween these two limits:
$\Rightarrow$ 'transrelativistic'
Magnetic energy $\Rightarrow$ kinetic (heating, acceleration, bulk motion)
Center for Visual computing, University of California Riverside
Magnetic tension drags away field lines at the Alfvén speed:
\[\begin{aligned} \frac{v_{\rm A}}{c} & = \sqrt{\frac{\sigma_{w}}{1+\sigma_{w}}}, {\rm where } \, \sigma_{w} \propto \frac{{\rm magnetic\,pressure}}{{\rm enthalpy\,density}} \end{aligned} \]
Occurs in magnetized plasmas, like the solar corona:
What would reconnection look like in black hole coronae?
How much magnetic energy is dissipated to electrons, ions?
TRISTAN-MP
2D in space, but track all 3 components of momentum
TRISTAN-MP
2D in space, but track all 3 components of momentum
"Plasma-beta," "magnetization," and "guide field strength" describe the initial state of the upstream plasma:
\[\begin{aligned} \beta_{\rm i} = \frac{n_{\rm i} k_{\rm B} T_{\rm i}}{B_{0}^{2}/8 \pi} & = \frac{\rm \color{#f6656c}{thermal\,pressure}}{\rm \color{#6ca0e8}{magnetic\,pressure}} \\ \sigma_{w} = \frac{B_{0}^{2}/4 \pi}{w} & = \frac{\rm \color{#6ca0e8}{magnetic\,pressure\,(\times 2)}}{\rm \color{#5AE769}{enthalpy\,density}} \\ b_{g} = B_{z}/B_{0} & = \frac{\text{Strength of out-of-plane } B}{\text{Strength of in-plane } B} \end{aligned} \]
Tune these in the upstream, measure heating in the downstream
How much are particles heated? Define heating fractions for e$^{-}$, p$^{+}$:
\[\begin{aligned} M_{T\rm e} & = \frac{\theta_{\rm e,down} - \theta_{\rm e,up}}{\frac{m_{\rm i}}{m_{\rm e}} \sigma_{\rm i}}, \, M_{T\rm i} = \frac{\theta_{\rm i,down} - \theta_{\rm i,up}}{\sigma_{\rm i}} \end{aligned} \]
Heating fraction can be written as:
$M_{T\rm e} = \frac{k_{\rm B} T_{\rm e,out} - k_{\rm B} T_{\rm e,in}}{m_{\rm i} v_{\rm A}^{2}}$
(slope of black line in this plot $\rightarrow$)
For wide range of inflow parameters, $M_{T\rm e}$ is constant
$M_{T\rm e} \approx 0.033$ for $m_{\rm i}/m_{\rm e}=25$. When extrapolated to realistic mass ratio $m_{\rm i}/m_{\rm e}=1836,$ $M_{T\rm e} \approx 0.017$
$$\beta_{\rm i}$$ | $$\sigma_{\rm i}$$ | $$\theta_{\rm e}$$ | $$\theta_{\rm i}$$ |
---|---|---|---|
4.9$~\times~$10$^{\rm -4}$ | 0.10 | 0.010 | 2.4$~\times~$10$^{\rm -5}$ |
0.031 | 0.10 | 0.041 | 0.0016 |
0.50 | 0.12 | 0.77 | 0.031 |
2.0 | 0.38 | 9.9 | 0.39 |
B-field initialized in Harris equilibrium:
$\mathbf{B} = B_{0} {\rm tanh}(y/\Delta)\mathbf{\hat{x}}$
B-field initialized in Harris equilibrium:
$\mathbf{B} = B_{0} {\rm tanh}(y/\Delta)\mathbf{\hat{x}}$
B-field initialized in Harris equilibrium:
$\mathbf{B} = B_{0} {\rm tanh}(y/\Delta)\mathbf{\hat{x}}$
B-field initialized in Harris equilibrium:
$\mathbf{B} = B_{0} {\rm tanh}(y/\Delta)\mathbf{\hat{x}}$
Use mixing as a criterion to ID cells where rec. has occured:
$r_{\rm down} < \frac{n_{\rm top}}{n_{\rm tot}}<1-r_{\rm down}$
Focus on 'island' region, where energy of bulk motion from outflows has thermalized:
When gas is compressed adiabatically, internal energy increases and entropy per particle remains constant
How much heating in the downstream is an effect of adiabatic-compression (no increase in entropy)?
\[\begin{aligned} \int_{u^{\rm up}}^{u^{\rm f}} \frac{1}{(\hat{\gamma}(u)-1)u} du - \log \left(\frac{n^{\rm f}}{n^{\rm up}}\right) = 0 \end{aligned} \]
Extract heating by measuring 'quasi-steady' values in the island region; time average heating over $\sim1$ Alfvénic crossing time:
Simulation parameters: $\beta_{\rm i}=0.0078, \sigma_{w}=0.1, T_{\rm e}/T_{\rm i}=0.1$
Simulation parameters: $\beta_{\rm i}=2, \sigma_{w}=0.1, T_{\rm e}/T_{\rm i}=0.1, {\color{lime}{ N_{\rm ppc}=64}}$
Simulation parameters: $\beta_{\rm i} = 0.03, \sigma_{w} =0.1, b_{\rm g}=0.3, 1, 6$
Simulation params: $\beta_{\rm i} = 2, \sigma_{w} =0.1, b_{\rm g}=0.3, 1, 6, {\color{lime}{ N_{\rm ppc}=64}}$
\[\begin{aligned} q_{u\rm e, irr} & = \frac{M_{u\rm e,irr}}{M_{u\rm e,irr} +M_{u\rm i,irr}} \end{aligned} \]
\[\begin{aligned} q_{u \rm e, irr, fit}(\beta_{\rm i}, b_{\rm g}, T_{\rm e}/T_{\rm i}, \sigma_{w}) = \frac{1}{2} (\tanh(0.33 b_{\rm g})-0.4) \\ \times 1.7 \tanh \left( \frac{(1-\beta_{\rm i}/\beta_{\rm i, max})^{1.5}}{(0.42 + T_{\rm e}/T_{\rm i})\sigma_{w}^{0.3}}\right) + \frac{1}{2} \end{aligned} \]
Particle heating in the presence of a guide field looks more similar to heating via turbulence; at small enough scales, turbulence is thought to be mediated by strong guide field reconnection
Track particles from upstream to downstream to assess relative importance of heating mechanisms
\[\begin{aligned} \frac{d \varepsilon_{\rm e}}{dt} = q \mathbf{E} \cdot (\mathbf{v}_{\parallel} + \mathbf{v}_{\rm c} + \mathbf{v}_{\nabla B}) + \frac{\mu}{\gamma} \frac{\partial B}{\partial t} \end{aligned} \]
Curvature heating tends to dominate for weak guide field, whereas $E$-parallel dominates for strong guide field (in this case, the magnetic field is almost straight in the out-of-plane direction)
Guide field suppresses reconnection rate, more variation at low-$\beta_{\rm i}$
KH instability is ubiquitous in nature; here, KH forms in clouds:
Velocity perturbation $\perp$ to shear velocity $\beta_{\rm sh}$ $\rightarrow$ unstable growth
In addition to shear velocity $\beta_{\rm sh}$, consider a magnetic field $B_{0}$ that permeates the plasma
A number of simulations suggest evidence for KH instability at the interface of the jet and wind/corona, which motivates a KH setup with:
Addressing the question of magnetic dissipation via magnetic reconnection requires fully-kinetic PIC simulation
Three-pronged attack to studying KH-induced reconnection:
To answer the first point, solve relativistic magnetohydrodynamic (RMHD) dispersion relation (i.e., polynomial in $\phi_{v_{\rm A}}$) of the form:
\[\begin{aligned} F(\phi_{v_{\rm A}}, \rm{params}) & = 0 \end{aligned} \]
$\phi_{v_{\rm A}} \equiv \omega/(k_{0} v_{\rm A})$ is the dimensionless growth rate of the mode
Im$(\phi_{v_{\rm A}})>0 \rightarrow $ unstable growth
Explicit dispersion relation:
$\beta_{{\rm i}x}$ | Plasma-beta in jet, computed with in-plane $B_{x0,\,\rm{j}}$ |
$\sigma_{wx}$ | Magnetization in jet, computed with in-plane $B_{x0,\,\rm{j}}$ |
$\beta_{\rm sh}$ | Speed of wind relative to jet |
$b_{\rm j}$ | $B_{z0,\,\rm{j}}/B_{x0,\,\rm{j}}$; jet guide field strength |
$b_{\rm w}$ | $B_{z0,\rm{w}}/B_{x0,\,\rm{j}}$; wind guide field strength |
$\rho_{0\rm j}$ | Mass density in jet |
$\rho_{0\rm w}$ | Mass density in wind |
$f$ | $k_{z}/k_{x}$; controls ang. of propagation of perturbation |
Time evolution of Fourier power (left) for RMHD simulation; parameters: $\beta_{\rm sh}=0.8$, $\sigma_{wx}=1$, $\beta_{\rm{i}x}=0.078$, $b_{\rm j}=3$, $b_{\rm w}=0.3$, $\rho_{0\rm{w}}/\rho_{0\rm{j}}=6.7$
Time evolution of density for PIC run; parameters: $\beta_{\rm sh}=0.8$, $\sigma_{wx}=1$, $\beta_{\rm{i}x}=0.078$, $b_{\rm j}=3$, $b_{\rm w}=0.3$, $\rho_{0\rm{w}}/\rho_{0\rm{j}}=6.7$
Time evolution of Fourier power (left) for PIC run; parameters: $\beta_{\rm sh}=0.8$, $\sigma_{wx}=1$, $\beta_{\rm{i}x}=0.078$, $b_{\rm j}=3$, $b_{\rm w}=0.3$, $\rho_{0\rm{w}}/\rho_{0\rm{j}}=6.7$
Dashed lines below correspond to 2D density profiles $\rightarrow$
Relativistic MHD simulations show good agreement with prediction:
Growth rate dependence on density ratio $\rho_{0\rm{w}}/\rho_{0\rm{j}}$ shows that even for $\rho_{0\rm{w}}/\rho_{0\rm{j}}\approx 10^3$, growth is reduced by only a factor of $3$–$4$
Convergence of growth rate with respect to ratio of transition width to box height
PIC simulations and analytical predictions show good agreement
Errorbars computed as standard deviation of measured slopes from five simulations with different random initial conditions
Growth rates for give simulations with different random initial seeds
Time evolution of density for 3D simulation; blue is jet, red is wind
parameters: $\beta_{\rm sh}=0.8$, $\sigma_{wx}=1$, $\beta_{\rm{i}x}=0.001$, $b_{\rm j}=3$, $b_{\rm w}=0.3$, $\rho_{0\rm{w}}/\rho_{0\rm{j}}=6.7$; box size is $250\times 750\times 250\, c/\omega_{\rm p e}$ (jet skin depths)
Comparison of predicted and measured growth rates from 3D run
Note that growth rate is largest for an angle slightly out of $xy$ plane; this angle in fact satisfies $k \cdot B_{0}\approx 0$
Time evolution of electron energy spectra shows evidence for particle energization up to Lorentz factors $\gamma \approx 100$
Exploration of guide field reconnection (see arXiv:1901.05438):
Studied KH instability analytically, in RMHD, and in PIC: