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Статті в журналах з теми "Airy's function"

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Автор: Grafiati

Опубліковано: 28 липня 2022

Оновлено: 8 серпня 2022

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1

Roy, Sabyasachi, N. S. Bordoloi, and D. K. Choudhury. "Isgur–Wise function within a QCD quark model with Airy's function as the wave function of heavy–light mesons." Canadian Journal of Physics 91, no. 1 (January 2013): 34–42. http://dx.doi.org/10.1139/cjp-2012-0165.

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We report a somewhat improved wave function for mesons by taking the linear confinement term in standard QCD potential as a parent and the coulombic term as a perturbation while applying quantum mechanical perturbation techniques in solving the Schrödinger equation with such a potential. We find that Airy's infinite series appears in the wave function of the mesons. We report our calculations on the Isgur–Wise function and its derivatives for heavy–light mesons within this framework.

2

ROY, SABYASACHI, and D. K. CHOUDHURY. "AN ANALYSIS OF THE ISGUR–WISE FUNCTION AND ITS DERIVATIVES WITHIN A HEAVY–LIGHT QCD QUARK MODEL." Modern Physics Letters A 27, no. 20 (June 28, 2012): 1250110. http://dx.doi.org/10.1142/s0217732312501106.

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In determining the mesonic wave function from QCD inspired potential model, if the linear confinement term is taken as parent (with Coulombic term as perturbation), Airy's function appears in the resultant wave function — which is an infinite series. In the study of Isgur–Wise function (IWF) and its derivatives with such a wave function, the infinite upper limit of integration gives rise to divergence. In this paper, we have proposed some reasonable cutoff values for the upper limit of such integrations and studied the subsequent effect on the results. We have also studied the sensitivity of the order of polynomial approximation of the infinite Airy series in calculating the derivatives of IWF.

3

THAPA, R. K., and GUNAKAR DAS. "A SIMPLE THEORY OF PHOTOFIELD EMISSION FROM THE SURFACE OF A METAL." International Journal of Modern Physics B 19, no. 19 (July 30, 2005): 3141–49. http://dx.doi.org/10.1142/s0217979205032000.

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A simple model calculation of photofield emission is presented in which the photofield emission current (PFEC) is calculated for metal W. The matrix element for photoexcitation is evaluated by using the free electron wavefunction. The transmission probability D(W) is deduced by solving Airy's differential equation. The variation of PFEC is studied as a function of parameters like the applied high electric field, the photon energy, the initial state energy with reference to the Fermi level. It is found that in addition to D(W), the matrix element Mfi also has effect on the photofield emission.

4

Bawankar, L. C., and G. D. Kedar. "MAGNETO-THERMOELASTIC PROBLEM WITH EDDY CURRENT LOSS OF A THERMOSENSITIVE CONDUCTIVE PLATE." Advances in Mathematics: Scientific Journal 10, no. 1 (January 22, 2021): 557–70. http://dx.doi.org/10.37418/amsj.10.1.55.

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In this paper a two dimensional magneto-thermoelastic problem of a thermosensitive finite conducting plate with eddy current loss is considered. It is assumed that the plate is influenced by a time-varying external magnetic field and that the heating is caused by Joule heat. The fundamental equations for magnetic field, heat conduction and elastic fields are formulated. Temperature dependent material properties and heat source as eddy current loss is considered in the heat conduction equation. Kirchhoff's variable transformation is employed to convert nonlinear to linear heat conduction equation. Integral transform technique is used to solve the magnetic field and temperature distribution. The stresses in a plane state are determined by using Airy's stress function. The numerical analysis is carried out and the results are graphically displayed.

5

Duc, Nguyen Dinh, Pham Dinh Nguyen, Nguyen Huy Cuong, Nguyen Van Sy, and Nguyen Dinh Khoa. "An analytical approach on nonlinear mechanical and thermal post-buckling of nanocomposite double-curved shallow shells reinforced by carbon nanotubes." Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 233, no. 11 (September 27, 2018): 3888–903. http://dx.doi.org/10.1177/0954406218802921.

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This work presents the nonlinear mechanical and thermal post-buckling of nanocomposite double-curved shallow shells reinforced by single-walled carbon nanotubes resting on elastic foundations based on the higher order shear deformation theory with geometrical nonlinearity in von Karman–Donnell sense. The composite shells are made of various amorphous polymer matrices: poly(methyl methacrylate) (PMMA) and poly{(m-phenylenevinylene)-co-[(2,5-dioctoxy-p-phenylene) vinylene]} (PmPV). The governing equations are solved by the Galerkin method and Airy's stress function to achieve mechanical and thermal post-buckling behaviors of nanocomposite double-curved shallow shells. Various types of distributions of carbon nanotubes, both uniform distributions, and functionally graded distributions are examined. The material properties of nanocomposite double-curved shallow shells are assumed to be temperature dependent. Detailed parametric studies are carried out on the effect of various types of distribution and volume fractions of carbon nanotubes, temperature increments, elastic foundations, edge to radius and edge to thickness ratios on the nonlinear mechanical and thermal post-buckling of nanocomposite double-curved shallow shells reinforced by CNTs.

6

ALIJANI, F., and M. AMABILI. "CHAOTIC VIBRATIONS IN FUNCTIONALLY GRADED DOUBLY CURVED SHELLS WITH INTERNAL RESONANCE." International Journal of Structural Stability and Dynamics 12, no. 06 (December 2012): 1250047. http://dx.doi.org/10.1142/s0219455412500472.

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Chaotic vibrations of functionally graded doubly curved shells subjected to concentrated harmonic load are investigated. It is assumed that the shell is simply supported and the edges can move freely in in-plane directions. Donnell's nonlinear shallow shell theory is used and the governing partial differential equations are obtained in terms of shell's transverse displacement and Airy's stress function. By using Galerkin's procedure, the equations of motion are reduced to a set of infinite nonlinear ordinary differential equations with cubic and quadratic nonlinearities. A bifurcation analysis is carried out and the discretized equations are integrated at (i) fixed excitation frequencies and variable excitation amplitudes and (ii) fixed excitation amplitudes and variable excitation frequencies. In particular, Gear's backward differentiation formula (BDF) is used to obtain bifurcation diagrams, Poincaré maps and time histories. Furthermore, maximum Lyapunov exponent and Lyapunov spectrum are obtained to classify the rich dynamics. It is revealed that the shell may exhibit complex behavior including sub-harmonic, quasi-periodic and chaotic response when subjected to large harmonic excitations.

7

Hamdan, M. H., S. Jayyousi Dajani, and D. C. Roach. "Asymptotic Series Evaluation of Integrals Arising in the Particular Solutions to Airy’s Inhomogeneous Equation with Special Forcing Functions." WSEAS TRANSACTIONS ON MATHEMATICS 21 (May 31, 2022): 303–8. http://dx.doi.org/10.37394/23206.2022.21.35.

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In this work, particular and general solutions to Airy’s inhomogeneous equation are obtained when the forcing function is one of Airy’s functions of the first and second kind, and the standard Nield-Kuznetsov function of the first kind. Particular solutions give rise to special integrals that involve products of Airy’s and Nield-Kuznetsov functions. Evaluation of the resulting integrals is facilitated by expressing their integrands in asymptotic series. Corresponding expressions for the Nield-Kuznetsov function of the second kind are obtained.

8

Hamdan, M. H., S. Jayyousi Dajani, and D. C. Roach. "Asymptotic Series Evaluation of Integrals Arising in the Particular Solutions to Airy’s Inhomogeneous Equation with Special Forcing Functions." PROOF 2 (June 6, 2022): 153–58. http://dx.doi.org/10.37394/232020.2022.2.19.

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In this work, particular and general solutions to Airy’s inhomogeneous equation are obtained when the forcing function is one of Airy’s functions of the first and second kind, and the standard Nield-Kuznetsov function of the first kind. Particular solutions give rise to special integrals that involve products of Airy’s and Nield-Kuznetsov functions. Evaluation of the resulting integrals is facilitated by expressing their integrands in asymptotic series. Corresponding expressions for the Nield-Kuznetsov function of the second kind are obtained.

9

Altan, BurhanettinS. "Airy's Functions in Nonlocal Elasticity." Journal of Computational and Theoretical Nanoscience 8, no. 11 (November 1, 2011): 2381–88. http://dx.doi.org/10.1166/jctn.2011.1873.

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10

Hamdan, M. H., S. Jayyousi Dajani, and M. S. Abu Zaytoon. "Higher Derivatives and Polynomials of the Standard Nield-Kuznetsov Function of the First Kind." International Journal of Circuits, Systems and Signal Processing 15 (December 8, 2021): 1737–43. http://dx.doi.org/10.46300/9106.2021.15.187.

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In this fundamental work, higher derivatives of the standard Nield-Kuznetsov function of the first kind, and the polynomials arising from this function and Airy’s functions, are derived and discussed. This work provides background theoretical material and computational procedures for the arising polynomials and the higher derivatives of the recently introduced Nield-Kuznetsov function, which has filled a gap that existed in the literature since the nineteenth century. The ease by which the inhomogeneous Airy’s equation can now be solved is an advantage offered by the Nield-Kuznetsov functions. The current analysis might prove to be invaluable in the study of inhomogeneous Schrodinger, Tricomi, and Spark ordinary differential equations.

11

Roach, D. C., and M. H. Hamdan. "Connecting Einstein Functions to the Nield-Kuznetsov and Airy’s Functions." EQUATIONS 2 (May 9, 2022): 48–53. http://dx.doi.org/10.37394/232021.2022.2.8.

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In this work, the problem of obtaining particular and general solutions to Airy’s inhomogeneous equation when the forcing function is one of Einstein’s functions is examined. Expressions for the particular solutions provide connections between the Nield-Kuznetsov and Einstein functions. Computations have been carried out using Wolfram Alpha.

12

Hamdan, M. H., and D. C. Roach. "Puiseux and Taylor Series of the Einstein Functions and Their Role in the Solution of Inhomogeneous Airy’s Equation." WSEAS TRANSACTIONS ON MATHEMATICS 21 (June 15, 2022): 395–402. http://dx.doi.org/10.37394/23206.2022.21.47.

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The Einstein functions in generalized Puiseux and Taylor series are used as forcing functions in Airy’s inhomogeneous equation, and particular and general solutions are obtained. Comparison are made with solutions obtained using the Nield-Kuznetsov functions’ approach. For each of the Einstein’s functions, the standard Nield-Kuznetsov function of the second kind is expressed in terms of Bessel functions. Computations and graphs in this work were produced using Wolfram Alpha.

13

Hamdan, M. H., S. M. Alzahrani, M. S. Abu Zaytoon, and S. Jayyousi Dajani. "Inhomogeneous Airy’s and Generalized Airy’s Equations with Initial and Bounday Conditions." International Journal of Circuits, Systems and Signal Processing 15 (September 20, 2021): 1486–96. http://dx.doi.org/10.46300/9106.2021.15.161.

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Inhomogeneous Airy’s and Generalized Airy’s equations with initial and boundary date are considered in this work. Solutions are obtained for constant and variable forcing functions, and general solutions are expressed in terms of Standard and Generalized Nield-Kuznetsov functions of the first- and second-kinds. Series representations of these functions and their efficient computation methodologies are presented with examples.

14

Keluskar, Yugesh C., Megha M. Navada, Chaitanya S. Jage, and Navin G. Singhaniya. "Implementation of Airy function using Graphics Processing Unit (GPU)." ITM Web of Conferences 32 (2020): 03052. http://dx.doi.org/10.1051/itmconf/20203203052.

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Special mathematical functions are an integral part of Fractional Calculus, one of them is the Airy function. But it’s a gruelling task for the processor as well as system that is constructed around the function when it comes to evaluating the special mathematical functions on an ordinary Central Processing Unit (CPU). The Parallel processing capabilities of a Graphics processing Unit (GPU) hence is used. In this paper GPU is used to get a speedup in time required, with respect to CPU time for evaluating the Airy function on its real domain. The objective of this paper is to provide a platform for computing the special functions which will accelerate the time required for obtaining the result and thus comparing the performance of numerical solution of Airy function using CPU and GPU.

15

Hamdan, M. H., S. Jayyousi Dajani, and M. S. Abu Zaytoon. "Recent Advances in the Nield-Kuznetsov Functions." WSEAS TRANSACTIONS ON SYSTEMS 20 (July 23, 2021): 178–86. http://dx.doi.org/10.37394/23202.2021.20.20.

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In this article, we discuss a class of functions known as the Nield-Kuznetsov functions, introduced over the past decade. These functions arise in the solutions to inhomogeneous Airy’s and Weber’s equations. Derivations of these functions are provided, together with their methods of computations

16

Hamdan, M. H., S. Jayyousi Dajani, and M. S. Abu Zaytoon. "Nield-Kuznetsov Functions: Current Advances and New Results." International Journal of Circuits, Systems and Signal Processing 15 (September 20, 2021): 1506–20. http://dx.doi.org/10.46300/9106.2021.15.163.

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In this article, we discuss a class of functions known as the Nield-Kuznetsov functions, introduced over the past decade. These functions arise in the solutions to inhomogeneous Airy’s and Weber’s equations. Derivations of these functions are provided, together with their methods of computations

17

Himeur, Mohammed, Abdesselam Zergua, and Mohamed Guenfoud. "A Finite Element Based on the Strain Approach Using Airy’s Function." Arabian Journal for Science and Engineering 40, no. 3 (January 3, 2015): 719–33. http://dx.doi.org/10.1007/s13369-014-1543-3.

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18

Jung, Soon-Mo. "Approximation of analytic functions by Airy functions." Integral Transforms and Special Functions 19, no. 12 (December 2008): 885–91. http://dx.doi.org/10.1080/10652460802321287.

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19

Albright, J. R., and E. P. Gavathas. "Integrals involving Airy functions." Journal of Physics A: Mathematical and General 19, no. 13 (September 11, 1986): 2663–65. http://dx.doi.org/10.1088/0305-4470/19/13/029.

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20

Ramkarthik, M. S., and Elizabeth Louis Pereira. "Airy Functions Demystified — II." Resonance 26, no. 6 (June 2021): 757–89. http://dx.doi.org/10.1007/s12045-021-1179-z.

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21

Ramkarthik, M. S., and Elizabeth Louis Pereira. "Airy Functions Demystified — I." Resonance 26, no. 5 (May 2021): 629–47. http://dx.doi.org/10.1007/s12045-021-1166-4.

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22

Xu, B. X., and M. Z. Wang. "The Fault in the Stress Analysis of Pseudo-Stress Function Method." Journal of Applied Mechanics 72, no. 4 (October 26, 2004): 615–16. http://dx.doi.org/10.1115/1.1935526.

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Although Peng Yafei and his co-workers discovered some faults with the pseudo-stress function method suggested by Y. S. Lee in 1987, the authors did not provide convincing arguments. We investigate the crucial assumption in Lee’s method by rewriting it as the form of real part and imaginary part. Through a specific counterexample, we point out that the crucial assumption in Lee’s theory is untenable. Namely, for given Airy’s stress function, it cannot be guaranteed that the pseudo-stress function Λ(x,y) exists. The root cause of the fault with Lee’s method is found in this paper.

23

Proskurin, Nikolai V. "Integral addition formulae for airy function." Integral Transforms and Special Functions 5, no. 1-2 (May 1997): 143–46. http://dx.doi.org/10.1080/10652469708819130.

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24

STÉPHAN, GUY. "AN AIRY FUNCTION FOR THE LASER." Journal of Nonlinear Optical Physics & Materials 05, no. 03 (July 1996): 551–57. http://dx.doi.org/10.1142/s0218863596000374.

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We give a general equation which allows the description of the evolution of the laser properties when the gain is varied from below to above threshold. The method is general in the context of the semi-classical theory of lasers. It is numerically illustrated in the simplest case of a single mode laser with a source term characterized by a very narrow spectrum. The transformation of the Airy line shape of the passive Fabry-Perot cavity into the laser line shape is described. A classical linewidth which complements the usual quantum limit can thus be introduced, together with its control parameters.

25

Deng, Dongmei, and Qi Guo. "Airy complex variable function Gaussian beams." New Journal of Physics 11, no. 10 (October 21, 2009): 103029. http://dx.doi.org/10.1088/1367-2630/11/10/103029.

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26

Krasikov, Ilia. "Approximations for the Bessel and Airy functions with an explicit error term." LMS Journal of Computation and Mathematics 17, no. 1 (2014): 209–25. http://dx.doi.org/10.1112/s1461157013000351.

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AbstractWe show how one can obtain an asymptotic expression for some special functions with a very explicit error term starting from appropriate upper bounds. We will work out the details for the Bessel function $J_\nu (x)$ and the Airy function ${\rm Ai}(x).$ In particular, we answer a question raised by Olenko and find a sharp bound on the difference between $J_\nu (x)$ and its standard asymptotics. We also give a very simple and surprisingly precise approximation for the zeros ${\rm Ai}(x).$

27

Vatannia, S., and G. Gildenblat. "Airy's functions implementation of the transfer-matrix method for resonant tunneling in variably spaced finite superlattices." IEEE Journal of Quantum Electronics 32, no. 6 (June 1996): 1093–105. http://dx.doi.org/10.1109/3.502388.

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28

Baldwin, P. "Zeros of generalized Airy functions." Mathematika 32, no. 1 (June 1985): 104–17. http://dx.doi.org/10.1112/s0025579300010925.

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29

Fernandez, Rahul N., V. S. Varadarajan, and David Weisbart. "Airy Functions Over Local Fields." Letters in Mathematical Physics 88, no. 1-3 (March 21, 2009): 187–206. http://dx.doi.org/10.1007/s11005-009-0311-x.

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30

MacLeod, Allan J. "Computation of inhomogeneous Airy functions." Journal of Computational and Applied Mathematics 53, no. 1 (July 1994): 109–16. http://dx.doi.org/10.1016/0377-0427(94)90196-1.

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31

Aghili, Arman. "Some Results Involving the Airy Functions and Airy Transforms." Tatra Mountains Mathematical Publications 79, no. 2 (December 1, 2021): 13–32. http://dx.doi.org/10.2478/tmmp-2021-0017.

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Abstract In the present work, the author studied some properties of the modified Bessel’s functions and Airy functions. It is worth mentioning that the Airy functions are used in many fields of physics. They are applied in many branches of classical and quantum physics. The author also studied certain properties of the Airy transform and derived some new integral relations involving the Airy functions. Non-trivial illustrative examples are provided as well. All the results are presented in lucid and comprehensible language.

32

Valle´e, Olivier. "Some Integrals Involving Airy Functions and Volterra μ-Functions". Integral Transforms and Special Functions 13, № 5 (1 січня 2002): 403–8. http://dx.doi.org/10.1080/10652460213531.

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33

Maiz, F., and S. AlFaify. "Quantum anharmonic oscillator: The airy function approach." Physica B: Condensed Matter 441 (May 2014): 17–20. http://dx.doi.org/10.1016/j.physb.2014.01.044.

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34

Miyamoto, Tadashi. "On an Airy function of two variables." Nonlinear Analysis: Theory, Methods & Applications 54, no. 4 (August 2003): 755–72. http://dx.doi.org/10.1016/s0362-546x(03)00102-0.

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35

Chen, Rui-Pin, Hong-Ping Zheng, and Chao-Qing Dai. "Wigner distribution function of an Airy beam." Journal of the Optical Society of America A 28, no. 6 (May 31, 2011): 1307. http://dx.doi.org/10.1364/josaa.28.001307.

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36

Katori, Makoto, and Hideki Tanemura. "Zeros of Airy Function and Relaxation Process." Journal of Statistical Physics 136, no. 6 (September 2009): 1177–204. http://dx.doi.org/10.1007/s10955-009-9829-7.

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37

Perelomov, A. M. "A remark on the matrix Airy function." Theoretical and Mathematical Physics 123, no. 2 (May 2000): 671–72. http://dx.doi.org/10.1007/bf02551400.

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38

Lakshmi, B. S., and M. V. Ramana Murty. "Airy function approximations to the Lorenz system." Chaos, Solitons & Fractals 33, no. 4 (August 2007): 1433–35. http://dx.doi.org/10.1016/j.chaos.2006.03.047.

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39

Menon, Govind. "The Airy Function is a Fredholm Determinant." Journal of Dynamics and Differential Equations 28, no. 3-4 (June 4, 2015): 1031–38. http://dx.doi.org/10.1007/s10884-015-9458-6.

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40

Zareian, N., P. Sarrafi, K. Mehrany, and B. Rashidian. "Differential-Transfer-Matrix Based on Airy's Functions in Analysis of Planar Optical Structures With Arbitrary Index Profiles." IEEE Journal of Quantum Electronics 44, no. 4 (April 2008): 324–30. http://dx.doi.org/10.1109/jqe.2007.912469.

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41

Hamdan, M. H., and M. T. Kamel. "On the Ni(x) integral function and its application to the Airy’s non-homogeneous equation." Applied Mathematics and Computation 217, no. 17 (May 2011): 7349–60. http://dx.doi.org/10.1016/j.amc.2011.02.025.

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42

Salmassi, Mohammad. "Inequalities Satisfied by the Airy Functions." Journal of Mathematical Analysis and Applications 240, no. 2 (December 1999): 574–82. http://dx.doi.org/10.1006/jmaa.1999.6620.

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43

Jia-chun, Li, and Zhao Da-gong. "Generalized airy functions with complex variables." Applied Mathematics and Mechanics 6, no. 11 (November 1985): 1053–59. http://dx.doi.org/10.1007/bf03250504.

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44

Torres-Vega, Go, A. Zúñiga-Segundo, and J. D. Morales-Guzmán. "Special functions and quantum mechanics in phase space: Airy functions." Physical Review A 53, no. 6 (June 1, 1996): 3792–97. http://dx.doi.org/10.1103/physreva.53.3792.

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45

YAMAKAWA, Tetsuo. "A FUNDAMENTAL INVESTIGATION ON THE BEAM THEORY BY USING THE ORTHOTROPIC PLATE (SCHEIBE) THEORY AND AIRY'S STRESS FUNCTIONS." Journal of Structural and Construction Engineering (Transactions of AIJ) 438 (1992): 117–26. http://dx.doi.org/10.3130/aijsx.438.0_117.

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MIYAMOTO, Tadashi. "On an Airy Function of Two Variables II." Tokyo Journal of Mathematics 26, no. 2 (December 2003): 549–68. http://dx.doi.org/10.3836/tjm/1244208608.

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Besieris, Ioannis M., and Amr M. Shaarawi. "Wigner distribution function of an Airy beam: comment." Journal of the Optical Society of America A 28, no. 9 (August 15, 2011): 1828. http://dx.doi.org/10.1364/josaa.28.001828.

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Ankalkhope, Suhas, Nilesh Jadhav, and Sunil Bhat. "Stress Solutions of some Axisymmetric and Non-Axisymmetric Cases with the Principles of Elasticity: A Review." Applied Mechanics and Materials 215-216 (November 2012): 1026–32. http://dx.doi.org/10.4028/www.scientific.net/amm.215-216.1026.

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Stress solutions are reviewed for some typical cases of axisymmetric and non-axisymmetric loads over a structural member with the principles of elasticity. A curved bar is chosen for the analysis. Tangential, radial and shear stress are determined analytically using Airy’s stress function. The curved bar is also modelled by finite element method to obtain numerical values of stress. Analytical and numerical results are in excellent agreement with each other.

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Varlamov, Vladimir. "Integrals involving products of Airy functions, their derivatives and Bessel functions." Journal of Mathematical Analysis and Applications 370, no. 2 (October 2010): 687–702. http://dx.doi.org/10.1016/j.jmaa.2010.05.004.

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Pressel, Kyle G., and William D. Collins. "First-Order Structure Function Analysis of Statistical Scale Invariance in the AIRS-Observed Water Vapor Field." Journal of Climate 25, no. 16 (August 15, 2012): 5538–55. http://dx.doi.org/10.1175/jcli-d-11-00374.1.

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Abstract The power-law scale dependence, or scaling, of first-order structure functions of the tropospheric water vapor field between 58°S and 58°N is investigated using observations from the Atmospheric Infrared Sounder (AIRS). Power-law scale dependence of the first-order structure function would indicate that the water vapor field exhibits statistical scale invariance. Directional and directionally independent first-order structure functions are computed to assess the directional dependence of derived first-order structure function scaling exponents (H) for a range of scales from 50 to 500 km. In comparison to other methods of assessing statistical scale invariance, the methodology used here requires minimal assumptions regarding the homogeneity of the spatial distribution of data within regions of analysis. Additionally, the methodology facilitates the evaluation of anisotropy and quantifies the extent to which the structure functions exhibit scale invariance. The spatial and seasonal dependence of the computed scaling exponents are explored. Minimum scaling exponents at all levels are shown to occur proximate to the equator, while the global maximum is shown to occur in the middle troposphere near the tropical–subtropical margin of the winter hemisphere. From a detailed analysis of AIRS maritime scaling exponents, it is concluded that the AIRS observations suggest the existence of two scaling regimes in the extratropics. One of these regimes characterizes the statistical scale invariance the free troposphere with H approximately = 0.55 and a second that characterizes the statistical scale invariance of the boundary layer with H approximately = ⅓.