Abstract
The word turbulence comes from late Latin “turbulentia” which means “full of commention.” It is defined as a “violent or unsteady movement of air or water, or of some other fluid” (Oxford Dictionary of English 2010). Thus, it is a process that dissipates or mixes. The antonyms are unity or homogeneity: they help us to understand more clearly what turbulence concretely means – turbulence mixes and disperses the medium in which it develops, and then it disappears once homogeneity returns.
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References
Abramowitz M, Segun IA (1972) Handbook of mathematical functions. Dover, New York edition
Abry P, Goncalves O, Flandrin P (1995) Wavelets, spectrum estimation and 1/f processes. In: Antoniadis A, Oppenheim G (eds) Wavelets and statistics. Lecture notes in statistics, vol 103. Springer, New York pp 15–30
Allan DW (1987) Time and frequency (time domain) characterization, estimation, and prediction of precision clocks and oscillators. IEEE Trans Ultrason Ferroelectr Freq Control UFFC-34(6):647–654
Armstrong JW Sramek RA (1982) Observations of tropospheric phase scintillations at 5 GHz on vertical paths. Radio Sci 17(6):1579–1586
Banta RM, Newsom RK, Lundquist JK (2002) Nocturnal low-level jet characteristics over Kansas during cases-99. Bound Layer Meteorol 105:221–252
Barnes J, Chi AR, Cutler LS, Healey DJ, Leeson DB, McGunigal TE, Mullen JA, Smith WL, Sydnor RL, Vessot RF, Winkler GM (1971) Characterization of frequency stability. IEEE Trans Instrum Meas 20:105120
Beutler G, Bauersima I, Gurtner W, Rothacher M (1987) Correlations between simultaneous GPS double difference carrier phase observations in the multistation mode: implementation considerations and first experiences. Manisc Geod 12(1):40–44
Bevis G, Businger S, Chiswell S, Herring TA, Anthes RA, Rocken C, Ware RH (1994) GPS meteorology: mapping zenith wet delays onto precipitable water. J Appl Meteorol Climatol 33(3):379–386
Bischoff W, Heck B, Howind J, Teusch A (2005) A procedure for testing the assumption of homoscedasticity in least-squares residuals: a case study of GPS carrier phase observations. J Geod 78(7–8):397–404
Boussinesq J (1877) Essai sur la theorie des eaux courantes, Memoires presentes par divers savants ‘a l’ Academie des Sciences XXIII, 1–680
Brunner FK, Hartinger H, Troyer L (1999) GPS signal diffraction modelling: the stochastic SIGMA-Dmodel. J Geod 73(5):259–267
Coulman CE (1990) Atmospheric Structure, Turbulence and Radioastronomical “Seeing”. Proceedings URSI/IAU Symposium on Radio Astronomical Seeing Beijing/Oxford, International Academic Publishers/Pergamon Press, pp. 11–20
Coulman CE, Vernin J (1991) Significance of anisotropy and the outer scale of turbulence for optical and radio seeing. Appl Opt 30(1):118–126
Cornish CR, Bretherton CS (2006) Maximal overlap wavelet statistical analysis with application to atmospheric turbulence. Bound Layer Meteorol 119(2):339–377
Cressie N (1993) Statistics for spatial data. Wiley, New York/Chichester/Toronto/Brisbane/ Singapore
Danielson EW, Levin J, Abrams E (2003) Meteorology, McGraw Hill, Boston
Davis JL (2001) Atmospheric water-vapor signals in GPS data: synergies, correlations, signals and errors. Phys Chem Earth 26(6–8):513–522
De Moor G (2006) Couche Limite atmosphérique et turbulence, les bases de la micro météorologie dynamique. Météo-France, Cours et manuels n ̈16, Toulouse
Domingues MO, Mendes O, Mendes da Costa A (2004) On wavelet techniques in atmospheric sciences. Adv Space Res 35(5):831–842
Dravskikh AF, Finkelstein AM (1979) Tropospheric limitations in phase and frequency coordinate measurements in astronomy. Astrophys Space Sci 60(2):251–265
Emardson TR, Jarlemark POJ (1999) Atmospheric modeling in GPS analysis and its effect on the estimated geodetic parameters. J Geod 73(6):322–331
El-Rabbany A (1994) The effect of physical correlations on the ambiguity resolution and accuracy estimation in GPS differential positioning. PhD thesis, Department of Geodesy and Geomatics Engineering, University of New Brunswick
Farge M (1992) Wavelet transform and their applications to turbulence. Ann Rev Fluid Mech 4:395–457
Fried DL (1967) Propagation of a spherical wave in a turbulent medium. J Opt Soc Am 57(2):175–180
Gage KS (1979) Evidence for a k to the -5/3 law inertial range in mesoscale two-dimensional turbulence. J Atmos Sci 36:1950–1954
Hagelberg CR, Gamage NKK (1994) Structure-Preserving wavelet decompositions of intermittent turbulence. Bound Layer Meteorol 70:217–246
Hartinger H, Brunner FK (1999) Variances of GPS phase observations: the SIGMA-\(\varepsilon\) model. GPS Solut 2(4):35–43
Hogg DC, Guiraud FO, Sweezy WB (1981) The short-term temporal spectrum of precipitable water vapor. Science 213(4512):1112–1113
Howind J (2005) Analyse des stochastischen Modells von GPS-Trägerphasenbeobachtungen. Deutsche Geodätische Kommission, Munich Heft Nr. 584
Howind J, Kutterer H, Heck B (1999) Impact of temporal correlations on GPS-derived relative point positions. J Geod 73(5):246–258
Ishimaru A (1984) Wave propagation and scattering in random media, vol II. Academic, New York
Jin SG, Luo O, Ren C (2010) Effects of physical correlations on long-distance GPS positioning and zenith tropospheric delay estimates. Adv Space Res 46(2):190–195
Kleijer F (2004) Tropospheric modeling and filtering for precise GPS leveling. PhD thesis, Netherlands Geodetic Commission, Publications on Geodesy 56
Koch KR, Kuhlmann H, Schuh WD (2010) Approximating covariance matrices estimated in multivariate models by estimated auto- and cross-covariances. J Geod 84(6):383–397
Kolmogorov NA (1941) Dissipation of energy in the locally isotropic turbulence. Proc USSR Acad Sci 32:16–18. (Russian), translated into English by Kolmogorov, 8 July 1991. “The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers”. Proc R Soc Lond Ser A Math Phys Sci 434(1980):15–17
Kolmogorov NA (1962) A refinement of previous hypotheses concerning the local structure of turbulence in a viscous incompressible fluid at high Reynolds number. Journal of Fluid Mechanics, 13, pp 82–85
Kraichnan RH (1967) Inertial ranges in two-dimensional turbulence. Phys Fluids 10:1417–1423
Laing D (1991) The earth system: an introduction to earth science, Wm. C. Brown Publishers, Dubuque University of California
Lay OP (1997) The temporal power spectrum of atmospheric fluctuations due to water vapor. Astron Astrophys Suppl Ser 122:535–545
Leandro R, Santos MC (2006) An Empirical Stochastic Model for GPS. International Association of Geodesy Symposia (Ed. C. Rizos), IAG, IAPSO and IABO Joint Assembly “Dynamic Planet”, Cairns, Australia, 22–26 August 2005, Springer, pp. 179–185.
Lesieur M (2008) Turbulence in fluids, 4th edn. Springer, Dordrecht
Lin, CC (1953) On Taylor’s hypothesis and the acceleration terms in the Navier-Stokes equation. Q Appl Maths 10:295–306
Lindsey WC, Chi CM (1976) Theory of oscillator instability based upon structure functions. Proc IEEE 64(12):1652–1666
Luo X (2013) GPS stochastic modelling – signal quality measures and ARMA processes. Springer theses: recognizing outstanding Ph.D. research. Springer, Berlin/Heidelberg
Mahrt L (1986) On the shallow motion approximations. J Atmos Sci 43:1036–1044
Mahrt L (1991) Eddy assymmetry in the sheared heated boundary layer. J Atmos Sci 48(3):472–492
Mathieu J, Scott J (2000) An introduction to turbulent flow. Cambridge University Press, Cambridge
Monin AS, Yaglom AM (1975) Statistical fluid mechanics, vol 2. MIT, Cambridge
Muzy JF, Bacry E, Arneodo A (1993) Multifractal formalism for fractal signals: the structure-function approach versus the wavelet-transform modulus-maxima method. Phys Rev E 47(2):875–884
Naudet CJ (1996) Estimation of tropospheric fluctuations using GPS data. TDA progress report, pp 42–126
Nastrom GD, Gage KS (1985) A climatology of atmospheric wavenumber spectra of wind and temperature observed by commercial aircrafts. J Atmos Sci 42(9):950–960
Nichols-Pagel GA, Percival DB, Reinhall PG, Riley JJ (2008) Should structure functions be used to estimate power laws in turbulence? A comparative study. Physica D Nonlinear Phenom 237(5):665–677
Nilsson T, Haas R (2010) Impact of atmospheric turbulence on geodetic very long baseline interferometry. J Geophys Res 115:B03407
Nilsson T, Haas R, Elgered G (2007) Simulations of atmospheric path delays using turbulence models. In: Böhm J, Pany A, Schuh H (eds) Proceedings of 18th European VLBI for Geodesy and Astrometry (EVGA) working meeting, Vienna University of Technology, Vienna, pp 175–180
Oxford Dictionary of English (2010) Oxford University Press, Auflage, 2nd edn., revised (11 Aug 2010)
Percival DB, Walden AT (2000) Wavelet methods for time series analysis. Cambridge University Press, Cambridge
Reddi SS (1984) Eigenvector properties of Toeplitz matrices and their application to spectral analysis of time series. Signal Process 7:45–56
Riley WJ (2008) Handbook of frequency stability analysis. NIST special publication 1065
Ripley BD (1981) Spatial Statistics. Wiley, pp. 252
Rutman J (1978) Characterization of phase and frequency instabilities in precision frequency sources: fifteen years of progress. Proc IEEE 66(9)1048–1075
Romero-Wolf A, Jacobs CS, Ratcli JT (2012) Effects of tropospheric spatio-temporal correlated noise on the Analysis of space geodetic data, IVS general meeting proceedings, Madrid, 5–8 Mar 2012
Satirapod C, Ogaja C, Wang J, Rozos C (2001) GPS analysis with the aid of wavelets. In: 5th international symposium on satellite navigation technology and applications, Canberra, 24–27 July, paper 39
Schön S, Brunner FK (2006) Modelling physical correlation of GPS phase observations: first results. Kahmen H, Chrzanowski A (eds) Proceedings of the 3rd IAG symposium on geodesy for geotechnical and structural engineering/12th FIG symposium on deformation measurements, Baden, 22–24 May 2006, pp PS-18.1–8
Schön S, Brunner FK (2007) Treatment of refractivity fluctuations by fully populated variance-covariance matrices. In: Proceedings of the 1st colloquium scientific and fundamental aspects of the galileo programme, Toulouse Okt
Schön S, Brunner FK (2008a) Atmospheric turbulence theory applied to GPS carrier-phase data. J Geod 82(1):47–57
Schön S, Brunner FK (2008b) A proposal for modeling physical correlations of GPS phase observations. J Geod 82(10):601–612
Seeber G (2003) Satellite geodesy. de Gruyter, Berlin
Stotskii A (1973) Concerning the fluctuation of characteristics of the Earth’s troposphere. Radiophys Quantum Electron 16(5):620–622
Stotskii A, Stotskaya IM (2001) Structure analysis of wet path delays in IRIS-S experiments. In: Behrend D, Rius A (eds) Proceedings of the 15th working meeting on European VLBI, Barcelona, 7–8 Sept 2001, p 154
Stotskii A, Elgered KG, Stotskaya M (2006) Structure analysis of path delay variations in the neutral atmosphere. Astron Astrophys Trans J Eurasian Astron Soc 17(1):59–68
Stull RB (1988) An introduction to boundary layer meteorology. Springer, Dordrecht
Tatarskii VI (1971a) Wave propagation in a turbulent medium. McGraw-Hill, New York
Tatarskii VI (1971b) The effects of the turbulent atmosphere on wave propagation. National Technical Information Service. Springfield VA VA22161
Taylor GI (1938) The spectrum of turbulence. Proc R Soc Lond Ser A CLXIV:476–490
Teunissen PJG, Kleusberg A (1998) GPS for Geodesy 2nd ed. Springer Verlag Berlin Heidelberg
Thomson MC Marler F, Allen K (1980) Measurement of the microwave structure constant profile. IEEE Trans Antennas Propag, 28(2):278–280
Thompson AR, Moran JM, Swenson GW (2004) Interferometry and synthesis in radio astronomy. Wiley, Hoboken
Tiberius C, Jonkman N, Kenselaar F (1999) The stochastics of GPS observables. GPS World 10(2):49–54
Treuhaft RN, Lanyi GE (1987) The effect of the dynamic wet troposphere on radio interferometric measurements. Radio Sci 22(2):251–265
Treuhaft RN, Lowe ST (1995) Vertical scales of turbulence at the Mt Wilson observatory. In: JPL TRS 1992+BEACON eSpace at Jet Propulsion Laboratory, California Institute of Technology
Van der Hoven I (1957) Power spectrum of horizontal wind speed in the frequency range from 0.0007 to 900 cycles per hour. J Meteorol 14:160–164
Vennebusch M, Schön S (2012) Generation of slant tropospheric delay time series based on turbulence theory. In: Kenyon S, Pacino M, Morti U (eds): Proceedings of geodesy for planet earth – IAG 2009, Buenos Aires, International association of geodesy symposia, vol 136, pp 801–808. Springer, New York/Berlin Heidelberg.
Vennebusch M, Schön S, Weinbach U (2011) Temporal and spatial stochastic behavior of high-frequency slant tropospheric delays from simulations and real GPS data. Adv Space Res 47(10):1681–1690
Vincent A, Meneguzzi M (1991) The spatial structure and statistical properties of homogeneous turbulence. J Fluid Mech 225:1–20
Voitsekhovich VV (1995) Outer scale of turbulence: comparison of different models. J Opt Soc Am A 12(6):1346–1353
Wang J, Satirapod C, Rizos C (2002) Stochastic assessment of GPS carrier phase measurements for precise static relative positioning. J Geod 76(2):95–104
Wheelon AD (2001) Electromagnetic scintillation part I geometrical optics. Cambridge University Press, Cambridge
Whichter B, Guttorp P, Percival D (2000) Wavelet analysis of covariance with application to atmospheric time series. J Geophys Res Atmos 105:14941–14962
Wieser A, Brunner FK (2000) An extended weight model for GPS phase observations. Earth Planet Space 52:777–782
Williams S (2003) The effect of coloured noise on the uncertainties of rates estimated from geodetic time series. J Geod 76(9–10):483–494. OU 2002?
Williams S, Bock Y, Fang P (1998) Integrated satellite interferometry: tropospheric noise, GPS estimates and implication for interferometric synthetic aperture radar products. J Geophys Res 103(B11):27051–27067
Wright MCH (1996) Atmospheric phase noise and aperture synthesis imaging at millimeter wavelengths. Publ Astron Soc Pac 108(724):520–534
Yaglom AM (1987) Correlation theory of stationary and related random functions I, II, 1st edn. Springer, New York/Berlin/Heilderberg
Acknowledgements
The authors thank FK Brunner for his introduction into turbulence theory. The stay at his institute at TU Graz, Austria, was funded by a Feodor Lynen Fellowship of Alexander von Humboldt Foundation, which is gratefully acknowledged. The first author thanks Dr. Markus Vennbusch for fruitful discussions and new development at IfE Hannover. The German Research Foundation (DFG) is thanked for the financial support to study the subject in the projects SCHO1314/1-1, 1-2 as well as SCHO1314/3-1.
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Schön, S., Kermarrec, G. (2015). Turbulence Theory. In: Freeden, W., Nashed, M., Sonar, T. (eds) Handbook of Geomathematics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-54551-1_77
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