Volume 49, Issue 19 e2022GL098926
Research Letter
Open Access

The Dependence of Tropical Cyclone Pressure Tendency on Size

Nathan Sparks

Corresponding Author

Nathan Sparks

Blackett Laboratory, Department of Physics, Imperial College, London, UK

Correspondence to:

N. Sparks,

[email protected]

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Ralf Toumi

Ralf Toumi

Blackett Laboratory, Department of Physics, Imperial College, London, UK

Department of Physics, Grantham Institute – Climate Change and Environment, Imperial College, London, UK

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First published: 01 October 2022
Citations: 2


Current theories of tropical cyclone (TC) intensification give little direct indication of the role of the TC size in intensity changes, although there are observations showing a relationship. We develop a new model of TC central pressure tendency where the pressure change can be expressed as exponential with a time constant determined by the ratio of radius maximum wind (Rmax) and the column inflow or outflow speed. An analysis of observations confirms the relationship which becomes more important for a larger pressure tendency and suggests an upper bound on pressure tendency for a given Rmax. The dependence of the pressure tendency on size poses a challenging constraint on the accurate forecasting of TCs in numerical weather prediction and climate models.

Key Points

  • A new analytical model for tropical central pressure tendency is developed and tested against observations

  • The new model predicts an inverse size-dependence of both intensification and decay rates of tropical cyclones

  • The model predictions are supported by analyses of historical best track size and central pressure estimates

Plain Language Summary

Tropical cyclone (TC) intensity may be measured by central pressure or maximum wind speed. More intense TCs have a lower central pressure and higher wind speed. Most current models of TC development do not explicitly identify TC size as a key factor affecting the rate at which a TC intensifies or decays. We develop a simple new model based on the recently published idea that the rate of change of TC intensity can be directly related to the mass flux into or out of the core of the TC via an inflow or outflow speed at a given radius. This leads to the prediction that the rate of change of TC intensity through this mechanism is inversely related to the size of the TC core. We test this finding using observations of TCs where both pressure and size information are available. We find that the observations confirm the predictions made by the model, smaller TCs change intensity more quickly than larger TCs, both when they are intensifying and decaying. This finding may have important implications for accurate forecasting of TC intensity changes.

1 Introduction

Understanding tropical cyclone (TC) intensity changes remains a challenging problem. Although observational studies have identified some measure of the TC size as an important factor controlling intensification, an explicit physical explanation has not been available.

Chen et al. (2011) and Carrasco et al. (2014) observed a significant dependence on size of the likelihood of rapid intensification (RI) occurring. Wu and Ruan (2021) observed a contraction of the radius of maximum winds (Rmax) prior to (and during) RI events. Xu and Wang (2018a) showed that the intensification rate over 24 hr was correlated with Rmax for Western North Pacific TCs and North Atlantic (NA) TCs (Xu & Wang, 2015). Numerical simulations of idealized TCs have also shown a similar inverse dependence of intensification rate on the radius of maximum wind (K. A. Emanuel, 1989; Xu & Wang, 2018b). These studies are all concerned with intensification and do not examine a potential dependence of decay rate on size and hence lack analysis and theory which unifies intensification and decay tendency.

Recently Shou et al. (2021) observed trends toward smaller core size in Western North Pacific TCs, tentatively linked to a simultaneous trend toward increasing intensity. However, in a global analysis Wang and Toumi (2021) inferred a stable outer size while the life time maximum intensity has increased. This points to different relationships between the inner and outer size with intensity which has indeed been found in observations (Chavas et al., 2016; Frank, 1977; Merrill, 1984), simulations (Rotunno & Bryan, 2012), and is captured by a physical model for the radial structure of the TC wind field (Chavas & Lin, 2016; Chavas et al., 2015). A theoretical framework suggests a local control on the outer size (Wang & Toumi, 2022a). The outer circulation may stay relatively constant during large changes of the inner-core structure, so that Rmax may be a logical size metric to link to the central pressure tendency.

TC size is included as a predictor in an empirical RI forecasting model (Knaff et al., 2018). However, size is not always considered in intensity forecasts (Tam et al., 2021). K. Emanuel (2017) proposes an intensity prediction algorithm without an explicit size dependence.

There are fewer studies examining the ocean decay. Wood and Ritchie (2015) identify several environmental conditions linked to probability of rapid weakening but TC size is not considered. Wang et al. (2020) observed an increase in the frequency of rapid weakening and intensification events of TCs across the globe but a possible connection to changed in TC size was not presented.

Gaining a theoretical understanding of TC intensification has been a major research goal for many decades. Montgomery and Smith (2014) provide an overview of four leading theories describing the physical mechanisms behind TC intensification which have emerged over the last 50 years or so. Simplifying somewhat, these theories propose convectively driven inflow acting to amplify tangential wind speeds under the conservation of angular momentum. An equilibrium state may be reached where the energy input to the TC core (convergence of moisture and momentum) is balanced by frictional dissipation. The size of the TC is not typically not considered in solutions of these models. Ventilation theory (Tang & Emanuel, 20102012) is known to be an important control on the intensification and decay of TCs and does have a form of size-dependence. A formulation by Ramsay et al. (2020) (without size-dependence) was shown to have real predictive skill for intensification rates. Theoretical decay models over the ocean have been few. Montgomery et al. (2001) and Wang and Toumi (2022b) propose turbulent frictional decay as an important driver, but the effect of size is not considered.

Sparks and Toumi (2022) developed a model for TC central pressure deficit decay at landfall based on mass conservation. This model was used to explain the dependence of landfall decay rate on TC size in a series of idealized numerical experiments. Here we generalize those ideas to develop a new model applicable to both TC intensification and decay and test it against observations of TC size central pressure changes.

2 Theory

Sparks and Toumi (2022) modeled the TC as an axisymmetric vortex, considering only horizontal motion, and showed that the tendency of the average surface pressure within a cylinder of radius r can be written as,
where P is the surface pressure with angled brackets denoting the areal average within radius r and urn:x-wiley:00948276:media:grl64949:grl64949-math-0002 is the density-weighted column mean radial wind velocity. In the vicinity of the inner core, urn:x-wiley:00948276:media:grl64949:grl64949-math-0003 is positive (directed away from the center) for intensification and negative for TC decay.
Testing this relationship requires size observations. The radius of maximum winds, Rmax, is now widely reported and chosen as a reference size here. We make the assumption that the average pressure within the radius of maximum wind, 〈P(Rmax)〉, and the pressure at Rmax, P(Rmax), may be treated as multiples (α, β) of the central pressure Pc,
Combining and substituting Equations 1-3 gives the central pressure tendency,
where χ is defined as urn:x-wiley:00948276:media:grl64949:grl64949-math-0007 at Rmax which we refer to as the “column speed,” and
While parameters α and β may change significantly during the course of a TC lifetime, they vary similarly, and it is their ratio, γ, which is important in this model. In theoretical radial pressure profiles, such as the λ model (Wang et al., 2015) and the Holland model (Holland, 1980), γ changes only from 1.025 to 1.001 over a central pressure range of 900–1,000 hPa with an environmental pressure of 1,010 hPa. This may seem counter-intuitive as the pressure deficit changes rapidly from the center to Rmax, but the absolute pressure has a much smaller fractional change. Since γ is not sensitive to changes in central pressure, the variability of χ and Rmax will therefore dominate the pressure tendency variability. Moreover, the presence of γ does not change the size dependence. As γ deviates from 1 by less than 2.5% over typically observed TC central pressures we treat γ as a constant equal to 1. This introduces a small error, estimated to be less than 2.5%, but permits a very simple expression of the central pressure tendency,
The above formula is for the pressure tendency, that is, the instantaneous rate of change of central pressure. However best track TC observations of central pressure (and radius of maximum wind) typically take the form of 6-hourly values rather than rates of change. To permit comparison with the observations we integrate Equation 6 under the assumption of constant χ and Rmax which yields
and thus
where Pc(t) is the central pressure at time t, Pc(0) its initial value, Rmax0 and χ0 are constant values of Rmax and χ over the integration period. Neither Rmax nor χ can reasonably be considered constant over the full duration of a TC lifecycle: χ must change sign from intensification to decay so certainly cannot be considered constant over long periods or a full TC lifecycle; Rmax is known to contract during the early stages of intensification (Shapiro & Willoughby, 1982; Stern et al., 2015) so a constant model for Rmax may also be inappropriate for timescales approaching TC lifetime. Hence Equation 8 is only appropriate for short periods during which these quantities may be considered approximately constant. We will primarily test this model over the shortest standard interval in the observation record, 6 hours, but will also test its utility over longer periods. We note that the presence of Rmax in Equation 8 doesn't necessarily imply a relationship with pressure tendency. For example, if χ were to scale linearly with Rmax, this would lead to a situation with no dependence of pressure tendency on size. Since χ is essentially an unobservable quantity, the above model, as a means of determining pressure tendency from TC parameters, is not fully defined. However with knowledge of pressure tendency and Rmax, χ maybe estimated. From Equation 7 a plot of the logarithm of the pressure ratio versus 1/Rmax0 will have a slope of −2χ0t. For short timescales, over which we assume Rmax and χ to be constant, central pressure changes exponentially with a time constant of Rmax0/2χ0.

3 Data

TC central pressure and size data were obtained from the International Best Track Archive for Climate Stewardship (IBTrACS and v04r00) World Meteorological Organization data (Knapp et al., 20102018). The National Hurricane Center provides the best track records in the NA and Eastern Pacific, and the best track data from the Joint Typhoon Warning Center are used in the West Pacific (WP), North Indian Ocean (NI), South Indian Ocean (SI), and South Pacific. Only records at the standard reporting times of 00, 06, 12, and 18 UTC were included. Size observations are challenging and were recently reviewed by Knaff et al. (2021). Storms before 2001 were excluded as before then Rmax data is not widely available. Rmax is treated less rigorously than other sizes. The NA is the only basin in which the TC size has been systematically directly observed in situ in at least some cases. In other basins we rely on more indirect observations and interpretation of satellite data. These differences could lead to quantitatively even qualitatively different results. Following Demuth et al. (2006) and Chavas and Knaff (2022) we identify the region in the NA south of 30°N and west of 50°W as a “high-confidence” area where direct observations of Rmax and central pressure are more frequent and refer to this subset as NA*. All storms which developed to at least a Category 1 on the Saffir-Simpson Hurricane scale (Simpson & Saffir, 1974) were included. Data points occurring after a given storm was designated as “extra-tropical” were excluded.

4 Results

We begin by analyzing the “high-confidence” NA* subset of NA best track data. We separate this into intensifying (Pc decreasing) and weakening (Pc increasing) storms and test the validity of Equation 7. Figure 1 shows a robust dependence of the pressure tendency on Rmax. Equation 7 gives a significant correlation for intensification cases (Pearson's: r = −0.41, p < 0.01) and significantly positively correlated for weakening cases (r = 0.22, p < 0.01). For both strengthening and weakening cases the smaller storms tend to change more quickly. The linear correlation shows that the Rmax variability does account for a significant fraction of the pressure tendency variability. A column speed across a range of radii can be inferred from the slope. In the intensification case, the typical column outflow speed is χ = 0.29 km day−1. In the weakening cases the typical column inflow speed is similar with χ = −0.24 km day−1.

Details are in the caption following the image

Natural logarithm of central pressure, P, after period t = 6 hr, normalized by initial central pressure, P0, against 1/Rmax for NA* cases of (a) intensification, ΔP < 0 and (b) decay, ΔP > 0. Box (median and quartiles) and whisker (0.05 and 0.95 quantiles) plots show ln P/P0 in bins of width 0.01 km−1. Solid lines are least squares regressions of Equation 7 to all data points in the figure where n is the sample size, r is the Pearson's correlation coefficient, and χ is the column speed (km day−1) inferred from Equation 7. Dashed lines (and associated r>75, χ>75) are regressions as above but to data points above the 75th percentile of | ln P/P0| in each Rmax−1 bin. χ99.9 is the 99.9th percentile of χ values inferred from all data points via Equation 7 and the dotted line shows the loci of points equivalent to this extreme χ.

We perform similar analysis on the full Global data and find broadly the same relationships as in the NA* data (Figure 2). For intensification cases, the correlation is slightly weaker in the Global set (r = −0.38, p < 0.01) than in the NA* (r = −0.41, p < 0.01). In the decay cases the correlation is stronger in the Global (r = 0.27, p < 0.01) compared to the NA* (r = 0.22, p < 0.01). Analysis by basin (Figure S1 in Supporting Information S1) reveals similar results in all basins.

Details are in the caption following the image

As in Figure 1 but for Global cases.

We now perform the above analysis on the Global data with a time interval, t, varying from 6 to 30 hr (Figure S2 in Supporting Information S1). The correlation of pressure change and inverse size for strengthening and weakening Global TCs for the different values of t are shown in Figure 3a. We find that the magnitude of the correlation is a maximum at t = 12 and t = 18 hr for intensification and decay cases, respectively. The correlation reduces further for longer integration times in both cases. However, the correlation still remains significant out to a 30 hr change. The magnitude of the column outflow/inflow, χ, decreases steadily with the time interval (Figure 3b). It decreases by about 30% from 6 to 30 hr. This pattern is seen in both intensification and decay cases.

Details are in the caption following the image

Magnitude of (a) Pearson's correlation coefficient, r, of ln(P/P0), with 1/Rmax and (b) the inferred column speed χ as a function of time, t.

To test the sensitivity of the relationship between the largest pressure tendencies and Rmax we repeat the analysis using only data from above the 75th percentile of bins of width 0.01 km−1 spanning the range of reciprocal Rmax for the NA* (Figure 1) and Global (Figure 2) cases. In NA*, when considering only the larger changes in pressure, the correlation with inverse size is substantially increased for intensification (r = 0.60, p < 0.01) and decay (r = 0.40, p < 0.01) compared to all the cases. The column speeds for the these larger pressure tendencies approximately double to χ = 0.61 and χ = −0.55 km day−1. Analysis of the largest changes of pressure in the Global data reveals similar increase in correlation strength and doubling of column speeds.

In both the intensifying and weakening cases there is a clear absence of cases of RI and weakening of large TCs, suggesting a physical limit on maximum pressure tendencies and column inflows/outflows across all TCs. We also show the line corresponding to the 99.9th percentile of column speed for the NA* (Figure 1) and Global (Figure 2) cases. The corresponding speeds are χ = 3.3 and χ = −3.6 km day−1 for the intensifying and weakening Global cases respectively. From this and Equation 6 we can then infer a maximum intensification pressure tendency equation (hPa/6 hr) as a function of the pressure, Pc, and Rmax,
and a maximum decay tendency equation,
for Rmax (km) larger than about 15 km by inspection of Figure 2. The maximum tendency at smaller radii is less clear from the data but may saturate.

5 Discussion and Conclusions

We derive a simple but useful new model of pressure tendency. Sparks and Toumi (2022) applied mass continuity in cylindrical coordinates to relate average pressure changes within a given radius to column radial wind speeds at that radius (Equation 1). From here our new model diverges. In their model assumptions were made about the radial distribution of radial wind speeds whose validity could only be demonstrated indirectly through idealized simulations. Here we take a different, simpler, approach relating the central pressure to the pressure at Rmax (Equation 3), and the average pressure within Rmax (Equation 2) via constants whose ratio is very close to unity. Ultimately this allows us to express the central pressure as an exponential for short timescales with a time constant that is proportional to Rmax and inversely proportional to the column radial speed, χ (Equation 7). This new form allows us to perform a comparison with observations of intensifying and decaying TCs.

The column radial speed, χ, is the radial wind speed representative of the whole (axisymmetric) vertical column at Rmax. The column integrates all the mass fluxes at different heights which add up to the net mass contribution to the central surface pressure tendency. χ is clearly a simplification of the many complex physical processes that affect radial flow at all heights. In practice, χ represents the small residual of the difference between inflows and outflows. Processes such as the strength and location of deep convective heating, the secondary circulation, the boundary layer frictional inflow, the upper level outflow are all important. They in turn are affected by environmental conditions such as the sea surface temperature and vertical wind shear. This simple model has nothing to say about the details of these processes, the TC dynamics and can not disentangle their relative contributions. In this sense, it is not a complete physical model. Indeed the observational analysis in our framework confirms the dominant role of the variability of outflow/inflow speed, χ, in driving the pressure tendency variability. The role of Rmax is however theoretically predicted and found to be robustly significant. This framework is a useful new approach as it allows us to understand the central pressure tendency in terms of just two quantities: Rmax, the size of the TC core, and χ, the speed of the net radial flow at Rmax into or out of the column. The fact that χ is an unobservable quantity limits the use of this model in its present form as a forecasting tool. However, a skillful parameterization of χ in terms of observable quantities may be possible.

We find a significant relationship between the observed TC central pressure change and size. This relationship holds in both intensifying and decaying TCs. The global correlation for intensifying TCs (r = −0.38; t = 6 hr) is somewhat larger than those reported, but not explained, by Xu and Wang (20152018a) of r = −0.17 and r = −0.21 for 24 hr intensification rates (notably in Vmax rather than Pc) for NA and WP TCs respectively. Their weaker relationships than found here may be due to several factors. First, we show theoretically that it is the reciprocal of Rmax rather than Rmax itself which is directly related to the pressure tendency (Equation 6). We also show that the logarithm of the pressure change is the correct transformation to diagnose tendencies. More generally, the central pressure may also be a more robust physical metric of intensity than the localized near-surface maximum wind speed. Finally we also show that the correlation between intensity change and size does tend to decrease for longer time scales (e.g., 6 hr vs. the 24 hr that they used). The results also hold true qualitatively for intensity tendency as measured by Vmax (not shown). This is to be expected given the pressure-wind relationship and observed Vmax tendency dependence on size (Carrasco et al., 2014). Our framework goes toward explaining their findings.

The inferred column speeds for intensifying and decaying cases are curiously similar. This is true of both typical changes and extremes. This may indicate similar balance of processes in both but only the net signs of the column speeds are reversed. This can also be seen in the energy balance approaches. For example, the wind speed tendency equation of Wang and Toumi (2022b) also points to a possible universal balance of heating and dissipation with either being dominant reversing the sign. The similar correlation between the central pressure change and reciprocal size for both decaying and intensifying TCs suggests that the geometric core size effect is one of the fundamental properties controlling rates of intensity change. This is in contrast to the suggestions made by Carrasco et al. (2014) and Xu and Wang (2018a) who invoke ad-hoc increased inertial stability for enhanced intensification specifically. Here we show that for a given column inflow or outflow, smaller cyclones will tend to both intensify or decay more quickly. This is not to say that invoking inertial stability is not relevant, rather this framework makes a clear theoretical link to size for both intensification and weakening.

We observe that our framework assumptions appear reasonable for timescales beyond 6 hr. This makes the case that the assumptions made in the model derivation are justifiable. The inferred column speed does decrease with time. This is to be expected as TCs do not intensify or decay exponentially for long periods; intensification reaches thermodynamic limits and when weakening the pressure does not increase beyond the environmental pressure. One can thus understand the observed reduced column speeds for longer time intervals.

It is worth noting that this framework may also hold for extra-tropical cyclones. Mass conversation is the basic principle. The assumption of the pressure radial profile shape, through γ, will likely apply universally and in any case does not change the pressure tendency dependence on geometric size. Here we only validate the model for TC data and Rmax, but we can see no reason why the predicted size dependence should not also hold in the extra-tropics. The framework applicability is perhaps most suggestive in the reported sensitivity of simulating extra-tropical bomb cyclones to model horizontal resolution (Jiaxiang et al., 2020).

We also show that large or fast changes in central pressure for both intensification and decay depend even more strongly on the size. A maximum pressure tendency equation dependent on the central pressure and Rmax is proposed and may be of operational or theoretical interest. An example of the application of these equations is that for a Category 1 storm with an initial pressure of Pc = 980 hPa and Rmax = 20 km would have a maximum potential intensification of about −39 hPa/6 hr.

Numerical weather prediction and climate models tend to systematically over estimate Rmax. Their coarse horizontal resolution can not adequately simulate smaller sizes (Bian et al., 2021; Wehner, 2021). Our framework clearly shows that this systematic large size bias will contribute to a systematic low bias of the pressure tendency (both positive and negative) and thus limit the skill of predicting the TC intensity evolution. The sensitivity to any large size bias becomes even more acute for RI and rapid weakening. It is interesting to note that this limitation can affect the early stage of cyclone formation where, for a given χ, small disturbances can either rapidly intensify to full TCs or just dissipate quickly. This may contribute to understanding the challenge of modeling genesis. Similarly, the strongest TCs typically have also undergone RI (Lee et al., 2016) and our model shows that RI is more likely to occur for initially smaller and weaker (larger Pc) storms. The simple framework presented here is useful in that it helps toward understanding why models need to simulate both the size and intensity accurately.


The research was supported by the UK Centre for Greening of Finance and Investment (UKRI-NE/V017756/1), the Singapore Green Finance Centre, and the Vodafone Foundation.

    Data Availability Statement

    Tropical cyclone data were taken from the International Best Track Archive for Climate Stewardship version 4: https://www.ncdc.noaa.gov/ibtracs/index.php?name=ib-v4-access (Knapp et al., 2018).