Atmospheric Chemistry and Physics Discussions www.atmos-chem-phys-discuss.net 1680-7367 1680-7375 9 4 2009 10.5194/acpd-9-16085-2009 http://www.atmos-chem-phys-discuss.net/9/16085/2009/ http://www.atmos-chem-phys-discuss.net/9/16085/2009/acpd-9-16085-2009.html http://www.atmos-chem-phys-discuss.net/9/16085/2009/acpd-9-16085-2009.pdf 16085 16129 2009-07-28 Advective mixing in a nondivergent barotropic hurricane model B. Rutherford rutherfo@math.colostate.edu G. Dangelmayr J. Persing W. H. Schubert M. T. Montgomery Department of Mathematics, Colorado State University, Fort Collins, CO 80523-1874, USA Department of Atmospheric Science, Colorado State University, Fort Collins, CO 80523-1371, USA Department of Meteorology, Naval Postgraduate School, Monterey, CA 93943-5114, USA This paper studies Lagrangian mixing in a two-dimensional barotropic model for hurricane-like vortices. Since such flows show high shearing in the radial direction, particle separation across shear-lines is diagnosed through a Lagrangian field, referred to as R-field, that measures trajectory separation orthogonal to the Lagrangian velocity. The shear-lines are identified with the level-contours of another Lagrangian field, referred to as S-field, that measures the average shear-strength along a trajectory. Other fields used for model diagnostics are the Lagrangian field of finite-time Lyapunov exponents (FTLE-field), the Eulerian Q-field, and the angular velocity field. Because of the high shearing, the FTLE-field is not a suitable indicator for advective mixing, and in particular does not exhibit clear ridges marking the location of finite-time stable and unstable manifolds. The FTLE-field is similar in structure to the radial derivative of the angular velocity. In contrast, distinct and persisting ridges and valleys can be clearly recognized in the R-field, and their propagation speed indicates that transport across shear-lines is caused by Rossby waves. A radial mixing rate derived from the R-field gives a time-dependent measure of flux across the shear-lines. On the other hand, a measured mixing rate across the shear-lines, which counts trajectory crossings, confirms the results from the R-field mixing rate, and shows high mixing in the eyewall region after the formation of a polygonal eyewall, which continues until the vortex breaks down. Frank, W M. and Ritchie, E A.: Effects of environmental flow upon tropical cyclone structure, Mon. Weather Rev., 127, 2044–2061, 1999. Frank, W M. and Ritchie, E A.: Effects of vertical wind shear on the intensity and structure of numerically simulated hurricanes, Mon. Weather Rev., 129, 2249–2269, 2001. Green, M A., Rowley, C W., and Haller, G.: Detection of Lagrangian coherent structures in 3-D turbulence, J. Fluid Mech., 572, 111–120, 2006. Haller, G.: Finding finite-time invariant manifolds in two-dimensioanl velocity fields, Chaos, 10, 99–108, 2000. Haller, G.: Lagrangian structures and the rate of strain in a partition of two-dimensional turbulence, Phys. Fluids, 13, 3365–3385, 2001. 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, Phys. Rev. E, 68, 056304–1–6, 2003. Haller, G. and Poje, A C.: 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. and Schubert, W H.: Transport and mixing in idealized barotropic hurricane-like vortices, Quart. J. Roy. Meteor. Soc., 135, 2009. Huber, M., McWilliams, J C., and Ghil, M.: A climatology of turbulent dispersion in the troposphere, J. Atmos. Sci., 58, 2377–2394, 2001. 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, Nonlinear Proc. Geoph., 9, 237–263, 2002. Joseph, B. and Legras, B.: Relation between kinematic boundaries, stirring, and barriers for the antarctic polar vortex, J. Atmos. Sci., 59, 1198–1212, 2001. Kossin, J P. and Schubert, W H.: Mesovortices, polygonal flow patterns, and rapid pressure falls in hurricane-like vortices, J. Atmos. Sci., 58, 2196–2209, 2001. Montgomery, M T. and Lu, C.: Free waves on barotropic vortices, Part I: Eigenmode structure, J. Atmos. Sci., 54, 1868–1885, 1997. Montgomery, M T., Bell, M M., Aberson, S D., and Black, M L.: Hurricane Isabel~(2003): New insights into the physics of intense storms, Part I: Mean vortex structure and maximum intensity estimates, Bull. Amer. Met. Soc., 87, 1349–1354, 2006. Nakamura, N.: Two-dimensional mixing, edge formation, and permeability diagnosed in area coordinates, J. Atmos. Sci., 53, 1524–1537, 1996. Ottino, J M.: The Kinematics of Mixing: Stretching, Chaos, and Transport, Cambridge University Press, 1989. Rotunno, R. and Emanuel, K.: An air-sea interaction theory for tropical cyclones, Part II: Evolutionary study using a nonhydrostatic axisymmetric numerical model, J. Atmos. Sci., 44, 542–561, 1987. Rutherford, B., Persing, J., Dangelmayr, G., Kirby, M., and Montgomery, M.: Lagrangian mixing in an axisymmetric hurricane model, preprint, submitted for publication, 2009. Sapsis, T. and Haller, G.: Inertial particle dynamics in a hurricane, preprint, submitted, 2009. Schubert, W H., Montgomery, M T., Taft, R K., Guinn, T A., Fulton, S R., Kossin, J P., and Edwards, J P.: Polygonal eyewalls, asymmetric eye contraction, and potential vorticity mixing in hurricanes, J. Atmos. Sci., 56, 1197–1223, 1999. Shadden, S C., Lekein, 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. Shuckburgh, E. and Haynes, P.: Diagnosing transport and mixing using a tracer-based coordinate system, Phys. Fluids, 15, 3342–3357, 2003. Willoughby, H E.: Tropical cyclone eye thermodynamics, Mon. Weather Rev., 126, 3053–3067, 2001.