NOVEMBER 2017 B Ü ELER AND PFAHL 3567

Potential Vorticity Diagnostics to Quantify Effects of Latent Heating in Extratropical . Part I: Methodology

DOMINIK BÜELER AND STEPHAN PFAHL Institute for Atmospheric and Climate Science, ETH Zurich,€ Zurich,

(Manuscript received 9 February 2017, in final form 31 July 2017)

ABSTRACT

Extratropical cyclones develop because of baroclinic instability, but their intensification is often sub- stantially amplified by diabatic processes, most importantly, latent heating (LH) through cloud formation. Although this amplification is well understood for individual cyclones, there is still need for a systematic and quantitative investigation of how LH affects intensification in different, particularly warmer and moister, climates. For this purpose, the authors introduce a simple diagnostic to quantify the contribution of LH to cyclone intensification within the potential vorticity (PV) framework. The two leading terms in the PV tendency equation, diabatic PV modification and vertical advection, are used to derive a diagnostic equation to explicitly calculate the fraction of a cyclone’s positive lower-tropospheric PV anomaly caused by LH. The strength of this anomaly is strongly coupled to cyclone intensity and the associated impacts in terms of surface weather. To evaluate the performance of the diagnostic, sensitivity simulations of 12 Northern Hemisphere cyclones with artificially modified LH are carried out with a numerical weather prediction model. Based on these simulations, it is demonstrated that the PV diagnostic captures the mean sensitivity of the cyclones’ PV structure to LH as well as parts of the strong case-to-case variability. The simple and versatile PV diagnostic will be the basis for future climatological studies of LH effects on cyclone intensification.

1. Introduction baroclinic instability as the adiabatic energy source re- quired for cyclone development (Charney 1947; Eady Extratropical cyclones are a key component of the 1949), latent heating (LH) due to the formation of global atmospheric circulation because of their transport clouds and precipitation can be an important diabatic of heat, moisture, and angular momentum (Held 1975; contribution affecting mainly cyclone intensification Chang et al. 2002). They contribute substantially to the (e.g., Kuo et al. 1991; Davis and Emanuel 1991; Davis synoptic weather variability in the midlatitudes on a daily 1992; Reed et al. 1993; Stoelinga 1996; Ahmadi-Givi to weekly time scale. When reaching high intensities, they et al. 2004; Marciano et al. 2015; Martínez-Alvarado can cause extreme winds and heavy precipitation with a et al. 2016). In the various studies investigating this huge impact on society, which makes them the third- contribution, potential vorticity (PV) has been estab- costliest natural disasters after tropical cyclones and lished as a key variable (Hoskins et al. 1985). Ertel’s PV earthquakes (e.g., Sigma Research 2016). A large fraction in Cartesian coordinates is defined as of wind extremes in are due to extratropical cy- clones, particularly during winter (e.g., Roberts et al. h =u 5 2014). Over 80% of precipitation extremes in the mid- Q r , (1) latitude -track regions, in the Mediterranean, and in parts of eastern Asia are associated with extratropical where h 5 = 3 u 1 2 V is the three-dimensional abso- cyclones (Pfahl and Wernli 2012). lute vorticity vector (with u denoting three-dimensional The intensity and, ultimately, impact of extratropical wind and 2 V Earth’s angular velocity), u is the potential cyclones depend on the strength of and interaction be- temperature, and r is density. The change of PV over tween their dynamical driving mechanisms: beside time can be derived from Eq. (1):

DQ 1 1 Corresponding author: Dominik Büeler, dominik.bueeler@env. 5 (h =u_) 1 (=u = 3 F). (2) ethz.ch Dt r r

DOI: 10.1175/JAS-D-17-0041.1 Ó 2017 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses). Unauthenticated | Downloaded 10/05/21 02:25 PM UTC 3568 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 74

Here, D/Dt 5 ›/›t 1 u = is the material derivative op- three anomalous components to be systematically stron- erator, u_ denotes the time derivative of the potential ger for cyclones with lower minimum sea level pressure temperature, and F is the three-dimensional frictional (SLP). Using a similar historical dataset, Binder et al. force per unit mass. The two terms on the right-hand (2016) showed that cyclones with stronger warm con- side of Eq. (2) reveal that PV is conserved (DQ/Dt 5 0) veyor belts (cf. Browning et al. 1973; Madonna et al. in adiabatic, frictionless flows but can be modified by 2014) near their center, and thus stronger diabatic diabatic (u_ 6¼ 0) and frictional processes (F 6¼ 0). The heating associated with more pronounced positive diabatic term of Eq. (2) thus allows for an explicit lower-tropospheric PV anomalies, often intensify more quantification of the contribution of LH (reflected in the strongly in terms of SLP deepening. These two findings potential temperature tendency u_) to the PV distribu- complement the various case studies by indicating the tion in a cyclone. PV distribution to be a good metric for both cyclone The link between the PV distribution in and the dy- intensification and intensity. Consequently, they prove namics of extratropical cyclones as well as the role of the PV concept to be beneficial also for investigating the embedded LH are well understood in a qualitative effect of LH on cyclone dynamics in a climatological sense. The cyclonic circulation is typically associated framework. Considering the importance of LH for present-day with three anomalous components: a positive upper- extratropical cyclones, the question of its role in a tropospheric PV anomaly, a positive potential temperature changing climate arises. Global warming and its associ- anomaly at the surface, and a positive lower-tropospheric ated increase of the atmospheric moisture content (e.g., PV anomaly (Davis and Emanuel 1991; Reed et al. 1992; Schneider et al. 2010) is going to increase the potential Campa and Wernli 2012). By applying PV inversion for LH. Predicting how this will change cyclone intensity techniques, the relative contributions of these anomalies in a future climate, however, remains a challenge for to the wind field and thus the intensity of the cyclone can general circulation models (GCMs). Many of them ex- be determined (Davis and Emanuel 1991). LH due to hibit problems in representing present-day storm tracks cloud and precipitation formation in the lower and (Chang et al. 2013), and there are large intermodel dif- middle troposphere is a crucial process influencing these ferences and thus uncertainties when projecting them anomalies. It mainly forms and maintains the lower- into the future (e.g., Ulbrich et al. 2009; Woollings 2010; tropospheric PV anomaly by diabatic PV generation as Chang et al. 2013; Shaw et al. 2016). This is partly due to expressed in Eq. (2). In addition, it can affect the upper- LH processes, which occur on rather small spatial scales tropospheric PV anomaly, which itself originates from and are thus difficult to capture by the coarse-resolution adiabatic PV advection from the stratosphere, in two GCMs (e.g., Willison et al. 2013; Colle et al. 2015; opposing ways: it can intensify the anomaly via the cy- Trzeciak et al. 2016). In particular, Willison et al. (2015) clonic wind field induced by the lower-tropospheric PV showed a strong amplification of the storm-track re- anomaly (e.g., Wernli et al. 2002) and at the same time sponse to global warming over the northeastern Atlantic weaken the anomaly slightly downstream via diabatic in regional climate simulations with higher-than-usual PV erosion (e.g., Grams et al. 2011). These two op- horizontal resolutions, which they attribute to diabatic posing mechanisms contribute to the typical westward processes. For understanding these diabatic processes tilt between the upper- and lower-tropospheric PV in a warmer and moister climate, the PV framework anomalies, which fosters strong cyclone intensification again provides a useful basis: Pfahl et al. (2015) and (e.g., Stoelinga 1996; Ahmadi-Givi et al. 2004). The- Marciano et al. (2015) found larger lower-tropospheric oretical underpinning for the importance of moist PV anomalies in intense cyclones in a warmer climate, processes for cyclone dynamics can also be obtained which they suggest result from enhanced LH. within the counterpropagating Rossby wave frame- Despite considerable progress in understanding LH work (Bretherton 1966; Heifetz et al. 2004): De Vries effects on extratropical cyclones and in demonstrating et al. (2010) showed that including moist processes in their increasing importance in a warming climate, there life cycle studies of baroclinic waves leads to additional is still a need for a more systematic and quantitative growth regimes that can be described as the interactions description: systematic in the sense that versatile di- of moist and dry Rossby waves. Thereby, the moist agnostic methods should be developed to investigate LH waves are defined based on diabatic PV. More recent effects on individual cyclones in various datasets [output studies applied the PV concept in a climatological of numerical weather prediction (NWP) and climate framework: Campa and Wernli (2012) analyzed the ver- models, as well as reanalysis data] and quantitative in tical PV distribution of a set of historical cyclones in the the sense that these methods need to be able to quantify Northern Hemisphere. They found the aforementioned the contribution of LH to cyclone dynamics, which in

Unauthenticated | Downloaded 10/05/21 02:25 PM UTC NOVEMBER 2017 B Ü ELER AND PFAHL 3569 turn allows for comparison between the different data- main assumptions and technical details of the PV sets and model simulations of different climates and diagnostic. Section 3 contains the results of the PV resolutions. Such methods could also help identify and diagnostic for the cyclone case study and the whole set eliminate deficiencies in GCMs, which would ultimately of cyclones. In section 4,wesummarizetheperfor- improve climate projections into the future (Shaw et al. mance of the PV diagnostic as well as its versatile 2016). Fink et al. (2012) developed such a systematic application. diagnostic based on the surface pressure tendency equation, which allowed them to quantify the relative 2. Method contribution of diabatic processes to the deepening of a. Model simulations cyclones in reanalysis data from a feature-based per- spective. Pirret et al. (2017) applied the same tool to a We perform hindcast simulations of extratropical large set of intense European windstorms and found a cyclones with the regional Consortium for Small-Scale strong case-to-case variability of the LH contribution. Modeling (COSMO) NWP model (Steppeler et al. Azad and Sorteberg (2014) developed a conceptually 2003) as a database to develop our PV diagnostic. The similar systematic diagnostic quantifying the diabatic model is initialized and driven at the boundaries with contribution to the lower-tropospheric geostrophic ERA-Interim (Dee et al. 2011) or ECMWF opera- vorticity tendency. They also found significant contri- tional analysis data (depending on the date of the butions from LH, particularly toward the end of the simulated cyclone and with horizontal resolutions intensification phase of a large set of intense North At- ranging from 0.258 to 18) and run on a rotated geo- lantic winter cyclones. In contrast to Pirret et al. (2017), graphical grid with a horizontal grid spacing of 0.1258 however, their case-to-case variability of the LH con- (;14 km) and 40 levels in the vertical. In addition to a tribution turned out to be relatively small compared to basic set of prognostic and diagnostic variables, the 2 other processes. No conceptually similar diagnostic instantaneous temperature tendencies from LH (K s 1) based on the PV instead of the surface pressure or are included in the output data written every 6 h. The geostrophic vorticity tendency equation is currently tendencies are separated into those from resolved mi- available. Chagnon et al. (2013) and Martínez-Alvarado crophysical processes and the Tiedtke (1989) convection et al. (2016) recently implemented an online diagnostic scheme. Temperature tendencies from the boundary using a set of numerical PV tracers in a regional NWP layer and the radiation scheme are not considered. To model, which allowed them to quantify the diabatic identify and track cyclones in the simulations, we use contributions to the dynamics of individual cyclones in a the SLP-based approach developed by Wernli and comprehensive way. However, even though such online Schwierz (2006). PV diagnostics foster a fundamental and quantitative In addition to these reference simulations (REFs), the understanding of the role of LH for cyclone dynamics, influence of LH on cyclone dynamics is investigated with their complex and model-specific implementation limits sensitivity experiments in which we artificially modify systematic applications to large climatological datasets. LH by scaling the physical latent heat constants of In this study, we develop a diagnostic approach that vaporization, fusion, and sublimation in the model code makes use of the PV framework and its advantages to with a factor a [similar to Stoelinga (1996)]. This factor explicitly investigate the link between LH and extra- a is set to 0.0 or 0.5, constituting two simulations of an dynamics. Our method allows us to environment without LH or with reduced LH, re- systematically quantify the contribution of LH to cy- spectively. This explicit simulation of the cyclones’ clone intensification and can be applied to output of sensitivity to LH serves as a reference for the evaluation NWP and climate models and to reanalysis data. It di- of our PV diagnostic. In doing so, we assume that the agnoses the diabatic PV fraction during a cyclone’s life changes in cyclone intensity and PV in these simulations cycle, focusing on the positive lower-tropospheric PV are caused solely by the altered LH and thus neglect anomaly as the key feature influenced by LH. The aim of possible nonlinear interactions with other processes. this paper is to introduce the concept of the PV di- (For instance, a weaker cyclone circulation due to re- agnostic and to demonstrate its performance by apply- duced LH may also affect the cyclone’s PV budget via ing it to a set of simulated historical midlatitude cyclones changes in surface friction. However, these nonlinear in the Northern Hemisphere. In section 2, we start with interactions are assumed to be less important than the the description of the model simulations and one specific direct effect of altered LH.) cyclone case that is used later to illustrate the use of the The reference and sensitivity simulations are run for PV diagnostic. Subsequently, some limitations of the PV 12 historical extratropical cyclones over the North At- tendency approach are discussed, which lead us to the lantic and North Pacific. One of these cases, winter

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FIG. 1. (a) Tracks and (b) temporal evolution of SLP minimum (hPa) of extratropical (2009). The reference simulation is shown in black, the one with a halved specific latent heat constant (a 5 0.5) in light blue, and the one with the specific latent heat constant set to zero (a 5 0.0) in dark blue. The yellow lines show the track of Klaus in ERA- Interim, which only starts 6 h later (as the initial cyclone size at 0000 UTC 23 Jan is too small to be captured by the identification algorithm). storm Klaus (2009) is used hereafter to introduce the caused by heavy rain (Liberato et al. 2011). Klaus was different assumptions and steps underlying the PV di- the most damaging storm in these regions, following agnostic. Therefore, its evolution is briefly described in Martin in December 1999 (Ulbrich et al. 2001). the following section. We initialize the COSMO simulations of Klaus at 0000 UTC 23 January, which is the start of the explosive b. Cyclone Klaus (2009) intensification phase. Figure 1 shows the track and Klaus formed as a small wave perturbation close to the development of the SLP minimum of Klaus from Bermuda on 21 January 2009 and moved northeastward the reference and the two sensitivity simulations as along a band of strong meridional equivalent potential well as from ERA-Interim. The intensification phase temperature gradients (Liberato et al. 2011). This gra- of the reference simulation spans over the first 24 h dient was associated with a strong ‘‘tropical moisture with the strongest drop during the first 12 h. In com- export’’ event (Knippertz and Wernli 2010) and allowed parison, ERA-Interim yields a slower intensification significant contributions of diabatic processes to the but reaches a slightly deeper SLP minimum in the storm’s intensification (Fink et al. 2012). Klaus in- end (967.5 hPa instead of 970 hPa). Also, the track of tensified explosively during 23 January [around 37 hPa the simulated cyclone is slightly shifted during the in 24 h, implying a Bergeron value of 1.3 according to intensification phase compared to ERA-Interim but Sanders and Gyakum (1980)] and reached its lowest SLP becomes more similar toward the time of maximum in- minimum of 965.9 hPa in the at 0000 tensity. In conclusion, although the reference simulation UTC 24 January (Liberato et al. 2011). From there, it shows some deviations from ERA-Interim, mainly propagated over southern and then into the during the first 12 h, it is able to capture the overall in- Mediterranean, where it started to weaken. Southern tensification and track reasonably well. France and northern reported the largest damage Figure 2 shows the development of some important 2 due to strong wind gusts (up to 198 km h 1) and floods variables associated with the explosive intensification of

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FIG. 2. Horizontal fields for the reference simulation of Klaus (2009) (a),(d),(g) in its initial stage (0000 UTC 23 Jan), (b),(e),(h) in its intensification phase (1200 UTC 23 Jan), and (c),(f),(i) at the time of its lowest SLP minimum (0000 UTC 24 Jan). In (a)–(c), equivalent potential temperature ue (K) at 850 hPa is shown in colors and PV (PVU) at 310 K in contours (2-PVU increment; purple line shows the 2 2-PVU isoline as an indicator of the dynamical tropopause). In (d)–(f), horizontal wind velocity (m s 1) at 250 hPa is shown in colors and SLP (hPa) in contours (5-hPa increments). In (g)–(i), PV (PVU) at 850 hPa is shown in colors and SLP (hPa) in contours. The star indicates the position of the cyclone center (SLP minimum).

Klaus in the reference simulation. At 0000 UTC 23 to form a ‘‘PV tower’’–like structure (Figs. 2c,i; Hoskins January, Klaus starts to form as a wave perturbation 1990; Rossa et al. 2000). As a result, Klaus reaches its located ahead of an upper-level trough and at the right most intense stage with an SLP minimum of 970 hPa at entrance of a zonal jet streak (Figs. 2a,d). These favor- 0000 UTC 24 January. This high intensity persists until able conditions for strong intensification persist during 0600 UTC 24 January, when the cyclone makes landfall the subsequent 12 h: the cyclone crosses the jet streak to in Europe (Fig. 1). its poleward side and ends up at its left exit (Fig. 2e) still Figure 3 compares the structure of Klaus in the ref- providing strong upper-level forcing for cyclone in- erence simulation to the simulations with reduced and tensification (e.g., Reed and Albright 1986). Its propa- without LH. It shows the two time steps after the first gation along the northern flank of a tongue of warm and 12 h of intensification (1200 UTC 23 January) and at the moist air in the lower troposphere (Figs. 2a,b) further time of maximum intensity (0000 UTC 24 January). The contributes to the intensification through the potential positive lower-tropospheric PV anomaly is weaker in for diabatic processes. The positive lower-tropospheric the simulation with reduced LH (Fig. 3e) and does not PV anomaly in the cyclone center reaching high values exist in the one without LH (Fig. 3f), in which Klaus is 2 2 2 of more than 5 PV units (1 PVU 5 10 6 Km2 kg 1 s 1; still a wave perturbation instead of a mature cyclone. Fig. 2h) most likely results from these processes (e.g., This exemplifies the strong contribution of LH in the Stoelinga 1996; Ahmadi-Givi et al. 2004). At 1200 lower troposphere in a qualitative way. Also at upper UTC 23 January, Klaus attains a mature stage: the levels, LH contributes to the conditions favoring the elongated cold and warm fronts start to occlude and roll explosive intensification: both the ridge (direct effect) up as a bent-back front around the cyclone center and the positive PV anomaly due to the descending (Figs. 2b,h). At upper levels, stratospheric high-PV air tropopause (indirect effect; see section 1) are sub- from the descending tropopause hooks into the cyclone stantially weaker in amplitude in the simulations with to its northwest and a ridge forms to the northeast reduced (Fig. 3b) and without LH (Fig. 3c). At 0000 (Fig. 2b). In the subsequent 12 h, Klaus propagates more UTC 24 January, this results in a weaker PV tower as- zonally but is still located in the left exit of the jet streak sociated with a less-pronounced SLP minimum in the (Fig. 2f). The upper-level ridge further amplifies simulation with reduced LH (Figs. 3h,k) and no lower- (Fig. 2c), and the lower- and upper-tropospheric PV tropospheric PV anomaly associated with a very weak anomalies within the cyclone gradually align vertically cyclone in the simulation without LH (Figs. 3i,l).

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FIG. 3. Horizontal fields for (a),(d),(g),(j) the reference simulation, (b),(e),(h),(k) the simulation with a halved specific latent heat constant (a 5 0.5), and (c),(f),(i),(l) the simulation with the specific latent heat constant set to zero (a 5 0.0) of Klaus (2009). The displayed time steps are (a)–(f) during the intensification phase (1200 UTC 23 Jan) and (g)–(l) at the time of the lowest SLP minimum (0000 UTC 24 Jan). (a)–(c),(g)–(i) Equivalent potential temperature ue (K) at 850 hPa in colors and PV (PVU) at 310 K in contours (2-PVU increment; purple line indicates the 2-PVU isoline as a measure for the dynamical tropopause). (d)–(f),(j)–(l) PV (PVU) at 850 hPa in colors and SLP (hPa) in contours. The star indicates the position of the cyclone center (SLP minimum).

Figure 1 shows how this distinct behavior in the c. PV tendency concept and its limitations sensitivity simulations translates into a much weaker As a basis for studying the impact of LH on cyclone drop of the SLP minima: the SLP minimum reached by dynamics in a quantitative and Eulerian way, we use the the cyclone is 7 hPa higher in the simulation with re- local PV tendency equation obtained by subtracting the duced LH and 20 hPa higher in the one without LH PV advection term from the material derivative of PV compared to the reference simulation. With respect [Eq. (2)]: to the track, the two sensitivity simulations behave similar to the reference. This seems surprising consid- ›Q 1 1 5 (h =u_) 2 (u =Q) 1 (=u = 3 F). (3) ering the stronger poleward deflection of extratropical ›t r r cyclone tracks in a moister environment found in pre- vious studies (e.g., Coronel et al. 2015; Tamarin and Mathematically, the contribution of LH to a cyclone’s Kaspi 2016). Understanding the even slightly opposite PV budget can be quantified by evaluating Eq. (3) at behavior in our case wouldrequireamorespecific every model grid point, averaging the result over a cy- investigation. clonic area of interest, and integrating these values over Our simulations reveal a strong sensitivity of the in- time. However, numerical limitations are inherent in tensification of Klaus to LH. In the same line, they this approach: calculating the terms on the right-hand demonstrate a qualitatively similar sensitivity of the side of Eq. (3) in model output of typical multihourly positive lower-tropospheric PV anomaly to LH. This time steps cannot fully reproduce the modeled total PV consistent behavior indicates the anomaly to be a good tendency [term on the left-hand side of Eq. (3); calcu- diagnostic measure to quantify the contribution of LH to lated by the PV difference between two consecutive cyclone intensification. In the following section, we will time steps]. The most important reasons for this are that study this in a more quantitative and systematic way. cyclones propagate over larger distances and that their

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21 FIG. 4. Instantaneous PV tendencies [PVU (280 s) ] on a model level in the lower troposphere (;850 hPa) for the reference simulation of Klaus (2009) during its intensification phase (1100 UTC 23 Jan). (a) Calculated diabatic PV tendency [DIAB; first term on the right-hand side of Eq. (3)], (b) calculated PV tendency due to advection [ADV; second term on the right-hand side of Eq. (3)], and (c) sum of (a) and (b), the total local PV tendency without the frictional term (DIAB 1 ADV). (d) Modeled local PV tendency [difference between the PV at 1100 UTC 1 280 s minus the one at 1100 UTC; term on the left-hand side of Eq. (3)] minus the sum of the calculated diabatic and advective tendencies (PVTEND 2 DIAB 2 ADV). The black contours show the SLP (hPa) field at the same time step (1100 UTC 23 Jan); the gray contours show the SLP field 280 s later. The circle in (a) indicates the area over which the vertical profiles in Fig. 5 are calculated. nonlinear evolution cannot be properly captured with a There are some differences close to the cyclone center in temporal discretization of hourly or longer time steps. regions of strong PV gradients, where numerical errors By going to shorter model output time steps (of associated with the advection term are largest. Figure 5a minutes), this discretization error can be reduced and shows the vertical profiles of the instantaneous tendency the PV budget nearly closed. To demonstrate this, we terms in Figs. 4a–c averaged within a radius of 200 km calculate the different terms of Eq. (3) for Klaus based (see Fig. 4a) around the cyclone center. The sum of di- on model output with a time step of 280 s [neglecting abatic and advective PV tendencies (DIAB 1 ADV; friction; i.e., the third term on the right-hand side of purple line) reproduces the modeled total PV tendency Eq. (3)]. Figure 4 shows the corresponding result on a (PVTEND; black line) reasonably well, in particular model level in the lower troposphere at a time step above the atmospheric boundary layer where friction during the strong intensification of Klaus. The sum of is less important. Hence, the instantaneous PV budget the two individually calculated diabatic (DIAB; Fig. 4a) averaged over the cyclone area can be closed fairly and advective (ADV; Fig. 4b) terms reproduces the well. However, there is a mismatch between these modeled PV tendency (PVTEND; calculated as the PV Eulerian PV tendencies (purple and black lines) and the field at the time t 1 280 s minus the one at time t) rea- Lagrangian PV tendency (gray line; calculated as the sonably well (PVTEND 2 DIAB 2 ADV; Fig. 4d). difference between the total PV profile at the cyclone

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FIG. 5. Vertical profiles of PV tendency terms averaged over a circle of 200-km radius (see Fig. 4a) around the center of Klaus (2009) during its intensification phase in the reference simulation. (a) Calculated instantaneous diabatic PV tendency [red; DIAB; first term on the right-hand side of Eq. (3)], advective tendency [yellow; ADV; second term on the right-hand side of Eq. (3)], and the sum of the two (purple; DIAB 1 ADV) at 1100 UTC 23 Jan. The black line shows the modeled Eulerian PV tendency [term on the left-hand side of Eq. (3)], calculated as the PV profile at time t 1 dt (with dt 5 280 s) and location Xt minus the profile at time t and location Xt. The gray line shows the modeled Lagrangian PV tendency, calculated as the PV profile at time t 1 dt and location Xt1dt minus the one at time t and location Xt. (b) Hourly PV tendency terms between 1100 and 1200 UTC 23 Jan, calculated as the sum of all in- stantaneous terms. The gray line in (b) is calculated as in (a), but using an hourly instead of a 280-s time step dt. location at time t 1 280 s and the profile at time t) re- PV and (vertical) advection has previously been dem- sulting from the cyclone’s propagation. When summing onstrated both theoretically (Tory et al. 2012; Martínez- up all the instantaneous tendencies between 1100 and Alvarado et al. 2016) and empirically (Persson 1995; 1200 UTC, as shown in Fig. 5b, these errors accumulate Lackmann 2002; Martínez-Alvarado et al. 2016). Since and yield a significant mismatch of up to 0.25 PVU be- the amplitude of the residual PV tendency is in the range tween the hourly Eulerian (purple and black lines) and of the aforementioned numerical error (purple or black Lagrangian PV tendency (gray line; calculated as the PV line minus gray line), integrating ›Q/›t over hours suf- profile at the cyclone location at 1200 UTC minus the fers from cancellation errors (see also Tory et al. 2012). profile at 1100 UTC). Hence, on an hourly time scale, Hence, integrating the contribution of the diabatic term the PV budget cannot be closed even when integrating to the evolution of the total PV over the whole cyclone instantaneous PV tendencies over time. Beside this nu- intensification phase is problematic. Going toward on- merical limitation of the PV tendency approach, Fig. 5b line PV tendency calculations (e.g., Chagnon et al. 2013; also reveals a limiting physical aspect: in the lower tro- Saffin et al. 2016; Martínez-Alvarado et al. 2016) mini- posphere of an intensifying cyclone, the diabatic and mizes this problem but only for simulations restricted to advective terms of Eq. (3) (red and yellow lines) are of short, (sub-)daily periods (Saffin et al. 2016; Martínez- similarly large amplitude but opposite sign. They thus Alvarado et al. 2016). In the following section, we sug- nearly cancel out each other and yield a small residual gest an alternative approach that does not suffer from PV tendency ›Q/›t. This quasi balance between diabatic such cancellation problems.

Unauthenticated | Downloaded 10/05/21 02:25 PM UTC NOVEMBER 2017 B Ü ELER AND PFAHL 3575 d. PV diagnostic the tropopause. The magnitudes of the diabatic and advective contributions, however, are typically Motivated by the aforementioned limitations of cal- larger. Furthermore, we mainly focus on the cyclone culating full three-dimensional PV tendencies [Eq. (3)] intensification phase, which is often located over and integrating them over time, we develop a PV di- the ocean, where surface roughness is relatively agnostic based on a simplified approach. Three main low. Therefore, PV modification due to friction is assumptions underlie this method, which are reasonable neglected. approximations for averages over a cyclone area but not on a gridpoint scale: Under these assumptions, the PV tendency equation [Eq. (3)] reduces to 1) PV is in steady state (›Q/›t 5 0): To first order, the › › small residual PV tendency Q/ t due to the cancel- 1 ›u_ ›u_ ›u_ ›Q h 1 h 1 h 5 w , (4) lation between the much larger diabatic PV modifi- r x›x y›y z›z ›z cation and PV advection terms allows us to assume lower-tropospheric PV to be in steady state. This h h h where x, y, and z are the x, y, and z components of assumption is fulfilled particularly well during the absolute vorticity and w denotes vertical wind velocity. phase of maximum intensity when the cyclone Integrating Eq. (4) vertically between a lower bound zl structure usually reaches its most balanced and (where Q is assumed to vanish) and the reference level symmetric stage. zr yields 2) There is no horizontal PV advection across the ð z _ _ _ boundaries of the cyclone area (u›Q/›x 1 r 1 ›u ›u ›u 1 Q (z ) 5 h 1 h 1 h dz. (5) y›Q/›y 5 0): The spatial average of the horizontal diab r r x›x y›y z›z w zl PV advection within a circular area around a per-

fectly symmetric and isolated cyclone would equal The left-hand-side term Qdiab(zr), hereafter referred zero. However, real cyclones are typically asymmet- to as diabatic PV (with ‘‘diabatic’’ referring to LH pro- ric, especially during their intensification. The defi- cesses only, excluding surface fluxes and radiation), nition of an effective cyclone area introduced in this describes the fraction of the PV at altitude zr within a study (see below) allows us to better account for this cyclone column resulting only from the assumed balance asymmetry by including all major structural features between diabatic PV modification and vertical advec- into the cyclone area. In this way, PV advection tion [Eq. (4)]. Lower-tropospheric regions of strong as- across the area boundaries can be significantly re- cent induce continuous condensation, freezing, and duced, which justifies the assumption of neglecting deposition of moisture and hence latent heat release. horizontal PV advection. This is only valid for the The resulting positive vertical gradients in the heating lower to middle troposphere, since in the upper rate lead to continuous generation of PV that is con- troposphere, quasi-horizontal PV advection from stantly transported upward. Thereby, the positive ten- the stratosphere becomes the main PV tendency dency from diabatic generation and the negative contribution. tendency from vertical advection balance each other to 3) There is no friction (F 5 0): Friction can modify PV first order (see also Persson 1995; Lackmann 2002; Tory mainly near the surface but also at transition zones et al. 2012; Martínez-Alvarado et al. 2016). such as the top of the boundary layer (e.g., Technically, Eq. (5) and hence our PV diagnostic are Adamson et al. 2006; Plant and Belcher 2007)and implemented as follows:

ð zr ›u_ ›u_ ›u_ h i 5 1 h 1 h 1 h 1 Qdiab(zr) effarea x y z dz . (6) r ›x ›y ›z w $ : 21 zl w 0 02 m s ,effarea

2 The integrand is first calculated on each grid point in the the PV diagnostic are fulfilled. The value of 0.02 m s 1 full three-dimensional domain and then averaged over has been chosen such that most of the ascent region is all grid points within the effective cyclone area (which included. Finally, the diabatic PV at a reference level zr will be introduced later) on each model level and at all is obtained by integration starting from the lowest model grid points with vertical wind velocity w larger than level zl. The resulting vertical profile of diabatic PV can 2 0.02 m s 1. In this way, only regions with significant as- then be compared to the total PV from the model output cent are considered where the balance assumptions of (averaged over the same effective cyclone area).

Unauthenticated | Downloaded 10/05/21 02:25 PM UTC 3576 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 74 e. Effective cyclone area occurs, following Berrisford (1988) and Wernli 2 (1995), the potential temperature tendency (K s 1) Cyclones typically have an asymmetric and non- is calculated as circular shape, particularly during the intensification phase. Defining the cyclone area simply with a circle of kuv L dqs a fixed radius thus often misses important regions of c p dT diabatic PV modification, for example, along strongly u_ 52 p [1 2 e(802f)/5], (7) L dq elongated fronts. Therefore, we introduce a more 1 1 s c dT cyclone-specific area—denoted effective cyclone area— p that accounts for this asymmetry to a certain degree, 2 where L (J kg 1) is the specific latent heat of vapor- partly based on the principle introduced by Flaounas 2 2 ization, c (J K 1 kg 1) is the specific heat capacity of et al. (2014). The definition makes use of the spatially p dry air at constant pressure, k (no units) is the ratio smoothed relative vorticity field on the 850-hPa level 21 21 Rd/cp [with Rd (J K kg ) being the gas constant of (we use a filter strength of 31 3 31 grid points; cf. 2 dry air], v (Pa s 1) is the vertical pressure velocity, Flaounas et al. 2014): all grid points above a specific 2 2 p dq dT 1 1 smoothed relative vorticity threshold (here chosen as (Pa) is the pressure, s/ (kg kg K )isthe 25 21 change of saturation specific humidity with tempera- 7 3 10 s ) and connected to the cyclone center are ture, and f (%) is the relative humidity. The Clausius– defined as the effective cyclone area. This method Clapeyron relationship, the change in saturation generally yields large effective areas for strong cy- specific humidity with respect to temperature, is ap- clones but also for large and weak cyclones (Flaounas et al. 2014). To also account for small and weak cy- proximated by clones,weassignacirclewitharadiusof200kmasa dq « L e minimum effective cyclone area. The main advantage s ’ s , (8) dT p R 2 is that this area definition captures all the main frontal y T features and thus minimizes the quasi-horizontal PV where « (no units) is the ratio of the gas constant of dry advection across the boundary of the effective cyclone 21 21 air and the gas constant of water vapor, Ry (J K kg ) area (which is required for our second assumption). As a is the gas constant of water vapor, e (Pa) is the satu- disadvantage, the varying size of this area limits the s ration vapor pressure, and T (K) is the temperature. In comparability of spatially averaged PV values among Eq. (7) u_ is set to zero at all grid points with either individual time steps or cyclones. Figure 6 shows the v $ 0% or f , 80%. The term in brackets on the right- development of the effective cyclone area for the ref- hand side of Eq. (7) accounts for subgrid-scale con- erence simulation of Klaus: at the beginning of the densation that can occur already at relative humidities strongest intensification phase (0000 UTC 23 January; below 100%. Fig. 6a), the area does not substantially exceed the Figure 7 compares the modeled and diagnosed minimum area of a circle with 200-km radius. After the potential temperature tendencies u_ in vertical cross explosive intensification (1200 UTC 23 January; sections through the cyclone center: during the in- Fig. 6b), the area is much larger and has a tail to the east tensification phase (1200 UTC 23 January), both the of the cyclone center, which accounts for the zonally vertical filament of strong heating along the occluded oriented warm front with high relative (and potential) bent-back front (400 km from point A in Figs. 7c and 7e) vorticity (cf. Fig. 2h). At the end of the intensification and the less intense and more shallow heating along (0000 UTC 24 January; Fig. 6c), the cyclone becomes the cold front (1000 km from point A in Figs. 7c and 7e) more circular and symmetric, which is again captured by are reasonably well captured by the diagnostic the effective cyclone area. method. Also, at the time of maximum intensity (0000 UTC 24 January), the diagnostic method reproduces f. Diagnosed potential temperature tendency several vertical filaments of heating in the cyclone re- Applying our PV diagnostic to model data requires gion with respect to amplitude and structure (Figs. 7d,f). instantaneous potential temperature tendencies u_ from However, the method, by , cannot re- model output. For cases in which these are not available, produce subgrid-scale convective heating (e.g., between we expand the diagnostic with the option to use a di- 500 and 550 km from point A in Figs. 7d and 7f), for agnosed potential temperature tendency. Based on the which the vertical motion is not resolved on the model simplifying assumptions that the ascent of air parcels grid. It does not account for diabatic cooling due to follows a moist adiabat, that no ice processes are in- evaporation of rain or melting and sublimation of ice volved, and only condensation to the liquid phase either (no negative values in Figs. 7e and 7f). While

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FIG. 6. Effective cyclone area for the reference simulation of Klaus (2009) (a) in its initial stage (0000 UTC 23 Jan), (b) in its intensification phase (1200 UTC 23 Jan), and (c) at the time of its lowest SLP minimum (0000 UTC 24 Jan). The colors show the vertical component of the 2 smoothed relative vorticity (s 1), with 7 3 1025 s21 being the minimum value for a grid point to be assigned to the effective cyclone area. The gray shading indicates the minimum effective cyclone area (circle of 200 km around the relative vorticity maximum). SLP (hPa) is shown in contours, and the cyclone center location (SLP minimum) is shown with a star. using this diagnosed potential temperature tendency be of second-order importance for the area-averaged thus neglects several microphysical processes that can lower-tropospheric PV anomaly. lead to distinct PV anomalies, particularly on small to The results of the PV diagnostic using the diagnosed mesoscales (e.g., Crezee et al. 2017), we assume the PV potential temperature tendency according to Eq. (7) are modification associated with these missing processes to discussed in section 3c. To test the sensitivity of these

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21 FIG. 7. Instantaneous potential temperature tendencies (K s ) for the reference simulation of Klaus (2009) (a),(c),(e) during its intensification phase (1200 UTC 23 Jan) and (b),(d),(f) at the time of its lowest SLP minimum (0000 UTC 24 Jan). (a),(b) Modeled potential temperature tendencies at 850 hPa [sum of tendencies from the resolved microphysics and the Tiedtke (1989) convection scheme; colors], SLP (hPa; contours), the cyclone center (star), and the line of the vertical cross section. (c),(d) Vertical cross sections along the lines in (a) and (b), respectively, with modeled potential temperature tendency in colors and PV (PVU) in contours (2-PVU increments; purple line indicates the 2-PVU isoline). (e),(f) Diagnosed potential temperature tendencies calculated with Eq. (7). results to the specific formulation of the potential tem- 3. Results perature tendency equation, we additionally apply the a. Cyclone Klaus (2009) PV diagnostic with the tendency equation introduced by Emanuel et al. (1987). Following Lackmann [2002; his Figure 8 shows vertical profiles of the total PV from Eq. (1)], we calculate this tendency at all grid points model output (Q; black line; hereafter denoted total with a vertical pressure velocity v smaller than zero PV) and the diagnosed diabatic PV from Eq. (6) ðQdiab; and a relative humidity higher than 70%. red line; hereafter denoted diabatic PV) over parts of

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FIG. 8. Total (Q; black) and diabatic [Qdiab, according to Eq. (6); red] PV profiles (PVU) of Klaus (2009) for (a)–(d) the reference simulation, (e)–(h) the simulation with reduced latent heating (a 5 0.5), and (i)–(l) the simulation with no latent heating (a 5 0.0) on pressure levels (hPa). The displayed time steps are (left to right) 1200 UTC 23 Jan, 1800 UTC 23 Jan, 0000 UTC 24 Jan, and 0600 UTC 24 Jan.

Unauthenticated | Downloaded 10/05/21 02:25 PM UTC 3580 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 74 the intensification phase and the phase of maximum in- profiles with height in Figs. 8e–h. We note, however, that tensity of Klaus. For simplicity, we mostly use the term diabatic PV in the middle and upper troposphere should PV anomaly hereafter even though the whole analysis is be interpreted with caution since the effective cyclone based on the absolute (total or diabatic) PV and no area definition does not account for any backward tilt of anomalies in a climatological sense are calculated. the vertical cyclone axis. Therefore, the PV diagnostic A positive total PV anomaly of up to 1.5 PVU builds up in does not capture the full cyclone system at these levels. the lower troposphere (.600 hPa) in the reference sim- Furthermore, the neglect of horizontal advection by the ulation (Figs. 8a–d). The diabatic PV profile reproduces PV diagnostic limits its applicability for studying the this anomaly remarkably well, particularly in the lowest structure at the tropopause level, where this term be- 200 hPa. In the middle troposphere (600–400 hPa), rela- comes dominant. tively high total PV values occur during the intensification To further demonstrate the strong sensitivity of the phase (Figs. 8a,b). During the phase of maximum in- lower-tropospheric PV anomaly to LH, Fig. 9 shows the tensity, these enhanced values gradually turn into a local mass-weighted vertical average of total PV Q and dia-

PV minimum sharpening the tropopause, which is in- batic PV Qdiab between 950 and 600 hPa for each simu- dicated by the kink in the total PV profile at 400 hPa lation. These pressure boundaries are arbitrary but (Figs. 8c,d). The diabatic PV profile has a different shape include most of the lower- to midtropospheric regions in in the middle troposphere during the intensification phase which substantial LH and thus PV generation occurs (Figs. 8a–c): it approaches zero with height and thus di- (e.g., Fig. 8d). At the same time, they are not too close to verges from the total PV profile. Only at the last time step the surface, where boundary layer processes dominate (Fig. 8d), the diabatic and total PV match almost perfectly the PV budget, and also do not extend too far into the up to the tropopause. The simulation with reduced LH upper troposphere, where the importance of PV ad- (Figs. 8e–h) shows a qualitatively similar behavior to the vection limits the use of our PV diagnostic. There are reference simulation. However, the amplitude of the three key features in Fig. 9: the slope of the Q line, the lower-tropospheric total PV anomaly is smaller by ap- similarity of the shapes of the Q and the Qdiab lines, and proximately 50%. The most striking difference in the the offset between the Q and the Qdiab lines. The slope middle troposphere is that the tropopause descends much of the Q line is a direct measure for the sensitivity to deeper (down to 500 hPa) within the effective cyclone LH: the steeper the Q line, the more sensitive the area than in the reference simulation. Overall, the cyclone’s lower-tropospheric PV anomaly is to LH. The diabatic PV again captures the lower-tropospheric total shape of the Qdiab line is similar to the shape of the Q line PV anomaly well and in the middle troposphere, as in at most of the time steps, indicating that the PV di- the reference simulation, approaches zero with height. agnostic is able to capture this sensitivity. Only at 0000 In the simulation without LH (Figs. 8i–l), the lower- UTC 24 January, this is not the case as the diagnostic tropospheric total PV anomaly is small and thus aligns underestimates diabatic PV generation in the reference again with the diabatic PV, which is zero by construction. simulation. The offset between the two lines represents The overall good agreement of diabatic and total PV the fraction of the lower-tropospheric PV anomaly that in the lower troposphere demonstrates the mainly dia- does not originate from LH-induced PV modification batic origin of the lower-tropospheric PV anomaly in the within the model domain and simulation period (as this reference simulation (Figs. 8a–d). The consistent result PV anomaly is also present in the simulation without in the simulations with reduced LH (Figs. 8e–h) and LH). It can originate from friction, radiation, or upper- without LH (Figs. 8i–l) indicates that the PV diagnostic tropospheric PV being advected downward. However, it is able to capture this sensitivity of Klaus to LH. In the can also be created by diabatic processes before the middle troposphere, the total and diabatic PV profiles initialization of the COSMO simulation or outside of the diverge mainly during the intensification phase, indi- model domain. cating a mixture of different air masses: the diabatic PV In summary, the three simulations demonstrate the profile represents the contribution from diabatic PV sensitivity of the lower-tropospheric PV anomaly of generation (in situ or advected from below), whereas the Klaus to LH. Applying our PV diagnostic to the simu- residual between total and diabatic PV indicates con- lations shows that the method is able to largely capture tributions from advection of high-PV air (e.g., because this sensitivity and quantify the contribution of LH to of the descending tropopause). In the simulations with the cyclone’s PV budget. reduced LH (Figs. 8e–h) and without LH (Figs. 8i–l), the b. Set of cyclones tropopause penetrates deeper, which indicates the re- duction of diabatic PV erosion at upper levels that is also To evaluate the PV diagnostic in a more robust and reflected in the weaker decrease of the diabatic PV systematic way, we apply it to a set of 12 historical

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FIG. 9. Mass-weighted vertical average of total (Q; black) and diabatic [Qdiab, according to Eq. (6); red] PV (PVU) between 950 and 600 hPa for the simulations without LH (a 5 0.0), with reduced LH (a 5 0.5), and the reference simulation. The displayed time steps are as in Fig. 8: (a) 1200 UTC 23 Jan, (b) 1800 UTC 23 Jan, (c) 0000 UTC 24 Jan, and (d) 0600 UTC 24 Jan. extratropical cyclones over the North Atlantic and sensitivity of a cyclone to LH (physical aspect) and the North Pacific. Most of them were explosively deepening performance of the PV diagnostic in reproducing this or very intense cyclones with severe impacts. However, sensitivity (verification aspect). The physical aspect is we also include cases of medium strength to test the illustrated by the distance of a cyclone marker from the performance of the PV diagnostic over a broad spec- origin: the more a marker is located at the top right, the trum. An additional selection criterion is that the cy- more sensitive the cyclone’s lower-tropospheric PV clone should not be located over land during its anomaly is to LH. This sensitivity is largest for cyclones intensification phase and time of maximum intensity. Xynthia, Berit, Klaus, Ulla, and Tini, followed by The reason for this is the larger importance of surface Emma, Dirk, Joe, and Kyrill. The smallest sensitivity is friction over land, which can significantly modify lower- found for Ted, Rob, and Ben. The sensitivity of the most tropospheric PV and thus violate the third assumption sensitive cyclone (Xynthia) is almost 4 times larger than underlying the PV diagnostic. Table 1 describes the set the one of the least sensitive (Ben). This demonstrates of simulated cyclones, including their region, date of the strong case-to-case variability in the strength of the maximum intensity (lowest SLP minimum), maximum diabatic contribution during cyclone intensification. The drop of the SLP minimum in 18 h, and categorization verification aspect of Fig. 10 is illustrated by the distance according to Binder et al. (2016) as discussed later. of a cyclone marker to the identity line (y 5 x). The PV The results of the PV diagnostic for this set of cyclones diagnostic underestimates the LH sensitivity for the are summarized in Fig. 10. It shows the sensitivity of cyclones above the identity line and overestimates it for total lower-tropospheric PV to LH as obtained from the the ones below. The scattering of the cyclone markers model experiments (Qref 2 Qa50:0) plotted against the yields a correlation coefficient of r 5 0.71 and a root- diagnosed diabatic PV (Qdiab,ref). The sensitivity values mean-square error of 0.15 PVU. This demonstrates that are averaged over the last three time steps before the the PV diagnostic does not capture the full variability of time of the lowest SLP minimum (see Table 1)to the modeled sensitivity. On average for the whole cy- provide a representative measure for the final phase of clone ensemble, however, the PV diagnostic reproduces the intensification. Figure 10 thus demonstrates both the the modeled LH sensitivity of the PV anomaly

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TABLE 1. Set of 12 simulated historical extratropical cyclones. Most of the cyclones have an official name issued by Free University of Berlin. For simplicity, we assigned arbitrary names to the following cyclones without an official name: Ben, Joe, Rob, and Ted. The region of intensification is either North Atlantic (NA) or North Pacific (NP). The time of lowest SLP minimum and the lowest SLP minimum are based on the COSMO simulation. As an intensification measure, the maximum drop of SLP minimum in 18 h (within the simulation period) is given. The category according to Binder et al. (2016) classifies the cyclones based on the strength of their intensification (measured in terms of SLP minimum) and of their WCB: C1: strong (explosive) intensification and strong WCB; C2: weak intensification and strong WCB; and C3: strong (explosive) intensification and weak WCB [see Binder et al. (2016) for details about the method].

Time of lowest Lowest SLP Max drop of SLP Category according to Name Region SLP min min (hPa) min in 18 h (hPa) Binder et al. (2016) Ben NP 0000 UTC 20 Jan 2012 984.1 6.7 None (weak WCB; weak cyclone) Berit NA 0600 UTC 25 Nov 2011 939.6 31.9 C1 Dirk NA 1200 UTC 24 Dec 2013 919.3 31.9 C3 Emma NA 0000 UTC 1 Mar 2008 957.0 17.1 None (between C1, C2, and C3) Joe NP 0600 UTC 6 Feb 2012 971.5 20.2 C3 Klaus NA 0000 UTC 24 Jan 2009 970.0 27.9 None (between C1 and C3) Kyrill NA 1800 UTC 17 Jan 2007 963.4 27.3 None (between C1 and C3) Rob NA 0000 UTC 21 Jan 2012 998.5 4.8 C2 Ted NP 0000 UTC 3 Feb 2012 966.1 15.1 None (between C1 and C3) Tini NA 1200 UTC 12 Feb 2014 951.6 29.4 C3 Ulla NA 0000 UTC 15 Feb 2014 951.0 32.3 C1 Xynthia NA 1200 UTC 27 Feb 2010 963.0 28.3 C1

remarkably well (blue cross in Fig. 10). The relative is able to indirectly identify lower-tropospheric PV position of the cyclone markers within the cyclone en- anomalies resulting from friction over land. For Emma, semble as well as the mean value (blue cross) in Fig. 10 the small LH-induced fraction of the total lower- do not change substantially if the pressure boundaries tropospheric PV anomaly rather reveals a case in (950–600 hPa) are changed to 900–600 or 950–700 hPa which our PV diagnostic struggles to capture the mod- (not shown). Also, the correlations stay reasonably high eled LH sensitivity: in Fig. 10, this cyclone is the one with when averaging between 900 and 600 (r 5 0.72) or be- the largest distance to the left of the identity line. In tween 950 and 700 hPa (r 5 0.59). addition, Fig. 11 shows that the lower-tropospheric PV An additional physical aspect not covered by Fig. 10 is due to adiabatic processes differs strongly between the the fraction of the lower-tropospheric PV anomaly three simulations (assuming that nonlinear effects of produced by LH in comparison to other processes (i.e., altered LH on the PV budget via altered frictional and the offset between the Q and the Qdiab lines shown in radiative processes are small, a perfect PV diagnostic Fig. 9 for Klaus). Such a decomposition into diabatic and would yield a similar adiabatic PV amount for all three adiabatic anomalies (where, in our case, the adiabatic LH sensitivity simulations). Emma is a cyclone with a anomalies also include PV generated through frictional relatively smooth field of small PV values in the lower and radiative processes) based on our PV diagnostic is troposphere and no pronounced fronts (not shown). The shown in Fig. 11. For most of the cyclones, LH con- effective cyclone area algorithm therefore struggles to tributes more than 50% to the total lower-tropospheric define an isolated cyclone, which probably leads to PV PV anomaly. Only for a few cyclones, namely, Ben, advection into the effective cyclone area. Emma, Rob, and Ted, this fraction is around 50% or c. PV diagnostic with diagnosed potential smaller. In the case of Rob, the lower-tropospheric PV temperature tendencies anomaly amounts to around 1.25 (1.15) PVU in the reference simulation (simulation with reduced LH), To allow for a more flexible application of our PV from which only around 40% (20%) is due to LH. The diagnostic, it can also be used based on diagnosed po- track of the cyclone provides a possible explanation for tential temperature tendencies [Eq. (7)] instead of u_ these small contributions from LH: in the 18 h before its from model output. Figure 12 shows the corresponding lowest SLP minimum, Rob moves over Newfoundland. result for the set of 12 simulated cyclones. In this setup, During this passage, the lower-tropospheric PV anom- the PV diagnostic overestimates the sensitivity to LH aly in the simulation without LH increases from around compared to the standard setup with u_ from model 0.75 to 1.1 PVU (not shown), which most probably re- output, which is indicated by the slight shift of the whole sults from friction because of the absence of diabatic PV cloud of cyclone markers and the mean value (blue modification. This demonstrates how the PV diagnostic cross) to the right in Fig. 12 compared to Fig. 10,

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FIG. 10. Mass-weighted vertical average between 950 and 600 hPa of the diabatic PV (PVU) in the reference simulation (Qdiab,ref; x axis) and the difference of the mass-weighted vertical averages of total PV (PVU) in the reference simulation and the simulation without latent heating (Qref 2 Qa50:0; y axis) for the 12 different cyclones (see Table 1). Temporal averages over the three last time steps of their intensification are shown (12 h before, 6 h before, and at the time of lowest SLP minimum). The colors of the cyclone markers indicate the maximum drop of the SLP minimum in hPa within 18 h in the reference simulation. The blue cross indicates the average over the whole cyclone ensemble, and the diagonal black line shows the identity line (y 5 x). The root-mean-square error (RMSE) and the Pearson correlation coefficient r are further shown. accompanied by an increase of the root-mean-square there is one outlier (cyclone Rob) for which the poten- error. This may be explained by the neglect of diabatic tial temperature tendencies and thus diabatic PV are cooling due to evaporation of rain or melting and sub- strongly overestimated. When this outlier is removed, limation of snow in Eq. (7). The linear correlation, the correlation between the sensitivity of diagnosed di- however, is still reasonably good in this modified setup abatic and modeled total PV is r 5 0.66, which is very (r 5 0.62). Also, the relative position of the cyclone similar to the correlation in Fig. 12. markers along the identity line (the sensitivity of the Overall, the PV diagnostic thus performs relatively lower-tropospheric PV anomaly to LH) does not change well also when using diagnosed potential temperature substantially. tendencies. This is largely independent of the specific If the potential temperature tendencies following parameterization of these tendencies, at least for the two Emanuel et al. (1987) and Lackmann (2002; section 2f) implementations tested here. are used instead of Eq. (7), the result of the PV di- d. Linking the lower-tropospheric PV anomaly to agnostic is qualitatively very similar as in Fig. 12 re- cyclone intensification and maximum intensity garding the relative position of the cyclone markers within the cyclone ensemble (not shown). However, Our PV diagnostic is able to systematically quantify diabatic PV tends to be overestimated slightly more, and the sensitivity of a cyclone’s lower-tropospheric PV

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FIG. 11. Mass-weighted vertical average of the lower-tropospheric PV (PVU) between 950 and 600 hPa for the simulation without latent heating (a 5 0.0; left bars), with reduced latent heating (a 5 0.5; middle bars), and the reference simulation (right bars). The dark bars show adiabatic PV (Qadiab 5 Q 2 Qdiab ), and the light bars show diabatic PV Qdiab. The fractions of Qdiab from total PV Q (%) are given at the top of the bars. The coloring of the bars indicates the intensification as in Fig. 10. Temporal averages are shown over the three last time steps of the intensification of every cyclone (12 h before, 6 h before, and at the time of lowest SLP minimum). anomaly to LH. We now analyze how this relates to the outlier Ulla is removed. The large sensitivity of the SLP sensitivity of cyclone intensification and maximum minimum of this cyclone most probably results from intensity to LH. The colors of the cyclone markers differences in the cyclone track between the simulations: in Fig. 10 (and Fig. 12) indicate the maximum drop the track has a much larger meridional component in the of the SLP minimum in hPa within 18 h in the reference reference simulation compared to the other two simu- simulation as one possible intensification measure. lations. The stronger meridional component leads to a The cyclones with a lower-tropospheric PV anomaly stronger SLP minimum drop because of the propagation being more sensitive to LH (markers at the top right) into higher latitudes with lower background SLP (e.g., experience a stronger intensification than the ones Sinclair 1995). This demonstrates one of the disadvan- with a less sensitive PV anomaly (markers at the bottom tages of using the SLP minimum along the cyclone track left). Figure 11 shows that cyclones with larger in- as an intensity measure. If, instead of the SLP minimum, tensification rates typically have higher relative contri- the vertical component of the relative vorticity at butions from LH to the total lower-tropospheric PV 850 hPa averaged over the effective cyclone area is used, anomaly. This good correlation between cyclone in- the correlation is also relatively high (r 5 0.77). If we use tensification and both the LH sensitivity and diabatic the vertically averaged diabatic PV based on modeled fraction of the lower-tropospheric PV anomaly indicates potential temperature tendencies Qdiab,ref instead of the potential importance of LH for strong cyclone in- total PV (Qref 2 Qa50:0), the correlation reduces to r 5 tensification. In addition, it demonstrates the lower- 0.45 (r 5 0.64 without Ulla) with the SLP minimum and tropospheric PV anomaly to be a good proxy for cyclone r 5 0.46 with relative vorticity. Using diagnosed instead intensification, consistent with previous climatological of modeled potential temperature tendencies as a basis studies (Campa and Wernli 2012). for diabatic PV, the correlations amount to r 5 0.32 (r 5 Figure 13 compares the LH sensitivity of the lower- 0.65 without Ulla) with the SLP minimum and r 5 0.57 tropospheric PV anomaly to the sensitivity of the SLP with relative vorticity. Because of the rather small minimum obtained from the model experiments. The sample size of 12 cyclones, these correlations have to be correlation between these sensitivities is relatively high interpreted with caution. Nevertheless, the tendency (r 5 0.81) and becomes almost perfect (r 5 0.94) if the toward smaller correlations when using diagnosed

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FIG. 12. As in Fig. 10, but when using the diagnosed potential temperature tendencies [according to Eq. (7)] instead of u_ from model output. diabatic instead of modeled total PV mainly reflects the Binder et al. (2016) is shown in Table 1. Their di- imperfectness of the PV diagnostic. Furthermore, the agnostic classifies Berit, Ulla, and Xynthia as strongly generally lower correlations with changes in the SLP diabatic and Ben, Emma, Joe, Kyrill, and Ted as minimum compared to changes in relative vorticity in- weakly to moderately diabatic cyclones, which agrees dicate that changes in the SLP minimum tend to be less relatively well with our PV diagnostic. Dirk, Klaus, and well explained by diabatic processes. Tini, however, are weakly diabatic according to Binder et al. (2016) but moderately to strongly diabatic ac- e. Comparison with other diagnostics cording to our PV diagnostic and vice versa for Rob. Binder et al. (2016) compared the intensification of Most likely, these mismatches result from the funda- over 5000 extratropical cyclones in Northern Hemi- mental differences between the two approaches sphere winter in ERA-Interim to the strength of their (number of WCB trajectories vs simplified Eulerian warm conveyor belts (WCBs; see their Fig. 1). Based on calculation of diabatic PV modification) but also be- this comparison, they defined three main cyclone clas- tween the two cyclone datasets from our COSMO ses: explosively intensifying cyclones with strong WCBs simulations and ERA-Interim (spatial resolution and (C1 cyclones), weakly intensifying cyclones with strong investigated intensification period). WCBs (C2 cyclones), and explosively intensifying Pirret et al. (2017) classified 58 historical European cyclones with weak WCBs (C3 cyclones). Since cloud cyclones in ERA-Interim according to their diabatic and formation and thus LH along a WCB is associated with baroclinic contributions using the surface pressure ten- low- to midtropospheric diabatic PV generation, WCB dency diagnostic developed by Fink et al. (2012). They strength can be regarded as an alternative and La- categorized Xynthia and Klaus as strongly, Dirk as in- grangian measure of the strength of LH in a cyclone. termediately, and Kyrill and Emma as weakly diabati- The categorization of our 12 cyclones according to cally (that means mainly baroclinically) driven, which

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FIG. 13. Difference of the mass-weighted vertical averages of total PV (PVU) in the ref- erence simulation and the simulation without latent heating (Qref 2 Qa50:0; x axis) and dif- ference of SLP minimum along the cyclone track between the simulation without latent heating and the reference simulation (SLPMINa50:0 2 SLPMINref; y axis). Temporal aver- ages are shown over the three last time steps of the intensification of every cyclone (12 h before, 6 h before, and at the time of lowest SLP minimum). The coloring of the cyclone markers is as in Fig. 10. The black line shows the linear regression line. The root-mean-square error (RMSE) and Pearson correlation coefficient r are further shown. corresponds relatively well with the results of our PV fulfilled (especially during the strongest intensification) and diagnostic (Fig. 10). our PV diagnostic thus typically overestimates diabatic PV. We further note that the evaluation of the results of our Similarly, the diagnostic can be limited during the decaying PV diagnostic, as presented in Figs. 10–13,differsfrom phase of a cyclone, when instantaneous diabatic PV gen- previous studies in the following sense: we average the ef- eration can be very weak, but remnants of diabatic PV from fects of LH temporally to obtain a representative sensitivity previous time steps may still be present. value for the final phase of the intensification, which is in In summary, our PV diagnostic yields similar results as contrast to Fink et al. (2012), who diagnosed the diabatic alternative methods to quantify the effects of LH on contribution at each time step of the cyclone lifetime, or cyclone intensification. However, there can be differ- Chagnon et al. (2013) and Martínez-Alvarado et al. (2016), ences for individual cases, which highlights the difficulty who integrated the diabatic contributions over time. Since in capturing the full case-to-case variability of LH ef- our PV diagnostic can be evaluated at any time step of the fects in cyclones with one single diagnostic method. analyzed dataset, it can in principle be used to also dem- onstrate the temporal evolution of diabatic processes over 4. Discussion and conclusions the cyclone lifetime, similar to Fink et al. (2012). However, such a temporal analysis is limited: there are time steps at We have presented a PV diagnostic to investigate the which the steady-state assumption (section 2d)isnotwell influence of latent heating (LH) on the dynamics and

Unauthenticated | Downloaded 10/05/21 02:25 PM UTC NOVEMBER 2017 B Ü ELER AND PFAHL 3587 intensity of extratropical cyclones in a quantitative, sys- effect. Despite the strong case-to-case variability, the PV tematic, and simple way. A set of hindcast simulations of diagnostic demonstrates that this first-order principle Northern Hemisphere cyclones with the COSMO model holds on average. Second, the lower-tropospheric PV has provided the basis for developing the diagnostic. Sen- anomaly of a cyclone rarely originates from anything sitivity experiments with altered LH have served as a ref- other than diabatic processes. This is valid in particular for erence to evaluate if the diagnostic is able to capture the explosively intensifying cyclones and is in agreement with sensitivity of the cyclone dynamics to LH. previous case studies (e.g., Reed et al. 1993; Stoelinga The PV diagnostic first identifies and tracks a cyclone 1996; Ahmadi-Givi et al. 2004). However, in certain based on its local SLP minimum and defines an effective cases with weak intensification (such as cyclone Rob), cyclone area, which includes all major frontal features friction also appears to significantly contribute to the and properly captures both small and large cyclone sizes. anomaly. Although our PV diagnostic is not able to The diagnostic makes use of the nonconservation of PV explicitly quantify PV modification due to friction, the in the presence of diabatic processes and explicitly cal- LH sensitivity experiments—in particular, the sim- culates the LH-induced fraction of the positive lower- ulations without LH yielding a significant lower- tropospheric PV anomaly averaged over the cyclone tropospheric PV anomaly—indirectly demonstrate that area. Thereby, it only accounts for the two most im- friction is a likely source of lower-tropospheric PV in portant terms in the PV tendency equation driving the these cases. lower-tropospheric PV budget of the cyclone to first Our main objective has been to design a diagnostic to order: the diabatic generation of PV and its vertical quantify the effects of LH on the PV structure, in- advection. Consequently, it assumes PV to be in steady tensification, and, ultimately, intensity of cyclones that state and neglects horizontal advection and friction can easily be applied to climatological datasets. The within the cyclone area. This assumed balance between simplifying assumptions made for this purpose are as- the diabatic and the vertical advection terms avoids sociated with certain drawbacks: the steady-state as- numerical errors due to their approximate cancella- sumption impedes the applicability of our PV diagnostic tion in the full equation (Tory et al. 2012). This is to periods during which a cyclone intensifies most an advantage also compared to alternative methods strongly and thus experiences a substantial temporal diagnosing the diabatic influence on cyclone intensifi- change of PV (›Q/›t 6¼ 0). The diagnostic does not ex- cation through the surface pressure tendency equation plicitly account for microphysical cooling processes such (Fink et al. 2012), in which diabatic and vertical motion as rain evaporation or ice melting and sublimation, terms are separated as well. Furthermore, the resulting which can potentially modify PV inside and below equation is purely diagnostic and can thus be applied to clouds (e.g., Joos and Wernli 2012; Dearden et al. 2016; individual cyclones in various datasets (NWP model or Crezee et al. 2017). In contrast to Lagrangian in- GCM output; reanalysis data) independent of their vestigations of different cyclonic airstreams [as done in temporal and spatial resolution and without the need to Binder et al. (2016)], the Eulerian characteristics of our perform dedicated model simulations (as opposed to PV diagnostic do not capture the full three-dimensional more complex PV tracer diagnostics; e.g., Chagnon et al. complexity of diabatic processes. Another drawback is 2013; Martínez-Alvarado et al. 2016). The largely similar the dependency of the diagnostic on some predefined performance of the diagnostic when using different pa- thresholds such as the minimum relative vorticity for rameterizations of the required potential temperature defining the effective cyclone area and the minimum tendencies allows for a robust application also to data- vertical wind velocity for selecting the grid points to be sets for which such tendencies are not available directly accounted for (see section 2). The need to adapt these as a model output. This versatile application can facili- parameters to the properties of the investigated dataset tate the comparison of LH effects on cyclone in- (such as the spatial resolution) limits the comparability tensification under different climate conditions and with of the quantitative results from the PV diagnostic between different model resolutions. different datasets. Moreover, the varying size of the ef- Our PV diagnostic not only serves as a useful meth- fective cyclone area complicates a comparison of spatially odological tool but also provides some physical insight averaged PV between individual time steps and cyclones. into the PV budget of intensifying cyclones. First, it Because of the approximations and simplifications shows that the lower-tropospheric PV anomaly is de- inherent to our PV diagnostic, it is not able to fully scribed reasonably well by two balancing processes: the represent the variability of LH effects on the lower- instantaneous diabatic generation of PV and its vertical tropospheric PV in our ensemble of 12 cyclones (cor- advection. The temporal change of the anomaly during relation of r 5 0.71 between diagnosed diabatic PV the intensification phase can be treated as a second-order anomalies and diabatic PV anomalies as obtained from the

Unauthenticated | Downloaded 10/05/21 02:25 PM UTC 3588 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 74 sensitivity experiments; see again Fig. 10). Nevertheless, the depression. Quart. J. Roy. Meteor. Soc., 99, 215–231, doi:10.1002/qj.49709942002. diagnostic accurately reproduces the ensemble mean dia- batic PV anomaly (0.55 PVU from the diagnostic and 0.53 Campa, J., and H. Wernli, 2012: A PV perspective on the vertical structure of mature midlatitude cyclones in the Northern PVU based on the sensitivity experiments). Accordingly, it Hemisphere. J. Atmos. Sci., 69, 725–740, doi:10.1175/ will be applied to study effects of LH on cyclone intensity JAS-D-11-050.1. in a climatological framework in a follow-up paper in which Chagnon, J. M., S. L. Gray, and J. Methven, 2013: Diabatic we will quantify the effect of LH on cyclones in a range of processes modifying potential vorticity in a North Atlantic very cold to very warm climates simulated with an idealized cyclone. Quart. J. Roy. 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