Magnetorotational instability

Magnetorotation instability ( MRI , magnetic rotation instability ) or Balbus-Hawley instability describes the phenomenon of instability of rotating fluids in the vicinity of small magnetic fields under certain conditions with the result that matter falls into the center.

The instability in astrophysical accretion disks causes, among other things, the formation of stars and black holes , but can also be observed in the laboratory. It follows from the basic equations of magnetohydrodynamics (MHD).

meaning

Both in star formation and in the formation of black holes, matter collects in an accretion disk and rotates around the center in which the star is formed.

The matter flows in a laminar manner and, due to Kepler's third law, the angular velocity decreases towards the outside. Since the state of the lowest energy consists in the fact that all mass collects in the center, but on the other hand the angular momentum of the entire system must be maintained, this leads to a large part of the mass moving inwards, which is caused by a small part that is far outside, is compensated.

Potential approaches to viscosity- generated turbulence are not proving efficient enough to cause star formation.

Steven A. Balbus and John F. Hawley showed in 1991 by analyzing the equations of magnetohydrodynamics that small magnetic fields lead to instabilities in the rotating disks. This leads to the fact that angular momentum is transported from the inside to the outside and that the inner layers of matter can fall into the center, so that mass accumulates there.

phenomenon

MRI is caused by the shear of the magnetic field in the plasma . The shear arises because the magnetic field follows the plasma and the inner layers move faster than the outer layers ( differential rotation ).

In a surface element sheared in this way, the magnetic field acts like a spring, which brakes the inner layer, thereby reducing its angular momentum in favor of the outer layers and bringing the inner layer to a lower orbit. This allows mass to accretion towards the center of gravity.

The MRI is therefore the cause of the high accretion rate and thus the high luminosity that is observed on various objects (e.g. AGN , quasars , microquasars ).

theory

The dispersion relation is obtained from the linear stability analysis of the MHD equations with differential rotation :

${\displaystyle {\begin{aligned}&\left[\omega ^{2}-\left({\vec {k}}\cdot {\vec {v}}_{\mathrm {A} }\right)^{2}\right]\cdot \left[\omega ^{4}-k^{2}\,(c_{\mathrm {s} }^{2}+v_{\mathrm {A} }^{2})\,\omega ^{2}+\left({\vec {k}}\cdot {\vec {v}}_{\mathrm {A} }\right)^{2}\,k^{2}c_{\mathrm {s} }^{2}\right]&\\-&\left[\kappa ^{2}\omega ^{4}-\omega ^{2}\left(\kappa ^{2}k^{2}\,(c_{\mathrm {s} }^{2}v_{\mathrm {A\phi } }^{2})+\left({\vec {k}}\cdot {\vec {v}}_{\mathrm {A} }\right)^{2}{\frac {\mathrm {d} \Omega ^{2}}{\mathrm {d} (\ln r_{\mathrm {zyl} })}}\right)\right]&\\-&k^{2}c_{\mathrm {s} }^{2}\left({\vec {k}}\cdot {\vec {v}}_{\mathrm {A} }\right)^{2}{\frac {\mathrm {d} \Omega ^{2}}{\mathrm {d} (\ln r_{\mathrm {zyl} })}}&=0\end{aligned}}}$

${\displaystyle \omega }$denotes the frequency and the wave vector of the disturbance, the Alfvén speed , the speed of sound , the epicyclic frequency and the rotational frequency of the disk. ${\displaystyle {\vec {k}}}$ ${\displaystyle {\vec {v}}_{\mathrm {A} }}$ ${\displaystyle c_{\mathrm {s} }}$ ${\displaystyle \kappa }$ ${\displaystyle \Omega }$

Instability occurs when becomes imaginary . Then one obtains an exponential growth in the wave equation of a perturbation . It turns out that the slow branch of the magnetosonic wave becomes unstable for sufficiently weak magnetic fields, i.e. H. Disturbances grow exponentially. The characteristic growth time of a disturbance is of the order of magnitude of the local rotation period. ${\displaystyle \omega }$ ${\displaystyle A\sim e^{\mathrm {i} ({\vec {k}}\cdot {\vec {x}}-\omega t)}}$

research

The effect was already described in 1959 by Evgeny Velikhov and in 1960 by Subrahmanyan Chandrasekhar in connection with stability considerations of Couette rivers , hence the name Velikhov-Chandrasekhar instability .

Steven A. Balbus and John F. Hawley applied the effect to astrophysical systems such as differentially rotating accretion disks. They demonstrated the effectiveness of MRI in angular momentum transport theoretically and in simulations. They thus provided the physical basis of the model of the standard disk designed by Nikolai I. Shakura and Rashid Sunyaev in 1973 (see accretion disk ).

Current research looks at e.g. B. the interaction of MRI with radiation , the MRI in resistive , the transition from optically dense to optically thin disks and the formation of jets from accretion disks.

It was proven experimentally in 2006 by Günther Rüdiger and Frank Stefani .

Web links

• Chapter in a diploma thesis on theory and simulation of MRI ( Memento from December 6, 2004 in the Internet Archive )
• Magnetorotational instability (PROMISE experiment). Helmholtz Center Dresden-Rossendorf 

Individual evidence

1. ↑ JE Pringle, ARA & A, 19 (1981) 137.
2. ^ ^{A b} Steven A. Balbus, John F. Hawley: A Powerful Local Shear Instability in Weakly Magnetized Disks. I - Linear Analysis. II - Nonlinear Evolution . In: The Astrophysical Journal, Part 1 . Vol. 376, July 20, 1991, ISSN  0004-637X , pp. 214-233 , doi : 10.1086 / 170270 , bibcode : 1991ApJ ... 376..214B .
3. ^ EP Velikhov, Stability of an Ideally Conducting Liquid Flowing Between Cylinders Rotating in a Magnetic Field , J. Exp. Theoret. Phys., 36: 1398-1404 (1959).
4. ↑ S. Chandrasekhar, The stability of non-dissipative Couette flow in hydromagnetics , Proc. Natl. Acad. Sci., 46: 253-257 (1960).
5. ↑ G. Rüdiger, F. Stefani u. a .: Experimental Evidence for Magnetorotational Instability in a Taylor-Couette Flow under the Influence of a Helical Magnetic Field, Phys. Rev. Lett., Vol. 97, 2006, p. 184502