Abstract
Motivated by important geophysical applications we study a non-linear model of geyser dynamics under the influence of external stochastic forcing. It is shown that the deterministic dynamics is substantially dependent on system parameters leading to the following evolutionary scenaria: (i) oscillations near a stable equilibrium and a transient tendency of the phase trajectories to a spiral sink or a stable node (pre-eruption regime), and (ii) fast escape from equilibrium (eruption regime). Even a small noise changes the system dynamics drastically. Namely, a low-intensity noise generates the small amplitude stochastic oscillations in the regions adjoining to the stable equilibrium point. A small buildup of noise intensity throws the system over its separatrix and leads to eruption. The role of the friction coefficient and relative pressure in the deterministic and stochastic dynamics is studied by direct numerical simulations and stochastic sensitivity functions technique.
Similar content being viewed by others
References
S. Hurwitz, R.A. Sohn, K. Luttrell, M. Manga, J. Geophys. Res. Solid Earth 119, 1718 (2014)
S. Hurwitz, A. Kumar, R. Taylor, H. Heasler, Geology 36, 451 (2008)
J. Vandemeulebrouck, R.A. Sohn, M.L. Rudolph, S. Hurwitz, M. Manga, M.J.S. Johnston, S.A. Soule, D. McPhee, J.M.G. Glen, L. Karlstrom, F. Murphy, J. Geophys. Res. Solid Earth 119, 8688 (2014)
J.A. Glennon, R.M. Pfaff, The GOSA Trans. 9, 184 (2004)
P.G. Silver, N.J. Valette-Silver, Science 257, 1364 (1992)
J.S. Rhinehart, Science 177, 342 (1972)
S. Rojstaczer, D.L. Galloway, S.E. Ingebritsen, D.M. Rubin, Geophys. Res. Lett. 30, 1953 (2003)
J.S. Rhinehart, Geysers and Geothermal Energy (Springer-Verlag, 1980)
S. Husen, R. Taylor, R.B. Smith, H. Healser, Geology 32, 537 (2004)
Old Faithful Virtual Visitor Center, http://mms.nps.gov/yell/ofvec/exhibits/eruption/prediction/
K.D. O’Hara, E.K. Esawi, GSA Today 23, 4 (2013)
P.S. Landa, D.A. Vlasov, Dokl. Math. 73, 623 (2007)
W. Horstemke, R. Lefever, Noise-Induced Transitions (Springer, Berlin, 1984)
D.V. Alexandrov, I.A. Bashkirtseva, L.B. Ryashko, Eur. Phys. J. B 88, 304 (2015)
Y.C. Lai, T. Tél, Transient Chaos: Complex Dynamics on Finite Time Scales (Springer, Berlin, 2011)
I. Bashkirtseva, G. Chen, L. Ryashko, Chaos 22, 033104 (2012)
L. Gammaitoni, P. Hänggi, P. Jung, F. Marchesoni, Rev. Mod. Phys. 70, 223 (1998)
M.D. McDonnell, N.G. Stocks, C.E.M. Pearce, D. Abbott, Stochastic Resonance: from Suprathreshold Stochastic Resonance to Stochastic Signal Quantization (Cambridge University Press, Cambridge, 2008)
B. Lindner, J. Garcia-Ojalvo, A. Neiman, L. Schimansky-Geier, Phys. Rep. 392, 321 (2004)
P. Hänggi, F. Marchesoni, Rev. Mod. Phys. 81, 387 (2009)
D.V. Alexandrov, I.A. Bashkirtseva, L.B. Ryashko, Eur. Phys. J. B 88, 106 (2015)
S.A. Menchon, H.S. Wio, Eur. Phys. J. B 87, 184 (2014)
J. Dowden, P. Kapadia, G. Brown, H. Rymer, J. Geophys. Res. Solid Earth 96, 18059 (1991)
Y. Sang, M. Dubé, M. Grant, Phys. Rev. E 77, 036123 (2008)
M.H. Müser, Phys. Rev. E 84, 125419 (2011)
S.E. Ingebritsen, S.A. Rojstaczer, J. Geophys. Res. Solid Earth 101, 21891 (1996)
L. Karlstrom, S. Hurwitz, R. Sohn, J. Vandemeulebrouck, F. Murphy, M.L. Rudolph, M.J.S. Johnston, M. Manga, R.B. McCleskey, J. Geophys. Res. Solid Earth 118, 4048 (2013)
E. Balcerak, C. Schultz, Eos 94, 296 (2013)
J. Garcia-Ojalvo, J. Sancho, Noise in Spatially Extended Systems (Springer, 1999)
S. Fedotov, I. Bashkirtseva, L. Ryashko, Phys. Rev. E 66, 066310 (2002)
S. Fedotov, I. Bashkirtseva, L. Ryashko, Phys. Rev. E 73, 066307 (2006)
M.I. Freidlin, A.D. Wentzell, Random Perturbations of Dynamical Systems (Springer, New York, 1984)
I.A. Bashkirtseva, L.B. Ryashko, Chaos Sol. Fract. 26, 1437 (2005)
I.A. Bashkirtseva, L.B. Ryashko, Chaos 21, 047514 (2011)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Alexandrov, D., Bashkirtseva, I. & Ryashko, L. Analysis of noise-induced eruptions in a geyser model. Eur. Phys. J. B 89, 62 (2016). https://doi.org/10.1140/epjb/e2016-60982-0
Received:
Published:
DOI: https://doi.org/10.1140/epjb/e2016-60982-0