This study proposes a framework for estimating how much tropical cyclone intensification could result from the amount of energy released inside of a convective burst. A convective burst is a sequence of vigorous convective cells occupying one portion of a tropical cyclone's eyewall for approximately 9 to 24 h. On the basis of Tropical Rainfall Measuring Mission (TRMM) satellite radar observations and previous modeling studies, a typical convective burst may release in 12 h an extra 6 × 1017 J of latent heat. TRMM observations suggest that this extra energy represents an increase of 25% or more in the rate that the eyewall releases latent heat prior to the convective burst. Previous studies suggest that 4.5% to 11% of this extra latent heat may be transformed, after a lag of several hours, into an increase in the kinetic energy of the tropical cyclone's inner-core tangential wind. On the basis of the H*wind analysis of aircraft and dropsonde observations, an increase in kinetic energy of this magnitude may be associated with an intensification of 9–16 m s−1 (17–31 kt) in a tropical cyclone's maximum surface wind. This conservative estimate takes into account the increase in ocean surface friction during the period of intensification and assumes that the associated increase in ocean surface enthalpy flux does not counteract any of the frictional loss. Despite sources of uncertainty, it still appears that significant intensification is possible from the amount of energy released inside of a typical convective burst.
Malkus and Riehl  and Riehl and Malkus  hypothesized that a tropical cyclone is maintained by the latent heat released inside of convective cells in the eyewall. A series of particularly vigorous cells in one portion of the eyewall is commonly called a “convective burst” [Hennon, 2006]. A number of studies have found that convective bursts persist for at least 9 h and rarely longer than 24 h [Steranka et al., 1986].
 Following the lead of studies such as Malkus and Riehl  and Montgomery et al. , this study hypothesizes that the energy to intensify the tropical cyclone passes through the convective burst. This idea likens the convective burst to an energy “pipeline.” As an alternative to the pipeline analogy, the convective burst might change the spatial distribution of kinetic energy already in the tropical cyclone. This second option treats the convective burst as a “reorganizing agent,” and evidence consistent with this option comes from this study's examination of the H*wind analysis.
 The goal of this study is to propose a framework for estimating the amount of intensification that might result from the energy released inside of a typical convective burst under the energy pipeline analogy. Previous studies support the energy pipeline analogy by invoking forced subsidence or vorticity. As shown schematically in Figure 1, vigorous convective cells in the eyewall have been observed to cause substantial subsidence, sometimes deep in the tropical cyclone's eye, which could warm the eye [Heymsfield et al., 2001, p. 1311; Schubert et al., 2007; Guimond et al., 2010, Figure 4; Molinari and Vollaro, 2010, p. 3882]. Following such a warming, the eye's surface pressure would drop through hydrostatic adjustment. The tangential wind outside of the eye would speed up to reestablish approximate gradient wind balance with the new, lower central pressure [Willoughby, 1998]. Another candidate mechanism is that a vigorous cell transports vorticity into the eyewall. Some of the additional vorticity later becomes mixed azimuthally around the eyewall, increasing intensity [Montgomery et al., 2006].
 This study puts together recent observations and modeling results to make a rough estimate of intensification using the three steps that are outlined below and in the block diagram (Figure 2). The first of these three steps is to estimate the amount of latent heat released inside of a convective burst in excess of the amount of latent heat normally released inside of the eyewall when no convective burst is present. Satellite radar observations during the past decade provide statistics on the amount of rainfall that falls out of a convective burst, which to a first approximation is proportional to the net condensation and therefore also proportional to the net latent heat release. The second step is to estimate what portion of the extra latent heat release may be converted into increased kinetic energy of the tangential wind. The recent modeling work of Nolan et al.  provides such an efficiency. The third step is to consider how the kinetic energy increase is distributed throughout the three-dimensional structure of the tropical cyclone's inner-core tangential wind field. This study uses the H*wind analysis of Powell et al. [1998, 2010] to map kinetic energy increase to intensity increase.
Section 2 constructs the equations that implement the three-step process described in the Introduction section and in the block diagram (Figure 2). These equations have parameters whose values are estimated in section 3 and are listed in Table 1.
Table 1. Input Parameters for Estimating the Intensification That Could Result From the Amount of Energy Released Inside of a Typical Convective Burst
The point value is used to estimate intensification in section 2. When two values are provided, the first is for a borderline category 1–2 tropical cyclone with 42 m s−1 (82 kt) initial intensity and the second is for a borderline category 4–5 tropical cyclone with 70 m s−1 (135 kt) initial intensity. When the units used in the calculation are different than the units used in the text, the value expressed in computational units is provided in parentheses.
The range is used in the error analysis in section 3.5.
Exponent relating intensity change to change in the kinetic energy of the tangential windg
The exponent relating intensity change to change in the frictional dissipation rate at the ocean's surfaceg
2.1. Released Energy
 Satellite observations of the surface rain rate provide a means to estimate the net column-integrated latent heat release inside of a convective burst. Latent heat is released in the eyewall even in the absence of a convective burst, so the quantity of interest is really the extra amount of latent heat released in the eyewall due to the presence of a convective burst.
Section 3.1 suggests that a typical convective burst increases the surface rain rate by approximately 40 mm h−1 when averaging over a 500 km2 area of the eyewall and over the duration of the convective burst. Even after averaging over the 500 km2 area, the enhancement to the surface rain rate will still vary with respect to time as the convective burst becomes more or less active. Any such fluctuations will not affect the results of this study, as long as 40 mm h−1 is the mean enhancement to the surface rain rate after averaging over both space and time.
 To calculate latent heat release, one must first convert to different units the convective burst's enhancement to the surface rain rate ΔR (mm h−1). To convert to rain mass flux Fprecip (g m−2 s−1), the partial canceling of a space factor of 1000 with a time factor of 3600 results in a coefficient of 5/18:
 Multiplying the rain mass flux by duration tburst (s) and ocean surface area Aburst (m2) gives the extra mass Mprecip (g) of rain that reaches the ocean surface:
 Rather than falling out as rain, some of the net condensation exits the convective burst as cloud ice in the upper-level outflow (Figure 1). This ice is observed as the large cirrus cloud shield that often obscures a satellite's view of the inner core during a convective burst [Gentry et al., 1970]. Section 3.1 estimates that only about 10% of the eyewall's net condensation exits the eyewall as cloud ice.
 One arrives at the extra latent heat Erelease (J) by multiplying the net condensate mass by the specific heat of vaporization Lv (2.5 × 103 J g−1). This calculation completes the first step in the block diagram (Figure 2). As equation (3) shows, either the precipitation mass can be scaled up by a factor fprecip of 1.1 to take into account the ∼10% of the net condensate that leaves as cloud ice or the cloud ice Mcloud (g) can be explicitly included in the equation:
 Using equation (3) and the parameters in Table 1 results in the convective burst releasing an extra 6.6 × 1017 J of latent heat during 12 h, beyond what the eyewall of a typical tropical cyclone releases when no convective burst exists. To put this value in context, section 3.2 analyzes Tropical Rainfall Measuring Mission (TRMM) satellite observations and finds that the convective burst's extra 6.6 × 1017 J of latent heat release represents a greater than 40% increase over the amount of latent heat that a weak tropical cyclone's eyewall would have released in 12 h and a greater than 25% increase over what a strong tropical cyclone's eyewall would have released in 12 h. In this text, “weak” means a minimal category 2 tropical cyclone and “strong” means minimal category 5 tropical cyclone on the Saffir-Simpson scale. By this criteria, weak and strong tropical cyclones have a 42 and 70 m s−1 (82 to 135 kt) intensity, respectively. Section 3 takes intensity to be the maximum wind reported in the 2-D surface field of the H*wind analysis, but the equations derived in section 2 are agnostic about where one obtains intensity estimates, either from H*wind, from National Hurricane Center best-track data, or from some other source.
Section 3.1 estimates the fraction fusable (unitless) of the extra latent heat release that may be converted into increased kinetic energy of the tropical cyclone's tangential wind. After multiplying the released energy by fusable, the result is the energy Eusable (J) that can be used to increase kinetic energy:
2.2. Intensification Ignoring the Increase in Friction
 The last step in the block diagram (Figure 2) is to map the increase in kinetic energy to an increase in intensity. Loosely following Maclay et al. [2008, equation (3)], this study uses a power law to match the fractional change in intensity to the fractional change in kinetic energy. The power law exponent e2 (unitless) maps the final and initial kinetic energy, KEf and KEi (J), to the final and initial intensity, If and Ii (m s−1):
 Kinetic energy is estimated from the H*wind surface wind field, and intensity is taken to be the maximum of the H*wind surface wind field. In section 3.3, exponent e2 is estimated by integrating out to a 70 km radius. To test the sensitivity of the integration limit, the integration was repeated with a 100 km radius and there was little change in the intensification. Nolan et al.  used 50% to 150% of the radius of maximum wind as the radial limit of integration and also found that the integration limit had little effect on the resulting intensification.
 Next, equation (5) is manipulated algebraically to create equation (6). Equation (6) is a better equation for determining the intensification that results from the amount of energy released inside of a convective burst:
Equation (6) expresses the change in inner-core kinetic energy ΔKE (J) using a ratio y (unitless) between final and initial intensity, defined as follows:
 It is easy to solve for the intensity change by replacing ΔKE in equation (6) with Eusable and then solving the resulting equation for y:
 The parameters in equation (8b) other than y have values that are estimated in section 3 and stated in Table 1. Substituting these values into equation (8b) gives an intensification of 14 or 27 m s−1 (27 or 52 knots) for a weak or strong tropical cyclone, respectively.
 The 12-h 14–27 m s−1 intensification proposed in this section would qualify, or at least comes close to qualifying, as rapid intensification even if no further intensification occurred during the rest of the 24-h period. A commonly cited definition of rapid intensification is 15.4 m s−1 in 24 h, which was first proposed by Kaplan and DeMaria [2003, p. 1106]. Kaplan and DeMaria chose this threshold because it was at the 95th percentile of 24-h intensity change in North Atlantic tropical cyclones.
2.3. Intensification Considering the Increase in Friction
 While section 2.2 and Nolan et al. [2007, p. 3380] omit this effect, ocean surface friction does increase when intensity begins to increase. This section considers that during the period that intensity is increasing, some of the added energy will end up lost due to the wind-induced increase in friction. This energy loss reduces the intensification from what was estimated in section 2.2.
Appendix A integrates the increase in frictional flux over the duration of the intensification period to come up with an estimate of the extra ocean surface frictional energy loss, ΔEfriction (J). Subtracting ΔEfriction from Eusable, as equation (9) does, is mathematically equivalent to reducing Eusable by lowering efficiency factor fusable in equation (4):
 This study chooses to handle friction by subtracting ΔEfriction so that this study can employ in equation (4) a value for the efficiency factor based on the constant friction results of Nolan et al. .
 The increasing-friction formulation (equation (9)) must be solved numerically to arrive at a value for y, the fractional change in intensity, because ΔEfriction is a function of y. Using the parameters in Table 1, the result is a weak tropical cyclone experiencing an intensification of 9 m s−1 (17 knots) from the amount of latent heat released inside of a typical convective burst during a 12 h period. A strong tropical cyclone experiences 16 m s−1 (31 knots) in 12 h from the same amount of latent heat release.
 This increasing-friction formulation actually overestimates the effect of the friction increase (and underestimates the intensification) because it ignores that the flux of enthalpy from the ocean surface into the boundary layer also immediately increases when the intensification period begins, just as friction immediately increases when intensification begins. Friction immediately works as a drag on wind speed, while enthalpy flux requires time (perhaps several hours) to work its way into free troposphere, be transformed into eye warming, and eventually help the tropical cyclone overcome the increased ocean surface friction.
 Another reason why the intensification may be underestimated in the increasing-friction formulation is that the increasing-friction formulation assumes that friction is purely detrimental to intensification. In contrast, Smith et al. [2009, p. 1323] suggest that increasing friction will increase the radial inflow of moisture into the eyewall which might invigorate convection, releasing more latent heat into the eyewall.
 In short, the 9–16 m s−1 increasing-friction estimate of this section and the 14–27 m s−1 constant friction estimate of the previous section might bracket the most realistic intensity increase. To be conservative, this study emphasizes the 9–16 m s−1 range.
3. The Observational and Modeling Basis for the Input Parameters' Values
3.1. Parameters Related to Latent Heat Release
 This section describes how observations and modeling studies are used to choose typical values for the latent heating parameters that are used in the equations in section 2.1 and that are listed at the top of Table 1.
 Each of 136 TRMM satellite overflights of tropical cyclone eyewalls is categorized as containing or not containing a convective burst based on whether the eyewall has a ≥4 × 104 km2 area of 11 μm infrared brightness temperature ≤203 K (−70°C) and a minimum 85 GHz horizontally polarized passive-microwave brightness temperature of ≤190 K [Hennon, 2006, p. 12]. While this study conceives of a convective burst as a period of 12 h of enhanced convection, this section uses a single overflight of the TRMM satellite to segregate burst from nonburst cases. If a separate study were undertaken, patching together observations from multiple satellites and factoring in any calibration issues between them, then a more-or-less continuous coverage of near-simultaneous microwave and infrared observations would be possible. Were such a study performed, the separation of burst from nonburst cases might differ for some of the cases considered here.
 In each of these overflights, the eyewall is located in the surface precipitation by manually choosing an inner and outer radius that encloses the eyewall's heavy surface precipitation. As will be described below, the TRMM radar reveals that most of the extra surface precipitation mass in burst versus nonburst cases occurs in the quarter of the eyewall (a 90° arc) with the heaviest average surface rainfall. Figure 3a shows the area of one quarter of the eyewall for the 42 convective burst cases. For these same cases, Figure 3c shows the surface rain rate averaged over this area. Figure 3d plots the corresponding rain rate average for the non-convective-burst cases, i.e., rain rate averaged over the 90° arc of the eyewall that has the highest average rain rate among all possible 90° arcs.
 The goal is to calculate the enhancement to the latent heat release in 12 h due to the presence of the convective burst, so the scatter of areas (Figures 3a–3b) and rain rate averages (Figures 3c–3d) are used to estimate the latent heat release (Figure 3f).
 Supporting a statement a few paragraphs earlier, Figure 3f shows that most of the extra latent heat released in the whole eyewall is actually released in just one quarter of the eyewall. To see this result, note that the median difference in latent heat release with and without a convective burst is similar whether one is looking at a quarter of the eyewall (black lines in Figure 3f) or one is looking at the entire eyewall (blue lines in Figure 3f). If one simplifies these results by taking the area of the convective burst to be 500 km2, as is done in Table 1 and section 2.1, then that requires that one use a rainfall rate enhancement of ∼40 mm h−1 in order to get a correct enhancement to the latent heat release.
 The eyewall rain rate estimated by the TRMM Precipitation Radar and shown in Figure 3 is higher than the rain rate estimated by the TRMM Microwave Imager (TMI). Averaging TMI observations of many tropical cyclones, Lonfat et al. [2004, Figure 11a] report that the average TMI rain rate is just 7 or 12 mm h−1 in the vicinity of the eyewall of a weak or strong tropical cyclone, respectively. Langousis and Veneziano [2009, p. 2] explain why the eyewall rain rate of the TRMM radar is likely to be more accurate than the lower rate estimated by TMI.
 As mentioned in section 2.1, it is difficult to improve on the estimate that ∼10% of the convective burst's net condensate ends up as cloud ice that leaves in the upper-level outflow (Table 1, fourth row). This 10% figure is based on the hurricane simulations of Braun [2006, p. 59] and the Doppler radar analysis of Gamache et al. [1993, p. 3239]. Other observations are in broad agreement with this 10% figure. For example, a large cloud shield might have 0.1 g kg−1 ice concentration, a 200 km radius, a 2 km depth, and thereby contain 2.5 × 1013 g of cloud ice, which is close to 10% of the 2.6 × 1014 g of precipitation mass estimated by equation (2). Sources of uncertainty in this estimate include the fact that the cloud-ice concentration in high-altitude, tropical, maritime/coastal clouds can be anything from 0.001–0.01 g kg−1 to ∼0.3 g kg−1 (low estimates: Heymsfield and Donner [1990, Figure 5], Knollenberg et al. [1993, Figure 16]; high estimates: Pueschel et al. [1995, Figure 2], Kelley et al. [2010, Figure 5i]). Also, cirrus cloud shields are often smaller than this hypothesized 200 km radius [Heymsfield et al., 2001, Figure 3; Guimond et al., 2010, Figure 5]. The smaller cloud shield size may be due to cloud ice being subject to a 0.1–1.0 m s−1 fall speed [Heymsfield and4 Iaquinta, 2000, Figure 14a; Schmitt and Heymsfield, 2009, Figure 3], slow subsidence [Kossin, 2002, Figure 8], and evaporation after mixing with background cloud-free air above and below the cloud shield [Houze, 1993, pp. 180–182].
3.2. Parameters Related to Initial Kinetic Energy and Friction
 This section describes how typical values were identified for the kinetic energy and ocean surface frictional flux (Table 1, sixth and seventh rows). The kinetic energy KE (J) in question is the three-dimensional kinetic energy of the tropical cyclone's inner-core tangential wind (equation (10a); Figure 4a). Frictional flux F (J s−1) is calculated by substituting the H*wind analysis into equation (10b), the bulk aerodynamic formula [Emanuel, 1997, equation (7); Emanuel, 2003, equation (5)], as shown in Figure 4b.
 The factors of the kinetic energy equation (equation (10a)) are listed in the order that they appear. Appendix B estimates s0 (unitless), the parameterization of the vertical variation in tangential wind speed. The mass m (kg m−2) of air in the tropospheric column from the surface to 12 km altitude is taken to be 80% of the total mass of a hydrostatic atmosphere with scale height H (8500 m) defined by Bohren and Albrecht [1998, p. 55–56]. In this study, the vertical integration is intentionally similar to the 2–12 km integration in the work of Nolan et al. [2007, p. 3401]. In this study, surface pressure under the eyewall is taken to be 970 hPa in both strong and weak tropical cyclones. Holland [1980, Figure 2a] suggests the 970 hPa value is approximately correct for moderate and strong tropical cyclones. The 970 hPa value is perhaps a ∼15 hPa underestimate of eyewall surface pressure for a weak tropical cyclone, but it has little effect on the results of this study. The area of each grid box of the H*wind analysis is Agrid (m2). The summation is over all the grid boxes within 70 km of the center of the eye, for reasons explained in the next section. The 700 hPa tangential wind speed v700hPa (m s−1) is here estimated as the H*wind surface wind speed divided by 0.85. Powell et al. [2009, Table 3] state that using a factor of 0.80 or 0.90 would overestimate or underestimate the difference between surface and 700 hPa wind speed, so this study uses a factor of 0.85.
 The factors of the frictional flux equation (equation (10b)) are listed in the order they appear. For the sake of simplicity, this study uses the same drag coefficient (cd = 2 × 10−3) for weak and strong tropical cyclones following Vickery et al. [2009, Figure 3] and Andreas and Emanuel [2001, p. 3746]. Other studies have suggested lower values for cd of 1.4 × 10−3 to 1.8 × 10−3 in weak tropical cyclones [Powell et al., 2003] or 0.6 × 10−3 to 2.2 × 10−3 for strong tropical cyclones [Soloviev and Lukas, 2010, Figure 4]. The surface air density under the eyewall is ρo (kg m−3). The surface wind speed as reported in the H*wind analysis is vsurf (m s−1).
3.3. Parameters Related to Change in Kinetic Energy and Friction
 This section describes how typical values were identified for the power law exponents that map change in intensity to change in either kinetic energy or ocean surface friction (Table 1, eighth and ninth rows). Kinetic energy and ocean surface friction are integrated within a 70 km radius measured from the storm center.
 To calculate the typical kinetic energy increase associated with an intensity increase, pairs of H*wind analyses are compared 12 h apart using equation (10a). For each 12 h interval, the power law exponent e2 is calculated using equation (5) as shown in Figure 4c. Table 1 (eighth row) reports the median value for the power law exponent during normal periods of intensification that are associated with increases in kinetic energy (the red symbols in Figure 4c).
 To calculate the typical increase in frictional flux associated with an intensity change, the analysis is repeated, this time using equation (10b) to calculate frictional flux and equation (A2) to calculate power law exponent e3. The scatter of values for e3 is shown in Figure 4d, with a typical value reported in Table 1 (ninth row).
 This study uses 27 twelve-hour intensification periods in the H*wind analysis to estimate the intensification that might result from the amount of energy released inside of a typical convective burst, conceiving of a convective burst as an energy pipeline. If changes in the spatial distribution of kinetic energy deviated from the typical, then much greater intensification could result. For example, five periods of intensification were excluded from the analysis because kinetic energy decreased (the green symbols in Figure 4c). Tables 2 and 3 and Figure 4g show that these five periods experienced greater intensification in the National Hurricane Center best track data than the intensification of the H*wind maximum wind. Either the kinetic energy is becoming more tightly concentrated at the radius of maximum winds or the overall area of the inner core is shrinking. The possibility of a shrinking radius of maximum winds is discussed by Vickery and Wadhera [2008, Figure 9], Willoughby and Rahn [2004, Figure 6], and Smith et al. [2009, Figure 7].
Table 2. North Atlantic Tropical Cyclones for Which Previous Observational Studies Provide Estimates of the Vertical or Horizontal Variation in Tangential Wind Speeda
Date and Time of Observation
First seven hurricanes were used to generate the vertical profiles in Figure B1a. The last seven were used to generate Figure 4.
Table 3. Twelve-Hour Periods of Atlantic Tropical Cyclones During Which the Maximum H*Wind Speed Increases but the H*Wind Inner Core Kinetic Energy Decreases
Date and Start Time of 12-h Period
12-h Change in Intensity (m s−1)
H*Wind Change in Max Wind
Best-Track Change in Intensity
1930 UTC 02 Sep 2007
1730 UTC 21 Sep 2005
1930 UTC 07 Nov 2008
0730 UTC 16 Jul 2007, 0130 UTC 17 Jul 2007
3.4. The Efficiency Parameter
 Linear perturbations in the tropical cyclone model of Nolan et al.  explicitly result in 4.5% of the latent heat that is released nonsymmetrically in one portion of a weak tropical cyclone's eyewall being transformed by the vortex into increased kinetic energy of the tropical cyclone's tangential wind (Table 1, fifth row). Future studies may include nonlinear effects not modeled by Nolan et al. and refine the value of this efficiency factor.
 The rest of this section extrapolates Nolan's results to a strong tropical cyclone. One could simply fit a power law to Nolan's results for weak tropical cyclones and extend the line to strong tropical cyclones. Such a purely mathematical extrapolation gives an ∼11% efficiency for strong tropical cyclones.
 A physics-based extrapolation gives close to the same answer as the purely mathematical extrapolation that was just described. One unplanned aspect of Nolan's model output for weak tropical cyclones is that the latent-heat-to-kinetic-energy conversion efficiency increases in proportion to the increase in the inertial stability of the tropical cyclone's vortex. If one assumes that such a proportionality is maintained in strong tropical cyclones, then the extrapolated efficiency for strong tropical cyclones (dotted line in Figure 5) is approximately the same as the previously stated 11% efficiency that is independent of any consideration of inertial stability.
 A rough proportionality between latent heat conversion and inertial stability is plausible, in a balanced axisymmetric vortex, because sufficiently gentle radial perturbations will give rise to waves that have a frequency that is less than or equal to the square root of the inertial stability [Kepert, 2010]. Incidentally, these waves are analogous gravity waves set up by vertical perturbations [Kepert, 2010, p. 5–6]. As inertial stability increases, the upper bound to the frequency of trapped radially forced waves increases, trapping a larger fraction of them. When a larger fraction of waves are trapped, one would expect more of the latent heat release to eventually be converted to inner-core kinetic energy.
 This study calculates density-weighted average 3-D inertial stability in the inner core for the H*wind analyses, as plotted in Figure 4f. Various studies use different formulas to calculate inertial stability. Kepert [2010, equation (19)] and Fudeyasu and Wang [2011, equation (4)] calculate inertial stability I2 (s−2) as the product of the absolute Coriolis parameter fabs (s−1) and the absolute vorticity ɛ (s−1):
Shapiro and Montgomery  provide an expression for fabs. In equation (11), f0 (s−1) is the planetary Coriolis parameter [Wallace and Hobbs, 2006, p. 277], vθ (m s−1) and vr (m s−1) are the tangential and radial components of the H*wind wind speed, δ/δθ and δ/δr are the tangential and radial derivatives, and r (m) is the distance from the center of an H*wind grid box to the center of the tropical cyclone's eye. Because the H*wind analysis is not axisymmetric, this study includes in ɛ the δ/δθ term of the vertical component of the curl of velocity, as does Kepert [2010, equation (15)]. In contrast, Nolan et al. [2007, equation (5.5)] can omit the δ/δθ term because their wind field is axisymmetric after it adjusts to the added latent heat.
3.5. Input Parameter Uncertainty
 One can propagate uncertainty in input parameters to the estimate of usable energy Eusable (J) and then propagate uncertainty to the estimated intensification ΔI (m s−1). Both Eusable and ΔI can be expressed as a function of their input parameters:
 The propagation of errors formula can be found in the work of Bevington and Robinson [1992, p. 43]. The rightmost column of Table 1 lists uncertainty in the input parameters that implies, through error propagation, a factor of 2.3 uncertainty in the intensification rate. Under the conservative estimate of increased friction and no effect from increased enthalpy flux, the previously calculated point estimate is 9 m s−1 for a weak tropical cyclone and 16 m s−1 for a strong tropical cyclone. Applying the 2.3 uncertainty factor, these point estimates become 4–21 m s−1 for a weak tropical cyclone and 7–37 m s−1 for a strong tropical cyclone.
 Since rapid intensification is usually defined as 15.4 m s−1 in 24 h, then “half a day's worth” of rapid intensification would consist of 7.7 m s−1 in 12 h. Most of the weak tropical cyclone's 4–21 m s−1 intensification range is above 7.7 m s−1 and virtually all of a strong tropical cyclone's 7–37 m s−1 intensification range is as well. In summary, a 12-h-long convective burst can cause half a day's worth of rapid intensification if it avoids the least favorable combinations of parameter values.
 This section explains why latent heat of vaporization is calculated in this study rather than other measures of energy. Only latent heat of vaporization (latent heat of condensation and evaporation) is calculated in the “released energy” equation, equation (3). In contrast, latent heat of fusion (latent heat released when water freezes) is omitted because heat of fusion is an insignificant fraction (∼1%) of the net latent heating inside of a convective burst. Only 10% of the condensate freezes and remains frozen (equation (3)). The rest of the net condensate never freezes or remelts before it falls to the ocean surface. The specific latent heat of fusion (Lf = 0.33 × 103 J g−1) is only ∼10% of the specific latent heat of vaporization (Lv = 2.5 × 103 J g−1). Combining these two factors of 10% result in the just-stated 1%.
 Even though latent heat of fusion is not explicitly included in the calculations, the authors acknowledge the great importance that freezing and melting can have on convective dynamics. Freezing and melting redistribute heat within the atmospheric column and do help convective cells to reach high altitude [Zipser, 2003]. In other words, latent heat of fusion is not explicitly included in the equations of this study, but it affects the efficiency factor fusable in equation (4).
Emanuel [2003, p. 84] describes a tropical cyclone as balancing enthalpy flux from the ocean with frictional loss at the ocean surface. Nonetheless, this study tracks latent heat instead of enthalpy (latent plus sensible heat) [Glickman, 2000]. The reason for preferring latent heat, in this context, is that this study uses the energy efficiency calculated by Nolan et al. . Nolan et al. use a vertical profile of heat release that is modeled after the vertical profile of latent heat release. As sections 2.2 and 2.3 have shown, the latent heat released by a convective burst appears, by itself, to be sufficient to cause significant intensification.
 Moist static energy is the sum of enthalpy and gravitational potential energy [Bohren and Albrecht, 1998, p. 290]. This study does not track intermediate exchanges between gravitational potential energy and other forms of energy as the released energy works its way toward becoming increased kinetic energy of tangential wind. Instead, intermediate transformations are summarized in this study by the use of an efficiency factor of latent heat release to kinetic energy increase, following the work of Nolan et al. .
 The framework developed in this study represents a convective burst as an energy pipeline that carries extra latent heat up from the boundary layer and releases the latent heat into the free troposphere. A convective burst is simply a sequence of vigorous convective cells that occupies one portion of a tropical cyclone's eyewall for a period of at least 9 h and rarely longer than 24 h.
 This study estimates that a typical convective burst releases an extra 6.6 × 1017 J of latent heat every 12 h. TRMM satellite radar observations suggest a roughly equal boost to the latent heat release regardless of whether the convective burst occurs in a weak or strong tropical cyclone. This extra latent heat release represents an increase of 25% or more, on average, in the rate that eyewall convection releases latent heat prior to the convective burst.
 In a strong tropical cyclone, a 12-h-long convective burst appears to be able to cause intensification of ≥16 m s−1 in about 12 h, which would qualify as rapid intensification even if no further intensification occurred during the rest of the 24-h period. A convective burst in a weak tropical cyclone appears able to cause intensification of ≥9 m s−1 in about 12 h (half of the 24-h rapid intensification threshold in half of a 24-h period). The same amount of latent heat release causes more intensification in a strong tropical cyclone because a strong tropical cyclone has about a 2.5-times greater efficiency at converting released latent heat into increased tangential kinetic energy.
 Ignoring boundary layer processes (increases in ocean surface enthalpy flux and friction), a 12-h-long convective burst would drastically increase intensity by 14–27 m s−1 (27–52 knots). This estimate, however, appears too high because it ignores the increase in ocean surface friction during the ∼12 h of intensification. More conservatively, the intensity increase would be smaller (9–16 m s−1) if one ignores the increase in enthalpy while considering the increase in friction during the intensification period. Friction's detrimental effect on intensity is immediate and relatively easy to estimate while the beneficial effect of increased enthalpy flux is harder to estimate and subject to a time lag. Owing to complications such as this one, it is left to future studies to propose more complete ways to handle increased friction and enthalpy flux.
 The intensification estimated in this study's “energy pipeline” framework is subject to a factor of 2.3 uncertainty due to uncertainty in the input parameters that define the convective burst and the tropical cyclone's inner core. This uncertainty could be reduced if future studies improve estimates of the drag at the ocean surface, estimates of the three-dimensional tangential wind field's variation with intensity, or estimates of the surface rain rate under the eyewall, among other factors.
Appendix A:: Increase in Friction During the Intensification Period
 This section calculates the extra energy loss due to the increase in ocean surface inner-core friction during a period of tropical cyclone wind intensification. In equation (A1), initial intensity Ii (m s−1) increases to I(t) after time t (s) elapses:
 The fractional increase in intensity at time t is y(t). The H*wind analysis is used to fit a power law exponent e3 relating the change in intensity to the change from initial frictional flux Fi (J s−1) to frictional flux F(t) (J s−1) at time t:
 Intensity is assumed to increase linearly with time to simplify the derivation. A constant rate of intensity increase is a simplification, even for a fairly short period such as 12 h. Hack and Schubert [1986, Figure 6] do show approximately constant intensification over 12 h in a model that shows significantly nonconstant intensification over 1 or 2 days. For simplicity, equation (A3) assumes that the period of intensification is 12 h long, i.e., equal in duration to the 12 h period during which latent heat is released inside of the convective burst:
 Next, the extra amount of frictional energy loss ΔEfriction (J) is calculated. The extra energy loss is the integral of frictional flux increase, F(t) − Fi, with respect to time. This integration is integration of the form xb − 1 with respect to t where x = 1 + at:
 The definite integral in equation (A5) is evaluated, like terms combined, and (y − 1)/tburst substituted for c/Ii. The result is equation (A6), in which y is shorthand for y(tburst):
 It is easy to calculate the 3-D inner-core kinetic energy from v700hPa (m s−1), the 700 hPa wind speed, if one assumes no altitude variation in the tangential wind. This rough estimate is designated KE700hPa (J).
 This appendix uses two methods to improve KE700hPa by calculating a scaling factor s0 (unitless) that parameterizes the effect of the vertical variation in tangential velocity:
The “improved” estimate of inner-core 3-D kinetic energy is designated KE3D (J).
 The first method estimates the scaling factor using several time steps of an MM5 simulation of Hurricane Bonnie (1998) and finds s0 to be fairly close to 1 (i.e., 0.8 to 0.9). These calculations are shown as diamonds in Figure B1a. This method takes into account the slope of the radius of maximum wind with altitude. This MM5 simulation is described by Braun .
 The second method uses vertical profiles from several studies of tropical cyclones and finds a similar range of 0.88 ± 0.05 for s0. These calculations are made for the tropical cyclones listed in Table 2 (first through seventh rows) and are shown as plus signs in Figure B1a. This second method does not take into account the outward slope of the radius of maximum wind, but it has the advantage that it can be applied to an individual vertical profile from a tropical cyclone rather than requiring a 3-D wind field. The vertical profiles used to calculate the plus signs in Figure B1a are shown in Figure B1b. For comparison, the simulated vertical variation in Hurricane Bonnie is shown in Figure B1c, and it agrees in its large-scale features with the observed profiles shown in Figure B1b.
 To calculate s0 with either of the two above-mentioned methods, one begins with the general definition of kinetic energy KE (J) for any velocity v (m s−1):
When a 3-D wind field is available from a model, one can calculate s0 via method 1:
When one has instead an observed vertical profile of velocity v(z) (m s−1), then s0 can be calculated via method 2:
 Before using equation (B4), each observed profile is expressed as the product of a unitless shape function s(z) times that profile's 700 hPa wind speed v700hPa (m s−1). The shape functions are plotted in Figure B1b for the observed profiles used in this study. The column mass from the surface to 12 km altitude is m (kg m−2).
 TRMM satellite data was provided by NASA and JAXA. H*wind analysis provided by NOAA's Hurricane Research Division. The first author would like to thank his Ph.D. advisor, Michael Summers, for providing feedback on early results related to this study. The first author was supported through NASA grant NNX07AJ22A. The second author was supported in part by NASA grant NNH08ZDA001N-HRSP. Both authors would like to thank Ramesh Kakar for his continued support of TRMM. Three anonymous reviewers provided suggestions that significantly strengthened this text. Computational facilities were provided by the Precipitation Processing System at NASA Goddard.