What Controls the Entrainment Rate of Dry Buoyant Thermals with Varying Initial Aspect Ratio?

Hugh Morrison aNational Center for Atmospheric Research, Boulder, Colorado

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Nadir Jeevanjee bGeophysical Fluid Dynamics Laboratory, Princeton, New Jersey

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Daniel Lecoanet cDepartment of Engineering Sciences and Applied Mathematics, Northwestern University, Evanston, Illinois

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John M. Peters dDepartment of Meteorology and Atmospheric Science, The Pennsylvania State University, University Park, Pennsylvania

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Abstract

This study uses theory and numerical simulations to analyze the nondimensional spreading rate α (change in radius with height) of buoyant thermals as they rise and entrain surrounding environmental fluid. A focus is on how α varies with initial thermal aspect ratio Ar, defined as height divided by width of the initial buoyancy perturbation. An analytic equation for thermal ascent rate wt that depends on α is derived from the thermal-volume-averaged momentum budget equation. The thermal top height when wt is maximum, defining a critical height zc, is inversely proportional to α. The height zc also corresponds to the thermal top height when buoyant fluid along the thermal’s vertical axis is fully replaced by entrained nonbuoyant environmental fluid rising from below the thermal. The time scale for this process is controlled by the vertical velocity of parcels rising upward through the thermal’s core. This parcel vertical velocity is approximated from Hill’s analytic spherical vortex, yielding an analytic inverse relation between α and Ar. Physically, this αAr relation is connected to changes in circulation as Ar is modified. Numerical simulations of thermals with Ar varied from 0.5 to 2 give α values close to the analytic theoretical relation, with a factor of ∼3 decrease in α as Ar is increased from 0.5 to 2. The theory also explains why α of initially spherical thermals from past laboratory and modeling studies is about 0.15. Overall, this study provides a theoretical underpinning for understanding the entrainment behavior of thermals, relevant to buoyantly driven atmospheric flows.

Significance Statement

Thermals, which are coherent, quasi-spherical regions of upward-moving buoyant fluid, are a key feature of many convective atmospheric flows. The purpose of this study is to characterize how thermals entrain surrounding fluid and spread out as they rise. We use theory and numerical modeling to explain why entrainment rate decreases with an increase in the initial thermal aspect ratio—the ratio of height to width. This work also explains why the nondimensional spreading rate (change in thermal radius with height) of initially spherical thermals from past laboratory and numerical modeling studies is about 0.15. Overall, this work provides a framework for conceptualizing the entrainment behavior of thermals and thus improved understanding of vertical transport in convective atmospheric flows.

© 2023 American Meteorological Society. This published article is licensed under the terms of the default AMS reuse license. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Hugh Morrison, morrison@ucar.edu

Abstract

This study uses theory and numerical simulations to analyze the nondimensional spreading rate α (change in radius with height) of buoyant thermals as they rise and entrain surrounding environmental fluid. A focus is on how α varies with initial thermal aspect ratio Ar, defined as height divided by width of the initial buoyancy perturbation. An analytic equation for thermal ascent rate wt that depends on α is derived from the thermal-volume-averaged momentum budget equation. The thermal top height when wt is maximum, defining a critical height zc, is inversely proportional to α. The height zc also corresponds to the thermal top height when buoyant fluid along the thermal’s vertical axis is fully replaced by entrained nonbuoyant environmental fluid rising from below the thermal. The time scale for this process is controlled by the vertical velocity of parcels rising upward through the thermal’s core. This parcel vertical velocity is approximated from Hill’s analytic spherical vortex, yielding an analytic inverse relation between α and Ar. Physically, this αAr relation is connected to changes in circulation as Ar is modified. Numerical simulations of thermals with Ar varied from 0.5 to 2 give α values close to the analytic theoretical relation, with a factor of ∼3 decrease in α as Ar is increased from 0.5 to 2. The theory also explains why α of initially spherical thermals from past laboratory and modeling studies is about 0.15. Overall, this study provides a theoretical underpinning for understanding the entrainment behavior of thermals, relevant to buoyantly driven atmospheric flows.

Significance Statement

Thermals, which are coherent, quasi-spherical regions of upward-moving buoyant fluid, are a key feature of many convective atmospheric flows. The purpose of this study is to characterize how thermals entrain surrounding fluid and spread out as they rise. We use theory and numerical modeling to explain why entrainment rate decreases with an increase in the initial thermal aspect ratio—the ratio of height to width. This work also explains why the nondimensional spreading rate (change in thermal radius with height) of initially spherical thermals from past laboratory and numerical modeling studies is about 0.15. Overall, this work provides a framework for conceptualizing the entrainment behavior of thermals and thus improved understanding of vertical transport in convective atmospheric flows.

© 2023 American Meteorological Society. This published article is licensed under the terms of the default AMS reuse license. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Hugh Morrison, morrison@ucar.edu
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