  The rotating shallow water equations with f-plane approximation and nonlinear
bottom drag are a prototypical model for mid-latitude geophysical flow that
experience energy loss through simple topography. Motivated by numerical
schemes for large-scale geophysical flow, we consider this model on the whole
space with horizontal kinetic energy backscatter source terms built from
negative viscosity and stabilizing hyperviscosity with constant parameters. We
study its interplay with linear and non-smooth quadratic bottom drag through
the existence of coherent flows. Our results highlight that backscatter can
have undesired amplification and selection effects, generating obstacles to
energy distribution. We find that decreasing linear bottom drag destabilizes
the trivial flow and generates nonlinear flows that can be associated with
geostrophic equilibria (GE) and inertia-gravity waves (IGWs). The IGWs are
periodic travelling waves, while the GE are stationary and can be studied by a
plane wave reduction. We show that for isotropic backscatter both bifurcate
simultaneously and supercritically, while for anisotropic backscatter the
primary bifurcation are GE. In all cases presence of non-smooth quadratic
bottom drag implies unusual scaling laws. For the rigorous bifurcation analysis
by Lyapunov-Schmidt reduction care has to be taken due to this lack of
smoothness and since the hyperviscous terms yield a lack of spectral gap at
large wave numbers. For purely smooth bottom drag, we identify a class of
explicit such flows that behave linearly within the nonlinear equations:
amplitudes can be steady and arbitrary, or grow exponentially and unboundedly.
We illustrate the results by numerical computations and present extended
branches in parameter space.
