Observational and modeling studies suggest that Earth's tropical belt has widened over the late 20th century and will continue to widen throughout the 21st century. Yet, estimates of tropical-width variations differ significantly across studies. This uncertainty, to an unknown degree, is partly due to the large variety of methods used in studies of the tropical width. Here, methods for eight commonly used metrics of the tropical width are implemented in the Tropical-width Diagnostics (TropD) code package in the MATLAB programming language. To consolidate the various methods, the operations used in each of the implemented methods are reduced to two basic calculations: finding the latitude of a zero crossing and finding the latitude of a maximum. A detailed description of the methods implemented in the code and of the code syntax is provided, followed by a method sensitivity analysis for each of the metrics. The analysis provides information on how to reduce the methodological component of the uncertainty associated with fundamental aspects of the calculations, such as monthly vs. seasonal averaging biases, grid dependence, sensitivity to noise, and sensitivity to threshold criteria.

Theoretical and climate modeling studies suggest that the tropics widen in
response to global warming

The standardized methodologies are implemented in the Tropical-width
Diagnostics (TropD;

PSI – the subtropical edge of the tropical circulation delineated by the meridional mass stream function,

TPB – the latitude of the subtropical tropopause break,

OLR – the subtropical latitude where outgoing longwave radiation crosses a certain threshold,

STJ – the latitude of the subtropical jet,

EDJ – the latitude of the midlatitude eddy-driven jet,

PE – the subtropical latitude where precipitation minus evaporation becomes positive,

UAS – the subtropical latitude where the zonal-mean near-surface wind becomes westerly, and

PSL – the latitude of the subtropical sea-level pressure maximum.

calculating the latitude of the zero crossing of a given field and

calculating the latitude of the maximum of a given field.

In Sect. 2, we provide technical guidelines for these two basic calculations
and provide general information on TropD. In Sect. 3, we provide technical
guidelines for each of the eight metric categories listed above. In Sect. 4,
we analyze the sensitivity of the metrics to the choice of methodology using
monthly zonal-mean data derived from the European Centre for Medium-Range
Weather Forecasts (ECMWF) interim reanalysis

ERAI and CMIP5 models' affiliations and the horizontal resolution of
the analyzed data (long

The code documentation below applies to the MATLAB version of TropD. The syntax for the Python version (PyTropD) follows the MATLAB syntax and is documented in the Python code package. Although some of the metrics presented here may be used on zonally varying fields, we stress that the methodologies described here are designed for use on zonal-mean fields (the code has not been tested on zonally varying fields). Calculations in the TropD software assume pressure–latitude (hPa, latitude degrees) coordinates where the pressure level closest to the top of the atmosphere and the latitude grid point nearest to the southern pole are the first elements in the vertical and meridional ordinates, respectively. To reduce sensitivity to format variations across datasets, this ordering is automatically enforced in TropD.

A depiction of the latitude of the zero crossing

TropD is divided into auxiliary calculation functions, generically named

The calculation of the zero-crossing latitude of some function can be generalized to the crossing of any cutoff value by raising or lowering the function by a constant. Therefore, all calculations involving cutoff criteria are translated in TropD to the basic operation of calculating the zero-crossing latitude of some field.

The following guidelines are implemented in calculations of the latitude of zero crossing:

Unless the zero crossing occurs at a grid point, the exact latitude of the zero crossing is calculated using linear interpolation between the two nearest data points on either side of the zero crossing.

In cases where multiple zero-crossing latitudes exist, the first zero crossing along the input interval is chosen.

In cases where multiple zero-crossing latitudes exist, the calculation can be defined as invalid if the latitudinal spacing between the first zero crossing along the input interval and the second zero crossing of the same sign change is smaller than some defined value.

The zero-crossing latitude is calculated in TropD using the following
syntax:

Example of the latitude of the maximum (

To account for potential noise in the data and to reduce grid dependence, the
latitude of the maximum

The dependence on

The latitude of the maximum is calculated in TropD using the following
syntax:

In this section, we provide technical guidelines for common methodologies in
each of the eight metric categories. We briefly introduce each of the
tropical-width metric categories below. For extended reviews of the physical
rationale and interrelations of these metrics in various datasets, see

The tropical mean meridional overturning circulation (i.e., the Hadley cells)
can be defined as the tropical circulation enclosed within the zero
streamlines of the zonal-mean meridional mass stream function

The meridional mass stream function satisfies the continuity equation such that

The most widely used

The stream function is calculated in TropD using the following syntax:

The PSI metric is calculated in TropD using the following syntax:

The tendency of the tropopause height to abruptly drop near the subtropical
jet (e.g., Fig.

The mean tropopause height (

Various tropopause-based methods for calculating the zonal-mean width of the
tropics are found in the literature. These generally include

the latitude of the largest negative poleward gradient in the tropopause
height

the most poleward latitude where the number of days per year with tropopause
heights above a certain altitude exceeds some threshold

the latitude at which the tropopause height drops below a certain fixed
threshold, or a threshold that depends on the mean properties of the tropical tropopause

the latitude of maximal difference between the potential temperature at the tropopause and at
the surface

Each of these methodologies present potential weaknesses. For example,
threshold-based metrics are sensitive to the choice of threshold values

The tropopause height is calculated in TropD using the following syntax:

The TPB metric is calculated in TropD using the following syntax:

The tropopause break latitude is calculated equatorward of 60

Due to variations in atmospheric absorption and surface temperature, the
longwave radiation emitted to space maximizes in the subtropics
(

Outgoing longwave radiation (OLR) at the top of the atmosphere in
CMIP5 models (green), ERAI (black), and the National Oceanic and Atmospheric
Administration (NOAA) interpolated OLR dataset

Common OLR-based methods for calculating the tropical width are

the most poleward latitude at which the zonal-mean OLR is equal to 250 W m

the first latitude poleward of the subtropical OLR maximum at which the zonal-mean OLR drops to 20 W m

For generality, several OLR metric methods are implemented in TropD, using
the following syntax:

The flexibility in the input parameters

In idealized theory, the subtropical jets form at the edges of the
poleward-moving upper tropospheric branches of the Hadley circulation in each
hemisphere

Accounting for the fact that the STJs exhibit significant variations in
longitude and altitude, the latitude of the STJ as an indicator of the
tropical width has been generally calculated in the literature as

the centroid of the upper-level zonal wind within a specified meridional band

the latitude of the maximum of the upper-level zonal wind

the latitude of the maximum of the upper-level minus lower-level zonal wind. As shown in Fig.

The zonally averaged annual-mean zonal wind at the 200 and 850 hPa
levels (

The STJ metric is calculated in TropD using the following syntax:

The macroturbulent eddy momentum fluxes in midlatitudes, which drive the
midlatitude jets, affect the zonal-mean overturning circulation and therefore
the tropical width

To reduce grid dependence, it is common practice to fit a quadratic
polynomial onto data from grid points surrounding the grid point of the
maximum, and define the position of the EDJ as the latitude of the maximum of
that polynomial

The EDJ metric is calculated in TropD using the following syntax:

The values of the smoothing parameter

The subtropical dry zones lie at the latitude bands of the descending
branches of the tropical meridional overturning circulation. The poleward
edges of the subtropical dry zones can therefore be used as indices of the
tropical width

Zonally averaged annual-mean values of

The edge of the tropics is characterized by a transition from surface
easterlies in the equatorward-flowing lower tropospheric branch of the Hadley
circulation to surface westerlies in midlatitudes (Fig.

The subtropical high-pressure belts form along the descending branches of the
tropical meridional overturning circulation. The latitude of maximum
sea-level pressure may therefore serve as a tropical width indicator

The syntax for the PSL metric is

In both the

We proceed with a method sensitivity analysis for the eight metrics
implemented in TropD. For clarity, we use

It is important to note that, since the basic operators (max finding and zero
crossing) applied in the metric calculations are not linear, the metric
calculations do not commute in space and in time. This is illustrated in
Fig.

Time series of the PSI metric for the

The annual means of

The agreement between dynamically consistent metrics is found to improve when
these are derived from seasonal means as opposed to monthly means

To study the grid dependence of the various metric methods, we examine the
relation of the interannual variability (the standard deviation of
annual-mean values during 1979–2004) and the latitudinal grid spacing of the
CMIP5 model output. Table

The Pearson coefficient of inter-model correlation between the
interannual variability (defined as the standard deviation of annual-mean
values for 1979–2004) of the metric and the latitudinal spacing of the model
output in the 34 CMIP5 models. Default methods and statistically significant
correlations (

The dependence of the mean metric latitude on the smoothing
parameter

As in Fig.

The mean SH

Given this sensitivity to grid spacing, method and model selection can play a
critical role in reducing the uncertainty in analyses of the tropical width.
For example, the interannual variability of the CMCC-CESM model, which has
the lowest latitudinal resolution of the CMIP5 models considered here
(3.68

In the presence of random noise, the

The mean latitude for the five available PSI metric methods, derived
from annual means of the zonal-mean meridional wind in the SH

The mean latitude for the

As in Fig.

As in Fig.

To demonstrate the sensitivity of the various methods to the smoothing
parameter

Due to the sharp gradient in the tropopause height at the tropopause break
(Fig.

The mean values of the default methods in each of the eight metrics
implemented in TropD are shown in Fig.

Figure

Similar candle plots for the TPB and OLR metrics are shown in Figs.

Figure

A similar sensitivity to subjective cutoff parameters is seen in the OLR
metric (i.e., the

As in Fig.

The subtraction of the 850 hPa wind from the upper-level zonal wind
distinguishes the signal of the subtropical jet from that of the eddy-driven
jet

The subtropical meridional profiles of the upper- and lower-level zonal wind
(Fig.

The TropD software package provides methodologies for eight commonly used metrics of the tropical width. TropD is designed to reduce, or aid in the assessment of, the methodological component of the uncertainty in studies of tropical width variations by

compiling the relevant methods for each metric category;

reducing all of the calculations in the metric methods to two basic operations: (i) finding the latitude of the zero crossing or (ii) finding the latitude of the maximum;

providing functions for calculating the meridional mass stream function and the tropopause height according to generally accepted guidelines;

providing consistent methods for implementing threshold criteria;

using consistent smoothing across methods; and

using consistent meridional limits for the various metrics.

In addition, TropD allows flexibility in the input parameters (e.g., in the cutoff and smoothing parameters) which enables consistent sensitivity testing of the metrics in various datasets, as well as testing new methods.

Our method sensitivity analysis highlights the importance of differentiating
between variations which arise from parameter choices and inconsistent
resolutions across datasets, as opposed to differences which arise from the
representation of physical processes in the datasets. The analysis suggests
that careful use of the metric methods can reduce some of the spurious
uncertainty. For example, using different metric methods for each hemisphere
can minimize the spurious uncertainty seen in inter-model variations of the
surface metrics (Fig.

Based on our inter-method analysis, the default methods and parameters for each metric category are optimized for the present climate. Nevertheless, the TropD code can be easily adapted for studies of past climates or perturbations of the present climate, which, as in studies of recent tropical width variations, suffer from unknown spurious uncertainty. Similarly, elements of the TropD code can be applied to a wide range of studies beyond calculations of the tropical width (e.g., the position of the intertropical convergence zone, tropopause height variations, and circulation intensity variations) where the use of standardized methodology can reduce spurious uncertainty.

The TropD (MATLAB) software, reference precalculated
metrics, and reference fields are freely available at

The supplement related to this article is available online at:

The text was written by OA with suggestions and editorial contributions from all of the co-authors. The MATLAB code was written by OA with algorithmic contributions from all of the co-authors. The Python code was translated from MATLAB by AM with help from PS and the TropD team.

The authors declare that they have no conflict of interest.

This work is part of the collaborative efforts of the International Space Science Institute (ISSI) Tropical Width Diagnostics Intercomparison Project and the US Climate Variability and Predictability Program (US CLIVAR) Changing Width of the Tropical Belt Working Group. The authors thank the members of these groups as well as the ISSI and US CLIVAR offices and sponsoring agencies (ESA, Swiss Confederation, Swiss Academy of Sciences, University of Bern, NASA, NOAA, NSF, and DOE) for their support. We acknowledge the World Climate Research Programme's Working Group on Coupled Modelling, which is responsible for CMIP, and we thank the climate modeling groups for producing and making available their model output. Ori Adam acknowledges support by the Israeli Science Foundation grant 1185/17. Alison Ming acknowledges funding from the NERC standard grant code NE/N011813/1.Edited by: Juan Antonio Añel Reviewed by: two anonymous referees