A Lagrangian approach to droplet condensation in turbulent clouds

1. A Lagrangian approach to droplet condensation in turbulent clouds Rutger IJzermans, Michael W. Reeks School of Mechanical & Systems Engineering Newcastle University, United Kingdom Ryan Sidin Department of Mechanical Engineering University of Twente, the Netherlands

2. Objective and motivation • Research question: How does turbulence influence • condensational growth of droplets? • Application: Rain initiation in atmospheric clouds • Objectives: - Gain understanding of rain initiation process, from cloud condensation nuclei to rain droplets - Elucidate role of turbulent macro-scales and micro-scales on condensation of droplets in clouds

3. Background: scales in turbulent clouds Turbulence: Large scales: L0~ 100 m, t0~ 103 s, u0~ 1 m/s, Small scales: h~ 1 mm, tk~ 0.04 s, uk~ 0.025 m/s. Droplets: Radius: Inertia: Settling velocity: Formation: rd~ 10-7 m, St = td/tk~ 2 × 10-6, vT/uk~ 3 × 10-5 Microscales: rd~ 10-5 m, St = td/tk~ 0.02, vT/uk~ 0.3 Rain drops: rd~ 10-3 m, St = td/tk~ 200, vT/uk~ 3000 CONDENSATION COLLISIONS / COALESCENCE Collisions / coalescence process vastly enhanced if droplet size distribution at micro-scales is broad

4. Droplet size distribution at microscales O(h) Twomey’s fluid parcel approximation is not allowed in turbulence Classic theory (Twomey (1959); Shaw (2003)): Fluid parcel, filled with many droplets of different sizes If parcel rises, temperature decreases due to adiabatic expansion, and supersaturation s increases: Droplet growth is given by: or: Size distribution PDF(rd) becomes narrower in time! • Problem: • Droplet size distribution in reality (experiments) becomes broader

5. Numerical model for condensation in cloud • Ideally, Direct Numerical Simulation of: • Velocity and pressure fields (Navier-Stokes) • Supersaturation and temperature fields • Computationally too expensive: O(L0/h)3~ 1015 cells • State-of-the-art DNS: 5123 modes, L0~ 70cm (Lanotte et al., J. Atm. Sci. (2008)) • Cloud turbulence modelled by kinematic simulation: • All relevant flow scales can be incorporated by choosing kn of appropriate length • Turbulent energy spectrum required as input

6. Energy spectrum in wavenumber space

7. Full condensation model Latent heat release Adiabatic cooling Vapour depletion Droplet modelled as passive tracer, contained within a moving air parcel: Along its trajectory (Lagrangian): Mixture of air & water vapour - rate-of-change of droplet mass md: volume Vp - rate-of-change of mixture temperature T: - rate-of-change of supersaturation s:

8. Simplified condensation model Air & water vapor Track droplets as passive tracers: Rate-of-change of droplet mass md: Temperature T and supersaturation s are assumed to depend on adiabatic cooling only:

9. Typical supersaturation profile Imposed mean temperature and supersaturation profiles: Focus on regions where supersaturation is close to zero

10. Computational strategy • Determine droplet size distribution: • Droplet population (Nd=8000) initially randomly • distributed in a plane at time t = te and height ze: • Droplet trajectories traced backward in time: t = te 0 • At t = 0 a monodisperse distribution is assumed: rd (0) = r0 = 10-7m • Condensation model equations are integrated forward in time to obtain droplet size distribution in the plane at t = te

11. Results: dispersion in 3D KS-flow field Flight of 2 particles initially separated by distance r0=h: Slope = 4.5, similar to [Thomson & Devenish, J.F.M. 2005] 1-particle statistics: Short times: <|x – x0|2> ~t2 Long times: <|x – x0|2> ~t 2-particle statistics: In agreement with Taylor (1921)

12. Time evolution of droplet position and size Forward tracing: t = 0  te Backward tracing: t = te 0 ze = 1355 m ; te =100 s ; size of sampling area = 1 x 1 cm2

13. Droplet evaporation in regions where s < 0 Number of droplets at various altitudes: Rapid initial evaporation Forward tracing: t = 0  tc

14. Droplet radius distributions in time Temporal evolution of radius distribution function (z0 = 1355 m):

15. Radius distributions after te = 100 s Influence of measurement altitude:(size of sampling area L = 500 m) Influence of sampling area width: (ze = 1350 m)

16. Effect of different scales in turbulence Droplet radius distribution in flow with:- Only large scales included (n=1-10)- Only small scales included(n=191-200)- Wide range of scales included(n=1-200) ze = 1350m, te = 100s, L = 0.01m

17. Results for two-way coupled model Eulerian evolution of droplet size distribution for nd = (5 η)-3 = 8.0 x 106 m-3: ze = 1350 m ze = 1380m

18. Results two-way coupled model: interpretation Equation for supersaturation s is: with: This can be rewritten into: Saturation of droplet radius distribution functionfollows from a balance between: - Adiabatic expansion (“forcing”)- Vapour depletion (“damping” with time scale ts)- Latent heat release (“damping” with time scale tL)

19. Results two-way coupled model: interpretation Dependence of vapour depletiontime scale ts on droplet radius rdand on droplet number density nd: Relative importance of the two damping terms, ts/tL,as a funciton of temperature:

20. Results two-way coupled model Influence of droplet number density nd: te = 100s,ze = 1350m,L=500m

21. Results two-way coupled model Influence of length of the sampling area L: te=100s,ze = 1350m,nd=(2h)-3 = 0.125 x 109 m-3

22. Conclusions • Droplet size distribution may become broader during condensation: • - Large scales of turbulent motion responsible for transport of droplets • to different regions of the flow, with different supersaturations • - Small scales of turbulent motion responsible for local mixing of large and small droplets • Broad droplet size distribution observed both in simplifiedcondensation model and in two-way coupled condensation model • Broadening of droplet size distribution enhanced by:- Higher flow velocities (more vigourous turbulence)- Lower droplet number density - Lower surrounding temperatures