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Long time numerical stability and asymptotic analysis for the viscoelastic Oldroyd flows
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 |
  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.
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. |
[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. |
[3] |
S. Rahmstorf, On the freshwater forcing and transport of the Atlantic thermohaline circulation, Clim. Dyn., 12 (1996), 799-811.
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), 351-365.
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), 1911-1920.
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), 2440-2453.
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), 326-336. |
[8] |
D. Applebaum, "Lévy Processes and Stochastic Calculus," Cambridge Studies in Advanced Mathematics, 93, Cambridge University Press, Cambridge, 2004. |
[9] |
K.-I. Sato, "Lévy Processes and Infinitely Divisible Distributions," Cambridge Studies in Advanced Mathematics, 68, Cambridge University Press, Cambridge, 1999. |
[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. |
[11] |
P. Glendinning, View from the pennines: Box models of the oceanic conveyor belt, Mathematics Today (Southend-on-Sea), 45 (2009), 230-232. |
[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. |
[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. |
[14] |
P. Imkeller and I. Pavlyukevich, First exit time of SDEs driven by stable Lévy processes, Stoch. proc. Appl., 116 (2006), 611-642. |
[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. |
[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. |
[17] |
H. Dijkstra, "Nonlinear Physical Oceanography," 2nd edition, Springer, 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. |
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. |
[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. |
[3] |
S. Rahmstorf, On the freshwater forcing and transport of the Atlantic thermohaline circulation, Clim. Dyn., 12 (1996), 799-811.
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), 351-365.
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), 1911-1920.
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), 2440-2453.
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), 326-336. |
[8] |
D. Applebaum, "Lévy Processes and Stochastic Calculus," Cambridge Studies in Advanced Mathematics, 93, Cambridge University Press, Cambridge, 2004. |
[9] |
K.-I. Sato, "Lévy Processes and Infinitely Divisible Distributions," Cambridge Studies in Advanced Mathematics, 68, Cambridge University Press, Cambridge, 1999. |
[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. |
[11] |
P. Glendinning, View from the pennines: Box models of the oceanic conveyor belt, Mathematics Today (Southend-on-Sea), 45 (2009), 230-232. |
[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. |
[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. |
[14] |
P. Imkeller and I. Pavlyukevich, First exit time of SDEs driven by stable Lévy processes, Stoch. proc. Appl., 116 (2006), 611-642. |
[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. |
[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. |
[17] |
H. Dijkstra, "Nonlinear Physical Oceanography," 2nd edition, Springer, 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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