1. Introduction

Thermals—coherent, isolated, quasi-spherical regions of upward-moving buoyant fluid—are a common feature of convective atmospheric flows. A key characteristic of thermals is the rate at which they increase in size as they ascend owing to entrainment of the surrounding fluid. Assuming thermal shape is self-similar (meaning that thermals do not change shape over time), dimensional analysis shows that thermal radius R is proportional to thermal top height z_t, that is, dR/dz_t is constant¹ (e.g., Scorer 1957). (Note that all symbols used in the paper are defined in the appendix.) Numerous laboratory and numerical modeling studies have supported this basic scaling (e.g., Scorer 1957; Richards 1961; Bond and Johari 2005; Zhao et al. 2013; Lai et al. 2015; Lecoanet and Jeevanjee 2019, hereinafter LJ2019; McKim et al. 2020; Morrison et al. 2021).

The rate of increase in R is closely related to the entrainment rate of thermals. From LJ2019, a thermal net fractional entrainment rate is defined as ϵ ≡ d(lnV)/dz_t, where V is the thermal volume. Combined with self-similarity, this gives ϵ = 3α/R, where α ≡ dR/dz_t. We emphasize that ϵ in this case is a net fractional entrainment rate because thermal volume is impacted by both entrainment (inflow of environmental fluid) and detrainment (outflow of thermal fluid). However, LJ2019 showed that detrainment is negligible for both laminar and turbulent dry, initially spherical thermals in a neutrally stable environment. Thus, ϵ provides a close approximation for total entrainment in such conditions. An entrainment efficiency can also be defined as e ≡ ϵR, which gives e = 3α for self-similar thermals.

Thermals entrain by a process of drawing in fluid mainly from below the thermal (e.g., LJ2019; Zhao et al. 2013; McKim et al. 2020; Morrison et al. 2021). As a thermal spins up, buoyancy becomes concentrated near the center of rotation in the thermal’s toroidal circulation (i.e., vortex ring core); see Fig. 1 for a schematic of thermal structure. As a result, there is baroclinic generation of buoyancy on the outside edge of the vortex ring and destruction on the inside edge that lead to a spreading of the vortex (McKim et al. 2020). Moreover, without buoyant fluid present along the thermal’s vertical axis, circulation is nearly constant. This implies a basic constraint on the spreading rate of thermals following the principle of momentum conservation (Turner 1957). Specifically, buoyant vortex rings (which form the core of thermals) must expand over time to conserve momentum, with the rate of spread determined by the thermal-integrated buoyant forcing and the circulation. McKim et al. (2020) combined the buoyant vortex ring argument of Turner (1957) with the thermal’s vertical momentum equation to derive an analytic model for the vertical velocity of thermal top w_t, R, and buoyancy B at any time past spinup that does not rely on empirically determined parameters, provided w_t, R, and B are known at the time when the thermal is spun up.

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Schematic diagram of a vertical cross section through the thermal center. The central vertical axis is indicated by the dashed line. Red X symbols mark the center of circulation comprising the vortex ring core with radius R_υ. After spinup, the region of nonzero buoyancy indicated by blue shading is confined to the vortex ring core. Baroclinic generation and destruction of vorticity associated with this buoyancy structure leads to outward spreading of the vortex ring structure and thermal as a whole as shown by the red arrows. Black curved lines illustrate streamfunction isolines (only shown for the right half of the thermal). The thermal boundary, which is also a streamfunction isoline, is indicated by the curved blue line. This boundary also defines the thermal radius R.

Citation: Journal of the Atmospheric Sciences 80, 11; 10.1175/JAS-D-23-0063.1

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While the basic mechanism of thermal entrainment and spreading is well understood, factors controlling the spreading rate are not. Lai et al. (2015) combined a relation between circulation, impulse (related to time-integrated buoyant forcing), and thermal spreading rate with an empirical power-law relation between normalized circulation and initial thermal aspect ratio A_r to predict α from A_r. They showed that variations in A_r for spheroidal thermals from ∼0.5 to 2 lead to substantial variability in α, from about 0.1 to 0.3. These results are consistent with laboratory experiments reporting a similar range of α (e.g., Scorer 1957; Escudier and Maxworthy 1973; Bond and Johari 2005, 2010; Zhao et al. 2013). A consensus from laboratory and numerical modeling studies is that α ≈ 0.12–0.18 for initially spherical thermals in an unstratified environment (e.g., LJ2019; Bond and Johari 2010; Zhao et al. 2013; Lai et al. 2015). Values are ∼0.2–0.3 for initially oblate thermals with A_r < 1 and smaller for prolate thermals with A_r > 1, ∼0.1–0.15 (see Fig. 17 of Lai et al. 2015). There is little sensitivity of α to initial aspect ratio for A_r > 2 (Bond and Johari 2005). Modifying other aspects of thermal initial conditions can also produce variability in α, such as having an initial circulation (Escudier and Maxworthy 1973). Note that α may also depend on the Reynolds number R_e of the flow, although LJ2019 showed with direct numerical simulation (DNS) that the basic mechanism of entrainment is the same for laminar and turbulent thermals (R_e of 630 and 6300, respectively), and α was only ∼20% higher for turbulent thermals. Their results indicate that turbulence is not necessary for entrainment and that the primary mechanism for entrainment is organized inflow controlled by the thermal’s buoyancy distribution.

The above discussion raises two important science questions. First, why do initially spherical thermals in an unstratified environment (and initially motionless) have α ≈ 0.15? Why this particular value, and what are the physical mechanisms explaining it? Second, why does the spreading rate of thermals as they rise (α) increase as their initial aspect (A_r) is decreased? To our knowledge, all previous studies have relied in some way on empirical constraint to obtain parameters, from either laboratory experiments or numerical modeling, at least during the spinup phase which is crucial for predicting α. In this study, we derive an expression for α as a function of A_r that does not rely on such empirically determined parameters. The goal is to predict α from A_r from the basic equations to provide a theoretical underpinning for understanding factors controlling the thermal spreading rate. The predicted values of α are compared to those obtained from numerical simulations of thermals over a range of A_r.

In the theoretical part, we first derive an analytic expression for thermal ascent rate w_t from the nondimensional thermal momentum budget equation. We then use this expression to derive an analytic relation between α and the thermal spinup height z_c (defined as the thermal top height when w_t reaches a maximum), valid over a range of initial thermal aspect ratios. We show that z_c also corresponds to the time for parcels initially near the thermal bottom to ascend through the thermal core to near the thermal top. This determines the time for buoyancy to be removed from the central thermal core by entrainment of nonbuoyant environmental fluid, after which circulation is nearly constant. This time scale for thermal spinup depends linearly on A_r and is determined by the thermal’s internal flow structure which is well modeled by Hill’s analytic spherical vortex (Hill 1894) even for nonspherical thermals. We show that the predicted values of w_t, α, and z_c are consistent with numerical simulations of buoyant thermals over a range of A_r from 0.5 to 2.

The paper is organized as follows. Section 2 provides a theoretical description of the problem and derivation of equations for w_t and α. Section 3 gives a description of the numerical model and experimental design. Section 4 presents results from the numerical simulations and comparison of these results with theory. A summary and conclusions are given in section 5.

2. Theoretical description

a. Thermal momentum budget

To derive our analytic thermal model and theoretical values for α, we first focus on the vertical momentum budget of thermals. This allows us to derive an analytic expression for the thermal ascent rate, from which we obtain expressions for thermal spreading rate α later. Defining the thermal as occupying some portion of space defined by Ω within the domain, we can integrate over Ω to obtain the thermal’s budget of vertical momentum ρ₀w, where w is the vertical velocity. Because Ω changes over time, we use Gauss’s theorem to relate the divergence of the momentum field over Ω to the flux of momentum across the surface of Ω (Romps and Charn 2015; Morrison et al. 2022):

$\frac{d}{dt}{\displaystyle {\int }_{\mathrm{\Omega }(t)}{\rho }_0}w{d}^{3}x=-{\displaystyle {\int }_{\mathrm{\Omega }(t)}\frac{\partial p}{\partial z}}{d}^{3}x+{\displaystyle {\int }_{\mathrm{\Omega }(t)}{\rho }_0}B{d}^{3}x+{\displaystyle {\int }_{\partial \mathrm{\Omega }(t)}(\widehat{n}\cdot {\mathbf{u}}_e)}{\rho }_{0}w{d}^{2}x,$(4)

where ∂Ω(t) is the two-dimensional boundary of Ω(t), $\widehat{n}$ is a unit vector normal to the thermal’s surface, and u_e is an effective entrainment velocity defined as the displacement rate of the thermal boundary u_b relative to flow velocity u, i.e., u_e = u_b − u.

Defining $\overline{w}\equiv V^{-1}{\int }_{\mathrm{\Omega }(t)}w{d}^{3}x$ as the thermal-averaged w (assumed to be equal to the thermal top ascent rate w_t) in (4), using the product rule d(w_tV)/dt = Vdw_t/dt + w_tdV/dt, dividing by thermal volume V, and rearranging terms, we express the thermal-averaged momentum budget as

${\rho }_0\frac{d{w}_t}{dt}=-F_d+E+{\rho }_0\overline{B},$(5)

where $F_d\equiv V^{-1}{\int }_{\mathrm{\Omega }(t)}(\partial p/\partial z)d^{3}x$ is the thermal-averaged pressure drag force, $E\equiv V^{-1}{\int }_{\partial \mathrm{\Omega }(t)}(\widehat{n}\cdot {\mathbf{u}}_e){\rho }_{0}w{d}^{2}x-V^{-1}(dV/dt)w_t{\rho }_0$ is the momentum entrainment, and $\overline{B}\equiv V^{-1}{\int }_{\mathrm{\Omega }(t)}B{d}^{3}x$ is the thermal-averaged buoyancy. Equation (5) expresses the vertical momentum budget of a thermal as an acceleration term on the left-hand side and a drag force arising from vertical pressure gradients, an entrainment “pseudoforce,” and a buoyant forcing term on the right-hand side.

We assume that detrainment is negligible and that thermal expansion incorporates fluid with w = 0. Thus, net entrainment is related only to the change in thermal volume (following the Boussinesq approximation). This assumption is well justified based on the DNS of dry thermals from LJ2019; see Morrison et al. (2022) for further discussion. It follows that we can write the entrainment term as $E\approx -V^{-1}(dV/dt)w_t{\rho }_0$, where $V^{-1}(dV/dt)=(d\text{ln}V/d{z}_t)w_t=ϵw_t$ using the chain rule. Thus, we can express $E\approx -ϵw_t^2{\rho }_0$.

We nondimensionalize the thermal momentum equation next. Based on a characteristic background fluid density ρ₀, thermal radius R₀, and thermal buoyancy B₀, we can define various scales including time $t_0=\sqrt{R_0/B_0}$, velocity $w_0=\sqrt{R_0{B}_0}$, and pressure p₀ = ρ₀B₀/R₀. From this, we write $t^{*}=t/t_0,z^{*}=z/R_0,w^{*}=w/w_0,F_d^{*}=(R_0/p_0)F_d,E^{*}=E/E_0=[R_0/({\rho }_0{w}_0^2)]E$, and $B^{*}=B/B_0$. Substituting these relations into the thermal-averaged momentum budget gives a nondimensional form of the equation:

$\frac{{\rho }_0{w}_0^2}{R_0}\frac{d{w}_t^{*}}{d{t}^{*}}=-\frac{p_0}{R_0}{F}_d^{*}+\frac{{\rho }_0{w}_0^2}{R_0}{E}^{*}+{\rho }_0{B}_0{\overline{B}}^{*}.$(6)

Multiplying (6) by $R_0/(w_0^2{\rho }_0)$, it can be expressed in terms of two nondimensional flow parameters: Froude number $F_r=w_0^2/(B_0{R}_0)$ and Euler number $E_u=p_0/({\rho }_0{w}_0^2)$. This gives

$\frac{d{w}_t^{*}}{d{t}^{*}}=-E_u{F}_{pB}^{*}-E_u{F}_{pD}^{*}+E^{*}+F_r^{-1}{\overline{B}}^{*},$(7)

where the nondimensional drag force $F_d^{*}$ is divided into two parts following the standard separation of perturbation pressure into buoyant and dynamic components: $F_d^{*}=F_{pB}^{*}+F_{pD}^{*}$. The buoyant part is approximated by $E_u{F}_{pB}^{*}\approx F_r^{-1}(1-C_{\upsilon }){\overline{B}}^{*}$. The term C_υ is a virtual mass parameter defined such that the sum of $F_r^{-1}{\overline{B}}^{*}$ and $-E_u{F}_{pB}^{*}$ is equal to $F_r^{-1}{C}_{\upsilon }{\overline{B}}^{*}$. C_υ depends on the structure of the buoyancy field. For example, C_υ = 2/3 for a spherical buoyancy perturbation (Tarshish et al. 2018). It follows that the nondimensional effective buoyancy—the sum of buoyancy and buoyant perturbation pressure forcing—is $B_{\text{eff}}^{*}=C_{\upsilon }{\overline{B}}^{*}$.

We approximate the dynamic pressure part of thermal drag using the standard drag equation divided by V to give a volume-averaged dynamic pressure drag force: $F_{pD}={\rho }_0{w}_t^2{C}_{d}A/(2V)$, where A is the cross-sectional thermal area perpendicular to the flow. Defining γ as the ratio AR/V, which is a constant for self-similar thermals (γ = 3/4 for a spherical thermal), we can express the thermal-averaged nondimensional dynamic pressure drag force as $F_{pD}^{*}=w_t^{*2}\gamma C_d/(2R^{*})$. Note that C_d ≈ 0 for initially spherical (A_r = 1) dry buoyant thermals (Morrison et al. 2022). Although C_d could in principle vary with A_r, the thermal simulations presented in section 4, with A_r varying from 0.5 to 2, all have small C_d (magnitude less than 0.1). Thus, the dynamic pressure drag is relatively unimportant in the thermal momentum budget. Nonetheless, we retain this term and C_d in the equations for generality.

Hereinafter, we take characteristic values of the physical scales ρ₀, R₀, and B₀ as unity so that F_r = 1 and E_u = 1 and drop the * indicating nondimensional quantities for convenience. It follows that we can write the nondimensional thermal-averaged vertical momentum budget equation given by (7) as

$\frac{1}{2}\frac{d{w}_t^2}{d{z}_t}+w_t^2ϵ+\frac{\gamma C_d{w}_t^2}{2R}-C_{\upsilon }\overline{B}=0,$(8)

where we have used the above relations for E and buoyant and dynamic pressure contributions to drag, and the chain rule to express the time derivative as a height derivative following the thermal top: d/dt = w_td/dz_t (z_t is the thermal top height).

If we assume α ≡ dR/dz_t is constant, consistent with recent numerical modeling studies (Morrison et al. 2021; LJ2019) and the simulations herein, we can write R = R₀ + αz_t = 1 + αz_t by integrating α from $z_t^{\prime }=0$ to $z_t^{\prime }=z_t$. Also, $\overline{B}=B_0/R^3=1/R^3$ (since we take B₀ = 1 as the initial buoyancy scale, and $\overline{B}$ scales inversely with the change in thermal volume as the thermal entrains nonbuoyant environmental fluid and expands) and ϵ = e/R (LJ2019). With these assumptions and relations, (8) may be written as

$\frac{1}{2}\frac{d{w}_t^2}{d{z}_t}+\left(e+\frac{\gamma C_d}{2}\right)\frac{w_t^2}{(1+\alpha z_t)}-\frac{C_{\upsilon }}{{(1+\alpha z_t)}^3}=0.$(9)

Assuming α, e, C_υ, γ, and C_d are constants, (9) represents a first-order linear differential equation with an exact solution given by

$w_t^2=\frac{C_{\upsilon }}{\left(e-\alpha +\frac{\gamma }{2}{C}_d\right){(1+\alpha z_t)}^2}+k_1{(1+\alpha z_t)}^{-(2e/\alpha )-(\gamma C_d/\alpha )},$(10)

where k₁ is a constant of integration. Given the boundary condition w_t = 0 at z_t = 0, we may solve for k₁ and use this in (10) to give

$w_t^2=\frac{C_{\upsilon }}{\left(e-\alpha +\frac{\gamma }{2}{C}_d\right){(1+\alpha z_t)}^2}-\frac{C_{\upsilon }}{\left(e-\alpha +\frac{\gamma }{2}{C}_d\right){(1+\alpha z_t)}^{(2e/\alpha )+(\gamma C_d/\alpha )}}.$(11)

Equation (11) is similar to Escudier and Maxworthy [1973, Eqs. (9)–(11) therein], except that we invoke the Boussinesq approximation, express w_t using a single analytic equation as a function of height z_t rather than t, and include an explicit dependence on α.

Solutions for w_t can be obtained from (11), provided values of α, e, γ, C_d, and C_υ are known. Past literature has suggested α ∼ 0.05–0.3 (Lai et al. 2015), e ≈ 3α (LJ2019), C_d ≈ 0 (Morrison et al. 2022), meaning that γ is not relevant, and C_υ ∼ 0.5–0.8 (Tarshish et al. 2018). Examples of solutions to the analytic w_t Eq. (11) using e = 3α, C_d = 0, C_υ = 2/3, and α ranging from 0.05 to 0.3 are shown in Fig. 2 (solid lines). With these parameter values, (11) gives a family of solutions that all exhibit a sharp increase of w_t with height initially corresponding to a “slippery” regime when upward buoyant forcing is primarily balanced by vertical acceleration, followed by a slower decrease of w_t corresponding to a “sticky” regime when weak buoyant forcing is balanced mainly by entrainment.² This shape of the w_t profile with the two distinct regimes of thermal evolution was discussed previously via analysis of numerical solutions (e.g., Wang 1971; Tarshish et al. 2018). In subsequent sections, we will determine constraints on values of α while also briefly exploring how e and C_υ vary with the initial thermal aspect ratio.

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Analytic solutions to the thermal w_t Eq. (11) as a function of nondimensional thermal top height z_t. Different color lines indicate assumed values of thermal expansion rate α as labeled. Black dotted and dashed black lines illustrate scalings from the asymptotic analysis corresponding to the “slippery” regime during thermal spinup and “sticky” regime after spinup, respectively. The * symbols indicate the height and magnitude of maximum w_t for each curve.

Citation: Journal of the Atmospheric Sciences 80, 11; 10.1175/JAS-D-23-0063.1

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We can understand scaling behaviors in the slippery and sticky regimes via asymptotic analysis and expansion of (11), similar to the asymptotic analysis of Escudier and Maxworthy (1973) applied to their equation set for w_t. For the slippery regime when z_t < 1, we can expand (11) using Taylor series about z_t = 0 and retain the first-order term to give $w_t\propto z_t^{1/2}$. This is equivalent to a scaling of z_t ∝ t² since w_t = dz_t/dt, which is consistent with the theory and numerical simulations in the slippery regime from Tarshish et al. (2018).

In contrast, when z_t ≫ 1 for the sticky regime, 1 + αz_t ≈ αz_t and the second term on the right-hand side of (11) is negligible compared to the first term (for 2e/α + γC_d/2 > 2, which is satisfied for typical values of 2e/α ≈ 6 and γC_d ≈ 0), implying a scaling of w_t with $z_t^{-1}$. This is equivalent to a scaling of z_t ∝ t^1/2, consistent with the theory and simulations of LJ2019 for the sticky regime. These scaling regimes well correspond to full solutions of the analytic w_t equation, Eq. (11), for small and large z_t as seen in Fig. 2 and are also consistent with the asymptotic analysis from Escudier and Maxworthy (1973).

c. Relationship between thermal spinup height and spreading rate α

We define a critical thermal top height z_c separating the “slippery” and “sticky” regimes when dw_t/dz_t = 0 and w_t is a maximum. During spinup, dw_t/dz_t > 0 as dΓ/dt > 0 owing to the presence of buoyant fluid along the thermal’s vertical axis. After buoyancy is eroded along the thermal’s vertical axis from entrainment of environmental fluid, Γ is constant following (13), and thus, dw_t/dz_t < 0 following the $w_t\propto z_t^{-1}$ scaling implied by (17). Taken together, the implication is that z_c corresponds to the thermal top height at the time when core buoyancy is eroded and thereafter dΓ/dt ≈ 0, which is supported by the numerical simulations presented in section 4.

The critical height z_c is obtained by taking d/dz_t of (11) and setting dw_t/dz_t = 0. We also introduce the parameter b ≡ e/α, the ratio of entrainment efficiency to thermal spreading rate. The resulting expression is solved analytically to yield the following relation between z_c, α, b, γ, and C_d:

$\left(b+\frac{\gamma C_d}{2\alpha }\right){(1+\alpha z_c)}^{-2b-(\gamma C_d/\alpha )+2}=1.$(18)

Figure 3 shows α calculated from (18) as a function of z_c for b equal to 2, 2.5, and 3 and C_d = 0 (Fig. 3a) and for C_d equal to −0.1, 0, and 0.1 and b = 3 (Fig. 3b). For these calculations, γ = 3/4 corresponding to spherical thermals for simplicity. A self-similar thermal shape implies b = 3, meaning that e = 3α and ϵ ≡ d(lnV)/dz_t = 3α/R (LJ2019). Deviations from b = 3, therefore, indicate the degree to which thermals are not self-similar. Zhao et al. (2013), while noting self-similarity of gross thermal characteristics (overall thermal shape and size) after spinup, found that internal vorticity and density structures evolved nonsimilarly in their experimental study. In the simulations presented later in the current paper, b is somewhat smaller than 3, ranging from ∼2.4 to 2.7 for A_r < 2. For the A_r = 2 simulation, b ∼ 2.1, indicating that initially vertically elongated thermals change their shape relatively more than the smaller A_r thermals. This may be related to the inability of thermals to take in all initially buoyant fluid when A_r ≥ 2, leaving a wake of buoyant fluid below the thermal, as discussed in Lai et al. (2015). Nonetheless, sensitivity of α to b over the range of 2–3 is fairly small, with a 24% decrease in α as b is increased from 2 to 3.

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Solutions to the analytic α–z_c relation (18) with (a) C_d = 0 and varying b as indicated and (b) b = 3 and varying C_d as indicated.

Citation: Journal of the Atmospheric Sciences 80, 11; 10.1175/JAS-D-23-0063.1

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The change in α for a given z_c as C_d is varied from −0.1 to 0.1 is small in magnitude, with α varying by up to 0.04 for the range of parameters shown in Fig. 3b. The relative change is greatest at small values of α (large z_c), up to ∼50% in Fig. 3b. For the thermal numerical simulations detailed later in the paper, with A_r varying from 0.5 to 2, mean C_d ranges from −0.06 to 0.08 [using the same method to calculate C_d from the simulated dynamic perturbation pressure field as in Morrison et al. (2022)]. Thus, dynamic pressure drag is relatively unimportant in (18), and hereinafter, we will assume C_d = 0. With this assumption, (18) can be rearranged to give

$\alpha =\frac{b^{1/(2b-2)}-1}{z_c}.$(19)

If we further assume a self-similar thermal shape, we can use (19) with b = 3 to obtain

$\alpha =\frac{3^{1/4}-1}{z_c}.$(20)

The one-to-one relationship between α and z_c means that if z_c is known, this uniquely constrains the value of α. We emphasize that z_c does not cause a particular value of α, but if z_c is known, then α can be predicted from it. A key point is that the expressions for α in (18)–(20) are independent of C_υ and thus expected to have little dependence on A_r (given that b does not vary much with A_r and C_d ≈ 0). Equation (20) gives consistent results with the analytic thermal w_t profiles shown in Fig. 2, which have z_c ranging from ∼1 to 6 for α of 0.05–0.3 (for b = 3 and C_d = 0).

3. Description of the numerical simulations

a. Model description and experimental design

We utilize the Cloud Model 1 (CM1) fluid flow model to numerically simulate thermals with varying initial A_r. CM1 is a nonhydrostatic model which has been widely used to simulate idealized atmospheric flows. Here, we use the incompressible Boussinesq configuration to solve the filtered Navier–Stokes equations similar to the large-eddy simulation (LES) configuration in Morrison et al. (2022). Prognostic variables are the 3D components of flow velocity and potential temperature perturbation θ′, although near-axisymmetry of the model fields is retained. Buoyancy B is obtained by gθ′/θ₀, where θ₀ is a constant background θ of the fluid environment. As noted by Morrison et al. (2022), in this framework, prognosing θ′ is equivalent to prognosing B itself. Simulations are nondimensionalized using a length scale equal to the radius of the initial thermal R₀ (the radius of the initial buoyancy perturbation) and a time scale given by $\sqrt{R_0/B_0}$, where B₀ is the initial thermal buoyancy. The density scale ρ₀ is equal to the constant background fluid density in this Boussinesq framework. All other quantities are nondimensionalized following these basic scales.

The initial A_r of thermals is varied from 0.5 to 2, similar to the range from Lai et al. (2015). As we show in section 4, this produces a wide spread of α (∼0.08–0.25). Thermals are initiated by adding a buoyancy perturbation B₀ uniformly within a spheroidal volume having a horizontal radius of R₀ and a vertical radius of A_rR₀. To minimize the impacts of boundary conditions, the initial buoyancy perturbations are centered at a height of 4R₀, and the horizontal domain width is ≥16R₀ and the vertical domain height is 64A_rR₀ (64 times the initial vertical thermal radius). The model grid is isotropic in all three directions with a grid spacing ΔL_m equal to 0.1A_rR₀. Since the initial A_r varies from 0.5 to 2, ΔL_m ranges from 0.05R₀ to 0.2R₀. The time step is 0.0362 times the time scale $\sqrt{R_0/B_0}$. Because the thermals expand as they ascend, the overall dynamical structure is well resolved with at least 10 grid points horizontally and 20 points vertically across the thermals. An additional set of simulations with A_r varying from 0.5 to 2 but ΔL_m = 0.1R₀ (thus at least 20 points horizontally and 10 points vertically across the thermals) was also run and analyzed. This set gives similar results to the first set, and thus, we only report the results of the first set of simulations in this paper. Other details of the model setup are given in Table 1.

Table 1.

Configuration details for the CM1 simulations presented in this paper.

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In this study, we use LES applied to the filtered Navier–Stokes equations instead of DNS to retain a close connection to atmospheric modeling, particularly modeling of dry and moist thermals in the planetary boundary layer and convective clouds in which DNS is not possible given the huge O(10⁹) Reynolds numbers involved. The LES framework is also consistent with our previous work on dynamic drag of dry buoyant thermals (Morrison et al. 2022) and similar to previous thermal simulations of Lai et al. (2015). The subgrid-scale (SGS) mixing follows a Smagorinsky-type approach as implemented by Stevens et al. (1999, see their appendix B, section b). The SGS mixing length is set to ΔL_m. Because the dissipation scale (the model’s filter scale) is a relatively large fraction of the thermals’ radii, the resolved scale flow is smooth and thus appears laminar. The resulting thermal evolution and internal flow structure of the simulated thermals is remarkably similar to the DNS of initially spherical laminar thermals in LJ2019 (R_e = 630). Results across the range of A_r are close to those of Lai et al. (2015), who also numerically solved the filtered (discretized) Navier–Stokes equations but using a k–δ turbulence closure (Launder and Spaulding 1974), where k is the resolved kinetic energy and δ is the energy dissipation rate. Our simulations are integrated forward in time until the thermal top (as defined in section 3b) reaches a height of 15R₀ above the initial thermal top (i.e., top of the initial buoyancy perturbation). To investigate the internal thermal flow characteristics, particularly the time for ascent of a parcel through the thermal, each simulation includes forward trajectories for a parcel placed at the thermal bottom at the initial time. We use the built-in parcel trajectory calculations in CM1 which are done during the model integration using linear interpolation of the flow field at each model time step.

4. Analysis of numerical simulations

Overall structure and evolution is similar for all of the simulated thermals. Starting from rest, rapid spinup ensues owing to vorticity generation from the thermals’ buoyancy distributions. The thermals spread outward as they rise and entrain the surrounding fluid. Spinup of the thermals (after which circulation is nearly constant) occurs when a parcel initially placed at the thermal bottom rises to near proximity of the thermal top. Here, we calculate the critical height z_c as the thermal top height when spinup is achieved, rather than directly from the height where dw_t/dz_t = 0 and w_t is maximum because dw_t/dz_t is rather noisy. Nonetheless, z_c calculated from the parcel trajectories matches well with broad maxima in w_t as shown later.

In accordance with the theory presented in section 2, z_c ranges from about 1 to 6 as A_r is varied from 0.5 to 2 (Table 2). After spinup, when z_t > z_c in the “sticky” regime, the thermals continue to expand by entraining environmental fluid, but their overall flow structure is fairly steady. The thermals undergo a slow deceleration (relative to the faster acceleration during spinup) with w_t roughly proportional to t^−1/2 (and thus also proportional to $z_t^{-1}$) in accordance with the classical similarity theory of Scorer (1957).

Table 2.

Time-averaged α ≡ dR/dz_t, b ≡ e/α, virtual mass parameter C_υ, ratio of parcel to thermal-time-averaged vertical velocity σ, circulation Γ, and critical spinup height z_c from the simulations with varying A_r. Note that b is obtained from the ratio of time-averaged e to time-averaged α. Because of some noise in calculating thermal velocity directly, σ is derived from (23) using z_c obtained from the simulations as described in the text. α, e, and C_υ are calculated as time averages over the full simulation period, whereas Γ values are time-averaged after spinup to the end of the simulations.

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Figure 4 shows vertical cross sections through the thermal center of B, w, horizontal vorticity in the y direction η_y, and streamfunction ψ after spinup, when thermal top height is at approximately z_c + 2R₀. Thermal flow features well documented by previous studies are seen in the figure. These include toroidal circulations with rotation centers near the thermal vertical midpoint, buoyancy concentrated near these rotation centers, and downward motion (in an absolute sense and relative to w_t) along the thermal periphery. Although buoyancy is almost entirely swept away from the thermal core (along the vertical axis at X = 0) for the A_r = 0.5 and 1 simulations, some positive buoyancy remains in the core when A_r = 2. There is also fluid with B > 0 and η_y ≠ 0 below the thermal in this simulation (Figs. 4e,f). This occurs because not all of the initially buoyant fluid is taken into the thermal’s vortex ring (toroidal circulation) when the aspect ratio is large, a result also noted by Lai et al. (2015). This behavior can be described by the “formation number” (Gharib et al. 1998), which is related to the maximum vorticity that can be incorporated into a vortex ring before it “pinches off” from a trailing stem. Earlier work showed a formation number of 4–5 for vortex rings (Gharib et al. 1998; Wang et al. 2009), whereas Lai et al. (2015) found a somewhat lower formation number of ∼2, consistent with our results.³ Despite the presence of a trailing stem of weakly buoyant fluid in the A_r = 2 simulation, buoyancy in the core is small relative to that near the rotation centers, and as detailed later, the theoretical relations between z_c, α, and A_r proposed in section 2 still well describe behavior of this simulation. We suspect that further increases in A_r would lead to greater deviation with the theory. Indeed, Lai et al. (2015) showed little change in α as A_r was increased beyond 2, likely because of the inability of such thermals to incorporate all of the initially buoyant fluid. The behavior of these thermals instead resembled a starting plume, consistent with the numerical results of Bond and Johari (2010).

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Vertical cross sections of (left) buoyancy (color contours) and vertical velocity (thin black solid lines for positive w and thin black dashed lines for negative w; contour values are ±0.1, 0.2, 0.6, and every 0.4 thereafter); (right) vorticity in the y plane η_y (color contours) and streamfunction ψ (contour lines). Thick black lines show thermal boundaries defined by the ψ = 0 isoline. Results are shown for (a),(b) A_r = 0.5, (c),(d) A_r = 1, and (e),(f) A_r = 2. Cross sections are shown at times when the thermal top is approximately 2R₀ above the critical height z_c for each simulation (see text).

Citation: Journal of the Atmospheric Sciences 80, 11; 10.1175/JAS-D-23-0063.1

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Differences in thermal aspect ratio with varying initial A_r persist beyond spinup, although these differences are reduced compared to the initial A_r. The thermals with initial A_r ≥ 1 become more flattened (smaller aspect ratio) during spinup. At the times shown in Fig. 4, the A_r = 2 simulation has an aspect ratio just slightly larger than 1, while that for A_r = 1 is about 0.75 and that for A_r = 0.5 is about 0.6. Different thermal aspect ratios among the simulations are reflected by variability in time-averaged values of C_υ (virtual mass parameter, see section 2); see Table 2. Here, C_υ is calculated at each model output time directly from the buoyancy and buoyant pressure forcing averaged over the thermal volume. Larger initial aspect ratios are associated with larger C_υ, consistent with results from Tarshish et al. (2018). There is also an overall decrease in C_υ over time during spinup as the thermals flatten, particularly for the simulations with A_r > 1. Changes in thermal shape during spinup also lead to deviation in b from the value for self-similar thermals (b = 3). The A_r = 2 thermal has the largest deviation, with b ≈ 2.06, which is consistent with it experiencing the greatest change in aspect ratio during spinup, whereas b ranges from ∼2.4 to 2.7 for the other simulations.

Thermal behavior during spinup is illustrated in Fig. 5, which shows vertical cross sections of B, w, η_y, and ψ in the same format as Fig. 4 except during the spinup period for the A_r = 1 simulation. Cross sections are shown in nondimensional time increments of 1.1 between t = 1.6 and 4.9. For context, the thermal top reaches z_c at t ≈ 3.7. The basic mechanism of spinup is similar for all the runs. Consistent with the discussion in section 2d, entrainment occurs as environmental fluid is swept into the thermal from below in the convergent flow. This appears as a “bite” taken from the buoyancy field from below and occurs because thermal-relative vertical velocities are strongest in the thermal core. Baroclinic vorticity generation is concentrated along the edge of the buoyancy field where there are large horizontal buoyancy gradients. Once the buoyancy field is deformed and starts to wrap around the vortex core (i.e., the center of rotation), baroclinic generation and destruction of vorticity drives a spreading of the thermal in the manner outlined by McKim et al. (2020) and Morrison et al. (2021). Flattening of the thermal during spinup is also evident in Fig. 5.

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As in Fig. 4, but for the A_r = 1 simulation during spinup at the times (t) indicated.

Citation: Journal of the Atmospheric Sciences 80, 11; 10.1175/JAS-D-23-0063.1

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In all simulations, the thermals’ internal flow structures consist of thermal-relative ascent in the core, with strongest ascent along the vertical axis and descent along the periphery. This flow pattern strongly resembles Hill’s analytic spherical vortex. To illustrate this point further, Fig. 6 compares w profiles along the thermals’ vertical axis from the simulations with A_r of 0.5, 1, and 2 with w profiles at the vertical axis from Hill’s vortex given by (26) and (27). This is similar to the comparison of w profiles from thermal simulations with Hill’s vortex in Lai et al. (2015, Fig. 12 therein). Simulation results here are shown at the time of spinup when the thermal top is at z_c. Profiles from the simulations are normalized by the maximum w with height normalized by the thermal depth; thermal bottom and top heights are set to −1 and 1, respectively. Correspondingly, a = 1 in (26) and (27) for the Hill’s vortex w profile. All of the simulations produce similar w profiles as Hill’s vortex, with the A_r = 1 being closest. There is also a close correspondence of the A_r = 2 simulation with Hill’s vortex, with greater deviation for A_r = 0.5. Overall, these results support the discussion in section 2e on the validity of approximating σ for thermals from Hill’s vortex.

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Comparison of vertical profiles of w from the simulated thermals (blue crosses at each model level) with that from Hill’s analytic spherical vortex (red lines). The thermal/vortex bottom and top heights are normalized to −1 and 1, respectively, and shown by the horizontal black lines. Profiles of nondimensional w are normalized such that the maximum value is 1. Simulation results are shown for (a) A_r = 0.5, (b) A_r = 1, and (c) A_r = 2 near the time of thermal spinup.

Citation: Journal of the Atmospheric Sciences 80, 11; 10.1175/JAS-D-23-0063.1

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Differences in spreading rate α ≡ dR/dz among the simulated thermals are seen in Fig. 7a, which shows thermal radius R as a function of z_t for the simulations with A_r of 0.5, 1, and 2. The increase of R with z_t is clearly greater as A_r is decreased, with α about 3 times larger in the A_r = 0.5 simulation compared to A_r = 2. Although R is somewhat noisy, the overall spreading rates are nearly constant with z_t (seen by the dotted lines) consistent with similarity theory.

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Vertical profiles of (a) thermal radius R and (b) ascent rate w for simulations with various A_r as indicated. Solid lines show results calculated directly from the simulations. Dotted lines in (a) show fit values of constant α ≡ dR/dz and in (b) show solutions to the analytic w Eq. (11) using C_d = 0 and mean values of C_υ, α, e from the simulations.

Citation: Journal of the Atmospheric Sciences 80, 11; 10.1175/JAS-D-23-0063.1

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A comparison of simulated w_t as a function of z_t with solutions to the analytic w_t equation, Eq. (11), is shown in Fig. 7b. The analytic w_t are obtained using mean values of C_υ, b, and α from each simulation (Table 2). The overall behavior of w_t is similar among the simulations, with a sharp increase during spinup followed by a slower decrease after spinup. The analytic w_t are close to the simulated values for each simulation (compared the dotted and solid lines in Fig. 7). Larger values of α in the A_r = 0.5 simulation correspond to faster spinup and lower height of maximum w_t (critical height z_c) compared to the A_r = 1 and especially A_r = 2 simulations with smaller α. Thus, z_c increases with A_r consistent with the theory in section 2d.

Thermal behavior during spinup for the simulations with A_r of 0.5, 1, and 2 is further illustrated in Fig. 8, which shows time series of thermal top height z_t and vertical velocity w_t, circulation Γ, and vertically integrated core buoyancy ($\int B_{c}dz$). Also shown in Fig. 8 are the height z_p and vertical velocity w_p of a parcel placed initially at the thermal bottom that rises relative to the thermal as a whole. Consistent with the discussion in previous sections, Γ increases during spinup owing to $\int B_{c}dz>0$ following (13), and this is accompanied by an increase in w_t. w_p increases relative to w_t as the parcel rises through the thermal core, with w_p reaching a maximum when the parcel is near the thermal’s vertical midpoint. As a result of this velocity difference, z_p increases faster than z_t. Since the parcel is initiated on the thermal edge at its bottom, this marks the upward advance of entraining fluid into the thermal core (see also vertical cross sections of B in Fig. 5). This leads to a decrease in $\int B_{c}dz$ and the rate of increase in Γ slows (i.e., dΓ/dt decreases). At the time when the parcel rises to near thermal top (z_p ≈ z_t), $\int B_{c}dz$ reaches steady values near 0 (though somewhat larger in the A_r = 2 simulation) and dΓ/dt ≈ 0. This point defines the time τ_c and height z_c of thermal spinup consistent with the discussion in section 2. The terms τ_c and z_c are calculated here as the time and height when z_p reaches within 2% of z_t. τ_c is denoted by the vertical black lines in Fig. 8. After spinup, z_p tracks closely to z_t and w_p remains close to w_t, while both decrease slowly. Overall, Γ (time-averaged past spinup) increases sharply as A_r is increased, from Γ ≈ 2.30 for A_r = 0.5 to Γ ≈ 18.02 for A_r = 2. Note that there is a small increase in Γ in the A_r = 2 simulation after spinup corresponding to a small but nonnegligible $\int B_{c}dz$ consistent with vertical cross sections of the B field (see Fig. 4c). This occurs because entrained fluid from below the thermal has B > 0 in this simulation; not all the initially buoyant fluid is taken up by the thermal initially when the aspect ratio is large so that some remains below the thermal’s circulation as discussed earlier. Values of $\int B_{c}dz$ reached in the A_r = 2 simulation after spinup appear to be nearly steady in time thereafter, and they are about an order of magnitude larger than in the other simulations after their spinup. It is expected that $\int B_{c}dz$ would eventually decrease in the A_r = 2 simulation as the thermal continued to rise and entrain, but investigating this would require longer simulations and thus a larger domain.

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Various nondimensional parcel and thermal properties as a function of time t during the thermals’ spinup for simulations with (a) A_r = 0.5, (b) A_r = 1, and (c) A_r = 2. Results are shown for thermal top height z_t, parcel height z_p, thermal ascent rate w_t, parcel ascent rate w_p, thermal circulation Γ, and vertically integrated buoyancy along the thermal’s central vertical axis B_c. The thermal top height at initial time (t = 0) is at z = 0. Parcels at t = 0 are centered horizontally at the thermal bottom and move upward through the thermal over time.

Citation: Journal of the Atmospheric Sciences 80, 11; 10.1175/JAS-D-23-0063.1

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Values of z_c from the simulations, estimated from z_p and z_t as described above, are compared to the theoretical linear z_c–A_r relation (23) using σ = 2 from Hill’s analytic vortex (see section 2e) and using the average σ = 1.80 from the simulations (Table 2) in Fig. 9a. The simulated and theoretical z_c values are similar, although the A_r = 2 simulation deviates more substantially from the theoretical relations. Reasons for this deviation are unclear but might be explained by the buoyant fluid entrained into the thermal from below in this simulation, leading to some buoyancy remaining along the thermal’s vertical axis even after spinup. The theoretical σ ≈ 2 derived from the Eulerian mean w of Hill’s vortex (see section 2e) is fairly close to σ values obtained directly from the simulations (within 10% except for the A_r = 2 simulation), though somewhat larger. The simulated values range from 1.80 to 1.95 for A_r < 2 but are slightly smaller (≈1.63) for A_r = 2 (see Table 2).

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(a) Critical height z_c and (b) thermal expansion rate α as functions of initial thermal aspect ratio A_r from the simulations and theory. α and z_c obtained directly from the simulations are shown by blue crosses. Green and red crosses in (b) show theoretical α values from (19) with b obtained from the simulations or 20 with b = 3, respectively, with z_c in both expressions obtained directly from the simulations. The theoretical z_c from (23) using σ = 2 from Hill’s analytic vortex and using σ = 1.8 (average σ from the simulations) are shown by the solid and dotted black lines, respectively. The solid and dotted black lines in (b) show theoretical α values from (25) using σ = 2 and 1.8, respectively.

Citation: Journal of the Atmospheric Sciences 80, 11; 10.1175/JAS-D-23-0063.1

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A direct comparison of the simulated and theoretical values of spreading rate α is shown in Fig. 9b. Theoretical values are obtained from 1) Eq. (19) using z_c and b derived from the simulations (Table 2), 2) Eq. (20) using z_c derived from the simulations and b = 3 following self-similarity, 3) Eq. (25) using the average σ = 1.80 from the simulations to predict z_c, and 4) Eq. (31) which calculates α from z_c predicted using σ = 2 from Hill’s analytic vortex. All of the theoretical calculations for α give similar results as the simulations. The simulations show a sharp decrease of α with increasing A_r that follows an approximate $A_r^{-1}$ dependence consistent with the theoretical expressions. Using b = 3 instead of b values obtained directly from the simulations leads to a small decrease in theoretical α. In this case, α values are somewhat smaller than simulated values for A_r ≥ 1, but closer to simulated values for A_r < 1. Using (31) to calculate α well describes the α–A_r relation but with ∼10% larger α compared to the simulations (solid line in Fig. 9b). This is consistent with the small overestimation of σ = 2 approximated from Hill’s vortex. Using the average σ from the simulations (σ = 1.80) to predict z_c, and in turn α following (25), gives a close correspondence to the simulated α over the range of A_r (dotted line in Fig. 9b).

5. Discussion

Overall, the simulations and theory are in reasonable agreement regarding thermal top height at spinup z_c and thermal spreading rate α and how they vary with initial aspect ratio A_r. Our results indicate a nearly linear relation between z_c and A_r (though with greater deviation for the A_r = 2 simulation) and an inverse relation between α and A_r ($\alpha \propto A_r^{-1}$). Qualitatively, this α–A_r relation is consistent with previous thermal studies (see Fig. 17 in Lai et al. 2015). Larger α is associated with greater fractional entrainment rate which leads to a lower critical height z_c, defined as the thermal top height when w_t is maximum. z_c also corresponds to the thermal top height when buoyant fluid along the thermal’s vertical axis is replaced by nonbuoyant environmental fluid entrained and advected upward through the thermal core, after which the thermal is spun up and dΓ/dt ≈ 0. The time scale for this process is controlled by how long it takes for parcels initially just below the thermal bottom to ascend through the thermal, which in turn depends on A_r. By relating to α to z_c, and z_c in turn to A_r, we obtained the inverse relation between α and A_r in section 2.