Energy Confinement for a Relativistic Magnetic Flux Tube in the Ergosphere of a Kerr Black Hole

Abstract

In the MHD description of plasma phenomena the concept of magnetic field lines frozen into the plasma turns out to be very useful. We present here a method of introducing Lagrangian coordinates into relativistic MHD equations in general relativity, which enables a convenient mathematical formulation for the behaviour of flux tubes. With the introduction of these Lagrangian, so-called "frozen-in" coordinates, the relativistic MHD equations reduce to a set of nonlinear 1D string equations, and the plasma may therefore be regarded as a gas of nonlinear strings corresponding to flux tubes. Numerical simulation shows that if such a tube/string falls into a Kerr black hole, then the leading portion loses angular momentum and energy as the string brakes, and to compensate for this loss, momentum and energy is radiated to infinity to conserve energy and momentum for the tube. Inside the ergosphere the energy of the leading part turns out to be negative after some time, and the rest of the tube then gets energy from the hole. In our simulations most of the compensated positive energy is also localized inside the ergosphere because 1 the inward speed of the plasma is approximately equal to the velocity of the MHD wave which transports energy outside. Therefore, an additional physical process has to be included which can remove energy from the ergophere.Magnetic reconnection seems fills this role releasing Maxwellian stresses and producing a relativistic jet.

The Origins of Causality Violations in Force-free Simulations of Black Hole Magnetospheres

“…The results of the MHD simulations of Koide et al (2002), Camenzind & Khanna (2000), and Semenov et al (2002) and the time stationary solution in Punsly (2001, pp. 251-313) indicate that this is the case.…”

“…This fact is apparent in all of the simulations above in which relativistic inertia imparted to the plasma by the gravitational field dominates the dynamics in the ergosphere, regardless of the degree of inertial dominance imposed in the initial state. The interaction drives very strong cross-field currents (note that these currents cannot be dismissed as transients in Punsly 2001, in which they are eternal, or in Semenov et al 2002, in which they are persistent for long simulations and in which the driving force never goes away). Not coincidentally, this is precisely the region in which unphysical waves emerge in the ffde simulation.…”

Thin filament simulations for Earth's plasma sheet: Interchange oscillations

“…In a paper aimed mainly at a theoretical interpretation of bursty bulk flows, Chen and Wolf [1999](hereinafter referred to as CW99) developed a model of the motion of thin, ideal‐MHD filaments moving through a stationary background. The thin filament approximation has been used in various plasma physical situations [e.g., Parker , 1981a, 1981b; Semenov , 2000; Semenov et al , 2002; Chen , 2007]. The results presented in the CW99 paper centered on a run involving a depleted filament (bubble) that initially crossed the equatorial plane at 40 R E .…”

Simulations of Jets Driven by Black Hole Rotation

“…11,12 In this treatment, a flux tube is thin by definition if the pressure variations across the flux tube is negligible compared to the total external pressure (gas plus magnetic), P, that represents the effects of the enveloping magnetized plasma (the magnetosphere). By concentrating the calculation on individual flux tubes in a magnetosphere, we can focus the computational effort on the physical mechanism of jet production (on all the field lines).…”