Destabilization of the thermohaline circulation by transient changes in the hydrological cycle

Abstract

We reconsider the problem of the stability of the thermohaline circulation as described by a two-dimensional Boussinesq model with mixed boundary conditions. We determine how the stability properties of the system depend on the intensity of the hydrological cycle. We define a two-dimensional parameters’ space descriptive of the hydrology of the system and determine, by considering suitable quasi-static perturbations, a bounded region where multiple equilibria of the system are realized. We then focus on how the response of the system to finite-amplitude changes in the surface freshwater forcings depends on their rate of increase. We show that it is possible to define a robust separation between slow and fast regimes of forcing. Such separation between slow and fast regimes is obtained by singling out an estimate of the critical rate of increase for the anomalous forcing. The critical rate of increase is of the same order of magnitude of the ratio between the typical intensity of the hydrological cycle and the advective time scale of the system. Basically, if the change of the forcing is faster than the estimated critical rate, the system responds similarly to the case of instantaneous changes of the same amplitude. Specularly, if the change of the forcing is slower than the critical rate, the behavior of the system resembles the response to quasi-static changes of the same amplitude. Furthermore, since the advective time scale is proportional to the square root of the vertical diffusivity, our analysis supports the conjecture that the efficiency of the vertical mixing might also be one of the key factors determining the response of the THC system to transient changes in the surface forcings. These results should be taken into account when engineering global warming scenario and paleoclimatic experiments with GCMs.

Appendix

1.1 Relevance of the vertical diffusivity ψ_ψψ

The vertical diffusivity K_V, or, equivalently, the diapycnal diffusivity K_d, is the critical parameter controlling the maximum THC strength Ψ_max in ocean models (Bryan 1987; Wright and Stocker 1992). On the other hand, an estimate of its value in the real ocean is a subject of current research (Gregg et al. 2003). Scaling theories proposing a balance between vertical diffusion and advection processes suggest, in the case of three-dimensional hemispheric model of the Atlantic ocean, a power law dependence $\Psi _{{\max }} \sim K^{{2/3}}_{v}$ (Zhang et al. 1999; Dalan et al. 2004). In the case of two-dimensional models, the expected dependence is $\Psi _{{\max }} \sim K^{{1/2}}_{v}$ (Knutti et al. 2000). This relation is verified in our model in the range 0.6 cm² s⁻¹ < K_V < 4.0 cm² s⁻¹ (Fig. 9). In this study, the value of K_V has been selected so that the corresponding northern sinking equilibrium state characterized by an hydrological cycle determined by Φ_N=Φ_S=Φ_av has an overturning circulation ~37 Sv (K_V=1.0 cm² s⁻¹). With this choice of K _V we can define an advective time scale τ for the system as τ=V/Ψ_max~350 year. Similarly, if we change K _V over the range shown in Fig. 8, the advective time scale would range over the interval 150 year<τ<400 year. Notice that by choosing a value for K _V , we also define a diffusive time scale τ_d=D²/K _V ~8,000 year, which is very long compared to the advective time scale. Given the parameters chosen for our simulations, our model integrations estimate Φ_inf~0.04 Sv and Φ_sup~0.73 Sv, so that Φ_av~0.39 Sv.

Fig. 9

Dependence of maximum value of the THC on the value of the vertical diffusivity for northern sinking pattern under symmetric boundary conditions. For this plot, we set the relaxation surface temperature by choosing $\bar T_o = 15^\circ {\text{C}},\;\Delta \bar T = 23.5^\circ {\text{C,}}$ and the freshwater flux by choosing Φ_N=Φ_S=Φ_av