Time-averages of fast oscillatory systems

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

Time-averages of fast oscillatory systems

Bin Cheng ^1, and Alex Mahalov ^1,

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

Abstract
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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.

Keywords: coherent structures., time-averages, Mathematical geosciences, waves.

Mathematics Subject Classification: Primary: 76U05; Secondary: 74J99, 35Q8.

Citation: Bin Cheng, Alex Mahalov. Time-averages of fast oscillatory systems. Discrete and 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, Birkhäuser Verlag, Basel, 2001.
[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), 291-300.
[3]	A. Babin, A. Mahalov and B. Nicolaenko, Global splitting and regularity of rotating shallow-water equations, European J. Mech. B Fluids, 16 (1997), 725-754.
[4]	H. Beirão da Veiga, On the barotropic motion of compressible perfect fluids, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 8 (1981), 317-351.
[5]	Bin Cheng, Singular limits and convergence rates of compressible Euler and rotating shallow water equations, SIAM J. on Mathematical Analysis, 44 (2012), 1050-1076. doi: 10.1137/11085147X.
[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), 48-58. doi: 10.1016/j.euromechflu.2012.06.001.
[7]	D. G. Ebin, The motion of slightly compressible fluids viewed as a motion with strong constraining force, Ann. of Math. (2), 105 (1977), 141-200.
[8]	David G. Ebin, Motion of slightly compressible fluids in a bounded domain. I. Comm. Pure Appl. Math., 35 (1982), 451-485. doi: 10.1002/cpa.3160350402.
[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), L13303.
[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), 83-98.
[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), 316-330.
[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), 225-240.
[13]	N. A. Maximenko, B. Bang and H. Sasaki, Observational evidence of alternating jets in the world ocean, Geophys. Res. Lett., 32 (2005), L12607.
[14]	, NASA/JPL/University of Arizona, http://photojournal.jpl.nasa.gov/catalog/PIA02873.
[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), 481-524. doi: 10.1002/cpa.3160340405.
[16]	H.-O. Kreiss, Problems with different time scales for partial differential equations, Comm. Pure Appl. Math., 33 (1980), 399-439. doi: 10.1002/cpa.3160330310.
[17]	T. Nozawa and S. Yoden, Formation of zonal band structure in forced two-dimensional turbulence on a rotating sphere, Phys. Fluids, 9 (1997), 2081-2093. doi: 10.1063/1.869327.
[18]	C. Porco, et al., Cassini imaging of Jupiter atmosphere, satellites and rings, Science, 299 (2003), 1541-1547.
[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), 12919-12939.
[20]	G. Roden, Flow and water property structures between the Bering Sea and Fiji in the summer of 1993, J. Geophys. Res., 105 (2000), 28595-28612.
[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), 124501.
[22]	G. Vallis and M. Maltrud, Generation of mean flows and jets on a beta plane and over topography, J. Phys. Oceanogr., 23 (1993), 1346-1362.

show all references

References:

[1]	W. Arendt, C. Batty, M. Hieber and F. Neubrander, "Vector-Valued Laplace Transforms and Cauchy Problems," Monographs in Mathematics, 96, Birkhäuser Verlag, Basel, 2001.
[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), 291-300.
[3]	A. Babin, A. Mahalov and B. Nicolaenko, Global splitting and regularity of rotating shallow-water equations, European J. Mech. B Fluids, 16 (1997), 725-754.
[4]	H. Beirão da Veiga, On the barotropic motion of compressible perfect fluids, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 8 (1981), 317-351.
[5]	Bin Cheng, Singular limits and convergence rates of compressible Euler and rotating shallow water equations, SIAM J. on Mathematical Analysis, 44 (2012), 1050-1076. doi: 10.1137/11085147X.
[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), 48-58. doi: 10.1016/j.euromechflu.2012.06.001.
[7]	D. G. Ebin, The motion of slightly compressible fluids viewed as a motion with strong constraining force, Ann. of Math. (2), 105 (1977), 141-200.
[8]	David G. Ebin, Motion of slightly compressible fluids in a bounded domain. I. Comm. Pure Appl. Math., 35 (1982), 451-485. doi: 10.1002/cpa.3160350402.
[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), L13303.
[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), 83-98.
[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), 316-330.
[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), 225-240.
[13]	N. A. Maximenko, B. Bang and H. Sasaki, Observational evidence of alternating jets in the world ocean, Geophys. Res. Lett., 32 (2005), L12607.
[14]	, NASA/JPL/University of Arizona, http://photojournal.jpl.nasa.gov/catalog/PIA02873.
[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), 481-524. doi: 10.1002/cpa.3160340405.
[16]	H.-O. Kreiss, Problems with different time scales for partial differential equations, Comm. Pure Appl. Math., 33 (1980), 399-439. doi: 10.1002/cpa.3160330310.
[17]	T. Nozawa and S. Yoden, Formation of zonal band structure in forced two-dimensional turbulence on a rotating sphere, Phys. Fluids, 9 (1997), 2081-2093. doi: 10.1063/1.869327.
[18]	C. Porco, et al., Cassini imaging of Jupiter atmosphere, satellites and rings, Science, 299 (2003), 1541-1547.
[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), 12919-12939.
[20]	G. Roden, Flow and water property structures between the Bering Sea and Fiji in the summer of 1993, J. Geophys. Res., 105 (2000), 28595-28612.
[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), 124501.
[22]	G. Vallis and M. Maltrud, Generation of mean flows and jets on a beta plane and over topography, J. Phys. Oceanogr., 23 (1993), 1346-1362.

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