Past exam of the geophysical fluid dynamics course of the University of Oxford / 2025 / 2 / d / Solution

With the same prescribed $\Delta T$, $S_0$, $\alpha$ and $\beta$, replacing $k$ by $\lambda k$ changes both transport coefficients:Hence the positive steady volume fluxes becomeand the negative steady volume flux isThe bifurcation diagram undergoes a particularly simple rescaling:Thus it contracts horizontally and vertically by the same factor. Its branch stability and saddle-node bifurcation structure persist, but the range of freshwater forcing that admits the stable positive circulation shrinks. At a fixed subcritical $E$, the upper stable positive branch weakens, the lower unstable positive branch moves upward, and the reversed branch decreases in magnitude. These changes reduce the separation of the positive equilibria and the margin against an overturning transition. The right panel of the figure in 2(b) shows this for $\lambda =0.7$ and $0.45$.

For the present $E=0.3\mathrm{Sv}$, a positive steady state exists only ifAt equality the positive branches merge at $q=\lambda A/2=5.6\mathrm{Sv}$; below it only the reversed steady state remains. If a discovery merely revises our estimate of $k$, it revises the inferred branch strengths and threshold, rather than itself causing an instantaneous physical transition. If the real effective $k$ were to decrease slowly through the critical value while $E$ stayed fixed, this model predicts loss of the positive steady circulation and transition to the reversed branch. Uncertainty in $k$ therefore translates directly into uncertainty in the model's freshwater threshold and its inferred proximity to a transition.