3D trajectories and velocities of rainfall drops in a multifractal turbulent wind field
 Hydrologie Meteorologie et Complexite (HM&Co), Ecole des Ponts ParisTech, ChampssurMarne, France
 Hydrologie Meteorologie et Complexite (HM&Co), Ecole des Ponts ParisTech, ChampssurMarne, France
Abstract. Weather radars measure rainfall in altitude whereas hydrometeorologists are mainly interested in rainfall at ground level. During their fall, drops are advected by the wind which affects the location of the measured field.
The governing equation of a rain drop motion relates the acceleration to the forces of gravity and buoyancy along with the drag force. It depends nonlinearly on the instantaneous relative velocity between the drop and the local wind; which yields to complex behaviour. Here, the drag force is expressed in a standard way with the help of a drag coefficient expressed as a function of Reynolds number. Corrections accounting for the oblateness of drops greater than 1–2 mm are suggested and validated through comparison of retrieved “terminal fall velocity” (i.e. without wind) with commonly used relationships in the literature.
An explicit numerical scheme then is implemented to solve this equation for 3+1D turbulent wind field, and hence analyse the temporal evolution of the velocities and trajectories of rain drops during their fall. It appears that multifractal features of the input wind are simply transferred to drop velocity with an additional fractional integration whose level depends on drop size, and a slight time shift. Using actual high resolution 3D sonic anemometer and a scale invariant approach to simulate realistic fluctuations of wind in space, trajectories of drop of various size falling form 1 500 m are studied. For a strong wind event, drops located within a radar gate in altitude during 5 min are spread on the ground over an area of size few kilometers. Spread for drops of a given diameter are found to cover few radar pixels. Consequences on measurements of hydrometeorological extremes which are needed to improve resilience of urban areas are discussed.
Auguste Gires et al.
Status: final response (author comments only)

RC1: 'Comment on amt2021434', Anonymous Referee #1, 08 Mar 2022
The paper models the 3D trajectories and velocities of discrete raindrops accounting for corrections for raindrop oblateness. The paper is original and results are sound, but there are some comments that need clarification before the paper is accepted for publication.
Section 2.1 needs improvement mainly on the definition/description of some of the variables/equations. For instance, the wind is a vector and has 3 components, but the definition of v_wind does not say in which direction (is it in the ‘z’ direction?). What’s “SA”? how did you come up with the equations shown in Lines 105 and lines 114 for MPA.
The force balance (FW = FB+ FD) is valid with the assumption that the particle reached terminal velocity and therefore any additional force due to acceleration is zero. Therefore, if the particle reached terminal velocity, should not dv/dt be equal to zero? However, Eq 1 shows that dv/dt is not zero. Could you please clarify? In addition, the force balance gives:
Fw = FB+FD
where Fw is the weight of particle, FB buoyancy and FD the drag. This equation leads to a wellknown expression in fluid mechanics for the terminal velocity of a single particle given by:
v^2 = 4*D*g/3/CD* (rho_p  rho_air)/rho_air
where rho_p and rho_air is the density of the particle and air respectively, CD the drag coefficient, g gravity, D particle’s diameter. So if the particle’s velocity v is equal to v_rel = v_wind  vp in your notation (assuming v_wind is in the z direction), then any change in v_wind over time will affect v_rel and CD (CD is a function of Re and Re a function of v_rel). So it is unclear how you came up with your Equation 1 without including the time derivatives of CD and v_wind. Perhaps I misunderstood something, but if you can elaborate please.
All the above does not account for raindrop breakup or aggregation and only applies for discrete particles that do not interact with each other. However, we know this is an important process in precipitation and this will affect v through the increase/decrease of D. Given the fact that you are using a more complex model to work out CD and account for raindrop oblateness, what are the implications of breakup/aggregation in your results?
Section 3.1. Recommendation: to use a different variable for C1 in Eq 7 to avoid confusion with ‘c1’ in Eq 3.
Section 3.2 The real part ‘Re’ might be confused with the Reynolds number ‘Re’.
Eq 11. be consistent with the variable definition. is ux, uy and uz the same as vx, vy and vz in Figs 5 and 6?
From the conclusions, it is clear that wind effects are important especially during strong winds. However, it is unclear how to correct radar rainfall estimates on the ground to account for this. Perhaps the authors can elaborate further.
From a practical point of view, what’s the difference in trajectories/displacements shown by this model and the one that assumes spherical particles (when computing CD). Is the additional complexity in the modelling adding any value? I would like to see the differences in terms of displacements as well.
Spelling mistakes:
“equivolumic” (line 193), “withing a voxel” (line 272).

RC2: 'Comment on amt2021434', Qiang Dai, 08 Apr 2022
The manuscript by Gires et al., developed an approach to simulate the 3D trajectory of raindrops by considering the fluctuation of smallscale wind in both space and time and drift patterns of nonspherical raindrops. This study overcomes the limitations of traditional coarse wind data and effectively corrects the rainfall observation under wind drift effect. The objectives of the study clearly explained in the introduction section, as well as the contribution of work. However, some figures in the article lack elaboration, the text also has obvious formatting problems. My detailed comments are provided below:
Despite some minor issues raised below, I think the paper would be a useful addition to the literature and recommend publication after the authors address the comments/questions below.
It’s not clear what is the difference between Ív_{rel} and v_{rel}? I cannot find any explanation about v_{rel} (v without underscore) in your manuscript. However, both appear frequently in subsequent equations (e.g. line 76, line 80, line135).
As for wind field generation part, there are still a few confusions. In line 222, what is the standard for distinguishing high and low wind speed? Is it the average speed of the three directions or something else? In line 232, what is the resolution of the 729 x 729 x 64 grid? Besides, what does 64 refer to? Time or altitude?
The manuscript lack of the description of study area and the 3D sonic anemometer instrument. Despite the author's citation of the data, I would still like to know how the instrument works and how the wind speed data is organized. Please improve this section by adding more detailed information.
Besides, I cannot find any description of Fig. 6 in the manuscript
Below are specific comments and suggestions:
Line 25: “km” in the brackets may be italicized.
Line 3637: “for example” appears consecutively.
Line 73/79/87/104...: Please remove spaces before colon characters.
Line 96: Please add a period at the end of the sentence.
Figure 1c: What is the difference between abscissa and ordinate? Which is the retrieved one?
Figure 1d: Please add ylabel.
Figure 1f: Please change “(e) Terminal fall velocity vs. equivolumic diameter” to “(f) Terminal fall velocity vs. equivolumic diameter”.
Line 107: Does “SA” refer to surface area?
Line 119: I wonder if the drag coefficient is C_{d}, C_{D} or c_{D} (in Eq. 4, Fig. 1e and line 76)?
Line 140: “â” may be “ât”.
Figure 2: Is D different from D_{eq}? Please clarify.
Line 198: Do you refer to 0.1 mm?
Line 200: “For the 2 mm drop, the scaling is slightly degraded but remains good (r2 = 0.95 for q = 1.5). α = 1.69, C1 = 0.14 and H = 0.79 is found.” I prefer to combine the two sentences into one.
Figure 4: Please check the numbering order.
Line 221/225: Please check format of the citations.
Line 242: Do you mean “at any point (x,y,z,t), a bi or trilinear interpolation…”ï¼
Line 258: Please list the specific parameter information of these 10 wind samples.
Line 275: Is the wind shift field here one of the 10 types of wind samples used in section 4.3? If yes, it is recommended to describe and highlight in Fig. 8 (e.g. in dot line); same recommend in Fig. 11.
Line 281: Please add units to the Δy numbers
Line 289: Please unify the number format, such as “1 500” or “1500”?
Line 295: “The increase ranges from 0.1 for 0.1 mm size drop to 0.8 for drops of size 11.5 mm.” change to “The increase ranges from 0.1 for 0.1 mm size drops to 0.8 for 11.5 mm size drops.”
Auguste Gires et al.
Auguste Gires et al.
Viewed
HTML  XML  Total  BibTeX  EndNote  

95  39  10  144  4  6 
 HTML: 95
 PDF: 39
 XML: 10
 Total: 144
 BibTeX: 4
 EndNote: 6
Viewed (geographical distribution)
Country  #  Views  % 

Total:  0 
HTML:  0 
PDF:  0 
XML:  0 
 1