Orientation dynamics of the ice crystal in a cloud: Effects of Turbulence and Electric Field

We investigate the orientation dynamics of an ice crystal in homogeneous isotropic turbulence and in the presence of an external electric field in a cloud. At the scale of the ice crystal, we assume that viscous effects dominate the flow, and thus, the dynamics can be studied in the Stokesian regime. Further, when the size of the ice crystal is smaller than the Kolmogorov scale, the flow field around the particle can be modeled locally as a stochastic linear flow. This approximation becomes particularly useful when studying the orientation dynamics of an ice crystal in homogeneous isotropic turbulence and when the orientation dynamics of the ice crystal is governed by the Jeffery equation, which involves the local fluctuating velocity gradient. The turbulent velocity gradient is obtained from the stochastic model given by Girimaji and Pope. The model uses the log-normal distribution of the pseudo-dissipation rate. In the presence of an external electric field, experiments performed in a laboratory cold chamber have revealed that the ice crystal aligns in the direction of the electric field. We study the competition due to the torque induced by the turbulent velocity gradient and the electric field. The orientation dynamics is analyzed by varying a non-dimensional parameter Σ, which is defined as a ratio of the Kolmogorov time scale and the electric relaxation time scale. For lower values of Σ, we show that the ice crystal exhibits an isotropic orientational distribution, whereas it fluctuates along the direction of the electric field at higher values of Σ. We calculate moments of the orientation distribution at large electric field limits using asymptotic methods and compare them with numerical calculations. A second-order moment in the orientation, which quantifies the fluctuations in the orientation, depends on Σ and the shape of an ice crystal. The fourth-order moment of the orientation, a measure of the non-Gaussian statistics of the orientation distribution, increases from its Gaussian value with the increase in Taylor-scale Reynolds number.