Parameterizing Explicit 3D Hydrometeor Momentum in Clouds

Two common assumptions when dealing with hydrometeors in the atmosphere is that they immediately fall at their terminal velocity and always advect horizontally with the wind. A drag force that is proportional to the square of the relative velocity of hydrometeors and air was used for the equations of condensate motion in a simplified atmosphere containing just friction and gravity to better understand the consequences of those assumptions. We show that the velocity evolution of either hydrometeors flowing through a prescribed velocity field or air accelerated by hydrometeors can be solved explicitly. These formal solutions match numerical simulations well. For small hydrometeors, like cloud droplets, the above assumptions are valid, as equilibration occurs in less than a second in both the horizontal and vertical. Thus, not much is lost during a short time step where the hydrometeor velocity is then set to the air or terminal velocity. However, for large hydrometeors, like large raindrops and hailstones, the equilibration time can be more than ten seconds in the vertical, and about twice that in the horizontal, which means that models run with short time steps need to consider the acceleration of these hydrometeors in all direction. Next, we show that coupled equations can also be solved. The coupled hydrometeor-air velocity solutions will be used to contextualize hydrometeor loading. Finally, we describe how an accurate bulk parameterization usable in models can be developed.