Electrohydrodynamic generation of atmospheric turbulence

Ionization produced by cosmic rays and atmospheric radioactivity creates charged short life-time aerosol droplets in the upper troposphere. Inhomogeneities in the spatial distribution of aerosol droplets lead to time-varying electric fields and space charge, which can often be amplified by more than three orders of magnitude in extreme conditions, such as thunderstorms. The nonlinear coupling between ionized air, charged aerosol droplets, and the background electric field can result in electrohydrodynamic body forces that augment atmospheric turbulence. In this paper, a theoretical and numerical study on the electrohydrodynamic generation of atmospheric turbulence under fair weather and thunderstorm conditions is presented. Linear stability shows that coupling between ionized air and a background electric field acts to increase turbulent kinetic energy (TKE) in the upper troposphere, albeit over long time durations. Direct simulations of charged droplets in homogeneous shear flow demonstrate a nonlinear feedback mechanism capable of accelerating the growth rate. Streamwise velocity gradients induce fluctuations in droplet concentration and electric potential, resulting in a body force that generates vertical velocity fluctuations. Pressure strain then transfers this energy to turbulent fluctuations in the streamwise direction and the process repeats. This feedback mechanism was found to augment TKE at late stages of the shear layer growth.

Figure 1

Fair weather atmospheric conditions [37] used to define the background state in the stability analysis.

Figure 2

Critical value of the electric Rayleigh number as a function of vertical span $L$ for different shear Reynolds numbers $S$. The system is unconditionally stable when $L\le 0.707$ km (denoted by the vertical dashed line).

Figure 3

Normalized growth of vertical velocity fluctuations under varying shear rates predicted by linear stability analysis under fair weather (solid fill) and thunderstorm (hatch fill) conditions.

Figure 4

Simulation configuration for ${\mathrm{\Phi }}_v=1\times {10}^{-4}$. Grayscale shows fluid velocity magnitude, ranging from 0 (black) to its maximum value (white). Particles are colored by charge: positive (red) and negative (blue).

Figure 5

Temporal evolution of the gas and particle phases with volume fraction increasing from top to bottom and time increasing from left to right. Color scheme same as Fig. 4.

Figure 6

Normalized TKE evolution for particle loading ${\mathrm{\Phi }}_v=0$ (–), $1\times {10}^{-4}$ ($\cdots$), $4\times {10}^{-4}$ ($-$-), $1\times {10}^{-3}$ ($-$), and $2\times {10}^{-3}$ (- -).

Figure 7

Evolution of the Reynolds stresses normalized by $w_0^2$ for (a) ${\mathrm{\Phi }}_v=0$, (b) $1\times {10}^{-4}$, (c) $4\times {10}^{-4}$, and (d) $1\times {10}^{-3}$.

Figure 8

Budget of (a) $\langle w^{\prime }{w}^{\prime }\rangle$, (b) $\langle u^{\prime }{w}^{\prime }\rangle$, and (c) $\langle u^{\prime }{u}^{\prime }\rangle$ normalized by $\mathrm{\Gamma }{w}_0^2$ for ${\mathrm{\Phi }}_v=1\times {10}^{-4}$. (d) $\langle u^{\prime }{u}^{\prime }\rangle$ normalized by $\mathrm{\Gamma }{w}_0^2$ in the absence of particles.

Figure 9

Energy transfer due to three-way coupling between turbulence, charged particles, and a background electric field. The dashed lines represent energy transfer when aerosol particles are present. Fluctuations in vertical velocity and space charge give rise to an EHD source term, ${\mathcal{E}}_{33}$, that produces TKE. In the absence of charged aerosol particles, TKE is eventually dissipated to heat. When particles are present, preferential concentration by the turbulence generates inhomogeneities in particle distribution and consequently gradients in electric potential. This further contributes to the EHD source term ${\mathcal{E}}_{33}$ in addition to ${\mathcal{E}}_{13}$ that generates TKE via mean gradient production.

Figure 10

Normalized TKE evolution for particle loading ${\mathrm{\Phi }}_v=0$ (–), $1\times {10}^{-4}$ ($\cdots$), $4\times {10}^{-4}$ ($-$-), $1\times {10}^{-3}$ ($-$), and $2\times {10}^{-3}$ (- -). Theoretical solution obtained from Eq. (A1) ($\circ$) is shown for comparison. The vertical dashed line indicates the turning point from upstream and downstream waves.