Thus, a meaningful physical interpretation can be made of Eqs. (1) and (2), but this interpretation differs from that of TL.
Comparing the numerical and UCNA results in Figs. 1 and 2, it is clear that for small values of τ, the stationary PDFs produced by numerical integration and by the UCNA are in close agreement. However, for τ ∼ O(1), there are marked differences. In particular, the PDFs produced by numerical integration do not display any nodes or new extrema. Instead, the result of raising τ for fixed σ is seen to be a shift of the PDF toward the right-hand peak. The differences between the numerical and UCNA approximations occur because of a breakdown of the validity of the UCNA for τ of O(1).
Calculating the stationary PDF of a system in a one-dimensional potential subject to colored noise remains an unsolved problem in physics. A number of different approximations have been proposed, but they are valid only in the limit of small or of very large τ (Horsthemke and Lefever 1984; Hänggi and Jung 1995). To obtain the stationary PDFs of y in the case where its timescale is of the same order of magnitude as ϵ, at present we must take recourse to numerical methods. We note that an essential conclusion of TL is unchanged, namely that increasing σ populates the left-hand peak of the stationary PDF of y at the expense of the right-hand peak, while increasing τ has the opposite effect.
Overall the idea is supported that changing noise characteristics might have a significant effect on the climatic mean states and their stability; this point is discussed in a more general framework in Palmer (2001). The nonlinear paradigm of noise-induced transitions is consistent with the findings recently reported by Aeberhardt et al. (2000) using a stochastically forced model of intermediate complexity. It might be interesting to study how the effect of noise-induced transitions of the THC translates to other more complex models.
REFERENCES
Aeberhardt, M., M. Blatter, and T. F. Stocker, 2000: Variability on the century time scale and regime changes in a stochastically forced zonally averaged ocean–atmosphere model. Geophys. Res. Lett., 27 , 1303–1306.
Cessi, P., 1994: A simple box model of stochastically forced thermohaline flow. J. Phys. Oceanogr, 24 , 1911–1920.
Gardiner, C. W., 1997: Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences. Springer, 442 pp.
Hänggi, P., and P. Jung, 1995: Colored noise in dynamical systems. Adv. Chem. Phys., 89 , 239–326.
Horsthemke, W., and R. Lefever, 1984: Noise-Induced Transitions: Theory and Applications in Physics, Chemistry, and Biology. Springer, 318 pp.
Jung, P., and P. Hänggi, 1987: Dynamical systems: A unified coloured-noise approximation. Phys. Rev., A35 , 4464–4466.
Kloeden, P. E., and E. Platen, 1992: Numerical Solution of Stochastic Differential Equations. Springer, 632 pp.
Lohmann, G., and J. Schneider, 1999: Dynamics and predictability of Stommel's box model: A phase space perspective with implications for decadal climate variability. Tellus, 51A , 326–336.
Maas, L. R. M., 1994: A simple model for the three-dimensional, thermally and wind-driven ocean circulation. Tellus, 46A , 671–680.
Nakamura, M., and Y. Chao, 2000: On the eddy isopycnal thickness diffusivity of the Gent–McWilliams subgrid mixing parameterization. J. Climate, 13 , 502–510.
Palmer, T. N., 2001: A nonlinear dynamical perspective on model error: A proposal for nonlocal stochastic–dynamic parameterisation in weather and climate prediction models. Quart. J. Roy. Meteor. Soc., 127 , 279–304.
Penland, C., 1996: A stochastic model of IndoPacific sea surface temperature anomalies. Physica D, 98 , 534–558.
Stommel, H. M., 1961: Thermohaline convection with two stable regimes of flow. Tellus, 13 , 224–230.
Timmermann, A., and G. Lohmann, 2000: Noise-induced transitions in a simplified model of the thermohaline circulation. J. Phys. Oceanogr., 30 , 1891–1900.
Stationary PDF of y from numerical integration (thick line) and from the UCNA approximation (thin line) for σ = 0.3.
Citation: Journal of Physical Oceanography 32, 3; 10.1175/1520-0485(2002)032<1112:CONITI>2.0.CO;2
As in Fig. 1 but for σ = 1
Citation: Journal of Physical Oceanography 32, 3; 10.1175/1520-0485(2002)032<1112:CONITI>2.0.CO;2