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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 in storm intensity. In contrast, the only mechanism to destabilize the atmosphere with
109 the Rayleigh damping scheme is surface fluxes and, as a result, the storm does not achieve
110 equilibrium, and there is little convection or variability in storm intensity (Hakim 2011).
111 Solar radiation is excluded in the interest of eliminating the complicating effects of the
112 diurnal and seasonal cycles.
1 113 The model uses the Thompson et al. (2008) microphysics scheme , aerodynamic formulae
114 for surface fluxes of entropy and momentum, with equal values for the exchange coefficients
◦ 115 for these quantities, and a constant sea-surface temperature of 26.3 C. The computational
116 grid is 1500 km in radius and 25 km in height, with 2 km horizontal resolution and 250 m
117 vertical resolution from 0–10 km, gradually increasing to 1 km at the model top. A 500-day
118 numerical solution from this model is obtained from a state of rest with full gridded data
119 sampled every 3 hours, and maximum azimuthal wind speed sampled every 15 minutes. The
120 mean state of the equilibrium solution is described in Hakim (2011).
121 3. Intrinsic Axisymmetric Variability
122 A radius–time plot of the azimuthal wind reveals substantial fluctuation of the lowest
−1 123 model-level wind (250 m elevation) about the time-mean value of 59 m s (Fig. 1, left
124 panel). Initial storm development occurs at small radii, and the radius of maximum wind
125 increases with time as the storm size increases (see Hakim (2011) for further details). After
126 approximately 25 days, a statistically steady storm is evident, with a time-mean radius of
−1 127 maximum wind around 30 km. Storm intensity fluctuates between about 40 and 70 m s ,
128 with bursts of stronger wind lasting for a few days. This behavior is more apparent in Figure
129 1 (right panel), which also shows that, in many cases, these bursts of stronger wind originate
1A coding error involving the microphysics scheme was discovered in the version of the model used here. Subsequent tests with the next version of the model (r15) yielded results quantitatively similar to those shown here.
5 130 more than 100 km from the storm center, and propagate inward. Moreover, these patterns
131 of stronger wind move radially inward more slowly as they reach the radius of maximum
132 wind. Finally, although not periodic, we note that the bursts of stronger wind appear on
133 average roughly every 5 days.
134 The temporal Fourier spectrum of the maximum wind speed at the lowest model level
135 (Fig. 2) exhibits two peaks: one near periods of 4–8 days, close to the range evident for the
136 convective bands noted in Figure 1, and the second near periods of three hours, which may
137 correspond to bursts of convection near the eyewall. Spectra for two single-lag autoregressive
138 [AR(1)] models are also shown by gray lines in Figure 2, one for the single-step (15-
139 minute) autocorrelation and the other for the single-step autocorrelation consistent with
140 the autocorrelation e-folding time (24.75 hours); the dashed lines give the upper-range of
141 the spectra at the 95% confidence interval based on a Chi-squared test (Wilks 2005, p. 397).
142 The single-step model provides an approximate fit of the observed spectrum for periods
143 shorter than about 12 hours, except for the peak near three hours, which suggests that the
144 peak is distinct from persistence forced by white noise. The e-folding-time AR(1) model
145 provides an approximate fit of the observed spectrum for periods longer than about one day,
146 except for the peak near 4–8 days, which suggest this peak is also distinct from persistence
147 forced by white noise.
148 The leading empirical orthogonal functions (EOFs) of the azimuthal wind field at the
149 lowest model level reveal two patterns that dominate the variance in this field (Fig. 3).
150 The leading pattern, which accounts for 40% of the variance, crosses zero near the radius
151 of maximum wind, and therefore suggests a radial shift of the radius of maximum wind.
152 The dipole structure of this EOF is asymmetric in radius, indicating that the response in
153 the azimuthal wind is largest inside the radius of maximum wind, where the radial gradient
154 in the time-mean field is largest. The second pattern, which explains 27% of the variance,
155 is largest near the radius of maximum wind, which suggests that this pattern is linked to
156 intensity change at the radius of maximum wind. Table 1 quantifies the relationship between
6 157 these EOFs and the radius of maximum wind and the azimuthal wind speed at this location
158 by projecting the azimuthal wind onto the EOFs to produce principle component (PC)
159 time series. The PC for the first EOF is uncorrelated with the maximum wind speed and
160 negatively correlated at −0.59 with the radius of maximum wind, so that increases in the
161 PC correspond to inward shifts of the radius of maximum wind. The second EOF PC time
162 series correlates at 0.82 with maximum wind, but there is also a weak negative correlation
163 of −0.37 with the radius of maximum wind, so that the second EOF is, on average, not
164 exclusively associated with intensity change at the mean radius of maximum wind.
165 Sample-mean structure of the azimuthal wind for times corresponding to the upper and
166 lower tercile for each PC reveal the storm structure associated with these dominant patterns
167 of variability for both large positive and negative EOF amplitude. For the first EOF, results
168 show that the positive composite is associated with an inward shift of the radius of maximum
169 wind and an increase in the radial gradient of azimuthal wind inside the eyewall, whereas
170 the negative composite is associated with an outward shift and a decrease in the gradient
171 (Fig. 4, top panel). In both cases, the composite storm is stronger than the mean, which
172 suggests that weaker storms are associated with smaller PC values. Results for composites of
173 the second EOF show that for positive (negative) events the storm is stronger (weaker) than
174 the mean, consistent with the interpretation suggested previously (Fig. 4, bottom panel).
175 The radius of maximum wind is diffuse, but displaced toward larger radius in the negative
176 sample, consistent with the weak negative correlation described for the entire PC time series;
177 the positive example exhibits little shift (Table 1).
178 The temporal evolution of variability associated with the leading two EOFs is described
179 through lag regression of the azimuthal wind onto the PC time series. Results for PC-1
180 indicate that this structure is linked to inward propagating bands of weaker and stronger
181 wind that originate more than 100 km from the storm center (Fig. 5); results for PC-2
182 are qualitatively similar (not shown). These bands travel radially inward at approximately
−1 183 1–2 m s until they reach about 50 km from the storm center, at which point they slow
7 −1 184 substantially to less than 0.5 m s . This pattern also appears for regression of the azimuthal
185 wind onto the wind speed at the radius of maximum wind, and is also evident in Figure 1.
186 Note that the dominant timescale of ∼6 days is also apparent in the regression field.
187 Lag regression of the azimuthal wind at locations outside the storm provides a link
188 between variability in the environment and near the storm. In the environment near the
2 189 storm, at a radius of 100 km, bands of stronger wind clearly propagate inward all the way
190 to the eyewall (Fig. 6, top panel) similar to Figure 5. In the outer environment, 600 km
−1 191 from the storm center, the bands propagate faster, at a rate of 5–10 m s , but vanish by a
192 radius of 400 km (Fig. 6, bottom panel).
193 The vertical structure of the bands at these locations is determined by regressing the
194 radial and vertical wind, water vapor mixing ratio, and azimuthal wind onto the 100 and
195 600-km azimuthal wind time series at the lowest model level (Fig. 7). At a radius of 100
196 km, results reveal that stronger azimuthal winds at this location are associated with a deep
197 layer of anomalously large mixing ratio and an overturning vertical circulation (Fig. 7a).
198 Specifically, the results suggest a convective band located near 75–100 km radius, with rising
199 air in the band and sinking air behind. Positive anomalous azimuthal wind is coincident with
200 the band over a deep layer, with a maximum near 2–3 km altitude (Fig. 7b). Radial inflow at
201 low levels near the region of positive azimuthal wind is expected to accelerate the azimuthal
202 wind as the band moves radially inward. It appears that the band is interacting with the
203 eyewall, enhancing the mixing ratio near a radius of about 25 km. Sinking motion is found
204 at smaller radii, and also above an altitude of 2 km outside the band. Near the eyewall,
205 subsidence is coincident with a narrow region of weaker azimuthal wind located radially
206 inward from the convective band. The vertical circulation is qualitatively in accord with
207 a balanced Eliassen secondary circulation, with a clockwise (counter-clockwise) circulation
208 where the radial gradient of anomalous mixing ratio, and presumably latent heating, is
209 negative (positive); one exception to this agreement is the radial inflow region near 6–10
2Although by linearity this discussion applies to either sign, we discuss only the positive case.
8 210 km altitude outside the band. Results for the regression onto lowest-model-level azimuthal
211 wind at a radius of 600 km reveal a similar vertical structure (Fig. 7c,d), except that the
212 convective band and associated vertical circulation are shallower and wider. The deeper
213 layer of anomalous mixing ratio ahead of the band and shallower layer behind the band is
214 suggestive of a convective band with trailing stratiform precipitation (Fig. 7c). A further
215 distinction from the results at a radius of 100 km is that the positive anomalous azimuthal
216 wind is more vertically aligned with two distinct maxima: one at the surface and another at
217 6 km altitude (Fig. 7d).
218 Another perspective on storm variability is provided by the time series of the radius of
219 maximum wind (Fig. 8). This field shows frequent jumps from around 20–30 km to much
220 larger values before returning again to smaller values. These jumps are objectively identified
221 after day 20 by finding times when the radius of maximum wind increase by at least 30
222 km over a 15-minute period. A second requirement is that any jump must be separated by
223 at least 6 hours from other events, which leaves a total of 131 events. Composite averages
224 of these events show that after the initial outward jump in the radius of maximum wind
225 by about 45 km, it becomes smaller at a decreasing rate, returning close to the original
226 value after about two days (Fig. 9, top panel). For two days prior to the time of the jump
227 in the radius of maximum wind, the storm slowly weakens, reaching minimum intensity
228 shortly after the radius of maximum wind jumps to larger radius (Fig. 9, bottom panel).
229 The maximum wind rapidly increases after reaching the minimum, returning after about
230 24 hours to values observed nearly two days before the jump in the radius of maximum
231 wind. This asymmetric response suggests slow weakening of a primary eyewall, followed by
232 a rapid intensification of its replacement, qualitatively similar to what is observed in eyewall
233 replacement cycles in real storms, although somewhat slower (e.g., Willoughby et al. 1982).
9 234 4. Predictability
235 Although there are many approaches to investigating the predictability of the axisym-
236 metric storm, we continue with the statistical theme of this paper by sampling from the
237 long equilibrium simulation to estimate predictability timescales and structures that control
238 predictability. One simple measure of predictability is the autocorrelation timescale. Near
239 the radius of maximum wind, the autocorrelation drops to zero around 42 h, reaching a
240 minimum around 4 days before becoming weakly positive again (Fig. 10, top panel). This
241 behavior is consistent with the average eyewall replacement cycle period of around 4-8 days.
242 The autocorrelation e-folding time reaches a maximum inside the eye around 33 h, with a
243 relative minimum of 15 h in the range 30–50 km radius, and an increase to a secondary
244 maximum around 21 h centered around a radius of 100 km (Fig. 10, bottom panel). A
245 similar behavior is obtained for the time when the lower bound on the autocorrelation 95%
246 confidence range reaches zero, except that outside the eyewall this timescale increases from
247 about 33 hours near the radius of maximum wind to 90–100 hours beyond a radius of about
248 140 km.
249 A more direct estimate of predictability time scales involves solving for error growth over
250 a large number of initial conditions. Here we construct a linear model for this purpose,
251 derived from the statistics of the long simulation (e.g., Penland and Magorian 1993). In
252 general, the linear model can be written
dx = Lx, (1) dt
253 where x is the state vector of model variables and L is a taken here as a constant matrix
254 that defines the system dynamics. The solution of (1) over the time interval t : t + τ is
x(t + τ) = M(t,t + τ) x(t), (2)
L τ 255 where M(t,t + τ) = e is the propagator matrix that maps the initial condition x(t) into
256 the solution vector x(t+τ). The propagator is determined empirically from the equilibrium
10 257 solution using the least-squares estimate
T T −1 M(t,t + τ) ≈ x(t + τ)x(t) x(t)x(t) . (3)
258 Superscript “T” denotes a transpose, and braces denote an expectation, which is estimated
259 by an average over a “training” sample consisting of 500 randomly chosen times out of a
260 total of 3760 during the last 470 days of the simulation. Once determined, the propagator
261 matrix is then applied to an independent sample consisting of 100 randomly chosen times
262 that do not overlap with the training sample. This process is repeated over 100 trials to
263 provide a bootstrap error estimate for the calculation.
264 Four prognostic variables are included in the linear model: azimuthal wind, v, radial
265 wind, u, potential temperature, θ, and cloud water mixing ratio, q. In order to reduce
266 dimensionality, the grid-point values of these variables are projected onto the leading 20
267 EOFs in each variable over all vertical levels from the origin to a radius of 200 km. In
268 terms of the fraction of the total variance accounted for by these EOFs, the leading 20 EOFs
269 capture 90% for v, 84% for θ, 73% for u, and 52% for q. This approach reduces M to an 80
270 by 80 matrix.
271 Forecast error is summarized for each forecast lead time τ by computing the error variance
272 in grid-point space
e = x(t + τ) − xT(t + τ), (4)
273 where xT(t + τ) is the projection of the true state onto the EOF basis. For each variable, at
274 each time, the spatial variance in e is computed, leaving a time series of error variance. This
275 error variance is normalized by the variance in xT, which means that skill is lost when the
276 normalized error variance reaches a value of one. The mean and standard deviation over the
277 100 trials as a function of τ are summarized in Figure 11. Results show fastest error growth
278 in cloud water mixing ratio, with error saturation at 15 hours (Fig. 11d). Error growth in
279 radial wind is similar for the first 6 hours, and then becomes much slower, reaching saturation
280 around 36 hours (Fig. 11a). The slowest error growth is observed in the azimuthal wind field,
11 281 which grows linearly for the first 18 hours, and then slowly approaches the saturation value
282 around 48 hours (Fig. 11b). A similar behavior is observed in the potential temperature
283 field, but the initial growth is larger in θ (Fig. 11c). The summary interpretation of these
284 results is that convective motions are most strongly evident in the cloud water and radial
285 wind fields, and these fields lose predictability the fastest, whereas the slow response is in the
286 azimuthal wind field, which responds through the secondary circulation. The azimuthal wind
287 and potential temperature fields should couple strongly through the thermal wind equation
288 except near convective and other unbalanced motions, so we hypothesize that the relatively
289 larger errors for θ compared to v at smaller τ are related to convection.
290 To test the effect of the linear assumption in these calculations, an independent pre-
291 dictability estimate is derived from solution divergence rates estimated by analogs in the
292 500-day simulation. The metric for comparing states is the summed squared difference in
293 azimuthal wind over all vertical levels from the storm center to a radius of 200 km. The
294 closest analog state is defined for each time by the state with minimum metric that is also
295 separated in time by at least 4 days. Taking one state to be the “truth” and the other to be
296 the “forecast” allows error statistics to be determined as defined above for the linear inverse
297 model. Results show that the initial errors are larger than for the linear inverse model,
298 reflecting the finite resource of available analogs, but the growth rate at this amplitude
299 is similar (Fig. 12). The analog results lack an error estimate, but it appears that the
300 predictability limit may be longer, by perhaps 12 hours, than in the linear inverse model.
301 In order to put these results in context, operational forecast errors from the National
302 Hurricane Center (NHC) are provided for comparison (Fig. 13), normalized by the errors
303 at 84 hours. Compared to the LIM results, the NHC errors follow a similar growth curve
304 with somewhat slower growth, and saturate approximately 12-24 hours later, similar to the
305 analog results. Although the close comparison between the LIM results and NHC forecast
306 errors suggests that intrinsic storm predictability may be an important factor in operational
307 intensity prediction, further research is needed to make this link more than circumstantial.
12 308 Another aspect of interest in predictability concerns the initial-condition errors that
309 evolve into structures that dominate forecast-error variance. Here we objectively identify
310 initial conditions that optimally excite the leading EOF of the forecast error variance in
311 the linear inverse model, subject to the constraint that the initial structure is drawn from a
312 distribution of equally likely disturbances that are consistent with short-term error statistics.
313 The theory for this calculation, described completely in Houtekamer (1995), Ehrendorfer and
314 Tribbia (1997), and Snyder and Hakim (2005), begins with a measure for the size of forecast
315 errors relative to initial condition errors:
x(t + τ), x(t + τ) t+τ λ = . (5) x(t), x(t) t
316 The initial-time inner product, , t, is defined by the probability density function for the
317 initial errors, p[x(t)], which we take to be Gaussian with mean zero and covariance matrix,
318 B, so that
−1 x(t), x(t) t = x(t) B x = −ln p[x(t)]. (6)
319 This relationship implies that the “size” of initial disturbances is determined by the covari-
320 ance matrix for the initial errors, so that disturbances of equal size have equal probability.
321 Making the substitution
x(t) = Eˆx(t), (7)
322 where E is a symmetric matrix that is a square root of B (i.e., B = EE), (5) becomes
T λ = EM ME ˆx , ˆx t+τ . (8)
323 Here we have assumed that the initial disturbances have equal probability. The eigenvector
T 324 of EM ME having the largest eigenvalue gives the initial disturbance that evolves into
325 the leading eigenvector of the forecast error covariance matrix; that is, the structure that
326 dominates the forecast error (Ehrendorfer and Tribbia 1997). For these calculations, the
327 leading forecast-error EOF accounts for 20% of forecast-error variance. We calculate the
328 leading eigenvector for a forecast lead time of 24 hours, and use as a proxy for B the 3-hour
13 329 forecast-error covariance; the “true” analysis-error covariance matrix requires an assimilation
330 system, and depends on observations, which is beyond the scope of the current paper.
331 For brevity, results are summarized for only the azimuthal wind, which shows an initial
332 structure with largest amplitude outside the storm, near 125 km radius and 5 km altitude
333 (Fig. 14a). A second region of enhanced azimuthal wind is found just outside the radius
334 of maximum wind above the surface. This second region grows rapidly in time from about
−1 −1 335 1ms to more that 24 m s by 24 hours (Fig. 14b–d) for the (arbitrary) amplitude
336 assigned to the initial condition. Further experimentation shows that the solution at 24 h is
337 due mostly to the initial structure of v and θ (not shown).
338 5. Summary and Conclusions
339 We have defined intrinsic variability of tropical cyclones as the structure and intensity
340 change independent of the environment in which the storm resides. Insight into intrinsic
341 variability is useful for determining the internal dynamics of the storm, and also for pre-
342 dictability research, since the intrinsic variability controls forecast errors in the absence of
343 environmental forcing. Here, leading-order estimates of intrinsic variability are estimated
344 from a 500-day numerical simulation of an axisymmetric tropical cyclone begun from a state
345 of rest. This framework is useful because, unlike simulations of real or idealized three-
346 dimensional tropical cyclones, the results are independent of the initial disturbance, and
347 large sample sizes are available to document variability and predictability.
348 Results for the axisymmetric solution show that variability in storm intensity, as measured
−1 349 by the maximum azimuthal wind speed, varies over 30 m s . This variability is controlled
350 by convective bands in the distant environment (r > 100 km) that propagate inward and
351 lead to eyewall replacement cycles. Bands in the distant environment are relatively shallow
352 compared to bands closer to the storm center. Moreover, the radial motion of the bands is
353 fastest in the distant environment, and slowest near the eyewall, consistent with expectations
14 354 based on the role of inertial stability on the strength of the Eliassen circulation forced by
355 radial gradients in latent heating. A composite of 131 eyewall replacement cycles indicates
356 that the secondary wind maximum exceeds the primary nearly 50 km from the primary radius
357 of maximum wind, and reaches the location of the primary maximum after approximately
−1 358 24–36 hours. The sample-mean maximum wind speed increases by about 13 m s during
359 this period where the secondary eyewall contracts.
360 A dominant period of 4–8 days is evident for the convective bands and eyewall replace-
361 ment cycles. We speculate that this time period is related to suppression of convection
362 outside the bands, due to subsidence from the balanced circulation forced by the band (see
363 Fig. 7). In this case, a scaling for the time period where convection is suppressed is given
364 by the ratio of a band-circulation length scale to the band speed, L Pband ∼ . (9) cband
365 Based upon the regression calculation shown in Fig. 7, subsidence extends about 400 km
−1 366 from the band, while cband is approximately 2 m s , giving Pband ≈ 55 hours. Beyond this
367 time, convection may begin to deepen and moisten the troposphere. Band formation in the
368 simulation likely involves an upscale aggregation of convection, the study of which is beyond
369 the scope of this paper.
370 Predictability timescales for the axisymmetric model are estimated by autocorrelation
371 times and by linear inverse modeling, with similar results. Autocorrelation e-folding times
372 for the azimuthal wind at the lowest model level are largest just inside the eyewall, at
373 around 33 hours, and smallest near and just outside the eyewall, at around 18 hours; a tail
374 of statistically significant, but small, autocorrelation hints at predictability timescales from
375 36 hours near the eyewall to 100 hours in the far-field. Linear inverse modeling provides a
376 more quantitative measure of predictability. The azimuthal wind is predictable, on average,
377 for at least 36–48 hours, with marginal predictability to about 3 days. The radial wind and
378 cloud water field are less predictable, with errors saturating around 24 hours. The optimal
379 initial condition that evolves into the leading EOF of the forecast-error covariance matrix
15 380 has a weak convective band just outside the radius of maximum wind that rapidly develops,
381 increasing 24-hour azimuthal-wind forecast errors by a factor of roughly 24 near the radius
382 of maximum wind. The optimal structure is also characterized by a maximum in azimuthal
383 wind near 125 km radius and 5 km altitude, which grows more slowly than wind errors near
384 the radius of maximum wind.
385 Although this work provides only a starting point from an axisymmetric framework,
386 results for idealized three-dimensional storms yield similar patterns of variability, and limits
387 of predictability (Brown and Hakim 2012). An important limitation of this research involves
388 the imposition of boundary conditions at finite radius. While the location of the boundary
389 is important for storm size (Chavas and Emanuel 2012), the sensitivity of variability and
390 predictability is unknown and the subject of future research.
391 Acknowledgments.
392 This research was support by grants awarded to the University of Washington from the
393 National Science Foundation (AGS-0842384) and the Office of Naval Research (N00014-
394 09-1-0436). Comments from Dave Nolan and three anonymous reviewers were helpful in
395 revision.
16 396
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19 461 List of Tables
462 1 Correlation between the first two EOF principle component time series and the
463 maximum wind speed (“max speed”) and radius of maximum speed (“RMW”). 21
20 Table 1. Correlation between the first two EOF principle component time series and the maximum wind speed (“max speed”) and radius of maximum speed (“RMW”).
field max speed RMW PC1 −0.02 −0.59 PC2 0.82 −0.37 PC1uppertercile 0.46 −0.44 PC1 lower tercile −0.19 −0.07 PC2uppertercile 0.84 0.18 PC2lowertercile 0.17 −0.40
21 464 List of Figures
−1 465 1 Azimuthalwind(ms ) as a function of radius and time for all 500 days (left)
466 andfordays300–400(right)ofthenumericalsimulation. 25
467 2 Fourier power spectrum of the time series of maximum wind speed at the
468 lowest model level (black line) and spectra for AR(1) processes having a single-
469 step correlation that matches the 15-minute data (lower solid gray line) and a
470 single-step correlation that matches the e-folding autocorrelation time (upper
471 gray solid line). Dashed gray lines give the 95% confidence bounds on the
472 AR(1)spectrabasedonaChi-squaredtest. 26
473 3 Leading EOFs of the azimuthal wind field at the lowest model level. The first
474 (second) EOF accounts for 40% (27%) of the variance, and is given by the
475 thick (thin) black line. A scaled version of the time-mean azimuthal wind is
476 given for reference in the thick gray line. 27
−1 477 4 Sample-mean radial profiles of the azimuthal wind (m s ) for upper and
478 lower terciles of the principle components of the first (second) EOF in the top
479 (bottom) panel. The sample-mean from the upper (lower) tercile are shown
480 in the solid (dashed) black lines, and the time-mean azimuthal wind is given
481 by the gray line. 28
482 5 Time-lag regression of the azimuthal wind at the lowest model level onto the
483 principle component time series of the first EOF (solid lines), scaled by one
484 standard deviation in the principle component. Positive values are shown by
−1 485 black lines with contours of .5, 1, 3, 5, and 10 m s ; negative values are
−1 486 shown by gray lines with contours of −.5, −1, and −3ms . 29
22 487 6 Time-lag regression of the azimuthal wind onto the azimuthal wind at the
488 lowest model level and a radius of 100 km (top panel) and 600 km (bottom
489 panel). Positive (negative) values are given by black (gray) lines, with contours
−1 −1 −1 490 every 1 m s (top panel) and every 0.2 m s starting at 0.4 m s (bottom
491 panel); zero contour omitted. Black dots denote the location of the base-point
492 for the regression. 30
493 7 (a) Regression of (a) radial and vertical wind (arrows) and water vapor mixing
494 ratio (colors; g/kg), and (b) the azimuthal wind, onto the azimuthal wind at
495 the lowest model level at a radius of 100 km. (c) and (d) as in (a) and (b)
496 except for regression onto the azimuthal wind at the lowest model level at
497 a radius of 600 km. Vectors are shown only at points where the regression
498 of both components of the vector are significant at the 95% confidence level.
−1 −1 499 Azimuthal wind contour interval is 0.25m s (bold ±0.5m s ) in (b) and
−1 −1 500 1m s (bold ±3m s )in(d);negativevaluesshowningray. 31
501 8 Timeseriesofradiusofmaximumwind(km). 32
502 9 Sample-mean temporal evolution of anomaly in the radius of maximum wind
503 (top panel) and anomalies of maximum wind (black line) and minimum central
504 pressure (gray line) (bottom panel). Anomalies are relative to the uncondi-
505 tioned mean in each field. The sample applies to 131 objectively identified
506 eyewallreplacementcycles;seetextfordetails. 33
507 10 (top panel) Temporal autocorrelation (solid line) at 40 km radius with 95%
508 confidence bounds (dashed lines). (bottom panel) e-folding autocorrelation
509 time (solid line) and zero-crossing autocorrelation time (dashed line) as a
510 function of radius. The zero-crossing time is defined when the lower bound
511 onthe95%confidenceintervalfirstcrosseszero. 34
23 512 11 Linear inverse model sample-mean (bold lines) and range of one standard
513 deviation (error bars) of forecast error variance as a function of lead time
514 (abscissa). Panels show (a) radial wind; (b) azimuthal wind; (c) potential
515 temperature; and (d) cloud water mixing ratio. Error variance is normalized
516 by the climatological value, so that skill is lost when the mean value reaches
517 unity, and standard deviation reaches values at large forecastleadtime. 35
518 12 Analog forecast error as a function of lead time, normalized by climatological
519 variance. 36
520 13 Comparison of the linear inverse model forecast errors for azimuthal wind
521 (gray line) with operation forecasts of tropical cyclone maximum wind from
522 the National Hurricane Center (NHC, black line). The NHC errors apply to
523 the Atlantic ocean and are normalized by the value at a forecast lead time of
524 84 hours. 37
525 14 Azimuthal wind evolution for the leading optimal mode of the linear inverse
526 model that explains the most forecast error variance (black lines). Thick
−1 −1 527 (thin) lines show positive (negative) values every 1 m s (a); 2 m s (b);
−1 528 and4ms (c) and (d); the zero contour is suppressed. Thin gray lines show
−1 −1 529 the time-mean azimuthal wind every 5 m s starting at 40 m s . 38
24 0 20 40 60 80
500 400
450 390
400 380
350 370
300 360
250 350
time (days) 200 340
150 330
100 320
50 310
0 300 0 50 100 150 200 250 0 50 100 150 200 250 radius (km) radius (km)
Fig. 1. Azimuthal wind (m s−1) as a function of radius and time for all 500 days (left) and for days 300–400 (right) of the numerical simulation.
25 Fig. ofiec onso h R1 pcr ae naCisurdt Chi-squared a on based spectra AR(1) the on bounds confidence ace h 5mnt aa(oe oi ryln)adasing a and line) gray havi solid the (lower processes data AR(1) 15-minute for the spectra matches and line) (black level model e fligatcreaintm uprga oi ie.Das line). solid gray (upper time autocorrelation -folding .Fuirpwrsetu ftetm eiso aiu ids wind maximum of series time the of spectrum power Fourier 2. 10 10 power 10 10 10 10 10 10 − − 5 3 2 1 2 1 4 0 21 8 16 32 4 0 1 2 eid(days) period 26 . 0 5 . 50 25 gasnl-tpcreainthat correlation single-step a ng ese orlto htmatches that correlation le-step e rylnsgv h 95% the give lines gray hed . 125 est. eda h lowest the at peed 0.5
0.4
0.3
0.2
0.1 amplitude
0
−0.1 mean eof1 40% eof2 27% −0.2 0 20 40 60 80 100 120 140 160 180 200 radius (km)
Fig. 3. Leading EOFs of the azimuthal wind field at the lowest model level. The first (second) EOF accounts for 40% (27%) of the variance, and is given by the thick (thin) black line. A scaled version of the time-mean azimuthal wind is given for reference in the thick gray line.
27 PC-1 composites 60
50
40
30
20 wind speed (m/s)
10
0 0 20 40 60 80 100
PC-2 composites 70
60
50
40
30
wind speed (m/s) 20
10
0 0 20 40 60 80 100 radius (km)
Fig. 4. Sample-mean radial profiles of the azimuthal wind (m s−1) for upper and lower terciles of the principle components of the first (second) EOF in the top (bottom) panel. The sample-mean from the upper (lower) tercile are shown in the solid (dashed) black lines, and the time-mean azimuthal wind is given by the gray line.
28 rnil opnn.Pstv ausaesonb lc lin s b black by m scaled shown 10 lines), are (solid values Positive EOF first component. principle the of series time component Fig. .Tm-a ersino h zmta ida h oetmo lowest the at wind azimuthal the of regression Time-lag 5.
− 065 − − − 1 eaievle r hw yga ie ihcnor of contours with lines gray by shown are values negative ; − 100 200 150 200 100 150 50 50 0 0100 50 0 zmta idrgesdot PC-1 onto regressed wind Azimuthal ais(km) radius 29 150 0 5 300 250 200 swt otuso 5 ,3 ,and 5, 3, 1, .5, of contours with es n tnaddvaini the in deviation standard one y e ee noteprinciple the onto level del − . 5 , − 1 , and − s m 3 − 1 . eoetelcto ftebs-on o h regression. the for base-point the of location the denote n vr . s m 0.2 every and Fig. oe ee n aiso 0 m(o ae)ad60k (botto km 600 and cont panel) with (top lines, km (gray) black 100 by of given radius are values a (negative) and level model .Tm-a ersino h zmta idot h azimut the onto wind azimuthal the of regression Time-lag 6.
time (hours) time (hours) − − − − − 100 100 100 100 1 50 50 50 50 triga . s m 0.4 at starting 0 0 0 400 300 0 50 − 1 500 ais(km) radius ais(km) radius bto ae) eocnoroitd lc dots Black omitted. contour zero panel); (bottom 30 0 5 200 150 100 0 0 800 700 600 useey1ms m 1 every ours a ida h lowest the at wind hal ae) Positive panel). m − 1 tppanel) (top 10 0.6
8 0.4
0.2 6
0 4
heig ht (km) − 0.2
2 − 0.4 .5 m/ s − 0.6 0 (a) 50 100 150 (b) 50 100 150 2 m /s radius (km) radius (km)
10
0.3 8 0.2
0.1 6
0 4 − 0.1 heig ht (km)
− 0.2 2
5 cm/ s − 0.3 0 300 400 500 600 700 800 300 400 500 600 700 800 2 m /s radius (km) (c) (d) radius (km)
Fig. 7. (a) Regression of (a) radial and vertical wind (arrows) and water vapor mixing ratio (colors; g/kg), and (b) the azimuthal wind, onto the azimuthal wind at the lowest model level at a radius of 100 km. (c) and (d) as in (a) and (b) except for regression onto the azimuthal wind at the lowest model level at a radius of 600 km. Vectors are shown only at points where the regression of both components of the vector are significant at the 95% confidence level. Azimuthal wind contour interval is 0.25m s−1(bold ±0.5m s−1) in (b) and 1m s−1(bold ±3m s−1) in (d); negative values shown in gray.
31 radius of maximum wind 140 120 100 80 60 radius (km) 40 20
50 100 150 200 250 300 350 400 450 500 time (days)
Fig. 8. Time series of radius of maximum wind (km).
32 apeapist 3 betvl dnie ywl replac eyewall uncondit identified the objectively minimu to 131 and relative to are line) applies Anomalies sample (black wind panel). maximum (bottom of line) anomalies and panel) Fig. .Sml-entmoa vlto faoayi h aiso radius the in anomaly of evolution temporal Sample-mean 9.
wind (m/s) & pres (hPa) anomalies
− radius max wind (km) − 30 50 10 20 40 15 10 10 − − 0 5 5 0 48 48 − − 36 36 − − 24 24 − − 12 ie(hours) time 201 43 48 36 24 12 0 12 ie(hours) time 33 22 648 36 24 12 0 mn yls e etfrdetails. for text see cycles; ement oe eni ahfil.The field. each in mean ioned eta rsue(gray pressure central m aiu id(top wind maximum f v autocorrelation 1.2
1
0.8
0.6
0.4 correlation 0.2
0
−0.2 0 50 100 150 200 lag (hours)
120
100
80
60
time (hours) 40
20
0 0 50 100 150 200 radius (km)
Fig. 10. (top panel) Temporal autocorrelation (solid line) at 40 km radius with 95% confidence bounds (dashed lines). (bottom panel) e-folding autocorrelation time (solid line) and zero-crossing autocorrelation time (dashed line) as a function of radius. The zero- crossing time is defined when the lower bound on the 95% confidence interval first crosses zero.
34 1 1 0.8 0.8 0.6 0.6 0.4 0.4
error variance 0.2 0.2 0 0 0 20 40 60 0 20 40 60 (a) (b)
1 1 0.8 0.8 0.6 0.6 0.4 0.4 error variance error variance 0.2 0.2 0 0 0 20 40 60 0 20 40 60 (c) forecast lead time (hours) (d) forecast lead time (hours)
Fig. 11. Linear inverse model sample-mean (bold lines) and range of one standard deviation (error bars) of forecast error variance as a function of lead time (abscissa). Panels show (a) radial wind; (b) azimuthal wind; (c) potential temperature; and (d) cloud water mixing ratio. Error variance is normalized by the climatological value, so that skill is lost when the mean value reaches unity, and standard deviation reaches values at large forecast lead time.
35 variance. Fig.
2 nlgfrcs ro safnto fla ie normaliz time, lead of function a as error forecast Analog 12. normalized error variance 0 1 0 0 0 0 ...... 9 1 5 7 8 1 6 02 04 50 40 30 20 10 0 oeatla ie(h) time lead forecast 36 60 08 90 80 70 db climatological by ed 1.4 NHC 2007-2011 idealized LIM 1.2
1
0.8
0.6
0.4 normalized error variance
0.2
0 0 10 20 30 40 50 60 70 80 forecast lead time (h)
Fig. 13. Comparison of the linear inverse model forecast errors for azimuthal wind (gray line) with operation forecasts of tropical cyclone maximum wind from the National Hurricane Center (NHC, black line). The NHC errors apply to the Atlantic ocean and are normalized by the value at a forecast lead time of 84 hours.
37 t=0h t=6h
15 15
10 10
5 5 height (km)
0 0 (a) 50 100 150 (b) 50 100 150
t=12h t=24h
15 15
10 10
5 5 height (km)
0 0 (c) 50 100 150 (d) 50 100 150 radius (km) radius (km)
Fig. 14. Azimuthal wind evolution for the leading optimal mode of the linear inverse model that explains the most forecast error variance (black lines). Thick (thin) lines show positive (negative) values every 1 m s−1 (a); 2 m s−1 (b); and 4 m s−1 (c) and (d); the zero contour is suppressed. Thin gray lines show the time-mean azimuthal wind every 5 m s−1 starting at 40 m s−1.
38