Inertial Runaway
89
several ways, but the one of greatest oceanographic relevance is the case in
which a southern region of downward Ekman pumping is balanced by a
northern region of the same size in which Ekman suction is equal and opposite
to that of the southern region. This balances the input of anticyclonic vorticity
in the southern (subtropical) gyre with an input of cyclonic vorticity in the
northern (subpolar) gyre. Thus, no net flux of vorticity is required to leave the
basin through side walls or bottom. This completely eliminates the need for the
boundary layers to do more than close the mass flux. The vorticity balance can
be achieved internally. Marshall (1984) carried out a very interesting numerical
study in which the Ekman pumping had the form:
. (2ny)
wE= -Wsm L ,
(2.14.10)
which naturally produces two gyres with an intergyre boundary, according to
Sverdrup theory, at y = L/2.
The dissipation in Marshall's model is a combination of bottom friction
and lateral friction. The latter is of an artificial form designed to be inconsequential on the scale of the circulation and introduced only to damp out very
small-scale fluctuations. The bottom friction was chosen small enough so that
the ratio bJ/ Ds is roughly 4.0, i.e., highly inertial. In the single-gyre model such
a parameter setting would lead, as we have seen, to the development of a steady
Fofonoff mode. In the two-gyre calculations reported by Marshall the circulation rapidly becomes unstable. A jet, which is identified with the eastwardly
flowing Gulf Stream after separation from the eastern boundary near y = L/2,
becomes an unstable, meandering current spawning a vigorous eddy field. The
eddies flux anticyclonic vorticity in an amount equal to that put in by the wind,
from the southern gyre to the northern gyre, where the wind-forced Ekman
pumping of the opposite sign removes it from the basin, establishing an
equilibrium. The time-averaged circulation is shown in Fig. 2.14.6. Panel a
shows the stream function for the time-averaged flow. Panel b shows the isolines of total vorticity, and panel c shows the superposition of the isolines of If;
and ( + {Jy. We see that in the time mean a small recirculation does exist, forced
by the advective transport of vorticity anomaly to the intergyre boundaries.
However, the region of recirculation is quite limited. The isolines of total
vorticity are distorted primarily in the region of the western boundary current
and the recirculation. Elsewhere they remain latitude circles, a sign of the
survival of the Sverdrup interior. The superposition of the isolines of the two
fields shows that in the western boundary layer, which is largely inertial, and in
which forcing and explicit dissipation are negligible, the mean (time-averaged)
flow nevertheless crosses isolines of mean total vorticity. This occurs only
because the eddy field provides sufficient flux of vorticity in this region to carry
away the potential vorticity anomaly that would otherwise be generated by the
advection of mean vorticity.
The singular character of this interesting calculation must be emphasized.
If the Ekman pumping is not exactly antisymmetric (as it is not in the natural
89
several ways, but the one of greatest oceanographic relevance is the case in
which a southern region of downward Ekman pumping is balanced by a
northern region of the same size in which Ekman suction is equal and opposite
to that of the southern region. This balances the input of anticyclonic vorticity
in the southern (subtropical) gyre with an input of cyclonic vorticity in the
northern (subpolar) gyre. Thus, no net flux of vorticity is required to leave the
basin through side walls or bottom. This completely eliminates the need for the
boundary layers to do more than close the mass flux. The vorticity balance can
be achieved internally. Marshall (1984) carried out a very interesting numerical
study in which the Ekman pumping had the form:
. (2ny)
wE= -Wsm L ,
(2.14.10)
which naturally produces two gyres with an intergyre boundary, according to
Sverdrup theory, at y = L/2.
The dissipation in Marshall's model is a combination of bottom friction
and lateral friction. The latter is of an artificial form designed to be inconsequential on the scale of the circulation and introduced only to damp out very
small-scale fluctuations. The bottom friction was chosen small enough so that
the ratio bJ/ Ds is roughly 4.0, i.e., highly inertial. In the single-gyre model such
a parameter setting would lead, as we have seen, to the development of a steady
Fofonoff mode. In the two-gyre calculations reported by Marshall the circulation rapidly becomes unstable. A jet, which is identified with the eastwardly
flowing Gulf Stream after separation from the eastern boundary near y = L/2,
becomes an unstable, meandering current spawning a vigorous eddy field. The
eddies flux anticyclonic vorticity in an amount equal to that put in by the wind,
from the southern gyre to the northern gyre, where the wind-forced Ekman
pumping of the opposite sign removes it from the basin, establishing an
equilibrium. The time-averaged circulation is shown in Fig. 2.14.6. Panel a
shows the stream function for the time-averaged flow. Panel b shows the isolines of total vorticity, and panel c shows the superposition of the isolines of If;
and ( + {Jy. We see that in the time mean a small recirculation does exist, forced
by the advective transport of vorticity anomaly to the intergyre boundaries.
However, the region of recirculation is quite limited. The isolines of total
vorticity are distorted primarily in the region of the western boundary current
and the recirculation. Elsewhere they remain latitude circles, a sign of the
survival of the Sverdrup interior. The superposition of the isolines of the two
fields shows that in the western boundary layer, which is largely inertial, and in
which forcing and explicit dissipation are negligible, the mean (time-averaged)
flow nevertheless crosses isolines of mean total vorticity. This occurs only
because the eddy field provides sufficient flux of vorticity in this region to carry
away the potential vorticity anomaly that would otherwise be generated by the
advection of mean vorticity.
The singular character of this interesting calculation must be emphasized.
If the Ekman pumping is not exactly antisymmetric (as it is not in the natural