American Institute of Mathematical Sciences

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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

Xiangjun Wang 1, , Jianghui Wen 1, , Jianping Li 2, and  Jinqiao Duan 3,

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.
Keywords: colored noise., mean exit time, Stommel box model, thermohaline circulation, $\alpha$-stable Lévy motion.
Mathematics Subject Classification: Primary: 60H10, 65C50; Secondary: 86A0.
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), 224-230. doi: 10.1111/j.2153-3490.1961.tb00079.x.  Google Scholar

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[4]

J. Scott, J. Marotzke and P. Stone, Interhemispheric thermohaline circulation in a coupled box model, J. Phys. Oceanogr., 29 (1999), 351-365. doi: 10.1175/1520-0485(1999)029<0351:ITCIAC>2.0.CO;2.  Google Scholar

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P. Cessi, A simple box model of stochastically-forced thermohaline flow, J. Phys. Oceanogr., 24 (1994), 1911-1920. doi: 10.1175/1520-0485(1994)024<1911:ASBMOS>2.0.CO;2.  Google Scholar

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S. M. Griffies and E. Tziperman, A linear thermohaline oscillator driven by stochastic atmospheric forcing, J. Climate, 8 (1995), 2440-2453. doi: 10.1175/1520-0442(1995)008<2440:ALTODB>2.0.CO;2.  Google Scholar

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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), 326-336. Google Scholar

[8]

D. Applebaum, "Lévy Processes and Stochastic Calculus," Cambridge Studies in Advanced Mathematics, 93, Cambridge University Press, Cambridge, 2004.  Google Scholar

[9]

K.-I. Sato, "Lévy Processes and Infinitely Divisible Distributions," Cambridge Studies in Advanced Mathematics, 68, Cambridge University Press, Cambridge, 1999.  Google Scholar

[10]

J. Brannan, J. Duan and V. Ervin, Escape probability, mean residence time and geophysical fluid particle dynamics, Physica D, 133 (1999), 23-33. doi: 10.1016/S0167-2789(99)00096-2.  Google Scholar

[11]

P. Glendinning, View from the pennines: Box models of the oceanic conveyor belt, Mathematics Today (Southend-on-Sea), 45 (2009), 230-232.  Google Scholar

[12]

N. Berglund and B. Gentz, Metastability in simple climate models: Pathwise analysis of slowly driven langevin equations, Stochastics and Dynamics, 2 (2003), 327-356. doi: 10.1142/S0219493702000455.  Google Scholar

[13]

J. Marotzke, Analysis of thermohaline feedbacks, in "Decadal Climate Variability: Dynamics and Predictability," (eds. D. L. T. Anderson and J. Willebrand), Springer-Verlag, 1996. Google Scholar

[14]

P. Imkeller and I. Pavlyukevich, First exit time of SDEs driven by stable Lévy processes, Stoch. proc. Appl., 116 (2006), 611-642.  Google Scholar

[15]

W. A. Woyczy\'nski, Lévy processes in the physical sciences, in "Lévy Processes" (eds. O. E. Barndorff-Nielsen, T. Mikosch and S. I. Resnick), Birkhäuser Boston, Boston, MA, (2001), 241-266.  Google Scholar

[16]

Z. Yang and J. Duan, An intermediate regime for exit phenomena driven by non-Gaussian Lévy noises, Stochastics and Dynamics, 8 (2008), 583-591. doi: 10.1142/S0219493708002469.  Google Scholar

[17]

H. Dijkstra, "Nonlinear Physical Oceanography," 2nd edition, Springer, 2005. Google Scholar

[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. Google Scholar

show all references

References:
[1]

H. Stommel, Thermohaline convection with two stable regimes of flow, Tellus., 13 (1961), 224-230. doi: 10.1111/j.2153-3490.1961.tb00079.x.  Google Scholar

[2]

L. Mass, A simple model for the three-dimensional, thermally and wind-driven ocean circulation, Tellus., 46 (1994), 671-680. doi: 10.1034/j.1600-0870.1994.t01-3-00008.x.  Google Scholar

[3]

S. Rahmstorf, On the freshwater forcing and transport of the Atlantic thermohaline circulation, Clim. Dyn., 12 (1996), 799-811. doi: 10.1007/s003820050144.  Google Scholar

[4]

J. Scott, J. Marotzke and P. Stone, Interhemispheric thermohaline circulation in a coupled box model, J. Phys. Oceanogr., 29 (1999), 351-365. doi: 10.1175/1520-0485(1999)029<0351:ITCIAC>2.0.CO;2.  Google Scholar

[5]

P. Cessi, A simple box model of stochastically-forced thermohaline flow, J. Phys. Oceanogr., 24 (1994), 1911-1920. doi: 10.1175/1520-0485(1994)024<1911:ASBMOS>2.0.CO;2.  Google Scholar

[6]

S. M. Griffies and E. Tziperman, A linear thermohaline oscillator driven by stochastic atmospheric forcing, J. Climate, 8 (1995), 2440-2453. doi: 10.1175/1520-0442(1995)008<2440:ALTODB>2.0.CO;2.  Google Scholar

[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), 326-336. Google Scholar

[8]

D. Applebaum, "Lévy Processes and Stochastic Calculus," Cambridge Studies in Advanced Mathematics, 93, Cambridge University Press, Cambridge, 2004.  Google Scholar

[9]

K.-I. Sato, "Lévy Processes and Infinitely Divisible Distributions," Cambridge Studies in Advanced Mathematics, 68, Cambridge University Press, Cambridge, 1999.  Google Scholar

[10]

J. Brannan, J. Duan and V. Ervin, Escape probability, mean residence time and geophysical fluid particle dynamics, Physica D, 133 (1999), 23-33. doi: 10.1016/S0167-2789(99)00096-2.  Google Scholar

[11]

P. Glendinning, View from the pennines: Box models of the oceanic conveyor belt, Mathematics Today (Southend-on-Sea), 45 (2009), 230-232.  Google Scholar

[12]

N. Berglund and B. Gentz, Metastability in simple climate models: Pathwise analysis of slowly driven langevin equations, Stochastics and Dynamics, 2 (2003), 327-356. doi: 10.1142/S0219493702000455.  Google Scholar

[13]

J. Marotzke, Analysis of thermohaline feedbacks, in "Decadal Climate Variability: Dynamics and Predictability," (eds. D. L. T. Anderson and J. Willebrand), Springer-Verlag, 1996. Google Scholar

[14]

P. Imkeller and I. Pavlyukevich, First exit time of SDEs driven by stable Lévy processes, Stoch. proc. Appl., 116 (2006), 611-642.  Google Scholar

[15]

W. A. Woyczy\'nski, Lévy processes in the physical sciences, in "Lévy Processes" (eds. O. E. Barndorff-Nielsen, T. Mikosch and S. I. Resnick), Birkhäuser Boston, Boston, MA, (2001), 241-266.  Google Scholar

[16]

Z. Yang and J. Duan, An intermediate regime for exit phenomena driven by non-Gaussian Lévy noises, Stochastics and Dynamics, 8 (2008), 583-591. doi: 10.1142/S0219493708002469.  Google Scholar

[17]

H. Dijkstra, "Nonlinear Physical Oceanography," 2nd edition, Springer, 2005. Google Scholar

[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. Google Scholar

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