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1. A Formula for the Maximum Vertical Velocity in Supercell Updrafts

   https://doi.org/10.1175/JAS-D-20-0103.1  

   Peters, John M.; Morrison, Hugh; Nowotarski, Christopher J.; Mulholland, Jake P.; Thompson, Richard L. (November 2020, Journal of the Atmospheric Sciences)

   null (Ed.)

   Abstract In supercell environments, previous authors have shown strong connections between the vertical wind shear magnitude, updraft width, and entrainment. Based on these results, it is hypothesized that the influences of entrainment-driven dilution on buoyancy and maximum updraft vertical velocity w in supercell environments are a predictable function of the vertical wind shear profile. It is also hypothesized that the influences of pressure perturbation forces on maximum updraft w are small because of a nearly complete offset between upward dynamic pressure forces and downward buoyant pressure forces. To address these hypotheses, we derive a formula for the maximum updraft w that incorporates the effects of entrainment-driven dilution on buoyancy but neglects pressure gradient forces. Solutions to this formula are compared with output from previous numerical simulations. This formula substantially improves predictions of maximum updraft w over past CAPE-derived formulas for maximum updraft w , which supports the first hypothesis. Furthermore, integrated vertical accelerations along trajectories show substantial offsets between dynamic and buoyant pressure forces, supporting the second hypothesis. It is argued that the new formula should be used in addition to CAPE-derived measures for w in forecast and research applications when accurate diagnosis of updraft speed is required. 

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2. Cumulonimbus Clouds Convert a Smaller Fraction of CAPE into Kinetic Energy in a Warmer Atmosphere

   https://doi.org/10.1175/JAS-D-24-0006.1  

   Peters, John M; Chavas, Daniel R; Lebo, Zachary J; Su, Chun-Yian (November 2024, Journal of the Atmospheric Sciences)

   Abstract This study investigates how entrainment’s diluting effect on cumulonimbus updraft buoyancy is affected by the temperature of the troposphere, which is expected to increase by the end of the century. A parcel model framework is constructed that allows for independent variations in the temperature (T), the entrainment rateε, the free-tropospheric relative humidity (RH), and the convective available potential energy (CAPE). Using this framework, dilution of buoyancy is evaluated withTand RH independently varied and with CAPE either held constant or increased with temperature. When CAPE is held constant, buoyancy decreases asTincreases, with parcels in warmer environments realizing substantially smaller fractions of their CAPE as kinetic energy (KE). This occurs because the increased moisture difference between an updraft and its surroundings at warmer temperatures drives greater updraft dilution. Similar results are found in midlatitude and tropical conditions when CAPE is increased with temperature. With the expected 6%–7% increase in CAPE per kelvin of warming, KE only increases at 2%–4% K⁻¹in narrow updrafts but tracks more closely with CAPE at 4%–6% in wider updrafts. Interestingly, the rate of increase in the KE withTbecomes larger than that of CAPE when the later quantity increases at more than 10% K⁻¹. These findings emphasize the importance of considering entrainment in studies of moist convection’s response to climate change, as the entrainment-driven dilution of buoyancy may partially counteract the influence of increases in CAPE on updraft intensity. Significance StatementCumulonimbus clouds mix air with their surrounding environment through a process called entrainment, which controls how efficiently environmental energy is converted into upward speed in thunderstorm updrafts. Our research shows that warmer temperatures will exacerbate the moisture difference between cumulonimbus updrafts and their surroundings, leading to greater mixing and less efficient conversion of environmental energy into updraft speeds. This effect should be considered in future research that investigates how climate change will affect cumulonimbus clouds. 

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3. Generalized Lapse Rate Formulas for Use in Entraining CAPE Calculations

   https://doi.org/10.1175/JAS-D-21-0118.1  

   Peters, John M.; Mulholland, Jake P.; Chavas, Daniel R. (March 2022, Journal of the Atmospheric Sciences)

   Abstract Common assumptions in temperature lapse rate formulas for lifted air parcels include neglecting mixing, hydrostatic balance, the removal of all condensate once it forms (pseudoadiabatic), and/or the retention of all condensate within the parcel (adiabatic). These formulas are commonly derived from the conservation of entropy, which leads to errors when nonequilibrium mixed-phase condensate is present. To evaluate these assumptions, a new general lapse rate formula is derived from an expression for energy conservation, rather than entropy conservation. This new formula incorporates mixing of the parcel with its surroundings, relaxes the hydrostatic assumption, allows for nonequilibrium mixed-phase condensate, and can be formulated for pseudoadiabatic or adiabatic ascent. The new formula is shown to exactly conserve entropy for reversible ascent. Predictions by the new formula are compared to that of older and less general formulas. The errors in previous formulas arise from the assumption of hydrostatic balance, which results in considerable warm biases due to the neglect of the energy sink from buoyancy. Predictions of ascent with entrainment using the new formula are then compared to parcel properties along trajectories in large eddy simulations. Simulated parcel properties are better predicted by the formula using a diluted analogy to adiabatic ascent, wherein condensate is diluted at the same rate as other parcel properties, than by the diluted analogy to pseudoadiabatic ascent, wherein all condensate is removed. These results suggest that CAPE should be computed with adiabatic, rather than pseudoadiabatic, parcel ascent. 

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4. The Influence of Shear on Deep Convection Initiation. Part II: Simulations

   https://doi.org/10.1175/JAS-D-21-0144.1  

   Peters, John M.; Morrison, Hugh; Nelson, T. Connor; Marquis, James N.; Mulholland, Jake P.; Nowotarski, Christopher J. (June 2022, Journal of the Atmospheric Sciences)

   Abstract This study evaluates a hypothesis for the role of vertical wind shear in deep convection initiation (DCI) that was introduced in Part I by examining behavior of a series of numerical simulations. The hypothesis states, “Initial moist updrafts that exceed a width and shear threshold will ‘root’ within a progressively deeper steering current with time, increase their low-level cloud-relative flow and inflow, widen, and subsequently reduce their susceptibility to entrainment-driven dilution, evolving toward a quasi-steady self-sustaining state.” A theoretical model that embodied key elements of the hypothesis was developed in Part I, and the behavior of this model was explored within a multidimensional environmental parameter space. Remarkably similar behavior is evident in the simulations studied here to that of the theoretical model, both in terms of the temporal evolution of DCI and in the sensitivity of DCI to environmental parameters. Notably, both the simulations and theoretical model experience a bifurcation in outcomes, whereby nascent clouds that are narrower than a given initial radius R 0 threshold quickly decay and those above the R 0 threshold undergo DCI. An important assumption in the theoretical model, which states that the cloud-relative flow of the background environment V CR determines cloud radius R , is scrutinized in the simulations. It is shown that storm-induced inflow is small relative to V CR beyond a few kilometers from the updraft edge, and V CR therefore plays a predominant role in transporting conditionally unstable air to the updraft. Thus, the critical role of V CR in determining R is validated. 

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5. Hopper flows of deformable particles

   https://doi.org/10.1039/d2sm01079h  

   Cheng, Yuxuan; Treado, John D.; Lonial, Benjamin F.; Habdas, Piotr; Weeks, Eric R.; Shattuck, Mark D.; O'Hern, Corey S. (November 2022, Soft Matter)

   Numerous experimental and computational studies show that continuous hopper flows of granular materials obey the Beverloo equation that relates the volume flow rate Q and the orifice width w : Q ∼ ( w / σ avg − k ) β , where σ avg is the average particle diameter, kσ avg is an offset where Q ∼ 0, the power-law scaling exponent β = d − 1/2, and d is the spatial dimension. Recent studies of hopper flows of deformable particles in different background fluids suggest that the particle stiffness and dissipation mechanism can also strongly affect the power-law scaling exponent β . We carry out computational studies of hopper flows of deformable particles with both kinetic friction and background fluid dissipation in two and three dimensions. We show that the exponent β varies continuously with the ratio of the viscous drag to the kinetic friction coefficient, λ = ζ / μ . β = d − 1/2 in the λ → 0 limit and d − 3/2 in the λ → ∞ limit, with a midpoint λ c that depends on the hopper opening angle θ w . We also characterize the spatial structure of the flows and associate changes in spatial structure of the hopper flows to changes in the exponent β . The offset k increases with particle stiffness until k ∼ k max in the hard-particle limit, where k max ∼ 3.5 is larger for λ → ∞ compared to that for λ → 0. Finally, we show that the simulations of hopper flows of deformable particles in the λ → ∞ limit recapitulate the experimental results for quasi-2D hopper flows of oil droplets in water. 

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