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    (a) Tracks and (b) temporal evolution of SLP minimum (hPa) of extratropical cyclone Klaus (2009). The reference simulation is shown in black, the one with a halved specific latent heat constant (α = 0.5) in light blue, and the one with the specific latent heat constant set to zero (α = 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).

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    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 (K) at 850 hPa is shown in colors and PV (PVU) at 310 K in contours (2-PVU increment; purple line shows the 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).

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    Horizontal fields for (a),(d),(g),(j) the reference simulation, (b),(e),(h),(k) the simulation with a halved specific latent heat constant (α = 0.5), and (c),(f),(i),(l) the simulation with the specific latent heat constant set to zero (α = 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 (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).

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    Instantaneous PV tendencies [PVU (280 s)−1] 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 + ADV). (d) Modeled local PV tendency [difference between the PV at 1100 UTC + 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 − DIAB − 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.

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    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 + 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 (with δt = 280 s) and location minus the profile at time t and location . The gray line shows the modeled Lagrangian PV tendency, calculated as the PV profile at time and location minus the one at time t and location . (b) Hourly PV tendency terms between 1100 and 1200 UTC 23 Jan, calculated as the sum of all instantaneous terms. The gray line in (b) is calculated as in (a), but using an hourly instead of a 280-s time step .

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    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 smoothed relative vorticity (s−1), with 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.

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    Instantaneous potential temperature tendencies (K s−1) 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).

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    Total (Q; black) and diabatic [, 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 (α = 0.5), and (i)–(l) the simulation with no latent heating (α = 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.

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    Mass-weighted vertical average of total (Q; black) and diabatic [, according to Eq. (6); red] PV (PVU) between 950 and 600 hPa for the simulations without LH (α = 0.0), with reduced LH (α = 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.

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    Mass-weighted vertical average between 950 and 600 hPa of the diabatic PV (PVU) in the reference simulation (; 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 (; 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 = x). The root-mean-square error (RMSE) and the Pearson correlation coefficient r are further shown.

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    Mass-weighted vertical average of the lower-tropospheric PV (PVU) between 950 and 600 hPa for the simulation without latent heating (α = 0.0; left bars), with reduced latent heating (α = 0.5; middle bars), and the reference simulation (right bars). The dark bars show adiabatic PV , and the light bars show diabatic PV . The fractions of 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).

  • View in gallery

    As in Fig. 10, but when using the diagnosed potential temperature tendencies [according to Eq. (7)] instead of from model output.

  • View in gallery

    Difference of the mass-weighted vertical averages of total PV (PVU) in the reference simulation and the simulation without latent heating (; x axis) and difference of SLP minimum along the cyclone track between the simulation without latent heating and the reference simulation (; y axis). 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). 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.

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

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  • 1 Institute for Atmospheric and Climate Science, ETH Zürich, Zurich, Switzerland
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Abstract

Extratropical cyclones develop because of baroclinic instability, but their intensification is often substantially 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 cyclone 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.

© 2017 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Dominik Büeler, dominik.bueeler@env.ethz.ch

Abstract

Extratropical cyclones develop because of baroclinic instability, but their intensification is often substantially 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 cyclone 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.

© 2017 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Dominik Büeler, dominik.bueeler@env.ethz.ch

1. Introduction

Extratropical cyclones are a key component of the global atmospheric circulation because of their transport of heat, moisture, and angular momentum (Held 1975; Chang et al. 2002). They contribute substantially to the synoptic weather variability in the midlatitudes on a daily to weekly time scale. When reaching high intensities, they can cause extreme winds and heavy precipitation with a huge impact on society, which makes them the third-costliest natural disasters after tropical cyclones and earthquakes (e.g., Sigma Research 2016). A large fraction of wind extremes in Europe are due to extratropical cyclones, particularly during winter (e.g., Roberts et al. 2014). Over 80% of precipitation extremes in the midlatitude storm-track regions, in the Mediterranean, and in parts of eastern Asia are associated with extratropical cyclones (Pfahl and Wernli 2012).

The intensity and, ultimately, impact of extratropical cyclones depend on the strength of and interaction between their dynamical driving mechanisms: beside baroclinic instability as the adiabatic energy source required for cyclone development (Charney 1947; Eady 1949), latent heating (LH) due to the formation of clouds and precipitation can be an important diabatic contribution affecting mainly cyclone intensification (e.g., Kuo et al. 1991; Davis and Emanuel 1991; Davis 1992; Reed et al. 1993; Stoelinga 1996; Ahmadi-Givi et al. 2004; Marciano et al. 2015; Martínez-Alvarado et al. 2016). In the various studies investigating this contribution, potential vorticity (PV) has been established as a key variable (Hoskins et al. 1985). Ertel’s PV in Cartesian coordinates is defined as
e1
where is the three-dimensional absolute vorticity vector (with u denoting three-dimensional wind and 2 Ω Earth's angular velocity), θ is the potential temperature, and ρ is density. The change of PV over time can be derived from Eq. (1):
e2
Here, is the material derivative operator, denotes the time derivative of the potential temperature, and F is the three-dimensional frictional force per unit mass. The two terms on the right-hand side of Eq. (2) reveal that PV is conserved (DQ/Dt = 0) in adiabatic, frictionless flows but can be modified by diabatic and frictional processes (F ≠ 0). The diabatic term of Eq. (2) thus allows for an explicit quantification of the contribution of LH (reflected in the potential temperature tendency ) to the PV distribution in a cyclone.

The link between the PV distribution in and the dynamics of extratropical cyclones as well as the role of embedded LH are well understood in a qualitative sense. The cyclonic circulation is typically associated with three anomalous components: a positive upper-tropospheric PV anomaly, a positive potential temperature anomaly at the surface, and a positive lower-tropospheric PV anomaly (Davis and Emanuel 1991; Reed et al. 1992; Čampa and Wernli 2012). By applying PV inversion techniques, the relative contributions of these anomalies to the wind field and thus the intensity of the cyclone can be determined (Davis and Emanuel 1991). LH due to cloud and precipitation formation in the lower and middle troposphere is a crucial process influencing these anomalies. It mainly forms and maintains the lower-tropospheric PV anomaly by diabatic PV generation as expressed in Eq. (2). In addition, it can affect the upper-tropospheric PV anomaly, which itself originates from adiabatic PV advection from the stratosphere, in two opposing ways: it can intensify the anomaly via the cyclonic wind field induced by the lower-tropospheric PV anomaly (e.g., Wernli et al. 2002) and at the same time weaken the anomaly slightly downstream via diabatic PV erosion (e.g., Grams et al. 2011). These two opposing mechanisms contribute to the typical westward tilt between the upper- and lower-tropospheric PV anomalies, which fosters strong cyclone intensification (e.g., Stoelinga 1996; Ahmadi-Givi et al. 2004). Theoretical underpinning for the importance of moist processes for cyclone dynamics can also be obtained within the counterpropagating Rossby wave framework (Bretherton 1966; Heifetz et al. 2004): De Vries et al. (2010) showed that including moist processes in life cycle studies of baroclinic waves leads to additional growth regimes that can be described as the interactions of moist and dry Rossby waves. Thereby, the moist waves are defined based on diabatic PV. More recent studies applied the PV concept in a climatological framework: Čampa and Wernli (2012) analyzed the vertical PV distribution of a set of historical cyclones in the Northern Hemisphere. They found the aforementioned three anomalous components to be systematically stronger for cyclones with lower minimum sea level pressure (SLP). Using a similar historical dataset, Binder et al. (2016) showed that cyclones with stronger warm conveyor belts (cf. Browning et al. 1973; Madonna et al. 2014) near their center, and thus stronger diabatic heating associated with more pronounced positive lower-tropospheric PV anomalies, often intensify more strongly in terms of SLP deepening. These two findings complement the various case studies by indicating the PV distribution to be a good metric for both cyclone intensification and intensity. Consequently, they prove the PV concept to be beneficial also for investigating the effect of LH on cyclone dynamics in a climatological framework.

Considering the importance of LH for present-day extratropical cyclones, the question of its role in a changing climate arises. Global warming and its associated increase of the atmospheric moisture content (e.g., Schneider et al. 2010) is going to increase the potential for LH. Predicting how this will change cyclone intensity in a future climate, however, remains a challenge for general circulation models (GCMs). Many of them exhibit problems in representing present-day storm tracks (Chang et al. 2013), and there are large intermodel differences and thus uncertainties when projecting them into the future (e.g., Ulbrich et al. 2009; Woollings 2010; Chang et al. 2013; Shaw et al. 2016). This is partly due to LH processes, which occur on rather small spatial scales and are thus difficult to capture by the coarse-resolution GCMs (e.g., Willison et al. 2013; Colle et al. 2015; Trzeciak et al. 2016). In particular, Willison et al. (2015) showed a strong amplification of the storm-track response to global warming over the northeastern Atlantic in regional climate simulations with higher-than-usual horizontal resolutions, which they attribute to diabatic processes. For understanding these diabatic processes in a warmer and moister climate, the PV framework again provides a useful basis: Pfahl et al. (2015) and Marciano et al. (2015) found larger lower-tropospheric PV anomalies in intense cyclones in a warmer climate, which they suggest result from enhanced LH.

Despite considerable progress in understanding LH effects on extratropical cyclones and in demonstrating their increasing importance in a warming climate, there is still a need for a more systematic and quantitative description: systematic in the sense that versatile diagnostic methods should be developed to investigate LH effects on individual cyclones in various datasets [output of numerical weather prediction (NWP) and climate models, as well as reanalysis data] and quantitative in the sense that these methods need to be able to quantify the contribution of LH to cyclone dynamics, which in turn allows for comparison between the different datasets and model simulations of different climates and resolutions. Such methods could also help identify and eliminate deficiencies in GCMs, which would ultimately improve climate projections into the future (Shaw et al. 2016). Fink et al. (2012) developed such a systematic diagnostic based on the surface pressure tendency equation, which allowed them to quantify the relative contribution of diabatic processes to the deepening of cyclones in reanalysis data from a feature-based perspective. Pirret et al. (2017) applied the same tool to a large set of intense European windstorms and found a strong case-to-case variability of the LH contribution. Azad and Sorteberg (2014) developed a conceptually similar systematic diagnostic quantifying the diabatic contribution to the lower-tropospheric geostrophic vorticity tendency. They also found significant contributions from LH, particularly toward the end of the intensification phase of a large set of intense North Atlantic winter cyclones. In contrast to Pirret et al. (2017), however, their case-to-case variability of the LH contribution turned out to be relatively small compared to other processes. No conceptually similar diagnostic based on the PV instead of the surface pressure or geostrophic vorticity tendency equation is currently available. Chagnon et al. (2013) and Martínez-Alvarado et al. (2016) recently implemented an online diagnostic using a set of numerical PV tracers in a regional NWP model, which allowed them to quantify the diabatic contributions to the dynamics of individual cyclones in a comprehensive way. However, even though such online PV diagnostics foster a fundamental and quantitative understanding of the role of LH for cyclone dynamics, their complex and model-specific implementation limits systematic applications to large climatological datasets.

In this study, we develop a diagnostic approach that makes use of the PV framework and its advantages to explicitly investigate the link between LH and extratropical cyclone dynamics. Our method allows us to systematically quantify the contribution of LH to cyclone intensification and can be applied to output of NWP and climate models and to reanalysis data. It diagnoses the diabatic PV fraction during a cyclone’s life cycle, focusing on the positive lower-tropospheric PV anomaly as the key feature influenced by LH. The aim of this paper is to introduce the concept of the PV diagnostic and to demonstrate its performance by applying it to a set of simulated historical midlatitude cyclones in the Northern Hemisphere. In section 2, we start with the description of the model simulations and one specific cyclone case that is used later to illustrate the use of the PV diagnostic. Subsequently, some limitations of the PV tendency approach are discussed, which lead us to the main assumptions and technical details of the PV diagnostic. Section 3 contains the results of the PV diagnostic for the cyclone case study and the whole set of cyclones. In section 4, we summarize the performance of the PV diagnostic as well as its versatile application.

2. Method

a. Model simulations

We perform hindcast simulations of extratropical cyclones with the regional Consortium for Small-Scale Modeling (COSMO) NWP model (Steppeler et al. 2003) as a database to develop our PV diagnostic. The model is initialized and driven at the boundaries with ERA-Interim (Dee et al. 2011) or ECMWF operational analysis data (depending on the date of the simulated cyclone and with horizontal resolutions ranging from 0.25° to 1°) and run on a rotated geographical grid with a horizontal grid spacing of 0.125° (~14 km) and 40 levels in the vertical. In addition to a basic set of prognostic and diagnostic variables, the instantaneous temperature tendencies from LH (K s−1) are included in the output data written every 6 h. The tendencies are separated into those from resolved microphysical processes and the Tiedtke (1989) convection scheme. Temperature tendencies from the boundary layer and the radiation scheme are not considered. To identify and track cyclones in the simulations, we use the SLP-based approach developed by Wernli and Schwierz (2006).

In addition to these reference simulations (REFs), the influence of LH on cyclone dynamics is investigated with sensitivity experiments in which we artificially modify LH by scaling the physical latent heat constants of vaporization, fusion, and sublimation in the model code with a factor α [similar to Stoelinga (1996)]. This factor α is set to 0.0 or 0.5, constituting two simulations of an environment without LH or with reduced LH, respectively. This explicit simulation of the cyclones’ sensitivity to LH serves as a reference for the evaluation of our PV diagnostic. In doing so, we assume that the changes in cyclone intensity and PV in these simulations are caused solely by the altered LH and thus neglect possible nonlinear interactions with other processes. (For instance, a weaker cyclone circulation due to reduced LH may also affect the cyclone’s PV budget via changes in surface friction. However, these nonlinear interactions are assumed to be less important than the direct effect of altered LH.)

The reference and sensitivity simulations are run for 12 historical extratropical cyclones over the North Atlantic and North Pacific. One of these cases, winter storm Klaus (2009) is used hereafter to introduce the different assumptions and steps underlying the PV diagnostic. Therefore, its evolution is briefly described in the following section.

b. Cyclone Klaus (2009)

Klaus formed as a small wave perturbation close to Bermuda on 21 January 2009 and moved northeastward along a band of strong meridional equivalent potential temperature gradients (Liberato et al. 2011). This gradient was associated with a strong “tropical moisture export” event (Knippertz and Wernli 2010) and allowed significant contributions of diabatic processes to the storm’s intensification (Fink et al. 2012). Klaus intensified explosively during 23 January [around 37 hPa in 24 h, implying a Bergeron value of 1.3 according to Sanders and Gyakum (1980)] and reached its lowest SLP minimum of 965.9 hPa in the Bay of Biscay at 0000 UTC 24 January (Liberato et al. 2011). From there, it propagated over southern France and then into the Mediterranean, where it started to weaken. Southern France and northern Spain reported the largest damage due to strong wind gusts (up to 198 km h−1) and floods caused by heavy rain (Liberato et al. 2011). Klaus was the most damaging storm in these regions, following Martin in December 1999 (Ulbrich et al. 2001).

We initialize the COSMO simulations of Klaus at 0000 UTC 23 January, which is the start of the explosive intensification phase. Figure 1 shows the track and the development of the SLP minimum of Klaus from the reference and the two sensitivity simulations as well as from ERA-Interim. The intensification phase of the reference simulation spans over the first 24 h with the strongest drop during the first 12 h. In comparison, ERA-Interim yields a slower intensification but reaches a slightly deeper SLP minimum in the end (967.5 hPa instead of 970 hPa). Also, the track of the simulated cyclone is slightly shifted during the intensification phase compared to ERA-Interim but becomes more similar toward the time of maximum intensity. In conclusion, although the reference simulation shows some deviations from ERA-Interim, mainly during the first 12 h, it is able to capture the overall intensification and track reasonably well.

Fig. 1.
Fig. 1.

(a) Tracks and (b) temporal evolution of SLP minimum (hPa) of extratropical cyclone Klaus (2009). The reference simulation is shown in black, the one with a halved specific latent heat constant (α = 0.5) in light blue, and the one with the specific latent heat constant set to zero (α = 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).

Citation: Journal of the Atmospheric Sciences 74, 11; 10.1175/JAS-D-17-0041.1

Figure 2 shows the development of some important variables associated with the explosive intensification of Klaus in the reference simulation. At 0000 UTC 23 January, Klaus starts to form as a wave perturbation located ahead of an upper-level trough and at the right entrance of a zonal jet streak (Figs. 2a,d). These favorable conditions for strong intensification persist during the subsequent 12 h: the cyclone crosses the jet streak to its poleward side and ends up at its left exit (Fig. 2e) still providing strong upper-level forcing for cyclone intensification (e.g., Reed and Albright 1986). Its propagation along the northern flank of a tongue of warm and moist air in the lower troposphere (Figs. 2a,b) further contributes to the intensification through the potential for diabatic processes. The positive lower-tropospheric PV anomaly in the cyclone center reaching high values of more than 5 PV units (1 PVU = 10−6 K m2 kg−1 s−1; Fig. 2h) most likely results from these processes (e.g., Stoelinga 1996; Ahmadi-Givi et al. 2004). At 1200 UTC 23 January, Klaus attains a mature stage: the elongated cold and warm fronts start to occlude and roll up as a bent-back front around the cyclone center (Figs. 2b,h). At upper levels, stratospheric high-PV air from the descending tropopause hooks into the cyclone to its northwest and a ridge forms to the northeast (Fig. 2b). In the subsequent 12 h, Klaus propagates more zonally but is still located in the left exit of the jet streak (Fig. 2f). The upper-level ridge further amplifies (Fig. 2c), and the lower- and upper-tropospheric PV anomalies within the cyclone gradually align vertically to form a “PV tower”–like structure (Figs. 2c,i; Hoskins 1990; Rossa et al. 2000). As a result, Klaus reaches its most intense stage with an SLP minimum of 970 hPa at 0000 UTC 24 January. This high intensity persists until 0600 UTC 24 January, when the cyclone makes landfall in Europe (Fig. 1).

Fig. 2.
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 (K) at 850 hPa is shown in colors and PV (PVU) at 310 K in contours (2-PVU increment; purple line shows the 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).

Citation: Journal of the Atmospheric Sciences 74, 11; 10.1175/JAS-D-17-0041.1

Figure 3 compares the structure of Klaus in the reference simulation to the simulations with reduced and without LH. It shows the two time steps after the first 12 h of intensification (1200 UTC 23 January) and at the time of maximum intensity (0000 UTC 24 January). The positive lower-tropospheric PV anomaly is weaker in the simulation with reduced LH (Fig. 3e) and does not exist in the one without LH (Fig. 3f), in which Klaus is still a wave perturbation instead of a mature cyclone. This exemplifies the strong contribution of LH in the lower troposphere in a qualitative way. Also at upper levels, LH contributes to the conditions favoring the explosive intensification: both the ridge (direct effect) and the positive PV anomaly due to the descending tropopause (indirect effect; see section 1) are substantially weaker in amplitude in the simulations with reduced (Fig. 3b) and without LH (Fig. 3c). At 0000 UTC 24 January, this results in a weaker PV tower associated with a less-pronounced SLP minimum in the simulation with reduced LH (Figs. 3h,k) and no lower-tropospheric PV anomaly associated with a very weak cyclone in the simulation without LH (Figs. 3i,l).

Fig. 3.
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 (α = 0.5), and (c),(f),(i),(l) the simulation with the specific latent heat constant set to zero (α = 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 (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).

Citation: Journal of the Atmospheric Sciences 74, 11; 10.1175/JAS-D-17-0041.1

Figure 1 shows how this distinct behavior in the sensitivity simulations translates into a much weaker drop of the SLP minima: the SLP minimum reached by the cyclone is 7 hPa higher in the simulation with reduced LH and 20 hPa higher in the one without LH compared to the reference simulation. With respect to the track, the two sensitivity simulations behave similar to the reference. This seems surprising considering the stronger poleward deflection of extratropical cyclone tracks in a moister environment found in previous studies (e.g., Coronel et al. 2015; Tamarin and Kaspi 2016). Understanding the even slightly opposite behavior in our case would require a more specific investigation.

Our simulations reveal a strong sensitivity of the intensification of Klaus to LH. In the same line, they demonstrate a qualitatively similar sensitivity of the positive lower-tropospheric PV anomaly to LH. This consistent behavior indicates the anomaly to be a good diagnostic measure to quantify the contribution of LH to cyclone intensification. In the following section, we will study this in a more quantitative and systematic way.

c. PV tendency concept and its limitations

As a basis for studying the impact of LH on cyclone dynamics in a quantitative and Eulerian way, we use the local PV tendency equation obtained by subtracting the PV advection term from the material derivative of PV [Eq. (2)]:
e3
Mathematically, the contribution of LH to a cyclone’s PV budget can be quantified by evaluating Eq. (3) at every model grid point, averaging the result over a cyclonic area of interest, and integrating these values over time. However, numerical limitations are inherent in this approach: calculating the terms on the right-hand side of Eq. (3) in model output of typical multihourly time steps cannot fully reproduce the modeled total PV tendency [term on the left-hand side of Eq. (3); calculated by the PV difference between two consecutive time steps]. The most important reasons for this are that cyclones propagate over larger distances and that their nonlinear evolution cannot be properly captured with a temporal discretization of hourly or longer time steps.

By going to shorter model output time steps (of minutes), this discretization error can be reduced and the PV budget nearly closed. To demonstrate this, we calculate the different terms of Eq. (3) for Klaus based on model output with a time step of 280 s [neglecting friction; i.e., the third term on the right-hand side of Eq. (3)]. Figure 4 shows the corresponding result on a model level in the lower troposphere at a time step during the strong intensification of Klaus. The sum of the two individually calculated diabatic (DIAB; Fig. 4a) and advective (ADV; Fig. 4b) terms reproduces the modeled PV tendency (PVTEND; calculated as the PV field at the time t + 280 s minus the one at time t) reasonably well (PVTEND − DIAB − ADV; Fig. 4d). There are some differences close to the cyclone center in regions of strong PV gradients, where numerical errors associated with the advection term are largest. Figure 5a shows the vertical profiles of the instantaneous tendency terms in Figs. 4a–c averaged within a radius of 200 km (see Fig. 4a) around the cyclone center. The sum of diabatic and advective PV tendencies (DIAB + ADV; purple line) reproduces the modeled total PV tendency (PVTEND; black line) reasonably well, in particular above the atmospheric boundary layer where friction is less important. Hence, the instantaneous PV budget averaged over the cyclone area can be closed fairly well. However, there is a mismatch between these Eulerian PV tendencies (purple and black lines) and the Lagrangian PV tendency (gray line; calculated as the difference between the total PV profile at the cyclone location at time t + 280 s and the profile at time t) resulting from the cyclone’s propagation. When summing up all the instantaneous tendencies between 1100 and 1200 UTC, as shown in Fig. 5b, these errors accumulate and yield a significant mismatch of up to 0.25 PVU between the hourly Eulerian (purple and black lines) and Lagrangian PV tendency (gray line; calculated as the PV profile at the cyclone location at 1200 UTC minus the profile at 1100 UTC). Hence, on an hourly time scale, the PV budget cannot be closed even when integrating instantaneous PV tendencies over time. Beside this numerical limitation of the PV tendency approach, Fig. 5b also reveals a limiting physical aspect: in the lower troposphere of an intensifying cyclone, the diabatic and advective terms of Eq. (3) (red and yellow lines) are of similarly large amplitude but opposite sign. They thus nearly cancel out each other and yield a small residual PV tendency . This quasi balance between diabatic PV and (vertical) advection has previously been demonstrated both theoretically (Tory et al. 2012; Martínez-Alvarado et al. 2016) and empirically (Persson 1995; Lackmann 2002; Martínez-Alvarado et al. 2016). Since the amplitude of the residual PV tendency is in the range of the aforementioned numerical error (purple or black line minus gray line), integrating over hours suffers from cancellation errors (see also Tory et al. 2012). Hence, integrating the contribution of the diabatic term to the evolution of the total PV over the whole cyclone intensification phase is problematic. Going toward online PV tendency calculations (e.g., Chagnon et al. 2013; Saffin et al. 2016; Martínez-Alvarado et al. 2016) minimizes this problem but only for simulations restricted to short, (sub-)daily periods (Saffin et al. 2016; Martínez-Alvarado et al. 2016). In the following section, we suggest an alternative approach that does not suffer from such cancellation problems.

Fig. 4.
Fig. 4.

Instantaneous PV tendencies [PVU (280 s)−1] 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 + ADV). (d) Modeled local PV tendency [difference between the PV at 1100 UTC + 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 − DIAB − 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.

Citation: Journal of the Atmospheric Sciences 74, 11; 10.1175/JAS-D-17-0041.1

Fig. 5.
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 + 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 (with δt = 280 s) and location minus the profile at time t and location . The gray line shows the modeled Lagrangian PV tendency, calculated as the PV profile at time and location minus the one at time t and location . (b) Hourly PV tendency terms between 1100 and 1200 UTC 23 Jan, calculated as the sum of all instantaneous terms. The gray line in (b) is calculated as in (a), but using an hourly instead of a 280-s time step .

Citation: Journal of the Atmospheric Sciences 74, 11; 10.1175/JAS-D-17-0041.1

d. PV diagnostic

Motivated by the aforementioned limitations of calculating full three-dimensional PV tendencies [Eq. (3)] and integrating them over time, we develop a PV diagnostic based on a simplified approach. Three main assumptions underlie this method, which are reasonable approximations for averages over a cyclone area but not on a gridpoint scale:

  1. PV is in steady state : To first order, the small residual PV tendency due to the cancellation between the much larger diabatic PV modification and PV advection terms allows us to assume lower-tropospheric PV to be in steady state. This assumption is fulfilled particularly well during the phase of maximum intensity when the cyclone structure usually reaches its most balanced and symmetric stage.
  2. There is no horizontal PV advection across the boundaries of the cyclone area : The spatial average of the horizontal PV advection within a circular area around a perfectly symmetric and isolated cyclone would equal zero. However, real cyclones are typically asymmetric, especially during their intensification. The definition of an effective cyclone area introduced in this study (see below) allows us to better account for this asymmetry by including all major structural features into the cyclone area. In this way, PV advection across the area boundaries can be significantly reduced, which justifies the assumption of neglecting horizontal PV advection. This is only valid for the lower to middle troposphere, since in the upper troposphere, quasi-horizontal PV advection from the stratosphere becomes the main PV tendency contribution.
  3. There is no friction (F = 0): Friction can modify PV mainly near the surface but also at transition zones such as the top of the boundary layer (e.g., Adamson et al. 2006; Plant and Belcher 2007) and the tropopause. The magnitudes of the diabatic and advective contributions, however, are typically larger. Furthermore, we mainly focus on the cyclone intensification phase, which is often located over the ocean, where surface roughness is relatively low. Therefore, PV modification due to friction is neglected.

Under these assumptions, the PV tendency equation [Eq. (3)] reduces to
e4
where , , and are the x, y, and z components of absolute vorticity and w denotes vertical wind velocity. Integrating Eq. (4) vertically between a lower bound (where Q is assumed to vanish) and the reference level yields
e5
The left-hand-side term , hereafter referred to as diabatic PV (with “diabatic” referring to LH processes only, excluding surface fluxes and radiation), describes the fraction of the PV at altitude within a cyclone column resulting only from the assumed balance between diabatic PV modification and vertical advection [Eq. (4)]. Lower-tropospheric regions of strong ascent induce continuous condensation, freezing, and deposition of moisture and hence latent heat release. The resulting positive vertical gradients in the heating rate lead to continuous generation of PV that is constantly transported upward. Thereby, the positive tendency from diabatic generation and the negative tendency from vertical advection balance each other to first order (see also Persson 1995; Lackmann 2002; Tory et al. 2012; Martínez-Alvarado et al. 2016).
Technically, Eq. (5) and hence our PV diagnostic are implemented as follows:
e6
The integrand is first calculated on each grid point in the full three-dimensional domain and then averaged over all grid points within the effective cyclone area (which will be introduced later) on each model level and at all grid points with vertical wind velocity w larger than 0.02 m s−1. In this way, only regions with significant ascent are considered where the balance assumptions of the PV diagnostic are fulfilled. The value of 0.02 m s−1 has been chosen such that most of the ascent region is included. Finally, the diabatic PV at a reference level is obtained by integration starting from the lowest model level . The resulting vertical profile of diabatic PV can then be compared to the total PV from the model output (averaged over the same effective cyclone area).

e. Effective cyclone area

Cyclones typically have an asymmetric and noncircular shape, particularly during the intensification phase. Defining the cyclone area simply with a circle of a fixed radius thus often misses important regions of diabatic PV modification, for example, along strongly elongated fronts. Therefore, we introduce a more cyclone-specific area—denoted effective cyclone area—that accounts for this asymmetry to a certain degree, partly based on the principle introduced by Flaounas et al. (2014). The definition makes use of the spatially smoothed relative vorticity field on the 850-hPa level (we use a filter strength of 31 × 31 grid points; cf. Flaounas et al. 2014): all grid points above a specific smoothed relative vorticity threshold (here chosen as 7 × 10−5 s−1) and connected to the cyclone center are defined as the effective cyclone area. This method generally yields large effective areas for strong cyclones but also for large and weak cyclones (Flaounas et al. 2014). To also account for small and weak cyclones, we assign a circle with a radius of 200 km as a minimum effective cyclone area. The main advantage is that this area definition captures all the main frontal features and thus minimizes the quasi-horizontal PV advection across the boundary of the effective cyclone area (which is required for our second assumption). As a disadvantage, the varying size of this area limits the comparability of spatially averaged PV values among individual time steps or cyclones. Figure 6 shows the development of the effective cyclone area for the reference simulation of Klaus: at the beginning of the strongest intensification phase (0000 UTC 23 January; Fig. 6a), the area does not substantially exceed the minimum area of a circle with 200-km radius. After the explosive intensification (1200 UTC 23 January; Fig. 6b), the area is much larger and has a tail to the east of the cyclone center, which accounts for the zonally oriented warm front with high relative (and potential) vorticity (cf. Fig. 2h). At the end of the intensification (0000 UTC 24 January; Fig. 6c), the cyclone becomes more circular and symmetric, which is again captured by the effective cyclone area.

Fig. 6.
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 smoothed relative vorticity (s−1), with 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.

Citation: Journal of the Atmospheric Sciences 74, 11; 10.1175/JAS-D-17-0041.1

f. Diagnosed potential temperature tendency

Applying our PV diagnostic to model data requires instantaneous potential temperature tendencies from model output. For cases in which these are not available, we expand the diagnostic with the option to use a diagnosed potential temperature tendency. Based on the simplifying assumptions that the ascent of air parcels follows a moist adiabat, that no ice processes are involved, and only condensation to the liquid phase occurs, following Berrisford (1988) and Wernli (1995), the potential temperature tendency (K s−1) is calculated as
e7
where L (J kg−1) is the specific latent heat of vaporization, (J K−1 kg−1) is the specific heat capacity of dry air at constant pressure, κ (no units) is the ratio [with (J K−1 kg−1) being the gas constant of dry air], ω (Pa s−1) is the vertical pressure velocity, p (Pa) is the pressure, (kg kg−1 K−1) is the change of saturation specific humidity with temperature, and ϕ (%) is the relative humidity. The Clausius–Clapeyron relationship, the change in saturation specific humidity with respect to temperature, is approximated by
e8
where (no units) is the ratio of the gas constant of dry air and the gas constant of water vapor, (J K−1 kg−1) is the gas constant of water vapor, (Pa) is the saturation vapor pressure, and T (K) is the temperature. In Eq. (7) is set to zero at all grid points with either or . The term in brackets on the right-hand side of Eq. (7) accounts for subgrid-scale condensation that can occur already at relative humidities below 100%.

Figure 7 compares the modeled and diagnosed potential temperature tendencies in vertical cross sections through the cyclone center: during the intensification phase (1200 UTC 23 January), both the vertical filament of strong heating along the occluded bent-back front (400 km from point A in Figs. 7c and 7e) and the less intense and more shallow heating along the cold front (1000 km from point A in Figs. 7c and 7e) are reasonably well captured by the diagnostic method. Also, at the time of maximum intensity (0000 UTC 24 January), the diagnostic method reproduces several vertical filaments of heating in the cyclone region with respect to amplitude and structure (Figs. 7d,f). However, the method, by construction, cannot reproduce subgrid-scale convective heating (e.g., between 500 and 550 km from point A in Figs. 7d and 7f), for which the vertical motion is not resolved on the model grid. It does not account for diabatic cooling due to evaporation of rain or melting and sublimation of ice either (no negative values in Figs. 7e and 7f). While using this diagnosed potential temperature tendency thus neglects several microphysical processes that can lead to distinct PV anomalies, particularly on small to mesoscales (e.g., Crezee et al. 2017), we assume the PV modification associated with these missing processes to be of second-order importance for the area-averaged lower-tropospheric PV anomaly.

Fig. 7.
Fig. 7.

Instantaneous potential temperature tendencies (K s−1) 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).

Citation: Journal of the Atmospheric Sciences 74, 11; 10.1175/JAS-D-17-0041.1

The results of the PV diagnostic using the diagnosed potential temperature tendency according to Eq. (7) are discussed in section 3c. To test the sensitivity of these results to the specific formulation of the potential temperature tendency equation, we additionally apply the PV diagnostic with the tendency equation introduced by Emanuel et al. (1987). Following Lackmann [2002; his Eq. (1)], we calculate this tendency at all grid points with a vertical pressure velocity ω smaller than zero and a relative humidity higher than 70%.

3. Results

a. Cyclone Klaus (2009)

Figure 8 shows vertical profiles of the total PV from model output (Q; black line; hereafter denoted total PV) and the diagnosed diabatic PV from Eq. (6) ; red line; hereafter denoted diabatic PV) over parts of the intensification phase and the phase of maximum intensity of Klaus. For simplicity, we mostly use the term PV anomaly hereafter even though the whole analysis is based on the absolute (total or diabatic) PV and no anomalies in a climatological sense are calculated. A positive total PV anomaly of up to 1.5 PVU builds up in the lower troposphere (>600 hPa) in the reference simulation (Figs. 8a–d). The diabatic PV profile reproduces this anomaly remarkably well, particularly in the lowest 200 hPa. In the middle troposphere (600–400 hPa), relatively high total PV values occur during the intensification phase (Figs. 8a,b). During the phase of maximum intensity, these enhanced values gradually turn into a local PV minimum sharpening the tropopause, which is indicated by the kink in the total PV profile at 400 hPa (Figs. 8c,d). The diabatic PV profile has a different shape in the middle troposphere during the intensification phase (Figs. 8a–c): it approaches zero with height and thus diverges from the total PV profile. Only at the last time step (Fig. 8d), the diabatic and total PV match almost perfectly up to the tropopause. The simulation with reduced LH (Figs. 8e–h) shows a qualitatively similar behavior to the reference simulation. However, the amplitude of the lower-tropospheric total PV anomaly is smaller by approximately 50%. The most striking difference in the middle troposphere is that the tropopause descends much deeper (down to 500 hPa) within the effective cyclone area than in the reference simulation. Overall, the diabatic PV again captures the lower-tropospheric total PV anomaly well and in the middle troposphere, as in the reference simulation, approaches zero with height. In the simulation without LH (Figs. 8i–l), the lower-tropospheric total PV anomaly is small and thus aligns again with the diabatic PV, which is zero by construction.

Fig. 8.
Fig. 8.

Total (Q; black) and diabatic [, 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 (α = 0.5), and (i)–(l) the simulation with no latent heating (α = 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.

Citation: Journal of the Atmospheric Sciences 74, 11; 10.1175/JAS-D-17-0041.1

The overall good agreement of diabatic and total PV in the lower troposphere demonstrates the mainly diabatic origin of the lower-tropospheric PV anomaly in the reference simulation (Figs. 8a–d). The consistent result in the simulations with reduced LH (Figs. 8e–h) and without LH (Figs. 8i–l) indicates that the PV diagnostic is able to capture this sensitivity of Klaus to LH. In the middle troposphere, the total and diabatic PV profiles diverge mainly during the intensification phase, indicating a mixture of different air masses: the diabatic PV profile represents the contribution from diabatic PV generation (in situ or advected from below), whereas the residual between total and diabatic PV indicates contributions from advection of high-PV air (e.g., because of the descending tropopause). In the simulations with reduced LH (Figs. 8e–h) and without LH (Figs. 8i–l), the tropopause penetrates deeper, which indicates the reduction of diabatic PV erosion at upper levels that is also reflected in the weaker decrease of the diabatic PV profiles with height in Figs. 8e–h. We note, however, that diabatic PV in the middle and upper troposphere should be interpreted with caution since the effective cyclone area definition does not account for any backward tilt of the vertical cyclone axis. Therefore, the PV diagnostic does not capture the full cyclone system at these levels. Furthermore, the neglect of horizontal advection by the PV diagnostic limits its applicability for studying the structure at the tropopause level, where this term becomes dominant.

To further demonstrate the strong sensitivity of the lower-tropospheric PV anomaly to LH, Fig. 9 shows the mass-weighted vertical average of total PV Q and diabatic PV between 950 and 600 hPa for each simulation. These pressure boundaries are arbitrary but include most of the lower- to midtropospheric regions in which substantial LH and thus PV generation occurs (e.g., Fig. 8d). At the same time, they are not too close to the surface, where boundary layer processes dominate the PV budget, and also do not extend too far into the upper troposphere, where the importance of PV advection limits the use of our PV diagnostic. There are three key features in Fig. 9: the slope of the Q line, the similarity of the shapes of the Q and the lines, and the offset between the Q and the lines. The slope of the Q line is a direct measure for the sensitivity to LH: the steeper the Q line, the more sensitive the cyclone’s lower-tropospheric PV anomaly is to LH. The shape of the line is similar to the shape of the Q line at most of the time steps, indicating that the PV diagnostic is able to capture this sensitivity. Only at 0000 UTC 24 January, this is not the case as the diagnostic underestimates diabatic PV generation in the reference simulation. The offset between the two lines represents the fraction of the lower-tropospheric PV anomaly that does not originate from LH-induced PV modification within the model domain and simulation period (as this PV anomaly is also present in the simulation without LH). It can originate from friction, radiation, or upper-tropospheric PV being advected downward. However, it can also be created by diabatic processes before the initialization of the COSMO simulation or outside of the model domain.

Fig. 9.
Fig. 9.

Mass-weighted vertical average of total (Q; black) and diabatic [, according to Eq. (6); red] PV (PVU) between 950 and 600 hPa for the simulations without LH (α = 0.0), with reduced LH (α = 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.

Citation: Journal of the Atmospheric Sciences 74, 11; 10.1175/JAS-D-17-0041.1

In summary, the three simulations demonstrate the sensitivity of the lower-tropospheric PV anomaly of Klaus to LH. Applying our PV diagnostic to the simulations shows that the method is able to largely capture this sensitivity and quantify the contribution of LH to the cyclone’s PV budget.

b. Set of cyclones

To evaluate the PV diagnostic in a more robust and systematic way, we apply it to a set of 12 historical extratropical cyclones over the North Atlantic and North Pacific. Most of them were explosively deepening or very intense cyclones with severe impacts. However, we also include cases of medium strength to test the performance of the PV diagnostic over a broad spectrum. An additional selection criterion is that the cyclone should not be located over land during its intensification phase and time of maximum intensity. The reason for this is the larger importance of surface friction over land, which can significantly modify lower-tropospheric PV and thus violate the third assumption underlying the PV diagnostic. Table 1 describes the set of simulated cyclones, including their region, date of maximum intensity (lowest SLP minimum), maximum drop of the SLP minimum in 18 h, and categorization according to Binder et al. (2016) as discussed later.

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].

Table 1.

The results of the PV diagnostic for this set of cyclones are summarized in Fig. 10. It shows the sensitivity of total lower-tropospheric PV to LH as obtained from the model experiments plotted against the diagnosed diabatic PV . The sensitivity values are averaged over the last three time steps before the time of the lowest SLP minimum (see Table 1) to provide a representative measure for the final phase of the intensification. Figure 10 thus demonstrates both the sensitivity of a cyclone to LH (physical aspect) and the performance of the PV diagnostic in reproducing this sensitivity (verification aspect). The physical aspect is illustrated by the distance of a cyclone marker from the origin: the more a marker is located at the top right, the more sensitive the cyclone’s lower-tropospheric PV anomaly is to LH. This sensitivity is largest for cyclones Xynthia, Berit, Klaus, Ulla, and Tini, followed by Emma, Dirk, Joe, and Kyrill. The smallest sensitivity is found for Ted, Rob, and Ben. The sensitivity of the most sensitive cyclone (Xynthia) is almost 4 times larger than the one of the least sensitive (Ben). This demonstrates the strong case-to-case variability in the strength of the diabatic contribution during cyclone intensification. The verification aspect of Fig. 10 is illustrated by the distance of a cyclone marker to the identity line (y = x). The PV diagnostic underestimates the LH sensitivity for the cyclones above the identity line and overestimates it for the ones below. The scattering of the cyclone markers yields a correlation coefficient of r = 0.71 and a root-mean-square error of 0.15 PVU. This demonstrates that the PV diagnostic does not capture the full variability of the modeled sensitivity. On average for the whole cyclone ensemble, however, the PV diagnostic reproduces the modeled LH sensitivity of the PV anomaly remarkably well (blue cross in Fig. 10). The relative position of the cyclone markers within the cyclone ensemble as well as the mean value (blue cross) in Fig. 10 do not change substantially if the pressure boundaries (950–600 hPa) are changed to 900–600 or 950–700 hPa (not shown). Also, the correlations stay reasonably high when averaging between 900 and 600 (r = 0.72) or between 950 and 700 hPa (r = 0.59).

Fig. 10.
Fig. 10.

Mass-weighted vertical average between 950 and 600 hPa of the diabatic PV (PVU) in the reference simulation (; 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 (; 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 = x). The root-mean-square error (RMSE) and the Pearson correlation coefficient r are further shown.

Citation: Journal of the Atmospheric Sciences 74, 11; 10.1175/JAS-D-17-0041.1

An additional physical aspect not covered by Fig. 10 is the fraction of the lower-tropospheric PV anomaly produced by LH in comparison to other processes (i.e., the offset between the Q and the lines shown in Fig. 9 for Klaus). Such a decomposition into diabatic and adiabatic anomalies (where, in our case, the adiabatic anomalies also include PV generated through frictional and radiative processes) based on our PV diagnostic is shown in Fig. 11. For most of the cyclones, LH contributes more than 50% to the total lower-tropospheric PV anomaly. Only for a few cyclones, namely, Ben, Emma, Rob, and Ted, this fraction is around 50% or smaller. In the case of Rob, the lower-tropospheric PV anomaly amounts to around 1.25 (1.15) PVU in the reference simulation (simulation with reduced LH), from which only around 40% (20%) is due to LH. The track of the cyclone provides a possible explanation for these small contributions from LH: in the 18 h before its lowest SLP minimum, Rob moves over Newfoundland. During this passage, the lower-tropospheric PV anomaly in the simulation without LH increases from around 0.75 to 1.1 PVU (not shown), which most probably results from friction because of the absence of diabatic PV modification. This demonstrates how the PV diagnostic is able to indirectly identify lower-tropospheric PV anomalies resulting from friction over land. For Emma, the small LH-induced fraction of the total lower-tropospheric PV anomaly rather reveals a case in which our PV diagnostic struggles to capture the modeled LH sensitivity: in Fig. 10, this cyclone is the one with the largest distance to the left of the identity line. In addition, Fig. 11 shows that the lower-tropospheric PV due to adiabatic processes differs strongly between the three simulations (assuming that nonlinear effects of altered LH on the PV budget via altered frictional and radiative processes are small, a perfect PV diagnostic would yield a similar adiabatic PV amount for all three LH sensitivity simulations). Emma is a cyclone with a relatively smooth field of small PV values in the lower troposphere and no pronounced fronts (not shown). The effective cyclone area algorithm therefore struggles to define an isolated cyclone, which probably leads to PV advection into the effective cyclone area.

Fig. 11.
Fig. 11.

Mass-weighted vertical average of the lower-tropospheric PV (PVU) between 950 and 600 hPa for the simulation without latent heating (α = 0.0; left bars), with reduced latent heating (α = 0.5; middle bars), and the reference simulation (right bars). The dark bars show adiabatic PV , and the light bars show diabatic PV . The fractions of 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).

Citation: Journal of the Atmospheric Sciences 74, 11; 10.1175/JAS-D-17-0041.1

c. PV diagnostic with diagnosed potential temperature tendencies

To allow for a more flexible application of our PV diagnostic, it can also be used based on diagnosed potential temperature tendencies [Eq. (7)] instead of from model output. Figure 12 shows the corresponding result for the set of 12 simulated cyclones. In this setup, the PV diagnostic overestimates the sensitivity to LH compared to the standard setup with from model output, which is indicated by the slight shift of the whole cloud of cyclone markers and the mean value (blue cross) to the right in Fig. 12 compared to Fig. 10, accompanied by an increase of the root-mean-square error. This may be explained by the neglect of diabatic cooling due to evaporation of rain or melting and sublimation of snow in Eq. (7). The linear correlation, however, is still reasonably good in this modified setup (r = 0.62). Also, the relative position of the cyclone markers along the identity line (the sensitivity of the lower-tropospheric PV anomaly to LH) does not change substantially.

Fig. 12.
Fig. 12.

As in Fig. 10, but when using the diagnosed potential temperature tendencies [according to Eq. (7)] instead of from model output.

Citation: Journal of the Atmospheric Sciences 74, 11; 10.1175/JAS-D-17-0041.1

If the potential temperature tendencies following Emanuel et al. (1987) and Lackmann (2002; section 2f) are used instead of Eq. (7), the result of the PV diagnostic is qualitatively very similar as in Fig. 12 regarding the relative position of the cyclone markers within the cyclone ensemble (not shown). However, diabatic PV tends to be overestimated slightly more, and there is one outlier (cyclone Rob) for which the potential temperature tendencies and thus diabatic PV are strongly overestimated. When this outlier is removed, the correlation between the sensitivity of diagnosed diabatic and modeled total PV is r = 0.66, which is very similar to the correlation in Fig. 12.

Overall, the PV diagnostic thus performs relatively well also when using diagnosed potential temperature tendencies. This is largely independent of the specific parameterization of these tendencies, at least for the two implementations tested here.

d. Linking the lower-tropospheric PV anomaly to cyclone intensification and maximum intensity

Our PV diagnostic is able to systematically quantify the sensitivity of a cyclone’s lower-tropospheric PV anomaly to LH. We now analyze how this relates to the sensitivity of cyclone intensification and maximum intensity to LH. The colors of the cyclone markers in Fig. 10 (and Fig. 12) indicate the maximum drop of the SLP minimum in hPa within 18 h in the reference simulation as one possible intensification measure. The cyclones with a lower-tropospheric PV anomaly being more sensitive to LH (markers at the top right) experience a stronger intensification than the ones with a less sensitive PV anomaly (markers at the bottom left). Figure 11 shows that cyclones with larger intensification rates typically have higher relative contributions from LH to the total lower-tropospheric PV anomaly. This good correlation between cyclone intensification and both the LH sensitivity and diabatic fraction of the lower-tropospheric PV anomaly indicates the potential importance of LH for strong cyclone intensification. In addition, it demonstrates the lower-tropospheric PV anomaly to be a good proxy for cyclone intensification, consistent with previous climatological studies (Čampa and Wernli 2012).

Figure 13 compares the LH sensitivity of the lower-tropospheric PV anomaly to the sensitivity of the SLP minimum obtained from the model experiments. The correlation between these sensitivities is relatively high (r = 0.81) and becomes almost perfect (r = 0.94) if the outlier Ulla is removed. The large sensitivity of the SLP minimum of this cyclone most probably results from differences in the cyclone track between the simulations: the track has a much larger meridional component in the reference simulation compared to the other two simulations. The stronger meridional component leads to a stronger SLP minimum drop because of the propagation into higher latitudes with lower background SLP (e.g., Sinclair 1995). This demonstrates one of the disadvantages of using the SLP minimum along the cyclone track as an intensity measure. If, instead of the SLP minimum, the vertical component of the relative vorticity at 850 hPa averaged over the effective cyclone area is used, the correlation is also relatively high (r = 0.77). If we use the vertically averaged diabatic PV based on modeled potential temperature tendencies instead of total PV , the correlation reduces to r = 0.45 (r = 0.64 without Ulla) with the SLP minimum and r = 0.46 with relative vorticity. Using diagnosed instead of modeled potential temperature tendencies as a basis for diabatic PV, the correlations amount to r = 0.32 (r = 0.65 without Ulla) with the SLP minimum and r = 0.57 with relative vorticity. Because of the rather small sample size of 12 cyclones, these correlations have to be interpreted with caution. Nevertheless, the tendency toward smaller correlations when using diagnosed diabatic instead of modeled total PV mainly reflects the imperfectness of the PV diagnostic. Furthermore, the generally lower correlations with changes in the SLP minimum compared to changes in relative vorticity indicate that changes in the SLP minimum tend to be less well explained by diabatic processes.

Fig. 13.
Fig. 13.

Difference of the mass-weighted vertical averages of total PV (PVU) in the reference simulation and the simulation without latent heating (; x axis) and difference of SLP minimum along the cyclone track between the simulation without latent heating and the reference simulation (; y axis). 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). 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.

Citation: Journal of the Atmospheric Sciences 74, 11; 10.1175/JAS-D-17-0041.1

e. Comparison with other diagnostics

Binder et al. (2016) compared the intensification of over 5000 extratropical cyclones in Northern Hemisphere winter in ERA-Interim to the strength of their warm conveyor belts (WCBs; see their Fig. 1). Based on this comparison, they defined three main cyclone classes: explosively intensifying cyclones with strong WCBs (C1 cyclones), weakly intensifying cyclones with strong WCBs (C2 cyclones), and explosively intensifying cyclones with weak WCBs (C3 cyclones). Since cloud formation and thus LH along a WCB is associated with low- to midtropospheric diabatic PV generation, WCB strength can be regarded as an alternative and Lagrangian measure of the strength of LH in a cyclone. The categorization of our 12 cyclones according to Binder et al. (2016) is shown in Table 1. Their diagnostic classifies Berit, Ulla, and Xynthia as strongly diabatic and Ben, Emma, Joe, Kyrill, and Ted as weakly to moderately diabatic cyclones, which agrees relatively well with our PV diagnostic. Dirk, Klaus, and Tini, however, are weakly diabatic according to Binder et al. (2016) but moderately to strongly diabatic according to our PV diagnostic and vice versa for Rob. Most likely, these mismatches result from the fundamental differences between the two approaches (number of WCB trajectories vs simplified Eulerian calculation of diabatic PV modification) but also between the two cyclone datasets from our COSMO simulations and ERA-Interim (spatial resolution and investigated intensification period).

Pirret et al. (2017) classified 58 historical European cyclones in ERA-Interim according to their diabatic and baroclinic contributions using the surface pressure tendency diagnostic developed by Fink et al. (2012). They categorized Xynthia and Klaus as strongly, Dirk as intermediately, and Kyrill and Emma as weakly diabatically (that means mainly baroclinically) driven, which corresponds relatively well with the results of our PV diagnostic (Fig. 10).

We further note that the evaluation of the results of our PV diagnostic, as presented in Figs. 1013, differs from previous studies in the following sense: we average the effects of LH temporally to obtain a representative sensitivity value for the final phase of the intensification, which is in contrast to Fink et al. (2012), who diagnosed the diabatic contribution at each time step of the cyclone lifetime, or Chagnon et al. (2013) and Martínez-Alvarado et al. (2016), who integrated the diabatic contributions over time. Since our PV diagnostic can be evaluated at any time step of the analyzed dataset, it can in principle be used to also demonstrate the temporal evolution of diabatic processes over the cyclone lifetime, similar to Fink et al. (2012). However, such a temporal analysis is limited: there are time steps at which the steady-state assumption (section 2d) is not well fulfilled (especially during the strongest intensification) and our PV diagnostic thus typically overestimates diabatic PV. Similarly, the diagnostic can be limited during the decaying phase of a cyclone, when instantaneous diabatic PV generation can be very weak, but remnants of diabatic PV from previous time steps may still be present.

In summary, our PV diagnostic yields similar results as alternative methods to quantify the effects of LH on cyclone intensification. However, there can be differences for individual cases, which highlights the difficulty in capturing the full case-to-case variability of LH effects in cyclones with one single diagnostic method.

4. Discussion and conclusions

We have presented a PV diagnostic to investigate the influence of latent heating (LH) on the dynamics and intensity of extratropical cyclones in a quantitative, systematic, and simple way. A set of hindcast simulations of Northern Hemisphere cyclones with the COSMO model has provided the basis for developing the diagnostic. Sensitivity experiments with altered LH have served as a reference to evaluate if the diagnostic is able to capture the sensitivity of the cyclone dynamics to LH.

The PV diagnostic first identifies and tracks a cyclone based on its local SLP minimum and defines an effective cyclone area, which includes all major frontal features and properly captures both small and large cyclone sizes. The diagnostic makes use of the nonconservation of PV in the presence of diabatic processes and explicitly calculates the LH-induced fraction of the positive lower-tropospheric PV anomaly averaged over the cyclone area. Thereby, it only accounts for the two most important terms in the PV tendency equation driving the lower-tropospheric PV budget of the cyclone to first order: the diabatic generation of PV and its vertical advection. Consequently, it assumes PV to be in steady state and neglects horizontal advection and friction within the cyclone area. This assumed balance between the diabatic and the vertical advection terms avoids numerical errors due to their approximate cancellation in the full equation (Tory et al. 2012). This is an advantage also compared to alternative methods diagnosing the diabatic influence on cyclone intensification through the surface pressure tendency equation (Fink et al. 2012), in which diabatic and vertical motion terms are separated as well. Furthermore, the resulting equation is purely diagnostic and can thus be applied to individual cyclones in various datasets (NWP model or GCM output; reanalysis data) independent of their temporal and spatial resolution and without the need to perform dedicated model simulations (as opposed to more complex PV tracer diagnostics; e.g., Chagnon et al. 2013; Martínez-Alvarado et al. 2016). The largely similar performance of the diagnostic when using different parameterizations of the required potential temperature tendencies allows for a robust application also to datasets for which such tendencies are not available directly as a model output. This versatile application can facilitate the comparison of LH effects on cyclone intensification under different climate conditions and with different model resolutions.

Our PV diagnostic not only serves as a useful methodological tool but also provides some physical insight into the PV budget of intensifying cyclones. First, it shows that the lower-tropospheric PV anomaly is described reasonably well by two balancing processes: the instantaneous diabatic generation of PV and its vertical advection. The temporal change of the anomaly during the intensification phase can be treated as a second-order effect. Despite the strong case-to-case variability, the PV diagnostic demonstrates that this first-order principle holds on average. Second, the lower-tropospheric PV anomaly of a cyclone rarely originates from anything other than diabatic processes. This is valid in particular for explosively intensifying cyclones and is in agreement with previous case studies (e.g., Reed et al. 1993; Stoelinga 1996; Ahmadi-Givi et al. 2004). However, in certain cases with weak intensification (such as cyclone Rob), friction also appears to significantly contribute to the anomaly. Although our PV diagnostic is not able to explicitly quantify PV modification due to friction, the LH sensitivity experiments—in particular, the simulations without LH yielding a significant lower-tropospheric PV anomaly—indirectly demonstrate that friction is a likely source of lower-tropospheric PV in these cases.

Our main objective has been to design a diagnostic to quantify the effects of LH on the PV structure, intensification, and, ultimately, intensity of cyclones that can easily be applied to climatological datasets. The simplifying assumptions made for this purpose are associated with certain drawbacks: the steady-state assumption impedes the applicability of our PV diagnostic to periods during which a cyclone intensifies most strongly and thus experiences a substantial temporal change of PV . The diagnostic does not explicitly account for microphysical cooling processes such as rain evaporation or ice melting and sublimation, which can potentially modify PV inside and below clouds (e.g., Joos and Wernli 2012; Dearden et al. 2016; Crezee et al. 2017). In contrast to Lagrangian investigations of different cyclonic airstreams [as done in Binder et al. (2016)], the Eulerian characteristics of our PV diagnostic do not capture the full three-dimensional complexity of diabatic processes. Another drawback is the dependency of the diagnostic on some predefined thresholds such as the minimum relative vorticity for defining the effective cyclone area and the minimum vertical wind velocity for selecting the grid points to be accounted for (see section 2). The need to adapt these parameters to the properties of the investigated dataset (such as the spatial resolution) limits the comparability of the quantitative results from the PV diagnostic between different datasets. Moreover, the varying size of the effective cyclone area complicates a comparison of spatially averaged PV between individual time steps and cyclones.

Because of the approximations and simplifications inherent to our PV diagnostic, it is not able to fully represent the variability of LH effects on the lower-tropospheric PV in our ensemble of 12 cyclones (correlation of r = 0.71 between diagnosed diabatic PV anomalies and diabatic PV anomalies as obtained from the sensitivity experiments; see again Fig. 10). Nevertheless, the diagnostic accurately reproduces the ensemble mean diabatic PV anomaly (0.55 PVU from the diagnostic and 0.53 PVU based on the sensitivity experiments). Accordingly, it will be applied to study effects of LH on cyclone intensity in a climatological framework in a follow-up paper in which we will quantify the effect of LH on cyclones in a range of very cold to very warm climates simulated with an idealized aquaplanet GCM [as used in Pfahl et al. (2015)].

With the PV diagnostic presented in this study and the envisaged applications to climate simulations, we provide new insights into the role of moisture for cyclones in a warmer climate, which is an important open research question (Shaw et al. 2016). The process understanding gained from our approach will ultimately help improve climate model parameterizations and hence storm-track predictions into the future.

Acknowledgments

We are grateful to Heini Wernli for his support and the helpful discussions throughout the project. We thank Hanin Binder for providing the warm conveyor belt classification for our set of cyclones, Lukas Papritz and Michael Sprenger for their technical support, Emmanouil Flaounas for his fruitful inputs on the cyclone area definition, and the three anonymous reviewers for their constructive comments to improve this publication. ECMWF and MeteoSwiss are acknowledged for granting access to ECMWF analysis data. D. Büeler acknowledges funding by the Swiss National Science Foundation (Project 200021_149140).

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