the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 10 Jun 2025
The Boundary Layer Dispersion and Footprint Model: A fast numerical solver of the Eulerian steady-state advection-diffusion equation
Abstract. Understanding how greenhouse gases and pollutants move through the atmosphere is essential for predicting and mitigating their effects. We present a novel atmospheric dispersion and footprint model: the Boundary Layer Dispersion and Footprint Model (BLDFM), which solves the three-dimensional steady-state advection-diffusion equation in Eulerian form using a numerical approach based on the Fourier method, the linear shooting method and the exponential integrator method. In contrast to analytical Gaussian plume or stochastic Lagrangian models, this novel numerical approach proves beneficial as it does not rely on any asymptotic assumptions or estimates. Furthermore, it is fully modular, allowing for the use of a variety of turbulence closure models in its implementation or direct usage of measured or simulated wind profiles. The model is designed to be flexible and can be used for a wide range of applications, including climate impact studies, industrial emissions monitoring and spatial flux attribution. We validate the model using an analytical test case. The numerical results show excellent agreement with the analytical solution. We also compare the model with the well-established Kormann and Meixner (Boundary-Layer Meteorology, 2001) footprint model (FKM) which is based on the analytical Gaussian plume. The results show overall good agreement but some differences in the fetch of the footprints, which are attributed to the neglect of streamwise turbulent mixing – being one of the aforementioned asymptotic assumptions – in the FKM model. Our results demonstrate the potential of the BLDFM model as a useful tool for atmospheric scientists, biogeochemists, ecologists, and engineers.
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Notice on discussion status
The requested preprint has a corresponding peer-reviewed final revised paper. You are encouraged to refer to the final revised version.
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Preprint
(1867 KB)
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The requested preprint has a corresponding peer-reviewed final revised paper. You are encouraged to refer to the final revised version.
- Preprint
(1867 KB) - Metadata XML
- BibTeX
- EndNote
- Final revised paper
Journal article(s) based on this preprint
Interactive discussion
Status: closed
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RC1: 'Comment on egusphere-2025-2415', Anonymous Referee #1, 04 Oct 2025
- AC1: 'Reply on RC1', Mark Schlutow, 18 Dec 2025
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RC2: 'Comment on egusphere-2025-2415', Anonymous Referee #2, 23 Nov 2025
The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2025/egusphere-2025-2415/egusphere-2025-2415-RC2-supplement.pdf
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AC2: 'Reply on RC2', Mark Schlutow, 18 Dec 2025
The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2025/egusphere-2025-2415/egusphere-2025-2415-AC2-supplement.pdf
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AC2: 'Reply on RC2', Mark Schlutow, 18 Dec 2025
Peer review completion
Interactive discussion
Status: closed
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RC1: 'Comment on egusphere-2025-2415', Anonymous Referee #1, 04 Oct 2025
This paper presents a linear model to simulate boundary layer diffusion of tracers and to estimate the source flux footprint of an observation point (BLDFM). The technique is mathematically and numerically straightforward, and the benefits are simplicity, computational speed and flexibility. The validity of the method is verified by selecting a case for which analytical solutions exist. Then an example for an unstable situation is shown where the wind and stability profiles obey Monin-Obukhov similarity. It is compared with a footprint model that uses exponential profiles(FKM). The results are obviously different given that wind profiles are different and that diffusion along the direction of the wind is neglected in FKM. The authors attribute the difference mainly to the missing downwind diffusion, about which I have doubts because diffusion in the direction of the wind is overwhelmed by advection.
In principle I welcome such a paper on the basis that it documents a practical piece of code that is made available to the wider scientific community. I looked at the web-pages and the documentation appears in good shape. The paper is well written and a clear description is given of what the code is doing.
As a scientific paper, publication would be harder to justify. In that case it would be necessary to simulate a range of cases with more emphasis on the results of the computational method.
In conclusion, I recommend publication as a documentation of useful and practical code that can be used by anyone interested.
Before publication and to show what can be done with the code, I suggest to present a few mores cases, e.g. a stable case and a case with wind turning in the stable boundary layer. Wing turning is an important component of dispersion. Vertical diffusion spreads the pollution in the vertical and wind turning amplifies the horizontal spreading of the plume by advection in different directions.
Comment about the "Well-mixed criterion" (line 45):
The mention of a "Well-mixed criterion" may be confusing here. In boundary layer meteorology "well mixed" is often used for the convective boundary layer where potential temperature and tracers are fairly constant in the vertical due to strong mixing. Steady state is appropriate near the surface because the mixing time scale is fast compared to evolution of the large scale forcing, often called "quasi-equilibrium". It means that the shape of the profiles are nearly instantaneously in equilibrium with the large scale forcing. The well-mixed criterion is mentioned a few times in the paper. Perhaps it is better to call it the "quasi-equilibrium" assumption throughout.On the conclusions:
To substantiate the conclusion that the difference between BLDFM and FKM is due to differences in diffusion along the wind vector, it would help to run BLDFM without diffusion in the wind direction to see the effect of down wind diffusion.Citation: https://doi.org/10.5194/egusphere-2025-2415-RC1 - AC1: 'Reply on RC1', Mark Schlutow, 18 Dec 2025
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RC2: 'Comment on egusphere-2025-2415', Anonymous Referee #2, 23 Nov 2025
The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2025/egusphere-2025-2415/egusphere-2025-2415-RC2-supplement.pdf
-
AC2: 'Reply on RC2', Mark Schlutow, 18 Dec 2025
The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2025/egusphere-2025-2415/egusphere-2025-2415-AC2-supplement.pdf
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AC2: 'Reply on RC2', Mark Schlutow, 18 Dec 2025
Peer review completion
Journal article(s) based on this preprint
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Mathias Göckede
The requested preprint has a corresponding peer-reviewed final revised paper. You are encouraged to refer to the final revised version.
- Preprint
(1867 KB) - Metadata XML
This paper presents a linear model to simulate boundary layer diffusion of tracers and to estimate the source flux footprint of an observation point (BLDFM). The technique is mathematically and numerically straightforward, and the benefits are simplicity, computational speed and flexibility. The validity of the method is verified by selecting a case for which analytical solutions exist. Then an example for an unstable situation is shown where the wind and stability profiles obey Monin-Obukhov similarity. It is compared with a footprint model that uses exponential profiles(FKM). The results are obviously different given that wind profiles are different and that diffusion along the direction of the wind is neglected in FKM. The authors attribute the difference mainly to the missing downwind diffusion, about which I have doubts because diffusion in the direction of the wind is overwhelmed by advection.
In principle I welcome such a paper on the basis that it documents a practical piece of code that is made available to the wider scientific community. I looked at the web-pages and the documentation appears in good shape. The paper is well written and a clear description is given of what the code is doing.
As a scientific paper, publication would be harder to justify. In that case it would be necessary to simulate a range of cases with more emphasis on the results of the computational method.
In conclusion, I recommend publication as a documentation of useful and practical code that can be used by anyone interested.
Before publication and to show what can be done with the code, I suggest to present a few mores cases, e.g. a stable case and a case with wind turning in the stable boundary layer. Wing turning is an important component of dispersion. Vertical diffusion spreads the pollution in the vertical and wind turning amplifies the horizontal spreading of the plume by advection in different directions.
Comment about the "Well-mixed criterion" (line 45):
The mention of a "Well-mixed criterion" may be confusing here. In boundary layer meteorology "well mixed" is often used for the convective boundary layer where potential temperature and tracers are fairly constant in the vertical due to strong mixing. Steady state is appropriate near the surface because the mixing time scale is fast compared to evolution of the large scale forcing, often called "quasi-equilibrium". It means that the shape of the profiles are nearly instantaneously in equilibrium with the large scale forcing. The well-mixed criterion is mentioned a few times in the paper. Perhaps it is better to call it the "quasi-equilibrium" assumption throughout.
On the conclusions:
To substantiate the conclusion that the difference between BLDFM and FKM is due to differences in diffusion along the wind vector, it would help to run BLDFM without diffusion in the wind direction to see the effect of down wind diffusion.