The intention of this study was to create a portable and scalable implementation of the front identification method of H98 as implemented by since that implementation has been successfully used in a number of other studies e.g.. However, one key methodological change was implemented regarding the order of operations when identifying front objects as lines. take what we will call a “mask-then-join” approach, illustrated in Fig. a. First they locate all those grid boxes that are that satisfy the TFP criterion in Eq. () to form a mask (the ABZ gradient criterion in Eq.  is not used). Zero points of the TFL in Eq. () are located by an exhaustive search using linear interpolation between only those grid boxes included in the TFP mask defined Eq. (). Finally, a line-joining algorithm is used to join the zero points of the TFL in Eq. () to form lines representing fronts. Points are joined with their nearest neighbour if the Euclidean distance calculated in degrees of longitude and latitude between two points is lower than a specified threshold. This requires the repeated calculation of the distance between the current point and all remaining un-joined points, making the algorithm computationally expensive. also apply a minimum front length criteria of 250 km.

In contrast, H98 originally proposed a “contour-then-mask” approach, which we adopt here and illustrate in Fig. b. We identify zero points in the complete TFL field defined by Eq. () and join them into lines using a contouring algorithm, specifically the contourLines() function in R. Zero points are again located by linear interpolation, but only zero points located in adjacent grid boxes are considered for joining into lines, reducing the computational expense compared to an exhaustive search and avoiding the need for repeatedly calculating the distance between a large number of points. We then interpolate the values of the fields defined by the TFP and ABZ criteria in Eqs. () and (), respectively, on the points located by the contouring algorithm. Only points that meet the TFP and ABZ criteria defined by Eqs. () and () are retained, leaving a set of pre-joined line segments representing fronts.

The two approaches are compared in Fig. . Zero points in Eq. () usually occur between grid points. That means that adjacent grid boxes meeting the TFP criteria in Eq. () are required in order to find zero points of the TFL using Eq. () by the mask-then-join approach. At or below the 0.75°×0.75° resolution of ERA-Interim, the region that satisfies the TFP criterion in Eq. () is often narrow, frequently only one grid box wide. Therefore, the mask-then-join approach frequently fails to locate front points. This behaviour can be seen in Fig. a where no front points are identified between 44.25 and 45.75° N since two zonally adjacent grid boxes would be required for successful interpolation of a zero point between two masked points given the orientation of the front. This may result in some features not being identified at all or, more frequently, gaps in what should be continuous features. The line-joining algorithm used by attempts to mitigate this using a search radius larger than one grid length, but this is only partially effective. In Fig. a, the search radius is effective in joining the southernmost located point but fails to bridge the gap from between 44.25 and 45.75° N to the region between 46.5 and 48.0° N, where multiple adjacent grid points might once again enable the location of zero points. The number of points located in that northern region is then too small to meet the minimum front length criteria on their own. The contour-then-mask approach originally proposed by H98 and demonstrated in Fig. b is able to successfully identify the whole front as a single object. The masked region is shown for illustration only; in practice, the masking variables are interpolated directly onto the potential front points located on the zero contour. Overall, the contour-then-mask approach results in more fronts and front points identified and fewer breaks, as can be seen in the examples in Fig. a and b and the climatologies in Fig. b and c. The expected decrease in the number of fronts due to there being fewer breaks is compensated by the number of new fronts located due to the increased sensitivity of the contour-then-mask approach to identifying potential front points. In some cases, these new fronts were missed completely by the mask-then-join approach; in others, they fail to meet the length criteria without additional points located by the contour-then-mask approach.

Although automated methods offer the promise of objective feature identification, it is still usually necessary to set some key parameters subjectively. For the front identification method of H98, there are three parameters that require tuning: the amount of smoothing applied to the θW field; the TFP threshold, K1; and the gradient threshold, K2. Some studies have compared outputs with manual analyses by meteorologists to calibrate the parameters. While comparing to charts is a necessary check of an objective algorithm, calibrating in this way is difficult and time-consuming and calibrates the algorithm to the subjective judgement of those meteorologists. Also, all three parameters depend on the resolution of the data. Therefore, the calibration must be repeated for each new dataset or datasets brought to a common resolution for comparison. Instead, we offer some suggestions for objective calibration criteria.

We first address the smoothing problem since the amount of smoothing applied to θW affects the choice of K1 and K2. The purpose of smoothing is to remove local minima and maxima that might break up otherwise continuous features. We particularly wish to avoid local extrema in the TFL field defined by Eq. (), which will appear as short closed contours of TFL=0Km-3. Therefore, it makes sense to examine the effect of smoothing on the average length of the contours. The noise in the TFL field will in part be due to the choice to use Eq. (), as implemented by , to define the location of the fronts rather than the method preferred by H98 (their Eq. 6), designed to quell the amplification of frontal curvature. Previous studies that applied the method of to ERA-Interim used n=2 passes of a simple five-point average to smooth the θW field. In testing on ERA-Interim data, it was found that the average length of the contours of TFL=0Km-3 initially increases rapidly with the number of passes of the five-point smoother, but after 6–10 passes, the effect of further smoothing diminishes (see Fig. S1 in the Supplement). Therefore, we settled on n=8 passes of a five-point smoother.

Equation () was retained for its simplicity and compatibility with and subsequent studies. However, the alternative method preferred by H98 may be made available as an option in future versions of the code associated with this study. classify all closed contours in the front locating field encircling an area smaller than a given threshold as being associated with (potential) local rather than synoptic fronts. Such a criterion introduces additional subjectivity but would effectively reduce the noise when identifying synoptic fronts, possibly allowing for less smoothing to be used and further distinguish fronts associated with orography and other local features. The issue of noise- and surface-driven gradients was also discussed in .

It is common to define weather phenomena as events exceeding some percentile of the climatological distribution. Therefore, we propose a quantile-based approach to setting the thresholds K1 and K2. The advantage of setting thresholds in terms of climatological quantiles is that the thresholds should be comparable between datasets of differing resolutions, while the actual values can differ quite widely. For example, used a threshold for K1 of -8×10-12Km-2 at 2.5° resolution in ERA-40 compared to the threshold of -5×10-11Km-2 at 0.75° resolution used in ERA-Interim by . In order to compute quantiles, we require climatologies of the TFP and the magnitude of the gradient obtained by evaluating Eqs. () and (), respectively, over an extended time period for the region of interest. The time period considered was 1979–2018 in ERA-Interim. Since most fronts occur in the extra-tropical regions, we will focus our attention there. We seek quantiles of the TFP and the magnitude of the gradient that produce continuous fronts in good agreement with published charts for the North Atlantic and Europe, focusing on January and July 2020. Combinations of quantiles of both the TFP and the magnitude of the gradient were systematically compared (see Figs. S2–S4 in the Supplement). We set the first threshold, K1, to the 25th percentile (0.25 quantile) of the climatological distribution of the TFP (see Fig. S2). In the Northern Hemisphere extra-tropics this is around -1.6×10-11Km-2. We set the second threshold, K2, to be equal to the 50th percentile (0.50 quantile) of the climatological distribution of the magnitude of the gradient of θW (see Fig. S3). In the Northern Hemisphere extra-tropics (23.4–66.6° N), this is around 7.5×-6Km-1. These choices are subjective, and an operational meteorologist might make other choices. However, in the absence of strong physical reasoning, these quantiles have a simple symmetry, i.e. each is approximately the 50th percentile of the allowed range (since K1<0Km-2 and, globally, the 50th percentile of TFP is approximately 0 Km-2), and produce continuous fronts in good agreement with published charts (Fig. S4).

Figure  illustrates the monthly and latitudinal climatological variation in the chosen quantiles of the TFP and the magnitude of the gradient of θW. The distributions of both the TFP and the magnitude of the gradient are very different in the tropics compared to the extra-tropics. The value of the TFP chosen for K1 is biased toward the upper latitudes of the Northern Hemisphere extra-tropics, where fronts are frequently observed and associated with extra-tropical cyclones. The chosen value of the magnitude of the gradient lies in the middle of the seasonal variation in the extra-tropics, which is fairly constant between around 35–65° N and 30–50° S, with greater spread in the Northern Hemisphere. The chosen values are broadly representative of the quantiles across the seasons in both the Northern and Southern Hemisphere extra-tropics. Given the relative insensitivity to reasonable values of K1 and K2 shown in the Supplement, the chosen values should be representative across the seasons and both hemispheres for both criteria.

For comparison, previous studies applying the method of to ERA-Interim used a threshold of K1=-5×10-11Km-2 after n=2 smoothing passes e.g.. The ABZ gradient threshold, K2, in Eq. () was not implemented by , which is equivalent to setting K2=0Km-1 since ∇θW≥0 by definition. Our threshold, K1, is higher primarily due to the additional smoothing, but the exclusion of the second threshold, K2, may have caused to choose a lower threshold for K1 in order to remove unwanted features that could more effectively have been eliminated by implementing the second threshold, K2.

When comparing analyses from different weather and climate datasets, the most common approach is to interpolate all the datasets to a common resolution, which is usually the lowest resolution among them. For some features, such as fronts that are more easily identified in higher-resolution data, this can be limiting. The objective calibration criteria described in Sect.  provide one route by which fronts could be identified at the native resolution of each dataset and then compared. The quantile-based criteria will identify the same fraction of grid boxes potentially containing front points for any reasonable resolution and number of smoothing passes. However, computing the required climatologies is time-consuming. An alternative is to keep the thresholds K1 and K2 constant and adjust the number of smoothing passes such that the climatological distributions of the TFP and the magnitude of the gradient are similar between datasets. Specifically, the quantiles used to set the thresholds should be similar. In testing, it was found that matching the threshold quantile of the TFP field provided a more consistent comparison than that of the gradient field. It is sufficient to compare the quantiles for only a small subset of the data, provided that the same subset is used for each dataset, avoiding the need to compute a long climatology in order to determine the thresholds. In testing, various lengths and spatial extents of training data were considered for comparing ERA-Interim and ERA5 – from 1 month to 30 years – for the Northern Hemisphere extra-tropics, Southern Hemisphere extra-tropics, or the whole globe. Finally, 1 month of data was found to be sufficient to consistently determine an appropriate number of smoothing passes. The procedure is not sensitive to either the month of the year or the spatial extent among those considered. In practice, we used January 2001 for the Northern Hemisphere extra-tropics, which is consistent with the examples in Figs.  and .

used repeated applications of a simple central finite-difference approximation to the first derivative to evaluate all the derivatives in Eqs. ()–() at each grid box. The simple approximation uses one grid box on either side of the box in question to approximate the first derivative to second-order accuracy. The zonal and meridional derivatives are evaluated separately using one box to the left and right or above and below, respectively. However, repeated applications of the approximation to the first derivative degrade the accuracy for higher derivatives. In contrast, we use an explicit central finite-difference approximation to the second derivatives required to evaluate ∇2θW=∂2θW∂x2+∂2θW∂y2 when computing the TFL in Eq. (), avoiding the need for repeated applications of the first derivative and maintaining second-order accuracy. The zonal and meridional terms, ∂2θW∂x2 and ∂2θW∂y2, respectively, are evaluated separately. The computation of both the first and the second derivatives was also updated to maintain second-order accuracy at the edges of the domain using forward and backward differences. The increased accuracy at the edges has no additional computational cost, and the improved approximation to the second derivative is actually more efficient than repeated applications of the first derivative.

Other numerical differences include updates to the computation of relative humidity and wet-bulb potential temperature (θW). Relative humidity is required to compute wet-bulb potential temperature. If only specific rather than relative humidity data are available, relative humidity can then be computed from temperature, specific humidity, and pressure (which is constant, at 850 hPa). The implementation by used the table-based approach built into the NCAR Command Language to compute relative humidity. The new implementation uses the mixed-phase parameterisation of relative humidity from the ECMWF Integrated Forecasting System Sect. 7.4.2. In the new implementation, the wet-bulb potential temperature (θW) is computed using the direct method of Eq. 3.8 rather than the iterative method implemented by . The final numerical difference between the two implementations is how short fronts are handled. In the original application, reject any fronts that are less than three points long. In later applications, this was updated to a great-circle distance-based criterion where fronts whose end points are less than 250 km apart are rejected. In our implementation, we sum the great-circle distance between all adjacent points in each front and reject fronts whose total length is shorter than 250 km. The length threshold of 250 km is a subjective choice that has been retained for approximate compatibility with previous studies.

Figure  illustrates the difference between the mask-then-join and contour-then-mask methods and the effect of the updated parameter choices (i.e. n, K1, and K2) in ERA-Interim on 1 January 2001, 00:00 UTC. The mask-then-join approach using the original parameters (Fig. a) clearly identifies fronts, but they are fractured with frequent gaps. The contour-then-mask (Fig. b) results in much smoother front features with fewer gaps and more fronts identified. Figure c shows the results of the updated parameters with more smoothing cycles and stronger thresholds. Figure d shows the fronts identified in ERA5 and is discussed further in Sect. . Compared to the original parameters, the front features are smoother, with fewer breaks, and many spurious local fronts have been removed. One feature that can be seen is a warm front running parallel to the (predominantly) cold front extending from the Azores across the south of the United Kingdom. Such features were noted by H98 and are associated with a warm conveyor belt running adjacent to the front. use a third masking criteria based on potential temperature rather than wet-bulb potential temperature that may be implemented in a future version of the code documented in this study.

Figure  compares the front frequency climatologies from three different implementations of the H98 algorithm applied to ERA-Interim with identical parameters (i.e. the same number of smoothing passes and thresholds K1 and K2): the implementation of used by (Fig. a), a version incorporating our numerical updates but using the original mask-then-join approach (Fig. b), and our final version using the contour-then-mask approach (Fig. c). Figure d shows that our numerical updates result in a slightly lower number of fronts identified in most of the Northern and Southern Hemisphere extra-tropics and a slightly higher number of fronts identified in the tropics. Figure e and f compare our final version with the implementation of and the version incorporating only the numerical updates. The numerical updates produce relatively modest differences in the number of fronts identified (Fig. d). Small decreases are seen in the extra-tropics, and larger increases are seen in the tropics. The move from the mask-then-join to the contour-then-mask approach has a greater effect in the extra-tropics (Fig. f). In the Northern and Southern Hemisphere storm tracks, the number of fronts being identified increases by between 40 % and 80 %. The increases are a combination of increases in the length of previously identified fronts and the addition of fronts that were not previously identified, as demonstrated in Figs.  and . The greatest relative increase in the number of fronts identified is seen in the tropics. Comparing Fig. e and f shows that this is due to a combination of the numerical updates and the move to the contour-then-mask approach. The relative increase in the tropics is very large due to the scarcity of fronts in the tropics making even a small increase seem large. The absolute number of fronts detected in the tropics remains small compared to the extra-tropics (Fig. c).

While changing the implementation of the front identification leads to an increase in the number of fronts identified, as shown in Fig. , the next aspect of the updated method is a change to the parameters used. Figure  compares the climatology of front frequency of our final version with updated parameters (i.e. smoothing passes and thresholds K1 and K2; shown in Fig. a) applied to ERA-Interim against the implementation by with the original parameters. The updated parameters result in slightly fewer fronts identified in almost all regions due to the increased smoothing, making the climatology more similar to earlier estimates but with the smoother individual fronts given by the contour-then-mask method. The greatest decreases are seen on the edges of the tropics, adjacent to regions with high front activity. This pattern is to be expected due to the rapid drop-off in the climatological quantile values of the masking parameters in the tropics in Fig. .

Figure  shows the climatology of front frequency of our final version with updated parameters applied to ERA-Interim, including the breakdown into cold, warm, and quasi-stationary fronts. Figure a allows for a direct comparison with the climatologies in Fig. , showing that while the updated parameters reduce the number of fronts identified compared only to the updated numerical implementation, overall, more fronts are still identified in almost all regions than in earlier versions. Figure b and c show that cold and warm fronts occur with similar frequencies in most extra-tropical regions, as previously shown in . Figure d shows that quasi-stationary fronts occur most often where winds are weaker, particularly in the horse latitudes close to 30° N and 30° S, the inter-tropical convergence zone (ITCZ) close to the Equator and adjacent to high orography, as expected.

Figure  breaks the classification of fronts down further, showing cold and warm fronts by season. Unsurprisingly, cold fronts in the Northern Hemisphere are most common at the beginning of the storm track regions of both Atlantic and Pacific oceans in northern winter (DJF; Fig. a). In contrast, warm fronts in northern summer (JJA; Fig. g) tend to outnumber cold fronts (Fig. c). In agreement with , the seasonal distribution of fronts in the Southern Hemisphere is much more stable. Cold fronts are slightly more common though less widely distributed in the Southern Hemisphere during southern summer (DJF; Fig. a) than in southern winter (JJA; Fig. c), which is consistent with and the climatology of frontogenesis by . The large number of warm fronts near Antarctica (JJA; Fig. f–h) is likely related to the strong temperature gradients between the sea surface and sea ice.

The ERA5 reanalysis has a higher resolution than ERA-Interim, with a grid spacing of 0.25°×0.25° compared to 0.75°×0.75° for ERA-Interim. For ERA5, a total of n=96 smoothing cycles were required to make the climatologies of the TFP and gradient similar to ERA-Interim. Figure d illustrates fronts identified over Europe and the North Atlantic on 1 January 2001, 00:00 UTC. As expected, the features are very similar to those identified in ERA-Interim against which ERA5 was calibrated (Fig. c). Figure  compares the frequency of fronts identified in ERA5 with that in ERA-Interim when fronts are identified in ERA5 at 0.25°×0.25° grid spacing with n=96 smoothing cycles but identical thresholds to those used for ERA-Interim and then aggregated to 0.75°×0.75° grid spacing for comparison with ERA-Interim. Aggregation is performed by counting individual fronts identified at the higher resolution passing through the lower-resolution grid. When aggregated to the same resolution, more fronts are identified almost everywhere in ERA5 than in ERA-Interim. Since aggregation is performed by counting individual fronts, this indicates that ERA5 is able to resolve more fronts due to its higher resolution. The pattern of increase broadly follows the general distribution of fronts, with more fronts seen where they were already common, particularly in the storm tracks where front frequency increases by between 20 % and 40 %. The greatest percentage increases are seen in the ITCZ region to the east and west of South America, where very few fronts were identified in ERA-Interim (and therefore represents a very small absolute increase in frequency) and is mostly associated with quasi-stationary fronts (Fig. d). Decreases in frequency are primarily associated with areas of high orography, which are in turn likely associated with the improved representation of the orography in the higher-resolution dataset.

Figure  shows the climatology of fronts by type identified in ERA5 at its native 0.25°×0.25° resolution. Due to the smaller grid boxes, the frequency is necessarily lower than for ERA-Interim in Fig.  and the aggregated data in Fig. . One ERA-Interim grid box contains nine ERA5 grid boxes. A perfectly straight front passing through one ERA-Interim grid box would typically pass through only three of the nine associated ERA5 grid boxes. Therefore, one might expect the front frequency in ERA5 at its native resolution to be approximately one-third of the frequency in ERA-Interim. Comparing Figs.  and  shows that this is approximately the case.

Figure  shows the seasonal breakdown of cold and warm fronts in ERA5, which is provided to be able to compare the most-up-to-date climatology from ERA5 with previous studies. In general, the maximum warm front frequency occurs at higher latitudes than the maximum cold front frequency due to the structure of extra-tropical cyclones and the associated poleward transport of warm air. During DJF especially, the high frequencies of atmospheric fronts that are influenced by the sea surface temperature (SST) fronts associated with the Gulf Stream in the North Atlantic and Kuroshio current in the North Pacific are clearly visible. The influence of the SST on the atmosphere is more marked for higher-resolution ocean and atmosphere .