Research article Topical Sections

Possible implications of self-similarity for tornadogenesis and maintenance

  • Received: 12 July 2018 Accepted: 28 September 2018 Published: 10 October 2018
  • MSC : 28A80, 76B47, 76D05, 76F06, 76F10, 76M55, 76U05, 86A10

  • Self-similarity in tornadic and some non-tornadic supercell flows is studied and power laws relating various quantities in such flows are demonstrated. Magnitudes of the exponents in these power laws are related to the intensity of the corresponding flow and thus the severity of the supercell storm. The features studied in this paper include the vertical vorticity and pseudovorticity, both obtained from radar observations and from numerical simulations, the tangential velocity, and the energy spectrum as a function of the wave number. Power laws for the vertical vorticity, pseudovorticity, and tangential velocity obtained from radar observations studied in the literature are summarized. Further support is given to the existence of a power law for vorticity by the analysis of data obtained from a numerical simulation of a tornadic supercell. A possible explanation for an increase in vorticity in a developing tornado is provided, as well as an argument that tornadoes have approximate fractal cross sections and negative temperatures. A power law that relates the increase of the approximate fractal dimension of the cross section of a negative temperature vortex to its energy content is derived, and the possible applicability of the box-counting method to determine this quantity from suitable radar images is demonstrated.

    Citation: Pavel Bělík, Brittany Dahl, Douglas Dokken, Corey K. Potvin, Kurt Scholz, Mikhail Shvartsman. Possible implications of self-similarity for tornadogenesis and maintenance[J]. AIMS Mathematics, 2018, 3(3): 365-390. doi: 10.3934/Math.2018.3.365

    Related Papers:

  • Self-similarity in tornadic and some non-tornadic supercell flows is studied and power laws relating various quantities in such flows are demonstrated. Magnitudes of the exponents in these power laws are related to the intensity of the corresponding flow and thus the severity of the supercell storm. The features studied in this paper include the vertical vorticity and pseudovorticity, both obtained from radar observations and from numerical simulations, the tangential velocity, and the energy spectrum as a function of the wave number. Power laws for the vertical vorticity, pseudovorticity, and tangential velocity obtained from radar observations studied in the literature are summarized. Further support is given to the existence of a power law for vorticity by the analysis of data obtained from a numerical simulation of a tornadic supercell. A possible explanation for an increase in vorticity in a developing tornado is provided, as well as an argument that tornadoes have approximate fractal cross sections and negative temperatures. A power law that relates the increase of the approximate fractal dimension of the cross section of a negative temperature vortex to its energy content is derived, and the possible applicability of the box-counting method to determine this quantity from suitable radar images is demonstrated.


    加载中
    [1] E. J. Adlerman and K. Droegemeier, A numerical simulation of cyclic tornadogenesis, 20th Conference on Severe Local Storms, Orlando, FL, American Meteorological Society, 2000.
    [2] J. C. André and M. Lesieur, Influence of helicity on the evolution of isotropic turbulence at high Reynolds number, J. Fluid Mech., 81 (1977), 187-207.
    [3] V. I. Arnold and B. A. Khesin, Topological Methods in Hydrodynamics, Applied Mathematical Sciences, Vol. 125, Springer, New York, 2 edition, 1998.
    [4] G. R. Baker and M. J. Shelley, On the connection between thin vortex layers and vortex sheet, J. Fluid Mech., 215 (1990), 161-194.
    [5] A. I. Barcilon, Vortex decay above a stationary boundary, J. Fluid Mech., 27 (1967), 155-157.
    [6] G. I. Barenblatt, Scaling, Self-similarity, and Intermediate Asymptotics, Cambridge University Press, 1996.
    [7] G. I. Barenblatt, Scaling, Cambridge University Press, 2003.
    [8] M. Barnsley, Fractals Everywhere, Academic Press, 1988.
    [9] S. J. Benavides and A. Alexakis, Critical transitions in thin layer turbulence, J. Fluid Mech., 822 (2017), 364-385.
    [10] T. B. Benjamin, Theory of the vortex breakdown phenomenon, J. Fluid Mech., 14 (1962), 593-629.
    [11] H. B. Bluestein, Severe Convective Storms and Tornadoes, Observations and Dynamics, Springer-Praxis books in Environmental Sciences, Springer, 2013.
    [12] H. B. Bluestein and A. L. Pazmany, Observations of tornadoes and other convective phenomena with a mobile, 3-mm wavelength, Doppler radar: The spring 1999 field experiment, B. Am. Meteorol. Soc., 81 (2000), 2939-2951.
    [13] R. Bohac, May 31, 2013 EF5 El Reno Tornado Showing Multiple Funnels/Sub Vortices Filmed from Dominator. Retrieved May 11, 2018. Available from: http://www.youtube.com/watch?v=C34EVyWRZbk.
    [14] G. H. Bryan and J. M. Fritsch, A benchmark simulation for moist nonhydrostatic numerical models, Mon. Weather Rev., 130 (2002), 2917-2928.
    [15] K. Bürger, M. Treib, R. Westermann, et al. Vortices within vortices: hierarchical nature of vortex tubes in turbulence, eprint arXiv: 1210.3325[physics.flu-dyn], Mar. 2013.
    [16] O. R. Burggraf and M. R. Foster, Continuation or breakdown in tornado-like vortices, J. Fluid Mech., 80 (1977), 685-703.
    [17] P. Bělík, D. Dokken, C. Potvin, et al. Applications of vortex gas models to tornadogenesis and maintenance, Open Journal of Fluid Dynamics, 7 (2017), 596-622.
    [18] P. Bělík, D. P. Dokken, K. Scholz, et al. Fractal powers in Serrin's swirling vortex model, Asymptot. Anal., 90 (2014), 53-82.
    [19] H. Cai, Comparison between tornadic and nontornadic mesocyclones using the vorticity (pseudovorticity) line technique, Mon. Weather Rev., 133 (2005), 2535-2551.
    [20] A. J. Chorin, Vorticity and Turbulence, Springer Verlag, New York, 1994.
    [21] A. J. Chorin and J. H. Akao, Vortex equilibria in turbulence and quantum analogues, Physica D: Nonlinear Phenomena, 52 (1991), 403-414.
    [22] C. R. Church, J. T. Snow and E. M. Agee, Tornado vortex simulation at Purdue University, B. Am. Meteorol. Soc., 58 (1977), 900-908.
    [23] G. P. Cressman, An operational objective in analysis system, Mon. Weather Rev., 87 (1959), 367-374.
    [24] R. Davies-Jones. A review of supercell and tornado dynamics, Atmos. Res., 158 (2015), 274-291.
    [25] J. W. Deardorff, Stratocumulus-capped mixed layer derived from a three-dimensional model, Boundary-Layer Meteorology, 18 (1980), 495-527.
    [26] D. C. Dowell and H. B. Bluestein, The 8 June 1995 McLean, Texas, storm. Part Ⅰ: Observations of cyclic tornadogenesis, Mon. Weather Rev., 130 (2002), 2626-2648.
    [27] D. C. Dowell and H. B. Bluestein, The 8 June 1995 McLean, Texas, storm. Part Ⅱ: Cyclic tornado formation, maintenance, and dissipation, Mon. Weather Rev., 130 (2002), 2649-2670.
    [28] R. E. Ecke, From 2D to 3D in fluid turbulence: unexpected critical transitions, J. Fluid Mech., 828 (2017), 1-4.
    [29] B. H. Fiedler and R. Rotunno, A theory for the maximum windspeeds in tornado-like vortices, J. Atmos. Sci., 43 (1986), 2328-2440.
    [30] R. Frehlich and R. Sharman, The use of structure functions and spectra from numerical model output to determine e_ective model resolution, Mon. Weather Rev., 136 (2008), 1537-1553.
    [31] T. T. Fujita, Tornadoes and downbursts in the context of generalized planetary scales, J. Atmos. Sci., 38 (1981), 1511-1534.
    [32] T. P. Grazulis, Significant Tornadoes Update, 1992-1995, Environmental Films, St. Johnsbury, VT, 1997.
    [33] A. Y. Klimenko, Strong swirl approximation and intensive vortices in the atmosphere, J. Fluid Mech., 738 (2014), 268-298.
    [34] A. N. Kolmogorov, The local structure of turbulence in incompressible viscous fluid for very large Reynolds number, Dokl. Akad. Nauk SSSR, 30 (1941), 9-13.
    [35] K. A. Kosiba and J.Wurman, Dow observations of multiple vortex structures in several tornadoes, In Preprints, 24th Conf. on Severe Local Storms, Vol. 3, 2008.
    [36] K. A. Kosiba, R. J. Trapp and J. Wurman, An analysis of the axisymmetric three-dimensional low level wind field in a tornado using mobile radar observations, Geophys. Res. Lett., 35 (2008).
    [37] S. P. Kuznetsov, Dynamical chaos and uniformly hyperbolic attractors: from mathematics to physics, Physics-Uspekhi, 54 (2011), 119-144.
    [38] D. C. Lewellen, W. S. Lewellen and R. I. Sykes, Large-eddy simulation of a tornado's interaction with the surface, J. Atmos. Sci., 54 (1997), 581-605.
    [39] D. K. Lilly, Tornado dynamics, NCAR Manuscript 69-117, 1969.
    [40] D. K. Lilly, Sources of rotation and energy in the tornado, Proc. Symp. on Tornadoes: Assessment of Knowledge and Implications for Man, pp 145-150, Lubbock, TX, 1976.
    [41] D. K. Lilly, The structure, energetics and propagation of rotating convective storms. Part Ⅰ: Energy exchange with the mean flow, J. Atmos. Sci., 43 (1986), 113-125.
    [42] D. K. Lilly, The structure, energetics and propagation of rotating convective storms. Part Ⅱ: Helicity and storm stablization, J. Atmos. Sci., 43 (1986), 126-140.
    [43] D. E. Lund and J. T. Snow, Laser Doppler velocimeter mesaurements in tornadolike vortices, The Tornado: Its Structure, Dynamics, Prediction, and Hazards, volume Monograph 79, pp 297-306, American Geophysiscal Union, 1993.
    [44] K. J. Mallen, M. T. Montgomery and B. Wang, Reexamining the near-core radial structure of the tropical cyclone primary circulation: Implications for vortex resiliency, J. Atmos. Sci., 62 (2005), 408-425.
    [45] B. B. Mandelbrot, The Fractal Geometry of Nature, W.H. Freeman and Company, 1983.
    [46] B. I. Miller, Characteristics of hurricanes, Science, 157 (1967), 1389-1399.
    [47] H. Morrison, J. A. Curry and V. I. Khvorostyanov, A new double-moment microphysics parametrization for application in cloud and climate models. Part Ⅰ: Description, J. Atmos. Sci., 62 (2005), 1665-1677.
    [48] L. Orf, R. Wilhelmson, B. Lee, et al. Evolution of a long-track violent tornado within a simulated supercell, B. Am. Meteorol. Soc., 98 (2017), 45-68.
    [49] PBS NOVA, Hunt for the Supertwister, Originally aired March 30, 2004. Retrieved May 11, 2018. Available from: http://www.pbs.org/wgbh/nova/earth/hunt-for-the-supertwister.html.
    [50] Y. Pesin and V. Climenhaga, Lectures on Fractal Geometry and Dynamical Systems, Vol. 52, American Mathematical Soc., 2009.
    [51] C. K. Potvin, A variational method for detecting and characterizing convective vortices in Cartesian wind fields, Mon. Weather Rev., 141 (2013), 3102-3115.
    [52] A. Pouquet and P. D. Mininni, The interplay between helicity and rotation in turbulence: implications for scaling laws and small-scale dynamics, Philos. T. R. Soc. A, 368 (2010), 1635-1662.
    [53] Y. K. Sasaki, Entropic balance theory and variational field Lagrangian formalism: Tornadogenesis, J. Atmos. Sci., 71 (2014), 2104-2113.
    [54] L. Schwartz, Lectures on disintegration of measures, Tata Institute of Fundamental Research, Vol. 50, 1976.
    [55] J. Serrin, The swirling vortex, Philos. T. R. Soc. A, 271 (1972), 325-360.
    [56] R. Timmer, Violent Minnesota wedge tornado intercept!!!, Jun 17, 2010. Retrieved May 11, 2018. Available from: http://www.youtube.com/watch?v=AvD2nDyXSQo.
    [57] R. J. Trapp, Observations of nontornadic low-level mesocyclones and attendant tornadogenesis failure during VORTEX 94, Mon. Weather Rev., 124 (1999), 384-407.
    [58] D. L. Turcotte, Fractals in fluid mechanics, Annu. Rev. Fluid Mech., 20 (1988), 5-16.
    [59] R. M. Wakimoto and H. Cai, Analysis of a nontornadic storm during VORTEX 95, Mon. Weather Rev., 128 (2000), 565-592.
    [60] R. M. Wakimoto, C. Liu and H. Cai, The Garden City, Kansas, storm during VORTEX 95. Part Ⅰ: Overview of the storm's life cycle and mesocyclogenesis, Mon.Weather Rev., 126 (1998), 372-392.
    [61] F. Waleffe, The nature of triad interactions in homogeneous turbulence, Phys. Fluids A, 4 (1992), 350-363.
    [62] J. Wurman, The multiple-vortex structure of a tornado, Weather and Forecasting, 17 (2002), 473-505.
    [63] J. Wurman and C. R. Alexander, The 30 May 1998 Spencer, South Dakota, storm. Part Ⅱ: Comparison of observed damage and radar-derived winds in the tornadoes, Mon. Weather Rev., 133 (2005), 97-119.
    [64] J.Wurman and S. Gill, Fine-scale radar observations of the Dimmitt, Texas (2 June 1995) tornado, Mon. Weather Rev., 128 (2000), 2135-2164.
    [65] N. Yokoi and A. Yoshizawa, Statistical analysis of the e_ects of helicity in inhomogeneous turbulence, Phys. Fluids A, 5 (1993), 464-477.
  • Reader Comments
  • © 2018 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(4530) PDF downloads(804) Cited by(0)

Article outline

Figures and Tables

Figures(16)  /  Tables(1)

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog