  In this paper we analyze the stability of equilibrium manifolds of hyperbolic
shallow water moment equations. Shallow water moment equations describe shallow
flows for complex velocity profiles which vary in vertical direction and the
models can be seen as extensions of the standard shallow water equations.
Equilibrium stability is an important property of balance laws that determines
the linear stability of solutions in the vicinity of equilibrium manifolds and
it is seen as a necessary condition for stable numerical solutions. After an
analysis of the hyperbolic structure of the models, we identify three different
stability manifolds based on three different limits of the right-hand side
friction term, which physically correspond to water-at-rest, constant-velocity,
and bottom-at-rest velocity profiles. The stability analysis then shows that
the structural stability conditions are fulfilled for the water-at-rest
equilibrium and the constant-velocity equilibrium. However, the bottom-at-rest
equilibrium can lead to instable modes depending on the velocity profile.
Relaxation towards the respective equilibrium manifolds is investigated
numerically for different models.
