  We study mixed finite element methods for the linearized rotating shallow
water equations with linear drag and forcing terms. By means of a strong energy
estimate for an equivalent second-order formulation for the linearized
momentum, we prove long-time stability of the system without energy
accumulation -- the geotryptic state. A priori error estimates for the
linearized momentum and free surface elevation are given in $L^2$ as well as
for the time derivative and divergence of the linearized momentum. Numerical
results confirm the theoretical results regarding both energy damping and
convergence rates.
