ACP Atmospheric Chemistry and Physics ACP 1680-7324 Copernicus GmbH Göttingen, Germany 10.5194/acp-12-5483-2012 Lagrangian coherent structures in tropical cyclone intensification Rutherford B. 1 Dangelmayr G. 2 Montgomery M. T. 1 Naval Postgraduate School, Monterey, CA, USA Department of Mathematics, Colorado State University, Fort Collins, CO, USA 22 06 2012 12 12 5483 5507 This is an open-access article ditributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. This article is available from http://www.atmos-chem-phys.net/12/5483/2012/acp-12-5483-2012.html The full text article is available as a PDF file from http://www.atmos-chem-phys.net/12/5483/2012/acp-12-5483-2012.pdf

Recent work has suggested that tropical cyclones intensify via a pathway of rotating deep moist convection in the presence of enhanced fluxes of moisture from the ocean. The rotating deep convective structures possessing enhanced cyclonic vorticity within their cores have been dubbed Vortical Hot Towers (VHTs). In general, the interaction between VHTs and the system-scale vortex, as well as the corresponding evolution of equivalent potential temperature (&theta;<sub>e</sub>) that modulates the VHT activity, is a complex problem in moist helical turbulence. <br><br> To better understand the structural aspects of the three-dimensional intensification process, a Lagrangian perspective is explored that focuses on the coherent structures seen in the flow field associated with VHTs and their vortical remnants, as well as the evolution and localized stirring of &theta;<sub>e</sub>. Recently developed finite-time Lagrangian methods are limited in the three-dimensional turbulence and shear associated with the VHTs. In this paper, new Lagrangian techniques developed for three-dimensional velocity fields are summarized and we apply these techniques to study VHT and &theta;<sub>e</sub> phenomenology in a high-resolution numerical tropical cyclone simulation. The usefulness of these methods is demonstrated by an analysis of particle trajectories. <br><br> We find that VHTs create a locally turbulent mixing environment. However, associated with the VHTs are hyperbolic structures that span between adjacent VHTs or adjacent vortical remnants and represent coherent finite-time transport barriers in the flow field. Although the azimuthally-averaged inflow is responsible for the inward advection of boundary layer &theta;<sub>e</sub>, attracting Lagrangian coherent structures are coincident with pools of high boundary layer &theta;<sub>e</sub>. Extensions of boundary layer coherent structures grow above the boundary layer during episodes of convection and remain with the convective vortices. These hyperbolic structures form initially as boundaries between VHTs. As vorticity aggregates into a ring-like eyewall feature, the Lagrangian boundaries merge into a ring outside of the region of maximal vorticity.

 References Bell, M. and Montgomery, M T.: Sheared deep vortical convection in pre-depression Hagupit during TCS08, Geophys. Res. Lett., 37, L06802, http://dx.doi.org/10.1029/2009GL042313doi:10.1029/2009GL042313, 2010. Bolton, D.: The computation of equivalent potential temperature, Mon. Weather Rev., 108, 1046–1053, 1980. Bui, H H., Smith, R K., Montgomery, M T., and~Peng, J.: Balanced and unbalanced aspects of tropical cyclone intensification, Q. J. Roy. Meteorol. Soc., 135, 1715–1731 2009. Chen, Y. and Yau, M K.: Asymmetric structures in a simulated landfalling hurricane, J. Atmos. Sci., 60, 2294–2312, 2003. Chen, Y.,~Brunet, G., and Yau, M K.: Spiral bands in a simulated hurricane part II: wave activity diagnostics, J. Atmos. Sci., 60, 1239–1256, 2003. Cohen, R A. and Schultz, D M.: Contraction rate and its relation to frontogenesis, the Lyapunov exponent, fluid trapping, and airstream boundaries, Mon. Weather Rev., 133, 1353–1369, 2005. d'Ovidio, F., Isern-Fontanet, J., Lopez, C., Hernndez-Garca, E., and Garca-Ladona, E.: Comparison between Eulerian diagnostics and finite-size Lyapunov exponents computed from altimetry in the Algerian basin, Deep Sea Res. Part I: Oceanographic Research Papers, 56, 15–31, 1993. Dudhia, J.: A non-hydrostatic version of the penn state-ncar mesoscale model: Validation tests and simulation of an atlantic cyclone and cold front, Mon. Wheather Rev., 8, 1493–1513, 1993. Dunkerton, T. J., Montgomery, M. T., and Wang, Z.: Tropical cyclogenesis in a tropical wave critical layer: easterly waves, Atmos. Chem. Phys., 9, 5587–5646, http://dx.doi.org/10.5194/acp-9-5587-2009doi:10.5194/acp-9-5587-2009, 2009. Emanuel, K A., Neelin, J D., and Bretherton, C S.: On large scale circulations in convecting atmosphere, Q. J. Roy. Meteorol. Soc., 120, 1111–1143, 1994. Fang , J and~Zhang, F.: Initial development and genesis of Hurricane Dolly (2008), J. Atmos. Sci., 67, 655–672, 2010. Grell, G.,~Dudhia, J., and Stauffer, D.: A description of the fifth generation Penn State-NCAR Mesoscale Model (MM5), NCAR Tech. Note NCAR/TN, 398, 1–138, 1995. Haller, G.: Finding finite-time invariant manifolds in two-dimensional velocity fields, Chaos, 10, 99–108, 2000. Haller, G.: Lagrangian coherent structures from approximate velocity data, Phys. Fluids, 14, 1851–1861, 2002. Haller, G and~Iacono, R.: Stretching, alignment, and shear in slowly varying velocity fields, Physical Review E, 68, 056304–1–6, 2003. Haller, G. and~Poje, A.: Finite time transport in aperiodic flows, Physica D, 119, 352–380, 1997. Haller, G. and~Yuan, G.: Lagrangian coherent structures and mixing in two-dimensional turbulence, Physica D, 147, 352–370, 2000. Hendricks, E A., Montgomery, M T., and Davis, C A.: On the role of vortical hot towers in tropical cyclone formation, J. Atmos. Sci., 61, 1209–1232, 2004. Holton, J R.: An introduction to dynamic meteorology, 4th edition, Elsevier, 2004. Ide, K., Small, D., and Wiggins, S.: Distinguished hyperbolic trajectories in time-dependent fluid flows: analytical and computational approach for velocity fields defined as data sets, Nonlin. Processes Geophys., 9, 237–263, http://dx.doi.org/10.5194/npg-9-237-2002doi:10.5194/npg-9-237-2002, 2002. Joseph, B. and Legras, B.: Relation between Kinematic Boundaries, Stirring, and Barriers for the Antarctic Polar Vortex, J. Atmos. Sci., 59, 1198–1212, 2002. Lapeyre, G.: Characterization of finite-time Lyapunov exponents and vectors in two-dimensional turbulence, Chaos, 12, 688–698, 2002. Levina, G. and Montgomery, M T.: A first examination of the helical nature of tropical cyclogenesis, Doklady Earth Sciences, 434, Part I, 1285–1289, 2010. Lilly, D K.: On the theory of disturbances in a conditionally unstable atmosphere, Mon. Weather Rev., 88, 1–17, 1960. Malhotra, N. and~Wiggins, S.: Geometric structures, lobe dynamics, and Lagrangian transport in flows with aperiodic time-dependence, with applications to Rossby wave flow, J. Nonlinear Sci., 8, 401–456, 1999. Melander, N J.,~Zabusky, M V., and McWilliams, J C.: Symmetric vortex merger in two dimensions – causes and conditions, J. Fluid Mech., 195, 303–340, 1988. Montgomery, M T. and Smith, R K.: Paradigms for tropical-cyclone intensification, Q. J. Roy. Meteorol. Soc., in review, 2012. Montgomery, M T., Nichols, M E., Cram, T A., and Saunders, A B.: A vortical hot tower route to tropical cyclogenesis, J. Atmos. Sci., 63, 355–386, 2006. Montgomery, M T., Nguyen, V S., Smith, R K., and~Persing, J.: Do tropical cyclones intensify by WISHE?, Q. J. Roy. Meteorol. Soc., 135, 1697–1714, 2009. Montgomery, M. T., Wang, Z., and Dunkerton, T. J.: Coarse, intermediate and high resolution numerical simulations of the transition of a tropical wave critical layer to a tropical storm, Atmos. Chem. Phys., 10, 10803–10827, http://dx.doi.org/10.5194/acp-10-10803-2010doi:10.5194/acp-10-10803-2010, 2010. Ngan, K. and Shepherd, T G.: A Closer Look at Chaotic Advection in the Stratosphere. Part I: Geometric Structure, J. Atmos. Sci., 56, 4134–4152, 1999. Nguyen, S A., Smith, R K., and Montgomery, M T.: Tropical-cyclone intensification and predictability in three dimensions, Q. J. Roy. Meterol. Soc., 134, 563–582, 2008. Nguyen, C M., Reeder, M J., Davidson, N E., Smith, R K., and Montgomery, M T.: Inner-core vacillation cycles during the rapid intensification of Hurricane Katrina, Q. J. Roy. Meteorol. Soc., 137, 829–844, 2011. Ottino, J M.: The kinematics of mixing: stretching, chaos, and transport, Annual Reviews of Fluid Mechanics, 22, 207–53, 1990. Pierrehumbert, R T.: Large-scale horizontal mixing in planetary atmospheres, Phys. Fluids A, 3, 1250–1260, http://dx.doi.org/10.1063/1.858053doi:10.1063/1.858053, 1991. Pierrehumbert, R T. and Yang, H.: Global chaotic mixing on isentropic surfaces, J. Atmos. Sci., 50, 2462–2480, 1993. Prieto, R., McNoldy, B. D., Fulton, S. R., and Schubert, W. H.: A classification of binary tropical-cyclone-like vortex interactions, Mon. Weather. Rev., 131, 2656–2666, 2003. Provenzale, A.: Transport by coherent barotropic vortices, Annual Review of Fluid Mechanics, 31, 55–93, 1999. Raymond, D. J. and López Carrillo, C.: The vorticity budget of developing typhoon Nuri (2008), Atmos. Chem. Phys., 11, 147–163, http://dx.doi.org/10.5194/acp-11-147-2011doi:10.5194/acp-11-147-2011, 2011. Reasor, P D., Montgomery, M T., and~Bosart, L.: Mesoscale observations of the genesis of Hurricane Dolly (1996), J. Atmos. Sci., 62, 3151–3171, 2005. Rutherford, B. and~Dangelmayr, G.: A 3d Lagrangian hurricane eye-eyewall computation, Q. J. Roy. Meteorol. Soc., 136, 1931–1944, 2010. Rutherford, B., Dangelmayr, G., Persing, J., Kirby, M., and Montgomery, M. T.: Lagrangian mixing in an axisymmetric hurricane model, Atmos. Chem. Phys., 10, 6777–6791, http://dx.doi.org/10.5194/acp-10-6777-2010doi:10.5194/acp-10-6777-2010, 2010a. Rutherford, B., Dangelmayr, G., Persing, J., Schubert, W. H., and Montgomery, M. T.: Advective mixing in a nondivergent barotropic hurricane model, Atmos. Chem. Phys., 10, 475–497, http://dx.doi.org/10.5194/acp-10-475-2010doi:10.5194/acp-10-475-2010, 2010b. Saffman, P G.: Dynamics of vorticity, J. Fluid Mechanics, 106, 49–58, 1981. Sapsis, T. and~Haller, G.: Inertial particle dynamics in hurricanes, J. Atmos. Sci., 66, 2481–2492, 2009. Schubert, W H. and Hack, J J.: Inertial stability and tropical cyclone development, J. Atmos. Sci., 39, 1687–1697, 1982. Scott, R K., Shuckburgh, E F., Cammas, J P., and~Legras, B.: Stretching rates and equivalent length near the tropopause, J. Geophys. Res., 108, 4394, http://dx.doi.org/10.1029/2002JD002988doi:10.1029/2002JD002988, 2003. Shadden, S.: A dynamical systems approach to unsteady flows, Ph.D. thesis, California Institute of Technology, 2006. Shadden, S C.,~Lekien, F., and Marsden, J E.: Definition and properties of Lagrangian coherent structures from finite-time Lyapunov exponents in two-dimensional aperiodic flows, Physica D, 212, 271–304, 2005. Shadden, S C., Dabiri, J O., and Marsden, J E.: Lagrangian analysis of fluid transport in empirical vortex rings, Phys. Fluids, 18, 047105–1–11, 2006. Shin, S. and Smith, R K.: Tropical cyclone intensification and predictability in a minimal three-dimensional model, Q. J. Roy. Meteorol. Soc., 134, 1385–1395, 2008. Siggia, E.: High Raleigh number convection, Annu. Rev. Fluid Mech., 26, 137–168, 1994. Sippel, J A., Nielsen-Gammon, J W., and Allen, S E.: The multiple- vortex nature of tropical cyclogenesis, Mon. Weather Rev., 134, 1796–1814, 2006. Smith, R K., Montgomery, M T., and~Vogl, S.: A critique of Emanuel's hurricane model and potential intensity theory, Q. J. Roy. Meteorol. Soc., 134–632, 551–561, 2008. Smith, R K., Montgomery, M T., and Nguyen, V S.: Tropical cyclone spin up revisited, Q. J. Roy. Meteorol. Soc., 134, 1385–1395, 2009. Tang, W., Mathur, M., Haller, G., Hahn, D. C., and Ruggiero, F. H.: Lagrangian coherent structures near a subtropical jet stream, J. Atmos. Sci., 67, 2307–2319, 2010. Truesdell, C.: The kinematics of vorticity, Indiana Uiversity Press, 1954. von Hardenberg, J., Mc~Williams, J C., Provenzale, A., Shchepetkin, A., and Weiss, J B.: Vortex merging in quasi-geostrophic flows, J. Fluid Mech., 412, 331–353, 2000. Velasco Fuentes, O.: Chaotic advection by two interacting finite-area vortices, Phys. of Fluids, 13-4, 901–912, 2001. Wang, Y.: Vortex rossby waves in a numerically simulated tropical cyclone. part ii: The role in tropical cyclone structure and intensity changes, J. Atmos. Sci., 59, 1239–1262, 2000. Weiss, J B. and~Provenzale, A.: Transport and mixing in geophysical flows, Springer, 2008. Wiggins, S.: The dynamical systems approach to Lagrangian transport in oceanic flows, Annual Review of Fluid Mechanics, 37, 295–328, 2005. \hack Willoughby, H E.: Forced secondary circulations in hurricanes, J. Geophys. Res., 84, 3173–3183, 1979.