Diabatic processes modifying potential vorticity in a North Atlantic cyclone

Authors


Abstract

Potential vorticity (PV) succinctly describes the evolution of large-scale atmospheric flow because of its material conservation and invertibility properties. However, diabatic processes in extratropical cyclones can modify PV and influence both mesoscale weather and the evolution of the synoptic-scale wave pattern. In this investigation, modification of PV by diabatic processes is diagnosed in a Met Office Unified Model (MetUM) simulation of a North Atlantic cyclone using a set of PV tracers. The structure of diabatic PV within the extratropical cyclone is investigated and linked to the processes responsible for it. On the mesoscale, a tripole of diabatic PV is generated across the tropopause fold extending down to the cold front. The structure results from a dipole in heating across the frontal interface due to condensation in the warm conveyor belt flanking the upper side of the fold and evaporation of precipitation in the dry intrusion and below. On isentropic surfaces intersecting the tropopause, positive diabatic PV is generated on the stratospheric side, while negative diabatic PV is generated on the tropospheric side. The stratospheric diabatic PV is generated primarily by long-wave cooling which peaks at the tropopause itself due to the sharp gradient in humidity there. The tropospheric diabatic PV originates locally from the long-wave radiation and non-locally by advection out of the top of heating associated with the large-scale cloud, convection and boundary layer schemes. In most locations there is no diabatic modification of PV at the tropopause itself but diabatic PV anomalies would influence the tropopause indirectly through the winds they induce and subsequent advection. The consequences of this diabatic PV dipole for the evolution of synoptic-scale wave patterns are discussed.

1. Introduction

Extratropical cyclones produce extensive regions of cloud and precipitation on scales ranging from intense convective storms to synoptic-scale cloud bands. Cloud and precipitation structures within extratropical cyclones are organized by synoptic and mesoscale structures such as warm and cold conveyor belts, dry intrusions and tropopause anomalies (see Browning, 1986, 2003, for a summary). Exchanges of heat due to diabatic processes in clouds are in turn likely to modify these flow structures, but the precise manner by which this takes place is not well understood. In addition, the parametrization of these processes in numerical weather prediction (NWP) models is uncertain and may be a major source of model error in forecasts (e.g. Ehrendorfer, 1997).

Potential vorticity (PV) provides a compact means to describe the dynamics of atmospheric flow on large scales. PV is also a convenient diagnostic for analysing the relationship between cloud-scale processes and the larger-scale flow. If the PV anomalies associated with non-conservative processes can be identified and located, then their impact on the structure and evolution of an extratropical weather system may be inferred (e.g. Stoelinga, 1996). Brennan et al. (2008) suggested that by examining non-conservative sources of PV due to latent heating in output from NWP models, forecasters could identify features that are associated with increased uncertainty. Piecewise PV inversion (PPVI) is a method that is often used to assess the influence of particular PV anomaly (e.g. Davis and Emanuel 1991; Pomroy and Thorpe, 2000). In PPVI, the total PV at a given time is divided spatially according to user-defined criteria. These criteria may, for example, isolate the PV generated at a given elevation. Inversion techniques may then be applied to the cut-away PV anomaly in order to establish which flow anomalies can be associated with it. PPVI does not directly indicate the origin of a given anomaly, nor does it unambiguously distinguish diabatically generated PV from the conservative background PV. A method for analysing the origin of diabatic PV using a set of PV tracers is presented in this paper. The technique is similar to that employed by Davis et al. (1993) and Stoelinga (1996) in which diabatic tendencies of PV are accumulated and advected. The approach based on diabatic PV clarifies the relationship between cloud-scale physical processes and larger-scale dynamics.

The tropopause is typically defined as a PV surface (except in the Tropics). There are two possible ways by which a diabatic process could alter the structure of the tropopause: (i) directly by heating or cooling in the vicinity of the tropopause generating PV at the tropopause level (i.e. mass converges across isentropic surfaces and dilutes PV substance at the tropopause); or (ii) indirectly by diabatically generated PV near the tropopause (e.g. in the upper troposphere within the downstream ridge) altering the winds and the gradient of PV, which in turn affects the development of waves on the jet stream. If these diabatic processes are not properly represented and the simulated PV anomaly is of insufficient magnitude or is in the wrong place, then the result is an error in the trough structure and potentially in the downstream forecast.

A particular bias that has been identified in global NWP systems concerns the structure of mid-latitude upper-level troughs on their downstream side. Didone (2006) demonstrated that simulated troughs have insufficient meridional slope and too weak a gradient of PV on their downstream side when compared to analysed troughs. The downstream side of a trough is typically located in the vicinity of the warm conveyor belt (WCB). The cloud and precipitation within the WCB generate significant heating in the mid troposphere. Heating applied against an environment having non-zero absolute vorticity provides a dipole in diabatic PV tendency (e.g. positive beneath the heating and negative above). Air mass trajectories flowing through WCBs experience an increase in PV below the maximum of latent heat release and decrease above, resulting in negative diabatic PV near the tropopause in the ridge downstream of the WCB (Pomroy and Thorpe, 2000; Grams et al., 2011).

The purpose of this paper is to isolate the influence of various diabatic processes on PV structure within a simulated extratropical cyclone using a PV tracer technique. The cyclone chosen was typical for the east Atlantic in October–intense but not exceptionally so. Simulations are performed in both a coarse-resolution global model and in a higher-resolution limited-area model. Each PV tracer is a three-dimensional prognostic field that is integrated online within the modelling system. At each time step, the tracers are advected, but they also accumulate tendencies attributable to various processes represented in the model. Each PV tracer therefore represents a history of the total PV accumulated along air mass trajectories contributed by a specific modelled process. The procedure not only allows one to identify precisely which PV structures are diabatically generated, but it also identifies the specific modelled processes that are responsible. In this paper, PV tracers are employed to answer the following questions:

  • (i)What is the structure of the PV that arises due to diabatic processes in an extratropical cyclone?
  • (ii)How does diabatic PV modify the structure of the tropopause?
  • (iii)Which specific modelled processes contribute to this PV anomaly?

The structure of the paper is as follows. Section 2 describes the methods of analysis, including the PV tracer method. Section 3 provides an overview of the case. Section 4 examines the diabatic PV in the simulated cyclone and presents a partitioning of PV into contributions from various physical processes. Section 5 provides a summary of results and discussion of their implications.

2. Methodology

The analysis of PV sources that is presented in this paper is performed in the context of a case study of an extratropical cyclone that passed over the British Isles on 20 October 2008. The case is summarized at the end of this section. Simulations of the case were performed using the Met Office Unified Model (MetUM) version 6.1, the details of which are provided below, along with a description of the PV tracer diagnostic.

2.1. Met Office Unified Model

MetUM is an operational numerical weather prediction system. Its dynamical core approximates solutions to the non-hydrostatic and fully compressible equations of motion on a sphere using a semi-Lagrangian technique. The semi-Lagrangian technique affords long time steps, but at a cost of introducing implicit diffusion. Solutions are obtained on an Arakawa-C grid in the horizontal with Charney–Phillips staggering in the vertical. The limited-area version of the model uses a rotated pole in order to minimize the variation in grid spacing across the domain. Davies et al. (2005) provide a comprehensive summary of the model's design. The MetUM employs a suite of parametrization schemes to represent those processes that are either not resolved or not represented within the dynamical core. These include the mass-flux convection scheme of Gregory and Rowntree (1990), the MOSES-II boundary layer scheme (Lock et al., 2000), the Edwards and Slingo (1996) radiation scheme and the cloud microphysics scheme of Wilson and Ballard (1999).

The case study described below was simulated in the MetUM (version 6.1) using two different model configurations. The first is a global model configuration (hereafter referred to as the GLOB). The GLOB grid has 640 × 480 grid points along the longitude and latitude dimensions, respectively, which implies a grid spacing of approximately 40 km in mid latitudes. The second is a limited-area model configuration (hereafter referred to as the LAM). The LAM is run in a domain consisting of 690 × 360 grid points that stretches across the North Atlantic and includes Western Europe and some of the eastern portion of North America. The LAM grid spacing is 0.11° (approximately 12 km). Both the LAM and GLOB use the same vertical grid levels, of which there are 38 non-constantly spaced levels between the surface and 39 km elevation. Lateral boundary conditions for the LAM simulation are supplied by the GLOB simulation. Both the LAM and GLOB simulations are initialized from a Met Office global analysis valid at 0000 UTC on 17 October 2008, which is 3.5 days before the validation time of most of the forecasts presented in this paper.

It is emphasized that the GLOB and LAM runs begin from the same initial conditions, and the LAM receives its lateral boundary conditions from the GLOB. Synoptic-scale waves that develop in the LAM run are constrained by the same larger-scale conditions as in the GLOB. Differences that arise between the two runs must therefore be due to processes that are represented differently in the two model configurations. Such processes may either be resolved explicitly on the LAM grid but not on the GLOB grid, or the parametrization of such processes may behave or interact with the resolved fields in a different way. The PV tracer technique described in the following subsection allows one to identify the contribution of a specific modelled process to a specific PV structure.

2.2. PV tracers

A set of PV tracers are integrated online in both the LAM and GLOB runs. The method presented here is similar to that developed by Davis et al. (1993) and applied in Stoelinga (1996) to the analysis of cyclogenesis in the West Atlantic. The PV tracer method has also been applied in the previous studies of Gray (2006) and Chagnon and Gray (2009) to the analysis of stratosphere–troposphere exchange and the PV of convective storms, respectively. The method works as follows. Each tracer ci(x,y,z,t) is advected by the model's semi-Lagrangian advection scheme, and is subject to a Lagrangian PV tendency Si from one process i:

equation image(1)

where D/Dt is the Lagrangian time derivative operator. In practice, the increment δSi is diagnosed every time step (i.e. the change in PV that takes place during the model call to process i). The increment δSi is constructed from the increments in velocity, potential temperature and density that are incurred during the call to process i, and is added to tracer ci. The tracers ci represent contributions to Ertel's PV, which is defined as

equation image(2)

where equation image is the absolute vorticity vector, θ is the potential temperature and ρ is the mass density of air. Ertel's PV is conserved following adiabatic frictionless flow.

In order to analyse diabatic effects on PV, consider the flux form of the PV conservation equation in isentropic coordinates, ignoring friction (after Haynes and McIntyre, 1987):

equation image(3)

where equation image is the velocity vector tangent to an isentropic surface, σ is the mass density in isentropic coordinates and

equation image(4)

is the non-advective PV flux associated with heating, where equation image is the Lagrangian rate of change of potential temperature (and also the vertical velocity in isentropic coordinates). In the absence of diabatic heating, the third term on the left-hand side of Eq. (3) is zero, and Eq. (3) reduces to the statement of material conservation of PV. Even with heating, equation image has no component across isentropic surfaces. Instead, changes in PV density are due exclusively to diabatic mass fluxes across isentropic surfaces which consequently dilute (or concentrate) the PV substance (Haynes and McIntyre, 1987). Furthermore, the diabatic PV must integrate to zero globally (except where the isentropic surface intersects the lower boundary). Although the Lagrangian tendency Si is not a source of ‘PV substance’ in the sense of a chemical constituent for the reasons given above, it is sometimes referred to as a ‘source’ for brevity.

All tracers, with one exception, are set to zero everywhere at the initial time. The subsequent evolution of ci may therefore be interpreted as a history of the total PV generated by process i. The one exception is referred to as the ‘advection only’ tracer. This tracer is initialized with the initial full PV field (i.e. diagnosed from the model velocity, density and potential temperature fields), and is advected conservatively (S = 0). The difference between the full PV and the advection-only PV tracer indicates the net ‘diabatic PV’ associated with the cumulative effect of non-conservative processes.

The numerical solution of Eq. (1) suffers from implicit diffusion associated with the semi-Lagrangian advection technique. Furthermore, estimates of the PV sources δSi suffer from interpolation-related inaccuracies associated with the estimation of relative vorticity and the gradients of potential temperature. If these problems did not exist, then the full PV field could be reconstructed exactly as the sum of all sources plus the advection-only tracer, i.e.

equation image(5)

The reliability of the tracer method was tested by comparing qcadv to equation image. After 5 days of simulation time, the residual difference between these two quantities was typically about 0.25 potential vorticity units (1PVU = 10−6 m2 K kg−1 s−1) in regions where the net diabatic PV was 1 PVU or larger. The net diabatic PV, qcadv, was typically of larger magnitude than equation image, most likely because the amplitude-reducing effect of implicit diffusion was magnified by the sum across multiple tracers. Critically, the structure of these two fields was similar. The application of the tracer partitioning technique is therefore reliable for short-term simulations, but should be applied with caution to simulations of duration 1 week or more.

The analysis presented in this paper will focus on those PV tracers that attain largest magnitude over the 3.5 days of simulation. These are the convection, boundary layer, cloud microphysics and long-wave radiation schemes. Tracers of other processes (e.g. short-wave radiation, gravity wave drag, explicit diffusion, among others) typically have lower magnitude and will not be shown here. The convection scheme PV tracer represents accumulated sources of PV due to either heating from condensation or mass transport that takes place during the call to the convection scheme (i.e. convective adjustment, leading to increments of potential temperature and momentum). The boundary layer heating PV tracer represents PV generated by heating due to phase changes in the boundary layer, turbulent transport, and sensible and latent heat fluxes communicated from the surface. The large-scale cloud PV tracer represents sources of PV due to the effects of evaporation, condensation and sublimation within volumes of air that are explicitly represented on the model grid (i.e. as opposed to such cloud processes associated with calls to the subgrid-scale boundary layer and convection schemes). Note that cloud microphysical processes can occur during the calls to the large-scale cloud, boundary layer and convection schemes separately.

The time of initialization and the 3.5-day duration of integration are chosen such that the PV tracers may accumulate tendencies during the life cycle of the cyclone and the amplification of the upper-level wave. The net diabatic PV structures that have accumulated after 3.5 days are those which may influence the structure and evolution of the cyclone and upper-level wave. Information concerning the time-scale and location of PV production is not made apparent by this procedure. A few interesting examples of the origin and time-scale of PV structures on the downstream side of the upper-level trough will be demonstrated in section 3.2 using backward trajectories.

3. Case overview

3.1. Comparing the model simulation with observed structure

A mature low-pressure system was located to the northwest of the British Isles at 0000 UTC on 20 October 2008. Figure 1(a) presents the Met Office surface analysis valid at this time, indicating that the occlusion of the primary warm and cold fronts had already taken place and was situated west of Norway. A secondary cold front had developed and was located to the south and west of the British Isles. Satellite water vapour channel imagery, shown in Figure 1(b), indicates that a cloud band extended for several thousand kilometres along and ahead of the cold front, which is the location of the WCB. The narrow band of dark shades on the back edge of the cloud band in Figure 1(b) marks the location of the dry intrusion. The cyclone analysed in this study was typically, but not exceptionally, intense for autumn in the east Atlantic. Total precipitation accumulation was 5–10 mm across most of England and Wales, with some higher amounts in western coastal areas.

Figure 1.

(a) Met Office surface analysisY (Crown copyright©) and (b) water vapour channel satellite imagery valid at 0000 UTC on 20 October 2008.

The structure of the tropopause is described in this paper in terms of the PV distribution, e.g the elevation of the 2 PVU surface. PV not only characterizes the larger-scale structure of the tropopause, but the sources of PV also indicate the role played by cloud-scale processes in the diabatic modification of the tropopause structure. Before considering these sources (see next section), an overview of the total PV in the MetUM simulations is presented in Figure 2. The total PV on a model level that intersects the tropopause (Figure 2(a,c) demonstrates that an upper-level trough was present to the west of the surface low. The meridional slope increases sharply at the base of the trough (see region just west of north Wales). This region on the downstream side of the trough is also located along the western edge of WCB. A deep tropopause fold extending below 4 km elevation is evident in a vertical section across this portion of the upper-level trough (Figure 2(b,d)). To the east of the fold, in the region of the WCB, the tropopause is elevated to altitudes exceeding 10 km. The vertical section also indicates smaller-scale PV structures in the troposphere, including negative PV on the eastern periphery of the tropopause fold in the LAM (see Figure 2(d)).

Figure 2.

PV valid at 1200 UTC on 20 October 2008 in the (a,b) GLOB and (c,d) LAM simulations on (a,c) a constant model level corresponding to an elevation of 8.4 km, and (b,d) along the vertical cross-section indicated by the dark line in panel c. The dark solid contours in (b,d) depict the 95 % relative humidity contour inside of which cloud is likely to have formed. The longitude and latitude indices on the axes correspond to grid points on the LAM mesh which are spaced 12 km apart.

To determine the locations in the cross-section where cloud has formed (and therefore diabatic processes have been active), the 95 relative humidity contour is shown in Figure 2(b,d). On the eastern side of the fold, the troposphere is characterized by high values of relative humidity extending from near-surface elevation to the tropopause within the WCB in both the LAM and GLOB runs. Dry stratospheric air resides within the interior of the fold. A shallow (∼1 km deep) layer of high relative humidity values is evident beneath and to the west of the fold in the upper troposphere.

The large-scale structure of the tropopause (e.g. tropopause location, trough and ridge shape) is quite similar in the LAM and the GLOB runs. This is in part due to the LAM run receiving its lateral boundary conditions from the GLOB run. Nevertheless, the similarities imply that resolution-related differences in the representation of cloud processes between the two simulations do not cause differences in the large-scale evolution of the trough. The primary differences between the LAM and GLOB runs are confined to the mesoscale. For example, in the LAM run the tropopause fold extends to lower elevation, is narrower, and is characterized by a sharper gradient of PV than in the GLOB run (compare Figure 2(b) and Figure 2(d)). The difference in the sharpness and elevation of the tropopause fold between the LAM and GLOB runs may be evaluated against observations from a vertically oriented wind-profiling Mesosphere–Stratosphere–Troposphere (MST) Doppler radar that is situated on the west coast of Wales at Aberystwyth (Vaughan, 2002). Time–height profiles of u and N2 (the zonal wind component and the dry static stability squared, respectively) were constructed at the grid points nearest to Aberystwyth in the LAM and GLOB runs. A large gradient in static stability indicates the tropopause elevation and is appropriate for comparison to the MST radar signal power. The signal power is largest where large gradients in radar refractive index exist. Figure 3 presents the comparison between the MST radar wind profiler, the LAM and the GLOB simulations. The shear of the horizontal velocity components, as well as the gradients in static stability and signal power, are of large magnitude in the middle troposphere between 4 and 6 km elevation after 1200 UTC in both the observations and simulations. This region of large gradient is indicative of the edge of the fold (Reid and Vaughan, 2004). The fold first appears in the profile at very low elevation, and thereafter ascends slowly. The profiles from the LAM run and MST are more qualitatively similar to one another than to the GLOB run in two important respects. First, the gradient of wind across the fold is larger in the LAM and MST profiles than in the GLOB profile, despite the vertical resolution of the LAM and GLOB runs being identical. Second, the elevation to which the fold descends is closer to the ground in the LAM and MST profile than in the GLOB profile.

Figure 3.

(a,b,c) Zonal component of velocity, (d) radar return signal power and (e,f) dry static stability in a time–height section from (a,d) the NERC Mesosphere–Stratosphere–Troposphere (MST) wind-profiling Doppler radar located at Aberystwyth in Wales, (b,e) the LAM and (c,f) GLOB simulations at grid points nearest to Aberystwyth. The white crosses in (d) depict the elevation of tropopause as diagnosed from the MST signal power, and the thick white line drawn in all panels depicts the height of the 2 PVU tropopause in the LAM. The time axis is reversed, increasing from right to left.

3.2. Analysis of net diabatic heating

A reverse-domain-filling (RDF) trajectory analysis was performed to infer diabatic heating, independently from the Unified Model simulation studied in the remainder of the paper. The RDF trajectory analysis is presented here to demonstrate the origin of diabatically heated air that arrives on the downstream side of the upper-level trough. The associated diabatic PV structures are analysed in section 4. Back-trajectories, released from a fine 3-D grid spanning the UK, were computed using winds from the European Centre for Medium Range Weather Forecasts (ECMWF) operational analyses (see Methven et al., 2003, for details). Potential temperature was interpolated on to the back-trajectories at 6 h intervals. Figure 4(b) shows the net changes in potential temperature computed over 1.75 days. Two distinct air masses that have experienced heating exist on the southeastern side of the trough–one in the middle troposphere centred at approximately 5 km elevation (marked W), and one above in the upper troposphere centred at approximately 9 km elevation. Back-trajectories from the centre of these two air masses are plotted in the left-hand column of Figure 4. The air mass that experienced the greatest heating arrives in the upper troposphere on the section via rapid advection across the Atlantic in the jet stream (Figure 4(a)). The ascent and latent heat release occurred in a WCB along the east coast of North America. The air mass labelled W is within the WCB ascending over the UK (Figure 4(d)).

Figure 4.

A reverse domain-filling (RDF) trajectory analysis of diabatic heating and cooling along particle trajectories arriving in section A–B (see Figure 2(c)), calculated from ECMWF analyses. (a) Trajectory paths integrated 1.75 days backwards arriving in the upper troposphere southeast of the tropopause fold. (b) Net change in potential temperature θ (K) along 1.75-day trajectories arriving on a fine grid on the vertical section shown in Figure 2(c). Positive values indicate integrated warming along trajectories. The bold line marks the tropopause using the 2 PVU contour obtained from RDF trajectories. (c) Trajectory paths arriving at W. (d) Net change in θ over 0.75 days, highlighting the WCB, which was still ascending over the UK (marked W).

The bold line in Figure 4(b) is the 2 PVU contour obtained by advecting PV conservatively along 1-day trajectories (the same as used for the heating estimates). Comparing it with the 2 PVU contour on the same sections through the GLOB and LAM simulations (Figure 2) shows that they are very similar. It can therefore be deduced that the 3.5-day forecast runs (initialized at 0000 UTC on 17 October 2008) do not suffer large displacement errors or structural differences with the analyses.* Therefore, in the paper it will be assumed that deductions relating diabatic PV structure to processes in the MetUM simulations are relevant to processes occurring in the atmosphere in this case.

4. Diabatic PV tracers

4.1. Net diabatic PV structure

The structure and origin of PV within the extratropical cyclone on 20 October 2008 are examined in this section. Figure 5 presents a comparison between the full PV and the advection-only PV tracer. During the first time step of the simulation, the advection-only tracer was initialized with the full PV, and was at subsequent times advected conservatively by the resolved winds (as described in section 2.2). The difference between the full PV and the advection-only PV (colour shading) represents the net effect of all processes that do not conserve the grid-resolved PV. For reference, the 2 PVU tropopause in both the full PV field (black, bold contour) and the advection-only PV tracer field (grey) is plotted in all panels of Figure 5. There is very little difference between the location of the 2 PVU contour in the full and advection-only PV fields. However, there are differences on either side of the 2 PVU tropopause. Diabatic processes are therefore active near the tropopause, but do not alter PV at the tropopause location itself.

Figure 5.

The net diabatic PV (computed as the difference between the full PV and the advection-only PV tracer) on model levels corresponding to altitudes of (a) 8.4 km and (b) 4.4 km, valid at 1200 UTC 20 October 2008. (c) Net diabatic PV in the vertical section indicated in panel (a). The solid black and grey lines depict the 2 PVU contour in the full PV and advection-only PV tracer, respectively.

Positive diabatic PV is generated in the lower stratosphere, while in the upper troposphere diabatic PV is predominantly negative (see Figure 5(a)). At lower tropospheric elevations (Figure 5(b)) narrow strips of both positive and negative diabatic PV stretch along the length of the front. Farther to the east (e.g. along the western shore of Iberia through Brittany to Denmark) strips of PV generated along a previously active front wrap about the periphery of the anticyclone over central Europe.

A vertical section (AB) across the tropopause fold, cold front and WCB (Figure 5(c)) highlights the change in sign of diabatic PV across the tropopause. The diabatic PV amplitude is greatest (>1 PVU) near the tropopause fold where the relative vorticity is also largest. The folding of the tropopause is also accompanied by advection of positive diabatic PV anomalies into the fold from the stratospheric side of the tropopause. However, the processes responsible for the negative diabatic PV flanking the tropopause fold and the other tropospheric structures are investigated below.

The dipole in diabatic PV across the tropopause is a striking and robust signal. To further emphasize this structure, Figure 6 presents the mean value of diabatic PV as a function of the advection-only PV tracer. In this figure, the advection-only PV tracer may be thought of as a PV-based vertical coordinate which characterizes the unheated background environment. The mean diabatic PV is zero where the advection-only PV tracer is approximately 2–3 PVU (i.e. at the tropopause level). The mean diabatic PV is negative beneath this level and positive above with amplitudes of 1 PVU, as discussed previously. To generate this plot, the grid-point values of advection-only PV tracer on each level were binned into intervals of 0.25 PVU. The mean value of net diabatic PV on the grid points contributing to each bin was computed (only grid points located within 50 grid points (i.e. 600 km) of the tropopause were included in the calculation, and three model levels intersecting the tropopause were used).

Figure 6.

The mean net diabatic PV as a function of advection-only PV tracer values on three model levels that intersect the tropopause.

4.2. Partition of diabatic PV by process

The contribution from individual parametrized processes in the MetUM to the diabatic PV is assessed by performing the partitioning of PV among sources using the tracer method described in section 2.2. Processes modelled by four schemes dominated the total diabatic PV production: the long-wave radiation, large-scale cloud, convection and boundary layer schemes. Figure 7 presents the partitioned PV tracers on the same model level (i.e. z = 8.4 km) as for the net PV sources in Figure 5(a).

Figure 7.

The diabatic PV tracers from the LAM run corresponding to the (a) long-wave radiation, (b) boundary layer heating, (c) large-scale cloud and (d) convection schemes at a model level corresponding to an altitude of 8.4 km. The solid black and grey lines in all panels depict the 2 PVU contour in the full PV and advection-only PV tracer, respectively.

Most of the positive diabatic PV on the stratospheric (poleward) side of the tropopause is generated in the long-wave radiation scheme (Figure 7(a)), for reasons discussed below. The long-wave radiation and the convection (Figure 7(d)) schemes contribute to the negative diabatic PV along the tropospheric side of the tropopause. The negative PV extends all the way along the tropopause from the eastern side of the upper-level trough back to the deep convective event off the east coast of North America. The large-scale cloud scheme (Figure 7(c)) also contributes some negative PV to the east side of the trough near the UK, but contributes some positive PV farther to the west at the base of the trough, as does the boundary layer scheme (Figure 7(b)). At this level, the diabatic PV generated by the boundary layer scheme has been transported out of the boundary layer by resolved winds; this is especially the case along the warm conveyor belt.

Figure 8 presents the partitioned PV tracers on a lower model level (i.e. z = 4.4 km) as shown in Figure 5(b). Here, the long-wave radiation scheme (Figure 8(a)) generates much less positive PV than at near-tropopause elevations (compare Figure 7(a)). Similarly, the negative PV generated by the convection scheme (Figure 8(d)) on the equatorward side of the trough is not as extensive at this lower elevation. The boundary layer (Figure 8(b)) and large-scale cloud (Figure 8(c)) schemes generate diabatic PV contributions that have opposing signs in many places. There are several plausible explanations for this behaviour within the boundary layer itself as an interaction between schemes in the time step within the model. For example, sensible heat fluxes cool the boundary layer resulting in condensation and latent heat release, or turbulent fluxes transfer moisture from the surface which condenses near the boundary layer top. The opposing anomalies in the boundary layer are obvious on the sections shown in Figure 9. Other mechanisms may operate to further modify the PV subsequent to modification by the boundary layer scheme. For example, the large-scale cloud scheme evaporates some of the cloud that is generated by the boundary layer scheme. Where this happens, the large-scale cloud scheme generates a cooling and the boundary layer scheme generates heating, which in turn is associated with PV sources of opposite sign.

Figure 8.

As Figure 7, but at an altitude of 4.4 km.

Figure 9.

The diabatic PV tracers from the LAM run corresponding to the (a) long-wave radiation, (b) boundary layer heating, (c) large-scale cloud and (d) convection schemes in the vertical cross-section depicted in Figure 5(c). The black solid lines depict the 2 PVU contour in the full PV.

Figure 9 presents the distribution of the four diabatic PV tracers on vertical section AB (see Figure 5(a)) across the tropopause fold and WCB. Four structures contributing to the net diabatic PV (see Figure 5(c)) stand out and are related to the parametrized processes as described below. Reasons for the diabatic PV structures are offered in the final discussion.

  • (i)PV dipole across the tropopause. The positive portion of the dipole located above the tropopause is primarily generated by long-wave radiation (Figure 9(a)). While the positive PV from this scheme stretches across the length of the cross-section, the largest amplitude anomalies are located within the tropopause trough. The long-wave radiation scheme also contributes to the negative portion of the dipole located beneath the tropopause, but some portions of the negative anomaly are generated by each of the three other schemes shown in Figure 9. For example, negative diabatic PV associated with the convection scheme and large-scale cloud is present beneath the tropopause to the east of the trough in the air mass that the trajectory analysis traced back via the jet stream to ascent near North America (Figure 4(a)). Therefore this diabatic PV was not generated locally but in the outflow of an upstream warm conveyor belt.
  • (ii)PV tripole across the tropopause fold. As discussed in item (i) above, the folding of the tropopause and associated folding of the diabatic PV dipole result in positive diabatic PV within the fold associated with long-wave cooling (Figure 9(a)). In addition, the fold is flanked by strong negative diabatic PV. On the eastern side it is generated in parts by all four schemes shown; on the western side it is generated mostly within the convection scheme. The extension of the tripole downward into the lower troposphere was not represented in the GLOB simulation (not shown).
  • (iii)Negative PV west of the surface cold front. The boundary layer scheme (Figure 9(b)) is primarily responsible for this region of negative PV near the ground below the tropopause trough. Warming of the air behind the cold front along the warm ocean surface leads to heating (with maximum at the surface) and negative PV above.
  • (iv)Bands of positive and negative PV in the mid-troposphere to the east of the trough sloping upwards and eastwards. The large-scale cloud scheme (Figure 9(c)) provides most of the PV comprising these structures. These bands appear to have been generated along the decaying occluded front on the western periphery of the anticyclone over central Europe. The tilt of the bands is consistent with the upwards and eastwards tilt of the occluded frontal interface.

5. Summary and discussion

The modification of PV by diabatic processes in a North Atlantic cyclone has been investigated using a set of PV tracers running online within the Met Office numerical weather prediction model (MetUM). The PV tendencies at every time step associated with individual model parametrization schemes were accumulated following air masses by adding them to a PV tracer. Each tracer field therefore provided a history of the total ‘diabatic PV’ generated by a particular scheme since the start of the simulation. To the extent that the advection scheme is perfect, the full Ertel PV can be decomposed into a sum of these diabatic PV tracers plus an advection-only tracer that equals PV at the initial time and is then advected conservatively. The residual error in this budget was shown to be smaller than the key diabatic tracers (even at grid points) over the 5-day MetUM simulation.

The cyclone examined in this study was typical for the North Atlantic in autumn. Among its features were a deep tropopause fold above a rearward-sloping cold front, a dry intrusion descending behind the front, and a warm conveyor belt and associated cloud band ahead of the front. The PV tracers were integrated online in simulations performed with the MetUM in limited area (LAM) and global (GLOB) configurations. The GLOB and LAM simulations were similar with respect to the larger-scale structure of the cyclone. The differences between the simulations were confined to the mesoscale, particularly in the vicinity of the tropopause fold. The LAM simulated a deeper fold that extended into the lower troposphere with structure very similar to that observed by a wind-profiling Doppler radar.

RDF trajectories calculated from ECMWF analyses showed two distinct air masses to the east of the trough with a history of recent heating. One set of trajectories was ascending within the warm conveyor belt (WCB) over the UK, centred above the cold front at 4 km altitude. At the time of the section across the UK, the second set of trajectories was located in the upper troposphere above the WCB. These trajectories had experienced strong heating in an upstream WCB, ascending along the east coast of North America, and then travelled rapidly in the jet stream across the Atlantic. This is an interesting example of communication between upstream and downstream cyclones, not via disturbance of the Rossby wave train, but instead by direct transport of air that has experienced diabatic modification.

The net diabatic PV was diagnosed as the difference between the full PV and the advection-only PV tracer. The net diabatic PV was found to be near zero along the 2 PVU tropopause surface, but large on either side. This implies that diabatic processes do not directly modify PV at the tropopause in this case, but may indirectly modify the structure of the tropopause by modifying the winds and PV gradient across the tropopause, as discussed below.

5.1. Linking results from PV tracers with baroclinic wave theory

De Vries et al. (2010) developed a linear theory of baroclinic waves with moisture which can be used to interpret the PV tracer results from the NWP model. In the theory, perturbations are linearized about a parallel shear flow (usually zonal) which assumes that they have small amplitude. PV anomalies are partitioned into three components:

equation image(6)

The qd component is related to meridional air parcel displacements, η, from the basic state by equation image where equation image represents the meridional PV gradient of the basic state. The moist PV, qm, is zero at the initial time and associated with PV generated by gradients in diabatic heating just as in Eq. (5). General initial conditions also contain PV, qp, that cannot be related to meridional displacements or heating (De Vries et al., 2009). The evolution of these PV components is governed by

equation image(7)
equation image(8)
equation image(9)

Meridional advection of the basic state PV only influences the displacement PV, heating only influences the moist PV and the qp component is advected as if passive by the zonal flow. Each PV component may be inverted and the velocities ‘induced’ by each component are additive:

equation image(10)

Although the diabatic PV tendency S is not present in Eq. (7), diabatic processes influence qd through advection by the velocity induced by the diabatic PV, vm. Similarly, qd ‘induces’ vertical velocity wd, which influences the heating term S and therefore qm. In general the dry and moist components interact indirectly via their induced velocities.

Dry baroclinic instability can be represented as an interaction between a pair of counter-propagating Rossby waves (CRWs), one with PV anomalies focused where equation image is positive and one where it is negative (e.g. Heifetz et al., 2004). Usually these locations are around the tropopause and the ground. The interaction and mutual growth of both waves are mediated through the meridional velocity that each CRW induces which extends beyond the PV anomalies of that wave to the location of the other wave. De Vries et al. (2010) demonstrated that moist baroclinic instability involves interactions between these waves and two additional diabatic PV components propagating along the top and bottom of a heated layer. These components propagate through an interaction between the vertical velocity they induce, the parametrized heating field and its effects on PV.

In the nonlinear model, the advection-only PV tracer contains all the PV of the initial conditions and is equivalent to the dry components qd + qp. The sum of diabatic PV tracers is equivalent to qm, but each one is advected by the full velocity. As with Eq. (10), each tracer could be inverted independently to yield ‘induced velocities’ (3-D) which would be additive if the inversion operator is linear in q.

The key finding in this paper is that qm is almost zero at the tropopause. However, it can influence tropopause shape through the velocity it induces, um, and subsequent advection just as in the linear theory. A Rossby wave train on the tropopause is depicted schematically in Figure 10 to draw out some of the generic features of the PV distribution. It is based on the sequence of PV maps from the global simulation of this case. T2 is the trough associated with the cyclone and the tropopause fold examined in detail (between T2 and R2). The solid curve indicates the tropopause itself (the 2 PVU contour on an isentropic surface). The shading denotes diabatic PV. To leading order the main effect of diabatic PV is to enhance the anomalies associated with meridional displacements (from a zonal background flow): the diabatic PV and dry PV anomalies are in phase.

Figure 10.

A schematic representation of the net diabatic PV (shaded) relative to large-amplitude waves on the tropopause (solid line). Positive diabatic PV associated with long-wave cooling is greatest in troughs T1, T2, T3, enhancing the PV anomalies associated with equatorward advection of stratospheric air. The negative diabatic PV enhances ridges R1, R2 and R3. The net diabatic effect is of greater amplitude in the Rossby wave along the tropopause.

Similar behaviour was found by De Vries et al. (2010), who showed that the Rossby wave at the top of the heated layer tends rapidly to lock into phase with the wave on the tropopause (see their Figure 10). Since the diabatic processes enhance upper-level PV wave amplitude, the growth rate of the baroclinic wave must also be greater. The enhancement from linear theory is not great and at most 10–20% greater than the fastest dry growth rate (depending on the heating parametrization used). The diabatic PV will also tend to help the tropopause wave to propagate westwards relative to the flow. If both the diabatic PV and the upper CRW lock into phase with the lower CRW (surface potential temperature wave), the resulting baroclinic wave will propagate more slowly towards the east as a result of the diabatic PV at tropopause level.

The stretching and wrapping of diabatic PV associated with nonlinear advection in large-amplitude waves will tend to make its influence on velocity greater as a result of the scale effect of PV inversion (broader PV anomalies have more influence on the induced winds). The positive diabatic PV on the stratospheric side of the tropopause is stretched around the poleward side of ridges, making it thin and ineffective (Figure 10), while it tends to wrap cyclonically in troughs so that it has a greater length scale and more impact on velocity. Similarly, the negative diabatic PV is stretched out on the equatorward side of troughs but tends to wrap anticyclonically within upper tropospheric ridges, giving it more weight in the induced wind field. Grams et al. (2011) presented a case study of a WCB cyclone in which several of the properties depicted in Figure 10 were observed. The outflow of negative diabatic PV material from a WCB led to the enhancement of a downstream ridge. Furthermore, the amplification of the downstream ridge in turn led to an elongation of the next downstream trough, which ultimately led to the formation of a PV streamer.

5.2. Physical processes responsible for diabatic PV

Several physical processes were responsible for generating the dipole in diabatic PV across the tropopause. Most of the positive diabatic PV located above the tropopause was generated by the long-wave radiation scheme. Cau et al. (2005) calculated radiative heating rates using a narrow-band radiative transfer model acting on vertical profiles with distinct dry intrusions, obtained from high-resolution radiosonde ascents in the Tropics. If clear sky is assumed, the cooling rate peaks just below the bottom edge of the dry layer. There is less long-wave cooling above, within the dry layer. If a cloud layer is inserted immediately below the dry layer (to represent capped convection) the cooling rate becomes a sharp spike at the cloud interface. Since humidity falls sharply at the tropopause we can expect similar behaviour with strongest long-wave cooling at the tropopause itself. This vertical structure in radiative heating would result in positive diabatic PV immediately above the tropopause and negative diabatic PV below, as observed in the model. Forster and Wirth (2000) modelled filamentary stratospheric intrusions and showed that the humidity contrast across the tropopause tends to enhance the PV contrast, in opposition to the radiative damping of temperature anomalies (and secondary circulation necessary to maintain thermal wind balance) which tend to reduce the PV contrast. The humidity effect dominates for dry intrusions with tall aspect ratios (relative to f/N) so the diabatic PV will increase more rapidly in deep tropopause troughs. Ozone gradients were found to be much less influential than humidity. In addition to influencing the PV contrast across the tropopause, Cavallo and Hakim (2012) have demonstrated that long-wave cooling and diabatic enhancement of positive PV anomalies lead to more frequent, longer-lived and more intense polar vortices on the tropopause.

The negative diabatic PV in the upper troposphere originated from several processes in addition to long-wave radiation: convection, large-scale cloud and boundary layer schemes. It arises primarily in the outflow of WCBs as trajectories pass the maximum latent heat release in the mid troposphere. These processes provide justification for simple parametrizations relating heating to the large-scale flow in baroclinic waves. For example, De Vries et al. (2010) considered two forms of heating parametrization. The first is ‘cumulus convection’, where the vertical heating profile is specified but the heating amplitude is closed on vertical velocity at a particular level (nominally cloud base). The second is ‘large-scale rain’, where the heating is proportional to vertical velocity at each level but also depends on an assumed water vapour profile. In both cases, the air is assumed to be saturated on ascent and the heating is allowed to feed back on vertical velocity by including it in the omega equation. The theory was made simpler by linearizing the heating parametrization in the sense that diabatic cooling occurs on descent. The evidence from diabatic PV in the MetUM simulations indicates that this is a reasonable approximation for the large-scale wave, since the positive diabatic PV in troughs (from long-wave cooling) has a similar magnitude (0.5–1 PVU) to the negative diabatic PV in the large-scale ridges.

5.3. Tripole of diabatic PV across tropopause folds

A tripole of diabatic PV formed along the tropopause fold and extended down to the surface cold front, with positive PV within the fold and negative diabatic PV on its flanks. The tripolar PV structure was present in the LAM but not the GLOB run. The schematic diagram in Figure 11 depicts the structure of the tripole relative to the cold frontal interface (the axis of the fold). In part, the positive PV is advected from above the tropopause where long-wave cooling creates it. However, the tripole is associated with a dipole in heating across the front. Latent heat release took place ahead of the fold within the cloudy air ascending the front in the WCB. Cooling took place behind the front as precipitation fell into the dry descending air beneath the frontal interface. The cooling is stronger at low levels near the surface front. Joos and Wernli (2012) also diagnosed a dipole of heating–cooling across a cold front in a WCB cyclone. They demonstrated that the amplitude of the corresponding PV was strongly moderated by the absolute vorticity of the environment against which the heating took place. The significance of evaporative cooling to the PV tripole is indicated by the contribution of the large-scale cloud scheme to the positive PV anomaly at the centre of the tripole. Clough and Franks (1991) demonstrated that the terminal fall speed of ice particles and their sublimation rates are critical parameters in determining the depth of the cooling layer. Chagnon and Gray (2009) demonstrated that horizontally oriented PV dipoles are generated by narrow rain bands in the presence of vertical shear of the horizontal wind. In this case study, the tripole appears instead to be associated with sloping heating gradients, but the role of vertical shear cannot be ruled out. For a north–south oriented cold front, a PV tripole aligned with the front as in Figure 11 would increase the poleward flow to the east and equatorward flow to the west of the front.

Figure 11.

A schematic cross-section through a rearward-sloping cold front and tropopause fold. Ascent in cloudy air ahead of the front and evaporation in descending air behind the front give rise to a dipole of heating and cooling, and a corresponding tripole of PV across the front.

Characterization of diabatic PV structures and their origin in a simulated extratropical cyclone is the first step towards understanding the interaction between between cloud-scale thermodynamical processes and larger-scale dynamics. The next step is to identify the consequences of these PV structures for system evolution. The combined application the PV tracer technique with PV inversion may bridge this gap. A longer-term ambition is to use diabatic PV tracers to link the cumulative effects of model processes to weather forecast errors and hence to use the tracers to back out the processes contributing to model error.

Acknowledgements

The lead author is supported by the National Centre for Atmospheric Science (NCAS) under the Weather directorate. The research is also supported by the Natural Environment Research Council project ‘Diabatic influences on mesoscale structures in extratropical storms’ (DIAMET) as part of the Storm Risk Mitigation programme, NERC Grant NE/I005196/1. The authors thank Willie McGinty of NCAS Computational Modelling Services (CMS) for providing data and support necessary to run the MetUM on the UK National Supercomputing Service. The authors also thank Sid Clough, Hylke de Vries and two anonymous reviewers, whose valuable advice and comments significantly improved this paper.

  • *

    An additional pair of LAM and GLOB simulations was initialized a day earlier at 0000 UTC on 16 October 2008. This set (not shown) simulated a large-scale trough that was very different from the analysis; the trough was elongated meridionally and narrowed in the zonal direction. The forecast error in the runs initialized a day earlier can be traced to the deep convective event off the east coast of North America (e.g. see area of high PV in this region in Figure 2(a,c)).

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