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Ocean–atmosphere dynamics and Rossby waves in fractal anisotropic media

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Abstract

We study some basic problems arising in ocean–atmosphere dynamics including the ocean surface waves and Rossby waves based on the concept of product-like fractal measure introduced recently by Li and Ostoja-Starzewski in their formulation of anisotropic media. We have derived the fractal fluid equations and we have analyzed several fundamental problems related to earth’s ocean and Rossby waves. This study demonstrates the mutual effects of eastward-propagating stationary barotropic Rossby waves and westward-propagating stationary barotropic Rossby waves in the atmosphere which are in agreement with some recent studies and satellite observations. Numerical estimates of fractal dimensions were also given.

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References

  • Alonso JJ, Villares P, Gonzalez MJ, Arias M, Marin B (2005) Ocean tides and fractal geometry: tidal station stability. Thalassas 21:21–16

    Google Scholar 

  • Aschwanden MJ, Aschwanden PD (2008a) Solar flare geometries. I. The area fractal dimension. Astrophys J 674:530–543

    Google Scholar 

  • Aschwanden MJ, Aschwanden PD (2008b) Solar flare geometries. II. The volume fractal dimension. Astrophys J 674:544–553

    Google Scholar 

  • Baines PG (1997) A fractal world of cloistered waves. Nature 388:518–519

    Google Scholar 

  • Balankin AS, Elizarraraz BE (2012) Map of fluid flow in fractal porous medium into fractal continuum flow. Phys Rev E 85:056314

    Google Scholar 

  • Balankin AS, Susarrey O, Mora Santos CA, Patiño J, Yoguez A, García EI (2011) Stress concentration and size effect in fracture of notched heterogeneous material. Phys Rev E 83:015101

    Google Scholar 

  • Balankin AS, Nena B, Susarrey O, Samayoa D (2017) Steady laminar flow of fractal fluids. Phys Lett A 381:623–638

    Google Scholar 

  • Bandyopadhyay G, Chattopadhyay (2008) A probe into the chaotic nature of total ozone time series by Correlation Dimension method. Soft Comp 12:1007–1012

    Google Scholar 

  • Barton CC, La Pointe PR (1995) Fractals in the earth sciences. Springer, New York

    Google Scholar 

  • Benilov ES, Shrira GVG (1992) Nonlinear interaction of a zonal jet and barotropic Rossby-wave turbulence: the problem of turbulent friction. Dyn Atmos Oceans 16:338–353

    Google Scholar 

  • Bershadskii AG (1990) Large-scale fractal structure in laboratory turbulence, astrophysics, and the ocean. Soviet Phys Uspekhi 33:1073

    Google Scholar 

  • Bingham RJ (2009) Nonlinear anisotropic diffusive filtering applied to the ocean’s mean dynamic topography. Remote Sens Lett 1:205–212

    Google Scholar 

  • Brown MG, Smith KB (1991) Stirring the oceans by chaotic advection. Phys Fluids A 3:1186–1192

    Google Scholar 

  • Chakraborty S, Chattopadhyay S (2021) Exploring the Indian summer monsoon rainfall through multifractal detrended fluctuation analysis and the principle of entropy maximization. Earth Sci Inform 14:1571–1577

    Google Scholar 

  • Chattopadhyay G, Chattopadhyay S, Midya SK (2021) Fuzzy binary relation based elucidation of air quality over a highly polluted urban region of India. Earth Sci Inform 14:1625–1631

    Google Scholar 

  • Corley S, Jacobson T (1996) Hawking spectrum and high frequency dispersion. Phys Rev D 54:1568

    Google Scholar 

  • Crossley DJ, Jensen OG (1989) Fractal velocity models in refraction seismology. In: Scholz CH, Mandelbrot BB (eds) Fractals in geophysics, pp 61–76

  • de Michelle M, Leprince S, Thiebot J, Raucoules D, Binet R (2012) Direct measurement of ocean waves velocity field from a single SPOT-5 database. Rem Sens Environ 119:266–271

    Google Scholar 

  • Demmie PN, Ostoja-Starzewski M (2011) Waves in fractal media. J Elasticity 104:187

    Google Scholar 

  • Dimri VP, Srivastava K (2015) Introduction for the special issue: the role of fractals in seismology. Nat Hazards 77:1–4

    Google Scholar 

  • Dimri VP (2005) Fractals in geophysics and seismology: an introduction. In: Dimri VP (eds) Fractal behaviour of the earth system. Springer, Berlin. https://doi.org/10.1007/3-540-26536-8_1

  • Dyski AV (2004) Continuum fractal mechanics of the earth’s crust. Pure Appl Geophys 161:1979

    Google Scholar 

  • Elgar S, Mayer-Kress G (1989) Observations of the fractal dimension of deep and shallow-water ocean surface gravity waves. Phys D Nonlinear Phenomena 27:104–108

    Google Scholar 

  • El-Nabulsi RA (2011a) The fractional Boltzmann transport equation. Comput Math Appl 62:1568–1575

    Google Scholar 

  • El-Nabulsi RA (2011b) The fractional white dwarf hydrodynamical nonlinear differential equation and emergence of quark stars. Appl Math Comput 218:2837–2849

    Google Scholar 

  • El-Nabulsi RA (2017) Nonlocal-in-time kinetic energy in nonconservative fractional systems, disordered dynamics, jerk and snap and oscillatory motions in the rotating fluid tube. Int J Nonlinear Mech 93:65–81

    Google Scholar 

  • El-Nabulsi RA (2018) Path integral formulation of fractionally perturbed Lagrangian oscillators on fractal. J Stat Phys 172:1617–1640

    Google Scholar 

  • El-Nabulsi RA (2019a) Emergence of quasiperiodic quantum wave functions in Hausdorff dimensional crystals and improved intrinsic Carrier concentrations. J Phys Chem Sol 127:224–230

    Google Scholar 

  • El-Nabulsi RA (2019b) Fractional Navier–Stokes equation from fractional velocity arguments and its implications in fluid flows and microfilaments. Int J Nonlinear Sci Numer Simul 20:449–459

    Google Scholar 

  • El-Nabulsi RA (2019c) Geostrophic flow and wind driven ocean currents based on dimensionality of the space medium. Pure Appl Geophys 176:2739–2750

    Google Scholar 

  • El-Nabulsi RA (2020) On a new fractional uncertainty relation and its implications in quantum mechanics and molecular physics. Proc R Soc A 476:20190729

    Google Scholar 

  • El-Nabulsi RA (2021a) On nonlocal fractal laminar steady and unsteady flows. Acta Mech 232:1413–1424

    Google Scholar 

  • El-Nabulsi RA (2021b) Thermal transport equations in porous media from product-like fractal measure. J Therm Stress 44:899–912

    Google Scholar 

  • El-Nabulsi RA (2021c) Superconductivity and nucleation from fractal anisotropy and product-like fractal measure. Proc R Soc A 477:20210065

    Google Scholar 

  • El-Nabulsi RA (2021d) Quantum dynamics in low-dimensional systems with position-dependent mass and product-like fractal geometry. Phys E Low Dimens Syst Nanostruct 134:114827

    Google Scholar 

  • El-Nabulsi RA (2021e) Quantization of Foster mesoscopic circuit and DC-pumped Josephson parametric amplifier from fractal measure arguments. Phys E Low Dimens Syst Nanostruct 133:114845

    Google Scholar 

  • El-Nabulsi RA (2021f) Position-dependent mass fractal Schrodinger equation from fractal anisotropy and product-like fractal measure and its implications in quantum dots and nanocrystals. Opt Quant Elect 53:503

    Google Scholar 

  • El-Nabulsi RA (2021g) Fractal neutrons diffusion equation: uniformization of heat and fuel burn-up in nuclear reactor. Nucl Eng Des 380:111312

    Google Scholar 

  • El-Nabulsi RA (2021h) Fractal Pennes and Cattaneo-Vernotte bioheat equations from product-like fractal geometry and their implications on cells in the presence of tumour growth. J R Soc Interface. https://doi.org/10.1098/rsif.2021.0564

    Article  Google Scholar 

  • Falconer KJ (2003) Fractal geometry-mathematical foundations and applications. Wiley, New York

    Google Scholar 

  • Fontan A, Cornuelle B (2015) Anisotropic response of surface circulation to wind forcing, as inferred from high-frequency radar currents in the southeastern Bay of Biscay. Adv Earth Space Sci 120:2945–2957

    Google Scholar 

  • Fu L, Chen Y, Yang H (2019) Time-space fractional coupled generalized Zakharov-Kuznetsov equations set for Rossby solitary waves in two-layer fluids. Mathematics 7:41

    Google Scholar 

  • Galperin B, Sukoriansky S, Dikovskaya N, Read P, Yamazaki Y, Wordsworth R (2006) Anisotropic turbulence and zonal jets in rotating flows with a β-effect. Nonlinear Process Geophys 13:83–98

    Google Scholar 

  • Haslam BD, Ronney PD (1995) Fractal properties of propagating fronts in a strongly stirred fluid. Phys Fluid 7:1931–1937

    Google Scholar 

  • Herzfeld UC, Kim II, Orcutt JA, Fox CG (1993) Fractal geometry and seafloor topography: theoretical concepts versus data analysis for the Juan de Fuca Ridge and the East Pacific Rise. Ann Geophys 11:532–541

    Google Scholar 

  • Herzfeld UC, Kim II, Orcutt JA (1995) Is the ocean floor a fractal? Math Geo 27:421–462

    Google Scholar 

  • Hirabayashi T, Ito K, Yoshi (1992) Multifractal analysis of earthquakes. Pure Appl Geophys 138:591–610

    Google Scholar 

  • Hirata T, Imoto M (1992) Multifractal analysis of spatial distribution of micro earthquakes in the Kanto region. Geophys Int J 107:155–162

    Google Scholar 

  • Holthuijsen LH (2007) Waves in oceanic and coastal waters. Cambridge University Press, Cambridge

    Google Scholar 

  • Hoskins BJ, Karoly K (1981) The steady response of a spherical atmosphere to thermal and orographic forcing. J Atmos Sci 38:1179–1196

    Google Scholar 

  • Hurst D, Vassilicos JC (2007) Scalings and decays of fractal-generated turbulence. Phys Fluids 19:035103

    Google Scholar 

  • Jiangting L, Shaofei Y, Lixin G (2017) Electromagnetic waves propagation in hypersonic turbulence using fractal phase screen method. J Electromagn Waves Appl 31:250–262

    Google Scholar 

  • Jumarie G (2008) Table of some basic fractional calculus formulae derived from a modified Riemann–Liouville derivative for non-differentiable functions. Appl Math Lett 22:378–385

    Google Scholar 

  • Jury MR, Huang B (2004) The Rossby wave as a key mechanism of Indian Ocean climate variability. Deep-Sea Res I51:2123–2136

    Google Scholar 

  • Kaneda Y, Holloway G (1994) Frequency-shifts of Rossby waves in geostrophic turbulence. J Phys Soc Jpn 63:2974–2985

    Google Scholar 

  • Keylock CJ, Nishimura K, Nemoto M, Ito Y (2012) The flow structure in the wake of a fractal fence and the absence of an "inertial regime’. Environ Fluid Mech 12:227–250

    Google Scholar 

  • Khan S, Noor A, Mughal MJ (2013) General solution for waveguide modes in fractional space. Prog Electromagn Res M 33:105

    Google Scholar 

  • Kim KY, Na H, Jhun JG (2012) Oceanic response to midlatitude Rossby waves aloft and its feedback in the lower atmosphere in winter Northern Hemisphere. J Geophys Res Atmos 117:1–13

    Google Scholar 

  • Kucharski F, Bracco A, Yoo J, Tompkins A, Feudale L, Ruti P, Dell’Aquila A (2009) A Gill–Matsuno-type mechanism explains the tropical Atlantic influence on African and Indian monsoon rainfall. Quart J R Meteorol Soc 135:569–579

    Google Scholar 

  • LaCasce J (2012) Atmosphere–ocean dynamics. Lectures given at University of Oslo, Department of Geosciences, Oslo

    Google Scholar 

  • Lanotte AS, Benzi R, Malapaka SK, Toschi F, Biferale L (2015) Turbulence as a fractal Fourier set. Phys Rev Lett 115:264502

    Google Scholar 

  • Lea Cox B, Wang JSY (1993) Fractal surfaces: measurement and applications in the earth sciences. Fractals 01:87–115

    Google Scholar 

  • Legras B (1979) Turbulent phase shift of Rossby waves. Geophys Astrophys Fluid Dyn 15:253–281

    Google Scholar 

  • Lenhardt WW (2000) Fractal concepts and their application to earthquakes in Austria. In: Lehner FK, Urai JL (eds) Aspects of tectonic faulting. Springer, Berlin. https://doi.org/10.1007/978-3-642-59617-9_4

  • Li J, Ostoja-Starzewski M (2009) Fractal solids, product measures and fractional wave equations. Proc R Soc A465:2521–2536

    Google Scholar 

  • Li J, Ostoja-Starzewski M (2011) Micropolar continuum mechanics of fractal media. Int J Eng Sci 49:1302

    Google Scholar 

  • Li J, Ostoja-Starzewski M (2020) Thermo-poromechanics of fractal media. Philos Trans R Soc A 378:20190288

    Google Scholar 

  • Ludwig FL (1989) Atmospheric fractals-a review. Environ Soft 4:9–16

    Google Scholar 

  • Luo Z, Liu C (2007) A validation of the fractal dimension of cloud boundaries. Geophys Res Lett 34:L02808

    Google Scholar 

  • Malinverno A (1995) Fractals and ocean floor topography: a review and a model. In: Barton CC, La Pointe PR (eds) Fractals in the earth sciences. Springer, Boston. https://doi.org/10.1007/978-1-4899-1397-5_6

  • Malyarenko A, Ostoja-Starzewski M (2017) Fractal planetary rings: energy inequalities and random field model. Int J Mod Phys B 31:1750236

    Google Scholar 

  • Mandal P, Mabawonku AO, Dimri VP (2005) Self-organized fractal seismicity of reservoir triggered earthquakes in the Koyna-Warna seismic zone. Western India Pure Appl Geophys 162:73–90

    Google Scholar 

  • Mandelbrot BB (1983) The fractal geometry of nature. W. H. Freeman and Company, New York

  • Mandelbrot BB (1989) Multifractal measures, especially for geophysicist. In: Scholz CH, Mandelbrot BB (eds) Fractals in geophysics, pp 5–42

  • Mareschal JC (1989) Fractal reconstruction of sea-floor topography. Pure Appl Geophys 131:197–210

    Google Scholar 

  • Marshall DP (2016) A theoretical model of long Rossby waves in the Southern ocean and their interaction with bottom topography. Fluids 1:17

    Google Scholar 

  • Mashayekhi S, Beerli P (2019) Fractional coalescent. Proc Nat Acad Sci 116:6244–6249

    Google Scholar 

  • Mashayekhi S, Sedaghat S (2021) Fractional model of stem cell population dynamics. Chaos Solitons Fractals 146:110919

    Google Scholar 

  • Mashayekhi S, Miles P, Hussaini MY, Oates WS (2008) Fractional viscoelasticity in fractal and non-fractal media: theory, experimental validation, and uncertainty analysis. J Mech Phys Solids 111:134–156

    Google Scholar 

  • Mashayekhi S, Hussaini MY, Oates WS (2019) A physical interpretation of fractional viscoelasticity based on the fractal structure of media: theory and experimental validation. J Mech Phys Solids 128:137–150

    Google Scholar 

  • Matsuzaki M (1994) Fractals in earthquakes. Philos Trans Phys Sci Eng 348:449–357

    Google Scholar 

  • Mendonça JT, Shukla PK, Bingham R (2009) Nonlinear excitation of zonal flows by Rossby wave turbulence. New J Phys 11:073038

    Google Scholar 

  • Meyers SD, Magnan JF, Brian JJ (1994) Fractal trajectories in a numerical model of the upper Indian ocean. Nonlinear Process Geophys 1:45–50

    Google Scholar 

  • Middleton GV (1991) Nonlinear dynamics, chaos and fractals, with applications to geological systems. Geological Association of Canada Short Course Notes, vol 9

  • Moharir PS (2000) Multifractals. In: Dimri VP (ed) Application of fractals in earth sciences. A.A. Balkema, USA Oxford IBH Pub Co, New Delhi, pp 46–57

  • Montagner JP, Maurya S, Burgos G (2018) Lithospheric and asthenospheric structure below oceans and continents from anisotropic tomography. Geophys Res Abstr 20:EGU2018-3508

  • Moore DW (1963) Rossby waves in ocean circulation. Deep-Sea Res 10:735–747

    Google Scholar 

  • Murray A, Barton CC (2007) How fractals are coastlines really? Observation and Theory. American Geophysical Union, Fall Meeting 2007, abstract id. U43B-1123

  • Omilion A, Turk J, Zhang W (2018) Turbulence enhancement by fractal square grids: effects of multiple fractal scales. Fluids 3:37

    Google Scholar 

  • Osborne AR, Kirwan AD, Provenzale A, Bergamasco L (1986a) A search for chaotic behavior in large and mesocale motions in the Pacific ocean. Phys D Nonlinear Dyn 23:75–83

    Google Scholar 

  • Osborne AR, Kirwan AD, Provenzale A, Bergamasco L (1986b) Fractal drifter trajectories in the Kuroshio extension. Tellus A41:416–435

    Google Scholar 

  • Osborne AR, Kirwan AD, Provenzale A, Bergamasco L (1989) Fractal drifter trajectories in the Kuroshio extension. Tellus 41A:416–435

    Google Scholar 

  • Ostoja-Starzewski M (2008) On turbulence in fractal porous media. Z Angew Math Phys 59:1111–1117

    Google Scholar 

  • Ostoja-Starzewski M (2009) Extremum and variational principles for elastic and inelastic media with fractal geometries. Acta Mech 205:161–170

    Google Scholar 

  • Ostoja-Starzewski M, Li J, Joumaa H, Demmie PN (2014) From fractal media to continuum mechanics. Zei Ang Math Phys 94:373–401

    Google Scholar 

  • Pedloski J (2003) Waves in the ocean and atmosphere. Introduction to wave dynamics. Springer, New York

    Google Scholar 

  • Ray SN, Chattopadhyay S (2021) Analyzing surface air temperature and rainfall in univariate framework, quantifying uncertainty through Shannon entropy and prediction through artificial neural network. Earth Sci Inform 14L:485–503

    Google Scholar 

  • Robinson IS (2010) Planetary waves and large-scale ocean dynamics. In: Discovering the ocean from space. Springer Praxis Books. Springer, Berlin. https://doi.org/10.1007/978-3-540-68322-3_6

  • Rychert CA, Harmon N, Kendall M (2018) Constraints on the anisotropic contributions to velocity discontinuities at about 60 km depth beneath the oceans with Pacific-Atlantic comparisons. In: 20th EGU General Assembly, EGU2018, Proceedings from the conference held 4–13 April, 2018 in Vienna, Austria, p 5199

  • Schaeffer N, Cardin P (2005) Rossby-wave turbulence in a rapidly rotating sphere. Nonlinear Process Geophys 12:947–953

    Google Scholar 

  • Scholze CH, Mandelbrot BB (1989) Fractals in geophysics. Springer Book Archive, Birkhäuser. https://doi.org/10.1007/978-3-0348-6389-6

    Book  Google Scholar 

  • Schwartz C, Baum G, Ribak EN (1994) Turbulence-degraded wave fronts as fractal surfaces. J Opt Soc Am A 11:444–451

    Google Scholar 

  • Shaman J, Tziperman E (2016) The superposition of Eastward and Westward Rossby waves in response to localized forcing. J Clim 29:7547–7557

    Google Scholar 

  • Shaman J, Esbensen SK, Maloney ED (2009) The dynamics of the ENSO–Atlantic hurricane teleconnection: ENSO related changes to the North African-Asian jet affect Atlantic basin tropical cyclogenesis. J Clim 22:2458–2482

    Google Scholar 

  • Shaman J, Samelson R, Tziperman E (2012) Complex wavenumber Rossby wave ray tracing. J Atmos Sci 69:2112–2133

    Google Scholar 

  • Silveira G, Strutzmann E (2002) Anisotropic tomography of the Atlantic ocean. Phys Earth Plant Int 132:237–248

    Google Scholar 

  • Sreenivasan KR (1994) Fractals in fluid mechanics. Fractals 02:253–263

    Google Scholar 

  • Steele JH (2008) Encyclopedia of ocean sciences, 2nd edn. Academic Press, Cambridge

    Google Scholar 

  • Stiassnie M, Agnon Y, Shemer L (1991) Fractal dimensions of random surfaces water. Phys D Nonlinear Phenomena 47:341–352

    Google Scholar 

  • Strong C, Magnusdottir G (2009) The role of tropospheric Rossby wave breaking in the Pacific Decadal Oscillation. J Clim 22:1819–1833

    Google Scholar 

  • Tajima T (2018) Computational plasma physics: with applications to fusion and astrophysics. CRC Press, Boca Raton

    Google Scholar 

  • Tao M, Zhang N, Gao D, Yang H (2018) Symmetry analysis for three-dimensional dissipation Rossby waves. Adv Differ Equ 2018:300

    Google Scholar 

  • Tarasov VE (2005) Continuous media model for fractal media. Phys Lett A 336:167

    Google Scholar 

  • Tarasov VE (2014) Flow of fractal fluid in pipes: non-integer dimensional space approach. Chaos Solitons Fractals 67:26–37

    Google Scholar 

  • Todoeschuck JP, Jensen OG (1989) Scaling geology and seismic deconvolution. PAGEOPH 131:273–287. https://doi.org/10.1007/BF00874491

  • Turcotte DL (1988) Fractals in fluid mechanics. Ann Rev Fluid Mech 20:5–16

    Google Scholar 

  • Turcotte DL (1997) Fractals and chaos in geology and geophysics. Second Edition, Cambridge University Press, Cambridge, p 398. https://doi.org/10.1017/CBO9781139174695

  • Wang Y, Yamazaki K, Fujiyoshi F (2007) The interaction between two separate propagations of Rossby waves. Mon Weather Rev 135:3521–3540

    Google Scholar 

  • Webber BGM, Matthews AJ, Heywood KJ, Stevens DP (2012) Ocean Rossby waves as a triggering mechanism for primary Madden–Julian events. Q J R Meteorol Soc 138:514–527

    Google Scholar 

  • Yu D, Fu L, Yang H (2021) A new dynamic model of ocean internal solitary waves and the properties of its solutions. Commun Nonlinear Sci Numer Simul 95:105622

    Google Scholar 

  • Yu L, Zou Z (2009) The Fractal Dimensionality of Seismic Wave. In: Yuan Y, Cui J, Mang HA (eds) Computational structural engineering. Springer, Dordrecht. https://doi.org/10.1007/978-90-481-2822-8_33

  • Zhang J, Zhang R, Yang L (2021) Coherent structures of nonlinear barotropic–baroclinic interaction in unequal depth two-layer model. Appl Math Comput 408:126347

    Google Scholar 

  • Zhuang Y, Tan H, Wang W, Li X, Guo Y (2019) Fractal features of turbulent/non-turbulent interface in a shock wave/turbulent boundary-layer interaction flow. J Fluid Mech 869:R6

    Google Scholar 

  • Zubair M, Mughal MJ, Naqvi QA (2010) The wave equation and general plane wave solutions in fractional space. Prog Electromagn Res Lett 19:137–146

    Google Scholar 

  • Zubair M, Mughal MJ, Naqvi QA (2011) An exact solution of the spherical wave equation in D-dimensional fractional space. J Electromagn Waves Appl 25:481–1491

    Google Scholar 

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Acknowledgements

The authors would like to thank the anonymous referees for their useful comments and valuable suggestions.

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The authors would like to thank Chiang Mai University for funding this research.

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Authors and Affiliations

  1. Research Center for Quantum Technology, Faculty of Science, Chiang Mai University, Chiang Mai, 50200, Thailand

    Rami Ahmad El-Nabulsi & Waranont Anukool

  2. Department of Physics and Materials Science, Faculty of Science, Chiang Mai University, Chiang Mai, 50200, Thailand

    Rami Ahmad El-Nabulsi & Waranont Anukool

  3. Mathematics and Physics Divisions, Athens Institute for Education and Research, 8 Valaoritou Street, Kolonaki, 10671, Athens, Greece

    Rami Ahmad El-Nabulsi

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  1. Rami Ahmad El-Nabulsi
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El-Nabulsi, R.A., Anukool, W. Ocean–atmosphere dynamics and Rossby waves in fractal anisotropic media. Meteorol Atmos Phys 134, 33 (2022). https://doi.org/10.1007/s00703-022-00867-9

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