  Several teams have reported peculiar frequency spectra for flows in a
spherical shell. To address their origin, we perform numerical simulations of
the spherical Couette flow in a dipolar magnetic field, in the configuration of
the DTS experiment. The frequency spectra computed from time-series of the
induced magnetic field display similar bumpy spectra, where each bump
corresponds to a given azimuthal mode number m. The bumps show up at moderate
Reynolds number (2 600) if the time-series are long enough (>300 rotations of
the inner sphere). We present a new method that permits to retrieve the
dominant frequencies for individual mode numbers m, and to extract the modal
structure of the full non-linear flow. The maps of the energy of the
fluctuations and the spatio-temporal evolution of the velocity field suggest
that fluctuations originate in the outer boundary layer. The threshold of
instability if found at Re_c = 1 860. The fluctuations result from two coupled
instabilities: high latitude B\"odewadt-type boundary layer instability, and
secondary non-axisymmetric instability of a centripetal jet forming at the
equator of the outer sphere. We explore the variation of the magnetic and
kinetic energies with the input parameters, and show that a modified Elsasser
number controls their evolution. We can thus compare with experimental
determinations of these energies and find a good agreement. Because of the
dipolar nature of the imposed magnetic field, the energy of magnetic
fluctuations is much larger near the inner sphere, but their origin lies in
velocity fluctuations that initiate in the outer boundary layer.
