417
relative vorticity is neglected (by a scale argument L2 ~ R2 which is
however not appropriate for a barotropic model), the resonance disappears
and the momentum balance of this barotropic model is linear in U. Then
multiple steady states do not exist. The resonance disappears as well when
the Rossby waves are filtered (f-plane approximation). In these cases the
form stress is very small compared to the forcing and the flow is directly
controlled by the bottom friction: we find U ~ T / (EH) (notice that we
have plotted S[U]/ E vs U in the Figure 3 so that both intersections of the
straight lines with the axes are equal to the frictional solution T / (EH)).
Significant sizes of the form stress can thus only arise if the topography
is sufficiently high, Rossby waves propagation is present and the forcing
is sufficiently strong. The three possible steady states which then exist
can be classified according to the size of the mean flow U compared to
the wave amplitudes: the high zonal index regime is frictionally controlled,
the flow is intense and and the wave amplitude is low; the low zonal index
regime is controlled by form stress, the mean flow is weak and the wave
is intense. The intermediate state is transitional, it is actually unstable
to perturbations. This 'form drag instability' works obviously when the
slope of the resonance curve is below the one associated with friction (E +
as/au> 0, see Figure 3) so that a perturbation must run away from the
steady state. Apparently the criterion of instability is always satisfied for
the intermediate state whereas the other two states are always stable.
In realistic atmospheric applications of the CdV model the parameter
window (topographic height, forcing and friction parameters) for multiple
solutions is quite narrow, maybe even nonexisting for more complex topographies (see e. g. the discussion and results of Tung and Rosenthal
1985). The most important feature revealed in the oceanic parameter
range, as displayed in Figure 3, is the absence of multiple states for realistic values of the forcing T and damping rate E. Oceanic velocities are
always considerably smaller than the barotropic Rossby wave speed at relevant topographic wavelengths, hence the resonance cannot be met and
a barotropic model is necessarily in the frictionally controlled state (see
e. g. Volker 1991). A different regime can be found when the topography
allows for blocked geostrophic contours (Krupitskyand Cane 1994) which
however is questionable for simple geometrical reasons: f / H -contours connecting the walls of a channel are an artifact of the model setup.
relative vorticity is neglected (by a scale argument L2 ~ R2 which is
however not appropriate for a barotropic model), the resonance disappears
and the momentum balance of this barotropic model is linear in U. Then
multiple steady states do not exist. The resonance disappears as well when
the Rossby waves are filtered (f-plane approximation). In these cases the
form stress is very small compared to the forcing and the flow is directly
controlled by the bottom friction: we find U ~ T / (EH) (notice that we
have plotted S[U]/ E vs U in the Figure 3 so that both intersections of the
straight lines with the axes are equal to the frictional solution T / (EH)).
Significant sizes of the form stress can thus only arise if the topography
is sufficiently high, Rossby waves propagation is present and the forcing
is sufficiently strong. The three possible steady states which then exist
can be classified according to the size of the mean flow U compared to
the wave amplitudes: the high zonal index regime is frictionally controlled,
the flow is intense and and the wave amplitude is low; the low zonal index
regime is controlled by form stress, the mean flow is weak and the wave
is intense. The intermediate state is transitional, it is actually unstable
to perturbations. This 'form drag instability' works obviously when the
slope of the resonance curve is below the one associated with friction (E +
as/au> 0, see Figure 3) so that a perturbation must run away from the
steady state. Apparently the criterion of instability is always satisfied for
the intermediate state whereas the other two states are always stable.
In realistic atmospheric applications of the CdV model the parameter
window (topographic height, forcing and friction parameters) for multiple
solutions is quite narrow, maybe even nonexisting for more complex topographies (see e. g. the discussion and results of Tung and Rosenthal
1985). The most important feature revealed in the oceanic parameter
range, as displayed in Figure 3, is the absence of multiple states for realistic values of the forcing T and damping rate E. Oceanic velocities are
always considerably smaller than the barotropic Rossby wave speed at relevant topographic wavelengths, hence the resonance cannot be met and
a barotropic model is necessarily in the frictionally controlled state (see
e. g. Volker 1991). A different regime can be found when the topography
allows for blocked geostrophic contours (Krupitskyand Cane 1994) which
however is questionable for simple geometrical reasons: f / H -contours connecting the walls of a channel are an artifact of the model setup.