Three-dimensional potential vorticity (PV) structures on the convective scale during extreme precipitation events are investigated. Using the high resolution COSMO-REA2 data set, 3D composites of the PV, with and without Coriolis parameter and related variables, are evaluated for different classes of precipitation intensity. The development of a significant horizontal dipole structure in the immediate vicinity of the precipitation maximum and the updraft can be explained by the twisting term in the vorticity equation. This is because the vorticity equation is proportional to the PV equation for strong convective processes. This theoretical is important on the convective scale without the consideration of the Coriolis effect, which is a typical characteristic on the synoptic scale. In accordance to previous studies, the horizontal PV dipole is statistically confirmed by 3D composites of the PV and corresponding variables. We show that the dipole structures are especially distinct for the relative PV without Coriolis parameter and the relative vorticity. On the convective scale, the thermodynamical sources and sinks of the potential vorticity indicate the diabatic processes that are related to conservative vortex dynamics via the proportionality of the diabatic heating and the vertical velocity. This work confirms that the PV equation is an important tool in atmospheric dynamics that unifies the thermodynamical processes as well as the dynamical processes into one scalar.

The potential vorticity is an important quantity in atmospheric dynamics that unifies the thermodynamical processes as well as the dynamical processes into one scalar. It was first derived by Ertel (

Especially on the meso- and synoptic scale the PV has been used to study atmospheric processes and phenomena. Using the semi-geostrophic theory, Thorpe and Emanuel (

On the convective scale, the PV has only rarely been studied. On this smaller scale, the sinks and the sources of the PV are more or less indispensable. The sources and sinks induced by diabatic processes and friction lead us to analyze the non-conservation of the PV. A fundamental diabatic process is the release of latent heat. It is accompanied by convective cells, and often related to the generation of strong precipitation. Chagnon and Gray (

In this context further questions arise, which will be tackled theoretically as well as statistically. How are the PV structures related to the maximal precipitation intensity? Can the horizontal PV dipole be detected and explained as phenomena on the convective scale without taking into account the Coriolis effect as important characteristic of the large scale dynamics? How does the spatial PV structure change during the development of convective cells? It is therefore necessary to analyze the relation of the time evolution of the PV with the time evolution of the precipitation in a statistical setting.

To answer these questions, this work is structured as follows. We will start with Ertel’s vorticity theorem as basis for the derivation of the potential vorticity conservation as well as the sources and sinks in Section 2. In Section 3, we will first summarize the classical explanations of the vertical and horizontal PV dipole structure on the synoptic and on the convective scale. There we will show the close relation of the spatial PV structure to the relative vorticity structure in greater detail, where the twisting term of the vorticity equation plays a crucial role. This close relationship results in a novel explanation of the horizontal PV dipole on the convective scale. The occurrence of the horizontal PV dipole for strong convective events will be confirmed statistically by the evaluation of 15 222 precipitation events using the COSMO-REA2 data set, described in Section 5. The variables are divided into different classes of precipitation intensity and evaluated for different time steps. The numerical scheme to identify such extreme events and to evaluate the three-dimensional PV structures and the corresponding variables is outlined in Section 6. In Section 7, the 3D structure of the absolute potential vorticity (Section 7.3) and the 3D structure of the relative potential vorticity without the Coriolis parameter (Section 7.1) confirm the theory, that the development and shape of the horizontal PV dipole structure depend on the Coriolis parameter, being more pronounced in the relative potential vorticity composites. Additionally, the structures of the potential temperature anomaly and the vertical velocity (Sections 7.4 and 7.5) are shown. The results will be discussed and finally summarized in Sections 8 and 9.

In 1942, H. Ertel published his pioneering work on the Lagrangian conservation of potential Vorticity (PV) (Ertel,

Ertel derived a seminal vortex theorem as a commutation-relation by introducing the arbitrary field variable

(see also Hollmann, _{a}

Further, the relative potential vorticity (PV) without Coriolis parameter will be denoted as Π with the PV-Unit 1 PVU

Small scale convective processes such as intense precipitation events are related to diabatic heating and to the release of latent heat above the boundary layer. Therefore, individual changes of the temperature (

(see Hoskins et al.,

Diabatic heating that accompanies convective events leads to the potential vorticity becoming non-conservative. Sources and sinks of the PV can be observed and PV dipole structures become visible. We will start this section with a summary of the current state of research on the vertical and horizontal PV dipoles. Then, we will analyze the horizontal potential vorticity dipole in more detail regarding the tilting term of the vorticity equation. In Section 7, we will confirm the theoretical results by evaluating cases of intense precipitation events statistically.

The vertical dipole of the non-reduced PV is typically explained by the diabatic heating term

Strong diabatic heating accompanied by background vorticity leads to a dominating term 3 in (6).^{1}

The horizontal PV dipole is characteristic for the smaller scale, where the Coriolis force can be neglected. We will show theoretically that on the smaller scale, the PV is strongly related to the relative vorticity. Davies-Jones (

Theoretically, the dipole can be explained by the formula of the individual time evolution of the potential vorticity (6), assuming a strong vertical wind shear of the horizontal wind and a horizontal change of the vertical wind. Furthermore, we assume that the gradient of the horizontal wind changes its sign in the center of the maximum updraft, as sketched in

A sketch of a convective cloud. The black arrows show the updraft.

A further theoretical explanation of the PV dipole on the storm scale was introduced by Chagnon and Gray (_{x}_{y}^{2}_{x}_{y}

(a) Shows a sketch of the vertical PV dipole caused by heating for a barotropic environment. (b) Indicates the tilt of the dipole caused by the horizontal gradient of the vertical velocity and by the vertical change of the horizontal velocity. The red region indicates the positive pole and the blue region shows the positive pole. The figures are adapted from Chagnon and Gray (

The following analysis of the horizontal PV dipole for strong convective processes is based on the twisting term of the vorticity equation. Here, we will focus on the convective scale, where the Coriolis parameter does not contribute to the equations of motion. We start with the equation for the vertical component of the vorticity for incompressible flows:
_{z}

The gradient _{z}

As initial state we consider a convective process with strong wind shear, as sketched in _{z}

As an example to illustrate the tilting mechanism, consider a strong convective updraft at _{0} that decreases along the

We recall that the idealized vorticity _{0} conversely to the sign of the twisting term:

Thus, in the case that the divergence term can be neglected, the horizontal dipole structure of the vorticity can be explained. In Sections 7.1 and 7.5, we will show that at the height, where the dipole of the relative PV is most pronounced, the vertical velocity is maximal. Using the continuity equation shows that the divergence term

Moreover, in the following we will show that in case of strong convective updrafts, the vorticity equation is directly related to the PV equation. Therefore, a significant dipole structure can also be observed in the 3D composites of the PV. To show this relation, we start with the Eulerian decomposition of the potential temperature:

We assume that the vertical component of the diabatic heating gradient is greater than the horizontal heating rates, i.e.

The anomaly composites of the potential temperature for intensity class D for the time

Inserting the Brunt Väisälä frequency

Therefore, for strong convective processes, the vertical velocity is proportional to the diabatic heating term up to a measure of hydrostatic stability:

We remark that Haynes and McIntyre (

The composites of the vertical velocity for intensity class D for the time steps _{max}

This means that in this approximation the time evolution of the potential vorticity is determined by the divergence term _{z}

Moreover, it follows that the vortex motion depends on the wind and can be expressed without a thermodynamical quantity. Therefore, the source term induces a vorticity equation for an ideal fluid. Thus, there is a possible relationship to conservation properties of numerical schemes through the energy-vorticity as introduced for the shallow water model by Sommer and Névir (

The proportionality (17) and its derivation also holds for the potential vorticity and the absolute vorticity with Coriolis parameter, as still as the vertical component of the diabatic heating gradient is greater than the horizontal heating rates. In Section 7, the statistical analysis will show that the horizontal PV dipole structure on the convective scale is especially distinct without Coriolis parameter. This can be explained by theoretical considerations. Moreover, similar to Pedlosky (

Under the condition of strong convection, we can expect similar spatial structures for the (relative) vorticity and (relative) potential vorticity. We have shown that the horizontal PV dipole can be explained by the twisting term of the vorticity equation, generated by diabatic heating and by a strong horizontal gradient of the vertical velocity accompanied by a strong vertical wind shear of the horizontal wind. The horizontal gradient of the vertical velocity and the vertical gradient of the horizontal wind are equally important. Thus, the dynamics should be considered in a three-dimensional isotropic space, which is characteristic for the convective scale. The composites in Sections 7.1 and 7.5 will show that at the height, where the dipole of the relative PV is most pronounced, the vertical velocity is maximal. Using the continuity equation, this leads to a vanishing divergence term. Thus, regarding (17) without considering the Coriolis parameter, the 3D twisting term plays a crucial role for convective processes. This theory can further be confirmed by a multiscale aymptotic analysis as proposed by Hittmeir and Klein (

To underline the importance of the dipole structure arising from the above discussed tilting process, we note that the tilting process is characterized by a dipole of a further important conserved quantity, the helicity. The dipole structure of the helicity for strong convective events is discussed e.g. by Lilly (

Consider the setup from our last example, sketched in _{0}. Then, the helicity density

Therefore, the helicity density changes its sign in the center of the updraft _{0} in

To summarize, the vertical wind shear of the horizontal wind on the synoptic scale and the embedded horizontal shear of the vertical wind on the smaller convective scale, are conditions for the generation of strong convective activity with heavy precipitation. These conditions lead to the tilt of the vorticity and a horizontal dipole structure of the relative vorticity is generated. Then, the here shown proportionality of the vorticity to the potential vorticity explains the potential vorticity dipole. In the following sections, the COSMO-REA2 data set will be applied to confirm statistically the horizontal dipole structures of the vorticity and the PV during strong precipitation events.

To analyze the PV structure with respect to strong convective events and to corroborate and extend the results of Weijenborg et al. (

In the following, the precipitation events of the months June, July and August during the time period 2007–2012 are analyzed. The 3D wind, the temperature, the pressure and the total precipitation of the COSMO-REA2 data set with a time resolution of one hour is used to calculate the density, the potential temperature, the vorticity and the potential vorticity. To neglect boundary effects, 60 grid points to the west, to the north, and to the east are excluded. Furthermore, to the south, the Alps are excluded to neglect the orographic effects. The evaluated domain is shown in

The domain of the COSMO-REA2 data set (gray box) and the data domain analyzed here (inner white box). The blue colorbar shows the precipitation (6 h-sum) from 29.06.2007 at 0:00 UTC with respect to the inner domain. The red colorbar shows the precipitation inside the gray box.

The intensity classes with respect to the precipitation intensity.

To study the PV structure during strong convective processes, first, the precipitation events are identified and classified into classes A-D. In this study, we will focus on the values in class D with a precipitation intensity greater than 29 mm/h, which are identified as extreme values. For a better evaluation of the PV, all variables defining the potential vorticity are calculated and rotated such that all precipitation cells move into the same direction. This procedure leads to an analysis in the Eulerian view. In the following, the steps are presented in detail, for a summary see

The steps to identify and calculate the 3D composites of the PV and the related variables during intense precipitation events are summarized. The categories of the precipitation intensities are listed in

All grid points with precipitation are identified and equally distributed into the four classes A (precipitation intensity 4.0–5.1 mm/h), B (intensity 10–10.6 mm/h), C (20–24 mm/h) and D (

Relative frequency (logarithmic profile) of a precipitation intensity larger than 0.1 mm/h (blue) and the percentiles of the intensity classes of

11 × 11 grid boxes around the grid boxes with precipitation intensities of classes A-D are considered. In this domain, the grid boxes of the minimum and maximum precipitation intensity are located. Finally, larger grids of size 27 × 27 (about 54 km) are centered at the precipitation maximum to achieve continuous structures. For each grid box in this domain the precipitation intensity, the date and the time are saved.

For the domains of the previous step, the same information is saved for four further time steps (one and two hours before, and one and two hours after the precipitation maximum is reached). In this way, the precipitation intensity, the date and the time is saved for five time steps.

It is checked that the precipitation maximum is the middle time step.

In each case the following variables are saved for all five time steps: the 3D wind vector components, the temperature, the pressure and the precipitation. Additionally, these data are saved for two further time steps before the five selected days, see

The 3D vorticity

Finally, all data fields are rotated so that all precipitation events are aligned propagating towards a common direction of propagation. This leads to a suitable representation to analyze the temporal change of the spatial structures of the variables. First, for each precipitation event and at each point of time the mean horizontal wind direction between 500 and 600 hPa is determined. Then, the angle of rotation is given by the angle between the mean horizontal wind direction and the

The variables given at the selected points of time. The maximum precipitation intensity is at _{max}

_{max}

The above steps lead to 336 composites (4 intensity classes × 14 variables × 6 time steps (

In this section, the composites of the potential vorticity and the related variables during extreme precipitation events (

We will start with the 3D composites of the relative PV (without Coriolis force), where a significant dipole structure can be observed. Additionally, we will show the 3D composites of the relative vorticity, the absolute PV, the potential temperature anomaly and the composites of the vertical velocity. In each _{max}

As shown in

Composites of the relative PV for intensity class D for the time one hour before the maximal precipitation intensity (_{max}

At time

At time _{max}

At the following points of time,

To summarize, a significant horizontal dipole of the relative PV close to the precipitation maximum can be observed. The dipole structure is characterized by a distinct positive pole and a negative pole, which has a smaller vertical extent. The absolute values as well as the vertical extent of both poles reach their maximum during the moment of maximal precipitation intensity.

The relative vorticity dominates the structure of the relative potential vorticity. In Section 4, we have shown theoretically that for strong convective updrafts, the temporal evolution of the relative potential vorticity is proportional to the temporal evolution of the vorticity. This theory is corroborated by the 3D composites of the relative vorticity in

The first column shows the composites of the relative vorticity _{max}

At

At _{max}

For the apparent influence of the relative vorticity on the relative PV in the precipitation intensity classes A-C and for further time steps in the intensity class D, see the

The composites of the absolute potential vorticity including the Coriolis parameter of the intensity class D at the times _{max}

The composites of the absolute PV for intensity class D for the time steps _{max}

The relative PV composites of 3785 extreme precipitation events for the time step (

The absolute PV composites at time

At the time of maximal precipitation _{max}

While a significant dipole structure of the relative PV without Coriolis parameter could be observed, the positive pole of the absolute PV with Coriolis force is much more distinct than the negative pole. We can conclude that the Coriolis effect supresses the development of a distinct dipole structure in the absolute PV growth rate.

The composites of the potential temperature anomalies of intensity class D are shown in _{max}

The red isosurfaces show the positive anomalies that indicate the heating of the air masses. The blue isosurfaces represent the negative anomalies that reflect the cooling of the air. The additional black contours in

At time

At time _{max}

Consider the process of diabatic heating in an unstable atmosphere. Warm air at the ground ascends towards the upper cold air. The process of convection has a stabilizing effect and leads to the generation of clouds. This is reflected in the composites of the potential temperature anomaly: In the geometric height of the latent heat release, a positive potential temperature anomaly can be recognized in the upper troposphere. In the lower troposphere on the other side, a negative temperature anomaly is generated.

Deep moist convection is always related to strong vertical updrafts. The composites of the significant vertical velocity are shown in

At the time _{max}

The vertical height of the updraft coincides with the height of the dipole of the relative potential vorticity. The strong relation of the precipitation intensity and the updraft can be recognized particularly well from the top views in the first row.

At the time of the maximal precipitation _{max}

Three-dimensional composites are used to analyze the structure of the PV with and without Coriolis parameter and the relative vorticity statistically, during extreme precipitation events for different time steps. This topic has been studied earlier in the framework of two case studies by Weijenborg et al. (

The first question we asked in the introduction was how the PV structure is related to the maximal precipitation intensity. To answer this question we focused on the class of extreme precipitation events with precipitation intensities larger than 29 mm/h. Thereby, all precipitation cells are rotated such that they have a common direction of propagation. The three-dimensional composites of the relative PV without Coriolis parameter show a distinct horizontal PV dipole at the time of maximal precipitation intensity, see

The dipole structures of the relative PV without Coriolis parameter are more visible than the pattern of the PV with Coriolis parameter. Considering the potential vorticity with Coriolis parameter, the vertical extent of the positive pole is much larger than the vertical extent of the negative pole, which is only rarely visible, see

The question arise, if the dipole structure can be explained by the dynamics on the convective scale alone, or if an interaction of the convective scale with the synoptic scale is required. To analyze the relevance of the larger scale Coriolis effect the maxima and minima of the potential vorticity with and without Coriolis parameter are determined and summarized in

The maximal and minimal values of the potential vorticity with and without Coriolis parameter from the ground up to a height of 8.5 km.

_{max}

The similarity of distinct dipoles of the vorticity and the PV at the time of maximum precipitation corroborates the strong relation of the potential vorticity with the relative vorticity, that we derived analytically in Section 4. The horizontal dipole structures of both variables can be explained as follows. Strong convective activity is characterized by a strong vertical wind shear of the horizontal wind and by an increasing wind with height, as sketched in

We have shown that the interaction of the large scale vertical gradient of the horizontal wind with the small scale horizontal gradient of the vertical wind determine the twisting term _{z}

The dominating twisting term leads to the vortex tilt and finally to the horizontal vortex dipole.

Furthermore, assuming strong convection, we show that the vertical velocity is proportional to the diabatic heating term (see also Haynes and McIntyre,

In contrast to the here observed horizontal dipole on the small scale, a vertical dipole can be observed on the large scale, see e.g. Raymond and Jiang (

In the introduction we asked the question, if the horizontal PV dipole can be detected and explained as phenomena on the convective scale without taking the influence of the Coriolis effect into account. The comparison of the PV structures with and without Coriolis parameter show that the Coriolis effect prevents the development of a strong horizontal dipole. Thus, the horizontal dipole seems to be an important characteristic on the convective scale directly explained by the dynamics of the convective scale. Therefore, we propose the use of the relative PV for further investigations restricted on the convective scale.

For a deeper insight into the PV structure, the composites of the variables defining the PV mathematically are also shown. The potential temperature anomaly and the vertical velocity, see

The third question we asked in the introduction was about the change of the PV structure during the development of convective cells. We found that the the structure of all analyzed variables is clearest during the time of maximal precipitation, see

The 3D composites of the potential vorticity (PV) with and without Coriolis parameter show a horizontal PV dipole structure on the convective scale during extreme precipitation events. The patterns of the PV with and without Coriolis parameter, as well as the patterns of the vorticity and of the vertical velocity are all spatially and temporally correlated with intense precipitation. The statistical composites confirm our theoretical explanation of the horiztontal PV dipole in Section 2, where we show the proportionality of the vorticity equation and the PV equation of motion in the case of strong convection. While in previous studies PV

On the convective scale, where the Coriolis force can be neglected, a horizontal PV dipole is a characteristic of strong convective motion. We have explained theoretically and investigated statistically that for strong convective events the 3D composites of the relative PV without Coriolis force and the relative vorticity, are characterized by coexisting dipole structures. The horizontal dipoles, with poles located to the right and left of the moving storm, were in particular reflected in the relative potential vorticity field and the relative vorticity field.

From the perspective of scale interaction, the generation of the horizontal PV dipole can be seen as the interplay of the vertical shear of the horizontal wind on the synoptic scale with the horizontal shear of the vertical wind on the convective scale. Theoretically, this can be explained by the twisting term in the vorticity equation and the proportionality of the time evolution of the PV and the vorticity.

Summarizing, we explained theoretically and corroborated statistically the horizontal PV dipole for extreme precipitation events on the convective scale. A three-dimensional isotropy is generated, if the horizontal and vertical wind shear have the same order of magnitude. This isotropy, characteristic for heavy precipitation events, identifies the relative potential vorticity as a useful quantity to diagnose and study atmospheric phenomena and processes on the convective scale.

The authors thank the unknown reviewer for helpful comments and suggestions. We also thank Tracy Kiszler for careful reading the manuscript and Martin Rudolph for reproducing

Supplemental data for this article can be accessed here

Recall the notation for the wind vector components

Chagnon and Gray (_{0}, _{0}), the heating length-scales in the cross-frontal and vertical directions _{x}_{z}