  The quasi-geostrophic equation or the Euler equation with dissipation studied
in the present paper is a simplified form of the atmospheric circulation model
introduced by Charney and DeVore [J. Atmos. Sci. 36(1979), 1205-1216] on the
existence of multiple steady states to the understanding of the persistence of
atmospheric blocking. The fluid motion defined by the equation is driven by a
zonal thermal forcing and an Ekman friction forcing measured by $\kappa>0$. It
is proved that the steady-state solution is unique for $\kappa >1$ while
multiple steady-state solutions exist for $\kappa<\kappa_{crit}$ with respect
to critical value $\kappa_{crit}<1$.
  Without involvement of viscosity, the equation has strong nonlinearity as its
nonlinear part contains the highest order derivative term. Steady-state
bifurcation analysis is essentially based on the compactness, which can be
simply obtained for semi-linear equations such as the Navier-Stokes equations
but is not available for the quasi-geostrophic equation in the Euler
formulation. Therefore the Lagrangian formulation of the equation is employed
to gain the required compactness.
