Astrophysicists have long inferred that accretion disks, orbiting
around such objects as forming protostars or black holes, are in a
turbulent state. The inferred accretion rates imply a turbulent outward
flux of angular momentum, which is necessary for accretion to occur.
But how do these disks become turbulent? The rotational (Keplerian)
equilibrium profile of a fluid disk bound by the gravity of a central
object produces a circumferential velocity that decays as r^-3/2, so
that this arrangement is linearly stable to the centrifugal
instability. Although there is some possibility of a nonlinear
(or subcritical) hydrodynamic transition to turbulence in these
systems, it is now commonly held that the transition to turbulence in
astrophysical accretion disks is via the magnetorotational instability
(MRI). Although the MRI was discovered some half-century ago, it came
to the attention of the astrophysical community relatively recently,
around 1990. The instability mechanism is essentially the coupling of
the Keplerian shear flow, which represents a gradient in kinetic
energy, by a magnetic field.
A rather simple mechanical analogue exists for
describing MRI that features the differential (i.e., sheared) movements
of parcels that are connected by a spring. See the picture
below.
This picture depicts two representative fluid parcels, m_i and m_o,
which are orbiting a central mass M_c. Originally m_i is slightly
farther in, and, due to the inherent shear, moving slightly faster. By
virtue of their connection, m_o ‘tugs’ on m_i, a process that speeds up
the former, and slows down the latter. Consequently the loss of energy
of mi causes it to ‘fall’ inward, while the energy gain of m_o cause it
to find a new equilibrium farther out. Depending on the spring strength
and rotation speeds, this could lead to a runaway process, leading to
the eventual infall of mass (i.e., accretion) onto the central object
M_c. In this analogy the spring strength is analogous to the magnetic
field strength. The strength of the magnetic field is important for
MRI: it must be high enough to be effective in communicating energy
before differential motion separates radially adjacent fluid parcels
(or else the spring is effectively not there), and also strong enough
so that the perturbation in the magnetic field does not diffuse away.
But the magnetic field strength must not be so strong as to restrict
relative motion (i.e., where the spring is like a rod, without
noticeable ‘springiness’, as would be the case with a perfectly
conducting fluid). Thus there is a maximum magnetic field that exists,
depending upon the shear rate, above which MRI is quenched.
These physical requirements described above may be
considered more quantitatively with a consideration of non-dimensional
parameters derived from the dynamical equations of motion. For MRI to
overcome the dissipative effects of hydrodynamic viscosity and magnetic
diffusivity, we should expect Reynolds numbers Re=UL/\nu and
Rem=UL/\eta, along with the Lundquist number S=U_A L/\eta, to
significantly exceed unity. (Here U is characteristic velocity, L
is characteristic size, \nu is kinematic viscosity, and \eta is
magnetic diffusibility.) These requirements set lower limits
for size, speed, and magnetic field strength for a fluid of given
viscosity and resistivity.