15 APRIL 2016 C H A V A S E T A L . 2923

Observed Tropical Size Revisited

DANIEL R. CHAVAS* AND NING LIN Department of Civil and Environmental Engineering, Princeton University, Princeton, New Jersey

WENHAO DONG AND YANLUAN LIN Ministry of Education Key Laboratory for Earth System Modeling, Center for Earth System Science, Tsinghua University, Beijing, China

(Manuscript received 14 October 2015, in final form 1 February 2016)

ABSTRACT

This work revisits the statistics of observed outer size in the context of recent advances in our 2 theoretical understanding of the storm field. The authors create a new dataset of the radius of 12 m s 1 based on a recently updated version of the QuikSCAT ocean wind vector database and apply an improved analytical outer wind model to estimate the outer radius of vanishing wind. The dataset is then applied to analyze the statistical distributions of the two size metrics as well as their dependence on environmental parameters, with a specific focus on testing recently identified parameters possessing credible theoretical relationships with tropical cyclone size. The ratio of the potential intensity to the Coriolis parameter is found to perform poorly in explaining variation of size, with the possible exception of its upper bound, the latter of which is in line with existing theory. The rotating radiative–convective equilibrium scaling of Khairoutdinov and Emanuel is also found to perform poorly. Meanwhile, mean storm size is found to increase systematically with the relative sea surface temperature, in quantitative agreement with the results of a recent study of storm size based on pre- cipitation area. Implications of these results are discussed in the context of existing tropical climate theory. Finally, an empirical dependence of the central pressure deficit on outer size is found in line with past work.

1. Introduction given maximum wind speed and latitude, is defined relative to some absolute radial length scale that rep- Though the bulk of research on tropical to resents storm size. The model was found to compare date has focused on the dynamics of storm intensity, as well with observations when defining size as the radius of measured by the maximum wind speed V , this quantity m maximum wind r . Alternatively, Chavas and Lin (2016, is but one component of the complete wind field, the m manuscript submitted to J. Atmos. Sci.) demonstrate broader understanding of which lags behind. Recent that, when defining size as the outer radius of vanishing work has begun to explore the size and structure of the wind r , the model successfully reproduces the charac- wind field in greater depth. In particular, Chavas et al. 0 teristic modes of wind-field variability in nature and (2015) developed a physical model for the radial struc- provides a credible prediction of r . As such, this model ture of the azimuthal-mean azimuthal wind that, for a m offers the potential to use outer-size information to infer inner size and structure. This outcome is significant given that rm is the relevant size metric for damage (Zhai Denotes Open Access content. and Jiang 2014; Irish and Resio 2010). Additionally, rm poses difficulties for observation because of the turbulent * Current affiliation: Department of Earth, Atmospheric, and nature of the inner core of a tropical cyclone, including Planetary Sciences, Purdue University, West Lafayette, Indiana. discontinuous jumps because of eyewall replacement cycles (Sitkowski et al. 2011). Moreover, in a numerical model, rm requires high radial resolution to be credibly Corresponding author address: Daniel R. Chavas, Department of # Earth, Atmospheric, and Planetary Sciences, Purdue University, represented ( 4 km, e.g., Chavas and Emanuel 2014), 550 Stadium Mall Drive, HAMP 3221, West Lafayette, IN 47907. which is of particular significance in the context of cli- E-mail: [email protected] mate models for which such resolutions are currently

DOI: 10.1175/JCLI-D-15-0731.1

Ó 2016 American Meteorological Society 2924 JOURNAL OF CLIMATE VOLUME 29 computationally expensive. In contrast, the relatively that links rm and r0. From r12, we use the model of outer quiescent outer circulation is much more stable in time wind structure of Emanuel (2004), incorporating im- relative to the turbulent inner core, including a near proved estimates of its input parameters, to calculate r0. independence of outer size on the maximum wind speed Ultimately, the objective is an understanding of the in both observations (Shoemaker 1989; Weatherford underlying physics that governs the length scale of the and Gray 1988; Merrill 1984; Chavas and Emanuel 2010; outer storm circulation. Such an understanding would Chavas et al. 2015) and idealized numerical models help explain storm size and its variation in nature, both (Rotunno and Bryan 2012; Chavas and Emanuel 2014). on short time scales for operational forecasting as well as Thus, we seek to improve our understanding of outer on longer time scales across climate states, including storm size in nature, which provides a key link to our global warming. Toward this end, recent work has pro- understanding of the radius of maximum wind and wind vided new theoretical insights into physical environ- field variability in general. mental parameters that govern this length scale. First, in How is outer storm size defined? Following the sem- rotating radiative–convective equilibrium (RCE), storm inal work of Frank (1977), as well as more recent work size has been demonstrated in both axisymmetric and by Chavas et al. (2015), the complete radial structure three-dimensional geometry to scale with the ratio of of a tropical cyclone may be considered to leading order the potential intensity to the Coriolis parameter Vp/f to be composed of a narrow inner convecting region that (Chavas and Emanuel 2014; Khairoutdinov and varies with intensity and a broad outer nonconvecting Emanuel 2013). This length scale corresponds to the region that does not vary with intensity, suggesting that intrinsic radial length scale in tropical cyclone theory any metric that lies sufficiently far outside of the con- [c.f. Table 1 of Emanuel (1995a)]; it also serves as a scaling vecting inner core may be suitable. More specifically, r0 for the theoretical upper bound on size [c.f. Equation 16 is fundamental to the theoretical outer wind structure of Emanuel (1995b)] that arises from the need for the model of Emanuel (2004), which has been shown to storm to do work to restore the angular momentum of credibly reproduce the radial structure of the outer wind outflow air parcels to its ambient value (Emanuel 1986). field in nature (Chavas et al. 2015). Unfortunately, r0 is Such a scaling is consistent with the finding of Knutson very difficult to estimate directly in observations or nu- et al. (2015) that global median storm size, as measured merical models because of the combination of the de- by r12, remains constant under a future global warming crease in data coverage and the increase in background scenario, given the anticipated increase in Vp under flow variability at large radii. To avoid this issue, greenhouse warming (Emanuel 1987) as well the in- Chavas and Emanuel (2010) employed version 3 of the crease in f associated with the poleward migration of QuikSCAT global near-surface wind vector database to tropical cyclone activity (Kossin et al. 2014). Addition- analyze the radius at which the azimuthal-mean azimuthal ally, Chavas and Emanuel (2014) demonstrated a sec- 21 wind is 12 m s (r12), from which they estimated r0 using ondary dependence of r0 on the nondimensional Ekman the analytical outer wind model of Emanuel (2004). suction rate, given by Wcool/(CdVp), where Wcool is the Alternative size metrics have also been analyzed, albeit clear-sky free-tropospheric radiative-subsidence rate 21 without linkage to r0, including the radius of 17 m s and Cd is the drag coefficient that arises from Ekman winds from QuikSCAT (Chan and Chan 2015, 2012), the balance in the outer descending region of the storm 2 radius of 15 m s 1 (Shoemaker 1989; Cocks and Gray circulation. Second, Khairoutdinov and Emanuel (2013) 2 2002), and the radius of 5-kt (2.572 m s 1) winds esti- proposed a scaling for the spacing between storms in mated from a combination of satellite infrared data three-dimensional rotating RCE, in which tropical cy- and a climatological linear model of the outer wind field clones fill the domain, that depends on the combination (Knaff et al. 2014). All such studies have reached similar of a thermodynamic efficiency and the length scale pffiffiffiffiffiffiffiffiffiffi conclusions regarding the large range of variability in Lyqb/f , where Ly is the latent heat of vaporization and storm size, in line with earlier work (Merrill 1984). qb is the boundary layer specific humidity. This scaling While all of these metrics are viable in their own right, was derived from global heat and energy balance con- here we focus again on r12, as this metric balances the straints under the assumption that frictional dissipation desire to maximize instrumental data coverage, to avoid by tropical cyclones dominates entropy production in the convecting inner core, and to readily separate the the domain. Third, Lin et al. (2015) demonstrated a signal from the noise of the background flow, as origi- strong observational relationship between mean storm nally discussed in Chavas and Emanuel (2010). More- size, defined by the overall precipitation area, and the over, r12 was found to provide an optimal prediction of relative sea surface temperature (SST), defined as the rm in the complete wind structure framework of Chavas difference between the local SST and its tropical mean and Lin (2016, manuscript submitted to J. Atmos. Sci.) value. Given that the relative SST in the tropics has been 15 APRIL 2016 C H A V A S E T A L . 2925 shown to covary closely with local midtropospheric en- climatology as well as updated information for tropical vironmental relative humidity (Stephens 1990; Emanuel cyclone risk analysis. et al. 1994), this result is consistent with the prior finding of an increase in storm size with increasing environmental 2. Data and methodology moisture in a numerical model (Hill and Lackmann 2009; a. Storm parameters Wang 2009). Here we investigate the extent to which each of these environmental parameters can explain variation in For analysis of outer storm size, we estimate the radius storm size in our observational dataset. at which the azimuthal mean of the azimuthal wind equals 21 In addition, quantitative information on storm size 12 m s (r12) from the QuikSCAT Tropical Cyclone Ra- variation is critical for tropical cyclone risk analysis dial Structure database (QSCAT-R; Chavas and Vigh given that r0 is closely linked to rm and, in turn, to storm- 2014, available at http://verif.ral.ucar.edu/tcdata/quikscat/ related hazards. Indeed, in a Monte Carlo–based storm dataset/). QSCAT-R contains radial profiles of the surge risk assessment, Lin et al. (2014) showed that the azimuthal-mean azimuthal wind calculated from the surge risk could be significantly underestimated if the NASA Jet Propulsion Laboratory (JPL) QuikSCAT average storm size, as opposed to its full statistical dis- tropical cyclone ocean near-surface (z 5 10 m) wind tribution, is applied to the simulated storms. This is par- vector database, which is a special version of version 3 ticularly the case when the size distribution is positively of the complete global QuikSCAT dataset that has been skewed, as has been observed in nature (Chavas and optimized specifically for tropical cyclones using a neu- Emanuel 2010; Chan and Chan 2015). Moreover, in addi- ral network algorithm to increase accuracy and reduce tion to potential changes in storm frequency and intensity contamination; the optimized dataset is described with climate change, change of storm size can also signifi- in Stiles et al. (2014) and is available online (http:// cantly change the wind and especially surge risk (Lin et al. tropicalcyclone.jpl.nasa.gov/). This global dataset spans 2012), although storm size has been found in some model the period 1999–2009 and has an approximate horizontal simulations to remain constant (Knutson et al. 2015)or resolution of 12.5 km. A simple estimate of the near- increase marginally (Kim et al. 2014)underglobalwarm- surface background-flow vector, taken to equal the storm ing. However, because of the limitation of current data and translation vector rotated 208 cyclonically and reduced the lack of a rigorous understanding of the physics of storm in magnitude by a factor of 0.55 (Lin and Chavas 2012), size and its variability in the tropics, the variation of size is removed for each case prior to calculating the radial from storm to storm and across climate states is yet to be profile. Because of the common occurrence of azi- accounted for in risk analysis (Lin and Emanuel 2016). In muthally periodic asymmetries (Uhlhorn et al. 2014; other applications where canonical wind–pressure rela- Reasor et al. 2000), azimuthal data coverage asymmetry tionships are used to estimate one from the other, infor- is a principal source of uncertainty in azimuthal-mean mation of storm size can also help to reduce estimation radial profile estimation. To quantify this uncertainty, uncertainties, as the three are physically correlated. QSCAT-R includes a data coverage asymmetry param- Thus, in an effort to update and improve upon the re- eter j representing the azimuthal coverage of data at a sults of Chavas and Emanuel (2010), here we first create given radius and whose value spans the range [0, 1]; new datasets of r12 and r0 based on a recently optimized a smaller value of j implies lower uncertainty such that version of the QuikSCAT database and an improved j 5 0 for data coverage with perfect azimuthal symmetry version of the analytical outer wind model (section 2). and j 5 1 in the case of a single data point. Following the

Second, we analyze the variations of r12 and r0 in geo- work of Chavas et al. (2015), we keep only those values of graphic space and develop statistical distributions of each r12 for which j # 0:5atr12,andvaluesofr12 , 50 km (i.e., size metric globally and for each basin (section 3). Then 4 times the horizontal grid spacing of the raw dataset) are we further investigate size variability in the context of ignored because of increasing uncertainty in the de- recent advances in our understanding of storm size and composition of flow direction at small radii. We include structure, with a specific emphasis on testing three envi- only cases where the storm center is over water where ronmental parameters that possess credible theoretical valid SST and potential intensity values can be estimated. relationships with storm size: Vp/f , the rotating RCE In addition to wind data, QSCAT-R also includes collo- scaling of Khairoutdinov and Emanuel (2013),andthe cated radial profiles of a quantity proportional to rain rate relative SST (section 4). Additionally, we explore empir- (denoted P*; not accurate for absolute rain rates), based ically the effect of storm size on the canonical wind– on data calculated from brightness temperature mea- pressure relationship (section 5). In the final section, we surements taken by the SeaWinds radiometer aboard the provide further discussion and synthesize key conclusions QuikSCAT satellite (Ahmad et al. 2005). Complete de- (section 6). This study provides new insights into storm size tails are provided in Chavas and Vigh (2014). 2926 JOURNAL OF CLIMATE VOLUME 29

From r12, we estimate the outer radius of vanishing minimum central pressure are also taken from QSCAT- azimuthal-mean azimuthal wind r0 using the analytical R. These values were interpolated from the NHC hur- model for the radial structure of the azimuthal-mean ricane database (HURDAT) and JTWC best-track data azimuthal wind in the outer, nonconvecting region first (extracted from the International Best Track Archive for described in Emanuel (2004). This model was demon- Climate Stewardship database v03r05) using a piecewise strated to compare very well to observations by Chavas cubic Hermite interpolating polynomial, a method that et al. (2015) and to idealized modeling simulations by eliminates overshoots between data points that can oc- Reed and Chavas (2015). The model is given by cur in traditional spline interpolation. The maximum wind speed in the best-track database is an estimate of ›M 2C (rV)2 5 d , (1) the maximum total wind speed at any point (denoted › 2 2 2 r W r0 r cool Vmax,BT) (i.e., including both storm and environmental flow effects), whereas the ideal intensity metric for this where Cd is the surface drag coefficient, Wcool is the magnitude of the radiative-subsidence rate in the free work is the maximum azimuthal-mean azimuthal wind y troposphere, and M is the absolute angular momentum, speed associated with the storm max because of its per unit mass, given by relevance to existing tropical cyclone theory. Un- fortunately, calculation of this quantity directly from 1 M 5 rV 1 fr2 , (2) QuikSCAT data is problematic because of reduced 2 precision at high wind speeds and high rain rates com- where r is the radius, V is the azimuthal wind speed, and mon to the storm inner core (Stiles et al. 2014). Instead, f is the Coriolis parameter. Following Chavas et al. we employ the dataset of maximum azimuthal-mean (2015), we specify Cd to be a function of wind speed fit to azimuthal wind speed of Chavas et al. (2015) calculated the data of Donelan et al. (2004) and the radiative- from the HWind near-surface wind field database (Powell subsidence rate equal to its estimated background value et al. 1998) to develop a simple method for estimating 21 of Wcool 5 2mms , and we solve for r0 using a shooting ymax from best-track data alone. First, following Lin and method. Note that, though Wcool is expected on theoretical Chavas (2012), we seek to remove the near-surface envi- grounds to remain relatively constant in space given weak ronmental flow by subtracting 55% of the translation temperature gradient considerations (Sobel and Bretherton vector magnitude Vtrans from the best-track wind speed; 2000), its true space–time variability in nature as well as a this yields an estimate of the maximum point wind speed 0 function of radius in the outer circulation remains unknown; associated with the storm Vmax,BT. The reduction factor is the value used here is taken simply as a characteristic mag- applied because the environmental flow at the surface is nitude that was found in Chavas et al. (2015) to perform weaker in magnitude than the column-integrated envi- well in modeling the outer wind field and that aligns with ronmental flow (i.e., the steering flow) that governs storm values from existing numerical modeling studies (Davis motion (e.g., Chan 2005). The linear relationship of Lin 2015; Reed and Chavas 2015). The Coriolis parameter is and Chavas (2012) is chosen for simplicity and consistency taken as its value at the storm center. This model was also with the methodology applied to the full wind fields from applied in Chavas and Emanuel (2010), though with a the QSCAT-R and HWind-based datasets; the nonlinear fixed value of Cd and a much larger, arbitrary value of relationship of Schwerdt et al. (1979),whichgivesa Wcool that translated to a very rapid decrease in wind comparable reduction factor magnitude for typical trans- speed with radius. Though in principle it is preferable to lation speeds in the tropics, was also tested and yields very estimate the outer radius using the complete wind field similar results. Second, we estimate the relationship be- model of Chavas et al. (2015), which merges this outer tween this maximum point wind speed and the maximum model with the inner model of Emanuel and Rotunno azimuthal-mean azimuthal wind speed using the HWind

(2011), rather than the outer model alone, the result is database. The result is shown in Fig. 1. On average, ymax is 0 identical except for a small subset of weak storms for approximately 80% of Vmax,BT at all intensities, with var- which r12 lies within the domain of the inner model; thus, iance that decreases steadily with increasing intensity. we focus on the outer region model here for simplicity. Thus, our estimate of the maximum azimuthal-mean azi- y Note that a nondimensional version of Eq. (1) is described muthal wind max* is given by in Chavas and Lin (2016, manuscript submitted to J. Atmos. y 5 : 2 : Sci.), where it is argued that r0 represents a fundamental max* 0 8(Vmax,BT 0 55Vtrans), (3) measure of storm size because of its existence as a param- eter in Eq. (1) itself. where the asterisk denotes this simple estimate of the Storm-center latitude and longitude position, local true value. We note that the conclusions presented be- storm-translation vector, maximum wind speed, and low are not sensitive to the use of this alternative 15 APRIL 2016 C H A V A S E T A L . 2927

ratio (Dee et al. 2011). The potential intensity is first calculated at all grid points globally for each month and then linearly interpolated in space and time to the storm-center position and day of year; monthly data are assumed to correspond to the middle of the month (e.g., 0000 UTC on day 16 for a 31-day month; leap year is ignored). The Coriolis parameter is taken as its value at the storm center. The scaling for storm size in rotating radiative– convective equilibrium of Khairoutdinov and Emanuel pffiffiffiffiffiffiffiffiffiffiffiffiffi (2013), given by «Lyq /f , is the product of the length pffiffiffiffiffiffiffiffiffiffi b scale Lyqb/f and the square root of a thermodynamic efficiency, « 5 (Ts 2 T0)/T0, where Ly is the latent heat of vaporization, qb is the boundary layer specific hu- midity, Ts is the sea surface temperature, and T0 is the inverse of the mean inverse temperature at which radi- ative cooling occurs in the system. The former was noted in Khairoutdinov and Emanuel (2013) to be propor- tional to the deformation radius in a moist adiabatic atmosphere, as the static stability, and to lesser degree FIG. 1. Ratio of HWind azimuthal-mean azimuthal wind ymax to 0 the tropospheric depth, increases with increasing q [cf. best-track intensity with translation effect removed Vmax,BT as b 0 afunctionofVmax,BT. Median (marker) and interquartile range (bar) Emanuel (1994), chapter 4; Khairoutdinov and Emanuel 5 0 displayed for data (dot; N 1314) binned by Vmax,BT in increments of (2013), their Fig. 4]. Here we take qb to be its value at 21 21 10 m s beginning at 15 m s ; final bin includes all data where 975 hPa from the ERA-Interim monthly mean database. 0 . 21 Vmax,BT 65 m s . Bin sample sizes are denoted at the bottom. We take T0 to be the outflow temperature calculated from the aforementioned potential intensity algorithm, intensity metric; its use enables a more precise explo- which thus defines « to be the tropical cyclone Carnot ration of physical relationships among our variables of efficiency including the effect of dissipative heating interest. For our analysis, we keep only cases for which pffiffiffiffiffiffiffiffiffiffiffiffiffi 2 (Bister and Emanuel 1998). The velocity scale «Lyq is y* $ 15 m s 1 (N 5 1794). Finally, in order to place our b max interpolated in time and space in the same manner as V , results in the context of the commonly used Saffir– p and the Coriolis parameter is taken as its value at the Simpson wind scale, we adjust the scale’s wind speed storm center. Note that the prediction given by thresholds using Eq. (3) with the translation effect fixed 2 Khairoutdinov and Emanuel (2013) for the absolute at 3 m s 1 based on the mean translation speed of 2 value of r includes a constant of proportionality given 5.38 m s 1 for the dataset shown in Fig. 1. The resulting 0 y by the product of two quantities: the ratio of the radius adjusted wind speed thresholds for max* for categories 1– 2 of maximum wind to the length scale V /f and the ratio 5 are 24.0, 32.0, 37.6, 44.8, and 53.6 m s 1, respectively. p of r0 to a characteristic dissipation length scale. Here we b. Environmental parameters fix this constant to a value of 0.5; we do not attempt to account for variation in these quantities because they To calculate the length scale V /f , the potential in- p are not readily specified from available data and because tensity V is calculated using the standard algorithm of p we seek environmental parameters that are independent Bister and Emanuel (2002) with the parameter specifi- of storm-specific characteristics, such as the radius of cations of Tang and Emanuel (2012)1 applied to the maximum wind. ERA-Interim monthly mean databases of SST, mean The relative SST is defined as the difference between sea level pressure, temperature, and water vapor mixing the local SST and the tropical mean SST, the latter taken to be the mean2 over the latitudinal band [308S, 308N] following the methodology of Lin et al. (2015) and past 1 Parameter specifications: Ck/Cd 5 0:7, pseudoadiabatic ascent, and including the effect of dissipative heating; no reduction to the z 5 10-m level is applied, as Vp is employed here as a thermody- namic parameter and is not being compared to observed near- 2 An unweighted mean is used following Lin et al. (2015); calcu- surface intensities. The outflow temperature T0 is defined as the lation using an area-weighted mean leads to a maximum single- temperature at the level of neutral buoyancy of a saturated parcel month difference in tropical mean SST of 0.099 K and has a negligible lifted from the surface at the radius of maximum wind. effect on the results. 2928 JOURNAL OF CLIMATE VOLUME 29 work on the effects of remote SST on tropical cyclone correlation between size and latitude. This effect is activity (e.g., Vecchi and Soden 2007; Camargo et al. particularly noticeable in the Atlantic basin, where the

2013). This quantity is calculated from the HadISST increase in size with latitude in r12 largely disappears in monthly SST dataset. Results are not sensitive to the r0. Also, because the primary region of eastern Pacific precise latitudinal band employed to calculate the storm activity lies at relatively low latitudes, size as tropical mean SST; very similar results are obtained for measured by r0 does not contrast as starkly with the rest [158S, 158N], [208S, 208N], and [258S, 258N]. Finally, for of the global distribution, though it clearly remains calculation of the storm central pressure deficit, the smaller than average. Moreover, medium and large environmental sea level pressure is defined as its value storm are more often found at relatively low latitudes in interpolated to the storm time and location from the western Pacific and thus appear even more extreme monthly ERA-Interim data in the same manner as the when measured by r0. These results are qualitatively above quantities. The use of monthly mean data acts to consistent with past work (Knaff et al. 2014; Chan and smooth out the contribution from the low pressure of the Chan 2015; Lee et al. 2010). More broadly, these ob- storm itself. servations highlight the complexity of examining dif- ferent metrics of storm size across latitudes given the dependence of the outer wind structure on f. 3. Size distributions Prior to further analysis, we first revisit the observed We first examine the geographic variation of storm dependence of storm size on intensity. Our focus here is size over the ocean basins, which is displayed in Fig. 2. to study r12 as a metric of the size of the outer circulation, Basins are defined as in Knutson et al. (2015) such that which would be expected to remain statistically steady each storm’s basin is assigned according to its location of as a function of storm intensity (Frank 1977). Figure 3 genesis in the best-track database. For r12, the interbasin displays observed r12 as a function of storm intensity. We variability in storm size in the tropics that has been focus on the tropics and filter the data to include only identified in past work (Chavas and Emanuel 2010; cases with latitude f # 308 (N 5 1538). There is, however, Knaff et al. 2014; Chan and Chan 2015) is clearly evi- a systematic increase in median size with intensity at rela- y # 21 dent. The western Pacific basin contains most of the tively low intensities, corresponding to max* 32 m s . largest storms in the dataset, though it contains some Linear regression of median r12 on intensity yields a best-fit 2 2 very small storms as well, indicative of both the very slope of 7.51 km (m s 1) 1 f95% confidence interval (CI) 5 : : y # 21 large storms that form from the monsoon gyre (Lander [5 5, 9 5]g over the range max* 32 m s . Meanwhile, 1994) as well as the ‘‘midget’’ typhoons that occasionally median r12 remains nearly constant with intensity form along the gyre periphery or else adjacent to the thereafter, with a linear best-fit slope that is not statis- subtropical high (Arakawa 1952; Brand 1972; Lee et al. tically significantly different from zero over the range # y # 21 5 2 : : 2010). The eastern Pacific basin contains almost exclu- 32 max* 50 m s (95% CI [ 2 3, 3 3]). Figure 3 sively smaller storms. The Atlantic basin displays pri- also displays the median QuikSCAT-estimated rain rate marily smaller storms evident at lower latitudes in the (proportional) at r12, [denoted as P*(r12)], as a function central and eastern Atlantic, a mix of sizes at in- of intensity, which is found to systematically decrease termediate latitudes, and large sizes at high latitudes, with intensity particularly at low intensities. Linear re- where storms are likely transitioning to extratropical gressions of median P*(r12) on intensity for the same systems (Hart and Evans 2001). The north Indian Ocean intensity ranges as above yield a best-fit slope for low basin is characterized by small and medium-sized intensities that is 5.5 times larger in magnitude than at storms, though this may simply reflect the confining high intensities, and the latter is not statistically signifi- coastline geometry of South Asia. In the Southern cantly different from zero at the 95% confidence level Hemisphere, both the south Indian Ocean and the South (not shown). In combination, this shift to stationarity is

Pacific do not appear to exhibit any characteristic size. likely indicative of the transition of r12 consistently into We may compare the results between our two storm the outer region, as at lower intensities r12 is increasingly size metrics in Fig. 2, which displays the same geographic likely to be found closer to rmax and thus to lie within the distribution for r0 as well. In terms of size variability, the inner region that is sensitive to intensity. Thus, for our 2 principal difference between these two quantities is the purposes, we consider 32 m s 1 to be an appropriate role of varying latitude: because the azimuthal wind in lower bound on intensity above which r12 is considered Eq. (1) decays more rapidly to zero for larger values of f to likely lie in the outer region of the storm; this value

(all else equal), the same value of r12 yields a smaller corresponds to the threshold for category 2 in our ad- value of r0 at higher latitude. Statistically, the implica- justed intensity scale. Moreover, focusing on storms of tion of this behavior is to dampen any positive at least moderate intensity is useful for risk analysis 15 APRIL 2016 C H A V A S E T A L . 2929

FIG. 2. Maps of spatial distribution of storm size: (top) r12 (km) estimated from the QSCAT-R database (1999– 2009) and (bottom) r0 (km) calculated from r12 using the outer region wind structure model [Eq. (1)]. Size values are shown on logarithmic scale. Gray lines denote associated storm tracks from best-track database. Dashed lines denote boundaries of basins employed in Fig. 4. applications, as the majority of damages are caused by typically found at higher latitudes. Figure 4 displays box- storms of at least category 2 intensity (Pielke Jr. and-whisker plots of r12 and r0 globally (N 5 578) and et al. 2008). within each ocean basin; statistics for these distributions Following from the results of Fig. 3, we focus our are also provided in Table 1. The global median values y $ 21 analysis on storms with intensities max* 32 m s (i.e., of r12 and r0 are 303 and 881 km with interquartile ranges category 2 or greater) and that are located in the tropics of [227, 400] and [741, 1054] km, respectively. Median (f # 308) to minimize potential biases introduced by storm size is largest in the western Pacific and smallest in extratropical interaction and rapidly translating storms the eastern Pacific, as has been found in past work 2930 JOURNAL OF CLIMATE VOLUME 29

FIG. 3. Dependence of r12 on the estimated maximum azimuthal- FIG. 4. Box-and-whisker plot of the distributions of r12 (black) and y mean azimuthal wind *max. Blue lines denote median (solid) and r0 (blue) in the tropics globally and by basin. Size values are shown interquartile range (dashed); red lines denote linear regression on on logarithmic scale. Dataset filtered to include cases for which y # 21 y 21 y $ 21 f # 8 median for max* 32 m s and max* 2 [32, 50] m s . Green lines max* 32 m s and latitude 30 ; sample sizes are listed at the denote median value of quantity proportional to rain rate esti- bottom. The box plot displays median (red line) and 95% confidence mated by QuikSCAT at r12 (maximum value rescaled to 100). Data interval on median (dark red lines), interquartile range [q1, q3] filtered for cases with latitude f # 308 (N 5 1538). Median values (black and blue fill), whiskers (black dashed lines), and outliers fgray 21 calculated in bins of 2 m s . (Two values of r12 exceeding 1000 km crosses; outside the range [q1 2 1:5(q3 2 q1), q3 1 1:5(q3 2 q1)] cal- are not shown.) culated from the logarithm of radiusg. Corresponding statistics and basin expansions are displayed in Table 1. (Chavas and Emanuel 2010; Knaff et al. 2014; Chan and Chan 2015); median r0 in the South Pacific is rejected for the global distribution (p 5 0.34) nor any comparable to the western Pacific. The western Pacific of the basin distributions (p 2 [0:36, 0:93]) at the 95% also exhibits the largest variance in size of any basin, confidence level. We note that the global median value though the largest coefficients of variation are found in of r12 is approximately 50% larger than that found in the eastern and western Pacific basins despite their Chavas and Emanuel (2010),likelybecauseofthemore starkly contrasting mean sizes (Table 1). The global rigorous filtering applied here to avoid both the low distributions of both r12 and r0 are very closely log- bias imposed by weaker storms and cases with higher normal, as was found in Chavas and Emanuel (2010); uncertainty because of poor data coverage j.The p values for the Kolmogorov–Smirnov (KS) goodness-of- values of r0 are dramatically larger (;100%) than the fit test (Massey Jr. 1951) between the standardized prior study given the larger values of r12 and also the use logarithm of the data and a standard normal distribu- of a proper estimate of the radiative-subsidence rate in tion indicate that the lognormal distribution cannot be Eq. (1), as discussed above.

TABLE 1. Statistics, including median, first and third quartiles, mean m, standard deviation s, and coefficient of variation, of distributions of r12 and r0, corresponding to Fig. 4.

r12 (km) r0 (km) Basin N Median 25% 75% msCV Median 25% 75% msCV Global 578 302.8 227.2 399.9 327.5 141.3 0.43 881.0 740.7 1054.4 909.4 248.5 0.27 North Atlantic (AL) 89 266.5 208.7 343.1 274.1 85.0 0.31 800.0 707.4 888.6 795.9 146.0 0.18 Eastern Pacific (EP) 103 214.7 160.6 279.2 235.0 101.2 0.43 755.5 613.5 870.3 759.3 189.5 0.25 Western Pacific (WP) 303 351.8 274.9 472.2 381.8 150.2 0.39 957.6 804.5 1161.0 993.5 262.7 0.26 North Indian Ocean(NI) 6 287.0 271.2 312.2 283.0 55.8 0.20 913.5 857.0 964.7 880.5 144.8 0.16 South Indian Ocean(SI) 36 279.9 198.5 356.1 275.4 92.8 0.34 893.6 724.9 1075.7 871.9 202.5 0.23 South Pacific (SP) 41 315.1 262.1 381.3 327.9 120.3 0.37 964.9 858.8 1060.5 948.1 222.1 0.23 15 APRIL 2016 C H A V A S E T A L . 2931

2000 For comparison purposes, Knaff et al. (2014) esti- (a) 1800 mated the wind speed at r 5 500 km at 850 hPa, denoted 1600 V500, from which they extrapolate outward using a simple linear decay model with fixed decay rate to esti- 1400 21 mate the radius of 5-kt winds r5kt (1 kt ’ 0:51 m s ). 1200 Absent filters for intensity or latitude, which provides 1000 the closest comparison with their methodology, the 2 median V500 is 6.3 m s 1 with an interquartile range of 800 2 radius [km] [4:4, 8:5]ms 1 (N 5 3939). These values are reasonably 600 close to their estimates of 6.61 and [5:09, 8:08] m s21, 400 respectively; their higher values may reflect the role of 200 friction in reducing wind speeds between 850 hPa and 0 the surface. Furthermore, we may predict r5kt, rather 50 60 70 80 90 100 than r0, from r12 using Eq. (1) which yields a median V [m/s] p value of 675 km with an interquartile range of 2000 5 (b) [497, 880] km absent filters for intensity or latitude (N 1800 2200). These values are quite a bit smaller than their 1600 respective estimates of 933 and [739, 1120] km (Knaff 1400 et al. 2014). This result suggests that their linear decay model underestimates the rate of wind field decay pre- 1200 dicted by our physical model in the outer region of the 1000 circulation, though the difference in analysis level likely 800 contributes to this discrepancy as well. radius [km] 600 400 4. Relationship to relevant environmental 200 parameters 0 Ideally we seek a physical explanation of the large 234567 variability in storm size evident in Fig. 4. Thus, as a first f [*10-5 s-1] 4000 step toward fully understanding variability in size, we (c) test three recently identified parameters with credible 3500 Cat1 (331) physical relationships with tropical cyclone size: Vp/f , Cat2 (183) the rotating RCE scaling of Khairoutdinov and Emanuel 3000 Cat3 (194) Cat4/5 (201) (2013), and the relative SST. We again focus our analysis Cat2-5 (578) y $ 21 2500 on cases from our database with max* 32 m s and f # 308, as was done in Fig. 4. 2000 a. Tropical cyclone length scale 1500 radius [km] We begin by testing the theoretical tropical cyclone 1000 length scale Vp/f . Figure 5 displays the direct relationship 500 between outer size and the length scale Vp/f as well as its constituent quantities. Results are stratified by intensity 0 0 1000 2000 3000 4000 according to Saffir–Simpson category adjusted as de- V / f [km] scribed in section 2. First, there is an overall positive re- p lationship between mean size and Vp (Fig. 5a), except at FIG. 5. Relationships between storm size and (a) V ,(b)f,and p the highest values of Vp. Note that a weak positive cor- (c) Vp/f . The bottom curves denote mean r12 andthetopcurvesde- relation between size and Vp was identified in Chavas and note mean r0; Circles and crosses denote all data for r12 and r0,re- spectively. The dataset is filtered to include cases for which f # 308. Emanuel (2010), yet this relationship was dismissed arbi- Results stratified by intensity according to the adjusted Saffir– trarily as being too small. Here it is evident that the mean Simpson category (see text for details), with sample sizes listed in the size varies systematically with Vp, even while there re- y $ 21 legend; the black curve denotes mean for all data with max* 32 m s mains substantial residual variance about that mean value. (i.e., categories 2–5). Error bars represent one standard error. For re- Though a positive relationship between size and lationship with Vp/f , a 1:1 line is shown (black solid). Vp exists, no comparably strong relationship emerges 2932 JOURNAL OF CLIMATE VOLUME 29 between size and f (Fig 5b). Indeed, r12 exhibits a slight increase with increasing f, consistent with past studies on size and latitude (Knaff et al. 2007; Mueller et al. 2006;

Kossin et al. 2007). On the other hand, r0 exhibits either no dependence or a slight decrease with increasing f for relatively small f, again an indication of how conclusions regarding the dependence of size on latitude may de- pend on the chosen metric. We note that there is perhaps weak evidence of a local maximum in mean r0 at in- termediate values of f, as has been found in an idealized model (Smith et al. 2011) and in observations (Chan and Chan 2015), in the vicinity of 5:5 3 1025 s21. Finally, Fig. 5c displays the direct relationship be- tween size and Vp/f . The relationship is weak, with a magnitude that is much smaller than that predicted by this length scale. However, the largest values of r0 do appear to scale with Vp/f for values of Vp/f less than approximately 2200 km, suggesting that this length scale may indeed serve as a scaling for the upper bound on storm size, as predicted by theory in Emanuel (1986). FIG.6.AsinFig. 5c, but for the relationship between storm size We note that the findings of idealized equilibrium and the rotating RCE scaling of Khairoutdinov and Emanuel modeling studies (Khairoutdinov and Emanuel 2013; (2013), rescaled by a constant of proportionality of 0.5 (see section Zhou et al. 2014; Chavas and Emanuel 2014; Reed and 2 for details). Chavas 2015) suggest that outer storm size at equilib- rium does approach this theoretical upper bound length mean SST (Ramsay and Sobel 2011). Results are quanti- scale. However, Fig. 5c indicates that the vast majority tatively similar when keeping « fixed as well as when re- of storms in nature exist at sizes well below this upper placing qb with the saturation mixing ratio at the sea bound; further discussion is provided in Section 6. Re- surface temperature, as was used in Khairoutdinov and sults are consistent across intensity bins and are quan- Emanuel (2013). titatively similar for the median in lieu of the mean. Note As noted above, the scaling of Khairoutdinov and that we may also account for the secondary dependence Emanuel (2013) applies to a world in which surface of r0 with the nondimensional Ekman suction rate, frictional dissipation within tropical cyclones dominates Wcool/(CdVp), with a scaling exponent of 20:33 found in entropy production in the system. Given that tropical Chavas and Emanuel (2014). Note that Cd in the outer cyclones typically do not cover a substantial fraction of region is not expected to vary from storm to storm, the tropical oceans, it seems likely that the underlying consistent with its specification in Eq. (1), while Wcool is assumptions of this theory may not apply in nature; in- expected to vary minimally in the tropics in accordance deed, the real world may lie closer to the unaggregated with weak temperature gradient theory (Sobel and RCE limit in which entropy production is dominated by Bretherton 2000). Thus, the result of this secondary irreversible phase changes and diffusion of water vapor 0:33 scaling is to multiply Vp/f by an additional factor of Vp , instead of frictional dissipation (Pauluis and Held 2002) which has little effect on the above results (not shown); and tropical cyclones impose only a weak influence on this finding is not sensitive to the value of the scaling the long-term mean state. This result is also in line with exponent. the finding of Chavas and Emanuel (2014) that equi- librium storm size in an axisymmetric model does not b. Rotating RCE scaling of Khairoutdinov and scale with the deformation radius. Moreover, as noted in Emanuel (2013) section 2, the complete theoretical prediction for r0 also WenexttesttherotatingRCEscalingofKhairoutdinov depends on the ratio of the radius of maximum wind to pffiffiffiffiffiffiffiffiffiffiffiffiffi and Emanuel (2013),givenby «Lyqb/f . The relationship the length scale Vp/f and the ratio of the characteristic between storm size and this scaling is shown in Fig. 6.The dissipation length scale to r0, both of which are assumed dependence is quite weak, similar to that of Fig. 5c.Thisis to be constant in Khairoutdinov and Emanuel (2013). not surprising, given that the numerator depends on the Though these assumptions may be justified in the rela- boundary layer specific humidity, which is closely tied to tive homogeneity of three-dimensional rotating RCE, in the SST and hence Vp in the absence of large changes in which potential intensity is spatially and temporally 15 APRIL 2016 C H A V A S E T A L . 2933

the relationship found for outer wind field size as noted above. Dynamically, one expects that the wind and precipitation fields are coupled (e.g., Shoemaker 1989), and recent idealized modeling work has demonstrated 21 that r12 and the radius of 1.0 mm h rainfall are indeed highly correlated (Reed and Chavas 2015). Thus, the logical next question is the following: Does wind field size also exhibit a dependence on relative SST? The relationship between storm size and relative SST is shown in Fig. 7. Indeed, a systematic dependence

on relative SST does exist for both mean r12 and mean r0; statistics are provided in Table 2. Furthermore, this dependence is consistent across intensity bins. Linear regression of mean size within each bin against mean relative SST for category 2–5 storms (i.e., the black 2 curve in Fig. 7) gives an increase of 47.7 and 97.0 km K 1 FIG.7.AsinFig. 5a, but for the relationships between storm size for r12 and r0, respectively; a similar trend is evident for and relative SST. Mean values calculated in integer bins of 1 K, category 1 storms as well. Trend magnitudes are not with the final bin including all data above 3 K. Corresponding sensitive to the latitude band used to define the tropical- statistics are displayed in Table 2. Two category 1 cases with rel- 8 8 8 8 8 2 mean SST: for [25 S, 25 N], [20 S, 20 N], and [15 S, ative SST values less than 2 K are not shown. 21 158N], the trend for r12 is 47.2, 47.1, and 43.7 km K ; and 21 for r0 it is 92.5, 91.9, and 84.3 km K , respectively, uniform and storms are free to attain their potential where relative SST bins have been successively shifted intensity uninhibited, they are likely to be less valid for downward by 0.5 K per 58 increment to account for the the real world, given its large heterogeneity. corresponding increase in mean SST. The trend for r12 is comparable to, though a bit larger than, the dependence c. Relative SST 2 of the radius of 0.5 mm h 1 rainfall on relative SST Based on large samples of storm size determined from identified in Lin et al. (2015) of approximately 2 rainfall fields, Lin et al. (2015) noted a strong de- 35 km K 1 from both TRMM data and output from an pendence of mean storm size on relative SST (i.e., SST idealized numerical model simulation. This dependence within the local environment of the storm minus the of size on relative SST likely explains the dependence on tropical mean SST). Notably, relative SST correlates its correlate Vp, shown in Fig. 5a. strongly with Vp and is often used as its surrogate (e.g., Additionally, note from Table 2 that the variance in- Vecchi and Soden 2007). As noted in section 1, the creases in conjunction with the mean such that the co- physical linkage of storm rainfall size with relative SST is efficient of variation (CV) remains relatively stable likely manifest through its effect on midtropospheric across the positive relative SST bins, particularly for r0 moisture. Moreover, Lin et al. (2015) found that the where the CV lies within the narrow range [0.25, 0.29]. precipitation-based size metric, as measured by, for ex- In the negative relative SST bins, the smaller CVs are ample, the radius where precipitation rate equals likely due to the small sample sizes: inclusion of the 2 0.5 mm h 1, is nearly insensitive to intensity, similar to category 1 cases for the two lowest relative SST bins

TABLE 2. Statistics of r12 and r0 binned by relative SST, corresponding to Fig. 7 for categories 2–5. Relative SST values (relSST) are the mean value within the bin.

r12 (km) r0 (km) relSST (K) N Median 25% 75% msCV Median 25% 75% msCV Global 578 302.8 227.2 399.9 327.5 141.3 0.43 881.0 740.7 1054.4 909.4 248.5 0.27 21.17 3 153.2 142.3 176.1 161.2 34.5 0.21 557.9 537.8 582.5 560.9 44.8 0.08 20.34 8 177.0 129.6 222.3 176.9 49.8 0.28 636.3 496.0 686.3 608.6 104.9 0.17 0.65 38 238.4 180.4 318.8 267.3 128.4 0.48 750.2 674.0 878.3 778.8 201.7 0.26 1.60 129 280.9 210.3 344.1 307.3 154.5 0.50 847.3 725.2 973.8 880.9 253.5 0.29 2.54 239 305.3 245.7 396.7 327.8 124.0 0.38 897.7 763.4 1072.1 912.9 224.8 0.25 3.34 161 354.1 266.9 463.9 368.3 147.1 0.40 969.2 808.8 1140.1 979.3 264.0 0.27 2934 JOURNAL OF CLIMATE VOLUME 29

y y FIG. 8. Joint dependence of central pressure deficit (color, hPa) on (left) max* and r12 and on (right) max* and r0. Data y $ 21 f # 8 5 are from the Atlantic basin only for cases with max* 32 m s and 30 (N 89).

2 2 ( 1.17 and 0.34 K) gives CVs for r12 of 0.43 and 0.44, where pmin and penv are the sea level pressures at the respectively, and CVs for r0 of 0.28 and 0.28, re- storm center and in the ambient environment, re- spectively, all of which align closely with their ranges for spectively, r is air density, r is radius, and y is the azi- positive relative SSTs. Last, KS tests applied to the size muthal component of the wind. Thus, dp depends on f distributions conditioned on relative SST for the bins in and the complete radial wind profile, which may be in- Table 2 indicate that the lognormal distribution cannot terpreted as a wind structure defined relative to the be rejected for any relative SST bin for both r12 and r0, maximum wind speed ymax and some absolute length with all p values exceeding 0.23. Results are similar scale [i.e., size (Chavas et al. 2015)]. Indeed, recent ob- when employing the full best-track wind speed Vmax,BT servational examples of anomalously large storms, such as the intensity metric, as was used in Lin et al. (2015) for as Hurricane Rita (2005), illustrate how large storms the rainfall field size, as well as when relative SST is can have very low central pressures despite relatively calculated using data from the ERA-Interim in lieu of modest maximum wind speeds (Stewart 2014). More HadISST. Results are also similar when including data generally, past observational studies have identified a for all latitudes as well as when calculating the median in dependence of central pressure or pressure deficit on lieu of the mean. A systematic investigation of the ob- various metrics of outer storm size (Knaff and Zehr served relationship between the storm rainfall field and 2007; Courtney and Knaff 2009; Shoemaker 1989; wind field will be explored in future work to corroborate Cocks and Gray 2002), as well as a role for variation in previous work based on a small subset of cases the radius of maximum wind in the context of second- (Matyas 2010). ary eyewall formation (Kieu et al. 2010). Moreover, in an idealized setting, Reed and Chavas (2015) demon- 5. Wind–pressure–size relationship strated in rotating RCE aquaplanet simulations at constant f that the combination of storm intensity and

One additional relationship of interest, particularly in outer size, as measured by r12, is sufficient to explain operations and risk analysis, is that between storm most of the variance in the central pressure for a large maximum wind speed and minimum central pressure: dataset of storms. that is, the ‘‘wind–pressure’’ relationship. As discussed Thus, here we perform a simple analysis using best- in Knaff and Zehr (2007), the storm central pressure track data to test empirically whether the effect of size deficit dp may be estimated from gradient wind bal- on central pressure deficit is discernible in our new-size ance as dataset. Figure 8 displays the joint dependence of min- ð   imum central pressure deficit on intensity and size. We r0 y2 5 2 5 r 1 y focus exclusively on data from the Atlantic basin, where dp penv pmin f dr, (4) 0 r independent estimates of wind intensity and minimum 15 APRIL 2016 C H A V A S E T A L . 2935 central pressure are most prevalent, with data filtered vanishing wind r0 estimated from r12 using an improved y $ 21 f # 8 5 for cases with max* 32 m s and 30 (N 89). analytical outer wind model. We then analyze geo- Though the dominant mode of variance in central graphic variation in size and calculate statistical distri- pressure is associated with variation in intensity as butions of each size metric globally and for each basin. expected, a dependence on size (r12 or r0) is also evident. Next, we explore the dependence of storm size on en- Indeed, for the data shown in Fig. 8, a standardized vironmental parameters as a first step toward un- y linear regression of pmin on the covariate max* explains derstanding size variability. We focus specifically on 74.5% of the variance (covariate coefficient 5 0.86; one three recently identified parameters that possess credi- standard error 560:054). Adding r12 as a linear co- ble physical relationships with tropical cyclone outer variate increases the explained variance to 81.4%. Re- size: 1) the theoretical tropical cyclone length scale gression coefficients are 0.81 (60.047) and 0.27 (60.047) given by the ratio of the potential intensity to the y for max* and r12, respectively, suggesting that the con- Coriolis parameter, which has been shown to govern tribution to central pressure deficit from intensity here is storm size in radiative–convective equilibrium; 2) the threefold larger than from size. Results are similar for r0, three-dimensional rotating radiative–convective equi- for which the joint model explains 79.3% of the variance librium scaling of Khairoutdinov and Emanuel (2013); with regression coefficients of 0.80 and 0.23, indicating a and 3) the sea surface temperature relative to the trop- 3.5-fold difference in the relative contributions of the ical mean, which was shown in a recent study to govern two covariates. Notably, the explained variance is a bit the size of the tropical cyclone precipitation area and higher when using the best-track maximum wind speed whose connection with storm size may be indirect via its y Vmax,BT in lieu of max* , for which a multiple linear re- modulation of environmental moisture. Finally, we test gression model explains 86.9% and 90.8% of the vari- the relationship between storm central-pressure deficit, ance without and with r12, respectively; for the latter intensity, and size identified in past research. case, the regression coefficient is 20.89 (60.027) for We find that both metrics of outer storm size are ap- Vmax,BT and 20.20 (60.027) for r12. This latter result proximately lognormally distributed, with global me- likely reflects the existence of cases within the dataset dian values of 303 km for r12 and 881 km for r0 and basin that lack independent estimates of Vmax,BT and pmin, for median values ranging from 215 to 352 km for r12 and which a standard wind–pressure relationship would have from 756 to 965 km for r0. The largest and smallest been applied where one intensity metric is directly values are found in the western Pacific and eastern Pa- specified by the other. cific Ocean basins, respectively, corroborating the re- Despite the caveats associated with the best-track sults of recent studies; the South Pacific basin was found dataset noted above, the dependence of minimum cen- to have a median r0 comparable to its value in the tral pressure on storm size is discernible from the com- western Pacific basin. Regarding environmental de- bination of the QuikSCAT and best-track datasets in pendence, we find that variation in storm size is not well line with prior work on this subject; indeed, cases for explained by the theoretical length scale Vp/f , though which standard wind–pressure relationships were ap- the absolute upper bound on size may be, in line with plied would introduce a low bias in our empirical esti- existing tropical cyclone theory; results are similar when mate of the dependence of dp on size. These results accounting for a secondary dependence on the non- motivate a deeper theoretical exploration of the wind– dimensional Ekman suction rate in the outer circulation. pressure–size relationship and comparison with in situ Size variation is also found to be poorly explained by the observational databases of independent wind and pres- scaling for a three-dimensional rotating RCE world, sure measurements, which is the subject of future work. likely an indication that its underlying assumptions may not be applicable in the real world. Closer analysis of the relationship V /f reveals that mean storm size varies 6. Discussion and conclusions p systematically with Vp but not f. This result suggests This work revisits the statistics of observed tropical that a correlate with Vp may be relevant instead. Indeed, cyclone size explored in Chavas and Emanuel (2010) mean storm size is shown to vary systematically with the using a recently updated version of the QuikSCAT relative SST, which typically covaries closely with Vp. satellite scatterometer-based ocean wind vector data- The dependence of r12 on relative SST is approximately 2 base and places the analysis in the context of recent 48 km K 1, which is comparable to, though a bit larger 2 advances in our theoretical understanding of storm size than, the dependence of the radius of 0.5 mm h 1 rainfall and structure. We first create new databases of the ra- identified in Lin et al. (2015). Meanwhile, the coefficient 2 dius of 12 m s 1 winds estimated from observations from of variation is found to remain nearly stable with relative the QSCAT-R database and of the outer radius of SST. These results, in combination with the experimental 2936 JOURNAL OF CLIMATE VOLUME 29 evidence of Lin et al. (2015), suggest that relative SST is scale; elucidating the precise physical length scale at indeed a more relevant parameter. We emphasize that play is the subject of future work. this does not necessarily imply that Vp/f is only relevant Additionally, we note that here we have sought ex- to the upper bound of the storm size distribution, just as planations for variance in storm size based strictly on Vp itself is highly relevant to the dynamics of the tropical local environmental parameters. However, the inability cyclone at all intensities (Tang and Emanuel 2010)de- of such parameters to explain large amounts of residual spite its strict definition as the upper bound on intensity. variance may also indicate an important role for storm- Importantly, for risk applications, the magnitude of specific parameters. First and foremost, the potential the variation of size with relative SST found here is existence of long time scales to equilibrium noted above substantial given the close relationship between r0 and may imply a long memory for size for a given storm, rm at fixed intensity and latitude (Chavas and Lin 2016, perhaps extending as far back as genesis. Indeed, the manuscript submitted to J. Atmos. Sci.). For example, length scale of the precursor disturbance is not explored we may test the sensitivity of rm to the median variations here, though it has been shown to correlate strongly with of r0 found here using the structural model of Chavas storm size at maturity in idealized modeling studies and Lin (2016, manuscript submitted to J. Atmos. Sci.): (Rotunno and Emanuel 1987) as well as in observations 21 taking ymax 5 40 m s , f 5 208N, and r0 5 600 km, a 50% in the western North Pacific basin (Cocks and Gray increase in r0, which corresponds to the approximate 2002; Lee et al. 2010). Second, past work has noted a range in median size between relative SST values of dependence of size simply on time since genesis (Kossin 0 and 3 K, translates to a 73% increase in rm.At et al. 2007), which would suggest some internal physical landfall, such an increase in the radius of maximum process to the tropical cyclone that encourages expan- wind would likely greatly enhance storm-related sion with time; the underlying physics of such behavior, hazards, particularly . Moreover, we however, have yet to be elucidated. Finally, an exami- demonstrate empirically that the central pressure nation of the direct physical relationship between storm deficit, which is often required for, for example, size and environmental moisture would be fruitful, storm surge modeling, is dependent on storm size, in though it is complicated by the strong time-dependent line with prior work. influence the overturning circulation of the storm itself Our results beg the following question: Why does Vp/f imposes on the local moisture field. Such questions not better explain storm size variation in nature? One warrant further research. existing hypothesis is that the equilibration time scale We have focused principally on storm size within the for storm size is very slow [O(10) days], and thus per- tropics, purposefully avoiding the role of extratropical turbations from this length scale decay too slowly rela- transition. The analyses of Fig. 2 appear to suggest that tive to the typical storm life cycle time scale in nature at middle and high latitudes, where extratropical tran- (Chavas and Emanuel 2014). Here we posit another sition is common, r12 expands with latitude in line with hypothesis that follows from the proposed physical prior research, yet outer storm size r0 may not, which is mechanism by which relative SST affects size: Vp/f is the perhaps a reflection of an expansion of the wind field relevant length scale in radiative–convective equilib- relative to a fixed outer radius, as was noted in Chavas rium, which is technically valid only globally in the and Lin (2016, manuscript submitted to J. Atmos. Sci.). tropics, yet in reality storms do not feel the global tropics However, our dataset is limited in this regard, particu- but rather feel their local environment. This local envi- larly given the characteristically low intensities of storms ronment is governed not by radiative–convective equi- at these high latitudes; a more detailed and/or idealized librium but by the dynamics of the weak temperature study is needed that focuses on storm size variability gradient (WTG) approximation (Sobel and Bretherton specifically in the context of extratropical transition. 2000), which dictates that the moist ascending branch of Moreover, it is worth noting that the resolution limita- thermally direct overturning circulations is found in re- tions of this observational dataset may inhibit us from gions of locally high SSTs [or, more precisely, boundary accounting for extraordinarily small storms in nature; layer entropy (Emanuel et al. 1994)]. This mean ascent indeed, no theoretical lower bound on storm size leads to a substantially moister local environment than currently exists. would be found in RCE. Hence, the RCE length scale Despite such limitations, the size distributions pre- may not be appropriate, as alternative length scales as- sented here are useful for risk analysis applications and sociated with the local WTG environment may act to as an observational benchmark for comparison with al- drive storm size away from the RCE solution. In this ternative observational size databases as well as output way, then, relative SST is representative of a distinct set from both idealized modeling studies (e.g., Knutson of non-RCE processes, though it is not itself a length et al. 2015) and operational weather forecasts. The 15 APRIL 2016 C H A V A S E T A L . 2937 analyses on the environmental parameters provide a ——, N. Lin, and K. Emanuel, 2015: A model for the complete basis for developing environment- and climate-dependent radial structure of the tropical cyclone wind field. Part I: models for storm size in the future. Finally, we note that Comparison with observed structure. J. Atmos. Sci., 72, 3647– 3662, doi:10.1175/JAS-D-15-0014.1. the outer radius translates to the surface area occupied Cocks, S. B., and W. M. Gray, 2002: Variability of the outer wind by a storm, and thus its dynamics may impose limits on profiles of western North Pacific typhoons: Classifications the number of storms that can form in a limited area at a and techniques for analysis and forecasting. Mon. Wea. given time. Thus, the results presented here may be rel- Rev., 130, 1989–2005, doi:10.1175/1520-0493(2002)130,1989: . evant for the statistics of tropical cyclogenesis in nature, VOTOWP 2.0.CO;2. Courtney, J., and J. A. Knaff, 2009: Adapting the Knaff and Zehr particularly in localized regions where conditions favor- wind–pressure relationship for operational use in Tropical able for genesis are spatially limited or where there is a Cyclone Warning Centres. Aust. Meteor. Oceanogr. J., 58, tendency for multiple storms to form in close proximity to 167–179. one another. Davis, C. A., 2015: The formation of moist vortices and tropical cyclones in idealized simulations. J. Atmos. Sci., 72, 3499– 3516, doi:10.1175/JAS-D-15-0027.1. Acknowledgments. Support for this work was pro- Dee, D., and Coauthors, 2011: The ERA-Interim reanalysis: Con- vided for DC and NL by the National Science Founda- figuration and performance of the data assimilation system. tion under Award AGS-1331362 and EAR-1520683 and Quart. J. Roy. Meteor. Soc., 137, 553–597, doi:10.1002/qj.828. for WD and YL by the Ministry of Science and Tech- Donelan, M., B. Haus, N. Reul, W. Plant, M. Stiassnie, H. Graber, nology of China Award 2014CB441303. All data in this O. Brown, and E. Saltzman, 2004: On the limiting aero- dynamic roughness of the ocean in very strong winds. Geo- analysis are publicly available upon request from the phys. Res. Lett., 31, L18306, doi:10.1029/2004GL019460. corresponding author. Emanuel, K. A., 1986: An air–sea interaction theory for trop- ical cyclones. Part I: Steady-state maintenance. J. Atmos. REFERENCES Sci., 43, 585–605, doi:10.1175/1520-0469(1986)043,0585: AASITF.2.0.CO;2. Ahmad, K. A., W. L. Jones, T. Kasparis, S. W. Vergara, I. S. ——, 1987: The dependence of hurricane intensity on climate. Adams, and J. D. Park, 2005: Oceanic rain rate estimates from Nature, 326, 483–485, doi:10.1038/326483a0. the QuikSCAT radiometer: A global precipitation mission ——, 1994: Atmospheric . Oxford University Press, pathfinder. J. Geophys. Res., 110, D11101, doi:10.1029/ 580 pp. 2004JD005560. ——, 1995a: The behavior of a simple hurricane model using a Arakawa, H., 1952: Mame-Taifu or midget typhoon. Geophys. convective scheme based on subcloud-layer entropy equi- Mag., 23, 463–474. librium. J. Atmos. Sci., 52, 3960–3968, doi:10.1175/ Bister, M., and K. A. Emanuel, 1998: Dissipative heating and hur- 1520-0469(1995)052,3960:TBOASH.2.0.CO;2. ricane intensity. Meteor.Atmos.Phys., 65, 233–240, doi:10.1007/ ——, 1995b: Sensitivity of tropical cyclones to surface exchange BF01030791. coefficients and a revised steady-state model incorporating ——, and ——, 2002: Low frequency variability of tropical cyclone dynamics. J. Atmos. Sci., 52, 3969–3976, doi:10.1175/ potential intensity. 2. Climatology for 1982–1995. J. Geophys. 1520-0469(1995)052,3969:SOTCTS.2.0.CO;2. Res., 107, 4621, doi:10.1029/2001JD000780. ——, 2004: Tropical cyclone energetics and structure. Atmo- Brand, S., 1972: Very large and very small typhoons of the western spheric Turbulence and Mesoscale Meteorology,E.Fedorovich, North Pacific Ocean. J. Meteor. Soc. Japan, 50, 332–341. R. Rotunno, and B. Stevens, Eds., Cambridge University Press, Camargo, S. J., M. Ting, and Y. Kushnir, 2013: Influence of local 165–192. and remote SST on North Atlantic tropical cyclone potential ——, and R. Rotunno, 2011: Self-stratification of tropical cyclone intensity. Climate Dyn., 40, 1515–1529, doi:10.1007/ outflow. Part I: Implications for storm structure. J. Atmos. Sci., s00382-012-1536-4. 68, 2236–2249, doi:10.1175/JAS-D-10-05024.1. Chan, J. C., 2005: The physics of tropical cyclone motion. Annu. Rev. ——, J. David Neelin, and C. S. Bretherton, 1994: On large-scale Fluid Mech., 37, 99–128, doi:10.1146/annurev.fluid.37.061903.175702. circulations in convecting atmospheres. Quart. J. Roy. Meteor. Chan, K. T., and J. C. Chan, 2012: Size and strength of tropical Soc., 120, 1111–1143, doi:10.1002/qj.49712051902. cyclones as inferred from QuikSCAT data. Mon. Wea. Rev., Frank, W. M., 1977: The structure and energetics of the tropical 140, 811–824, doi:10.1175/MWR-D-10-05062.1. cyclone. I: Storm structure. Mon. Wea. Rev., 105, 1119–1135, ——, and ——, 2015: Global climatology of tropical cyclone size as doi:10.1175/1520-0493(1977)105,1119:TSAEOT.2.0.CO;2. inferred from QuikSCAT data. Int. J. Climatol., 35, 4843–4848, Hart, R. E., and J. L. Evans, 2001: A climatology of the extratropical doi:10.1002/joc.4307. transition of Atlantic tropical cyclones. J. Climate, 14, 546–564, Chavas, D. R., and K. A. Emanuel, 2010: A QuikSCAT climatol- doi:10.1175/1520-0442(2001)014,0546:ACOTET.2.0.CO;2. ogy of tropical cyclone size. Geophys. Res. Lett., 37, L18816, Hill, K. A., and G. M. Lackmann, 2009: Influence of environmental doi:10.1029/2010GL044558. humidity on tropical cyclone size. Mon. Wea. Rev., 137, 3294– ——, and ——, 2014: Equilibrium tropical cyclone size in an ide- 3315, doi:10.1175/2009MWR2679.1. alized state of axisymmetric radiative–convective equilibrium. Irish, J. L., and D. T. Resio, 2010: A hydrodynamics-based surge J. Atmos. Sci., 71, 1663–1680, doi:10.1175/JAS-D-13-0155.1. scale for hurricanes. Ocean Eng., 37, 69–81, doi:10.1016/ ——, and J. Vigh, 2014: QSCAT-R: The QuikSCAT tropical cy- j.oceaneng.2009.07.012. clone radial structure dataset. NCAR Tech. Note TN- Khairoutdinov, M., and K. Emanuel, 2013: Rotating radiative- 5131STR, 27 pp. convective equilibrium simulated by a cloud-resolving model. 2938 JOURNAL OF CLIMATE VOLUME 29

J. Adv. Model. Earth Syst., 5, 816–825, doi:10.1002/ Matyas, C. J., 2010: Associations between the size of hurricane rain 2013MS000253. fields at landfall and their surrounding environments. Meteor. Kieu,C.Q.,H.Chen,andD.-L.Zhang,2010:Anexaminationof Atmos. Phys., 106, 135–148, doi:10.1007/s00703-009-0056-1. the pressure–wind relationship for intense tropical Merrill, R. T., 1984: A comparison of large and small tropical cyclones. Wea. Forecasting, 25, 895–907, doi:10.1175/ cyclones. Mon. Wea. Rev., 112, 1408–1418, doi:10.1175/ 2010WAF2222344.1. 1520-0493(1984)112,1408:ACOLAS.2.0.CO;2. Kim, H.-S., G. A. Vecchi, T. R. Knutson, W. G. Anderson, T. L. Mueller, K. J., M. DeMaria, J. Knaff, J. P. Kossin, and T. H. Vonder Delworth, A. Rosati, F. Zeng, and M. Zhao, 2014: Tropical Haar, 2006: Objective estimation of tropical cyclone wind

cyclone simulation and response to CO2 doubling in the GFDL structure from infrared satellite data. Wea. Forecasting, 21, CM2.5 high-resolution coupled climate model. J. Climate, 27, 990–1005, doi:10.1175/WAF955.1. 8034–8054, doi:10.1175/JCLI-D-13-00475.1. Pauluis, O., and I. M. Held, 2002: Entropy budget of an atmosphere Knaff, J. A., and R. M. Zehr, 2007: Reexamination of tropical cy- in radiative–convective equilibrium. Part I: Maximum work clone wind–pressure relationships. Wea. Forecasting, 22, 71– and frictional dissipation. J. Atmos. Sci., 59, 125–139, doi:10.1175/ 88, doi:10.1175/WAF965.1. 1520-0469(2002)059,0125:EBOAAI.2.0.CO;2. ——, C. R. Sampson, M. DeMaria, T. P. Marchok, J. M. Gross, and Pielke, R. A., Jr., J. Gratz, C. W. Landsea, D. Collins, M. A. C. J. McAdie, 2007: Statistical tropical cyclone wind radii Saunders, and R. Musulin, 2008: Normalized hurricane dam- prediction using climatology and persistence. Wea. Fore- ages in the United States: 1900–2005. Nat. Hazards Rev., 9, 29– casting, 22, 781–791, doi:10.1175/WAF1026.1. 42, doi:10.1061/(ASCE)1527-6988(2008)9:1(29). ——, S. P. Longmore, and D. A. Molenar, 2014: An objective Powell, M. D., S. H. Houston, L. R. Amat, and N. Morisseau- satellite-based tropical cyclone size climatology. J. Climate, 27, Leroy, 1998: The HRD real-time hurricane wind analysis 455–476, doi:10.1175/JCLI-D-13-00096.1; Corrigendum, 28, system. J. Wind Eng. Ind. Aerodyn., 77–78, 53–64, doi:10.1016/ 8648–8651, doi:10.1175/JCLI-D-15-0610.1. S0167-6105(98)00131-7. Knutson, T., J. Sirutis, M. Zhao, R. Tuleya, M. Bender, G. Vecchi, Ramsay, H. A., and A. H. Sobel, 2011: Effects of relative and ab- G. Villarini, and D. Chavas, 2015: Global projections of in- solute sea surface temperature on tropical cyclone potential tense tropical cyclone activity for the late twenty-first century intensity using a single-column model. J. Climate, 24, 183–193, from dynamical downscaling of CMIP5/RCP4.5 scenarios. doi:10.1175/2010JCLI3690.1. J. Climate, 28, 7203–7224, doi:10.1175/JCLI-D-15-0129.1. Reasor, P. D., M. T. Montgomery, F. D. Marks Jr., and J. F. Kossin, J. P., J. A. Knaff, H. I. Berger, D. C. Herndon, T. A. Cram, Gamache, 2000: Low-wavenumber structure and evolution C. S. Velden, R. J. Murnane, and J. D. Hawkins, 2007: Esti- of the hurricane inner core observed by airborne dual- mating hurricane wind structure in the absence of aircraft doppler radar. Mon. Wea. Rev., 128, 1653–1680, doi:10.1175/ reconnaissance. Wea. Forecasting, 22, 89–101, doi:10.1175/ 1520-0493(2000)128,1653:LWSAEO.2.0.CO;2. WAF985.1. Reed, K. A., and D. R. Chavas, 2015: Uniformly rotating global ——, K. A. Emanuel, and G. A. Vecchi, 2014: The poleward mi- radiative–convective equilibrium in the Community Atmo- gration of the location of tropical cyclone maximum intensity. sphere Model, version 5. J. Adv. Model. Earth Syst., 7, 1938– Nature, 509, 349–352, doi:10.1038/nature13278. 1955, doi:10.1002/2015MS000519. Lander, M. A., 1994: Description of a monsoon gyre and its effects Rotunno, R., and K. A. Emanuel, 1987: An air–sea interaction on the tropical cyclones in the western North Pacific during theory for tropical cyclones. Part II: Evolutionary study using a August 1991. Wea. Forecasting, 9, 640–654, doi:10.1175/ nonhydrostatic axisymmetric numerical model. J. Atmos. 1520-0434(1994)009,0640:DOAMGA.2.0.CO;2. Sci., 44, 542–561, doi:10.1175/1520-0469(1987)044,0542: Lee, C.-S., K. K. Cheung, W.-T. Fang, and R. L. Elsberry, 2010: AAITFT.2.0.CO;2. Initial maintenance of tropical cyclone size in the western ——, and G. H. Bryan, 2012: Effects of parameterized diffusion on North Pacific. Mon. Wea. Rev., 138, 3207–3223, doi:10.1175/ simulated hurricanes. J. Atmos. Sci., 69, 2284–2299, doi:10.1175/ 2010MWR3023.1. JAS-D-11-0204.1. Lin, N., and D. Chavas, 2012: On hurricane parametric wind and Schwerdt, R. W., F. P. Ho, and R. R. Watkins, 1979: Meteorological applications in storm surge modeling. J. Geophys. Res., 117, criteria for standard project hurricane and probable maximum D09120, doi:10.1029/2011JD017126. hurricane windfields, Gulf and East Coasts of the United ——, and K. A. Emanuel, 2016: Grey swan tropical cyclones. Nat. States. NOAA Tech. Rep. NWS 23, 317 pp. Climate Change, 6, 106–111, doi:10.1038/nclimate2777. Shoemaker, D. N., 1989: Relationships between tropical cyclone ——, ——, M. Oppenheimer, and E. Vanmarcke, 2012: Physically deep convection and the radial extent of damaging winds. based assessment of hurricane surge threat under climate University of Colorado Department of Atmospheric Science change. Nat. Climate Change, 2, 462–467, doi:10.1038/ Paper 457, 109 pp. [Available online at http://hdl.handle.net/ nclimate1389. 10217/78813.] ——, P. Lane, K. A. Emanuel, R. M. Sullivan, and J. P. Donnelly, Sitkowski,M.,J.P.Kossin,andC.M.Rozoff,2011:Intensityand 2014: Heightened hurricane surge risk in northwest Florida structure changes during hurricane eyewall replacement revealed from climatological-hydrodynamic modeling and cycles. Mon. Wea. Rev., 139, 3829–3847, doi:10.1175/ paleorecord reconstruction. J. Geophys. Res., 119, 8606–8623, MWR-D-11-00034.1. doi:10.1002/2014JD021584. Smith, R. K., C. W. Schmidt, and M. T. Montgomery, 2011: An Lin, Y., M. Zhao, and M. Zhang, 2015: Tropical cyclone rainfall investigation of rotational influences on tropical-cyclone size area controlled by relative sea surface temperature. Nat. and intensity. Quart. J. Roy. Meteor. Soc., 137, 1841–1855, Commun., 6, 6591, doi:10.1038/ncomms7591. doi:10.1002/qj.862. Massey, F. J., Jr., 1951: The Kolmogorov–Smirnov test for good- Sobel, A. H., and C. S. Bretherton, 2000: Modeling tropical pre- ness of fit. J. Amer. Stat. Assoc., 46, 68–78, doi:10.1080/ cipitation in a single column. J. Climate, 13, 4378–4392, 01621459.1951.10500769. doi:10.1175/1520-0442(2000)013,4378:MTPIAS.2.0.CO;2. 15 APRIL 2016 C H A V A S E T A L . 2939

Stephens, G. L., 1990: On the relationship between water vapor and relationships to motion and environmental shear. Mon. over the oceans and sea surface temperature. J. Climate, 3, 634–645, Wea. Rev., 142, 1290–1311, doi:10.1175/MWR-D-13-00249.1. doi:10.1175/1520-0442(1990)003,0634:OTRBWV.2.0.CO;2. Vecchi, G. A., and B. J. Soden, 2007: Effect of remote sea surface Stewart, S., 2014: National Hurricane Center annual summary: temperature change on tropical cyclone potential intensity. 2012 Atlantic hurricane season. National Hurricane Center Nature, 450, 1066–1070, doi:10.1038/nature06423. Tropical Cyclone Rep., 11 pp. [Available online at http://www. Wang, Y., 2009: How do outer spiral rainbands affect tropical cy- nhc.noaa.gov/data/tcr/summary_atlc_2012.pdf.] clone structure and intensity? J. Atmos. Sci., 66, 1250–1273, Stiles,B.W.,R.E.Danielson,W.L.Poulsen,M.J.Brennan, doi:10.1175/2008JAS2737.1. S. Hristova-Veleva, T.-P. Shen, and A. G. Fore, 2014: Weatherford, C., and W. Gray, 1988: Typhoon structure as re- Optimized tropical cyclone winds from QuikSCAT: A vealed by aircraft reconnaissance. Part I: Data analysis and neural network approach, 52, 7418–7434, doi:10.1109/ climatology. Mon. Wea. Rev., 116, 1032–1043, doi:10.1175/ TGRS.2014.2312333. 1520-0493(1988)116,1032:TSARBA.2.0.CO;2. Tang, B., and K. Emanuel, 2010: Midlevel ventilation’s constraint Zhai, A. R., and J. H. Jiang, 2014: Dependence of U.S. hurricane on tropical cyclone intensity. J. Atmos. Sci., 67, 1817–1830, economic loss on maximum wind speed and storm size. Environ. doi:10.1175/2010JAS3318.1. Res. Lett., 9, 064019, doi:10.1088/1748-9326/9/6/064019. ——, and ——, 2012: A ventilation index for tropical cyclones. Bull. Amer. Zhou, W., I. M. Held, and S. T. Garner, 2014: Parameter study of Meteor. Soc., 93, 1901–1912, doi:10.1175/BAMS-D-11-00165.1. tropical cyclones in rotating radiative–convective equilibrium Uhlhorn, E. W., B. W. Klotz, T. Vukicevic, P. D. Reasor, and R. F. with column physics and resolution of a 25-km GCM. Rogers, 2014: Observed hurricane wind speed asymmetries J. Atmos. Sci., 71, 1058–1069, doi:10.1175/JAS-D-13-0190.1.