A Generated using version 3.1.2 of the official AMS LTEX template 1 The variability and predictability of axisymmetric hurricanes 2 in statistical equilibrium ∗ 3 Gregory J. Hakim University of Washington, Seattle, WA ∗Corresponding author address: Gregory J. Hakim, Department of Atmospheric Sciences, Box 351640, University of Washington, Seattle, WA, 98195. E-mail: [email protected] 1 4 ABSTRACT 5 The variability and predictability of axisymmetric hurricanes is determined from a 500-day 6 numerical simulation of a tropical cyclone in statistical equilibrium. By design, the solution 7 is independent of the initial conditions and environmental variability, which isolates the 8 “intrinsic” axisymmetric hurricane variability. 9 Variability near the radius of maximum wind is dominated by two patterns: one asso- 10 ciated primarily with radial shifts of the maximum wind, and one primarily with intensity 11 change at the time-mean radius of maximum wind. These patterns are linked to convective 12 bands that originate more than 100 km from the storm center and propagate inward. 13 Bands approaching the storm produce eyewall replacement cycles, with an increase in storm 14 intensity as the secondary eyewall contracts radially inward. A dominant time period of 15 4–8 days is found for the convective bands, which is hypothesized to be determined by the 16 timescale over which subsidence from previous bands suppresses convection; a leading-order 17 estimate based on the ratio of the Rossby radius to band speed fits the hypothesis. 18 Predictability limits for the idealized axisymmetric solution are estimated from linear 19 inverse modeling and analog forecasts, which yield consistent results showing a limit for 20 the azimuthal wind of approximately three days. The optimal initial structure that excites 21 the leading pattern of 24-hour forecast-error variance has largest azimuthal wind in the 22 midtroposphere outside the storm, and a secondary maximum just outside the radius of 23 maximum wind. Forecast errors grow by a factor of 24 near the radius of maximum wind. 24 1. Introduction 25 Forecasts of tropical cyclone position have steadily improved in the past decade (e.g., 26 Elsberry et al. 2007) along with the global reduction in forecast error for operational models. 27 If tropical cyclone intensity were mainly determined by the storm environment (e.g., Frank 28 and Ritchie 1999; Emanuel et al. 2004) such as through variability in sea-surface temperature 1 29 and large-scale vertical wind shear, then one might expect improvements in tropical cyclone 30 intensity forecasts similar to those for track. The fact that tropical cyclone intensity forecasts 31 from numerical models have been resistant to these environmental improvements suggests 32 that the challenge to improve these forecasts lies with the dynamics of the tropical cyclone 33 itself. While increasing observations and novel data assimilation strategies may reduce 34 forecast errors, it is unclear what improvements one should expect, even in the limit of 35 vanishing initial error, since the fundamental predictability of tropical cyclone structure and 36 intensity is essentially unknown. We take a step in the direction of addressing this problem 37 here by examining the variability and predictability of idealized axisymmetric numerical 38 simulations of a tropical cyclone in statistical equilibrium. By eliminating variability due to 39 the environment, asymmetries, and storm motion we isolate what we define as the “intrinsic” 40 variability. A 500 day equilibrium solution allows this variability to be estimated with 41 sample sizes much larger than can be obtained from observed data or from the simulation 42 of individual storms. 43 Since the structure of tropical cyclones is dominated by features on the scale of the vortex, 44 primarily the azimuthal mean, a starting hypothesis for predictability and variability in storm 45 structure and intensity is that it is controlled by the axisymmetric circulation. However, it 46 is clear that although asymmetries have much smaller amplitude compared to the azimuthal 47 mean, they may affect storm intensity through, for example, wave–mean-flow interaction 48 (e.g., Montgomery and Kallenbach 1997; Wang 2002; Chen et al. 2003). A prominent 49 example of structure and intensity variability involves eyewall replacement cycles, where 50 the eyewall is replaced by an inward propagating ring of convection (e.g., Willoughby et al. 51 1982; Willoughby 1990). While this process may be initiated by asymmetric disturbances, 52 it is unclear whether such variability requires asymmetries. Here we explore variability 53 associated with only the symmetric circulation, which provides a basis for comparison to 54 three-dimensional studies that include asymmetries (Brown and Hakim 2012). 55 In terms of predictability, most research studies focus on forecasts of the location and 2 56 intensity of observed storms (e.g., DeMaria 1996; Goerss 2000). Since there are a wide 57 variety of environmental influences on observed storms, it is very difficult to determine the 58 intrinsic predictability independent of these influences. Given the aforementioned resistance 59 of hurricane intensity forecasts to improvements in the larger-scale environment, the intrinsic 60 variability provides a starting point for understanding how the environment adds, or removes, 61 predictive capacity. Ultimately, we wish to know the limit of tropical cyclone structure and 62 intensity predictability, which may be approached by forecasts as a result of improvements 63 to models, observations, and data assimilation systems. Again, the present study provides 64 a starting point for the limiting case of axisymmetric storms in statistical equilibrium. 65 One dynamical perspective on tropical cyclone predictability views the vortex as analo- 66 gous to the extratropical planetary circulation, where growth of the leading Lyapunov vectors 67 is determined mostly by balanced perturbations developing on horizontal and vertical shear 68 (e.g., Snyder and Hamill 2003). Some initial conditions are more sensitive than others, 69 and the upscale influence of moist convection may accelerate large-scale error growth in 70 some cases (e.g., Zhang et al. 2002), but these appear to be exceptional. An alternative 71 perspective, given the central role of convection in tropical cyclones, is that upscale error 72 growth from convective scales may be relatively more important for tropical cyclones as 73 compared to extratropical weather predictability. For example, Van Sang et al. (2008) 74 randomly perturb the low-level moisture distribution of a ten-member ensemble for a three- 75 dimensional numerical simulation of tropical cyclogenesis, and find maximum azimuthal- −1 76 mean wind speed differences among the members as large as 15 m s . Based on these 77 simulations, Van Sang et al. (2008) conclude that inner-core asymmetries are “random and 78 intrinsically unpredictable.” If hurricane predictability is indeed more closely linked to the 79 convective decorrelation time, then intrinsic predictability may be limited to a few hours 80 or less. While we cannot completely answer this question here, we take the first step of 81 acknowledging its fundamental importance to understanding tropical cyclone structure and 82 intensity predictability, and make estimates of predictive timescales and structures that 3 83 control forecast errors in an idealized axisymmetric model. 84 Using a radiative–convective equilibrium (RCE) modeling approach to generate samples 85 for statistical analysis has a long history in tropical atmospheric dynamics. For exam- 86 ple, (Held et al. 1993) employ this approach to examine the statistics of simulated two- 87 dimensional tropical convection for a uniform sea-surface temperature, and show that in 88 the absence of shear the convection aggregates into a subset of the domain. A similar, 89 three-dimensional, study by Bretherton et al. (2005) finds upscale aggregation of convection 90 and, when ambient rotation is included, tropical cyclogenesis. Nolan et al. (2007) use RCE 91 to examine the sensitivity of tropical cyclogenesis to environmental parameters, and also 92 find the spontaneous development of tropical cyclones. The current paper uses RCE of a 93 specific state, a mature tropical cyclone, to examine the variability and predictability of 94 these storms in an axisymmetric framework; results for three-dimensional simulations are 95 presented in Brown and Hakim (2012). 96 The remainder of the paper is organized as follows. Section 2 describes the axisymmetric 97 numerical model used to produce a 500 day solution in statistical equilibrium. The intrin- 98 sic axisymmetric variability of this solution is explored in section 3, and predictability is 99 discussed in section 4. Section 5 provides a concluding summary. 100 2. Experiment Setup 101 The axisymmetric numerical model used here is a modified version of the one used by 102 Bryan and Rotunno (2009) as described in Hakim (2011). The primary modification involves 103 the use of the interactive RRTM-G radiation scheme (Mlawer et al. 1997; Iacono et al. 2003) 104 as a replacement for the Rayleigh damping scheme used by Bryan and Rotunno (2009). 105 Interactive radiation allows for the generation of convection by reducing the static stability 106 as the atmosphere cools by emitting infrared radiation to space. Radiation allows the storm 107 to achieve statistical equilibrium, maintain active convection, and produce robust variability 4 108