Head-On and Long Range Beam-Beam Interactions in the LHC: Effective Tune Spread and Beam Stability Due to Landau Damping

We discuss the Landau damping of coherent instabilities in the presence of betatron tune spread. This tune spread can originate from dedicated non-linear magnets such as octupoles, or through the beam-beam interaction. In the latter case we have to distinguish the contribution from head- on and parasitic beam-beam interactions and the collision pattern of different bunches plays an important role. The interplay of these sources of tune spread and the resulting stability is discussed for the case of the LHC.