# American Institute of Mathematical Sciences

October  2013, 6(5): 1151-1162. doi: 10.3934/dcdss.2013.6.1151

## Time-averages of fast oscillatory systems

 1 School of Mathematical and Statistical Sciences, Arizona State University, Wexler Hall (PSA), Tempe, Arizona, 85287-1804, United States, United States

Received  December 2011 Revised  March 2012 Published  March 2013

Time-averages are common observables in analysis of experimental data and numerical simulations of physical systems. We describe a straightforward framework for studying time-averages of dynamical systems whose solutions exhibit fast oscillatory behaviors. Time integration averages out the oscillatory part of the solution that is caused by the large skew-symmetric operator. Then, the time-average of the solution stays close to the kernel of this operator. The key assumption in this framework is that the inverse of the large operator is a bounded mapping between certain Hilbert spaces modular the kernel of the operator itself. This assumption is verified for several examples of time-dependent PDEs.
Citation: Bin Cheng, Alex Mahalov. Time-averages of fast oscillatory systems. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1151-1162. doi: 10.3934/dcdss.2013.6.1151
##### References:
 [1] W. Arendt, C. Batty, M. Hieber and F. Neubrander, "Vector-Valued Laplace Transforms and Cauchy Problems,", Monographs in Mathematics, 96 (2001). Google Scholar [2] A. Babin, A. Mahalov and B. Nicolaenko, Global splitting, integrability and regularity of 3D Euler and Navier-Stokes equations for uniformly rotating fluids,, European J. Mechanics B Fluids, 15 (1996), 291. Google Scholar [3] A. Babin, A. Mahalov and B. Nicolaenko, Global splitting and regularity of rotating shallow-water equations,, European J. Mech. B Fluids, 16 (1997), 725. Google Scholar [4] H. Beirão da Veiga, On the barotropic motion of compressible perfect fluids,, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 8 (1981), 317. Google Scholar [5] Bin Cheng, Singular limits and convergence rates of compressible Euler and rotating shallow water equations,, SIAM J. on Mathematical Analysis, 44 (2012), 1050. doi: 10.1137/11085147X. Google Scholar [6] Bin Cheng and Alex Mahalov, Euler equations on a fast rotating sphere-time-averages and zonal flows,, European J. Mech. - B/Fluids, 37 (2013), 48. doi: 10.1016/j.euromechflu.2012.06.001. Google Scholar [7] D. G. Ebin, The motion of slightly compressible fluids viewed as a motion with strong constraining force,, Ann. of Math. (2), 105 (1977), 141. Google Scholar [8] David G. Ebin, Motion of slightly compressible fluids in a bounded domain. I., Comm. Pure Appl. Math., 35 (1982), 451. doi: 10.1002/cpa.3160350402. Google Scholar [9] B. Galperin, H. Nakano, H. Huang and S. Sukoriansky, The ubiquitous zonal jets in the atmospheres of giant planets and Earth oceans,, Geophys. Res. Lett., 31 (2004). Google Scholar [10] B. Galperin, S. Sukoriansky, N. Dikovskaya, P. L. Read, Y. H. Yamazaki and R. Wordsworth, Anisotropic turbulence and zonal jets in rotating flows with a $\beta$-effect,, Nonlinear Processes in Geophysics, 13 (2006), 83. Google Scholar [11] E. Garcýa-Melendo and A. Sánchez-Lavega, A study of the stability of Jovian zonal winds from HST images: 1995-2000,, Icarus, 152 (2001), 316. Google Scholar [12] H.-P. Huang, B. Galperin and S. Sukoriansky, Anisotropic spectra in two-dimensional turbulence on the surface of a rotating sphere,, Phys. Fluids, 13 (2001), 225. Google Scholar [13] N. A. Maximenko, B. Bang and H. Sasaki, Observational evidence of alternating jets in the world ocean,, Geophys. Res. Lett., 32 (2005). Google Scholar [14] , NASA/JPL/University of Arizona,, , (). Google Scholar [15] Sergiu Klainerman and Andrew Majda, Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids,, Comm. Pure Appl. Math., 34 (1981), 481. doi: 10.1002/cpa.3160340405. Google Scholar [16] H.-O. Kreiss, Problems with different time scales for partial differential equations,, Comm. Pure Appl. Math., 33 (1980), 399. doi: 10.1002/cpa.3160330310. Google Scholar [17] T. Nozawa and S. Yoden, Formation of zonal band structure in forced two-dimensional turbulence on a rotating sphere,, Phys. Fluids, 9 (1997), 2081. doi: 10.1063/1.869327. Google Scholar [18] C. Porco, et al., Cassini imaging of Jupiter atmosphere, satellites and rings,, Science, 299 (2003), 1541. Google Scholar [19] G. Roden, Upper ocean thermohaline, oxygen, nutrients, and flow structure near the date line in the summer of 1993,, J. Geophys. Res., 103 (1998), 12919. Google Scholar [20] G. Roden, Flow and water property structures between the Bering Sea and Fiji in the summer of 1993,, J. Geophys. Res., 105 (2000), 28595. Google Scholar [21] S. Sukoriansky, B. Galperin and N. Dikovskaya, Universal spectrum of two-dimensional turbulence on a rotating sphere and some basic features of atmospheric circulation on giant planets,, Phys. Rev. Lett., 89 (2002). Google Scholar [22] G. Vallis and M. Maltrud, Generation of mean flows and jets on a beta plane and over topography,, J. Phys. Oceanogr., 23 (1993), 1346. Google Scholar

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##### References:
 [1] W. Arendt, C. Batty, M. Hieber and F. Neubrander, "Vector-Valued Laplace Transforms and Cauchy Problems,", Monographs in Mathematics, 96 (2001). Google Scholar [2] A. Babin, A. Mahalov and B. Nicolaenko, Global splitting, integrability and regularity of 3D Euler and Navier-Stokes equations for uniformly rotating fluids,, European J. Mechanics B Fluids, 15 (1996), 291. Google Scholar [3] A. Babin, A. Mahalov and B. Nicolaenko, Global splitting and regularity of rotating shallow-water equations,, European J. Mech. B Fluids, 16 (1997), 725. Google Scholar [4] H. Beirão da Veiga, On the barotropic motion of compressible perfect fluids,, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 8 (1981), 317. Google Scholar [5] Bin Cheng, Singular limits and convergence rates of compressible Euler and rotating shallow water equations,, SIAM J. on Mathematical Analysis, 44 (2012), 1050. doi: 10.1137/11085147X. Google Scholar [6] Bin Cheng and Alex Mahalov, Euler equations on a fast rotating sphere-time-averages and zonal flows,, European J. Mech. - B/Fluids, 37 (2013), 48. doi: 10.1016/j.euromechflu.2012.06.001. Google Scholar [7] D. G. Ebin, The motion of slightly compressible fluids viewed as a motion with strong constraining force,, Ann. of Math. (2), 105 (1977), 141. Google Scholar [8] David G. Ebin, Motion of slightly compressible fluids in a bounded domain. I., Comm. Pure Appl. Math., 35 (1982), 451. doi: 10.1002/cpa.3160350402. Google Scholar [9] B. Galperin, H. Nakano, H. Huang and S. Sukoriansky, The ubiquitous zonal jets in the atmospheres of giant planets and Earth oceans,, Geophys. Res. Lett., 31 (2004). Google Scholar [10] B. Galperin, S. Sukoriansky, N. Dikovskaya, P. L. Read, Y. H. Yamazaki and R. Wordsworth, Anisotropic turbulence and zonal jets in rotating flows with a $\beta$-effect,, Nonlinear Processes in Geophysics, 13 (2006), 83. Google Scholar [11] E. Garcýa-Melendo and A. Sánchez-Lavega, A study of the stability of Jovian zonal winds from HST images: 1995-2000,, Icarus, 152 (2001), 316. Google Scholar [12] H.-P. Huang, B. Galperin and S. Sukoriansky, Anisotropic spectra in two-dimensional turbulence on the surface of a rotating sphere,, Phys. Fluids, 13 (2001), 225. Google Scholar [13] N. A. Maximenko, B. Bang and H. Sasaki, Observational evidence of alternating jets in the world ocean,, Geophys. Res. Lett., 32 (2005). Google Scholar [14] , NASA/JPL/University of Arizona,, , (). Google Scholar [15] Sergiu Klainerman and Andrew Majda, Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids,, Comm. Pure Appl. Math., 34 (1981), 481. doi: 10.1002/cpa.3160340405. Google Scholar [16] H.-O. Kreiss, Problems with different time scales for partial differential equations,, Comm. Pure Appl. Math., 33 (1980), 399. doi: 10.1002/cpa.3160330310. Google Scholar [17] T. Nozawa and S. Yoden, Formation of zonal band structure in forced two-dimensional turbulence on a rotating sphere,, Phys. Fluids, 9 (1997), 2081. doi: 10.1063/1.869327. Google Scholar [18] C. Porco, et al., Cassini imaging of Jupiter atmosphere, satellites and rings,, Science, 299 (2003), 1541. Google Scholar [19] G. Roden, Upper ocean thermohaline, oxygen, nutrients, and flow structure near the date line in the summer of 1993,, J. Geophys. Res., 103 (1998), 12919. Google Scholar [20] G. Roden, Flow and water property structures between the Bering Sea and Fiji in the summer of 1993,, J. Geophys. Res., 105 (2000), 28595. Google Scholar [21] S. Sukoriansky, B. Galperin and N. Dikovskaya, Universal spectrum of two-dimensional turbulence on a rotating sphere and some basic features of atmospheric circulation on giant planets,, Phys. Rev. Lett., 89 (2002). Google Scholar [22] G. Vallis and M. Maltrud, Generation of mean flows and jets on a beta plane and over topography,, J. Phys. Oceanogr., 23 (1993), 1346. Google Scholar
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