The plume rise (Δh) calculation in GEM-MACH is driven by nine variables: stack height (hs), exit temperature at the stack outlet (Ts), stack emission volumetric flow rate (V), air temperature at stack height (Ta), wind speed at stack height (U), surface temperature (Tsurface), boundary-layer height (H), friction velocity (u*), and Obukhov length (L). These input parameters are used to generate the rise in the plume above the stack height (Δh), as well as the upper and lower boundaries of the plume after it has risen to equilibrium. In models such as GEM-MACH, buoyant transport of emissions through that region is assumed to be instantaneous. The emitted mass is distributed through the given region under the assumption that the buoyant plume has reached equilibrium. Here, all of these variables are obtained from observations (either directly or via the use of the appropriate formulae with observed quantities).

The algorithm makes use of derived quantities (the buoyancy flux, Fb; the stability parameter, s; and the convective velocity, H*) with different formulae for plume rise corresponding to neutral, stable, and unstable atmospheric conditions. The buoyancy flux is calculated from Briggs (1984, equivalent to their Eq. 8.35) as Fb=gπVTs-TaTs,Ts>Ta,0,Ts≤Ta, where g=9.81 m s-2 is the gravitational acceleration. The stability parameter is calculated from Briggs (1984, combining their Eqs. 8.8 and 8.14) as S=gTadTadz+gcp, where z is the height coordinate and cp=1005 J K-1 kg-1. The temperature gradient is calculated from the temperature difference over the stack height (dT/dz=Ta-Tsurface/hs), with a minimum value set at -5 K km-1 (i.e. S≥ 0.047/Ta). We note that calculating the temperature difference between the stack height and the surface may underestimate the temperature gradient above the stack height, where the plume rises. The extent of this effect is tested later using temperature gradients throughout the boundary layer (Sect. 2.2). Finally, the convective velocity (H*=-2.5u*3/L) is defined in Briggs (1985).

The atmosphere is considered neutral if L>2hs or L<-0.25hs (i.e. -4<hsL<0.5). These values are suggested in Briggs (1984) and the sensitivity of the results to these values is tested in Sect. 4. The plume rise in neutral conditions is taken as the minimum of two formulations of Briggs outlined in Sharf et al. (1993) and Byun and Ching (1999) as Δh=min⁡39Fb3/5U,1.2Fbu*2U3/5hs+1.3Fbu*2U2/5. The atmosphere is considered stable at the plume height if either 0<L<2hs (stable conditions) or hs≥H (direct emission above the boundary layer). From Briggs (1984, their Eq. 8.71), the plume rise is calculated as Δh=2.6FbSU13 The atmosphere is considered unstable if -0.25hs<L<0. In the unstable case, the plume rise is taken as the minimum value of two formulations of Briggs outlined in Byun and Ching (1999), Δh=min⁡3FbU35H*-25,30FbU35. This effectively places a lower limit on the magnitude of the convective velocity in determining plume rise as H*>0.00316 m2 s-3 (from H*-2/5<10) gives the example of clear summer conditions as H*=0.007 m2 s-3.

The only difference between Eqs. (3), (4), and (5) and the plume rise parameterizations used in SMOKE (described in Bieser et al., 2011, and Houyoux, 1998) is the option of the minimum values in unstable and neutral conditions. In the SMOKE model, only the second parameterizations within the minima of Eqs. (3) and (5) are used. Both of the approaches used in GEM-MACH and SMOKE are investigated in the following analysis.

Plume rise is also modified for situations where the stack height is less than the boundary-layer height (hs<H), but the plume rises high enough to penetrate the boundary-layer height to some degree (hs+Δh>H). This is referred to as “bumping” (Briggs, 1984). The vertical plume depth is assumed to be equal to the plume rise so that the plume is bound by the height range hs+0.5Δh<z<hs+1.5Δh. If any portion of the plume is above H, the plume rise is calculated (from Briggs, 1984) as Δh=0.62+0.38pH-hs, where p is the fraction of the plume above H (i.e. p=0 if hs+1.5Δh=H and p=1 if hs+0.5Δh=H).

While the above formulae are used in GEM-MACH and other models, we also examine a layer-based approach suggested by Briggs, described below, and the companion paper, Akingunola et al. (2018), examines the impact of this approach within the GEM-MACH model itself.

In addition to the parameterization discussed above, Briggs (1984) suggests a layer-based approach to calculate plume rise for complex stability profiles. In this approach, the plume buoyancy (F) is modified as it passes through each discrete layer as F=Fj-0.053SjUjzc3-zj3, where Fj is the buoyancy flux at the bottom of layer j, Sj is the layer stability calculated using Eq. (2), Uj is the wind speed, and zj is the layer height above the stack height. The wind speed in the original Briggs formulation is taken as constant with height, while here we use an average wind speed for each layer. The lower boundary of the first layer is the stack height (zj=0=0). The value of F is determined sequentially for each layer at the top of each layer (with zc=zj+1) until it becomes negative. For the layer where F becomes negative, Eq. (7) is solved to give the plume height zc for which F=0. Plume rise is calculated as Δh=zc. Layer thickness will depend on the vertical model or measurement resolution. Layer thickness for this analysis is discussed in detail in Sect. 2.6.

Equation (7) is intended for use with stable (S>0) or neutral (S=0) layers. For unstable layers we follow the approach outlined in our companion paper (Akingunola et al., 2018), in which the plume rises through the unstable layer without gaining or losing buoyancy or momentum (equivalent to S=0 in Eq. 7). As is discussed below (Sect. 4.1), the majority of layer temperature profiles (>90 %) measured by the aircraft were stable or neutral, so this assumption should not have a significant effect on the resulting plume rise. However, we also found that the stability was spatially heterogeneous in the study region, with significant differences in stability noted from the different sources of meteorological information.

While the Briggs parameterization discussed in Sect. 2.1 is driven by surface (or near-surface) observations, the layered method (Eq. 7) is driven by observations up to the height of the plume. The observed plume centreline heights (Sect. 2.7) vary between approximately 100 and 1000 m above the surface. Hence, the layered method can be used with the elevated observations from an aircraft measurement platform and an acoustic profiler (Sect. 2.4).

As part of the Continuous Emission Monitoring System (see CEMS, 1998), measurements of 19 stacks in the region of study with valid hourly measurements of SO2 and average effluent velocity and temperature were obtained from Alberta Environment and Parks. Stacks that emit primarily NOx and no reported SO2 are not used in this analysis. A key requirement for our evaluation is that the stacks selected for comparison have sufficient levels of SO2 emissions to be easily discernable from the aircraft observations. For stacks without reported CEMS SO2 emission rates, the average rates determined from the Cumulative Environmental Management Association inventory for the year 2010 (see CEMA, 2012) were used to eliminate stacks from the analysis that would not emit enough SO2 to be observed by the aircraft-based instrumentation. It is assumed that the emission profiles of SO2 in 2013 are not significantly different from 2010. Stacks from the Imperial Oil Kearl facility are not in the CEMA inventory because those stacks started operation later than 2010. A comparison of observed plume locations, as outlined below in Sect. 2.7, demonstrates that the Kearl and Firebag stacks produce no discernable SO2 plumes. Based on this comparison, there are 7 stacks that emit significant (more than 0.050 kg s-1) SO2. The 12 non-SO2 emitting stacks all report less than 0.005 kg s-1.

A flaring stack at the CNRL facility was added to the list (CNRL2) because daily reports indicated a large amount of SO2 emissions were released from the flaring stacks for a 1-week period during the field study. However, by their nature (a high temperature flame at the top of the stack is used to lift pollutants upwards), CEMS monitoring of flare stacks is not possible with current technology, and hence emissions rates and stack parameters for this source are engineering estimates. The stack parameters for this flaring stack were parameterized using effluent velocity and temperature based on annual NPRI inventory values (NPRI ID 23275, NPRI website; see ECCC and AEP, 2016).

Although NRPI data are available for the CNRL flaring stack, the other CNRL stack used here (a “sulfur recovery unit”) has both CEMS and NPRI data available. This allows for a test of the variability in Ts and ws through comparison of NPRI data (where annual average values are reported) and CEMS data (hourly) for this period and stack. For stack CNRL1 the annual average NPRI values were Ts=811 K and ws=17 m s-1, and the CEMS data averages for the study period are Ts=851 K and ws=4.1 m s-1 (a 5 % temperature difference and more than a factor of 4 difference in flow rate). Hence, there may be significant differences between data reported through both methods; by extension the CNRL2 values (for the 1-week period it is active) should be considered only approximations.

All eight stacks are listed in Table 1 and the locations of these eight stacks are shown in Fig. 1. For comparison, average effluent velocities (calculated from flow rate and stack diameter as ws=4V/πds2) and temperatures were calculated for each stack over the 84 h of research aircraft flight time (with the exception of CNRL2, which is based on annual NPRI inventory values). These averages are shown for comparison only; plume rise in the analysis that follows is calculated using hourly CEMS data concurrent with the time of plume observations. Plume observations and the aircraft flight campaign are discussed in more detail in the following sections.

The relatively high flow rates and diameters of some stacks may lead to plume rise due to momentum alone, especially under stable conditions. Briggs also developed similar equations for rise due to momentum (see Briggs, 1984). These equations are typically used when Fb=0, and the plume is assumed to be either a vertical jet (momentum driven) or a bent-over plume (buoyancy driven). The potential effect of momentum on the plume rise is discussed in Sect. 4.4.

Wind speed (U), wind direction (θ), and temperature (Ta) data at the stack height and at the surface were estimated based on measurements made at either of the following: one of two meteorological towers in the study region (WBEA: AMS03 and AMS05), or a radio-acoustic sounding system (wind RASS, Scintec). Figure 1 demonstrates the sites of the WBEA meteorological towers, and the radio-acoustic sounding system (RASS).

The AMS03 tower measures wind speed, wind direction, and temperature at heights of 20, 45, 100, and 167 m (all heights above ground level). The AMS05 tower measures wind speed and direction at heights of 20, 45, 75, and 90 m and temperature at heights of 2, 20, 45, and 75 m. Tower measurements are reported as 1 h averages. The RASS measures wind speed and temperature (among other variables) between a minimum height of 40 m and a maximum height that varies depending on wind conditions (Cuxart et al., 2012). During the aircraft flight period, the maximum RASS measurement height varied from 130 to 800 m, with an average of 336 m. The RASS measurements are 15 min averages.

As part of JOSM, aircraft-based measurements were made in the Athabasca oil sands region between 13 August and 7 September 2013. The project included 22 flights, which were flown in some combination of either box formations (circumnavigating a facility at variable heights in order to determine facility pollutant emissions), screen formations (flown perpendicular to the plume centreline axis to characterize the transformation of the plumes), spiral ascent and descent (to characterize boundary-layer structure), or horizontal area coverage (to verify satellite observations over a larger spatial extent). Figure 1 shows all these flight formations. Within the 22 flights, there were 16 box-flight formations and 21 screens used for this analysis. Aircraft flight times varied from approximately 2.5 h to over 5 h, typically in the mid-afternoon, for a total of 84 h. Wind speeds and temperatures were measured from the aircraft with a Rosemount 858 probe, sampled at 32 Hz and averaged to 1 Hz. For details of the aircraft measurements, see Li et al. (2017), Liggio et al. (2016), and Gordon et al. (2015). The aircraft flew at a minimum height of 150 m a.g.l. The maximum height of box formations varied from 500 to 1300 m a.g.l., while the maximum height of screen formations ranged from 350 to 2000 m a.g.l.

Tower, RASS, and aircraft measurements were compared over the 84 flight hours. The RASS was not operational until 17 August (thus missing 3 flights); hence, RASS data are compared for a reduced period. For comparison to the tower measurements, the 15 min RASS and 1 s aircraft measurements were averaged to concurrent 1 h values. For comparison to the RASS, the 1 s aircraft measurements were averaged to 15 min values. The resulting correlation coefficients are listed in Table 2. The aircraft wind and temperature measurements are also compared with the highest tower (AMS03) and the RASS. For comparison to aircraft measurements, the RASS measurements at a height of 90 m were compared to all concurrent aircraft measurements below 200 m. In the case of AMS03, the measurement at a height of 167 m was compared to all concurrent aircraft measurements below 200 m. The wind speed comparisons are best between the two towers (R2=0.80). Wind direction compares well for the towers and the RASS (R2>0.84). Temperature compares well for all measurement platforms (R2>0.78). Generally, comparisons with the aircraft give the lowest correlation values. We note that the correlations of Table 2 do not show potential local offsets in magnitude, and that the aircraft observations are averages over a larger region that may not be spatially co-located with the towers. We also note from Fig. 1b that towers AMS03 and AMS05 are less than 10 km apart, while the RASS is approximately 20 km from the two towers. The correlations between AMS03 and AMS05 are higher than between either of these towers and the more distant RASS, and that correlations with the aircraft have the lowest values, implying that some of the lower correlations may reflect local heterogeneity in meteorological conditions.

We note that the Athabasca oil sands region is centred on the Athabasca River valley, with over 500 m of vertical relief within 60 km of the facilities; the flow within the valley may be complex, with frequent observations of shear between plumes from stacks at different elevations under stable conditions. The low correlations between the stations and between the stations and the aircraft reflect this variation in local meteorological conditions. We examine this possibility through the use of a high-resolution GEM-MACH simulation in our companion paper (Akingunola et al., 2018).

Stability, boundary-layer height, and friction velocity were all determined from the observations using wind speed and temperature profiles from multiple height measurements. Anemometers and temperature sensors on the towers, mounted at variable heights between 2 and 167 m, are within the surface layer and are best suited for these estimations. The RASS, which has a minimum measurement height of 40 m, may not capture the surface layer effectively. As the aircraft did not fly below a height of 150 m, aircraft-based measurements cannot be used to estimate the stability, boundary-layer height, and friction velocity. For our analysis, we calculate L, H and u* to drive the Briggs parameterization (Eqs. 1–6) using observations from the two towers (AMS03 and AMS05) and the RASS.

The atmospheric stability is determined using the Bulk Richardson Number, which is defined (Garratt, 1994) as Ri=gzhθΔθΔU2. Here Δθ and ΔU are the potential temperature and wind speed differences over the height range (zh). The height range is determined as the difference in height between the highest measurement location and the lower measurement location. For example, zh=147 m for AMS03, zh=55 m for AMS05, and zh is variable for the RASS. The Richardson number is then related to the stability parameter (Kaimal and Finnigan, 1994) as zL=RiforRi<0,Ri1-Ri/Ricfor0<Ri<Ric,+∞forRi>Ric. Here Ric=0.25 is the critical Richardson number, chosen as the mid-range of reported values (0.2, 0.25, or 0.5; Mahrt, 1981). For Ri>Ric there is no solution, so this is modelled as an extremely stable boundary layer with L slightly larger than zero (to satisfy the stability condition L>0). The Obukhov length is calculated from the stability parameter as L=zmax⁡/(z/L), where zmax⁡ is the highest measurement height of 167, 90, or up to 800 m for AMS03, AMS05, and the RASS respectively.

Boundary-layer height can be parameterized for stable and unstable conditions following Mahrt (1981) as H=RiTsurgU(H)2θ(H)-θsurface, where Ri is the bulk Richardson number and U(H) and θ(H) are the respective wind speed and potential temperature at the boundary-layer height and θsurface is the potential temperature at the surface. Since measurements at the boundary-layer height may not be available, we approximate the ratio of wind speed to temperature gradient in Eq. (10) as Uzmax⁡2/θzmax⁡-θsurface.

The boundary-layer height derived from Eq. (10) can be compared to the boundary-layer height estimated from in situ aircraft measurements of the CH4 mixing ratio during vertical profile flight formations. These CH4 profiles demonstrate a well-defined background level above a given height, with elevated CH4 mixing ratios below this height. The boundary-layer heights determined by the aircraft measurements range from 340 to 1790 m with an average of 1180 m. The values of H derived from Eq. (10) using the AMS03 tower data for the same time periods as the flights range from 460 to 3050 m, with an average of 1160 m.

The friction velocity (u*) was determined from the wind speed profile (Garratt, 1994) as uz=u*kln⁡zzo-Φ, where zo is the roughness length, k=0.4, and the stability parameter is Φ=2ln⁡121+xo+ln⁡121+xo2forzL<0,-2atanxo+π2,-5zL,forzL>0, with xo=1-16z/L1/4. A least-squares method is used for each hourly profile to determine an appropriate zo for the measurement location, which is taken as the median value of all the hourly fits. This median zo value calculated using this method varies considerably by location (1.5 m for AMS03, 0.75 m for AMS05, 10.1 m for RASS). The median zo values were then used to calculate u* using the hourly wind speed measured at the highest location. The calculation of u* with the RASS may be inaccurate due to the lack of measurements between the surface and a height of 40 m. However, the large difference in values of zo may be also due to the different environment surrounding the measurement locations, since the towers are surrounded by forest and the RASS is located in the town of Fort McKay.

It is noted that parameterizing stability without a measurement of heat flux and estimating boundary-layer height based on near-surface measurements may lead to significant uncertainties in these values. This will also affect the estimation of u*, and may be evident in the median zo values for the RASS, which are very large even for a town with two or three-story buildings. Tests to determine the sensitivity of the calculated plume rise to these variables (L,H,u*) are discussed in Sect. 4.2.

To drive the layered method discussed in Sect. 2.2, profiles of temperature and wind speed were derived for each box and each screen using RASS and aircraft observations. RASS layers were 10 m thick to match the instruments resolution. The lowest RASS measurement is at a height of 40 m, well below the lowest stack height (76 m). Because the maximum observation height of the RASS varies (with an average of 336 m), it was necessary to extrapolate temperature and wind speed above the maximum measurement height in some cases. This was done by assuming a constant wind speed and a constant temperature gradient, based on measurements in the highest 100 m of observations.

For aircraft observations, the box and screen flights were designed to approximate 100 m vertical spacing between each box circuit or screen pass. Based on this resolution we use a layer thickness of 100 m for the layered method driven by aircraft observations. Testing demonstrates that the algorithm is not sensitive to the layer thickness. Flight measurements of wind (U) and temperature (T) for each box and screen are averaged in vertical layers within the 100 m spacing. Since there are no measurements below a height of 150 m a.g.l., the temperature at the lowest layer (0<z<100 m) is extrapolated by assuming a constant lapse rate and stability below 200 m (i.e. Sj=1=Sj=0). There are no cases of calculated plume height based on the layered method exceeding the maximum aircraft measurement height and hence no need for upward extrapolation of the measurements.

Our temperature profiles for the layered method thus have the following as key assumptions: (1) that the profiles at the RASS location and derived from the aircraft are representative of conditions at the stacks, and (2) that the extrapolations and vertical resolution used here provide a reasonable representation of the atmospheric temperature profile.

The aircraft measured numerous pollutants, of which SO2 is used here to define the stack plume locations since approximately 95 % of the SO2 emissions in the region originate in stacks (Zhang et al., 2018). The SO2 analyzer (Thermo Fisher Scientific, model 43i) on the aircraft measured at a rate of 1 Hz. The flight paths were designed to create a 100 m spacing between measurement points (in both horizontal, s, and vertical, z) in order to optimize interpolation of the measurements. The measurements were interpolated in s and z using simple kriging as outlined in the Topdown Emission Rate Retrieval Algorithm (TERRA; Gordon et al., 2015). This creates two-dimensional images of SO2 mixing ratio. For box flights, which circumnavigate the facilities, the s coordinate is the distance along the box in the counter-clockwise direction from the southeast corner. For screens, s is the lateral distance along the screen, generally perpendicular to the wind direction. Below the lowest flight path (at 150 m a.g.l.), no interpolation is performed and the screen is left blank between this level and the ground. Figures 2 and 3 show example box and screen flight paths in both horizontal (Fig. 2) and vertical (Fig. 3) profiles.

A semi-empirical approach was used to match each stack to the observed plume locations. The wind direction measured from the aircraft was averaged for the duration of each box or screen. Tower or RASS-based wind direction measurements were not used, as an initial comparison of wind directions and observed plume locations demonstrated that the aircraft measurements are a better representation of the wind direction associated with plume transport than surface measurements. This agreement is most likely due to the consistent proximity of the aircraft to the stack sources; the towers and RASS locations can often be much further away (Fig. 1).

The average wind directions were then used to predict the direction of plume transport downwind of each stack. The intercept of each plume's predicted path with the box or screen (sint) was calculated based on this forward trajectory from the stack source to the box or screen intercept. Example box and screen flight paths, forward trajectories, and observed plume locations are shown in Fig. 2 for the flights on 29 August (Fig. 2a) and 15 August (Fig. 2b). This simple forward trajectory methodology ignores the local effects of topography, vertical winds, and the variability of the wind during the box or screen segment of each flight (typically less than 2 h of flight time). Some screens were flown up to 150 km from the eight stacks (see Fig. 1). Since other stratification, topography, and diffusion effects may influence a plume height at such a large distance from the plume origin, we restrict our analysis to box walls and screens within 50 km of the plume stack sources.

Plume rise (Δh) was calculated for each stack based on the Briggs parameterization, the observed meteorological conditions at the tower or RASS locations (or RASS and aircraft data, for the layered approach), and the CEMS stack parameters, all averaged for the duration of the box or screen flight periods. This calculation also defined the estimated plume centreline location at each box or screen as (sint, zh), where zh=zsurface+hs+Δh and zsurface is the surface elevation (a.m.s.l.) at the intercept.

The flight path observations are converted to two-dimensional (s, z) images by kriging interpolation following the method outlined in Gordon et al. (2015). Example interpolated images from both a box and a screen flight are shown in Fig. 3. A disadvantage of kriging interpolation of the aircraft data is that the maxima of the plumes will always be fixed at a flight measurement location. To improve the resolution of observed plume height from the interpolated images, the aircraft measurements within a 100 m wide window (i.e. s±50 m) are fitted to a Gaussian vertical profile. Example profiles are shown in Fig. 3b and d, which correspond to the windows shown as thick black lines through the maximum SO2 locations (the plume centres) in Fig. 3a and c. The maxima of the Gaussian fits for each identified plume are then used to identify the prominent plume locations as (sp, zp). The identified plume locations are visually compared to the predicted Briggs plume locations based on the forward trajectories for each box or screen (sint, zh).

Non-stationarity of the wind speed, wind direction, and plume buoyancy during the measurements is a potential source of uncertainty as each flight circuit (or pass) around the facility can take between 10 and 15 min. This effect is discussed in Gordon et al. (2015) for this flight campaign. Although this can have significant effect on the calculation of emissions, the effect on the estimation of plume height should be less than the vertical distance between passes (∼ 100 m). Further, some flights were flown from bottom to top, while others were from top to bottom, so there should be no directional bias on average.

Each calculated plume location (sint, zh) was paired with each nearby observed plume location (sp, zp) to maximize the correlation of calculated and observed plume heights. For example, the calculated plume rise from three stacks would be paired with three observed plume heights by matching the lowest calculated plume height to the lower observed plume height, the middle calculated plume height to the middle observed plume height, and the highest calculated plume height to the highest observed plume height. This gave the highest correlation between predicted values and observations. For a single plume observation and multiple SO2-emitting upwind stacks, the stack plumes were assumed to have merged and the calculated plume height for each stack was paired to the same observed plume height. The merging of plumes is supported by visual observation by the authors during the field study, especially far downwind of the stack locations.

For the example of the 15 August screen flight (Figs. 2b, 3c, d), the forward trajectory and Briggs algorithm model intercept the flight screen approximately 2 km further south, and 140 m higher, than the observed plume centre, indicating the possibility of more complex wind flow than a simple trajectory. In the example of the 29 August box flight (Figs. 2a, 3a, b), there are two observed plumes along the northwest–southeast oriented wall of the box. The forward trajectory model places the plume intercept between these two plumes, closer to the vertically higher and more northern observed plume at the horizontal location given by s=58 km. There are four stacks within the box, two of which have calculated intercept heights near zh= 540 m and two of which have calculated intercept heights near zh=430 m. All four calculated values are clearly well below the observed intercept heights (zp=650 m and 880 m). This demonstrates some ambiguity and subjectivity in this analysis, as four calculated plume locations must be matched to two observed plumes. As described above and for the purposes of statistical comparisons, we match the highest two modelled plumes (near heights of 540 m) with the highest observed plume (880 m) and the lower two modelled plumes (near heights of 430 m) with the lower observed plume (650 m).

The topography of the Athabasca oil sands region can be generally described as a north–south river valley approximately 1 to 5 km in width, within a larger and more gradually sloped north–south valley between 10 and 50 km in width, and up to 500 m of vertical relief (Fig. 1a). Local surface wind patterns can be heterogeneous, especially within the valley. The AMS03 and AMS05 towers are in the vicinity of the Suncor stacks and the Syncrude stacks (Table 1), while the RASS is nearly equidistant to the eight stacks used for this analysis (Fig. 1b).

As an approximate measure of the uncertainty associated with local meteorology, plume rise values from the eight stacks are compared using the Briggs parameterization (Eqs. 1–6) with all three meteorological measurement platforms (i.e. AMS03, AMS05, and RASS) as well using the layered method (Eq. 7) with both RASS and aircraft measurements. This comparison was done for all concurrent times during which the aircraft was flying in box or screen patterns. There were approximately 26 h during which the aircraft flew in a box pattern and 20 h during which the aircraft flew in a screen formation, for a total of more than 46 h. The resulting distributions of calculated plume heights for these 46 h of flight time for the eight stacks are compared in Fig. 4.

The distributions of plume rise heights are similar for the Briggs parameterization with the three fixed, near-surface measurement platforms. Approximately 90 % of the plume rise values calculated with the AMS tower and RASS measurements are below approximately 250 m, with half or more below 75 m. With the layered method, the plume heights calculated with the RASS measurements are similar to those calculated with aircraft measurements. As with the Briggs parameterization, approximately 90 % of the plume rise values are below 250 m; however, more than half of the plume rise heights calculated with the layered method are above 125 m.

The plume rise was calculated for each flight for each stack with the Briggs parameterization for each input (towers, RASS) as well as with the layered method (RASS, aircraft). These plume rises were then paired with the measured plume locations following the method described in Sect. 2.7. For simplicity, the parameterized plume rise is described as hB=Δh, and the observed plume rise is described as hM=zp-zsurface-hs. Results of this comparison are shown in Fig. 5. The analysis resulted in 82 stack-to-observed plume pairings, for each measurement platform. (Note that a smaller number of pairings were possible for the RASS, which was not in operation for 4 of the 22 flight days.) Table 3 compares the results for each measurement method. The low slopes (b<0.5), significant intercepts (44<a<107 m), and low correlation coefficients (r2≤0.2) demonstrate that the Briggs parameterization of plume rise was a poor predictor of actual plume rise. For 95 % confidence (calculated from the standard error of the slopes) none of these slopes is significantly different from zero.

Using the tower or RASS measurements with the standard Briggs parameterization suggests an average underestimation (based on the average ratio) between 18 % (RASS) and 45 % (AMS03). The layered method using the RASS and aircraft-based measurements predicts a plume rise that is, on average, nearly half (47 %–49 %) of the observed value. In all cases, more than half of the plume rise values are underestimated by more than a factor of 2, and between 22 % and 42 % of predicted plume rise values are within a factor of 2 of the observations.

Table 4 lists the frequency of each stability class during box and screen flight times according to each measurement platform as determined by the sign and magnitude of the Obukhov length (L). Stable classification is separated as either due to small positive values of 0<L<2hh, or stack height above the boundary-layer height (hs>H). The RASS and the two towers give similar, predominantly (70 % to 94 %) neutral, stability during the flights, with RASS indicating the highest frequency (94 %) of neutral conditions. Of these three measurement platforms, only the measurements of AMS05 predict plume rise through unstable conditions. We also note that AMS03 and AMS05 are in close spatial proximity to each other (less than 10 km), suggesting substantial local changes in stability, again arguing for heterogeneity in the local conditions.

Based on previous studies summarized in VDI (1985), the authors suggested a reduction of the Briggs parameterization by 30 % in neutral conditions. Although the atmospheric stability is predominantly classified as neutral in our analysis, we are seeing an underestimation by the Briggs parameterization, in contrast to the previous studies.

Stability was determined using the RASS and aircraft temperature profile measurements based on a comparison of the temperature profile to the adiabatic lapse rate (Γ=g/cp=0.0098 K m-1). The temperature profiles were derived from measurements between the minimum aircraft height of 150 and 300 m a.g.l. The profile was considered neutral if -dT/dz was within 20 % of Γ. Because the RASS profiles demonstrated very different lapse rates near the surface compared to further aloft, these data were separated into near-surface (<100 m) and higher (>100 m). The profile measurements used for the layered method give a much different indication of stability class, with predominantly stable conditions for between 53 % and 89 % of the time. The RASS measurement profiles demonstrate a higher frequency of stable conditions near the surface (based on comparison to the lapse rate). For the RASS measurements, there is a significant difference between stability classifications based on Obukhov length compared to stability classifications based on the temperature lapse rate, suggesting either that these two methods are not directly comparable, or that significant spatial heterogeneity exists within the region (as is also implied by the comparison in stability classes noted at AMS03 and AMS05). The layered approach of Eq. (7) is based on the assumption of neutral or stable conditions. For unstable conditions we follow the assumptions outlined in Akingunola et al. (2018) and assume S=0. Since there is a relatively low frequency of unstable conditions in all cases (4 % to 13 %), any error caused by the assumption of S=0 during unstable conditions is likely small.

A comparison is also made using the Pasquill–Gifford (P-G; Turner and Schulze, 2007) stability class, based on cloud cover and the wind speed at 10 m (U10m). The P-G stability class specifies that during moderate daytime radiation (“a summer day with few broken clouds with the sun 25–60∘ above the horizon”), the atmosphere will be unstable (Classes A, B, or C) for wind speeds U10m<5 m s-1. For days with some cloud and U10m>5 m s-1 or for completely overcast days, the atmosphere will be neutral (Class D). According to the Pasquill–Gifford system, stable conditions (Classes E, F) will only occur at night (all flights were during daylight hours). Here U10m is determined from the lowest tower measurements (20 m) and Eq. (11), and cloud conditions are estimated from photographs taken during the flights. This results in predominantly unstable and neutral conditions, as shown in the first two rows of Table 4.

Hence all three methods produce a different prominent stability class: the Obukhov length calculation predicts mostly neutral conditions; the lapse rate predicts mostly stable conditions; and the Pasquill–Gifford stability classes predict an approximately equal occurrence of unstable and neutral conditions. Both the Obukhov length and Pasquill–Gifford class approaches show a substantial difference in the frequency of occurrence of unstable conditions between towers AMS03 and AMS05, underscoring the local variability that may exist in temperature profiles. In light of this disagreement, we test the change in results with different stability classification schemes in Sect. 4.4 in order to estimate the extent to which the average plume rise depends on the stability classification.

The above analysis suggests the potential for substantial variability between measurement locations, which may be due to heterogeneity of the terrain and surface conditions in the area. Here we perform a simple test of the sensitivity of the Briggs algorithm to uncertainties in input variables due to this variability between measurement platforms. Input variables are modified based on differences between the AMS03 and AMS05 measurement platforms. First, the average plume rise is calculated for the box and screen flight times for the eight stacks used in the analysis using AMS03 measurements as input. The input variables were then modified by the ratio of the average absolute difference between stations to the mean value (i.e. X03-X05‾/X‾, where X03 and X05 are the measurements variables at AMS03 and AMS05 towers respectively, and X‾ is the mean value from both stations combined). Instead of modifying the surface temperature (Tsurface) directly, the difference between the air temperature at stack height and surface temperature (ΔT=Ta-Tsurface) is modified by a fraction, as it is the difference that drives the parameterization (through Eqs. 2, 8, and 10). The average plume rise was then recalculated with the modified variables to determine the resulting change in average plume rise relative to the average plume rise calculated with unmodified input variables.

Average percentage changes in the plume rise for each modification for each measurement platform are listed in Table 5. The largest differences between the two measurement locations are boundary-layer height (H, 71 %) and Obukhov length (L, 165 %). This is expected as the parameterizations of Eqs. 9 and 10 are known to be unreliable without heat-flux or upper-air measurements. A decrease in boundary-layer height values by 71 % leads to an average decrease in the plume rise of 27 %, while an increase in boundary-layer height by 71 % leads to an average increase in plume rise of 6.7 %. Although the average difference in wind speeds between measurement stations is relatively low (14 %), this has a considerable impact on the plume rise, ranging from a 23.1 % increase to a 15.6 % decrease in average plume rise. This is in contrast to air temperature (Ta), temperature difference (ΔT), and friction velocity (u*), which all result in an average change in plume rise of less than 8 %.

The table identifies the variables with the largest impact on the parameterization results, and hence which variables require the greatest accuracy when obtained from a meteorological model forecast. These results also help explain the low correlation coefficients of the observation-driven plume rise height comparisons (Table 3), as uncertainty in the estimation of these derived quantities will lead to uncertainty in individual plume rise estimations.

If the stacks are physically close enough to the interception of the plume with the box walls or screens, it may be the case that the plumes have not travelled a sufficient distance to reach the maximum plume rise that is parameterized by the Briggs algorithms. Briggs (1984) also developed parameterizations of downwind distance to maximum plume rise. A plume in stable conditions will reach its final rise (Briggs, 1984) at xe=4.7US. A plume in neutral conditions will reach its final rise (Briggs, 1975) at xe=49Fb5/8forFb<55m4s-3,119Fb2/5forFb>55m4s-3. In unstable conditions, the plume fumigates and is evenly distributed in concentration between the surface and a height of 1.5Δh, based on the assumption that the half-width of the plume is 0.5Δh. Although no parameterization has been developed for the distance required to reach maximum plume rise in unstable conditions, Briggs (1984) provides a parameterization of the average horizontal distance to fumigation (contact of the plume with the surface) as xf=Uwhs+0.5Δh, where the average downdraft speed is w=0.8u*, following Briggs (1984).

Using the AMS03 input data as an example, none of the 87 matched plumes have distance from stack to measurement location (xd) less than the horizontal distance to reach maximum plume rise (xd<xe) in neutral or stable cases, and there are no unstable cases (Table 4). As discussed above, the analysis is limited to plume sources that are within 50 km of the box walls or screens. The distances between stacks and box walls (following the forward trajectories) range from 4 to 16 km, while the distances between stacks and screens range from 3 km to more than 150 km. There are 8 screens located within 40 km of the stack sources and 12 screens located more than 60 km from the stack sources (there are none in the 40–60 km range). Tests demonstrate (discussed in the next section) that including the 12 screen plume observations beyond 60 km from the sources in the analysis results in lower correlations and poorer performance of the Briggs parameterizations, as expected.

Given that the observed plume rise is generally much higher than the calculated plume rise, it should also be the case that distance to maximum plume rise is also underestimated. If it is assumed that the plume reaches its maximum height at the measurement location and the predicted plume rise (hb) is less than the measured plume rise (hM), then the actual distance to maximum plume rise can be estimated as xe′=xehM/hb. Using this modified distance to plume rise, 13 % of the plumes have distance to maximum plume rise greater than the distance between stack and screen (or box wall). This indicates that for these plumes, the assumption that xe′=xd is incorrect and the maximum rise for these plumes is higher than hM. Hence, the parameterized plume rise may underestimate the actual plume rise in some cases due to the measured plumes not reaching their maximum height. This magnitude of the underestimation is investigated as one of the modifications discussed below.

To investigate the underestimation of plume rise by the parameterization, we recalculate the predicted plume rise with a number of modifications. For ease of comparison, we use only the AMS03 tower data to drive the algorithm. Table 6 lists the results of these modifications. The “base case” is the analysis as described in the preceding sections with no modifications. The base case statistics are reprinted in Table 6 (case 0) from the first line of Table 3 in order to facilitate comparison. The results are presented as scatter plots for each case in the Supplement. Each of the comparison studies presented as different cases in Table 6 are described in more detail in the sub-sections that follow.

Our focus within this work was the use of the available measurement data as a proxy for the meteorological conditions at the stack locations themselves. However, significant differences could be seen in the data between the different measurement platform locations (see Table 2). In subsequent work in our companion paper (Akingunola et al., 2018, this issue), high-resolution meteorological model forecast simulations for the region were carried out. These suggested the presence of significant spatial heterogeneity in the meteorological parameters used to drive both the Briggs parameterization and the layered method. Predicted meteorological parameters at the meteorological measurement platform locations were substantially different from those at stack locations. When tested using the model-predicted at-stack meteorological values, and NPRI stack emissions data, the Briggs parameterization and the layered approach resulted in very different plume rise behaviour. Predicted surface SO2 concentration performance was substantially improved across all metrics when the layered approach was used, and aircraft SO2 comparisons improved for all metrics aside from bias. For the predicted plume heights, the slope of the model observation line was -0.16 for the Briggs parameterization and 0.97 for the layered approach, with the former under-predicting and the latter over-predicting the aircraft-observation-estimated plume height. The reader is directed to Akingunola et al. (2018) for a discussion of these issues, which suggests that accuracy of estimates of the driving meteorological parameters at the stack locations has a controlling influence on the performance of the layered approach, and with the layered approach recommended for future development.