Simple 1D models have been used to assess the influence of several parameters or processes on microphysical or observed quantities (e.g., precipitation rates, polarimetric variables) and to test new parameterizations . These models are much simpler than full 3D models (like ICON-LEM) and are therefore also referred to as rain-shaft models. Because we apply such a simple model to ice microphysics we use the term “snowshaft” model. In these simple models, the atmospheric variables (e.g., temperature gradient, relative humidity) are predefined and feedback mechanisms from microphysics to thermodynamic and thus dynamic variables are neglected. These simplifications allow the analysis of selected processes and their sensitivity to a range of parameters without having to consider the full range of complexity. Another advantage of the snowshaft model is the low computational effort, which allows testing a large number of parameter combinations and process formulations.

The snowshaft model has 250 layers and the temperature spans the range from 0 to -40 ∘C, which covers the most relevant range for precipitating ice clouds. The temperature profile is linear with a gradient of 0.0062 K m-1. Consequently, the top of the model is at 6450 m. The relative humidity with reference to ice (RHi) is constant for h>3000 m and increases linearly until it reaches RH = 100 % (RH is the relative humidity with reference to water; Fig. ). The thermodynamic variables are constant in time and there is no air motion. These simplifications can be justified by the nearly stationary nature of many clouds and the low vertical velocity seen in the dataset of D18.

At the top of the model, a gamma distribution (following the size distribution parameter as described in Table ) is initialized for cloud ice and snow. Together with the size distribution parameter, the mass density Q and the number density N completely define the size distribution at the model top. Below the model top, the size distribution evolves through the following microphysical processes: sedimentation, depositional growth, and aggregation. These processes are considered dominant below the cloud top (where nucleation is especially important) and above temperatures near the melt layer, where riming rates increase sharply . The values of RHi, Q, and N are chosen in Sect.  to match profiles of observables with substantial precipitation.

In our simulations, we use a small domain setup of ICON-LEM. This setup has been shown to be both computationally efficient and to represent clouds well in various conditions . The domain is circular with a radius of 110 km, and the observational site Jülich Observatory for Cloud Evolution Core Facility JOYCE-CF; is in the center. At JOYCE the TRIple-frequency and Polarimetric radar Experiment for improving process observation of winter precipitation (Tripex, D18) took place, which we use in the model–observation comparison. The horizontal grid spacing of the simulations is ca. 400 m, and the vertical grid spacing ranges from 20 m at the surface to 380 m at the model top. With a total of 150 vertical layers, the atmosphere is simulated up to a height of 21 km. Initial and lateral boundary conditions are taken from the ECMWF Integrated Forecasting System (IFS). Initialization is carried out each day at 00:00 UTC. IFS data are incorporated as forcing on the lateral boundary every hour.

The SB06 scheme is used in the snowshaft simulations (Sect. ) and as the microphysics scheme in the ICON-LEM simulations (Sect.  and ). The SB06 scheme is a two-moment scheme that simulates the evolution of the number (N=M(0)) and mass density (L=M(1)) from which the mixing ratio (Q=L⋅ρair-1) can be easily derived. ρair is the air density and Mn (Eq. ) represents the moments of the mass distribution (Eq. ).

2M(n)=∫0∞mnf(m)dm

The scheme simulates six different hydrometeor classes (cloud water, cloud ice, rain, snow, graupel, and hail). The conversion from one class to another is in general associated with a specific microphysical process. For example, if cloud ice forms aggregates, Q and N of cloud ice are converted to snow (Sect. ). Therefore, it is consistent to assume properties of monomers for cloud ice and properties of aggregates for snow. The predefined particle properties of the default setting of the scheme are listed in Table  for each hydrometeor, along with the properties of the cloud ice and snow class alternatives proposed in Sect. .

In the SB06 scheme, aggregation rates are the product of collision rates and Estick because Ecoll is assumed to be 1. In the scheme, the variance approximation (SB06), based on the work of , provides a computationally feasible analytical solution of bulk collision rates. The variance approximation of avoids the usage of pre-calculated lookup tables and, unlike , is able to estimate collision rates of self-collection, i.e., aggregation within a particle class. The latter is made possible by considering the square root of the second moment of the velocity differences, which also has the advantage over the approximation by that the collision rates between different particles are nonzero even if their bulk velocities are equal. The default SB06 scheme assumes power-law relations for the vt–size relation in the calculation of the collision rates. The extension of the variance approximation of , which allows using Atlas-type vt–size relations (Sect. ), is applied in the SB06 scheme for the first time in this study.

Details of the components of the aggregation process considered in the SB06 scheme can be found in Sect.  and Appendix .

The Passive and Active Microwave radiative TRAnsfer tool PAMTRA; is used to simulate synthetic radar observations. Microphysical properties are represented consistently in the SB06 scheme and PAMTRA (Table ).

Throughout the study, we adopt the same scattering assumptions for each of the hydrometeor classes in the SB06 default scheme (“SB cloud ice”, “SB snow”, “cloud droplet”, “rain”, “graupel”, and “hail” in Table ). As in O20, we apply the self-similar Rayleigh–Gans approximation SSRGA; and coefficients derived from 3D models of aggregates of plates for cloud ice and aggregates of needles for snow. In O20, the coefficients used for the snow class were slightly adjusted to closely match the observed triple-frequency signature. The SSRGA parameters of aggregates of plates are also used for the new cloud ice categories (“column” and “needle” in Table ). For Mix2, SSRGA parameters derived from the same 3D models used for the determination of particle properties K20 are available . Since we find little influence of SSRGA parameters in Sect. , we use the adjusted SSRGA properties of the aggregates of needles from O20 for the Mix2 aggregates throughout the study to be consistent with 020, although using the SSRGA parameters derived from the same 3D aggregate models would be most physically consistent.

Like O20, we use multi-frequency observations to derive information about the aggregation process. Multi-frequency observations are useful to distinguish the size of particles, since the ratio of wavelength and particle size along with the particle density are the main factors that determine their scattering properties. The scattering of particles much smaller than the wavelength can be approximated well by the Rayleigh approximation. For larger particles, however, the interference of waves scattered from different parts of the particles must be considered , which leads to differential scattering among the various frequencies.

The ratio between the reflectivities of two radars with operating wavelengths λ1 and λ2 (λ1<λ2), 3DWRλ1,λ2=Ze(λ1)Ze(λ2)=λ14λ24∫σb(m,λ1)f(m)dm∫σb(m,λ2)f(m)dm, quantifies the amount of differential scattering. DWR is called the dual-wavelength ratio, Ze is the reflectivity, and σb is the backscattering cross section (all variables in linear units). Although differential attenuation can also contribute to DWR , we did not include this effect in Eq. () because the processing of D18 already corrects for the impact of differential attenuation on DWR. D18 evaluated the absolute calibration of the observed Ze values from the Ka-band radar using disdrometer measurements during rainfall. After correcting differential attenuation due to gases at all three frequencies, the Ka-band radar was then used as a reference for estimating calibration biases and differential attenuation effects due to hydrometeors by comparing the three Ze values at cloud top. The DWRs caused by differential scattering are usually close to 0 dB for small ice particles present at the cloud top. Calibration biases can be identified as DWR biases which are relatively constant over time; differential attenuation effects due to supercooled liquid water, rainfall, or the melting layer vary more strongly on shorter timescales (minutes to hours). The path-integrated differential attenuation estimated at cloud top was then used to correct the DWRs in the entire profile. A more in-depth discussion of various correction methods for multi-frequency radar observations is provided in . If differential scattering effects are the only contributor to DWR, it correlates well with the mean mass of the distribution f(m) (Sect. ), as can be seen from Eq. (). For small particles, the Rayleigh approximation is valid for all frequencies and σb scales with the mass squared. However, for larger particles and shorter wavelengths, σb is smaller than predicted by the Rayleigh approximation and σb(m,λ2) is smaller than σb(m,λ1). As a result, particle populations that contain larger particles, e.g., due to their large mean mass, have larger DWRs than particle populations with smaller mean masses. and others have shown that not only the mean mass, but also the shape of the distribution, the particle density, and the internal structure of the particles (through σb) can substantially affect the DWRs. Given the radars available in D18, we investigate the sensitivity of aggregation by analyzing DWRX,Ka and DWRKa,W. The subscripts W, Ka, and X denote the radar bands and, more specifically, the wavelengths of 3.3, 8.6, and 31 mm. Each combination of wavelengths is sensitive to a different range of particle sizes. For example, DWRKa,W is most sensitive to mean particle sizes of unrimed cloud ice and snow between 0.5 and 3 mm, and DWRX,Ka is sensitive between 1.5 and 15 mm (O20). Outside this sensitivity range, DWRs are zero (small mean size) or asymptotically approach (saturate) a DWR value (large mean sizes) that depends on the scattering properties of the particles present. More detailed information on the approach and its sensitivities can be found in O20.

Moreover, D18 reported that strong riming is rare in their dataset, so aggregation is the main contributor to particle growth and thus the increasing DWRs from cloud top to cloud bottom.

To interpret the following sensitivity experiments, we describe which parameters need to be considered in the simulation of aggregation in a bulk scheme, which parameters and process formulations are currently used in the SB06 scheme, and how the assumptions could be updated with recently published parameterizations.

The stochastic collection equation (SCE) describes how the particle distribution (PSDm) changes with time under the action of aggregation . 4∂f(mi)∂t=∫0mi/2f(mj)f(mi-mj)K(mi-mj,mj)dmj-∫0∞f(mi)f(mj)K(mi,mj)dmj Here, f(m) is the particle distribution as a function of mass and K is the aggregation kernel described in Sect. . The first term of Eq. () describes the gain of particles of mass mi by aggregation of particles with masses mj and mi-mj. The second term considers the loss of particles of mass mi by aggregation with particles of mass mj (illustrated in Fig. a and b). In general, PSDs cannot be perfectly described by simple functional relationships (e.g., gamma distribution) but can have complex shapes (Fig. a). Thus, explicit prediction of the evolution of PSDs must take into account the full SCE.

Bulk schemes, however, can only account for the evolution of the PSD in a simplified form. The tendencies of the moments in the SB06 scheme (mass density: ∂Q/∂t, number density: ∂N/∂t) can be calculated by considering only the loss term. The reason for this can be further explained by Fig. c–h, where the collision events are separated among the ice (monomers) and snow (aggregates) classes. In fact, because of the mass conservation, the total mass of particles gained (integral of the first term) has to match the total mass of particles lost (integral of the second term). Since it is assumed that within one time step a particle can participate only in one collision event, only one snow particle results from the collision of two ice particles (number of arrows in Fig. c and d). The same applies for the ice–snow and snow–snow collisions, but here there is no conversion of N from one category to another but only a loss of Ni or Ns. Thus, it is sufficient to calculate only one collision rate for each of the three considered collision scenarios (ice–ice, ice–snow, snow–snow) and moments (N and Q).

The snowshaft model (Sect. ) allows us to test the influence of the particle properties, the formulation of the collision kernel, Estick, and the size distribution on the aggregation rates with low computational effort and with reduced complexity. In Sect.  we showed how these parameters affect aggregation. We not only examine the influence of the parameters on the predicted model variables but also on the radar observables. After carefully setting up the model, the comparison in radar space enables us to directly contrast the statistics of the simulation and the observations, as given in O20 and D18. Since we compare the statistics of the model and observations over a relatively long time range this analysis already attempts to select a combination of parameters that can reproduce the observational statistics well. The optimal parameter combinations found in the snowshaft simulations will then be applied in the 3D model to simulate a case study (Sect. ) before we use it to rerun simulations for the whole time period of the Tripex campaign (Sect. ).

This comparison between the model and observation benefits from the simultaneous consideration of multiple model parameters and multiple observables. When looking at a single observable only, one might reduce a bias by an adjustment of a single process or parameter, even though this might just compensate for an inaccurate choice in another parameter, introducing compensating errors. As the number of independent observables increases, this problem is reduced as the inaccurate choice of a parameter might be detectable in one of the remaining observables. In other words, the larger degree of freedom in the observations helps to better constrain the parameters by comparison with the model when several observables are considered. We focus our comparison on the DWRs (as a measure of particle size) and the MDV (as a measure of vt). These two quantities constrain the strength of aggregation and the assumed vt–size relationship, and the statistical comparison in O20 also revealed the largest differences between observations and the model in these variables.

In the snowshaft simulations (Sect. ) we had to use several idealized assumptions. ICON-LEM (Sect. ) contains additional processes (e.g., advection, nucleation, varying humidity field) and therefore simulates a more realistic representation of the atmosphere. In this section, we investigate the impact of the various parameters studied in the sensitivity analysis in a more complex case study with an ICON-LEM simulation. Furthermore, the ICON-LEM simulations provide an opportunity to extend the analysis to various conditions (e.g., nonstationary regime during the frontal passage, sublimation layers).

The case study of interest was 3 January 2016, when a low-pressure area over the British Isles and an accompanying frontal system over western and central Europe determined the synoptic situation over the modeled domain. Shallow mixed-phase clouds are present in the morning and dissipate around noon (Fig. a). The passage of a warm front manifests itself at 10:00 UTC, first in high clouds and then in sinking cloud bases. These frontal clouds start to precipitate at 18:00 UTC. The selected case is particularly interesting because it contains clouds in different regimes and precipitation of weak to moderate intensity.

The observed and simulated ZeKa fields match relatively well for all simulations in terms of cloud structure and precipitation (Fig. ). Both the shallow mixed-phase clouds and the frontal cloud are very well-captured in terms of temporal and spatial structure.

ZeKa exhibits strong differences between the observations and the simulation only in the rain and ice slightly above the melting temperature in the period from 19:00 to 23:00 UTC. The sharp decrease in the observed ZeKa indicates strong sublimation. The presence of sublimation is also revealed by the model showing subsaturated air in this time range (Fig. ). There are three main reasons that explain why the model does not accurately represent the sharp decrease in ZeKa in this sublimation scenario. First, the humidity could be overestimated in the model, e.g., due to inaccurate forcing data. Second, particle sizes could be overestimated due to processes in microphysics that weaken the effect of sublimation. We cannot completely rule out the humidity mismatch, but we found good agreement between the model and radiosonde data when available. Unfortunately, there was no radiosonde launched on the considered day. Thus, we are confident in the general ability of the model to accurately simulate the humidity field, but we cannot rule out the possibility that inaccuracies in the simulated humidity field contribute to the bias in ZeKa. Lastly, the parameterization of sublimation could also be an error source. For example, the evolution of the PSD during sublimation is challenging to represent in a two-moment scheme . Since all of these reasons might be able to explain the mismatch in ZeKa, we should be cautious in assessing the validity of the assumptions of the individual model settings based on this sublimation feature. However, regardless of the accuracy of the model in predicting the humidity or simulating sublimation, the following differences in ZeKa of the model simulations underscore the importance of accurate prediction.

While the NEW (Fig. c) and most sensitivity runs show a slight decrease in ZeKa due to sublimation in the time period when the air is subsaturated, the sublimation is barely seen in ZeKa of some other simulations (e.g., CTRL, colMix2_Dkernel; Fig. b and g). The differences between the simulations are caused by the differences in the particle size indicated by DWRX,Ka (Fig. ). Similar to the snowshaft simulations, DWRX,Ka is strongly overestimated in colMix2_Dkernel and the CTRL run. In contrast, DWRX,Ka is well-matched closely above the melting temperature in the NEW simulation. The hydrometeor populations with realistic particle sizes are more strongly affected by the subsaturated air and sublimate quickly, whereas particles that are too large sublimate less and therefore retain more mass. Thus, the overestimated particle size leads to overestimated precipitation. Between 18:00 and 24:00 UTC, 1.40 mm of accumulated rain was observed, 8.91 mm simulated by the default simulation and 2.29 mm by colMix2_Akernel. While this represents an overestimates of 536 % by the CTRL run during this time period, we emphasize the overall good agreement between modeled and observed precipitation reported by O20 for the entire campaign. While Estick appeared to be important for the simulated DWRKa,W in the snowshaft simulations (Fig. ), the differences between the simulation with the old (colMix2_Akernel_LinCot; Fig. d) and the new Estick parameterization (colMix2_Akernel; Fig. c) are relatively small. In the ICON-LEM simulation, the weaker growth of the particles in colMix2_Akernel_LinCot at lower temperatures might be partly compensated for by advection or nucleation.

Besides the DWRs, MDV provides valuable information about the microphysical properties. As also reported by O20, MDV is overestimated in the SB06 default simulation, especially in regions where the particle sizes are overestimated (Fig. ). MDV is often used to distinguish rimed from unrimed particles e.g.,. Using this method, we detect some smaller episodes in which rimed particles dominate at about 04:00, 18:00, and 22:00 UTC. At other times, the observations indicate unrimed or only slightly rimed particles. In the SB06 default simulation, high MDVs are obtained in the whole time range after 18:00. Since the profiles of the hydrometeors show only very little mass of rimed particles during this period, the larger predicted MDV can be attributed to the overestimation of the unrimed snow particle vt.

The new simulations, all using the new particle properties, have significantly lower values of MDV at all temperatures. This reduction of MDV compared to the SB06 default setup constitutes a significant reduction of the bias in MDV at temperatures below -10 ∘C. For T>-10 ∘C, MDV is even slightly underestimated. Considering that Fig.  shows good agreement of MDV between the observations and the vt–size relation of Mix2, we assume that the underestimation of MDV is not caused by the underestimation of the vt–size relation of aggregates. Since DWRX,Ka also matches well at these temperatures, processes other than aggregation and sedimentation of unrimed aggregates most probably cause this underestimation of MDV. One could speculate that riming rates are underestimated or that the vertical air motion is not well-simulated.

Most of the findings from the snowshaft simulations (e.g., the strong reduction of MDV and DWR at temperatures close to the melting temperature) are confirmed by the ICON-LEM simulation of this case study. However, the ICON-LEM simulations reveal that the influence of Estick seems to be overestimated in the snowshaft simulations. Moreover, accurate modeling of particle sizes and vt in the presence of a sublimating layer is critical. The simulations with the new particle properties showed a slight underestimation of the MDV. This underestimation most likely does not arise from an inaccurate representation of the particle properties or the aggregation rates but is caused by another process (e.g., riming, vertical air motion). In previous analyses of the SB06 default setup, this underestimation could not have been detected because it was masked by the overestimation of the aggregate's vt. Because errors can be specific to the chosen day, such as a particular mismatch of the relative humidity, relying on only one case to detect a discrepancy in the microphysical properties is prone to error. Therefore, we analyze the statistics of a multi-month simulation in the next section.

After evaluating the choices of the new scheme in the snowshaft model and in a case study with ICON-LEM, we perform ICON-LEM simulations for the entire Tripex time period. By comparing observed and modeled histograms of DWR and MDV as a function of temperature, we can evaluate the new scheme. Since we additionally contrast the histograms of the NEW and CTRL simulations, we can test whether the reduction in the bias of DWRs and MDV found in Sect.  is specific to the selected case or rather a consistent feature of the model changes. As DWRs are related to the mean particle size, we can assess whether the chosen parameter combination can accurately simulate aggregation in various weather situations present in the simulated days. The same applies to MDV profiles, which are especially valuable in evaluating the suitability of the assumed vt–size relationship.

The observed and synthetic radar profiles are filtered in the same way for comparability. For example, the first 6 h of simulation and observation are not considered because the model output could contain artifacts during this spin-up time. Moreover, we include only profiles in which the rain rate RR exceeds 1 mm h-1. The latter filter enables us to focus on the most relevant cases for precipitation. Interestingly, O20 found the discrepancy between the model and observations to be especially obvious for these profiles. For a detailed description of the processing, we refer to O20.

To quantify the agreement between the histograms of the simulations and the observation, the Hellinger distance H is used. H can be defined for two distributions P =(p1, …, pk) and Q =(q1, …, qk) as 17H(P,Q)=12∑i=1kpi-qi2. H is zero for two identical distributions and 1 if the distributions do not overlap at all.

The medians and larger quantiles of the observed distributions of DWRs indicate a strong increase in particle size around -15 ∘C (most evident in DWRKa,W; Fig. a) and just above the melting temperature (most evident in DWRX,Ka; Fig. e). Both of these characteristic increases in the particle sizes are found to some extent in CTRL (panels b and f in Fig. ) and NEW (panels c and g in Fig. ). The increase in particle sizes between -15 and -10 ∘C happens in the new simulations at slightly lower temperatures, and the different profiles reveal a greater variability (visible, e.g., in the difference of DWRKa,W between the lower and upper decile). H indicates a slightly better match by CTRL in this temperature range. For T>-10 ∘C the mean and higher quantiles of DWRX,Ka increase very rapidly in CTRL and more slowly in NEW and the observation. The increase in particle sizes as simulated by NEW is in much better agreement with the observed profiles. The upper quartile of DWRX,Ka only slightly exceeds 5 dB in the observations and NEW but is higher than 10 dB in CTRL for T>-1 ∘C. This better match is also indicated by H (Fig. h), which is about 5 times larger for CTRL compared to NEW.

Besides the overestimation of DWRX,Ka closely above the melting temperature, O20 also highlighted the overestimation of MDV by CTRL. This overestimation is present at all temperatures (compare panels a and b in Fig. ) and can be attributed to the overestimation of the vt–size relationship of the snow class as reported in and the overestimated particle sizes for the higher temperatures. The overestimation of MDV by CTRL is most pronounced for T>-15∘C. In this temperature range, CTRL cannot reproduce the asymptotic approach because of the power-law vt–size relationship (Sect. ). For example, the median of MDVKa at -5 ∘C is 1 m s-1 in the observations and 1.3 m s-1 in CTRL. In contrast, the new simulations agree better with the observations and H is about half as large as for CTRL. The new scheme setup is more accurate in this temperature range because the Atlas-type vt–size relationship of the Mix2 particles (Fig. ) correctly considers the asymptotic approach to 1 m s-1 at large sizes. However, MDV is slightly underestimated by NEW for T>-10 ∘C. Values substantially above 1 m s-1 occur in the observations and the new simulations only closely above the melting temperature, where rain is present. At temperatures below -15 ∘C, both simulations perform similarly, with H ranging from 0.2 to 0.5. While CTRL exhibits a continuous overestimation of MDV, the new simulations lack the observed increase in MDV for T<-20 ∘C. At these temperatures, the selected PSD width (Sect.) and cloud ice particle properties (Sect. ) may not be ideal.

The statistical comparison shows that the changes we made to the model could eliminate the most striking biases, namely the overestimation of DWRX,Ka and MDV closely above the melting temperatures. The match of these quantities is important for accurate simulation of precipitation, as exemplified in the case study in Sect. . Some discrepancies remain, namely the overly strong increase in the DWRs at temperatures between -15 and -10 ∘C and the overestimation (underestimation) of MDV temperatures below -25 ∘C (above -10 ∘C). These discrepancies can be caused by several model errors (inaccurate simulation of, e.g., PSD shape, Estick, degree of riming, variability in cloud ice properties) that cannot be fully deciphered by this observational setup and could benefit from advances in laboratory measurements, observational setup, and representation of cloud ice habits and riming degree in the model.