  Shallow flow models are used for a large number of applications including
weather forecasting, open channel hydraulics and simulation-based natural
hazard assessment. In these applications the shallowness of the process
motivates depth-averaging. While the shallow flow formulation is advantageous
in terms of computational efficiency, it also comes at the price of losing
vertical information such as the flow's velocity profile. This gives rise to a
model error, which limits the shallow flow model's predictive power and is
often not explicitly quantifiable.
  We propose the use of vertical moments to overcome this problem. The shallow
moment approximation preserves information on the vertical flow structure while
still making use of the simplifying framework of depth-averaging. In this
article, we derive a generic shallow flow moment system of arbitrary order
starting from a set of balance laws, which has been reduced by scaling
arguments. The derivation is based on a fully vertically resolved reference
model with the vertical coordinate mapped onto the unit interval. We specify
the shallow flow moment hierarchy for kinematic and Newtonian flow conditions
and present 1D numerical results for shallow moment systems up to third order.
Finally, we assess their performance with respect to both the standard shallow
flow equations as well as with respect to the vertically resolved reference
model. Our results show that depending on the parameter regime, e.g. friction
and slip, shallow moment approximations significantly reduce the model error in
shallow flow regimes and have a lot of potential to increase the predictive
power of shallow flow models, while keeping them computationally cost
efficient.
