• Adler, R. F., and E. B. Rodgers, 1977: Satellite-observed latent heat release in a tropical cyclone. Mon. Wea. Rev., 105, 956963, https://doi.org/10.1175/1520-0493(1977)105<0956:SOLHRI>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Andersson, E., and M. Masutani, 2010: Collaboration on observing system simulation experiments (Joint OSSE). ECMWF Newsletter, No. 123, ECMWF, Reading, United Kingdom, 14–16, http://www.emc.ncep.noaa.gov/research/JointOSSEs/publications/JOSSE-Publication-files/Andersson_JOSSE_ECMWF_News_No123.pdf.

  • Braun, S. A., 2006: High-resolution simulation of Hurricane Bonnie (1998). Part II: Water budget. J. Atmos. Sci., 63, 4364, https://doi.org/10.1175/JAS3609.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Cecil, D. J., and E. J. Zipser, 1999: Relationships between tropical cyclone intensity and satellite-based indicators of inner-core convection: 85-GHz ice-scattering signature and lightning. Mon. Wea. Rev., 127, 103123, https://doi.org/10.1175/1520-0493(1999)127<0103:RBTCIA>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Dudhia, J., and Coauthors, 2008: Prediction of Atlantic tropical cyclones with the Advanced Hurricane WRF (ARW) Model. 28th Conf. on Hurricanes and Tropical Meteorology, Orlando, FL, Amer. Meteor. Soc., 18A.2, https://ams.confex.com/ams/pdfpapers/138004.pdf.

    • Search Google Scholar
    • Export Citation
  • Dunion, J. P., C. D. Thorncroft, and C. S. Velden, 2014: The tropical cyclone diurnal cycle of mature hurricanes. Mon. Wea. Rev., 142, 39003919, https://doi.org/10.1175/MWR-D-13-00191.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Dunion, J. P., C. D. Thorncroft, and D. S. Nolan, 2019: Tropical cyclone diurnal signals in a hurricane nature run. Mon. Wea. Rev., 147, 363388, https://doi.org/10.1175/MWR-D-18-0130.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Emanuel, K. A., 1991: The theory of hurricanes. Annu. Rev. Fluid Mech., 23, 179196, https://doi.org/10.1146/annurev.fl.23.010191.001143.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Fairall, C. W., E. F. Bradley, J. E. Hare, A. A. Grachev, and J. B. Edson, 2003: Bulk parameterization of air–sea fluxes: Updates and verification to the COARE algorithm. J. Climate, 16, 571591, https://doi.org/10.1175/1520-0442(2003)016<0571:BPOASF>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Gamache, J. F., R. A. Houze, and F. D. Marks, 1993: Dual-aircraft investigation of the inner core of Hurricane Norbert. Part III: Water budget. J. Atmos. Sci., 50, 32213243, https://doi.org/10.1175/1520-0469(1993)050<3221:DAIOTI>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Guimond, S. R., G. M. Heymsfield, and F. J. Turk, 2010: Multiscale observations of Hurricane Dennis (2005): The effects of hot towers on rapid intensification. J. Atmos. Sci., 67, 633654, https://doi.org/10.1175/2009JAS3119.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Guimond, S. R., M. A. Bourassa, and P. D. Reasor, 2011: A latent heat retrieval and its effects on the intensity and structure change of Hurricane Guillermo (1997). Part I: The algorithm and observations. J. Atmos. Sci., 68, 15491567, https://doi.org/10.1175/2011JAS3700.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Guimond, S. R., G. M. Heymsfield, P. D. Reasor, and A. C. Didlake Jr., 2016: The rapid intensification of Hurricane Karl (2010): New remote sensing observations of convective bursts from the Global Hawk platform. J. Atmos. Sci., 73, 36173639, https://doi.org/10.1175/JAS-D-16-0026.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Hack, J. J., and W. H. Schubert, 1986: Nonlinear response of atmospheric vortices to heating by organized cumulus convection. J. Atmos. Sci., 43, 15591573, https://doi.org/10.1175/1520-0469(1986)043<1559:NROAVT>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Hazelton, A. T., R. E. Hart, and R. F. Rogers, 2017: Analyzing simulated convective bursts in two Atlantic Hurricanes. Part II: Intensity change due to bursts. Mon. Wea. Rev., 145, 30953117, https://doi.org/10.1175/MWR-D-16-0268.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Hendricks, E. A., M. T. Montgomery, and C. A. Davis, 2004: The role of “vortical” hot towers in the formation of Tropical Cyclone Diana (1984). J. Atmos. Sci., 61, 12091232, https://doi.org/10.1175/1520-0469(2004)061<1209:TROVHT>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Hong, S.-Y., and J.-O. J. Lim, 2006: The WRF single-moment 6-class microphysics scheme (WSM6). J. Korean Meteor. Soc, 42, 129151.

  • Hong, S.-Y., Y. Noh, and J. Dudhia, 2006: A new vertical diffusion package with an explicit treatment of entrainment processes. Mon. Wea. Rev., 134, 23182341, https://doi.org/10.1175/MWR3199.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Hong, S.-Y., K.-S. S. Lim, Y.-H. Lee, J.-C. Ha, H.-W. Kim, S.-J. Ham, and J. Dudhia, 2010: Evaluation of the WRF double-moment 6-class microphysics scheme for precipitating convection. Adv. Meteor., 2010, 707253, https://doi.org/10.1155/2010/707253.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Hou, A. Y., R. K. Kakar, S. Neeck, A. A. Azarbazin, C. D. Kummerow, M. Kojima, K. Nakamura, and T. Iguchi, 2014: The Global Precipitation Measurement Mission. Bull. Amer. Meteor. Soc., 95, 701722, https://doi.org/10.1175/BAMS-D-13-00164.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Houze, R. A., 1997: Stratiform precipitation in regions of convection: A meteorological paradox?. Bull. Amer. Meteor. Soc., 78, 21792196, https://doi.org/10.1175/1520-0477(1997)078<2179:SPIROC>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Huffman, G. J., D. T. Bolvin, and E. J. Nelkin, 2017: Integrated Multi-satellitE Retrievals for GPM (IMERG) Technical Documentation. NASA Tech. Doc., 54 pp., https://pmm.nasa.gov/sites/default/files/document_files/IMERG_technical_doc_3_22_17.pdf.

  • Iacono, M. J., J. S. Delamere, E. J. Mlawer, M. W. Shephard, S. A. Clough, and W. D. Collins, 2008: Radiative forcing by long-lived greenhouse gases: Calculations with the AER radiative transfer models. J. Geophys. Res., 113, D13103, https://doi.org/10.1029/2008JD009944.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Jiang, H., C. Tao, and Y. Pei, 2019: Estimation of tropical cyclone intensity in the North Atlantic and northeastern Pacific basins using TRMM satellite passive microwave observations. J. Appl. Meteor. Climatol., 58, 185197, https://doi.org/10.1175/JAMC-D-18-0094.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Kaplan, J., M. DeMaria, and J. A. Knaff, 2010: A revised tropical cyclone rapid intensification index for the Atlantic and Eastern North Pacific basins. Wea. Forecasting, 25, 220241, https://doi.org/10.1175/2009WAF2222280.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Kelley, O. A., and J. B. Halverson, 2011: How much tropical cyclone intensification can result from the energy released inside of a convective burst? J. Geophys. Res., 116, D20118, https://doi.org/10.1029/2011JD015954.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Kelley, O. A., J. Stout, and J. B. Halverson, 2004: Tall precipitation cells in tropical cyclone eyewalls are associated with tropical cyclone intensification. Geophys. Res. Lett., 31, L24112, https://doi.org/10.1029/2004GL021616.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Klotz, B. W., and D. S. Nolan, 2019: SFMR surface wind undersampling over the tropical cyclone life cycle. Mon. Wea. Rev., 147, 247268, https://doi.org/10.1175/MWR-D-18-0296.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Landsea, C. W., and J. L. Franklin, 2013: Atlantic hurricane database uncertainty and presentation of a new database format. Mon. Wea. Rev., 141, 35763592, https://doi.org/10.1175/MWR-D-12-00254.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Miyamoto, Y., and D. S. Nolan, 2018: Structural changes preceding rapid intensification in tropical cyclones as shown in a large ensemble of idealized simulations. J. Atmos. Sci., 75, 555569, https://doi.org/10.1175/JAS-D-17-0177.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Moon, Y., and D. Nolan, 2015: Spiral rainbands in a numerical simulation of Hurricane Bill (2009). Part I: Structures and comparisons to observations. J. Atmos. Sci., 72, 164190, https://doi.org/10.1175/JAS-D-14-0058.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Morrison, H., G. Thompson, and V. Takarskii, 2009: Impact of cloud microphysics on the development of trailing stratiform precipitation in a simulated squall line: Comparison of one- and two-moment schemes. Mon. Wea. Rev., 137, 991137, https://doi.org/10.1175/2008MWR2556.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Nolan, D. S., 2011: Evaluating environmental favorableness for tropical cyclone development with the method of point-downscaling. J. Adv. Model. Earth Syst., 3, M08001, https://doi.org/10.1029/2011ms000063.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Nolan, D. S., and M. G. McGauley, 2012: Tropical cyclogenesis in wind shear: Climatological relationships and physical processes. Cyclones: Formation, Triggers and Control, K. Oouchi and H. Fudeyasu, Eds., Nova Science Publishers, 1–34.

    • Search Google Scholar
    • Export Citation
  • Nolan, D. S., and C. A. Mattocks, 2014: Development and evaluation of the second hurricane nature run using the joint OSSE nature run and the WRF model. 31st Conf. on Hurricanes and Tropical Meteorology, San Diego, CA, Amer. Meteor. Soc., P91.

    • Search Google Scholar
    • Export Citation
  • Nolan, D. S., Y. Moon, and D. P. Stern, 2007: Tropical cyclone intensification from asymmetric convection: Energetics and efficiency. J. Atmos. Sci., 64, 33773405, https://doi.org/10.1175/JAS3988.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Nolan, D. S., R. Atlas, K. T. Bhatia, and L. R. Bucci, 2013: Development and validation of a hurricane nature run using the joint OSSE nature run and the WRF model. J. Adv. Model. Earth Syst., 5, 382405, https://doi.org/10.1002/jame.20031.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Okamoto, K., K. Aonashi, T. Kubota, and T. Tashima, 2016: Experimental assimilation of the GPM core observatory DPR reflectivity profiles for Typhoon Hailong (2014). Mon. Wea. Rev., 144, 23072326, https://doi.org/10.1175/MWR-D-15-0399.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Riehl, H., and J. Malkus, 1961: Some aspects of Hurricane Daisy, 1958. Tellus, 13, 181213, https://doi.org/10.3402/tellusa.v13i2.9495.

  • Rios Gaona, M. F., G. Villarini, W. Zhang, and G. A. Vecchi, 2018: The added value of IMERG in characterizing rainfall in tropical cyclones. Atmos. Res., 209, 95102, https://doi.org/10.1016/j.atmosres.2018.03.008.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Rodgers, E., W. S. Olson, V. M. Karyampudi, and H. F. Pierce, 1998: Satellite-derived latent heating distribution and environmental influences in Hurricane Opal (1995). Mon. Wea. Rev., 126, 12291247, https://doi.org/10.1175/1520-0493(1998)126<1229:SDLHDA>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Rodgers, E., W. S. Olson, J. Halverson, J. Simpson, and H. Pierce, 2000: Environmental forcing of Supertyphoon Paka’s latent heat structure. J. Appl. Meteor., 39, 19832006, https://doi.org/10.1175/1520-0450(2001)040<1983:EFOSPS>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Rogers, R. F., P. D. Reasor, and J. A. Zhang, 2015: Multiscale structure and evolution of Hurricane Earl (2010) during rapid intensification. Mon. Wea. Rev., 143, 536562, https://doi.org/10.1175/MWR-D-14-00175.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Rogers, R. F., J. A. Zhang, J. Zawislak, H. Jiang, G. R. Alvey III, E. J. Zipser, and S. S. Stevenson, 2016: Observations of the structure and evolution of Hurricane Edouard (2014) during intensity change. Part II: Kinematic structure and the distribution of deep convection. Mon. Wea. Rev., 144, 33553376, https://doi.org/10.1175/MWR-D-16-0017.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Shimada, U., K. Aonashi, and Y. Miyamoto, 2017: Tropical cyclone intensity change and axisymmetricity deduced from GSMaP. Mon. Wea. Rev., 145, 10031017, https://doi.org/10.1175/MWR-D-16-0244.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Simpson, J., C. Kummerow, W.-K. Tao, and R. Adler, 1996: On the Tropical Rainfall Measuring Mission. Meteor. Atmos. Phys., 60, 1936, https://doi.org/10.1007/BF01029783.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Simpson, J., J. B. Halverson, B. S. Ferrier, W. A. Petersen, R. H. Simpson, R. Blakeslee, and S. L. Durden, 1998: On the role of “hot towers” in tropical cyclone formation. Meteor. Atmos. Phys., 67, 1535, https://doi.org/10.1007/BF01277500.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Stephens, G. L., and Coauthors, 2002: The CloudSat Mission and the A-Train: A new dimension of space-based observations of clouds and precipitation. Bull. Amer. Meteor. Soc., 83, 17711790, https://doi.org/10.1175/BAMS-83-12-1771.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Tang, B., and K. Emanuel, 2010: Midlevel ventilation’s constraint on tropical cyclone intensity. J. Atmos. Sci., 67, 18171830, https://doi.org/10.1175/2010JAS3318.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Tao, W.-K., and Coauthors, 2006: Retrieval of latent heating from TRMM Measurements. Bull. Amer. Meteor. Soc., 87, 15551572, https://doi.org/10.1175/BAMS-87-11-1555.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Thompson, G., R. M. Rasmussen, and K. Manning, 2004: Explicit forecasts of winter precipitation using an improved bulk microphysics scheme. Part I: Description and sensitivity analysis. Mon. Wea. Rev., 132, 519542, https://doi.org/10.1175/1520-0493(2004)132<0519:EFOWPU>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Thompson, G., P. R. Field, R. M. Rasmussen, and W. D. Hall, 2008: Explicit forecasts of winter precipitation using an improved bulk microphysics scheme. Part II: Implementation of a new snow parameterization. Mon. Wea. Rev., 136, 50955115, https://doi.org/10.1175/2008MWR2387.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Tourville, N., G. Stephens, M. DeMaria, and D. Vane, 2015: Remote sensing of tropical cyclones. Bull. Amer. Meteor. Soc., 96, 609622, https://doi.org/10.1175/BAMS-D-13-00282.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Vigh, J. L., and W. H. Schubert, 2009: Rapid development of the tropical cyclone warm core. J. Atmos. Sci., 66, 33353350, https://doi.org/10.1175/2009JAS3092.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Wadler, J. B., R. F. Rogers, and P. D. Reasor, 2018: The relationship between spatial variations in the structure of convective bursts and tropical cyclone intensification as determined by airborne Doppler radar. Mon. Wea. Rev., 146, 761780, https://doi.org/10.1175/MWR-D-17-0213.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Wang, Y., and J. Xu, 2010: Energy production, frictional dissipation, and maximum intensity of a numerically simulated tropical cyclone. J. Atmos. Sci., 67, 97116, https://doi.org/10.1175/2009JAS3143.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Wilks, D. S., 2006: Statistical Methods in the Atmospheric Sciences: An Introduction. 2nd ed. International Geophysics Series, Vol. 100, Academic Press, 648 pp.

    • Search Google Scholar
    • Export Citation
  • Wu, S.-N., and B. J. Soden, 2017: Signatures of tropical cyclone intensification in measurements of ice and liquid water content. Mon. Wea. Rev., 145, 40814091, https://doi.org/10.1175/MWR-D-17-0046.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Zagrodnik, J. P., and H. Jiang, 2014: Rainfall, convection, and latent heating distributions in rapidly intensifying tropical cyclones. J. Atmos. Sci., 71, 27892809, https://doi.org/10.1175/JAS-D-13-0314.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Zhang, D., Y. Liu, and M. K. Yau, 2002: A multiscale numerical study of Hurricane Andrew (1992). Part V: Inner-core thermodynamics. Mon. Wea. Rev., 130, 27452763, https://doi.org/10.1175/1520-0493(2002)130<2745:AMNSOH>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • View in gallery

    Intensities, tracks, and sizes for the five hurricane simulations used in this study: (a) location of minimum surface pressure, (b) azimuthal-mean tangential wind speed at 2 km height, and (c) radius of maximum winds at the surface (10 m). Note that the two idealized hurricanes are simulated on an f-plane channel with no topography. They are overlaid here with their positions defined by arbitrary latitudes and longitudes, for comparison purposes. In all three panels data are plotted every 30 min using data from the 3 km grid.

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    Intensities of the 270 members of the MN18 ensemble, in terms of maximum azimuthal-mean tangential wind at z = 2 km. Red colors indicate ensemble members that have a period of rapid intensification.

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    Intensity (defined by maximum azimuthal-mean tangential wind at 2 km height), and total moist diabatic heating and total condensate within a specified radius for HNR1: (a) inside radius 111 km (equivalent to 1.0°), (b) inside 55 km, (c) inside 222 km, (d) inside the radius of maximum winds (RMW) at each output time, (e) inside 1.5 × RMW, and (f) inside 2 × RMW.

  • View in gallery

    As in Fig. 3, but for (a) HNR2 inside 139 km (1.25°), (b) Bill2 inside 83 km (0.75°), (c) Ideal5 inside 83 km, and (d) Ideal3 inside 111 km.

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    Condensate budgets in specified cylinders moving with HNR1: (a) inside 111 km, from the surface to the top of the domain, and (b) inside 111 km, from z = 10 km to the top of the domain. See text for details.

  • View in gallery

    Scatterplots of total moist diabatic heating vs total condensate, in specified cylinders moving with the TC centers, over all output times for all 5 simulations: (a) inside radius 111 km, (b) inside radius 55 km, (c) inside radius 111 km, and above 10 km, (d) inside radius 55 km, and above 10 km, (e) inside radius 111 km, and above 12 km, and (f) inside radius 55 km, and above 12 km. The correlation and the slope of the linear fit line are also shown for each plot.

  • View in gallery

    Scatterplots of current or near-future intensity vs total condensate inside a specified radius: (a) current intensity inside radius 55 km, (b) inside 111 km, (c) inside the RMW, (d) inside 1.75 × RMW, (e) intensity 12 h later vs current condensate inside 111 km, and (f) intensity 12 h later vs condensate inside 1.75 × RMW.

  • View in gallery

    Scatterplots of subsequent intensity change vs total condensate inside specified radii and above specified altitudes: (a) 12 h intensity change vs condensate inside 55 km and above 10 km, (b) 12 h change vs condensate inside 55 km and above 12 km, (c) 12 h change vs condensate inside the RMW and above 10 km, (d) 12 h change vs condensate inside the RMW and above 12 km, (e) 24 h intensity change vs condensate inside 55 km and above 12 km, and (f) as in (e), but for 36 h.

  • View in gallery

    Scatterplots of heating, intensity, and intensity change for the MY18 ensemble: (a) moist heating vs condensate inside 111 km, (b) intensity vs condensate inside 111 km, (c) intensity change vs condensate inside 111 km and above 12 km, and (d) intensity change vs condensate inside 55 km and above 12 km.

  • View in gallery

    Results for Ideal5 and Ideal3 simulations with different microphysics schemes: (a) intensity for the each simulation using WDM6, Thompson, and Morrison, (b) RMW for each simulation, (c) heating vs condensate inside 166 km with the Thompson scheme, (d) heating vs condensate inside 166 km with Morrison, (e) current intensity vs condensate with Thompson, and (f) as in (e), but with Morrison.

  • View in gallery

    Azimuthal-mean total condensate (kg kg−1) and moist diabatic heating rates (K h−1), time-composited over the 6 h up to t = 4.5 day in the Ideal5 simulations: (a) condensate for WDM6, (b) moist heating for WDM6, (c) condensate for Thompson, (d) moist heating for Thompson, (e) condensate for Morrison, and (f) moist heating for Morrison.

  • View in gallery

    Correlations between rain rate and TC intensity: (a) for the 5 simulations, scatterplot of current intensity vs mean rain rate inside 111 km, (b) scatterplot of intensity vs rain rate inside 1.25 × RMW, (c) using GPM rain rate data, mean rain rate inside 83 km alongside the best track intensity of Hurricane Jose (2017), and (d) rain rate inside 83 km alongside intensity of Hurricane Florence (2018).

  • View in gallery

    Two methods for estimating mean condensate in the TC inner core from a single narrow swath. The gray area indicates a hurricane eyewall. In the left rectangle, the total vertically integrated condensate in the swath is integrated over the swath, divided by the rectangular area, and used as an estimate of the area-mean value in a circle with radius r = yout. On the right, the values of the vertically integrated condensate at each location of yj and yj are averaged and then used an estimate for the mean value in the red circle. These values are integrated radially, weighted by 2πrΔr, where Δr is calculated from the difference between rj and rj−1, and then also divided by the total area in the annulus.

  • View in gallery

    Total condensate estimates based on the two narrow swath methods shown in Fig. 13: (a) with the swaths passing directly over the TC center for every sample, and (b) with the swaths passing within random locations evenly distributed within −111 to + 111 km of the center.

  • View in gallery

    Scatterplots of actual total condensate inside a specified radius vs the estimated condensate from single swaths: (a) within 55 km using the SS method, (b) within 55 km using the RIS method, (c) within 111 km using SS, (d) within 111 km using RIS, (e) within 166 km using SS, and (f) within 166 km using RIS.

  • View in gallery

    Scatterplots of TC intensity vs condensate estimated from the two single-swath methods: (a) intensity vs condensate inside 111 km, using the SS method, (b) intensity vs condensate inside 111 km using RIS, (c) intensity vs condensate inside 166 km using SS, (d) intensity vs condensate inside 166 km using RIS, (e) intensity vs condensate inside 111 km and above 6 km, using SS, and (f) intensity vs condensate inside 166 km and above 6 km using SS.

  • View in gallery

    Scatterplots of 12 h intensity change vs condensate from single-swath estimates: (a) inside 111 km using SS, (b) inside 111 km using RIS, (c) inside 55 km using SS, and (d) inside 55 km using RIS.

  • View in gallery

    Current intensity from real TCs (defined by estimated maximum surface wind) vs total condensate computed from a CloudSat swath that passed within 111 km of the TC center: (a) for a swath from −111 to +111 km of the point of closest approach, and (b) for −166 to + 166 km.

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On the Correlation between Total Condensate and Moist Heating in Tropical Cyclones and Applications for Diagnosing Intensity

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  • 1 Rosenstiel School of Marine and Atmospheric Science, University of Miami, Miami, Florida
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Abstract

While the transfer of moist enthalpy from the ocean to the atmosphere is the fundamental energy source for tropical cyclones, the release of latent heat in moist convection is the mechanism by which this energy is converted into the kinetic and potential energy of these storms. Most observational estimates of this heat release rely on satellite estimates of rain rate. Here, examination of five high-resolution numerical simulations of tropical cyclones reveals that there is a close correlation between the total condensate and the total heating, even though the former quantity is an amount and the latter is a rate. This relationship is due to the fact that for condensate to be sustained at large values, it must be rapidly replaced by new condensate and associated latent heating. Total condensate and total rain rate within fixed radial distances such as 111 km also show good correlations with the current intensity of the storm, but surprisingly, high values of condensate at high altitudes and close to the storm center are not good predictors of imminent intensification. These relationships are confirmed with an additional ensemble of 270 idealized simulations of tropical cyclones with varying sizes and intensities. Finally, simulated measurements of total condensate are computed from narrow swaths modeled after the cloud profiling radar on the CloudSat satellite. Despite their narrow footprint and the fact that they rarely cut through the exact center of the cyclone, these estimates of total condensate also show a useful correlation with current intensity.

Current affiliation: Faculty of Environment and Information Studies, Keio University, Kanagawa, Japan.

© 2019 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: David S. Nolan, dnolan@rsmas.miami.edu

Abstract

While the transfer of moist enthalpy from the ocean to the atmosphere is the fundamental energy source for tropical cyclones, the release of latent heat in moist convection is the mechanism by which this energy is converted into the kinetic and potential energy of these storms. Most observational estimates of this heat release rely on satellite estimates of rain rate. Here, examination of five high-resolution numerical simulations of tropical cyclones reveals that there is a close correlation between the total condensate and the total heating, even though the former quantity is an amount and the latter is a rate. This relationship is due to the fact that for condensate to be sustained at large values, it must be rapidly replaced by new condensate and associated latent heating. Total condensate and total rain rate within fixed radial distances such as 111 km also show good correlations with the current intensity of the storm, but surprisingly, high values of condensate at high altitudes and close to the storm center are not good predictors of imminent intensification. These relationships are confirmed with an additional ensemble of 270 idealized simulations of tropical cyclones with varying sizes and intensities. Finally, simulated measurements of total condensate are computed from narrow swaths modeled after the cloud profiling radar on the CloudSat satellite. Despite their narrow footprint and the fact that they rarely cut through the exact center of the cyclone, these estimates of total condensate also show a useful correlation with current intensity.

Current affiliation: Faculty of Environment and Information Studies, Keio University, Kanagawa, Japan.

© 2019 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: David S. Nolan, dnolan@rsmas.miami.edu

1. Introduction

Tropical cyclones (TCs) are naturally occurring heat engines that convert moist enthalpy transferred from the ocean to the atmosphere into kinetic energy, in the form of wind, and into available potential energy, in the form of a balanced warm temperature anomaly in the troposphere. After the initial evaporation of water into the atmospheric surface layer, the next key thermodynamic process is the condensation of water vapor in the rising updrafts that surround the TC center. This condensation releases heat which is then converted to kinetic and potential energy through a dynamical adjustment process (Hack and Schubert 1986; Nolan et al. 2007).

For a TC in an approximately steady state, the energy gained from the ocean must balance the energy lost due to frictional dissipation and other processes (Emanuel 1991; Wang and Xu 2010). However, actual measurement of this transfer of energy would be very difficult: even assuming that standard bulk-transfer formulas are accurate (e.g., Fairall et al. 2003; Dudhia et al. 2008), it would require knowledge of both surface wind speed and near-surface humidity in a broad region around the TC to account for all of the evaporation into near-surface parcels as they travel in long spiral paths toward the center of the cyclone. It would also be difficult to account for the energy lost to updrafts into rainbands, and to what extent this loss may be compensated by increased fluxes caused by cooler and drier downdrafts reaching the surface in the rainband-active regions.

Therefore, direct or inferred measurement of the conversion of moist energy into heat energy at the time of condensation is more straightforward, especially due to the simultaneous production of liquid or frozen condensate. Close relationships between the production of condensate, the release of latent heat, and subsequent precipitation are the physical basis for a long history of efforts to diagnose TC intensity and intensity change from satellites. Many previous studies have found that reduced brightness temperatures observed by microwave imagers from precipitating clouds can be correlated with precipitation rates and thus with TC intensity (Adler and Rodgers 1977; Rodgers et al. 1998, 2000; Cecil and Zipser 1999). Most recently, Jiang et al. (2019) showed that somewhat skillfull analyses of current intensity could be derived from 85 GHz brightness temperatures, and the skill could be substantially improved if rainfall rates from the Tropical Rainfall Measuring Mission (TRMM; Simpson et al. 1996) were added to the predictors.

Surface precipitation is generally a good estimate of the total moist heating in the overlying atmosphere (i.e., condensation minus evaporation). However, due to the advection of falling hydrometeors, it may not accurately represent the horizontal distribution of where latent heating occurs, and it cannot provide direct information about its vertical distribution. Several dynamical studies have found that the “efficiency” of the conversion of heat energy into kinetic and available potential energy is not only much greater for heat released near the TC center, but it is also greater for heat released at higher altitudes, such as 8–10 km (Hack and Schubert 1986; Nolan et al. 2007; Vigh and Schubert 2009). These papers suggest that knowledge of the vertical distribution of heat release near the TC center could provide additional diagnostic and predictive power for remote sensing of TC intensity.

This dynamical significance of high-altitude heating aligns with a long-standing interest in the tallest, most intense convective cells near TC centers, which can be traced back to early studies such as Riehl and Malkus (1961) and more recently Simpson et al. (1998). A dynamical component to this view was added by Hendricks et al. (2004), who proposed that in addition to their elevated latent heating, deep convective towers generate vorticity anomalies that eventually merge together, leading to TC intensification. Many past and recent observational studies have found strong evidence that these “hot towers” or “convective bursts” either precede or occur concurrently with periods of significant intensification (Kelley et al. 2004; Kelley and Halverson 2011; Guimond et al. 2010, 2016; Rogers et al. 2015, 2016; Hazelton et al. 2017; Wadler et al. 2018).

An alternate view takes a broader, more “mesoscale” perspective, advocating that the total amount of heat released inside the RMW is what matters most. In this view, unusually deep convective events have no special role other than their net positive impact on the total heating. The highest towers are often short lived and cover small areas, suggesting that wide-spread, steadier heat release could have a larger impact. The rapid intensification (RI) forecasting model of Kaplan et al. (2010) and observational studies such as Zagrodnik and Jiang (2014) and Shimada et al. (2017) have found that broader, more symmetric, and more persistent convection around the storm center is better correlated with intensification than the appearance of localized clusters of very cold brightness temperatures that indicate very high cloud tops. The two views outlined above are not necessarily contradictory to each other, as deep convective towers are the first stage of evolution the convective life cycle, and usually precede longer-lasting and more widespread stratiform precipitation structures (Houze 1997).

Therefore, detailed knowledge of the distribution of latent heat release in TCs is needed to advance our understanding of the competing or complementary roles of net moist heating versus the horizontal and vertical distribution of that heating. Satellite-based cloud radars such as the TRMM Precipitation Radar (Simpson et al. 1998; Tao et al. 2006), and its replacement, the Dual-frequency Precipitation Radar (DPR) on the Global Precipitation Measurement (GPM) mission Core Observatory satellite (Hou et al. 2014; Okamoto et al. 2016), can provide vertical profiles of precipitating hydrometeors across fairly large swaths (215 and 245 km, respectively) with resolutions of about 5 km; such measurements are used to estimate surface precipitation as well. While these precipitation estimates have been widely used in studies of tropical precipitation, including TCs, the vertical distribution of reflectivity has been less frequently exploited.

In contrast to the precipitation radars, a cloud-profiling radar such as exists on the CloudSat satellite (Stephens et al. 2002) can provide detailed vertical structure of cloud ice and snow at higher altitudes, which is important for identifying the vertical extent and ice mass content of the tallest updrafts near TC centers (Tourville et al. 2015). Given its very narrow footprint (1.7 km), useful CloudSat passes over tropical cyclones are infrequent, but over time more than 1500 overpasses within 300 km of TC centers have been accumulated. Wu and Soden (2017) used this data to find significant differences in the ice water content around intensifying versus weakening TCs, extending out several hundred kilometers from the center. However, the higher frequency (94 GHz) of the CloudSat W-band radar leads to significant attenuation in heavy precipitation, such that its utility for retrieving the total vertically integrated condensate or precipitation rate may be quite limited.

In this study, we will consider whether the total amount and vertical distribution of condensate–including small particles that may not be seen by previous spaceborne precipitation radars–could be used to diagnose latent heating in TCs, and possibly TC intensity and future intensity change. The output of high-resolution, high-quality numerical simulations of both realistic and idealized TCs will be used to show that there is in fact an excellent correlation between total condensate and concurrent latent heat release. It is does not appear to be necessary to use the radar returns to diagnose rain rate, and from that to infer latent heating (and possibly its vertical structure; Tao et al. 2006). Rather, the retrieved condensate values themselves can be linearly correlated with net moist heating rates, and they can also be linearly correlated with TC intensity.

The goal of this paper is to assess the extent to which instruments that directly measure condensate in tropical cyclones, either in broad or narrow swaths, can also be used to estimate TC intensity and intensity change. Section 2 describes the numerical simulations used to generate synthetic datasets of observed condensate. Section 3 demonstrates the close relationships between total condensate, rain rate, and moist diabatic heating in TCs. Sections 4 and 5 explore to what extent these relationships can be used to diagnose TC intensity and TC intensity change. Conclusions are provided in section 6.

2. Simulated tropical cyclones

In this study we use output from two distinct sets of numerical simulations of hurricanes. The first set consists of five simulations modeled after the framework of the “hurricane nature run” originally presented by Nolan et al. (2013). Three of these five simulations were first introduced in the study by Klotz and Nolan (2019), and all five are described in full detail in that paper. For convenience, their important features are described below. The second set consists of an ensemble of 270 idealized simulations with varying rates of intensification produced for the study by Miyamoto and Nolan (2018; hereafter MN18).

a. Simulations modeled after the hurricane nature run

The hurricane nature run is a regionally downscaled simulation of an Atlantic hurricane that occurs in the joint OSSE nature run (JONR) (Andersson and Masutani 2010) which was created from a free-running simulation of global weather using the ECMWF forecast model from 2005, forced only by observed sea surface temperatures (SSTs). This first hurricane nature run (hereafter HNR1) was produced with the Weather Research and Forecasting (WRF) Model, version 3.2.1, using nested grids with 27, 9, 3, and 1 km grid spacing; the three latter grids follow the storm as it moves across the tropical Atlantic. HNR1 used the Yonsei University planetary boundary layer scheme (YSU; Hong et al. 2006), the WRF double-moment 6-class microphysics scheme (WDM6; Hong et al. 2010), and the RRTMG longwave and shortwave radiation schemes (Iacono et al. 2008).

The first of the five simulations is HNR1 itself. HNR1 provides 13 days of data. The second simulation is the second hurricane nature run, HNR2, which reproduces a second hurricane simulated by the JONR (Nolan and Mattocks 2014). It was produced with version 3.4.1 of WRF, but otherwise used the same parameters and parameterizations as HNR1, and provides 8 days of data. Unlike HNR1, this storm has substantial land interactions, passing directly over Hispaniola, Cuba, and Florida.

The third simulation is a reproduction of the simulation of Hurricane Bill (2009) produced by Moon and Nolan (2015), except that the simulation was repeated with version 3.4.1 of WRF, and with all the same parameterizations, grid sizes for the moving nests, and vertical levels reconfigured to be identical to those of HNR1 and HNR2. It provides 3.5 days of data as Hurricane Bill turned northward, passing well east of the U.S. coastline, and is referred to as Bill2.

The fourth and fifth simulations are idealized simulations of hurricanes produced using the modeling framework of Nolan (2011) and Nolan and McGauley (2012). The domain for each is a large zonally periodic channel, initialized with a purely zonal flow that varies with height. The flow is balanced with meridional gradients of pressure and temperature that are computed from an iterative scheme. The SST, initial vortex, mean flow, and wind shear were selected so as to produce two different TCs: one that rapidly intensifies to a fairly small category 5 hurricane (hereafter “Ideal5”), and one that develops more slowly into a larger and more asymmetric category 3 hurricane (“Ideal3”). Further details on the production of the Ideal5 and Ideal3 simulations are provided in appendix A of Klotz and Nolan (2019).

Ideal5 and Ideal3 were both integrated for 10 days. In Ideal5, the storm develops quickly and begins RI at the start of day 3, with peak surface winds exceeding 70 m s−1 from days 5 to 8. The cyclone initially contracts to an RMW of about 18 km and then slowly increases in size. In Ideal3, the development is slower and more episodic, with the surface winds first exceeding 50 m s−1 on day 6. For the first 7 days the storm is much larger and much more asymmetric than Ideal5. Ideal3 becomes a hurricane on day 4 with an RMW of 40 km. Later the RMW expands to 70 km with an eyewall replacement cycle that occurs on day 6. After this event, the storm intensifies further and the RMW contracts to 30 km near the end of the simulation.

Figure 1 shows the tracks, intensities, and sizes of the 5 TCs. The tracks for the two idealized storms are included for comparison, but as these storms exist in idealized channel domains, the latitude and longitude values do not correspond to real locations. Rather than in terms of the peak surface wind speed, the intensities are shown as the maximum azimuthal-mean tangential wind speed at 2 km height, which is more stable and more representative of the overall strength of the storm. The maximum surface winds for these storms are shown in Fig. 2 of Klotz and Nolan (2019). For TC size, we show the RMW at the surface, rather than at 2 km, as the latter is actually more variable than its surface counterpart, because it can jump outward to local wind maxima associated with long rainbands [an example of this is shown for HNR1 in Fig. 17b of Nolan et al. (2013)]. Even when defined in this manner, the RMW of 3 of the 5 TCs are highly variable in time.

Fig. 1.
Fig. 1.

Intensities, tracks, and sizes for the five hurricane simulations used in this study: (a) location of minimum surface pressure, (b) azimuthal-mean tangential wind speed at 2 km height, and (c) radius of maximum winds at the surface (10 m). Note that the two idealized hurricanes are simulated on an f-plane channel with no topography. They are overlaid here with their positions defined by arbitrary latitudes and longitudes, for comparison purposes. In all three panels data are plotted every 30 min using data from the 3 km grid.

Citation: Monthly Weather Review 147, 10; 10.1175/MWR-D-19-0010.1

b. The Miyamoto and Nolan ensemble

MN18 studied structural changes preceding rapid intensification in an ensemble of 270 idealized simulations. These simulations used WRF 3.7.1 with grid spacings of 18, 6, and 2 km, and 40 vertical levels up to 20 km altitude. The YSU boundary layer scheme and the WRF single-moment 6-class scheme were used (WSM6; Hong and Lim 2006), but radiation was not included. The ensemble was created by using 5 different values for surface wind speed ranging from 2.5 to 12.5 m s−1, 6 values of 850 to 200 hPa wind shear from 0 to 12.5 m s−1, 3 values of the initial vortex size from 90 to 210 km, and 3 values of the initial vortex strength from 10 to 20 m s−1 (see Table 2 of MN18). Figure 2 shows the intensities of the ensemble members.

Fig. 2.
Fig. 2.

Intensities of the 270 members of the MN18 ensemble, in terms of maximum azimuthal-mean tangential wind at z = 2 km. Red colors indicate ensemble members that have a period of rapid intensification.

Citation: Monthly Weather Review 147, 10; 10.1175/MWR-D-19-0010.1

3. Total condensate and moist heating

a. Time series of integrated heating and condensate

The initial inspiration for this study was provided by plots like Fig. 3a, which was generated from HNR1. The blue curve shows the intensity of the cyclone as measured by the maximum azimuthally averaged tangential wind (Vt) at z = 2 km height above the surface. The thick and thin red curves show the area-mean, vertically integrated moist diabatic heating for a cylinder of radius 111 km (equal to 1° latitude) around the cyclone center. To be consistent with many earlier studies on satellite estimates of TC intensity that used degrees of latitude to define distance (e.g., Adler and Rodgers 1977; Cecil and Zipser 1999), radial distances in this study will correspond to commonly used distances in latitude, such as 55 km ~0.5°, 111 km ~1.0°, and so on. The thin red curve shows total heating above 10 km altitude. Not surprisingly, there are several “bursts” of latent heat release during the early development of the storm, leading up to RI after t = 4 days, which is matched with a large and then permanent increase in total latent heating.

Fig. 3.
Fig. 3.

Intensity (defined by maximum azimuthal-mean tangential wind at 2 km height), and total moist diabatic heating and total condensate within a specified radius for HNR1: (a) inside radius 111 km (equivalent to 1.0°), (b) inside 55 km, (c) inside 222 km, (d) inside the radius of maximum winds (RMW) at each output time, (e) inside 1.5 × RMW, and (f) inside 2 × RMW.

Citation: Monthly Weather Review 147, 10; 10.1175/MWR-D-19-0010.1

The green curves show the area-averaged vertically integrated total condensate (rain, cloud water, cloud ice, graupel, and snow), both for the entire column, and again for above 10 km. It is apparent that the rate of heat release and the amount of total condensate in the cylinder out to r = 111 km are very highly correlated (actual correlation coefficients will be shown below). This is true for both the entire column and for the volume above 10 km. Plots using other altitudes, such as 6 and 8 km, showed similar correlations.

Figures 3b and 3c show the same results for HNR1, but averaging over cylinders of radius 222 and 55 km, respectively. The correlation between latent heating and total condensate is even greater for the larger cylinder. In Fig. 3b, the oscillations in heating and condensate between t = 5 and 9 days are indicative of the TC diurnal cycle (Dunion et al. 2014), which has been found to be well represented in HNR1 (Dunion et al. 2019). Within a radius of 55 km, the heating and condensate time series are considerably noisier, but the correlation is still apparent.

As previously discussed, one might expect that total latent heat release should be well correlated with TC intensity, and also that latent heat released inside the RMW and at higher altitudes should be indicative of an impending increase in intensity, or perhaps a greater intensity for the same amount of heating. The data in Fig. 3 only partly support these ideas. In Fig. 3a there are bursts of increased heating during the development of HNR1 from t = 0 to 4 days. After 4 days a series of bursts of heating within 111 km radius appear to be well correlated with the RI of this storm. However, it is difficult to say from these figures alone if these bursts precede (and cause) RI, or are simply emblematic of it. Nonetheless, between 4 and 8 days the total heating roughly parallels the storm intensity. After this time, the total heating within 111 km declines more rapidly than the intensity of the storm.

The size of the inner core of HNR1 increases during this period, and it could be that the 111 km cylinder becomes too small to be representative. It might seem that better results would be found when the size of the averaging cylinder is scaled to match the size of the inner core, taken here to be measured by the RMW at the surface, as shown in Fig. 1. Figures 3d–f show similar plots with the averaging cylinder taken to be 1.0, 1.5, and 2.0 × RMW computed at each model output time. However, scaling by the RMW does not lead to much improvement in the correlation between total heating and current intensity, nor does it show a better correlation between heating and intensity change. In fact, time series using radii proportional to the RMW are generally noisier than those using fixed volumes.

Figure 4 shows similar plots for the other four simulations. In each case the radius has been chosen that shows the best combined correlation between condensate and heating, and heating and intensity.

Fig. 4.
Fig. 4.

As in Fig. 3, but for (a) HNR2 inside 139 km (1.25°), (b) Bill2 inside 83 km (0.75°), (c) Ideal5 inside 83 km, and (d) Ideal3 inside 111 km.

Citation: Monthly Weather Review 147, 10; 10.1175/MWR-D-19-0010.1

b. Condensate budgets

Several previous studies have presented budgets for water vapor and condensate in TCs. Gamache et al. (1993) computed a budget using airborne Doppler radar reflectivity and velocity data obtained in Hurricane Norbert (1984). They used three-dimensional velocity fields and two rather different methods for computing the amounts of cloud condensate and precipitating condensate (qc and qp) to compute the rates of condensation, evaporation, and horizontal transports of qc and qp. Their budget was confined to a cylinder of 37.5 km within the TC center. Surprisingly, they found very weak inward and outward transports of both water vapor and condensate, which may be related to the fact that Norbert was weakening and fairly asymmetric at that time.

Zhang et al. (2002) and later Braun (2006) computed water budgets around their simulations of Hurricane Andrew (1992) and Hurricane Bonnie (1998), respectively. In both papers, the budgets were accumulated from frequent model output (5 and 3 min intervals) over a period of just one hour. Zhang et al. (2002) found that the production of condensate closely matched the diabatic heat source and was also closely correlated with vertical motion, with diabatic cooling and downward motion occurring along the eye–eyewall interface. Somewhat like Gamache et al. (1993), Braun (2006) separated the condensate budget into the nonprecipitating and precipitating types qc and qp, and integrated their production rates in cylinders out to 70 and 200 km. While Braun found much more substantial inward and outward transports of water vapor, the radial transports of condensate were very small compare to the net condensation, evaporation, and precipitation rates.

Guimond et al. (2011) attempted to more directly diagnose the latent heating from airborne radar observations in Hurricane Guillermo (1997) by correlating regions of saturation with upward motion, and thereby explicitly computing the production of condensate and thus heating. The method was tested by applying the same procedure to the output of the same simulation of Hurricane Bonnie (1998) analyzed by Braun (2006). Both Braun and Guimond et al. noted that qc was converted into qp almost as fast as it was produced (i.e., the “residence time” of water in cloud form was short compared to its time as precipitation).

Neither Gamache et al. (1993), Zhang et al. (2002), nor Braun (2006) made an explicit statement about a correlation between the condensate amount and net latent heat release, and given that their budgets were essentially snapshots, they did not have the opportunity to observe or empirically test this relationship. The close correlation between total heating and condensate is emblematic of a system where the rates of removal of the two condensate types are proportional to the amounts of those types. As above, let us consider the radially and vertically integrated values of such quantities as latent heat release H, condensate Qc, and precipitation Qp. The source of Qc is determined by the dynamics of the TC and corresponds to latent heat release:
Sc=HLHc,
where LHc is the latent heat of condensation while the loss of Qc is proportional to Qc, and equal to the source of Qp:
Lc=αQc=Sp,
and finally the loss of Qp also proportional to Qp and equal to the precipitation at the surface:
Lp=βQp=P.
In steady state, when the volume-integrated values of Qc and Qp are constant, and if there is no significant gain or loss of condensate by radial advection (as found by Gamache et al. and Braun), then Sc = αQc = βQp = P. The relative values of Qc and Qp are determined by their conversion rates, so that:
Qp=αβQc,
where the fast conversion of Qc to Qp discussed by Braun and Guimond means that αβ and QpQc.
In this regime, it is actually not necessary to differentiate between the two condensate types. Qc and Qp are related by (3.4), and the total condensate can be related to either; for example,
Qtot=(1+αβ)Qc,
and
H=LυSc=LHcα1+α/βQtot=LHcβ1+β/αQtot.
The above discussion relies on a balance between condensate production and precipitation, with both the rate of change of condensate in the cylinder (the “storage” term) and the outward transport of condensate being small. To show that these assumptions are valid, at least in our simulations, we compute condensate budgets in cylinders around the TC centers. The quantities and their fluxes are computed from azimuthally averaged fields, with fluxes computed from azimuthal means of the products; for example,
Fr=0ztop2πrρuqtot¯dz,
where qtot = qc + qp, and the overbar indicates the azimuthal mean, so that eddy fluxes are naturally included, and ztop is the top of the model domain. Figure 5a shows an example of such a budget computed from HNR1. The magenta curve shows the production of condensate inferred from the volume-integrated diabatic heating produced from the microphysics scheme. (In simulations with WRF 3.2.1 and 3.4.1, time-step heating rates could be saved as output, but changes in microphysics species could not.) The cyan and black curves show the storage term and the net outward flux of condensate, respectively. The green curve shows the loss due to precipitation at the surface as reported by the model output, while the thick black curve shows the precipitation estimated as the residual of the other terms.
Fig. 5.
Fig. 5.

Condensate budgets in specified cylinders moving with HNR1: (a) inside 111 km, from the surface to the top of the domain, and (b) inside 111 km, from z = 10 km to the top of the domain. See text for details.

Citation: Monthly Weather Review 147, 10; 10.1175/MWR-D-19-0010.1

It is apparent from the figure that 1) the time rate of change of total condensate is small compared to the production and loss terms, 2) the vertically integrated outward flux is also very small, 3) the production and precipitation terms nearly balance, and 4) they are both well correlated with the total condensate in the cylinder. There is some discrepancy between the precipitation rate produced by the model as compared to what is estimated from closing the budget, with the explicit precipitation being as much as 14% less than the residual. This may be explained by the fact that all of the simulations set both negative values of water vapor and values of condensate less than 1.0 × 10−10 kg kg−1 to zero. Braun (2006) noted that such practices can have a large effect on the water budget for a simulated TC. The effect here is not as nearly as large as found by Braun, perhaps due to the improved numerical methods that have been developed since that time. Furthermore, while eliminating negative water vapor does not affect the condensate budget (as condensation is treated as a source), zeroing out condensate would tend to decrease the condensate within the volume to a value smaller than all the sources, leading to less precipitation at the surface. Regardless, the two precipitation curves follow each other quite closely.

Figure 5b shows a similar budget, computed for the same cylinder, but only for the volume above z = 10 km. In this case, we must add the net vertical transport of condensate into the upper-level cylinder, which is shown as the red curve. As a contributor to condensate, it is nearly equal to the production term. The vertical transport does not include the loss due to sedimentation across z = 10 km, which can only be computed from the budget residual. Nonetheless, the main conclusions are the same: outward transport and storage are very small, and there is a good correlation between the total amount of condensate and the production term.

4. Total condensate and intensity

a. Total condensate as a predictor of total heating

To make more definitive statements about how well total condensate correlates with diabatic heating, we generate scatterplots of heating versus condensate from data accumulated over all 5 simulations. The quantities are radially and vertically integrated from azimuthal-mean fields computed every 30 min on the 3 km grid of each simulation. All output times from the 5 simulations are used, except for the first 12 h of Bill2, as this simulation is initialized as a “cold start” from a hurricane-strength vortex leading to a violent adjustment during the first few hours. To begin, Fig. 6a shows total diabatic heating out to r = 111 km versus total condensate, with the best-fit line and the correlation coefficient1 (r), equal to 0.99 in this case, included. As suggested by the time series shown above there is a very high correlation between total condensate and total heating. Figure 6b shows the same information for a cylinder out to r = 55 km. The correlation is also very high. In each scatterplot, the slope a of the linear fit y = ax + b is also shown.

Fig. 6.
Fig. 6.

Scatterplots of total moist diabatic heating vs total condensate, in specified cylinders moving with the TC centers, over all output times for all 5 simulations: (a) inside radius 111 km, (b) inside radius 55 km, (c) inside radius 111 km, and above 10 km, (d) inside radius 55 km, and above 10 km, (e) inside radius 111 km, and above 12 km, and (f) inside radius 55 km, and above 12 km. The correlation and the slope of the linear fit line are also shown for each plot.

Citation: Monthly Weather Review 147, 10; 10.1175/MWR-D-19-0010.1

If the volume of interest is restricted to higher altitudes, the correlations decrease. The same results are shown in Fig. 6 for condensate and heating above 10 km and above 12 km. Inside 111 km, r decreases from 0.99 to 0.96 for condensate above 10 km, and then further to 0.84 for condensate above 12 km. Inside 55 km, r decreases from 0.98 to 0.87, and then to 0.73.

Results for larger radii, such as 166 and 222 km, show even closer correlations between condensate and heating (not shown). Calculations based on the RMW are also very similar, although the correlations are reduced, especially for smaller radii (not shown).

b. Statistical significance

Using model output every 30 min from all 5 simulations, but eliminating the first 12 h of the Bill2 simulation, results in 2117 data points in each of the above scatterplots. However, it is evident from Figs. 35 that time series of intensity, diabatic heating, and condensate have strong serial correlations for each individual simulation. Using autocorrelation functions, we estimated the decorrelation time scale for time series of intensity and condensate from the longer simulations (i.e., HNR1, Ideal5, and Ideal3, which each provide 624, 480, and 480 output times, respectively). The decorrelation periods for each of these simulations are approximately 100, 74, and 70 output intervals (50, 37, and 30 h), respectively. Using the longest of these periods and the total number of data points (2117) suggests that there are effectively about 20 degrees of freedom in the sample, requiring a correlation coefficient r equal to approximately 2/(20)1/2 = 0.44 for statistical significance at the 95% level (see, e.g., Wilks 2006, chapter 5). Treating each of these simulations independently would require the correlation to increase to values of r = 0.80, 0.78, and 0.76, respectively. With either approach, all of the correlations shown above in Fig. 6 between total condensate and moist diabatic heating are significant, with the exception of the correlation for condensate and heating inside 55 km radius and above 12 km. In the remainder of this paper, we use the more liberal condition of r = 0.44 to indicate significance at the 95% level.

c. Total condensate and current intensity

As TCs may be considered heat engines (Emanuel 1991; Wang and Xu 2010) where the conversion of heat energy to work constantly replaces the loss of energy due to friction, then the total diabatic heating should be well correlated with the current intensity. One might expect considerable scatter about this relationship, since it does not account for differences in storm size, or factors that contribute to a reduced efficiency in the conversion of heat to wind energy, such as wind shear (Tang and Emanuel 2010).

Figure 7 shows scatterplots of total condensate in cylinders with radii of 55 and 111 km and the simultaneous intensities of each storm, defined as above by Vt at z = 2 km. The correlation coefficients are 0.74 and 0.92, respectively, the latter being the highest over all cylinder sizes. The bottom panels show similar plots for storm-size-dependent cylinders that are proportional to the current RMW of each cyclone, for 1.0 × RMW and 1.75 × RMW, with r values of 0.74 and 0.90, respectively.

Fig. 7.
Fig. 7.

Scatterplots of current or near-future intensity vs total condensate inside a specified radius: (a) current intensity inside radius 55 km, (b) inside 111 km, (c) inside the RMW, (d) inside 1.75 × RMW, (e) intensity 12 h later vs current condensate inside 111 km, and (f) intensity 12 h later vs condensate inside 1.75 × RMW.

Citation: Monthly Weather Review 147, 10; 10.1175/MWR-D-19-0010.1

In their study of the relationship between TC intensity and reduced 85 GHz brightness temperatures, Cecil and Zipser (1999) actually found slightly greater correlations with TC intensity 12 and 24 h after the time of observation. Scatterplots of total condensate versus intensity 12 h into the future are also shown in Fig. 7 for cylinders with radii of 111 km and 1.75 × RMW; for these two cases, the correlations are nearly identical, but not greater as found by Cecil and Zipser.

d. Total condensate and intensity change

As previously discussed, a common theme in research on RI is the notion that an increase in the amount, intensity, or peak altitudes of deep convection towers near and inside the eyewall is a precursor of intensity increase. To see if this is true in our simulations, we plot total condensate above certain altitudes, such as 10 or 12 km, versus the subsequent 12 h intensity change. 24 and 36 h intensity changes were also considered.

Surprisingly, the correlations between increased quantities of elevated condensate and subsequent intensification are only moderate at best. As shown in Fig. 8, the strongest correlation is found for condensate above 12 km within a radius of 55 km, with r = 0.42, just below the 95% confidence level. Correlations between intensity change and condensate above 10 km, or within cylinders equal to 1 × RMW, also show modest correlations. Correlations were less for even smaller cylinders [e.g., with radius 27 km or 0.75 × RMW (not shown)], and they also decreased quickly to zero and became even slightly negative correlations for larger cylinders (not shown). For the optimal case of condensate above 12 km and within a radius of 55 km, the correlation was equally strong for a 24 h intensity change. However, it was less for a 36 h intensity change and for longer lead times (not shown).

Fig. 8.
Fig. 8.

Scatterplots of subsequent intensity change vs total condensate inside specified radii and above specified altitudes: (a) 12 h intensity change vs condensate inside 55 km and above 10 km, (b) 12 h change vs condensate inside 55 km and above 12 km, (c) 12 h change vs condensate inside the RMW and above 10 km, (d) 12 h change vs condensate inside the RMW and above 12 km, (e) 24 h intensity change vs condensate inside 55 km and above 12 km, and (f) as in (e), but for 36 h.

Citation: Monthly Weather Review 147, 10; 10.1175/MWR-D-19-0010.1

e. Results for the Miyamoto and Nolan ensemble

Most of the analyses shown above were also computed using model output from the 270 members of the MN18 ensemble. Hourly output was used from the ends of day 1 to the ends of day 5 for all members, resulting in 26 190 data points for each comparison. Figure 9 shows results for the correlations between total condensate and total heating, total condensate and current intensity, and total condensate above 12 km altitude and 12 h intensity change. Each was computed for cylinders out to 111 km radius, with an additional plot for condensate above 12 km and within 55 km versus intensity change. These results and many others not shown, such as within larger and small cylinders for each type of correlation, are similar to those computed above using the 5 high-resolution simulations. The best correlations of condensate versus current intensity are slightly less than what was found above, while the best correlations between elevated condensate and future intensity change are slightly greater.

Fig. 9.
Fig. 9.

Scatterplots of heating, intensity, and intensity change for the MY18 ensemble: (a) moist heating vs condensate inside 111 km, (b) intensity vs condensate inside 111 km, (c) intensity change vs condensate inside 111 km and above 12 km, and (d) intensity change vs condensate inside 55 km and above 12 km.

Citation: Monthly Weather Review 147, 10; 10.1175/MWR-D-19-0010.1

f. Some results with different microphysics schemes

It is natural to consider to what extent the results depend on the choice of microphysics parameterization. The two idealized simulations, Ideal3 and Ideal5, were repeated using the Thompson microphysics scheme (Thompson et al. 2004, 2008) and the Morrison scheme (Morrison et al. 2009), as they were implemented into WRF version 3.4.1. The WDM6 scheme that is primarily used in this study is in fact only a “two-moment” scheme (predicting both number concentration and mixing ratio) for cloud water and rain. The Thompson scheme in WRF 3.4.1 uses two moments for cloud ice and for rain, while the Morrison scheme predicts two moments for all five condensate types: cloud water, rain, cloud ice, snow, and graupel.

Figure 10 shows the intensities of these TCs, along with the original WDM6 results for comparison. Both schemes make Ideal3 and Ideal5 considerably stronger after their initial intensification phases, but then bring them back closer to the WDM6 results for the last few days of each simulation. However, for both Ideal3 and Ideal5, the Thompson and Morrison schemes lead to steadily increasing TC sizes, with the RMW of the Ideal3–Morrison case reaching 150 km by t = 10 day, and Ideal3–Thompson showing an eyewall replacement cycle just before day 10, leading to nearly equal size.

Fig. 10.
Fig. 10.

Results for Ideal5 and Ideal3 simulations with different microphysics schemes: (a) intensity for the each simulation using WDM6, Thompson, and Morrison, (b) RMW for each simulation, (c) heating vs condensate inside 166 km with the Thompson scheme, (d) heating vs condensate inside 166 km with Morrison, (e) current intensity vs condensate with Thompson, and (f) as in (e), but with Morrison.

Citation: Monthly Weather Review 147, 10; 10.1175/MWR-D-19-0010.1

Scatterplots from the results combined over these additional simulations are shown in Fig. 10. Due to the larger sizes of these TCs, we use data for cylinders out to 166 km, rather than 111 km, as 3 of the 4 new cases become large enough for some or all of the outward-sloping eyewalls to extend beyond 111 km. Interestingly, the correlations between total condensate and latent heating are significantly degraded for both Thompson and Morrison as compared to previous results for WDM6 (see Fig. 6a). The same scatterplot for WDM6 data out to 166 km (not shown) was nearly identical to that for 111 km, also with r = 0.99, and with slope a = 1.3 × 103 and a near-zero intercept. For Thompson and Morrison, the correlations are reduced to 0.87 and 0.96, respectively.

Correlations between condensate and current intensity are also somewhat degraded, being reduced to 0.89 and 0.85, as compared to 0.91 for WDM6 (using 166 km). Both cases also show a fair number of data points with large condensate values and fairly low intensities. These points are associated with bursts of convection that produce very large amounts of condensate and precipitation near each TC center around the time that intensity begins to increase, t = 2 days for both cases. Eliminating these data points improves the correlations slightly, but they still do not reach the same values as for WDM6.

Despite these differences, the slopes of all four best-fit lines shown in Fig. 10 are quite similar to their respective values for WDM6 (a = 1.3 × 103 and a = 7.7). This is perhaps even more surprising when we look at the distributions of condensate produced for the three schemes. Figure 11 shows azimuthal-mean total condensate, time-composited from the half-hourly output from t = 4.25 to t = 4.5 days for Ideal5, when all three simulations had about the same TC size and intensity. WDM6 has a fairly limited region of condensate, with values exceeding 4 g kg−1 only in the eyewall and below 6 km altitude. The Thompson scheme has a vast region of higher concentrations, all above 8 km height and extending out beyond 60 km radius. The distribution for the Morrison scheme is somewhat between the two, but closer to the Thompson scheme. However, the ranges of total condensate, normalized by area to units of kg m−2 as shown in the scatterplots, are very similar across all three schemes, as are the latent heating rates. In fact, the distributions of the moist heatings rates, also shown in Fig. 11, are quite similar for all three schemes, although the Thompson and Morrison schemes do show greater heating rates in the upper eyewall. Perhaps one way to understand the larger variability of the condensate versus heating relationships for the Thompson and Morrison schemes is to note that far more of their condensate is in the form of ice, and that their scatterplots look quite similar to the WDM6 scatterplots for heating and condensate above 10 and 12 km (Figs. 6c,e).

Fig. 11.
Fig. 11.

Azimuthal-mean total condensate (kg kg−1) and moist diabatic heating rates (K h−1), time-composited over the 6 h up to t = 4.5 day in the Ideal5 simulations: (a) condensate for WDM6, (b) moist heating for WDM6, (c) condensate for Thompson, (d) moist heating for Thompson, (e) condensate for Morrison, and (f) moist heating for Morrison.

Citation: Monthly Weather Review 147, 10; 10.1175/MWR-D-19-0010.1

g. TC intensity and rain rate

Although both the TRMM and GPM missions have precipitation radars that can see the vertical structure of condensate, their most widely used products are their rain rates, which are produced in combination with simultaneous microwave radiometer measurements. Therefore, although we have argued above that total condensation provides the most direct measurement of total heating, we should consider how well precipitation rates around TCs are correlated with intensity.

Figure 12 shows correlations between area-averaged precipitation rates and TC intensity over all storms, using either the fixed averaging radius of 111 km or variable radii equal to 1.25 × RMW; these show correlations of r = 0.90 and 0.84, respectively. Not surprisingly, these are nearly identical to the equivalent correlations between condensate and intensity.

Fig. 12.
Fig. 12.

Correlations between rain rate and TC intensity: (a) for the 5 simulations, scatterplot of current intensity vs mean rain rate inside 111 km, (b) scatterplot of intensity vs rain rate inside 1.25 × RMW, (c) using GPM rain rate data, mean rain rate inside 83 km alongside the best track intensity of Hurricane Jose (2017), and (d) rain rate inside 83 km alongside intensity of Hurricane Florence (2018).

Citation: Monthly Weather Review 147, 10; 10.1175/MWR-D-19-0010.1

Given these good correlations, we compared the intensity of two real TCs to their GPM precipitation rates, averaged in circles around the storm centers. These were obtained from the level 3 data provided by Integrated Multi-Satellite Retrievals for GPM (IMERG; Huffman et al. 2017). The comparisons were made for Hurricane Jose (2017) and Hurricane Florence (2018). These were both long-lived storms that reached major hurricane status, had little interaction with land, and each had two distinct intensity maxima separated by several days. Time series of their best track intensity and area-averaged precipitation rates are shown in Fig. 12. These results used an averaging radius of 83 km (0.75°), as this produced slightly higher correlations for both storms (0.63 and 0.77, respectively) than using 111 km (0.59 and 0.75). Due to the small number of data points and the high temporal correlations, none of these are statistically significant. Subjectively, the Florence result appears promising, while the Jose result does not. Rios Gaona et al. (2018) computed rain rates as a function of radius around 166 TCs from 2014 to 2016, also using the IMERG dataset. They found a strong increase in azimuthal-mean rain rates for TCs of increasing categories, but they did not correlate against actual intensities.

5. Results with narrow-swath measurements

a. Motivation and framework

In the future, there may be new cloud-profiling radars similar to CloudSat, and there may be more of them, such that overpasses near TC centers may be more common. Could condensate estimates from narrow-swath, cloud-profiling radars be useful for assessing intensity or intensity change?

To answer this question, we simulated narrow-swath estimates of total condensate by computing the condensate in 3-km wide paths running from north to south through our simulated storms. Of course, low-orbit satellites do not move exactly from north to south, but rather at some angle as they cross the equator (e.g., about 8° to the west of due north for CloudSat). However, this distinction is not important because the main issue is how close the narrow swath passes to the center of the cyclone. Flight paths directly over the eye of the storm are extremely rare, and paths even within 111 km (1° latitude) of the center are also quite rare. For example, in the CloudSat record from 2006 to 2015 of over 8000 “TC overpasses” (defined as flights that pass within 1000 km of any TC), only about 500 of these passed within 111 km of the cyclone center. Narrow-swath overpasses more than 111 km from the center would probably not be considered useful for diagnostic analysis in an operational setting.

Therefore, we attempt to determine how useful narrow-swath measurements of total condensate would be when they pass within 111 km of the TC center. For this purpose, we compute total condensate in the full column, and also above various altitudes, in north–south stripes of model output that are randomly located with equal probability between −111 and + 111 km of the TC center. We use model output from the 3 km grids, and the stripes are exactly 1 grid point wide, representing a 3-km-wide footprint. Based on the results above that showed useful correlations, the lengths of the stripes are either from 55 km south to 55 km north of the TC center, or from 111 km south to 111 km north, or from 166 km south to 166 km north.

The condensate is estimated in two ways. First, the mean condensate per unit area in the vertical slab of the atmosphere (3 km wide, within 111 km of the zonal position of the TC, and ranging meridionally from −55 to + 55 km, −111 to + 111 km, or −166 to + 166 km) is computed and used as a proxy for mean condensate in a 55, 111, or 166 km cylinder, without considering the displacement of the cylinder from the TC center. Given the inherent errors in operational estimates of TC location (Landsea and Franklin 2013), this displacement may not be accurately known in real time. This estimate is illustrated by rectangle on the left side of Fig. 13, and will be referred to as the single-slice (SS) method.

Fig. 13.
Fig. 13.

Two methods for estimating mean condensate in the TC inner core from a single narrow swath. The gray area indicates a hurricane eyewall. In the left rectangle, the total vertically integrated condensate in the swath is integrated over the swath, divided by the rectangular area, and used as an estimate of the area-mean value in a circle with radius r = yout. On the right, the values of the vertically integrated condensate at each location of yj and yj are averaged and then used an estimate for the mean value in the red circle. These values are integrated radially, weighted by 2πrΔr, where Δr is calculated from the difference between rj and rj−1, and then also divided by the total area in the annulus.

Citation: Monthly Weather Review 147, 10; 10.1175/MWR-D-19-0010.1

Second, assuming that this displacement could be known accurately in real time, and using the circular geometry of TCs, each single vertical column is considered to be representative of the entire circular band in which it intersects. In fact, there will be two columns that intersect each circle, so their values are averaged. This is illustrated in the rectangle on the right side of Fig. 13. Then these values are multiplied by 2πrΔr so that the data along the north–south overpass is used to estimate the total condensate in the annulus defined from r = xdisp to rout=(xdisp2+yout2)1/2, that is,
Qann=xdisproutπ[zbotztopρ(y,z)Qtot(y,z)dz+zbotztopρ(y,z)Qtot(y,z)dz]rdr,
where Qann is the total condensate in the annulus, vertically integrated from zbot to ztop. Qann is then divided by the area of the annulus to give a mean condensate per unit area, as computed for the cylinders above. This method will be called the radially integrated slice (RIS).

b. Results

How well do these two methods—either naively using the single swath (SS), or weighting by radius (RIS)—correlate with the actual total condensate inside a given cylinder? Figure 14 shows time series of estimates from the two methods for Ideal3. In the top panel, the slices were set to actually pass directly over the TC center. Both methods for estimating the total condensate (the blue and red lines) are reasonably close to the true total condensate except between t = 2 and 3 days, when the storm was reorganizing, causing a large displacement between the surface center and the majority of the convection (not shown). From t = 4 to 6 days, the SS method tended to underestimate the total condensate, while the RIS method appears to have fairly small bias. For the remainder of the simulation, both methods produce values that are close to each other and to the actual total condensate.

Fig. 14.
Fig. 14.

Total condensate estimates based on the two narrow swath methods shown in Fig. 13: (a) with the swaths passing directly over the TC center for every sample, and (b) with the swaths passing within random locations evenly distributed within −111 to + 111 km of the center.

Citation: Monthly Weather Review 147, 10; 10.1175/MWR-D-19-0010.1

Figure 14b shows the same result with the swaths passing randomly within 111 km of the zonal position of the center. This adds substantial errors to the estimates. The biases are also somewhat different, with both methods showing very small mean biases from t = 4 to 6 days, while later in the simulation the SS method develops positive bias and the RIS method develops negative bias.

Using the random slice displacements of within 111 km of the TC centers, Fig. 15 shows scatterplots of the actual versus estimated total condensate within 55, 111, and 166 km for all output times. The results are virtually identical for both methods (SS in the left column and RIS on the right), with nearly identical correlation values for each radius. The estimates for within 55 km are significantly less accurate (correlations of about 0.6) than for 111 and 166 km (correlations of about 0.8).

Fig. 15.
Fig. 15.

Scatterplots of actual total condensate inside a specified radius vs the estimated condensate from single swaths: (a) within 55 km using the SS method, (b) within 55 km using the RIS method, (c) within 111 km using SS, (d) within 111 km using RIS, (e) within 166 km using SS, and (f) within 166 km using RIS.

Citation: Monthly Weather Review 147, 10; 10.1175/MWR-D-19-0010.1

Of course, what matters is whether these estimates have any utility for diagnosing intensity or intensity change. Figure 16 shows the correlations between slice-estimated mean condensate within 111 and 166 km and the current TC intensity using both methods. The correlations are modestly good, with values ranging from 0.74 to 0.77.

Fig. 16.
Fig. 16.

Scatterplots of TC intensity vs condensate estimated from the two single-swath methods: (a) intensity vs condensate inside 111 km, using the SS method, (b) intensity vs condensate inside 111 km using RIS, (c) intensity vs condensate inside 166 km using SS, (d) intensity vs condensate inside 166 km using RIS, (e) intensity vs condensate inside 111 km and above 6 km, using SS, and (f) intensity vs condensate inside 166 km and above 6 km using SS.

Citation: Monthly Weather Review 147, 10; 10.1175/MWR-D-19-0010.1

As noted in Tourville et al. (2015) and Wu and Soden (2017), there can be considerable attenuation of the CloudSat radar return in deep convection, such that reflectivity signals below 5–6 km may be diminished or even lost completely. In this case, the wisest course would be to neglect reflectivity below some altitude for which the condensate estimates are not reliable. Therefore we also computed correlations between current intensity and total condensate in vertical slices above 6 km. These are shown in Figs. 16e and 16f, only for the SS method. This reduces the correlations slightly, to 0.69 and 0.70.

Finally, we consider to what extent high-altitude condensate estimated from vertical slices can predict intensity change. Figure 17 shows scatterplots of slice-estimated condensate above 12 km height versus 12-h intensity change, for 55 and 111 km, for SS (left) and RIS (right). The linear fit lines have a small positive slope for 55 km and a small negative slope for 111 km, and were more negative for 166 km (not shown). Regardless, the correlations are very small.

Fig. 17.
Fig. 17.

Scatterplots of 12 h intensity change vs condensate from single-swath estimates: (a) inside 111 km using SS, (b) inside 111 km using RIS, (c) inside 55 km using SS, and (d) inside 55 km using RIS.

Citation: Monthly Weather Review 147, 10; 10.1175/MWR-D-19-0010.1

c. Analogous estimates with real CloudSat profiles

Just as the model-derived area-averaged precipitation rates indicate some utility for diagnosing TC intensity (section 4g), these simulated narrow-swath estimates of condensate also suggest some potential utility for estimating TC intensity. Thus, we briefly consider the correlations between slice-estimated condensate from real CloudSat profiles and actual TC intensities. For this purpose we took advantage of the CloudSat Tropical Cyclone (CSTC) dataset produced by Tourville et al. (2015). Figure 18a shows correlations between area-average total condensate estimated from 555 actual CloudSat profiles that passed within 111 km of TC centers, using data along the flight path from −111 to +111 km from the point of closest approach, whereas Fig. 18b shows from −166 to +166 km. Both show a positive trend, with correlation values coincidentally equal to 0.47. Unlike with the model output, these data points are completely independent, because they were derived from different days and mostly different TCs. Therefore, this correlation, while weak, is also statistically significant.

Fig. 18.
Fig. 18.

Current intensity from real TCs (defined by estimated maximum surface wind) vs total condensate computed from a CloudSat swath that passed within 111 km of the TC center: (a) for a swath from −111 to +111 km of the point of closest approach, and (b) for −166 to + 166 km.

Citation: Monthly Weather Review 147, 10; 10.1175/MWR-D-19-0010.1

6. Conclusions

While the number of satellite-based methods for diagnosing TC intensity continues to increase, their accuracies (e.g., in terms of RMS error) are generally similar, with the subjective Dvorak technique still regarded as the most accurate (Jiang et al. 2019). However, remote sensing methods based on different wavelengths and/or different physical processes may provide independent information, and thus will increase the accuracy of consensus methods, provided they are weighted correctly. As Jiang et al. (2019) argued for intensity estimates based on microwave-sensed rain rates and wide-swath precipitation radars, an intensity algorithm based on cloud-profiling radars could also add to consensus skill.

The numerical simulations used in this study show a very close correlation between total condensate and total moist diabatic heating in the same volume. They also show good correlations between these quantities and current TC intensity. It is surprising, however, that the vertical distribution of this heating is not important, or more specifically, moist heating above certain altitudes is not a reliable indicator of imminent intensity change. While positive correlations were found, they mostly were not statistically significant, even using our lower threshold for 95% confidence, as described in section 4b.

It is also surprising that the results were not particularly sensitive to the radius of the cylinder over which the correlations were made. While a radius of 111 km generally showed the strongest correlations, correlations for slightly smaller and slightly larger radii were not substantially reduced. Furthermore, radii that were proportional to the current RMW of each cyclone also did not consistently produce higher correlations than simply using 111 km. These weak dependencies suggest compensating factors in the relationship between total heating within a given radius and TC intensity. For example, a very small TC may produce less moist heating inside a radius of 111 km than a larger storm, but it may use that energy more efficiently (Nolan et al. 2007), or there may be correspondingly less loss of energy due to friction due to its smaller wind field. Similarly weak dependencies on the area of consideration were found in previous studies (Cecil and Zipser 1999; Jiang et al. 2019).

This paper does not advocate development of a real-time diagnosis scheme for TC intensity based on the CloudSat radar. CloudSat overpasses over real TCs are too infrequent and irregular to be useful for forecasting centers, and the attenuation of its signal in heavy precipitation is limiting. Conceivably, there may be a larger number of cloud-profiling, narrow-swath radars in orbit someday, perhaps even with less attenuation. The far more frequent return periods of wide-swath precipitation radars such as the TRMM-PR and GPM-DPR suggest that either their rain rate estimates, or their estimates of total condensate based on observed reflectivity, could be used to estimate TC intensity. While our initial examination of this relationship was not very promising, further effort is warranted.

Acknowledgments

The authors acknowledge Dr. Tim DelSole for his guidance on statistics, and three anonymous reviewers whose comments led to significant improvements of this paper. This work was supported by the NASA CloudSat/CALIPSO Science Team Program under Grant NNXW16AP19G. Y. Miyamoto was supported by JSPS Scientific Research 26-358 for the Japanese Society for the Promotion of Science fellowship program for overseas researchers. All simulations for this study were produced at the University of Miami Center for Computational Sciences.

REFERENCES

  • Adler, R. F., and E. B. Rodgers, 1977: Satellite-observed latent heat release in a tropical cyclone. Mon. Wea. Rev., 105, 956963, https://doi.org/10.1175/1520-0493(1977)105<0956:SOLHRI>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Andersson, E., and M. Masutani, 2010: Collaboration on observing system simulation experiments (Joint OSSE). ECMWF Newsletter, No. 123, ECMWF, Reading, United Kingdom, 14–16, http://www.emc.ncep.noaa.gov/research/JointOSSEs/publications/JOSSE-Publication-files/Andersson_JOSSE_ECMWF_News_No123.pdf.

  • Braun, S. A., 2006: High-resolution simulation of Hurricane Bonnie (1998). Part II: Water budget. J. Atmos. Sci., 63, 4364, https://doi.org/10.1175/JAS3609.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Cecil, D. J., and E. J. Zipser, 1999: Relationships between tropical cyclone intensity and satellite-based indicators of inner-core convection: 85-GHz ice-scattering signature and lightning. Mon. Wea. Rev., 127, 103123, https://doi.org/10.1175/1520-0493(1999)127<0103:RBTCIA>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Dudhia, J., and Coauthors, 2008: Prediction of Atlantic tropical cyclones with the Advanced Hurricane WRF (ARW) Model. 28th Conf. on Hurricanes and Tropical Meteorology, Orlando, FL, Amer. Meteor. Soc., 18A.2, https://ams.confex.com/ams/pdfpapers/138004.pdf.

    • Search Google Scholar
    • Export Citation
  • Dunion, J. P., C. D. Thorncroft, and C. S. Velden, 2014: The tropical cyclone diurnal cycle of mature hurricanes. Mon. Wea. Rev., 142, 39003919, https://doi.org/10.1175/MWR-D-13-00191.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Dunion, J. P., C. D. Thorncroft, and D. S. Nolan, 2019: Tropical cyclone diurnal signals in a hurricane nature run. Mon. Wea. Rev., 147, 363388, https://doi.org/10.1175/MWR-D-18-0130.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Emanuel, K. A., 1991: The theory of hurricanes. Annu. Rev. Fluid Mech., 23, 179196, https://doi.org/10.1146/annurev.fl.23.010191.001143.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Fairall, C. W., E. F. Bradley, J. E. Hare, A. A. Grachev, and J. B. Edson, 2003: Bulk parameterization of air–sea fluxes: Updates and verification to the COARE algorithm. J. Climate, 16, 571591, https://doi.org/10.1175/1520-0442(2003)016<0571:BPOASF>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Gamache, J. F., R. A. Houze, and F. D. Marks, 1993: Dual-aircraft investigation of the inner core of Hurricane Norbert. Part III: Water budget. J. Atmos. Sci., 50, 32213243, https://doi.org/10.1175/1520-0469(1993)050<3221:DAIOTI>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Guimond, S. R., G. M. Heymsfield, and F. J. Turk, 2010: Multiscale observations of Hurricane Dennis (2005): The effects of hot towers on rapid intensification. J. Atmos. Sci., 67, 633654, https://doi.org/10.1175/2009JAS3119.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Guimond, S. R., M. A. Bourassa, and P. D. Reasor, 2011: A latent heat retrieval and its effects on the intensity and structure change of Hurricane Guillermo (1997). Part I: The algorithm and observations. J. Atmos. Sci., 68, 15491567, https://doi.org/10.1175/2011JAS3700.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Guimond, S. R., G. M. Heymsfield, P. D. Reasor, and A. C. Didlake Jr., 2016: The rapid intensification of Hurricane Karl (2010): New remote sensing observations of convective bursts from the Global Hawk platform. J. Atmos. Sci., 73, 36173639, https://doi.org/10.1175/JAS-D-16-0026.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Hack, J. J., and W. H. Schubert, 1986: Nonlinear response of atmospheric vortices to heating by organized cumulus convection. J. Atmos. Sci., 43, 15591573, https://doi.org/10.1175/1520-0469(1986)043<1559:NROAVT>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Hazelton, A. T., R. E. Hart, and R. F. Rogers, 2017: Analyzing simulated convective bursts in two Atlantic Hurricanes. Part II: Intensity change due to bursts. Mon. Wea. Rev., 145, 30953117, https://doi.org/10.1175/MWR-D-16-0268.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Hendricks, E. A., M. T. Montgomery, and C. A. Davis, 2004: The role of “vortical” hot towers in the formation of Tropical Cyclone Diana (1984). J. Atmos. Sci., 61, 12091232, https://doi.org/10.1175/1520-0469(2004)061<1209:TROVHT>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Hong, S.-Y., and J.-O. J. Lim, 2006: The WRF single-moment 6-class microphysics scheme (WSM6). J. Korean Meteor. Soc, 42, 129151.

  • Hong, S.-Y., Y. Noh, and J. Dudhia, 2006: A new vertical diffusion package with an explicit treatment of entrainment processes. Mon. Wea. Rev., 134, 23182341, https://doi.org/10.1175/MWR3199.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Hong, S.-Y., K.-S. S. Lim, Y.-H. Lee, J.-C. Ha, H.-W. Kim, S.-J. Ham, and J. Dudhia, 2010: Evaluation of the WRF double-moment 6-class microphysics scheme for precipitating convection. Adv. Meteor., 2010, 707253, https://doi.org/10.1155/2010/707253.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Hou, A. Y., R. K. Kakar, S. Neeck, A. A. Azarbazin, C. D. Kummerow, M. Kojima, K. Nakamura, and T. Iguchi, 2014: The Global Precipitation Measurement Mission. Bull. Amer. Meteor. Soc., 95, 701722, https://doi.org/10.1175/BAMS-D-13-00164.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Houze, R. A., 1997: Stratiform precipitation in regions of convection: A meteorological paradox?. Bull. Amer. Meteor. Soc., 78, 21792196, https://doi.org/10.1175/1520-0477(1997)078<2179:SPIROC>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Huffman, G. J., D. T. Bolvin, and E. J. Nelkin, 2017: Integrated Multi-satellitE Retrievals for GPM (IMERG) Technical Documentation. NASA Tech. Doc., 54 pp., https://pmm.nasa.gov/sites/default/files/document_files/IMERG_technical_doc_3_22_17.pdf.

  • Iacono, M. J., J. S. Delamere, E. J. Mlawer, M. W. Shephard, S. A. Clough, and W. D. Collins, 2008: Radiative forcing by long-lived greenhouse gases: Calculations with the AER radiative transfer models. J. Geophys. Res., 113, D13103, https://doi.org/10.1029/2008JD009944.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Jiang, H., C. Tao, and Y. Pei, 2019: Estimation of tropical cyclone intensity in the North Atlantic and northeastern Pacific basins using TRMM satellite passive microwave observations. J. Appl. Meteor. Climatol., 58, 185197, https://doi.org/10.1175/JAMC-D-18-0094.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Kaplan, J., M. DeMaria, and J. A. Knaff, 2010: A revised tropical cyclone rapid intensification index for the Atlantic and Eastern North Pacific basins. Wea. Forecasting, 25, 220241, https://doi.org/10.1175/2009WAF2222280.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Kelley, O. A., and J. B. Halverson, 2011: How much tropical cyclone intensification can result from the energy released inside of a convective burst? J. Geophys. Res., 116, D20118, https://doi.org/10.1029/2011JD015954.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Kelley, O. A., J. Stout, and J. B. Halverson, 2004: Tall precipitation cells in tropical cyclone eyewalls are associated with tropical cyclone intensification. Geophys. Res. Lett., 31, L24112, https://doi.org/10.1029/2004GL021616.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Klotz, B. W., and D. S. Nolan, 2019: SFMR surface wind undersampling over the tropical cyclone life cycle. Mon. Wea. Rev., 147, 247268, https://doi.org/10.1175/MWR-D-18-0296.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Landsea, C. W., and J. L. Franklin, 2013: Atlantic hurricane database uncertainty and presentation of a new database format. Mon. Wea. Rev., 141, 35763592, https://doi.org/10.1175/MWR-D-12-00254.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Miyamoto, Y., and D. S. Nolan, 2018: Structural changes preceding rapid intensification in tropical cyclones as shown in a large ensemble of idealized simulations. J. Atmos. Sci., 75, 555569, https://doi.org/10.1175/JAS-D-17-0177.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Moon, Y., and D. Nolan, 2015: Spiral rainbands in a numerical simulation of Hurricane Bill (2009). Part I: Structures and comparisons to observations. J. Atmos. Sci., 72, 164190, https://doi.org/10.1175/JAS-D-14-0058.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Morrison, H., G. Thompson, and V. Takarskii, 2009: Impact of cloud microphysics on the development of trailing stratiform precipitation in a simulated squall line: Comparison of one- and two-moment schemes. Mon. Wea. Rev., 137, 991137, https://doi.org/10.1175/2008MWR2556.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Nolan, D. S., 2011: Evaluating environmental favorableness for tropical cyclone development with the method of point-downscaling. J. Adv. Model. Earth Syst., 3, M08001, https://doi.org/10.1029/2011ms000063.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Nolan, D. S., and M. G. McGauley, 2012: Tropical cyclogenesis in wind shear: Climatological relationships and physical processes. Cyclones: Formation, Triggers and Control, K. Oouchi and H. Fudeyasu, Eds., Nova Science Publishers, 1–34.

    • Search Google Scholar
    • Export Citation
  • Nolan, D. S., and C. A. Mattocks, 2014: Development and evaluation of the second hurricane nature run using the joint OSSE nature run and the WRF model. 31st Conf. on Hurricanes and Tropical Meteorology, San Diego, CA, Amer. Meteor. Soc., P91.

    • Search Google Scholar
    • Export Citation
  • Nolan, D. S., Y. Moon, and D. P. Stern, 2007: Tropical cyclone intensification from asymmetric convection: Energetics and efficiency. J. Atmos. Sci., 64, 33773405, https://doi.org/10.1175/JAS3988.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Nolan, D. S., R. Atlas, K. T. Bhatia, and L. R. Bucci, 2013: Development and validation of a hurricane nature run using the joint OSSE nature run and the WRF model. J. Adv. Model. Earth Syst., 5, 382405, https://doi.org/10.1002/jame.20031.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Okamoto, K., K. Aonashi, T. Kubota, and T. Tashima, 2016: Experimental assimilation of the GPM core observatory DPR reflectivity profiles for Typhoon Hailong (2014). Mon. Wea. Rev., 144, 23072326, https://doi.org/10.1175/MWR-D-15-0399.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Riehl, H., and J. Malkus, 1961: Some aspects of Hurricane Daisy, 1958. Tellus, 13, 181213, https://doi.org/10.3402/tellusa.v13i2.9495.

  • Rios Gaona, M. F., G. Villarini, W. Zhang, and G. A. Vecchi, 2018: The added value of IMERG in characterizing rainfall in tropical cyclones. Atmos. Res., 209, 95102, https://doi.org/10.1016/j.atmosres.2018.03.008.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Rodgers, E., W. S. Olson, V. M. Karyampudi, and H. F. Pierce, 1998: Satellite-derived latent heating distribution and environmental influences in Hurricane Opal (1995). Mon. Wea. Rev., 126, 12291247, https://doi.org/10.1175/1520-0493(1998)126<1229:SDLHDA>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Rodgers, E., W. S. Olson, J. Halverson, J. Simpson, and H. Pierce, 2000: Environmental forcing of Supertyphoon Paka’s latent heat structure. J. Appl. Meteor., 39, 19832006, https://doi.org/10.1175/1520-0450(2001)040<1983:EFOSPS>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Rogers, R. F., P. D. Reasor, and J. A. Zhang, 2015: Multiscale structure and evolution of Hurricane Earl (2010) during rapid intensification. Mon. Wea. Rev., 143, 536562, https://doi.org/10.1175/MWR-D-14-00175.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Rogers, R. F., J. A. Zhang, J. Zawislak, H. Jiang, G. R. Alvey III, E. J. Zipser, and S. S. Stevenson, 2016: Observations of the structure and evolution of Hurricane Edouard (2014) during intensity change. Part II: Kinematic structure and the distribution of deep convection. Mon. Wea. Rev., 144, 33553376, https://doi.org/10.1175/MWR-D-16-0017.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Shimada, U., K. Aonashi, and Y. Miyamoto, 2017: Tropical cyclone intensity change and axisymmetricity deduced from GSMaP. Mon. Wea. Rev., 145, 10031017, https://doi.org/10.1175/MWR-D-16-0244.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Simpson, J., C. Kummerow, W.-K. Tao, and R. Adler, 1996: On the Tropical Rainfall Measuring Mission. Meteor. Atmos. Phys., 60, 1936, https://doi.org/10.1007/BF01029783.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Simpson, J., J. B. Halverson, B. S. Ferrier, W. A. Petersen, R. H. Simpson, R. Blakeslee, and S. L. Durden, 1998: On the role of “hot towers” in tropical cyclone formation. Meteor. Atmos. Phys., 67, 1535, https://doi.org/10.1007/BF01277500.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Stephens, G. L., and Coauthors, 2002: The CloudSat Mission and the A-Train: A new dimension of space-based observations of clouds and precipitation. Bull. Amer. Meteor. Soc., 83, 17711790, https://doi.org/10.1175/BAMS-83-12-1771.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Tang, B., and K. Emanuel, 2010: Midlevel ventilation’s constraint on tropical cyclone intensity. J. Atmos. Sci., 67, 18171830, https://doi.org/10.1175/2010JAS3318.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Tao, W.-K., and Coauthors, 2006: Retrieval of latent heating from TRMM Measurements. Bull. Amer. Meteor. Soc., 87, 15551572, https://doi.org/10.1175/BAMS-87-11-1555.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Thompson, G., R. M. Rasmussen, and K. Manning, 2004: Explicit forecasts of winter precipitation using an improved bulk microphysics scheme. Part I: Description and sensitivity analysis. Mon. Wea. Rev., 132, 519542, https://doi.org/10.1175/1520-0493(2004)132<0519:EFOWPU>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Thompson, G., P. R. Field, R. M. Rasmussen, and W. D. Hall, 2008: Explicit forecasts of winter precipitation using an improved bulk microphysics scheme. Part II: Implementation of a new snow parameterization. Mon. Wea. Rev., 136, 50955115, https://doi.org/10.1175/2008MWR2387.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Tourville, N., G. Stephens, M. DeMaria, and D. Vane, 2015: Remote sensing of tropical cyclones. Bull. Amer. Meteor. Soc., 96, 609622, https://doi.org/10.1175/BAMS-D-13-00282.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Vigh, J. L., and W. H. Schubert, 2009: Rapid development of the tropical cyclone warm core. J. Atmos. Sci., 66, 33353350, https://doi.org/10.1175/2009JAS3092.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Wadler, J. B., R. F. Rogers, and P. D. Reasor, 2018: The relationship between spatial variations in the structure of convective bursts and tropical cyclone intensification as determined by airborne Doppler radar. Mon. Wea. Rev., 146, 761780, https://doi.org/10.1175/MWR-D-17-0213.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Wang, Y., and J. Xu, 2010: Energy production, frictional dissipation, and maximum intensity of a numerically simulated tropical cyclone. J. Atmos. Sci., 67, 97116, https://doi.org/10.1175/2009JAS3143.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Wilks, D. S., 2006: Statistical Methods in the Atmospheric Sciences: An Introduction. 2nd ed. International Geophysics Series, Vol. 100, Academic Press, 648 pp.

    • Search Google Scholar
    • Export Citation
  • Wu, S.-N., and B. J. Soden, 2017: Signatures of tropical cyclone intensification in measurements of ice and liquid water content. Mon. Wea. Rev., 145, 40814091, https://doi.org/10.1175/MWR-D-17-0046.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Zagrodnik, J. P., and H. Jiang, 2014: Rainfall, convection, and latent heating distributions in rapidly intensifying tropical cyclones. J. Atmos. Sci., 71, 27892809, https://doi.org/10.1175/JAS-D-13-0314.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Zhang, D., Y. Liu, and M. K. Yau, 2002: A multiscale numerical study of Hurricane Andrew (1992). Part V: Inner-core thermodynamics. Mon. Wea. Rev., 130, 27452763, https://doi.org/10.1175/1520-0493(2002)130<2745:AMNSOH>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
1

The variable r is used for both radius and for correlation coefficient, but their meanings are hopefully clear by context in each case.

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