Landau damping with two-dimensional betatron tune spread

We discuss Landau damping of the rigid dipole oscillations for a beam with amplitude-dependent betatron tunes. In particular, we derive analytic formulae for the stability limit when the detunings are linear combinations of the two betatron action variables. Such linear dependence represents the lowest contribution of magnetic octupoles to the detuning with amplitude and is of special interest for the stabilization of transverse oscillation modes at collision energy in the LHC. When the detuning coefficients have opposite signs, we find that the case of two-dimensional betatron spread is qualitatively different from the one-dimensional case: for a Gaussian distribution in the two transverse planes, the beam transfer function has tails both in the positive and negative tune directions and collective rigid dipole oscillations can be Landau damped for any real coherent tune shift caused by the impedance. The stability limit exhibits complicated pathologies for a truncated Gaussian distribution and, to obtain quantitative results concerning the loss of Landau damping for a given real coherent tune shift, we discuss the case of a `quasi-parabolic'' distribution in the two transverse planes.