Parameterizing the vertical downward dispersion of ship exhaust gas in the near-field

Estimating the impact of ship emissions on local air quality is a topic of high relevance, especially in large harbor cities. For chemistry transport modeling studies, the initial plume rise and dispersion play a crucial role for the distribution of pollutants into vertical model layers. This study aims at parameterizing the vertical downward dispersion in the near-field of a 10 prototype cruise ship, depending on several meteorological and technical input parameters. By using the micro-scale transport and stream model MITRAS, a parameterization scheme was developed to calculate the downward dispersion, i.e. the fraction of emissions, which will be dispersed below stack height. This represents the local concentration in the vicinity of the ship. Cases with and without considering the obstacle effect of the ship have been compared. Wind speed and ship size were found to be the strongest factors influencing the downward dispersion, which can reach values up to 55 % at high wind speed and 15 lateral wind. This compares to 31 % in the case where the obstacle effect was not considered and shows the importance of obstacle effects when assessing the ground-level pollution situation in ports.

effects of pollutant downward dispersion behind the obstacle, i.e. the vessel, by lowering the effective plume height and adjusting dilution parameters in the model, this effect has yet to be applied to large ships.
The aim of this study is to quantify the main factors that have an impact on the downward dispersion process for a large cruise ship. For this purpose, an Eulerian micro-scale model is used to calculate the downward dispersion in close vicinity to the ship.
Furthermore, a parameterization is developed for the downward dispersion in dependence of the crucial meteorological 70 influences and the technical specifications of the ship. Finally, it will be shown under which conditions the obstacle-effect on the downward dispersion needs to be considered.

Methodology
The dispersion of an exhaust plume is affected by several meteorological and technical parameters (Fig. 1). The upward movement, i.e. the plume rise, is mainly determined by the initial temperature of the exhaust and its exit velocity, which can 75 be calculated by dividing the gas volume flow by the stack diameter. The stack angle describes whether the exhaust flow is directed vertically, horizontally or at an angle. The stack height only has an indirect effect on the plume rise, as higher emitted gases experience a stronger wind speed inside the boundary layer.
Turbulence enhances the plume dispersion, leading to dilution of the embedded gases by entrainment of ambient air into the plume. The dispersion increases with the wind speed. It depends also on the ship geometry and the flow direction of the wind 80 emission cell has a constant temperature, which corresponds to a given exhaust temperature and a vertically directed exhaust velocity. The wind field is affected by Coriolis force and friction force, which cause the wind to slightly turn counterclockwise according to an Ekman spiral. Furthermore, the flow field is modified by the obstacle itself, the high temperature of the exhaust and the exit velocity. No deposition occurs in the model domain, the surface is a mirror source which reflects the concentration 100 when the lowest model layer is reached.

Meteorological data
Idealized meteorological conditions are used to investigate effects of single variations of input parameters on the dispersion process. The range of input values is listed in Table 1. One input parameter per model run was varied while the other meteorological and technical parameters were fixed at predefined default values. 105 The ambient temperature is set to 15 °C at the surface. It changes with altitude according to the given ambient temperature gradient, which represents the atmospheric stability. The value of ambient temperature itself has a negligible small effect on the plume dispersion compared to the plume temperature and was therefore not varied in this study.
The atmospheric stability is varied in a range of different lapse rates, covering one unstable condition (-1.2 K · 100 m -1 ), one neutral condition (-0.98 K · 100 m -1 ) and several stable conditions including inversions (up to +0.5 K · 100 m -1 ). 110 The wind speed is investigated in a range of 2-15 m s -1 . The limits were chosen according to hourly wind speed data from Hamburg weather mast in 2018 (see Appendix A) and can also be seen as representative for other large North Europe ports including Rotterdam and Antwerp. The value 2 m s -1 is close to the 5 th percentile and 15 m s -1 corresponds to the 95 th percentile at 280 m measurement height. This covers most of the naturally occurring scenarios. The selected default value is 5 m s -1 which fits well with the mean wind speed in Hamburg at a height of 50 m, which is close to the stack height. 115 The effect of wind direction is relevant in correspondence to the orientation of the ship. Frontal wind is herein defined at an angle of 0° and lateral wind at 90°. Oblique wind conditions lie between these values.

Ship characteristics
This study represents a cruise ship prototype. From an online database (Port of Hamburg, 2020) the average length and width of cruise ships that were visiting Hamburg harbor during the years 2018 to 2019 has been calculated. The stack height was 120 approximated from freely available photos (e.g. Vesseltracker, 2020). The ship prototype has a length of 246 m, a width of 30 m and a stack height of 52 m (see Table 1 and Fig. 3). This corresponds to a typical cruise ship that can carry between 1000 and 2500 passengers. A non-moving source is assumed, i.e. a hoteling ship at berth.
The study goes beyond a case study. A loaded container ship of similar size and exhaust characteristics would deliver similar results because its shape is comparable. On top of that, for all investigated input characteristics, the results of stack-only cases 125 are presented as well. Therefore, one can assume that results for smaller ships lie between these two cases.
The exhaust gas temperature depends on technical parameters of the ship's engine and can be found in engine data sheets provided by manufacturers like Caterpillar (CAT, 2020), Wärtsilä (Wärtsilä, 2020) and MAN (MAN, 2020) on their websites. https://doi.org/10.5194/acp-2020-753 Preprint. Discussion started: 1 September 2020 c Author(s) 2020. CC BY 4.0 License.
For large cruise vessels, it ranges between approximately 300 °C and 400 °C, depending on the used engine power. However, the exhaust temperature can be lowered by 75-100 °C when a heat exchanger which generates electric energy from the excess 130 heat is in operation (Murphy et al., 2009). Therefore, the temperature effects are investigated for 200 °C, 300 °C and 400 °C plumes to cover a realistic spectrum. Similarly, the exit velocity was assumed from these data sheets. It depends on the engine type (main engine or auxiliary engine) and the used engine power and was investigated in a range of 4-12 m s -1 .

Plume dispersion in different regimes
When investigating plume dispersion, one needs to separate two regimes: the momentum-driven regime and the buoyancy-135 driven regime. In the momentum-driven regime the movement of the plume is affected by (a) the initial plume rise due to both, the exit velocity and the high-temperature convective upward transport and (b) the dispersion due to turbulence generated by the obstacle (i.e. the ship) inside the wind field. In the buoyancy-driven regime, the movement of the plume is determined by the wind field and turbulence generated by the ambient conditions (e.g. orography effects and surface roughness). Here, the plume temperature is equal to the ambient temperature. The micro-scale model MITRAS can investigate plume behavior in 140 both regimes on a high resolution.
MITRAS is used to capture the initial plume rise and turbulence effects in the momentum driven regime. The vertical concentration profiles are calculated at a distance outside of the momentum-driven regime, i.e. when the buoyancy-driven regime is reached. Then, the concentration profiles are calculated on a 100 m x 100 m area column with layer-mean values ( Fig. 4). The calculation of these column values has two benefits. First, it covers the mean behavior of the whole plume better 145 than single values of 2 m x 2 m x 2 m grid sizes, since the movement of the plume can be highly variable. Second, the concentration profiles can then also be transferred into larger-scale models which usually have a much coarser grid (e.g. 100 m x 100 m horizontally). However, the coupling of MITRAS results into a larger-scale model will be part of a future study and is not covered here.
Since the plume needs to have cooled down to ambient temperatures to be considered outside the momentum-driven regime, 150 test simulations have been performed to find a distance at which this condition is met (see Appendix B). This was the case at a distance of 100 m downwind of the ship. Therefore, all concentration profiles are calculated as 100 m x 100 m columns with average concentration per layer at a distance of 100 m downwind of the ship.
In the following, the term downward dispersion D is defined as the relative proportion of the total concentration column in the layers below the stack height. 155 where htop is the altitude of the highest model layer (500 m), hstack is the stack height (52 m) and c is the total concentration. A mean downward dispersion is calculated for the described 100 m x 100 m column at a distance of 100 m downwind from the https://doi.org/10.5194/acp-2020-753 Preprint. Discussion started: 1 September 2020 c Author(s) 2020. CC BY 4.0 License.
stack. From an application perspective, this downward dispersion parameter is an indicator for the pollution situation in the vicinity of the ship and useful to evaluate the level of pollution inside of a harbor. 160 For single regression analyses, downward dispersion values are investigated depending on the variation of one single input parameter at a time while the others remain at default settings (Table 1). To assess the sensitivity of the downward dispersion to each input parameter, an effective range r is calculated. It is defined as the difference between the highest and the lowest downward dispersion value for one regression: where i is the individual input parameter that is varied while the other remain at default setting. The effective range describes how strongly one parameter can change the downward dispersion and helps to evaluate which input parameter has the strongest impact in the given range of values. The following subsections describe the results of single-and multi-parameter regressions that were performed in order to describe the relationship between the downward dispersion and the input parameters. From the multi-parameter regression, a parameterization is derived that covers all input parameters in the investigation range. A bootstrapping procedure is presented 175 to test how well the parameterization results match with the MITRAS model results. The obstacle effect is evaluated and, finally, some limitations of the modeling approach are discussed.

Results of single-parameter regressions
Single-parameter regressions are performed after basic statistic formulae (see Appendix C) to investigate the impact of 180 individual input parameters, i.e. wind speed, exit velocity, wind direction, plume temperature and atmospheric stability on the downward dispersion.

Effect of wind speed and exit velocity
The dependence of the downward dispersion from wind speed was modelled in the range of 2-15 m s -1 at the uppermost model layer, which is set as the input parameter. It is slightly lower at stack height following the 185 logarithmic vertical wind profile. See Table C1 for the exact values at stack height.
https://doi.org/10.5194/acp-2020-753 Preprint. Discussion started: 1 September 2020 c Author(s) 2020. CC BY 4.0 License. Figure 6 presents results of a single linear regression for the dependence of downward dispersion on varying wind speeds with and without obstacle effect. Other input parameters remained constant at default values (Table 1). A linear relationship with correlation coefficients R² of 0.98 was found for both runs with and without ship, respectively. At high wind speeds, the turbulence behind the obstacle causes strong downward dispersion. Under these settings, the wind speed has an effective range 190 on the downward dispersion of 40.3 % with and 21.1 % without ship, making the wind speed a crucial factor influencing the downward dispersion ( Fig. 6 and Table 2).
A similarly strong linear relationship has been found between the exit velocity of the exhaust gas and the downward dispersion ( Fig. 7 and Table 2) with regression R² of 1.00 for cases with and without obstacle. It is, however, a negative dependence, because higher exit velocities transport the plume into higher altitudes and consequently the downward dispersion is lower. 195 The effective range is much smaller than for the wind speed with only 3.7 % with and 2.1 % without obstacle, respectively.

Effect of wind direction
The strength of the downward dispersion was investigated depending on different wind directions in relation to the orientation of the ship. Frontal wind (angle 0°) hits the short side of the vessel, which has a width of 30 m, whereas lateral wind (angle 90°) has to be lifted over the 246 m length of the ship. Therefore, a stronger distortion of the flow during lateral wind has been 200 observed.
The downward dispersion correlates linearly with the cosine of the flow angle φ (Fig. 8). A regression coefficient R² of 0.98 was calculated. At default settings a downward dispersion ratio of 7.0 % and 16.6 % was found under frontal and lateral wind conditions, respectively. This results in an effective range of 9.6 %. The corresponding downward dispersion under no obstacle condition is 2.3 %. There is no effective range for no-obstacle conditions, because here a single symmetrical stack is assumed, 205 where the downward dispersion values are the same for both, frontal and lateral wind. However, very small differences between these conditions can occur during the modeling (see Table C1), which result from an asymmetry in the numerical grid.

Effect of exhaust plume temperature
The exhaust plume temperature depends on technical parameters like the engine power and the use of a heat exchanger and, therefore, a range of possible temperatures (200 °C-400 °C) was investigated. Figure 9 presents results of the single linear 210 regression for the downward dispersion at varying exhaust temperatures with and without obstacle effect.
Once again, a strong linear relationship with correlation coefficients R² of 0.98 and 0.99 was found for results with and without ship, respectively. At higher exhaust temperatures, the plume reaches higher altitudes by convective upward movement, which results in lower downward dispersion ratios. The effective range under default settings is 6.9 % with and 2.9 % without obstacle effect (Table 2).

Effect of atmospheric stability
The effect of atmospheric stability Γ on the downward dispersion was investigated in a range from unstable (-1.2 K · 100 m -1 ) to very stable (+0.5 K · 100 m -1 ) vertical temperature gradients. Under default settings, linear regression resulted in correlation coefficients of R² = 0.90 and 0.94 with and without ship, respectively (Fig. 10a). Since the R² coefficient was low compared to the other investigated input parameters, a linear dependence would deliver poorer results for this parameter. Therefore, a 220 quadratic dependence was calculated as well.
Since the square of a negative vertical temperature gradient would result in a positive value, a sign function was applied. The mathematical expression is: Then, the correlation between downward dispersion and sgn(Γ)Γ² is calculated (Fig. 10b). It shows better agreement in the 225 cases considering obstacle effects (R² = 0.99) and slightly better agreement in cases without ship (R² = 0.96), as well. It is a negative correlation, because higher temperature gradients correspond to a higher stability which thermodynamically prevents the plume to disperse vertically and therefore lowers the downward dispersion ratio. The effective range of the temperature gradient on the downward dispersion is 6.6 % for ship cases and 3.8 % for stack-only cases.

Result of the multiple regression 230
Multiple regression is performed according to the equations in Appendix C2. The downward dispersion ratio depends linearly on all investigated input parameters, their cosine (in case of the angle of wind direction) or their squares (in case of atmospheric stability). With that in mind, a training data set for the multiple regression was created. Here, all independent input parameters are varied at the same time (but in the given range) and the downward dispersion ratio is calculated with MITRAS. For a set of 39 different combinations (Table C1) of input parameters with obstacle effect and 27 without, the estimation coefficients 235 ̂ for individual parameters i (wind speed, exit velocity, etc.) are calculated with the multiple regression. The number of simulated cases without obstacle effects are lower, because in these cases the wind direction has been varied which will not show differences in case of stack-only conditions. The resulting formulae for the parameterization read [%] = 13.03 + 3.45 − 1.01 − 0.026 ℎ − 3.81 sgn(Γ)Γ 2 + 6.13 cos (ϕ),

Bootstrapping 245
A bootstrapping procedure is applied to estimate how well the parameterization can represent the model data. For this purpose, downward dispersion ratios were calculated with the parameterization formulae (Eq. 4 and 5) and compared to the original MITRAS results for all investigated cases and ranges. Results of the individual parameterization results are listed in Table C1 and Table 3 gives the overall results of the bootstrapping procedure.
With a mean absolute error of 1.9 ± 1.6 % for cases with ship and 1.2 % ± 0.9 % without ship the parameterization delivers 250 very similar results to the model runs. The maximum absolute errors were found to be 6.1 % in cases with ship and 4.0 % in cases without ship.

Assessment of the obstacle effect
Another aim was to investigate under which conditions the strongest downward dispersion occurs and which effect the consideration of the obstacle has on the downward dispersion. 255 From the single-parameter regressions, it is assumed that the strongest downward dispersion occurs at high wind speed (15 m s -1 ) with lateral wind (90°), low exit velocity (4 m s -1 ), low plume temperature (200 °C) and during unstable atmospheric conditions (-1.2 K · 100 m -1 ). This is displayed in Fig. 11 with the ship as an obstacle (panel a) and under stack-only conditions (panel b).
The calculated downward dispersion ratio for this condition is 54.9 % and 31.1 % with and without obstacle effect, respectively. 260 This means that a significant proportion of nearly 25 % of the emission can be dispersed downwards only by taking into account the turbulence caused by the ship.

Discussion of limitations
Despite efforts to represent real conditions as best as possible, the results are subject to a few limitations or uncertainties that will be discussed in the following section. 265 One factor that is not considered in this study is relative humidity. Here, a distinction must be made between the relative humidity of the ambient air and the relative humidity of the exhaust. By using the Lagrangian concept based on the so-called projected area entrainment (Lee and Cheung, 1990), Affad et al. (2006) stated that the relative humidity of the ambient air has only a slight impact on the plume rise, diameter and temperature for values between 20 and 90 %. It can have an impact on particle growth, but as this study focusses on a passive gaseous tracer, this effect is neglected. On the other hand, the humidity 270 of the exhaust gas might have a larger impact on the plume rise. Since water vapor has a lower density than air, an exhaust gas mixture of high humidity will show a stronger plume rise. Furthermore, as the gas will quickly condense, it will release latent heat and rise further. However, the data base on humidity of ship exhaust is sparse. It could play a role in case of vessels using a scrubber to wash out SO2 from the exhaust. During this process, the exhaust is cooled down significantly and therefore will https://doi.org/10.5194/acp-2020-753 Preprint. Discussion started: 1 September 2020 c Author(s) 2020. CC BY 4.0 License.
show a weaker plume rise (Murphy et al., 2009). It is unclear, if the additional buoyancy can compensate for the lower exhaust 275 temperatures. Due to these uncertainties and lack of data, the relative humidity has not been included in this study.
Second, the emission is assumed to occur in the grid cell above the stack, which has a size of 2 m x 2 m x 2 m. This corresponds to a stack with a square cross section of 4 m² and is a limitation connected to the chosen grid size. Real stacks are usually round and have a smaller diameter. The real exit velocity could therefore differ slightly. However, by comparing the effective ranges for exit velocity against all other input factors (Table 2), it can be seen that this parameter has the smallest overall 280 impact of the downward dispersion and therefore, this uncertainty factor has a low impact on the overall performance.
Another assumption was that the ship has been considered as a non-moving source, i.e. a hoteling ship. However, the results can be applied to a moving ship by calculating the vector sum of the wind and the vessel speed. It is difficult to account for complex maneuvers, though, as the resulting wind vector may change quickly and the technical conditions like exhaust temperature and exit velocity may also vary with the speed of the ship. 285 The chosen model surface is water but assuming a hoteling ship, the land surface effects may play a role for the dispersion.
This effect has not been part of this study, as this is a highly variable parameter that depends on the structure of the harbor, the city and the orography. These effects need to be covered by a larger scale model.

Conclusion
A ship plume modelling study was performed with the micro-scale numerical model MITRAS to investigate the downward 290 dispersion of the exhaust in close vicinity to a modelled cruise ship (i.e. in the momentum-driven regime). A set of 39 different scenarios with varying meteorological and technical input parameters were analyzed. A multiple regression algorithm was used to estimate a parameterization function for the downward dispersion. This parameterization has been tested against the MITRAS model results through a bootstrapping procedure.
From single-parameter regressions a positive linear relationship of the downward dispersion from wind speed and negative 295 linear relationships from exit velocity, plume temperature and the cosine of the angle of wind direction was found. The downward dispersion ratio was larger in case of lateral wind than in case of frontal wind. In case of atmospheric stability, the downward dispersion showed a squared dependence from the vertical temperature gradient multiplied by the sign function.
From all these input parameters, the wind speed shows the largest effect on the downward dispersion in the investigated range (2-15 m s -1 ). 300 A comparison of the model results and the parameterization from multiple regression shows a good agreement with a mean absolute error of 1.9 ± 1.6 % for cases with ship and 1.2 ± 0.9 % without ship. For the case of strongest downward dispersion, the difference was calculated between downward dispersion with (54.9 %) and without considering the obstacle effect (31.1 %), which was almost 25 %. This shows how important it is to consider the effects of the downward dispersion in the momentum-driven regime when one wants to evaluate the air pollution situation in harbor areas.
The parameterization functions can also be used for container ships of similar size. It may also be applied to different emission situations like industrial stacks.
In a future study, other plume parameters will be derived from the vertical concentration profiles in a similar way as the downward dispersion. This includes the height of the plume axis and the shape of the vertical plume profile, which may deviate from the often assumed Gaussian distribution. These results can further be used in a city-scale model, which only calculates N, 10° 06′ 10.3″ E). Hourly data from the full year 2018 were statistically analyzed at five different measurement heights (Fig.   A1, Table A1). Figure B1 presents results for maximum temperatures in the MITRAS domain for one case with the highest temperature (400 °C). Ambient temperatures (15 °C) are reached at a horizontal distance of approximately 100 m from the stack. 445

C: Regressions
This section describes the general application of linear and multiple regression on the model results.

C.1: Single linear regression
A simple approach to estimate a target variable Y (e.g. the downward dispersion) from one single independent variable X (e.g. the wind speed) is a linear regression in the form of 450 where 0 and 1 are the ordinate axis intersection and the slope, respectively, and the circumflex (^) describes an estimated parameter. 0 and 1 are calculated with the least squares method, minimizing the quadratic deviation between model result values and estimated values ̂. The required function Q reads: https://doi.org/10.5194/acp-2020-753 Preprint. Discussion started: 1 September 2020 c Author(s) 2020. CC BY 4.0 License.
Minimizing is done by applying partial derivations from Q to 0 and 1 . This results in where X ̅ and Y ̅ are the mean values of the corresponding dataset.

C.2: Multiple regression 460
The variable Y can depend on more than one independent input variable (X1, X2, …, Xp). Then, a multiple regression can be applied and in the case of linear dependencies, the corresponding regression is called multilinear. The multilinear estimation for ̂ reads: Again, the minimum distance between and ̂ can be calculated by the least squares method, similar to the case of linear 465 regression, by minimizing the function Q:      Atmospheric stability -1.2 to 0.5 K · 100 m -1 -0.65 K · 100 m -1 6.6 % 3.8 % Table 3: Results of the bootstrapping procedure for cases with and without considering the ship-induced obstacle effect.

With Ship Without Ship
Number of training cases 39 27 Mean absolute error 1.9 % 1.2 % Standard deviation 1.6 % 0.9 %