  Quasi-Periodic Oscillations (QPOs) observed during Soft Gamma Repeaters giant
flares are commonly interpreted as the torsional oscillations of magnetars. The
oscillatory motion is influenced by the strong interaction between the shear
modes of the crust and Alfven-like modes in the core. We study the dynamics
which arises through this interaction, and present several new results: (1) We
show that global {\it edge modes} frequently reside near the edges of the core
Alfven continuum. (2) We compute the magnetar's oscillatory motion for
realistic axisymmetric magnetic field configurations and core density profiles,
but with a simplified model of the elastic crust. We show that one may
generically get multiple gaps in the Alfven continuum. One obtains discrete
global {\it gap modes} if the crustal frequencies belong to the gaps. (3) We
show that field tangling in the core enhances the role of the core discrete
Alfven modes and reduces the role of the core Alfven continuum in the overall
oscillatory dynamics of the magnetar. (4) We demonstrate that the system
displays transient and/or drifting QPOs when parts of the spectrum of the core
Alfven modes contain discrete modes which are densely and regularly spaced in
frequency. (5) We show that if the neutrons are coupled into the core Alfven
motion, then the post-flare crustal motion is strongly damped and has a very
weak amplitude. Thus magnetar QPOs give evidence that the proton and neutron
components in the core are dynamically decoupled and that at least one of them
is a quantum fluid. (6) We show that it is difficult to identify the
high-frequency 625 Hz QPO as being due to the physical oscillatory mode of the
magnetar, if the latter's fluid core consists of the standard
proton-neutron-electron mixture and is magnetised to the same extent as the
crust. (Abstract abridged)
