Fig 1.
A low-latitude box of ocean water with temperature T1 and salinity S1 is connected via a capillary tube and surface channel to a high-latitude box with temperature T2 and salinity S2. Baths with temperatures ,
and salinities
,
represent surrounding conditions at low and high latitudes.
Fig 2.
Solution trajectories for Eq (3) with , R = 2, and
are plotted as thin black lines. The system has three equilibria: the stable node A, saddle B, and stable spiral C. The stable manifold of B (heavy dashed line) is the separatrix between the basin of A (blue) and the basin of C (light yellow). The hatched pattern indicates the region of positive circulation and no pattern indicates negative circulation. Combinations of basin and circulation direction partition state space into four regions labeled (i)-(iv).
Fig 3.
Basin and circulation outcomes.
Panel (a) categorizes disturbance space according to the long-term behavior of flow-kick trajectories that start at equilibrium A of Stommel’s model (3) and receive kicks
every τ time units. Trajectories can stabilize in the basin of A (blue) or in the basin of C (light yellow). Furthermore, the long-term circulation direction can be reversed (hatched), mixed (dotted) or preserved (no pattern) relative to A. Numerals (i)-(iv) designate the region(s) of state space from Fig 2 that the flow-kick trajectory occupies in the long term. Panels (b)–(e) illustrate four qualitatively different combinations of behaviors in state space for the corresponding
parameter combinations shown in panel a.
Fig 4.
(a) Three flow-kick fixed points exist for : stable fixed points
and
and unstable fixed point
.
(b) As τ decreases from 9, the branches of fixed points maintain their stability, but and
undergo a saddle-node bifurcation at
and
, while the branch
persists and crosses into the original basin of A at
and
. Light orange curves illustrate flows and kicks at
,
(compare Fig 3c),
, and
(compare Fig 3b).
Fig 5.
Bifurcation diagram showing the transition of equilibria as disturbance ratio (r) increases.
A saddle-node bifurcation at marks the collapse from three equilibria to a single equilibrium. The vertical axis represents the y coordinate of the equilibria.
Fig 6.
Bifurcation diagram for a hypothetical system.
Intensifying disturbance first destroys an attracting fixed point A via a saddle-node bifurcation (i→ii) and then shifts an alternative attractor C into A’s original basin (ii→iii). Temporary, moderate disturbance can cause hysteresis (panel a), while temporary, high-intensity disturbance can avoid hysteresis (panel b). Further, increasing disturbance intensity from moderate to high can avert hysteresis (panel c). Double arrows indicate rapid parameter changes while single arrows indicate evolution of state. Arabic numerals indicate event order.
Table 1.
Flow-kick parameter estimates for two geophysical processes that perturb North Atlantic salinity.