This section provides a detailed description and record (including relevant citations) of the PI algorithm with its thermodynamic/meteorological origins, assumptions, and computations.

Potential intensity may be derived following (which is largely based on the formulation of ) idealizing a tropical cyclone as a Carnot heat engine e.g., and assuming the following: (1) the work done against friction by the outflow is ignored, (2) when the storm intensity reaches its maximum, the anticyclone at the top of the storm is fully developed, and (3) the gradient wind may be approximated by cyclostrophic wind at the RMW. Under these conditions, the Carnot cycle formulation yields an expression for the maximum potential intensity (which roughly scales with the approximated PI expression, Eq. ; ): 2(Vmax)2=TsT0CkCD(CAPE*-CAPEenv)|RMW, where CAPE* is the convective available potential energy of saturated air lifted from sea level to the outflow level referencing the environmental profile, and CAPEenv is the convective available potential energy of the environment. Because the final term is evaluated at the RMW and CAPE* is pressure dependant, an expression for the surface pressure at the RMW is needed. Following (cf. also , their Eq. 6), the minimum pressure of the tropical cyclone at the RMW, pm, is found with

Equation (4) in mistakenly replaces Rd with cp. pyPI includes the correct factor of Rd.

The pressure dependence of CAPE requires solving Eqs. () and () with numerical iteration, which pyPI performs with individual PI and CAPE modules. Algorithm 1 summarizes how pyPI computes maximum potential intensity by modeling a tropical cyclone as a Carnot heat engine. Algorithm inputs and outputs are provided in Table 1 and described in Sect. ; meteorological constants are provided in the Appendix (Table ). We begin by describing the CAPE calculation, which is used throughout the pyPI algorithm.

CAPE is defined as the sum of positive and negative areas of buoyancy energy of a lifted parcel on a sounding (e.g., , their Eq. 6.3.6, discussed more below) and is calculated by pyPI with the procedure in Algorithm 2.

Given an initial surface parcel temperature, pressure, and mixing ratio, the procedure begins by finding the parcel's reversible entropy, which is conserved as it is lifted on the sounding. The parcel's water vapor pressure is found via the ideal gas law (e.g., , their Eq. 16): 4e=r×pϵ+r. Saturation vapor pressure (in hPa) is given empirically as a function of T in degrees Celsius by , their Eq. (10), following the Clausius–Clapeyron relation: 5es(T)=6.112×exp⁡17.67×TT+243.5. Then fractional relative humidity is defined as RH≡ees≤1.0. Assuming the temperature dependence of specific heats is negligible over the range of temperatures in the tropical atmosphere and integrating Kirchhoff's equation (e.g., , Eqs. 4.4.3–4.4.4), the temperature dependence of the latent heat of vaporization is 6Lv=Lv0+(cpv-cℓ)×T, with T in degrees Celsius. Finally, we are equipped to calculate the parcel's reversible total specific entropy (per unit mass of dry air), s, which is conserved as the parcel is lifted along the sounding (, their Eq. 4.5.9): 7s=(cpd+rTcℓ)log⁡(T)-Rdlog⁡(p)+LvrT-rRvlog⁡(RH), where rT is the total water content mixing ratio, which is identical to the parcel mixing ratio at the surface.

Having determined the parcel's initial moisture properties, we next find the lifting condensation level (LCL) of the parcel, in order to partition the upcoming buoyancy calculation between saturated and unsaturated regions of the profile. The pressure of the LCL, pLCL, is found empirically with

This is likely derived empirically from and was developed for (Kerry Emanuel, personal communication, 2020). Modern calculations of pLCL are made following exact expressions from ; cf. Sect. .

The PI module begins by checking to ensure that the input atmospheric profile is appropriate for the PI calculation. If not, missing values are returned by the algorithm (see Sect. ). Water vapor mixing ratios above the boundary layer do not influence the PI calculation (they are redundant in CAPE*-CAPEenv), so any missing r values above the surface are replaced with 0 gkg-1.

Following Algorithm 2 described above, pyPI computes CAPEenv, assuming the environmental air parcel is lifted from the lowermost input level in p.

Next, pyPI iteratively solves Eqs. () and () (with objective

When reduced by an order of magnitude to 0.05 hPa, Vmax values increase by <0.3 ms-1, while computation times increase by >25 %.

pyPI includes six adjustable parameters that may be set in the module call, with the caveat that each should be chosen within the defined “Values” column of Table 1. Parameters set outside these values could result in syntax errors or logical errors in the output or may give rise to unphysical PI estimates.

Mirroring the output flag convention of the BE02 MATLAB code, IFL = 1 when the PI algorithm successfully returns valid potential intensity outputs, IFL = 0 when the algorithm fails because the input data are improper for a PI calculation (e.g., if Ts<5∘C), and IFL = 2 when the algorithm fails to converge.

One major difference between pyPI and the BE02 MATLAB algorithm is the handling of missing data and the (related) flag provided in the output. By convention, missing input variables in pyPI are assigned Python's “Not a Number”, NaN, to avoid errors. The BE02 MATLAB code default is that profiles may contain missing values (specifically temperatures on pressure levels), and the algorithm computes PI over the remaining valid levels.

Because missing values may sometimes be found at the surface – and the primary CAPE calculation (Sect. ) relies heavily on the assumption of lifting the parcel within the storm and environment from that level – errors could arise from estimating PI when ignoring near-surface buoyancy. In principle, PI should be calculated only over data points with existent sea-surface temperatures and lowest profile level temperatures. In practice, missing data may arise at the lowest profile level, which would lead to errant PI calculations if these profiles are input to the BE02 MATLAB code.

pyPI addresses this challenge in three ways. First, an adjustable parameter (miss_handle) is implemented to allow the user to specify how pyPI handles the missing values. If miss_handle = 0, the code attempts to handle missing values akin to the way that the BE02 MATLAB code did, although there still remain some differences in the outputs between pyPI and the MATLAB algorithm. Specifically, pyPI's CAPE calculation proceeds as normal only as long as there are no missing values between the lowest valid (non-missing) level and the OTL; otherwise, CAPE module outputs (and hence PI module outputs) are returned as missing. Second, if miss_handle = 1, then the CAPE function will automatically interpret temperature profiles with missing data as invalid and return missing values to the PI algorithm, resulting in the PI outputs being set to missing in the return. Third, a new output flag value (IFL = 3) is introduced in pyPI which is returned when missing values in the temperature profile results in a missing output return from the PI module (i.e., in either of the two cases described above), which aids in interpreting pyPI output.

Figure 2 shows an example of the output algorithm status flags from pyPI calculations with a single month's mean environment (September 2004) from MERRA2 data (Sect. ), and with the default miss_handle = 1. The figure illustrates the few global points which had at least some missing data, resulting in a missing PI return from pyPI (IFL = 3; red grid points). In contrast, these locations have an output (but likely errant) PI from the BE02 MATLAB code. The majority of locations where missing input data results in missing output PI are near land (e.g., the Caribbean and Indo-Pacific), where missing values arise as an artifact of the differences between the sample data and the land–sea mask applied (Sect. ). Missing values in the sample data (which has lowest data pressure level of 1000 hPa) could also be in locations where the monthly average pmsl is below 1000 hPa, resulting in T(1000hPa)=NaN.

In the example pyPI calculation (Fig. 2) there are no inputs for which the algorithm does not converge (cf. ). One final complication is that BE02 MATLAB code occasionally returns Vmax=0 as an output. In these cases, pyPI instead returns Vmax=NaN.

pyPI outputs valid (non-missing) potential intensities over ∼65.6 % of the ocean grid points in the global 2004 sample dataset – compared with ∼64.41 % returned by the BE02 MATLAB code. In addition to how missing data are handled, output differences may also arise from slight variations in numerical computation between Python and MATLAB. pyPI validation tests (Sect. ) are computed over all spatiotemporal locations for which both algorithms have non-missing/non-zero potential intensities (and with miss_handle = 1).

In addition to updating the original BE02 algorithm, pyPI is designed with the intention to undergo further scientific developments as requested or created by the community. In addition to including alternative intensity indices e.g.,, potential pyPI improvements generally fall into two broad categories.

Increased model flexibility. Such improvements would alter the code to provide users more parameter choices when the PI module is called, through either input flags or additional variables. For instance, users may want to select between the CAPE definition used to estimate pmin in Eq. () (i.e., CAPE|RMW or CAPEenv) or explore alternative outflow temperature definitions (e.g., setting TLNB during the computation of CAPEenv compared with the default CAPE* calculation).

Incorporation and/or development of fundamental or incremental scientific advances. As the scientific understanding of tropical cyclone intensity and potential intensity increases, such knowledge could be brought into pyPI. For instance, showed that the assumption of moist neutrality in calculating PI may not be appropriate, even when a cyclone is mature. Multiplying Eq. () by a factor of (1-αΓ×Γ) in the pyPI codebase, where Γ is the environmental lapse rate and αΓ is an empirical parameter, would enable further exploration of how stratification affects PI.

A specific goal for future pyPI development is to replace the current empirical estimate of pLCL with a modern formulation e.g.,. Other opportunities for improvement might be more involved, requiring significantly more research to implement. For example, the magnitude of CkCD is nonlinearly related to wind speed ; a mathematical relationship between wind speed and this ratio could in principle be incorporated into potential intensity theory, but this would require revisiting PI theory and further theoretical advancements before inclusion within pyPI.

These examples are not exhaustive but show the range of possibilities for future pyPI development and the potential future value of this model in the tropical meteorology community.

The pyPI sample data are monthly means of state variables from the second Modern-Era Retrospective Analysis for Research and Applications (MERRA2, ) in 2004, interpolated onto a 2.5∘×2.5∘ global grid. Note that for these example pyPI calculations the water vapor mixing ratio, r, is approximated by substituting in the reanalysis specific humidity, q (as q≡r1+r≈r, because r≪1).

Potential intensity calculations are generally linear (i.e., mean potential intensities may be estimated as a function of mean environmental variables): E[Vmax(Ts,pmsl,p,T,r)]15≈Vmax(E[Ts],E[pmsl],E[p],E[T],E[r]), where E[⋅] is the expected value of a function or variable. Using monthly mean environmental conditions – instead of 6-hourly observations – to compute climatological monthly means of potential intensity (and the algorithm's other output variables) generates a small bias of <1 ms-1 globally and <0.5 ms-1 in the tropics. PI's linearity property is convenient, because it reduces the scale of data needed to compute PI: daily or hourly data are not needed for monthly or longer (climatological) applications. Applications on shorter (e.g., operational or daily) timescales, however, should use appropriately shorter frequency inputs to the pyPI algorithm.

The sample data uses the land–sea mask from the European Centre for Medium-Range Weather Forecasts Interim (ERA-Interim, ) on a 2.5∘×2.5∘ global grid. By definition, Vmax≡0 where Ts=NaN (i.e., over land); in some cases (e.g., if skin temperatures valid over land are used in lieu of sea-surface temperatures) PI may be mistakenly calculated over land with the PI module. In these cases, users should assign all PI algorithm outputs over land to the missing value in post-processing. As an alternative, in this pyPI example input variables over land are set to missing in a pre-processing step. Note that the mismatch between using the ERA-I land–sea mask and MERRA2 data in this example results in a set of minor output artifacts caused by missing (MERRA2 land grid points) input data over ERA-I defined ocean grid points. This artifact provides a useful demonstration of the missing data flag employed in pyPI (Sect. ).

Accompanying the environmental conditions in the sample data are outputs from the BE02 MATLAB code written by Kerry Emanuel . Potential intensities calculated over September 2004 with pyPI and the extensively used BE02 MATLAB code are compared in Fig. 3 over the globe; their difference is computed and plotted in Fig. 4. There is excellent agreement between the two algorithms; 98.5 % of output potential intensities have absolute differences <0.01 ms-1. Potential intensities calculated with the Python algorithm exhibit a slightly negative bias relative to the MATLAB calculations, but these differences are negligible compared with other uncertainties in the PI calculation, such as the ratio of surface exchange coefficients (Table 1).

Figure 5 shows the scatter between all potential intensity values calculated with the two algorithms over the sample data, plotted against a 1 : 1 line (values lying on this line exhibit perfect agreement). The R2 of this comparison is R2≈1.0 to seven significant digits, such that the calculations are nearly identical. All other output variables (cf. Table 1) from the two algorithms have similarly strong levels of agreement.

A minor PI difference between pyPI and the BE02 MATLAB algorithm arises when the pyPI-found outflow level has a higher pressure and warmer temperature than found by the BE02 code. The outflow property differences result from pyPI's correction of a minor error that was present in the BE02 algorithm, where ϵ was defined as 0.622 rather than directly calculated as RdRv. The small rounding error results in a handful of profiles with lower altitude outflow and lower PI values calculated by pyPI. In the absence of correcting this error, the correlation between the two calculations is R2≈1.0 to 13 significant digits, and the absolute maximum difference anywhere is 4.1×10-5 ms-1.

I conclude that the PI calculations made with the pyPI algorithm are adequately validated against the BE02 MATLAB code, and that pyPI is sufficiently accurate for use in research applications.

Figure 6 shows 2004 annual mean sea-surface temperatures, as well as pyPI-calculated potential intensities, outflow temperatures, and outflow temperature levels. The familiar pattern of warm SSTs in the tropics corresponds with high Vmax values, suggesting that on an annual timescale PI is strongly influenced by Ts. These warm and high-PI regions are accompanied by outflow temperature levels with annual pressures below 100 hPa, deep in the tropical tropopause region (e.g., ). Near the tropical tropopause, annual mean outflow temperatures are remarkably cold, around 200 K. On average, the coldest outflow temperatures are found in the western North Pacific basin, where consistent deep convection and stratospheric circulation act to keep tropopause temperatures very cold and highly variable (e.g., ).

A slightly more sophisticated application of pyPI is the calculation of potential intensity seasonal cycles. Reproducing the methodology of with pyPI calculations over 2004, Fig. 7 shows the seasonal cycles of sea-surface temperatures, as well as outflow temperatures, outflow temperature levels, and potential intensities in 2004 averaged over tropical cyclone main development regions (defined in , their Table 1).

The seasonal cycles of PI are known to be quite robust year over year and exhibit clear differences between regions. The western North Pacific has a nearly flat seasonal cycle of PI, while the other basins are more intraseasonally variable. While the muted sea-surface temperatures certainly play an important role in this damped cycle, the outflow temperature pattern is typical of the cold-point tropopause seasonal cycle (e.g., ) – which the OTLs are reaching – which acts to damp the seasonal cycle further by decreasing PI in the boreal summer and increasing PI in the boreal winter. As a result, tropical cyclones in the western North Pacific have higher speed limits during the boreal winter months. Consistent with this finding, historical observed typhoons show intense wind speeds during the winter and spring months . For example, in early April 2004 Typhoon Sudal reached Category 4 strength, ∼67 ms-1, when the co-located monthly average PI was about 75 ms-1.

The seasonal cycles of each basin illustrate the complex relationship between sea-surface temperatures, OTLs, and outflow temperatures. Figure 8 diagrams this relationship in more detail, showing how one assumption in the pyPI algorithm impacts the output PI values. pyPI assumes that the outflow temperature and its level are derived by finding the LNB assuming a saturated parcel lifted from the sea-surface with temperature, Ts. This implies that, following a moist adiabat, the level of the neutral buoyancy is a function of only Ts and the environmental temperature profile, T. Given a fixed temperature profile, a 3 ∘C increase in Ts (e.g., from SST1 → SST2 in Fig. 8) requires that the associated OTL will be found at a higher altitude (OTL1 → OTL2 in Fig. 8), and the associated outflow temperature will likewise change. As the atmosphere's stratification increases into the lower stratosphere, increases in Ts become less effective at changing the OTL and its temperature, with the effect nearly saturating when the outflow reaches the cold-point tropopause (e.g., 100 hPa in Fig. 8). At this point, T0 variability is almost completely decoupled from Ts variability. Instead these T0 values become influenced by tropopause region variability (e.g., ) which is controlled by radiation, dynamics, and deep convection (e.g., ).

These properties are borne out in the example 2004 seasonal cycles computed in Fig. 7. In the North Atlantic basin sea-surface temperature and OTL seasonal cycles are inversely proportional: colder sea-surface temperatures have higher-pressure OTLs and warmer outflow temperatures found in the upper troposphere (where dTdz<0) in all months except August–September. In these late summer months the OTL reaches near the cold-point tropopause, and the outflow temperature slightly increases, following the seasonal cycle of warmer tropopause temperatures (e.g., ). A contrasting pattern is observed in the western North Pacific, where OTLs have almost no seasonal cycle: in this basin the calculated outflow always reaches the lowermost stratosphere (OTL ≤90 hPa). Accordingly, the outflow temperature seasonal cycle perennially follows the seasonality of lowermost stratospheric temperatures, which minimize in the boreal winter and maximize in the boreal summer . Comparing with Eq. (), this T0 seasonality leads to relatively increased PI values in the boreal winter and relatively decreased PI values in the boreal summer. Overall, the PI seasonal cycle in the western North Pacific is damped over the year, a pattern that is observed in real-world tropical cyclone intensities (cf. ).

The relative contributions to potential intensity may be mathematically derived by decomposing Eq. (). Taking the natural logarithm of both sides, 2×log⁡(Vmax)=log⁡CkCD+log⁡Ts-T0T016+log⁡(ho*-h*). Then PI variability is related to variability in either tropical cyclone efficiency (Ts-T0T0) or thermodynamic disequilibrium (ho*-h*); recall that CkCD is taken as a constant. As an example, Eq. () is applied to pyPI-calculated 2004 seasonal cycles of potential intensity (from Fig. 7). pyPI calculates PI directly, and efficiency may be directly computed from input Ts and output T0; following the disequilibrium term is taken as a residual from Eq. ().

After finding each term in Eq. () over each basin and seasonal cycle, the amplitude (defined as the annual range evaluated with monthly observations) of each seasonal cycle is plotted in Fig. 9. By convention, a negative amplitude indicates the approximately sinusoidal seasonal cycle reached its maximum in the boreal winter and minimum in the boreal summer.

In all basins, the disequilibrium term drives the largest portion of the seasonal amplitude. This is consistent with sea-surface temperature seasonal cycles which dominate the disequilibrium variance (Fig. 7). The efficiency term is smaller, and in each basin it follows the same cycle as thermodynamic disequilibrium, with the exception of the western North Pacific (where the efficiency seasonal cycle maximizes in the boreal winter and minimizes in the boreal summer). This opposite-signed seasonality between disequilibrium and efficiency in the western North Pacific is directly related to the influential seasonality of the near-tropopause outflow temperatures found with the pyPI calculations (Sect. ; see full discussions in ). Notably, the Southern Hemisphere shares this outflow temperature seasonality, which actually amplifies the efficiency seasonal cycle through both sea-surface temperatures and outflow temperature intraseasonal variability. In all other basins, outflow temperature seasonality is offset by the sea-surface temperature seasonality, which acts to mute the efficiency term and further contribute to disequilibrium dominating their seasonal cycles. The decomposition in Fig. 9 illustrates how the roles of environmental conditions in PI seasonality are basin dependant.

These simple examples show how pyPI may be used to study tropical cyclone intensities and likewise demonstrate pyPI's ability to produce findings similar to those previously computed with the BE02 MATLAB code.