This paper presents a novel methodology to use direct numerical simulation (DNS) to study the impact of isotropic homogeneous turbulence on the condensational growth of cloud droplets. As shown by previous DNS studies, the impact of turbulence increases with the computational domain size, that is, with the Reynolds number, because larger eddies generate higher and longer-lasting supersaturation fluctuations that affect growth of individual cloud droplets. The traditional DNS can only simulate a limited range of scales because of the excessive computational cost that comes from resolving all scales involved, that is, from large scales at which the turbulent kinetic energy (TKE) is introduced down to the Kolmogorov microscale, and from following every single droplet. The novel approach is referred to as the “scaled-up DNS”. The scaling up is done in two parts, first by increasing both the computational domain and the Kolmogorov microscale and second by using super-droplets instead of real droplets. To ensure proper dissipation of TKE and scalar variance at small scales, molecular transport coefficients are appropriately scaled up with the grid length. For the scaled-up domains, say, meters and tens of meters, one needs to follow billions of real droplets. This is not computationally feasible, and so-called super-droplets are applied in scaled-up DNS simulations. Each super-droplet represents an ensemble of identical real droplets, and the number of real droplets represented by a super-droplet is referred to as the multiplicity attribute. After simple tests showing the validity of the methodology, scaled-up DNS simulations are conducted for five domains, the largest of

The impact of turbulence on the growth of cloud droplets is an
important and still poorly understood aspect of cloud physics. This
is because of the wide range of spatial scales that affect droplet
growth, from the Kolmogorov microscale (about a millimeter for
typical atmospheric turbulence levels) to the scale of the entire
cloud or cloud system. Cloud droplets grow by the diffusion of water
vapor and by gravitational collision/coalescence, with the former
dominating growth until droplets are large enough so that the collisional growth can be initiated and eventually lead to drizzle and rain
formation. For the gravitational collision/coalescence, the frequency
of droplet collisions depends on the droplet spectrum width. It
follows that understanding processes leading to the observed droplet
spectra is important for the understanding of the rain onset.
Observations of natural droplet spectra go back to the early days
of aircraft cloud studies

Homogeneous isotropic turbulence simulations of

To this end, we propose to use what we refer to as the “scaled-up
DNS” approach. Since the largest eddies are the key for the
condensational growth, one would like to apply the DNS technique
in simulations with domains much larger than currently possible.
For instance, taking a

The paper is organized as follows. The next section presents the model and modeling setup. Section 3 presents a general methodology of the scaled-up DNS and discusses numerical tests of this approach. Cloud droplets are added to scaled-up DNS simulations in Sect. 4 applying the super-droplet method. Section 5 compares DNS and scaled-up DNS supersaturation fluctuations with those obtained from a simple stochastic model. Concluding discussion is the focus of Sect. 6.

The numerical code used here is that of

Two modifications have been made to the code to carry out the present study. First, we included an additional source/sink term in the temperature equation that was missing in the original code. The term describes evolution of temperature fluctuations affected by the vertical velocity. This effect is incorporated in the DNS through the source/sink term

Second, we modified the way condensation rate is calculated for a single droplet. The analytic formulation applied originally has the form

The coupling of the Eulerian fields and the droplets is done using trilinear interpolation. The condensation rate is calculated for each droplet by interpolating the values of

The modeling setup follows one of the simulations discussed in

The intensity of turbulence is typically expressed by the turbulent kinetic energy (TKE) dissipation rate

We used a DNS with

Figure 1 shows energy spectra for the real 25.6 cm DNS and the three scaled-up DNS. The black dashed lines represent the

Comparison of energy spectra for real and scaled-up DNS.

TKE in the scaled-up simulations for the same TKE dissipation rate

Evolution of TKE (upper panels) and TKE dissipation rate (lower panels) for four simulations mentioned in the text. The dashed lines are theoretical values. Bottom panels show nondimensional time using eddy turnover time; see Eq. (13) in Sect. 5.

The simulations shown in Figs. 1 and 2 feature the same dynamic range, that is, the same Reynolds number and the

The scaling Eq. (10) is illustrated in Fig. 3 that shows the spectra in simulations with the domain size of either 0.512 or 1.024 m and applying either DNS or scaled-up DNS. The spectra are obtained at final simulation times. The red lines represent spectra for the real DNS, and green and blue lines show spectra for scaled-up DNS. Scaling up accurately predicts the energy at the largest scales, but some energy at smaller scales, still far from the dissipation, is lost. This means that the total TKE for a scaled-up DNS is slightly lower than the real DNS within the same volume. For the simulations shown in Fig. 3, TKE for

Energy spectrum comparison of real DNS and scaled-up DNS. Panels

For a scaled-up DNS, one needs to follow a significantly larger number of droplets when compared to DNS. For instance, for the droplet concentration of 130 cm

At the onset of simulations, super-droplets are inserted into the computational domain in the same way as regular droplets; that is, they are randomly positioned inside the domain and subsequently followed in space and time as regular droplets. The condensation rate for a super-droplet is calculated as in Eq. (3) except for an additional multiplicity factor

The super-droplet approach is first tested in the real DNS. Figure 4
shows evolutions of the standard deviation of the supersaturation
spatial fluctuations

Supersaturation statistics in DNS and scaled-up DNS are calculated using fluid flow grid data and not the supersaturation interpolated to droplet positions. Limited tests suggest that the differences between the two methods are small (not shown). Supersaturation statistics for the stochastic model in Sect. 5 are for the vicinity of a droplet. Discussion in Appendix A of Vaillancourt et al. (2001) is pertinent to this issue.

for real DNS ofStandard deviation of supersaturation fluctuations for

Number of super-droplets and their multiplicity for real DNS domains of volume

In general, the multiplicity value should be decided on carefully because too large multiplicity results in too many grid boxes without droplets when compared to real droplets, and this may cause undesirable effects in the mean supersaturation and its spatial variability. In the two DNS cases, slight deviations in the mean supersaturation are present, although the simulations are not long enough to document the impact with confidence. For the scaled-up DNS, the number of droplets is in billions, and we have to select higher multiplicity values to make computations feasible.

The evolution of the radius squared (

Evolutions of the radius squared standard deviation (

After the super-droplet technique is tested in DNS, the same method is used in scaled-up DNS. In general, one may expect that if the multiplicity is increased beyond a certain value, the results will start to deviate from those with a low multiplicity featuring a larger number of super-droplets. However, high multiplicity is desirable to reduce the number of super-droplets that need to be followed. For the DNS, the number of super-droplets was shown to be relatively low to maintain similarity between real droplet and super-droplet solutions (see Figs. 4 and 5; as low as one super-droplet in a few dozen grid volumes). With scaled-up DNS, one might expect a different requirement because of a stronger local forcing of the supersaturation due to higher TKE and thus larger vertical velocities.

For the scaled-up DNS study, we apply a

Details of DNS and scaled-up DNS. From left to right: domain length

Figures 6 and 7 present evolutions of the mean supersaturation and standard deviation of its spatial distribution for the scaled-up simulations from Table 2. The five scaled-up domains shown in the table and figures correspond to the domain size

Evolution of the mean supersaturation for various scaled-up domains. Colors represent different domain sizes; different line styles correspond to different multiplicities. The additional simulation of 10 super-droplets per grid volume is only shown for

Evolution of standard deviation of supersaturation fluctuations for different domain sizes. Colors represent different domain sizes; different line styles correspond to different multiplicities. The additional simulation of 10 super-droplets per grid volume is only shown for

To further study the impact of the multiplicity, additional scaled-up DNS simulations are run with

Number of super-droplets and multiplicity for different

Results obtained from these simulations are shown in Fig. 8, with some results already shown in Fig. 7. As the figure shows, only the largest multiplicity with a super-droplet in 1 out of 50 grid volumes differs significantly from other simulations. The highest multiplicity simulation also results in the non-zero mean supersaturation (not shown). Note that for real DNS (Table 2 and Fig. 4), having a droplet in one of several dozens of grid volumes still results in supersaturation fluctuations in agreement with real droplets. This suggests that the maximum multiplicity that can be used in scaled-up DNS depends on the domain size. This perhaps should not be surprising because the magnitude of the vertical velocity perturbation and thus the supersaturation forcing increases with the domain size. Results for the largest domain considered in the current study (

Standard deviation of supersaturation fluctuations for the

As shown in Fig. 9, evolutions of the radius squared standard deviation

Evolutions of the radius squared standard deviation

We apply the stochastic model similar to that in

The stochastic model used here applies 1000 realizations, each starting from a random velocity perturbation (i.e.,

Figure 10 shows the standard deviation of supersaturation fluctuations (

Standard deviation of supersaturation fluctuations in DNS and scaled-up DNS (color circles) and the stochastic model (black stars). Vertical lines for the stochastic model represent variability among individual realizations. For DNS and scaled-up DNS, the variability comes from different multiplicity for super-droplets; it is not shown as it is smaller than the symbol size. Red circles in the left half are for DNS with three different multiplicities as in Fig. 4. Blue circles are for

Overall, the stochastic model seems to reasonably represent the scale dependence of the supersaturation fluctuations. At small scales (i.e.,

There are a few reasons for the discrepancy between the stochastic model and scaled-up DNS. First, the stochastic model uses TKE obtained from the scaled-up DNS. However, scaled-up DNS features reduced TKE when compared to the real DNS as documented in Sect. 3. Allowing more TKE on input for the stochastic model would shift the stochastic model results upwards, that is, closer to the scaled-up DNS. But increasing the Reynolds number in the scaled-up DNS increases

This study presents a novel modeling methodology that extends the traditional technique to simulate homogeneous isotropic turbulence, the direct numerical simulation (DNS). DNS is typically used for small-scale simulations applying grid lengths on the order of the Kolmogorov microscale, that is, about a millimeter for typical levels of atmospheric turbulence. Such a choice allows proper dissipation of the turbulent kinetic energy (TKE) that cascades through the inertial range from large scales where TKE is introduced. To reach domain sizes of about 1 m

This paper presents such an approach. The key idea is simple: rather than assuming that the dynamic model grid length is the Kolmogorov microscale

For simulations targeting growth of cloud droplets in homogeneous isotropic turbulence, the scaled-up DNS faces the problem of a large number of droplets that need to be followed inside the computational domain. For instance, a cube volume with

The scaled-up DNSs starting from unimodal droplet distribution with no mean ascent

Finally, we also consider supersaturation fluctuations in a simple stochastic model of a droplet ensemble

The scaled-up DNS methodology presented here was developed with diffusional growth of cloud droplets in mind. The next step can be to apply this approach in a rising parcel simulations as in

Data supporting the study are available at

LT ran simulations and performed data analysis under the supervision of GW and BK. GW and LT developed the idea of scaled-up DNS. All three authors were involved in preparing the manuscript. BK helped in accessing and using the HPC system of IITM that was used to run the DNS.

The authors declare that they have no conflict of interest.

DNS and scaled-up DNS were performed on HPC system Aaditya at the Indian Institute of Tropical Meteorology, Pune, India. Bipin Kumar is grateful for support from the Ministry of Earth Sciences, Government of India, for providing HPC facilities to conduct the required simulations for this work. Lois Thomas is grateful for NCAR's Advanced Study Program and Mesoscale and Microscale Meteorology Laboratory support for her 6-month visit to NCAR during which most of the research described here was completed. NCAR is sponsored by the National Science Foundation. Part of this work is supported by the National Center of Meteorology, Abu Dhabi, UAE, under the UAE Research Program for Rain Enhancement Science. The lead author is grateful for fruitful interactions with Sisi Chen, Peter Sullivan, Gustavo C. Abade, and Lian-Ping Wang during the course of this work.

This research has been partially supported by the U.S. Department of Energy, Atmospheric System Research (grant no. DE-SC0020118; PI: Wojciech W. Grabowski) and by the Ministry of Earth Sciences, Government of India.

This paper was edited by Timothy Garrett and reviewed by three anonymous referees.