# American Institute of Mathematical Sciences

July  2012, 17(5): 1575-1584. doi: 10.3934/dcdsb.2012.17.1575

## Impact of $\alpha$-stable Lévy noise on the Stommel model for the thermohaline circulation

 1 School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, 430074, China, China 2 State Key Laboratory of Atmospheric Sciences and Geophysical Fluid Dynamics, Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing, 100029, China 3 Department of Applied Mathematics, Illinois Institute of Technology, Chicago, IL 60616

Received  November 2011 Revised  January 2012 Published  March 2012

The Thermohaline Circulation, which plays a crucial role in the global climate, is a cycle of deep ocean due to the change in salinity and temperature (i.e., density). The effects of non-Gaussian noise on the Stommel box model for the Thermohaline Circulation are considered. The noise is represented by a non-Gaussian $\alpha$-stable Lévy motion with $0<\alpha < 2$. The $\alpha$ value may be regarded as the index of non-Gaussianity. When $\alpha=2$, the $\alpha$-stable Lévy motion becomes the usual (Gaussian) Brownian motion.
Dynamical features of this stochastic model is examined by computing the mean exit time for various $\alpha$ values. The mean exit time is simulated by numerically solving a deterministic differential equation with nonlocal interactions. It has been observed that some salinity difference levels remain in certain ranges for longer times than other salinity difference levels, for different $\alpha$ values. This indicates a lower variability for these salinity difference levels. Realizing that it is the salinity differences that drive the thermohaline circulation, this lower variability could mean a stable circulation, which may have further implications for the global climate dynamics.
Citation: Xiangjun Wang, Jianghui Wen, Jianping Li, Jinqiao Duan. Impact of $\alpha$-stable Lévy noise on the Stommel model for the thermohaline circulation. Discrete & Continuous Dynamical Systems - B, 2012, 17 (5) : 1575-1584. doi: 10.3934/dcdsb.2012.17.1575
##### References:
 [1] H. Stommel, Thermohaline convection with two stable regimes of flow,, Tellus., 13 (1961), 224. doi: 10.1111/j.2153-3490.1961.tb00079.x. [2] L. Mass, A simple model for the three-dimensional, thermally and wind-driven ocean circulation,, Tellus., 46 (1994), 671. doi: 10.1034/j.1600-0870.1994.t01-3-00008.x. [3] S. Rahmstorf, On the freshwater forcing and transport of the Atlantic thermohaline circulation,, Clim. Dyn., 12 (1996), 799. doi: 10.1007/s003820050144. [4] J. Scott, J. Marotzke and P. Stone, Interhemispheric thermohaline circulation in a coupled box model,, J. Phys. Oceanogr., 29 (1999), 351. doi: 10.1175/1520-0485(1999)029<0351:ITCIAC>2.0.CO;2. [5] P. Cessi, A simple box model of stochastically-forced thermohaline flow,, J. Phys. Oceanogr., 24 (1994), 1911. doi: 10.1175/1520-0485(1994)024<1911:ASBMOS>2.0.CO;2. [6] S. M. Griffies and E. Tziperman, A linear thermohaline oscillator driven by stochastic atmospheric forcing,, J. Climate, 8 (1995), 2440. doi: 10.1175/1520-0442(1995)008<2440:ALTODB>2.0.CO;2. [7] G. Lohmann and J. Schneider, Dynamics and predictability of Stommel's box model: A phase space perspective with implications for decadal climate variability,, Tellus., 51 (1998), 326. [8] D. Applebaum, "Lévy Processes and Stochastic Calculus,", Cambridge Studies in Advanced Mathematics, 93 (2004). [9] K.-I. Sato, "Lévy Processes and Infinitely Divisible Distributions,", Cambridge Studies in Advanced Mathematics, 68 (1999). [10] J. Brannan, J. Duan and V. Ervin, Escape probability, mean residence time and geophysical fluid particle dynamics,, Physica D, 133 (1999), 23. doi: 10.1016/S0167-2789(99)00096-2. [11] P. Glendinning, View from the pennines: Box models of the oceanic conveyor belt,, Mathematics Today (Southend-on-Sea), 45 (2009), 230. [12] N. Berglund and B. Gentz, Metastability in simple climate models: Pathwise analysis of slowly driven langevin equations,, Stochastics and Dynamics, 2 (2003), 327. doi: 10.1142/S0219493702000455. [13] J. Marotzke, Analysis of thermohaline feedbacks, in "Decadal Climate Variability: Dynamics and Predictability,", (eds. D. L. T. Anderson and J. Willebrand), (1996). [14] P. Imkeller and I. Pavlyukevich, First exit time of SDEs driven by stable Lévy processes,, Stoch. proc. Appl., 116 (2006), 611. [15] W. A. Woyczy\'nski, Lévy processes in the physical sciences, in "Lévy Processes", (eds. O. E. Barndorff-Nielsen, (2001), 241. [16] Z. Yang and J. Duan, An intermediate regime for exit phenomena driven by non-Gaussian Lévy noises,, Stochastics and Dynamics, 8 (2008), 583. doi: 10.1142/S0219493708002469. [17] H. Dijkstra, "Nonlinear Physical Oceanography,", 2nd edition, (2005). [18] T. Gao, J. Duan, X. Li and R. Song, Mean exit time and escape probability for dynamical systems driven by Lévy noise,, submitted, (2011).

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##### References:
 [1] H. Stommel, Thermohaline convection with two stable regimes of flow,, Tellus., 13 (1961), 224. doi: 10.1111/j.2153-3490.1961.tb00079.x. [2] L. Mass, A simple model for the three-dimensional, thermally and wind-driven ocean circulation,, Tellus., 46 (1994), 671. doi: 10.1034/j.1600-0870.1994.t01-3-00008.x. [3] S. Rahmstorf, On the freshwater forcing and transport of the Atlantic thermohaline circulation,, Clim. Dyn., 12 (1996), 799. doi: 10.1007/s003820050144. [4] J. Scott, J. Marotzke and P. Stone, Interhemispheric thermohaline circulation in a coupled box model,, J. Phys. Oceanogr., 29 (1999), 351. doi: 10.1175/1520-0485(1999)029<0351:ITCIAC>2.0.CO;2. [5] P. Cessi, A simple box model of stochastically-forced thermohaline flow,, J. Phys. Oceanogr., 24 (1994), 1911. doi: 10.1175/1520-0485(1994)024<1911:ASBMOS>2.0.CO;2. [6] S. M. Griffies and E. Tziperman, A linear thermohaline oscillator driven by stochastic atmospheric forcing,, J. Climate, 8 (1995), 2440. doi: 10.1175/1520-0442(1995)008<2440:ALTODB>2.0.CO;2. [7] G. Lohmann and J. Schneider, Dynamics and predictability of Stommel's box model: A phase space perspective with implications for decadal climate variability,, Tellus., 51 (1998), 326. [8] D. Applebaum, "Lévy Processes and Stochastic Calculus,", Cambridge Studies in Advanced Mathematics, 93 (2004). [9] K.-I. Sato, "Lévy Processes and Infinitely Divisible Distributions,", Cambridge Studies in Advanced Mathematics, 68 (1999). [10] J. Brannan, J. Duan and V. Ervin, Escape probability, mean residence time and geophysical fluid particle dynamics,, Physica D, 133 (1999), 23. doi: 10.1016/S0167-2789(99)00096-2. [11] P. Glendinning, View from the pennines: Box models of the oceanic conveyor belt,, Mathematics Today (Southend-on-Sea), 45 (2009), 230. [12] N. Berglund and B. Gentz, Metastability in simple climate models: Pathwise analysis of slowly driven langevin equations,, Stochastics and Dynamics, 2 (2003), 327. doi: 10.1142/S0219493702000455. [13] J. Marotzke, Analysis of thermohaline feedbacks, in "Decadal Climate Variability: Dynamics and Predictability,", (eds. D. L. T. Anderson and J. Willebrand), (1996). [14] P. Imkeller and I. Pavlyukevich, First exit time of SDEs driven by stable Lévy processes,, Stoch. proc. Appl., 116 (2006), 611. [15] W. A. Woyczy\'nski, Lévy processes in the physical sciences, in "Lévy Processes", (eds. O. E. Barndorff-Nielsen, (2001), 241. [16] Z. Yang and J. Duan, An intermediate regime for exit phenomena driven by non-Gaussian Lévy noises,, Stochastics and Dynamics, 8 (2008), 583. doi: 10.1142/S0219493708002469. [17] H. Dijkstra, "Nonlinear Physical Oceanography,", 2nd edition, (2005). [18] T. Gao, J. Duan, X. Li and R. Song, Mean exit time and escape probability for dynamical systems driven by Lévy noise,, submitted, (2011).
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