Weather radars measure rainfall in altitude, whereas hydro-meteorologists 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's motion relates the acceleration to the forces of gravity and buoyancy along with the drag force. It depends non-linearly on the instantaneous relative velocity between the drop and the local wind, which yields complex behaviour. Here, the drag force is expressed in a standard way with the help of a drag coefficient expressed as a function of the Reynolds number. Corrections accounting for the oblateness of drops greater than 1–2 mm are suggested and validated through a comparison of the retrieved “terminal fall velocity” (i.e. without wind) with commonly used relationships in the literature.

An explicit numerical scheme is then implemented to solve this equation for a 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 the drop velocity with an additional fractional integration whose level depends on the drop size, and a slight time shift. Using an actual high-resolution 3D sonic anemometer and a scale invariant approach to simulate realistic fluctuations of wind in space, trajectories of drops of various sizes falling form 1500 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 the size of a few kilometres. The spread for drops of a given diameter is found to cover a few radar pixels. Consequences on measurements of hydro-meteorological extremes that are needed to improve the resilience of urban areas are discussed.

During their fall, drops are advected by wind. Quantitative rainfall estimation with the help of weather radars is affected by this issue since drops can be displaced horizontally between their measurement location in altitude and their ground impact location, which is of interest for hydro-meteorologists. This effect is usually called wind drift in the literature and sometimes wind advection. The potential bias and uncertainty introduced in radar measurements is stronger at higher resolution, i.e. typically with pixel sizes smaller than 1–2 km

Most correction schemes rely on the use of 4D wind profiles derived from numerical prediction models

Wind effects on rainfall drops is also reported to generate discrepancies between the vertical velocities measured and expected terminal fall velocities. For example,

Turbulence is found to have contradictory effects on the distribution of the fall velocity. Indeed, increasing the turbulence level in windy and rainfall conditions will yield more collision and breakup, resulting in smaller drops inheriting the speed of larger parent drops, and hence observations of super-terminal velocities. On the other hand, turbulence is said to yield a decrease in fall velocities because drops (especially ones of

Such findings on the discrepancies between observed and expected fall velocities have effects on the relation between rainfall and kinetic energy, i.e. the erosivity “power” of rainfall

The studies previously mentioned basically do not account for small-scale wind fluctuations in both space and time. In this paper, we suggest studying the behaviour of individual rainfall drops of various sizes in a high-resolution turbulent wind field. The variability of the wind is accounted for through the framework of universal multifractals (UM) (see

The paper is organised as follows. In Sect. 2, a deterministic equation for the fall of individual oblate drops in a 3D field is derived and validated through the comparison of the terminal fall velocity obtained with commonly used formulas. In Sect. 3, the framework of universal multifractals is described briefly. Then, the drops are subjected to simulated multifractal fields as the wind input and multifractal behaviour of the horizontal drop velocity is assessed. Finally, in Sect. 4, 3D wind is reconstructed from high-resolution 3D sonic anemometer data and strong scaling assumptions. This field is used to study the trajectories of drops between falling from 1500 m to the ground.

Let us denote

The drop is subjected to three forces:

The gravity equal to

The buoyancy equal to

The drag, which is commonly written as

As a consequence, the equation of motion of the falling particle is given by Newton's second law, which equals the mass times the acceleration to the net force (here, it was divided by the mass):

Before discussing how the drag coefficient is determined, it should be recalled that the rainfall drops considered in this paper are not spherical. Indeed, drops greater than typically 1.5 mm become oblate in their fall. This oblateness increases with size. A very commonly used model consists in an ellipsoid with an axis ratio varying depending on the size.

This shape corresponding to a solid of revolution around the

Illustration of how the volume (Vol), the surface area (SA) and the mean projected longitudinal cross-sectional area (MPA

For non-spherical shapes, it is quite tricky to compute the corresponding drag coefficient as a function of the Reynolds number. The literature about this issue is quite abundant, and the interested reader is referred to Chap. 4 of the PhD dissertation of

The sphericity

The crosswise sphericity

The lengthwise sphericity

The evolution of these parameters as a function of

The evolution of

In order to validate the developed equation, the retrieved terminal fall velocity is assessed for each equivolumic diameter. It corresponds to the velocity of the permanent regime with no wind, i.e. the drag plus the buoyancy exactly compensate the gravity. Computations are carried out with

The relation between the terminal fall velocity obtained vs. the equivolumic diameter is displayed in red in Fig.

Equation (

It is outside the scope of the paper to introduce the framework of universal multifractals (UM) in detail. Hence, only the most important elements are recalled here, and interested readers are referred to the references mentioned or to a recent review by Schertzer and Tchiguirinskaia

Let us consider a field

For UM, we have:

A non-conservative field (

The first step of a multifractal analysis usually consists in a spectral analysis. For multifractal fields, the power spectra (

In this section, the scaling behaviour of the horizontal drop velocity is assessed using numerical simulations. Working with such input whose features are fully known is helpful to understand how drops react to wind.

More precisely, a horizontal input

Figure

In order to quantify this qualitative behaviour more precisely, a multifractal analysis on the retrieved ensembles was performed. Figure

Scaling behaviour of the simulated drop velocity for

Figure

Summary of the multifractal analysis performed on the ensembles of simulated horizontal drop velocities using a wind input with

The purpose of this section is to investigate where drops falling from a height of 1500 m reach the ground. Given the time step of 0.01 s used in the equation and the fact that drops move in space during their fall, this means that it is necessary to have high-resolution space-time 3D wind data over an area of the typical size of a few kilometres to fully address the issue. Such data is unfortunately not available. Hence, we suggest here to reconstruct a somehow realistic wind from a punctual measurement relying on previous findings on turbulence.

More precisely, we use 100 Hz 3D sonic anemometer data collected at by a device installed at 78 m on a meteorological mast located on the Pays d'Othe wind farm within the framework of the ANR RW-Turb project. The wind farm is roughly 120 km south-east of Paris on a slightly sloppy area. More details can be found in the data paper under discussion at ESSD

Temporal evolution of the 100 Hz wind data from a 3D sonic anemometer for the low (top) and strong (bottom) wind events used in this paper.

In this section, we discuss how to stochastically generate a turbulent field reproducing the physics of such a flow constraint as well as possible to have the empirical velocity values

It is quite obvious that gravity has such a strong impact on drop trajectories and dynamics of drops that classical scaling approaches just fail because they presuppose isotropy. On the contrary, the anisotropy between the vertical and the horizontal induced by gravity is so ubiquitous in geophysics that it has led to the general concept and framework of “generalized scale invariance” for the analysis and simulation of anisotropic fields

Equation (

This confirms that

Practical difficulties only occur with discrete scales because they cannot easily deal with arbitrary

To get around these difficulties, it was tentatively proposed to consider the following 3D model:

Equalities in distribution are replaced by deterministic equalities, which oversimplify and trivialise the dynamics for each realisation.

The flux density

The introduction of

The scale of the flux densities

Moreover, the density values are arbitrarily taken at the locations (

All time steps are fully independent, with the exception of the empirical velocity values

The limitations of this model are, therefore, extremely strong. Many of them would have been resolved with the help of the scalar anisotropic cascades recalled above (see the previous paragraph). But to fully overcome them would require us to consider their extension to vector fields

The fields

The fields

Finally, at any point

In order to illustrate the suggested process, let us consider a 0.5 mm drop during the low wind event. Its initial position is

The actual total wind perceived by the drop (i.e. the input in Eq.

(Top) Temporal evolution with 0.01 s time steps of the wind data from 3D sonic anemometer, (middle) the wind shift and (bottom) the total wind perceived by the 0.5 mm drop falling from the position

Trajectory (solid line) of a 0.5 mm drop in a turbulent wind field for the low wind event. The dotted lines correspond to the trajectory projected on the

The process to generate an estimation of a 3D wind field is actually stochastic through the UM fields

The projected trajectories for the low wind event, are displayed in Fig.

The same as Fig.

In this last section, initial investigations toward understanding the consequence of previous work on quantitative rainfall measurement with weather radars are carried out. Indeed, weather radar measure rainfall at a given altitude, while hydro-meteorologists are interested in rainfall at ground level. During their fall, significant shifts can occur.
In order to study it, the following process is implemented. For 5 min, one rainfall drop is dropped every 15 s from a random position within a voxel of size 100 m centred on

Figure

The same as in Fig.

As was previously pointed out, this spread is due to the fact that smaller drops spend more time in the atmosphere and are more sensitive to wind fluctuations. Indeed, the duration of a fall from 1500 m to the ground at 0 m is equal to 716, 378, 238 and 192 s for drops of size 0.5, 1, 2 and 3 mm, respectively. Given that high-resolution radar pixels are typically of the size of a few hundred metres, one should note that drops within a given voxel at measurement height can reach the ground within an area of size 3 km

In this paper, we have aimed for a better understanding of the behaviour of individual rainfall drops falling from typically 1500 m. In a first step, we developed a new approach to compute the drag coefficient accounting for drop oblateness and findings in fluid mechanics. This was validated for drops of equivolumic size of up to 4 mm through the comparison between the retrieved terminal fall velocity and the commonly used formula.

Then the temporal evolution of the horizontal drop velocity under turbulent wind constraints was studied. It appears that multifractal features of the input wind are simply transferred to drop velocity with an additional fractional integration and slight time shift. The UM parameters

Finally, the trajectories of drops of various sizes falling form 1500 m was studied as a proof of concept. For this, 100 Hz anemometer data was used, and an approach to simulate realistic fluctuations of wind in space was developed. It notably enables to analyse how drop shift during their fall between their location measurement by weather radars and ground impact. For a strong wind event, drops located within a radar gate in altitude for 5 min are spread on the ground over an area of a few kilometres. The spread for drops of a given diameter is found to cover a few radar pixels.

In order to further explore the consequences of these findings on quantitative rainfall estimation with weather radars, further investigations are needed. More precisely, (i) the model to simulate wind fluctuations should be improved, notably to use vector simulations and tune the prefactors according to local wind conditions; (ii) space-time outputs of numerical weather prediction models could also be tested to retrieve wind fields; (iii) the actual drop size distribution should be used to better assess the impact for the ground estimation of precipitation, which implies making some simulations for a much larger number of drops; (v) a longer period of time should be tested to investigate where the water volume (i.e. all the drops) of a given radar gate fall during an event. For the two last points, data is available within the RW-Turb project. Such step would then need to be repeated over various radar gates to derive updated radar maps. Given the limited computation power that will not allow us to simulate the trajectories of all the drops, some statistical behaviour according to each radar gate and wind conditions would need to be designed and then computed. It should also be stressed that only individual drops are currently being handled. This means that the methodology developed does not account for either collision, aggregation between drops or for breakup. Such processes are also known to affect drop velocities by changing their size and shape. Future investigations should also aim at accounting for them. Finally, it should also be stressed that the method developed stochastically simulates wind fluctuations at small scales. This means that the output will not be a deterministic radar measurement but an ensemble of possible realistic outputs, out of which a probability distribution could be derived. Such a probabilistic approach is discussed in

The algorithms used in this paper were developed in Python programming language and can be obtained from the authors on request.

Wind data used in this paper can be found at

All authors designed the structure and main content of the paper. AG performed the numerical simulations and wrote most of the text. DS and IT identified and highlighted the limitations of the current modelling and outlined perspectives to overcome them. All authors contributed to the revision of the paper.

The contact author has declared that none of the authors has any competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The authors gratefully acknowledge partial financial support from the Chair “Hydrology for Resilient Cities” (endowed by Veolia) of Ecole des Ponts ParisTech, the Île-de-France region RadX@IdF Project.

This research has been supported by the ANR JCJC RW-Turb project (grant no. ANR-19-CE05-0022).

This paper was edited by Alexis Berne and reviewed by Miguel Angel Rico-Ramirez and one anonymous referee.