A Global Quantification of the Physical Processes Leading to Near‐Surface Cold Extremes

Kinematic backward trajectories are used to globally quantify the contributions of temperature advection, adiabatic compression and diabatic processes to near‐surface temperature anomalies (hereafter T′ ${T}^{\prime }$ ) during the coldest day of each year (TN1day events) based on ERA5. Diabatic cooling dominates TN1day anomalies in the climatologically coldest regions, while advection forms TN1day anomalies over most ocean regions. Over most extratropical land masses, TN1day anomalies arise from a combination of both processes. The mean age and formation distance of TN1day anomalies vary strongly in space, from one to 8 days, and 500–5,500 km, respectively. Five distinct types of TN1day events are identified from these physical and spatio‐temporal characteristics, and their geographical occurrence is investigated. Furthermore, advective, adiabatic and diabatic contributions typically cancel each other partially, but less so for the most intense TN1day events, which occur when the atmosphere's ability to dampen near‐surface temperature anomalies is limited.


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Diabatic cooling, that is, the reduction of an air parcel's potential temperature, may occur due to a wide range of cloud microphysical and radiative processes, whose relative importance and interactions depend on the air parcel's thermodynamic properties and the environment in which it is embedded (Curry, 1983;Emanuel, 2008;Turner & Gyakum, 2011;Turner et al., 2013). In polar continental regions during winter, near-surface air is cooled through the radiative transfer between the (usually snow-covered) surface and the lowermost atmosphere (Wexler, 1936). However, the formation of deep, continental cold airmasses requires the interaction between surface and cloud-top radiative cooling and microphysical processes, in particular ice crystal formation and ice fogs near the surface (Curry, 1983;Emanuel, 2008;Turner & Gyakum, 2011). During the cooling process, these airmasses subside, often by several kilometers, which counteracts the cooling via adiabatic compression and the associated warming (e.g., Papritz, 2020;Walsh et al., 2001). With typical cooling rates on the order of 1 K day −1 , the formation of cold airmasses is a slow process compared to their erosion by diabatic heating that can well exceed 10 K day −1 , for example, when the cold air is advected over a warm ocean surface (Iwasaki et al., 2014;Papritz & Spengler, 2017). Thus, the diabatic formation of anomalously cold air requires sheltering of the air from diabatic heat sources over several days.
In summary, previous studies have documented the general relevance of cold air advection and diabatic cooling for cold extreme formation in specific regions. However, a global quantification of their most likely spatially varying relative importance is lacking so far.
We address this research gap by employing a novel Lagrangian temperature anomaly decomposition introduced by Röthlisberger and Papritz (2023), who quantified the contributions of advection, adiabatic warming and diabatic heating to temperature anomalies during near-surface hot extremes. Here, we apply their methodology to near surface temperature anomalies during the coldest days of each year (hereafter TN1day events) globally in the period 1979-2020 using ERA5 reanalyzes (Hersbach et al., 2020, see below). In the following, we briefly introduce the concept of the temperature anomaly decomposition followed by a description of its technical implementation in Section 2. For a more in-depth description the reader is referred to Röthlisberger and Papritz (2023).
The Lagrangian temperature anomaly decomposition uses the Lagrangian temperature anomaly equation Where is the horizontal wind, ∇ℎ the horizontal gradient, = = 0.286 , is pressure ( 0 = 1,000 hPa), is the vertical velocity in pressure coordinates, and the potential temperature. Equation 1 describes the change of an air parcel's temperature anomaly ′ = − relative to a reference climatology along its trajectory.
Based on Equation 1, any temperature anomaly ′ ( , ) of an air parcel at location and time can be decomposed by integrating Equation 1 along the air parcel's backward trajectory ( ( ), ) from the time when the temperature anomaly of this air parcel was last zero [this time is hereafter referred to as "genesis time" of the anomaly ′ ( , ) ] to time , that is, Terms on the r.h.s. of Equation 2 denote from left to right, ′ due to changes in the temperature climatology over time, ′ arising from horizontal advection of the air parcel in the direction of the climatological temperature gradient, ′ resulting from vertical motion, and ′ arising from diabatic processes along the trajectory. As in Röthlisberger and Papritz (2023), these terms are hereafter referred to as seasonality ′ , advective ′ , adiabatic ′ and diabatic ′ , respectively.
According to Equation 2 there are only few ways how large negative near-surface temperature anomalies (i.e., TN1day events) can come about. First, for TN1day anomalies away from major orographic features, the adiabatic ′ necessarily needs to be non-negative (as ascent of the respective air parcel since is not possible). Over elevated terrain, however, adiabatic ′ can in principle be negative. Second, on the time scale of TN1day anomaly formation changes only little and thus the seasonality ′ is usually negligible (see below). Consistent with other diagnostic approaches (e.g., Tamarin-Brodsky et al., 2019) this only leaves advective ′ and diabatic ′ as possible causes of TN1day anomalies. Furthermore, in the climatologically coldest regions, advective ′ can only 3 of 10 be positive, and therefore it directly follows from Equation 2 that in these regions TN1day anomalies must form through diabatic cooling. Elsewhere, the relative importance of advective and diabatic ′ for TN1day anomalies needs to be investigated, which we shall do in this study.

Data
We use data from the European Centre for Medium-Range Weather Forecasts (ECMWF) ERA5 reanalysis (Hersbach et al., 2020), interpolated to a 0.5° latitude by 0.5° longitude grid and at three-hourly temporal resolution. For three-dimensional atmospheric fields, the original hybrid-sigma model levels are used. For each grid point and year, the respective TN1day events are identified as the days when the daily mean 2-m temperature is the lowest in a given year. As temperature climatology we use the same transient temperature climatology on model levels as in Röthlisberger and Papritz (2023), which takes the long-term warming trend as well as the seasonal and daily cycles into account. More specifically, for any date, is given as the average over all timesteps with the same time of the day in a 21-day window centered around the given calendar day and within ± 4 years.

Trajectory Calculations
To evaluate Equation 2 for each TN1day event at each grid point, we use LAGRANTO 2.0 (Sprenger & Wernli, 2015) to compute 15-day backward trajectories from the respective grid point. Trajectories are started in three-hourly intervals between 00 and 21 UTC of the respective day from 10, 30 and 50 hPa above ground, yielding 24 trajectories for each TN1day event. Along any trajectory ( ( ), ) , we trace , , , , and , whereby the latter two quantities are computed as first order finite differences. Hereafter, only average statistics of the 24 trajectories per event will be considered and referred to as, for example, the temperature anomaly ′ of a particular TN1day event.

Lagrangian ′ Decomposition and Characteristics of ′ Formation
Any ′ ( , ) at time and location is decomposed in a two-step approach. First, the respective backward trajectory ( ( ), ) starting at ( ( ), ) is used to find the genesis time and genesis location ( ) of the anomaly. Hereby, is identified by following ′ ( ( ), ) backward in time and corresponds to the last trajectory time step at which ′ ( ( ), ) has the same sign as ′ ( , ) . Then, the terms on the r.h.s. of Equation 2 are evaluated by integrating them along ( ( ), ) , from to , as detailed in Röthlisberger and Papritz (2023).
The seasonality ′ is usually small (Figure S1b in Supporting Information S1) and thus, in good approximation ′ is given by the sum of advective, adiabatic and diabatic ′ . However, in the computation of the r.h.s. terms in Equation 2, two further residuals appear for numerical reasons. The residual res1 corresponds to the temperature anomaly at anomaly genesis, that is, res1 = ′ ( ( ), ) , and arises because ′ is never exactly zero along a trajectory with discrete time steps. The second numerical residual, res2 , arises from numerical inaccuracies in the computation of derivatives in Equation 2 and is negligible everywhere (Figure S1d in Supporting Informa tion S1). As in Röthlisberger and Papritz (2023), we compute an overall residual res as the sum of the seasonality ′ and the two numerical residuals, which is presented in Figure 1e.
Furthermore, evaluating Equation 2 allows quantifying three important spatio-temporal characteristics of the anomaly formation for any ′ ( , ) : (a) The Lagrangian "age" of the anomaly ′ ( , ) , computed as − , (b) the Lagrangian formation distance of ′ ( , ) , computed as the great circle distance between the location of anomaly genesis, ( ) , and ( ) , as well as (c) the vertical displacement (∆ ) of the respective air parcel since anomaly genesis, that is, ∆ = ( ) − ( ) .
In some regions a non-negligible fraction of anomalies contributing to TN1day events are older than 15 days (up to 20%, Figure 2). This precludes assessing the true age, formation distance and ∆ of these anomalies with 15-day backward trajectories, and only allows decomposing the part of the temperature anomaly that forms within the 15-day period of the trajectory. In such cases we set = − 15 d , and thus assign an age of 15 days, a formation distance corresponding to the great circle distance between ( − 15 d) and ( ) , and a 4 of 10 ∆ = ( ) − ( − 15 d) . Moreover, in such cases res1 = ′ ( ( ), ) can be considerable, but averaged across all TN1day events res1 is typically less than 1.5 K in magnitude even in these regions ( Figure S1c in Supporting Information S1).

TN1day Anomaly Decomposition
We begin by discussing the TN1day anomaly decomposition (Figure 1), which shows values averaged across all 42 TN1day events at the respective grid point. The globally most intense TN1day events (i.e., those with the largest magnitudes of TN1day ′ ) occur in central North America, Siberia and along the coast of Antarctica (Figure 1a), where TN1day ′ reaches values as low as −16 K. Over Southern Hemisphere continents, the largest TN1day ′ magnitudes are found in central South America as well as in southern Africa, with values of −12 and −8 K, respectively. The globally least intense TN1day events occur over tropical oceans. In panel (f) all grid points are classified according to their TN1day ′ composition. Categories denote (from left to right along the bottom colorbar): Green: only advective ′ < 0. Turquoise: multiple processes contribute negatively, but advective ′ is at least twice as large as diabatic ′ and adiabatic ′ . Brown: multiple processes contribute negatively, but neither is twice as important as the second most important one. Light blue: multiple processes contribute negatively but diabatic ′ is at least twice as large as advective ′ and adiabatic ′ . Dark blue: only diabatic ′ < 0. Red: remaining grid points where neither advective nor diabatic dominate due to substantial negative adiabatic ′ . 10.1029/2022GL101670

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The advective ′ is highly variable in space, but negative everywhere except in eastern Siberia, and over Antarctica and Greenland (Figure 1b). Hereby, the advective ′ across all extratropical oceans far exceeds TN1day ′ in magnitude, and is particularly large in both hemisphere's storm track regions (advective ′ < −24 K) as well as in the southern Indian Ocean, where the advective ′ < −32 K. In Europe and southeastern North America the advective ′ is roughly as large as the TN1day ′ (Figure 1b). Hatching and stippling in all panels depicts grid points where more than 10% and 15% of the TN1day anomalies are older than 15 days, respectively.

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The diabatic ′ too is negative over mid-to high-latitude continents and over Arctic and Antarctic sea ice. The largest negative values (up to −16 K) are found in Siberia and Antarctica, but also in the Canadian Arctic (up to −12 K, Figure 1d). Hereby, particularly large negative diabatic ′ occur near the climatologically coldest regions that feature only moderate negative or even positive advective ′ . Over extratropical oceans, the diabatic ′ is positive, reaching values of up to +24 K in the Gulf Stream region and along the North Atlantic storm track. This suggests that during the formation of TN1day ′ in these regions, turbulent heat fluxes from the ocean to the atmosphere during cold air advection (e.g., Ogawa & Spengler, 2019) far overcompensate radiative cooling, leading to a net positive diabatic ′ . As a result, diabatic heating partly, in some regions almost completely (see below), compensates the negative advective ′ . Finally, over the deep tropical oceans diabatic ′ is small.
As anticipated above, the adiabatic ′ is positive away from major orography and shows very prominent peaks around Antarctica and Greenland (Figure 1c). In the Northern Hemisphere, adiabatic ′ is particularly large over subtropical oceans (reaching +8 to +12 K). Moreover, it is almost zero in the regions with the globally largest TN1day ′ magnitudes in central North America. The only regions with negative adiabatic ′ are the most elevated parts of the ice sheets of Greenland and Antarctica. Figure 1f summarizes the results of the TN1day ′ decomposition by classifying each grid point according to its average TN1day ′ contributions. Over most extratropical oceans, advection is the only negative contributor to TN1day ′ , while diabatic and adiabatic ′ partly compensate negative ′ generated through advection. That is, after anomaly genesis, air parcels contributing to TN1day events over extratropical oceans thus move from climatologically colder regions to warmer regions, thereby they subside but also experience net diabatic heating. The same is true for TN1day events in most of Africa, Australia, large parts of South and Central America. Over North America and Eurasia, though, negative diabatic ′ contributes significantly to TN1day ′ too. For TN1day events in most of the US and western Europe, this contribution is subtle, as the advective ′ is still at least twice as large as the diabatic ′ in these events. For TN1day events in central Canada, eastern Europe, Scandinavia, and large parts of central Eurasia, however, advective ′ and diabatic ′ contribute with comparable magnitudes to the respective TN1day ′ . Finally, over the climatologically coldest regions in northern Canada and northern Siberia (cf. Figure 1b and Figure S3 in Supporting Information S1), diabatic ′ dominates over advective ′ . While there is some case-to-case variability in TN1day anomaly composition at each grid point ( Figure S2 in Supporting Information S1), Figure 1f reveals a gradual transition from diabatically driven TN1day events in the climatologically coldest regions to more and more advectively driven TN1day events in climatologically warmer regions.

Spatiotemporal Characteristics: Age, Formation Distance and Vertical Displacement
We next discuss the age and formation distance of TN1day ′ , as well as ∆ (Figure 2). TN1day anomalies are oldest in Siberia, the high Arctic and the Labrador Sea, where their average age is roughly 8 days. In most of Eurasia and North America, the mean age of TN1day anomalies is 5-7 days, while in the Southern Hemisphere extratropics it is mostly 4-5 days only, with slightly older anomalies in Australia and in central South America (Figure 2a). The globally youngest TN1day anomalies occur in the deep tropics, where on average temperature anomalies contributing to TN1day events form within less than 2 days.
The formation distance of TN1day anomalies shows a latitudinal stratification (Figure 2b). Shortest average formation distances are found in the tropics (often <500 km) with a rapid transition to much larger formation distances in the subtropics, where they exceed 5,000 km at some grid points. Further poleward, formation distances of TN1day anomalies gradually decrease again to values below 1,600 km in polar regions of both hemispheres. Interestingly, the age and formation distances of TN1day ′ far exceed those of near surface hot extremes (compare Figures 2a and 2b with Figure 4 in Röthlisberger & Papritz, 2023).
By definition, the spatial pattern of ∆ closely resembles that of the adiabatic ′ and, moreover, is also similar to the spatial pattern of the formation distance, albeit modulated by major orography. In the tropics ∆ is typically less than 100 hPa, with large regions featuring an average ∆ of even less than 50 hPa. Peak values with ∆ 250 hPa are reached for TN1day events in the subtropics as well as in coastal seas around Antarctica. In the regions of largest TN1day ′ magnitudes, in the North American and Eurasian mid-to-high latitudes, ∆ is again less than 100 hPa. Finally, on the Greenland and Antarctic ice sheets, ∆ is negative, revealing that TN1day anomalies there form in ascending air parcels. 7 of 10

Key Physical Aspects of TN1day Formation and the Globally Most Intense TN1day Events
We next elucidate a key physical characteristic of TN1day formation, which helps to understand a pivotal aspect of the globally most intense TN1day events (Figure 1a). Figure 1 reveals that in most regions the contribution of advective, adiabatic and diabatic ′ oppose each other and the resulting TN1day anomaly is often a comparatively small residual. Here, we quantify this cancellation explicitly by the cancellation factor Γ (Figure 3a), defined as whereby The cancellation factor Γ quantifies how much of the negative ′ generated through one or several of the three processes is compensated by the remaining processes. If no cancellation occurs, then all three contributions to ′ are negative and therefore Γ = 0 . Moreover, Γ approaches 1 for completely canceling contributions.  Figures 1a and 1f, and sorts the Earth's surface into 100 bins according to their TN1day ′ (x-axis, from the 1% of the Earth's surface with the largest TN1day ′ magnitude (left) to the 1% with the smallest TN1day ′ magnitude (right)) and displays for each of the six categories introduced in Figure 1f the area fraction of the respective bin falling into the respective category. Panel (c) shows the average TN1day ′ (blue, left y-axis, i) and Γ (black, right y-axis) for each percentile.

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Over extratropical oceans (where advective ′ is the only contributor, Figure 1f), typically more than 70% of the negative advective ′ is canceled by positive adiabatic and diabatic ′ (in some subtropical regions even more than 90%). Over extratropical land regions, Γ is much lower but still considerable, with values of roughly 0.5 in Europe and between less than 0.2 in central North America and roughly 0.3 in central Russia.
Interestingly, the globally most intense TN1day events feature comparatively small Γ , that is, occur in situations where these cancellations are not particularly strong. Moreover, the most intense TN1day events do not occur where advective and diabatic ′ are largest individually (Figure 1). Rather, they occur in situations where both advective and diabatic ′ are negative and thus act in concert to amplify negative near-surface temperature anomalies. Figures 3b and 3c make this finding explicit by combining data displayed in Figures 1a and 1f: First, we sort the Earth's surface into percentiles according to TN1day ′ (i.e., from the 1% of the Earth's surface with the largest TN1day ′ magnitude (left in Figure 3b) to the 1% with the smallest TN1day ′ magnitude (right in Figure 3b)). Then, we quantify for each of the six categories introduced in Figure 1f what fraction of the area in the respective percentile falls into the respective category (y-axis). Figure 3b further underlines the importance of advection for TN1day formation at a global scale, as on 67% of Earth's surface only advection generates the negative ′ during TN1day events, while diabatic and adiabatic ′ oppose the negative advective ′ (class "advective only"). However, this is not the case for the most intense TN1day events. For instance, considering the tenth of the Earth surface with largest TN1day ′ , 88% of that area falls into classes where advective and diabatic ′ jointly contributed to TN1day ′ . Moreover, this tenth of the Earth surface features an average Γ of 0.43, which is lower than for any other tenth of the Earth surface ( Figure 3c).

Discussion and Conclusions
The magnitude and composition of TN1day ′ strongly vary in space, nevertheless five archetypal types of TN1day events emerge from the results of this study: I. Diabatic: This type is present in the climatologically coldest regions in Siberia, northern Canada, and high-elevation regions of Antarctica (see also Figure S3 in Supporting Information S1). There, TN1day ′ forms diabatically over short distances but comparatively long time periods in near surface air (little descent), with very little negative (or even positive) advective ′ . The TN1day formation of this type thus bears close resemblance to the classical "Arctic air mass formation" (Curry, 1983;Emanuel, 2008). II. Advective: This is the dominant type globally and occurs over most subtropical land regions and all ice-free ocean regions except in the deep tropics. In this type, ′ forms through advective ′ in equatorward moving, often descending air parcels. Hereby, the vast majority of the advective ′ is canceled through adiabatic and diabatic ′ . This type is thus characterized by particularly large cancellations with Γ values of up to 0.95 and the largest formation distances globally. We hypothesize that particularly rapid descent is required for this type of TN1day events especially over ocean, because air parcels horizontally crossing gradients on near surface levels would be too strongly diabatically heated, for instance by surface turbulent heat fluxes, to ultimately contribute to a TN1day event. Hence, dynamically induced descent in dry intrusions (Raveh-Rubin, 2017) may be particularly relevant for advective TN1day events in the storm track regions and the subtropics. III. Extratropical advective and diabatic: This type occurs across much of the Northern Hemisphere land masses and is perhaps best described as continental cold air outbreak (Smith & Sheridan, 2020). The aforementioned eastern European cold wave in January and February 2012 (Demirtaş, 2017) likely falls into this type, in which both advection and diabatic cooling contribute to TN1day ′ . It occurs predominantly over land because there the anomalously cold near-surface air parcels cannot be heated diabatically as rapidly as over the oceans, due to the lower heat capacity and the resulting cold surface. The globally most intense TN1day events largely fall into this type, which features particularly small Γ , often comparatively small descent (contrary to the advective type II), and relatively old anomalies. IV. Tropical advective and diabatic: This is the dominant type of TN1day events in the deep tropics, in particular in tropical Africa, the tropical Indian Ocean, Indonesia and parts of tropical South America. It is clearly distinct from types (I-III) in that it features very small (often > −1 K) TN1day ′ forming advectively and diabatically in near-surface air (descent of less than 50 hPa), quasi locally (within less than 1,000 km), and within less than 2 days. V. Katabatic: This type occurs in the seas surrounding Antarctica and Greenland and arises from air parcels subsiding along the respective ice sheets in katabatic flows (e.g., Parish & Bromwich, 1991) before 9 of 10 contributing to TN1day events in coastal seas. It shares characteristics with both the advective (II) and extratropical advective and diabatic types (III), as it is characterized by large negative advective ′ , which is largely canceled by adiabatic ′ (similarity to type (II)). Also, both advective and diabatic ′ are negative (similarity to (III)), which, however, makes it distinct from type (II). Furthermore, the large positive adiabatic ′ for TN1day events of this type goes along with relatively large descent, which contrasts type (III).
A key finding of this study is that in most regions, TN1day ′ are relatively small residuals of large but opposing contributions from the three processes. These cancellations occur between all three processes and can be related to the principle that the atmosphere acts to dampen temperature contrasts whenever possible in order to minimize available potential energy (e.g., Tailleux, 2013). That is, an air parcel that is cooled radiatively (thereby acquiring negative diabatic ′ ) will develop negative buoyancy and subside (if it is not located at the surface) and thereby reduce its negative temperature anomaly adiabatically. Similarly, an air parcel accumulating negative ′ through horizontal advection across climatological temperature gradients will subside too, provided it is located away from the surface, yielding a cancellation between advective and adiabatic ′ . Likewise, an anomalously cold near surface air parcel, for example, with large negative advective ′ , will be heated diabatically through surface heat fluxes and thereby reduce its negative ′ . This Lagrangian thinking implies that unusually large negative ′ (e.g., the globally most intense TN1day events) have a propensity to form when the atmosphere has no efficient option to dampen these anomalies, that is, in situations when the cancellations between the processes are weak. A region where this line of arguments appears particularly adequate is central North America, where the globally most intense TN1day events form both diabatically and advectively, in near surface air (little cancellation through adiabatic ′ ) and where in winter surface sensible heat fluxes can be expected to be weak (little cancellation through diabatic heating).
In summary, this study for the first time quantifies the contributions of advection and diabatic cooling to near-surface cold extremes globally, as well as the Lagrangian time-and spatial scales over which TN1day anomalies form. The average age of TN1day anomalies ranges from roughly 1 day in the deep tropics to 8 days in Russia and the high Arctic, while average TN1day anomaly formation distances range from less than 500 km in tropical Africa to more than 5,000 km over subtropical ocean regions. Moreover, our results underline the dominant role of advection for TN1day events at a global scale and also reveal that the most intense TN1day events do not occur in regions with particularly large diabatic or advective ′ , but rather in regions where little cancellation occurs between advective, adiabatic, and diabatic contributions. earlier draft of this study. Moreover, MR acknowledges funding of the INTEXseas project from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant agreement no. 787652). The authors thank the two reviewers for their constructive comments.