American Institute of Mathematical Sciences

Advanced
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.  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.  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.  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.  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.  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.  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.   Google Scholar

[8]

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

[9]

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

[15]

W. A. Woyczy\'nski, Lévy processes in the physical sciences, in "Lévy Processes", (eds. O. E. Barndorff-Nielsen, (2001), 241.   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.  doi: 10.1142/S0219493708002469.  Google Scholar

[17]

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

[8]

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

[9]

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

[15]

W. A. Woyczy\'nski, Lévy processes in the physical sciences, in "Lévy Processes", (eds. O. E. Barndorff-Nielsen, (2001), 241.   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.  doi: 10.1142/S0219493708002469.  Google Scholar

[17]

H. Dijkstra, "Nonlinear Physical Oceanography,", 2nd edition, (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

[1]

Badr-eddine Berrhazi, Mohamed El Fatini, Tomás Caraballo, Roger Pettersson. A stochastic SIRI epidemic model with Lévy noise. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2415-2431. doi: 10.3934/dcdsb.2018057
[2]

Leanne Dong. Random attractors for stochastic Navier-Stokes equation on a 2D rotating sphere with stable Lévy noise. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020352
[3]

Ziheng Chen, Siqing Gan, Xiaojie Wang. Mean-square approximations of Lévy noise driven SDEs with super-linearly growing diffusion and jump coefficients. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 4513-4545. doi: 10.3934/dcdsb.2019154
[4]

Yong-Kum Cho. On the Boltzmann equation with the symmetric stable Lévy process. Kinetic & Related Models, 2015, 8 (1) : 53-77. doi: 10.3934/krm.2015.8.53
[5]

Martin Redmann, Melina A. Freitag. Balanced model order reduction for linear random dynamical systems driven by Lévy noise. Journal of Computational Dynamics, 2018, 5 (1&2) : 33-59. doi: 10.3934/jcd.2018002
[6]

Chao Xing, Ping Zhou, Hong Luo. The steady state solutions to thermohaline circulation equations. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3709-3722. doi: 10.3934/dcdsb.2016117
[7]

Hongjun Gao, Jinqiao Duan. Dynamics of the thermohaline circulation under wind forcing. Discrete & Continuous Dynamical Systems - B, 2002, 2 (2) : 205-219. doi: 10.3934/dcdsb.2002.2.205
[8]

Jiao Song, Jiang-Lun Wu, Fangzhou Huang. First jump time in simulation of sampling trajectories of affine jump-diffusions driven by $ \alpha $-stable white noise. Communications on Pure & Applied Analysis, 2020, 19 (8) : 4127-4142. doi: 10.3934/cpaa.2020184
[9]

David Lipshutz. Exit time asymptotics for small noise stochastic delay differential equations. Discrete & Continuous Dynamical Systems - A, 2018, 38 (6) : 3099-3138. doi: 10.3934/dcds.2018135
[10]

Adam Andersson, Felix Lindner. Malliavin regularity and weak approximation of semilinear SPDEs with Lévy noise. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 4271-4294. doi: 10.3934/dcdsb.2019081
[11]

Tomasz Kosmala, Markus Riedle. Variational solutions of stochastic partial differential equations with cylindrical Lévy noise. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020209
[12]

Kexue Li, Jigen Peng, Junxiong Jia. Explosive solutions of parabolic stochastic partial differential equations with lévy noise. Discrete & Continuous Dynamical Systems - A, 2017, 37 (10) : 5105-5125. doi: 10.3934/dcds.2017221
[13]

Justin Cyr, Phuong Nguyen, Sisi Tang, Roger Temam. Review of local and global existence results for stochastic pdes with Lévy noise. Discrete & Continuous Dynamical Systems - A, 2020, 40 (10) : 5639-5710. doi: 10.3934/dcds.2020241
[14]

Chaman Kumar, Sotirios Sabanis. On tamed milstein schemes of SDEs driven by Lévy noise. Discrete & Continuous Dynamical Systems - B, 2017, 22 (2) : 421-463. doi: 10.3934/dcdsb.2017020
[15]

Justin Cyr, Phuong Nguyen, Roger Temam. Stochastic one layer shallow water equations with Lévy noise. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 3765-3818. doi: 10.3934/dcdsb.2018331
[16]

Jonathan C. Mattingly, Etienne Pardoux. Invariant measure selection by noise. An example. Discrete & Continuous Dynamical Systems - A, 2014, 34 (10) : 4223-4257. doi: 10.3934/dcds.2014.34.4223
[17]

Josselin Garnier, George Papanicolaou, Tzu-Wei Yang. Mean field model for collective motion bistability. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 851-879. doi: 10.3934/dcdsb.2018210
[18]

Xueqin Li, Chao Tang, Tianmin Huang. Poisson $S^2$-almost automorphy for stochastic processes and its applications to SPDEs driven by Lévy noise. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3309-3345. doi: 10.3934/dcdsb.2018282
[19]

Markus Riedle, Jianliang Zhai. Large deviations for stochastic heat equations with memory driven by Lévy-type noise. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 1983-2005. doi: 10.3934/dcds.2018080
[20]

Chaman Kumar. On Milstein-type scheme for SDE driven by Lévy noise with super-linear coefficients. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1405-1446. doi: 10.3934/dcdsb.2020167

2019 Impact Factor: 1.27

Tools

Metrics

  • PDF downloads (32)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences