ERA5 monthly averaged ensemble member data from March 1979 through February 2022 were acquired for use in this study. March is selected as the starting month as meteorological winter spans multiple calendar years, meaning that March 1979 is the start of the first complete season, while February 2022 is chosen as the final month to allow for the maximum number of complete seasons. ERA5 contains nine ensemble members and one control member. Because the control member is produced through a more thorough data assimilation process, involving slightly more precision, the control was excluded from the ensemble analysis. The ensemble members are created by perturbing the observations and model tendencies. As these ensemble members only represent a portion of the actual uncertainty, compared to the interannual variation and inter-reanalysis variation these differences are small. Because the ensemble spread in ERA5 is smaller than the actual uncertainty, we will primarily focus on the relative uncertainties between the different metrics. Acquired pressure-level data include the meridional and zonal wind fields, while surface-level data include 10 m zonal wind, sea-level pressure, precipitation, and evaporation. These data are used to compute the tropical extent for the six metrics described in the next subsection. Data from these nine ensemble members are regridded using a first-order conservative regridding from a 0.5∘ resolution to a 1∘ resolution using the Climate Data Operator (CDO) “remapcon” function (Schulzweida, 2022). The 1∘ and 0.5∘ resolution data produced similar tropical extents. All ensemble members are then zonally and seasonally averaged into March–April–May (MAM), June–July–August (JJA), September–October–November (SON), and December–January–February (DJF). Separately annual means are calculated for each year from March–February.

Six different tropical-width estimation methods are used, each covering different atmospheric regions near the HC edge or using different atmospheric variables. Each method is computed using the Python version of TropD (PyTropD), a module which estimates HC width from zonally averaged atmospheric data, yielding a latitude estimate for the poleward extent of the tropics in each hemisphere (Adam et al., 2018). PyTropD uses spline interpolation to produce more latitudinally precise estimates of tropical extent than the 1∘ resolution. All tropical-width metrics use the standard configuration of TropD.

To quantify the HC trend uncertainty, trends are calculated for each ensemble member using all six metrics over the period from 1979–2022. From these trends, kernel density estimates are computed in each season and as an annual mean, for all six metrics in the Northern Hemisphere (NH) (Fig. 1) and Southern Hemisphere (SH) (Fig. 2). A brief description of kernel density estimates is presented in Sect. 2.2.7.

While in individual seasons the trends vary substantially between metrics, the annual-mean trends are typically more similar. The NH annual-mean trend for SF is roughly 0.05∘ per decade, while other metrics estimate 0.05 to 0.3∘ per decade. The SH annual-mean trend ranges between 0.1 and 0.3∘ per decade, with most metrics (including SF) estimating 0.2∘ per decade. The near-zero NH tropical expansion rate and weak SH expansion are similar to the generally weak expansion found across multiple reanalysis datasets in Grise et al. (2019). It is also of note that the rate of expansion for the annual mean of P - E in the NH is larger than all other metrics.

On seasonal timescales, the trends and their uncertainty vary substantially between the different metrics, as shown by the kernel density estimates and the ensemble trend SD; a table of the latter is presented in the Supplement (Table S1 in the Supplement). P - E features the largest uncertainty, particularly in the NH and in JJA where the uncertainty is roughly 2 orders of magnitude greater than other metrics. Over the NH, SF exhibits the second least certain trend in all seasons, particularly in JJA, where the range of estimates includes both poleward and equatorward trends and is nearly 1 order of magnitude less certain than other metrics. Considering that the ensemble spread underestimates the actual uncertainty, the presence of both positive and negative trends indicates that in JJA in the NH, SF is not a reliable metric. Over the SH, SF-estimated tropical expansion is more robust, as shown by the consistency of the negative sign in all seasons as well as the smaller uncertainty. While the uncertainty over the SH is small for STJ and EDJ in most seasons, STJ in DJF and EDJ in JJA are substantial outliers to the trend, estimating tropical contraction while all other metrics record expansion.

We hypothesize that data uncertainty is a major contributor to the larger uncertainties in SF and P - E. SF is based on poorly observed and poorly constrained meridional wind, and P - E uses imperfectly modeled evaporation and precipitation data. On the other hand, STJ and EDJ are based on better-observed and constrained (through the thermal-wind relationship) zonal-wind data, and UAS and PSL are also well observed and constrained quantities.

We next examine the HC extent uncertainty in individual years. In doing so, the impact of improved observations over time can be seen along with the potential influences of internal climate variability on tropical-width estimates. The yearly ensemble SD of tropical extent is calculated for each metric in the NH and SH (Fig. 3). P - E has the greatest SD in nearly all seasons and is particularly unreliable in the NH JJA, where the uncertainty is roughly 2 orders of magnitude greater than the four reliable metrics. SF uncertainty is typically 1 order of magnitude larger than the four most reliable metrics and is particularly unreliable in the NH JJA. Over the SH, the SF extent uncertainty is largest relative to the other metrics. We also note that there is not much interannual variation in SD for the six metrics in either hemisphere, indicating that internal climate variability does not have a substantial impact on uncertainty. Counter to this observation are SF and P - E in the NH in JJA, which are even undefined in some years, resulting from P - E not recording a zero crossing near the poleward edge of the tropics and SF not recording a closed circulation cell. Examples of the absence of a closed HC in 2019 and 2020 are included in the Supplement (Fig. S1 in the Supplement). As a result, in these years the SD is calculated excluding undefined ensemble member values, resulting in imperfect estimates of variation. It is interesting that 2019 and 2020 are the only 2 years with some members not reproducing closed NH summer circulation cells, and it remains to be seen whether this is part of a systematic longer-term trend.

Most notably, Fig. 3 shows a reduction in uncertainty over the 40-year period in most seasons and for nearly every metric, likely due to the increased quantity and quality of observations assimilated into the reanalysis. Because the ensemble members are created by perturbing the model parameters along with observations, both a lack of observations and the presence of lower-quality observations increase the data uncertainty. The metrics based on u wind and sea-level pressure (STJ, EDJ, UAS, PSL) have consistently small uncertainty in all seasons, years, and hemispheres, though the variation for PSL especially in the SH is the smallest of all metrics. Despite the presence of more conventional observations in the NH, particularly in the earlier decades, the extent uncertainty is smallest in the SH for all metrics, indicating that both observations and regional dynamics are relevant. The greater uncertainty in the NH may be due to the abundance of ocean in the SH and topographic variability in the NH, both of which impact the atmospheric processes which are measured by the HC extent metrics.

Because SF is the most widely used metric, the following sections will take a closer look at the magnitude and impact of data error for just SF. To quantify the spatial uncertainty in SF data, the ensemble SD of SF is calculated for all years using zonally and annually averaged meridional-wind data. These annual values are then averaged over two 20-year periods and normalized as described in the Methods subsection. This results in the annual-mean SF normalized ensemble SD over two time periods (Fig. 4). The two time periods are 1979–2000 and 2001–2021, comparing the period with fewer observations to the better-observed period following the deployment of many remote sensors (Hersbach et al., 2020). The normalized ensemble SD during the earlier period is clearly larger than during the later period, indicating the role of improved observations in the reduced SF extent uncertainty in recent decades seen in Fig. 3. As the SF metric used here calculates the tropical extent using the 500 hPa SF values between the Hadley and Ferrel cells, only the improvements in these regions are impactful, while the large decrease in normalized SD in other regions is not directly relevant. However, the improvements in observations in these regions, particularly the inner tropics, constrain the meridional circulation, likely contributing to reduced uncertainties in other regions of the Hadley circulation in recent decades. As can be seen from Fig. 4, the normalized SD improves from roughly 6 % to 4 % at 500 hPa at the tropical edges in both hemispheres between the two time periods, though these changes are not statistically significant at the 95 % confidence level due to the 0.25 fractional uncertainty in SD as described in Sect. 2.2.7. While the ensemble variation is generally less than 10 % of the interannual variation, the meridional-wind uncertainty is nearly 1 order of magnitude larger than in the zonal wind used by STJ and EDJ (Fig. S2). Larger normalized uncertainty in the data underlying the SF metric causes larger error in SF-derived HC expansion rates.

As shown earlier in this study, in Sect. 3.2, the SF extent uncertainty is greater in the NH than SH and is particularly pronounced in the NH in JJA. Here, we aim to investigate the impact of the HC structure on SF extent uncertainty. In order to calculate the tropical extent, the latitude where the circulation changes from clockwise to counterclockwise at 500 hPa must be determined. Observational errors will have a larger impact on SF extent estimates if the region of near-neutral circulation between the Hadley and the Ferrel cell is large, as small variations between the ensemble members can cause the zero crossing to occur over a larger latitude range. We define Δ in each hemisphere as the width in degrees latitude of the region in the vicinity of the HC edge (poleward of 20∘ and equatorward of 50∘) at 500 hPa where SF is very weak (± 5 × 109 kgs-1). This Δ is larger in the NH than SH and is particularly large in NH JJA, coinciding with the seasons and hemispheres where the SF trend and extent uncertainty are greatest (Fig. 5).

As shown by Fig. 5, the HC over the NH has a smaller latitudinal extent and features a distinctive narrowing in the middle and upper troposphere. This narrowing coincides with a large Δ, which is not present in other seasons or in the SH. The wide Δ likely represents the impacts of large landmasses and greater longitudinal variation in meridional circulation in the NH (Hoskins et al., 2020). In both hemispheres, however, the region of near-zero circulation in summer is larger than in winter, suggesting a possible relationship between the weak summer HC and meridional gradients at the poleward edge of the HC. We speculate that Δ is impacted by both the zonal variation in circulation and the strength of the meridional circulation. Hence, in JJA in the NH, the weak meridional circulation and large zonal variation in circulation combine to create a persistently large Δ which causes the SF metric to perform poorly. It is also worth noting that the latitudinal width of the near-zero circulation region does not change significantly during the 40-year period, suggesting that this is a typical and quasi-permanent feature of the atmosphere and is unrelated to measurement errors.

In the previous sections, the SF extent uncertainty was shown to be related to both data error and weak meridional gradients. We next determine the impact of these two factors on seasonal and annual HC estimates. The large NH JJA and SON tropical-extent uncertainty coincide with large Δ values, suggesting that Δ plays a significant role when using SF to estimate tropical width (Fig. 6a). Here, Δ is computed from the 1∘ regridded ERA5 data, resulting in relatively coarse latitudinal estimates. Although data uncertainty, as shown earlier (Fig. 4), partially explains the poor performance of SF relative to other metrics (Figs. 1 and 2) as well as the improvement over time (Fig. 3), no significant relationship was found between the climatological averages of uncertainties in SF data and SF-derived HC extent; a scatterplot showing the poor correlation is included in the Supplement (Fig. S3).

However, when looking at individual annual means, the uncertainty in the SF-based HC extent is well approximated as a linear function of the average of the SF ensemble SD in the vicinity (±2∘ latitude) of the HC edge at 500 hPa (Fig. 6b). In the NH, the extent uncertainty is larger than in the SH, likely due to the aforementioned complicating influences of the NH landmasses. The extent uncertainty is well correlated with the SF uncertainty and poorly correlated with Δ, indicating that the reduction in data error is the main reason for the decrease in SF extent uncertainty seen before (Fig. 3); a scatterplot showing the poor fit between Δ and the annual-mean SF extent uncertainty is included in the Supplement (Fig. S4). The improved observations in recent decades thus result in more precise tropical-width estimates relative to earlier decades.

The uncertainty in the SF-derived HC trend has no discernable relationship with the SF data uncertainty when comparing seasons (Fig. S5). However, the trend uncertainty is greatest in seasons and hemispheres where Δ is largest, though this is mostly noticeable in NH JJA (Fig. S6).

Following this analysis, we conclude that Δ represents a persistent factor for the uncertainty in the SF metric, which varies primarily by season and hemisphere and is nearly constant over the years. As discussed earlier, Δ can be seen as a sensitivity of the HC extent to the data uncertainty, as a larger Δ results in a larger range of latitudes where data error could cause SF to be zero. Δ in individual seasons and hemispheres does not vary much from year to year as it is mostly a consequence of relatively fixed meteorological and geographical factors that control the structure of the HC. As a result, Δ represents a persistent amplification of data error, resulting in greater extent uncertainty in certain seasons and hemispheres, in particular JJA in the NH.