We have analyzed the amplification of slow magnetosonic (or pseudo-Alfvénic) waves (SMW) in incompressible shear flow. As found here, the amplification depends on the component of the wave-vector perpendicular to the direction of the shear flow. Earlier numerical results are consistent with the general analytic solution for the linearized magnetohydrodynamic equations, derived here for the model case of pure homogeneous shear (without Coriolis force). An asymptotically exact analytical formula for the amplification coefficient is derived for the case when the amplification is sufficiently large.

We investigate the conditions under which the magnetohydrodynamic (MHD) modes in a cylindrical magnetic flux tube moving along its axis become unstable against the Kelvin-Helmholtz (KH) instability. We use the dispersion relations of MHD modes obtained from the linearized Hall MHD equations for cool (zero beta) plasma by assuming real wave numbers and complex angular wave frequencies/complex wave phase velocities. The dispersion equations are solved numerically at fixed input parameters and varying values of the ratio l Hall /a, where l Hall = c/ω pi (c being the speed of light, and ω pi the ion plasma frequency) and a is the flux tube radius. It is shown that the stability of the MHD modes depends upon four parameters: the density contrast between the flux tube and its environment, the ratio of external and internal magnetic fields, the ratio l Hall /a, and the value of the Alfvén Mach number defined as the ratio of the tube axial velocity to Alfvén speed inside the flux tube. It is found that at high density contrasts, for small values of l Hall /a, the kink (m = 1) mode can become unstable against KH instability at some critical Alfvén Mach number (or equivalently at critical flow speed), but a threshold l Hall /a can suppress the onset of the KH instability. At small density contrasts, however, the magnitude of l Hall /a does not affect noticeably the condition for instability occurrence-even though it can reduce the critical Alfvén Mach number. It is established that the sausage mode (m = 0) is not subject to the KH instability.

Propagation of three dimensional magnetosonic waves is considered for a homogeneous shear flow of an incompressible fluid. The analytical solutions for all magnetohydrodynamic variables are presented by confluent Heun functions. The problem is reduced to finding a solution of an effective Schrodinger equation. The amplification of slow magnetosonic waves is analyzed in great details. A simple formula for the amplification coefficient is derived. The velocity shear primarily affects the incompressible limit of slow magnetosonic waves. The amplification is very strong for slow magnetosonic waves in the long-wavelength limit. It is demonstrated that the amplification of those waves leads to amplification of turbulence. The phenomenology of Shakura-Sunyaev for the friction in accretion disks is derived in the framework of the Koknogorov turbulence. The presented findings may be the key to explaining the anomalous plasma heating responsible for the luminosity of quasars. It is suggested that wave amplification is the keystone of the self-sustained turbulence in accretion disks.

Abstract. We have derived the full set of MHD equations for incompressible shear flow of a magnetized fluid and considered their solution in the wave-vector space. The linearized equations give the famous amplification of slow magnetosonic waves and describe the magnetorotational instability. The nonlinear terms in our analysis are responsible for the creation of turbulence and self-sustained spectral density of the MHD (Alfvén and pseudo-Alfvén) waves. Perspectives for numerical simulations of weak turbulence and calculation of the effective viscosity of accretion disks are shortly discussed in k-space.

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