  In the two-layer quasi-geostrophic model, the friction between the flow at
the lower layer and the surface boundary layer, placed beneath the lower layer,
is modeled by the Ekman term, which is a linear dissipation term with respect
to the horizontal velocity at the lower layer. The Ekman term appears in the
governing equations asymmetrically; it is placed at the lower layer, but does
not appear at the upper layer. A variation, proposed by Phillips and Salmon,
uses extrapolation to place the Ekman term between the lower layer and the
surface boundary layer, or at the surface boundary layer. We present
theoretical results that show that in either the standard or the extrapolated
configurations, the Ekman term dissipates energy at large scales, but does not
dissipate potential enstrophy. It also creates an approximately symmetric
stable distribution of potential enstrophy between the two layers. The behavior
of the Ekman term changes fundamentally at small scales. Under the standard
formulation, the Ekman term will unconditionally dissipate energy and also
dissipate, under very minor conditions, potential enstrophy at small scales.
However, under the extrapolated formulation, there exist small "negative
regions", which are defined over a two-dimensional phase space, capturing the
distribution of energy per wavenumber between baroclinic energy and barotropic
energy, and the distribution of potential enstrophy per wavenumber between the
upper layer and the lower layer, where the Ekman term may inject energy or
potential enstrophy.
