Numerical Simulation cloud-clear air interfaces

Direct numerical simulations (DNS) of a decaying turbulent layer at a cloud top reveal enhanced diffusivity in the mixing region between cloudy and clear air. In this research, we build upon the foundational work of Lewis F. Richardson (1926), who introduced a distance-neighbour distribution function, Q(ℓ, t), to describe the dispersion of particles over a separation distance ℓ in turbulent flows. Due to its definition, the evolution of Q can be modeled by a diffusion-type equation:

∂Q/∂t = ∂/∂ℓ ( F(ℓ, t) ∂Q/∂ℓ )

A central challenge is identifying the correct form of the diffusion function F(ℓ, t). Richardson originally proposed a scaling of the form F(ℓ) ~ ℓ^4/3, which laid the foundation for the well-known Richardson–Obukhov law (ℓ² ~ t³) applicable in homogeneous, isotropic turbulence. This theory has been extensively revisited and refined, including significant contributions by Obukhov and others.

Our work extends this classical framework into a more complex and realistic setting through direct numerical simulations (DNS) of a time-decaying turbulent, shear-free layer. This simulated domain represents a localized segment of a top warm cloud boundary, modeled as a multiphase system involving air, water vapor, and liquid water droplets.

A key focus is the interfacial region between the cloud and surrounding clear air—a highly anisotropic and intermittently turbulent zone. In this region, we anticipate that the diffusion function F should reflect the unique microphysical and turbulent properties at play. Based on our DNS data, we propose two alternative scaling laws for F(ℓ, t), each incorporating relevant thermodynamic and turbulence parameters: 

Proposed Scaling Laws

• Scaling Law 1: F(ℓ, t) ~ ε^2/3(t) <κ_s(t)> τ_p ℓ^2/3
• Scaling Law 2: F(ℓ, t) ~ ε^2/5(t) <κ_s(t)> τ_p^1/5 ℓ^6/5

Here:

ε(t): turbulent kinetic energy dissipation rate
<κ_s(t)>: kurtosis of supersaturation fluctuations
τ_p: phase relaxation time, characterizing droplet response to supersaturation

These scaling models are consistent in order of magnitude with values originally observed by Richardson and reported in historical boundary layer data (e.g., Schmidt 197, Akerblom 1908, Taylor 1914, Hesselberg & Sverdrup 1915).

Observed Parameter Ranges from Simulations

Within the temporal window t/τ₀ ∈ [1, 6], the DNS results show:

• Energy dissipation rate: ε(t) ∈ [10, 100] cm²/s³
• Supersaturation kurtosis: <κ_s(t)> ≈ 3 in the cloud; 15–30 in the mixing region
• Phase relaxation time: τ_p ≈ 10 in the cloud; 20–80 in the mixing region

Our results demonstrate a significant increase in turbulent diffusivity within the mixing region, aligning well with empirical observations from early atmospheric studies. This highlights the need to incorporate local microphysical variability in turbulence diffusion models, especially in cloud-edge regions where phase changes and thermodynamic fluctuations are prominent.

The figure below show the computed two turbulent diffusion trends and the large diffusion increase in the mixing region.