Idealized Cumulus Cloud-Scale Motions and the Dynamics of Isolated and Coupled Flows

TL;DR

This work introduces the KRoNUT model as a low-dimensional, kinematic representation of cloud-scale updrafts and develops the DoNUT reduced-order dynamics via circulation-density moment reduction. By performing relaxation experiments that remove baroclinic forcing, it reveals that isolated clouds evolve toward a stable fixed circulation set by turbulent diffusion, with a fixed-line in the dimensionless phase space given by $(\alpha^*,R^*) \approx (2.205,11.845)$ and $\Gamma^* \approx 26.24\nu$. It further shows that cross-interactions between neighboring clouds can either attract or repel them depending on geometry and separation, and provides a scalable framework for incorporating cloud-scale dynamics into parameterizations. The methodology, applicable to additional forcings and more complex multi-cloud configurations, offers a path toward improved, physically grounded convection schemes.

Abstract

Developing an understandable theory for the dynamic evolution of the morphology of clouds remains intractable. To break this deadlock, we introduce a new conceptual model for cloud-scale motions named the Kinematics Representation of Non-rotating Updraft Tori (KRoNUT) model, where non-rotating reflects the absence of motion in the azimuthal direction. Using this model, we conduct a series of relaxation experiments whereby we ``turn off'' the baroclinic term associated with a pre-existing cloud-scale circulation. We then implement a moment reduction technique to generate a system of differential equations named the Dynamics of Non-rotating Updraft Tori (DoNUT) equations, which describe the temporal evolution of a cloudy circulation under various combinations of forcings, namely turbulent diffusion, self-advection, and cross-advection from a neighboring cloud-scale flow. The solutions of the DoNUT equations show that all single KRoNUT configurations either start at or evolve toward a specific steady state circulation. The cloud-scale motions represented by the current KRoNUT model always grow vertically but may narrow, due to advection, or widen, due to diffusion. Meanwhile, invigoration or enervation of the vertical velocity may result from advection or diffusion processes, with short, wide KRoNUTs more likely to invigorate and tall, narrow KRoNUTs likely to enervate. Our study of the coupled KRoNUTs provides insight into clouds' tendencies to attract or repel one another. Important results of the coupled KRoNUT analysis include a scaled metric for interaction, ranges of specific height ratios that induce the most meaningful interaction, and circulation parameters that alter the location and stability of a steady KRoNUT.

Figures (14)

• Figure 1: A simple schematic illustrating KRoNUT parameters and their relationship to coupled cloud-scale motions. Here, a cloud's streamlines are indicated by navy blue or red, the geometric KRoNUT parameters, $L$ and $H$ are denoted by orange and cyan dashed lines, respectively, while a KRoNUT's intensity parameter, $w_*$, is represented by the magenta arrow. The separation between KRoNUTs, $d$, is marked by a solid yellow line, and a dashed green line represents the vorticity line at the $r=L$ and $z=H$.
• Figure 2: Example KRoNUT fields. The left column shows the $y=0$ cross-sections of the streamlines. The right column shows the $z=H$ cross-sections of the vortex lines. The first two rows are individual KRoNUT circulations, and the bottom row shows the superposition KRoNUTs at a given separation. In all plots, color indicates the magnitude and field line lengths are arbitrary.
• Figure 3: Horizontal slices of the structural change in a KRoNUT's velocity field at an instant in time for various heights. Here, the left column corresponds to the vertical velocity, and the right column coincides with the horizontal velocity. The initial conditions for this KRoNUT are $w_{*g}=3 m s^{-1}$, $L_g=1000 m$, $H_g=1000m$, $x_g=0m$, and $y_g=0m$.
• Figure 4: Three-hour trajectory evolutions of the Sh. cloud type test cases projected onto the $\alpha_n$-$R_n$ subspace of the phase space. Here the pink-shaded quadrilateral corresponds to the $\alpha_n$-$R_n$ region of the phase space occupied by Sh. convection.
• Figure 5: SK explicit temporal evolution of implicit and explicit parameters and quantities of interest associated with the Sh. cloud-type test cases.