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## Abstract

The Davies-Jones formulation of effective buoyancy is used to define inertial and buoyant components of vertical force and to develop an intuition for these components by considering simple cases. This decomposition is applied to the triggering of new boundary layer mass flux by cold pools in a cloud-resolving simulation of radiative–convective equilibrium (RCE). The triggering is found to be dominated by inertial forces, and this is explained by estimating the ratio of the inertial forcing to the buoyancy forcing, which scales as *H*/*h*, where *H* is the characteristic height of the initial downdraft and *h* is the characteristic height of the mature cold pool’s gust front. In a simulation of the transition from shallow to deep convection, the buoyancy forcing plays a dominant role in triggering mass flux in the shallow regime, but the force balance tips in favor of inertial forcing just as precipitation sets in, consistent with the RCE results.

## Abstract

The Davies-Jones formulation of effective buoyancy is used to define inertial and buoyant components of vertical force and to develop an intuition for these components by considering simple cases. This decomposition is applied to the triggering of new boundary layer mass flux by cold pools in a cloud-resolving simulation of radiative–convective equilibrium (RCE). The triggering is found to be dominated by inertial forces, and this is explained by estimating the ratio of the inertial forcing to the buoyancy forcing, which scales as *H*/*h*, where *H* is the characteristic height of the initial downdraft and *h* is the characteristic height of the mature cold pool’s gust front. In a simulation of the transition from shallow to deep convection, the buoyancy forcing plays a dominant role in triggering mass flux in the shallow regime, but the force balance tips in favor of inertial forcing just as precipitation sets in, consistent with the RCE results.

## Abstract

By introducing an equivalence between magnetostatics and the equations governing buoyant motion, we derive analytical expressions for the acceleration of isolated density anomalies (thermals). In particular, we investigate buoyant acceleration, defined as the sum of the Archimedean buoyancy *B* and an associated perturbation pressure gradient. For the case of a uniform spherical thermal, the anomaly fluid accelerates at 2*B*/3, extending the textbook result for the induced mass of a solid sphere to the case of a fluid sphere. For a more general ellipsoidal thermal, we show that the buoyant acceleration is a simple analytical function of the ellipsoid’s aspect ratio. The relevance of these idealized uniform-density results to turbulent thermals is explored by analyzing direct numerical simulations of thermals at a Reynolds number (Re) of 6300. We find that our results fully characterize a thermal’s initial motion over a distance comparable to its length. Beyond this buoyancy-dominated regime, a thermal develops an ellipsoidal vortex circulation and begins to entrain environmental fluid. Our analytical expressions do not describe the total acceleration of this mature thermal, but they still accurately relate the buoyant acceleration to the thermal’s mean Archimedean buoyancy and aspect ratio. Thus, our analytical formulas provide a simple and direct means of estimating the buoyant acceleration of turbulent thermals.

## Abstract

By introducing an equivalence between magnetostatics and the equations governing buoyant motion, we derive analytical expressions for the acceleration of isolated density anomalies (thermals). In particular, we investigate buoyant acceleration, defined as the sum of the Archimedean buoyancy *B* and an associated perturbation pressure gradient. For the case of a uniform spherical thermal, the anomaly fluid accelerates at 2*B*/3, extending the textbook result for the induced mass of a solid sphere to the case of a fluid sphere. For a more general ellipsoidal thermal, we show that the buoyant acceleration is a simple analytical function of the ellipsoid’s aspect ratio. The relevance of these idealized uniform-density results to turbulent thermals is explored by analyzing direct numerical simulations of thermals at a Reynolds number (Re) of 6300. We find that our results fully characterize a thermal’s initial motion over a distance comparable to its length. Beyond this buoyancy-dominated regime, a thermal develops an ellipsoidal vortex circulation and begins to entrain environmental fluid. Our analytical expressions do not describe the total acceleration of this mature thermal, but they still accurately relate the buoyant acceleration to the thermal’s mean Archimedean buoyancy and aspect ratio. Thus, our analytical formulas provide a simple and direct means of estimating the buoyant acceleration of turbulent thermals.

## Abstract

Entrainment in cumulus convection remains ill understood and difficult to quantify. For instance, entrainment is widely believed to be a fundamentally turbulent process, even though Turner pointed out in 1957 that dry thermals entrain primarily because of buoyancy (via a dynamical constraint requiring an increase in radius *r*). Furthermore, entrainment has been postulated to obey a 1/*r* scaling, but this scaling has not been firmly established. Here, we study the classic case of dry thermals in a neutrally stratified environment using fully resolved direct numerical simulation. We combine this with a thermal tracking algorithm that defines a control volume for the thermal at each time, allowing us to directly measure entrainment. We vary the Reynolds number (Re) of our thermals between laminar (Re ≈ 600) and turbulent (Re ≈ 6000) regimes, finding only a 20% variation in entrainment rate *ε*, supporting the claim that turbulence is not necessary for entrainment. We also directly verify the postulated *ε* ~ 1/*r* scaling law.

## Abstract

Entrainment in cumulus convection remains ill understood and difficult to quantify. For instance, entrainment is widely believed to be a fundamentally turbulent process, even though Turner pointed out in 1957 that dry thermals entrain primarily because of buoyancy (via a dynamical constraint requiring an increase in radius *r*). Furthermore, entrainment has been postulated to obey a 1/*r* scaling, but this scaling has not been firmly established. Here, we study the classic case of dry thermals in a neutrally stratified environment using fully resolved direct numerical simulation. We combine this with a thermal tracking algorithm that defines a control volume for the thermal at each time, allowing us to directly measure entrainment. We vary the Reynolds number (Re) of our thermals between laminar (Re ≈ 600) and turbulent (Re ≈ 6000) regimes, finding only a 20% variation in entrainment rate *ε*, supporting the claim that turbulence is not necessary for entrainment. We also directly verify the postulated *ε* ~ 1/*r* scaling law.

## Abstract

Atmospheric radiative cooling is a fundamental aspect of Earth’s greenhouse effect, and is intrinsically connected to atmospheric motions. At the same time, basic aspects of longwave radiative cooling, such as its characteristic value of 2 K day^{−1}, its sharp decline (or “kink”) in the upper troposphere, and the large values of CO_{2} cooling in the stratosphere, are difficult to understand intuitively or estimate with pencil and paper. Here we pursue such understanding by building simple spectral (rather than gray) models for clear-sky radiative cooling. We construct these models by combining the cooling-to-space approximation with simplified greenhouse gas spectroscopy and analytical expressions for optical depth, and we validate these simple models with line-by-line calculations. We find that cooling rates can be expressed as a product of the Planck function, a vertical emissivity gradient, and a characteristic spectral width derived from our simplified spectroscopy. This expression allows for a pencil-and-paper estimate of the 2 K day^{−1} tropospheric cooling rate, as well as an explanation of enhanced CO_{2} cooling rates in the stratosphere. We also link the upper-tropospheric kink in radiative cooling to the distribution of H_{2}O absorption coefficients, and from this derive an analytical expression for the kink temperature *T*
_{kink} ≈ 220 K. A further, ancillary result is that gray models fail to reproduce basic features of atmospheric radiative cooling.

## Abstract

Atmospheric radiative cooling is a fundamental aspect of Earth’s greenhouse effect, and is intrinsically connected to atmospheric motions. At the same time, basic aspects of longwave radiative cooling, such as its characteristic value of 2 K day^{−1}, its sharp decline (or “kink”) in the upper troposphere, and the large values of CO_{2} cooling in the stratosphere, are difficult to understand intuitively or estimate with pencil and paper. Here we pursue such understanding by building simple spectral (rather than gray) models for clear-sky radiative cooling. We construct these models by combining the cooling-to-space approximation with simplified greenhouse gas spectroscopy and analytical expressions for optical depth, and we validate these simple models with line-by-line calculations. We find that cooling rates can be expressed as a product of the Planck function, a vertical emissivity gradient, and a characteristic spectral width derived from our simplified spectroscopy. This expression allows for a pencil-and-paper estimate of the 2 K day^{−1} tropospheric cooling rate, as well as an explanation of enhanced CO_{2} cooling rates in the stratosphere. We also link the upper-tropospheric kink in radiative cooling to the distribution of H_{2}O absorption coefficients, and from this derive an analytical expression for the kink temperature *T*
_{kink} ≈ 220 K. A further, ancillary result is that gray models fail to reproduce basic features of atmospheric radiative cooling.

## Abstract

The cooling-to-space (CTS) approximation says that the radiative cooling of an atmospheric layer is dominated by that layer’s emission to space, while radiative exchange with layers above and below largely cancel. Though the CTS approximation has been demonstrated empirically and is thus fairly well accepted, a theoretical justification is lacking. Furthermore, the intuition behind the CTS approximation cannot be universally valid, as the CTS approximation fails in the case of pure radiative equilibrium. Motivated by this, we investigate the CTS approximation in detail. We frame the CTS approximation in terms of a novel decomposition of radiative flux divergence, which better captures the cancellation of exchange terms. We also derive validity criteria for the CTS approximation, using simple analytical theory. We apply these criteria in the context of both gray gas pure radiative equilibrium (PRE) and radiative–convective equilibrium (RCE) to understand how the CTS approximation arises and why it fails in PRE. When applied to realistic gases in RCE, these criteria predict that the CTS approximation should hold well for H_{2}O but less so for CO_{2}, a conclusion we verify with line-by-line radiative transfer calculations. Along the way we also discuss the well-known “*τ* = 1 law,” and its dependence on the choice of vertical coordinate.

## Abstract

The cooling-to-space (CTS) approximation says that the radiative cooling of an atmospheric layer is dominated by that layer’s emission to space, while radiative exchange with layers above and below largely cancel. Though the CTS approximation has been demonstrated empirically and is thus fairly well accepted, a theoretical justification is lacking. Furthermore, the intuition behind the CTS approximation cannot be universally valid, as the CTS approximation fails in the case of pure radiative equilibrium. Motivated by this, we investigate the CTS approximation in detail. We frame the CTS approximation in terms of a novel decomposition of radiative flux divergence, which better captures the cancellation of exchange terms. We also derive validity criteria for the CTS approximation, using simple analytical theory. We apply these criteria in the context of both gray gas pure radiative equilibrium (PRE) and radiative–convective equilibrium (RCE) to understand how the CTS approximation arises and why it fails in PRE. When applied to realistic gases in RCE, these criteria predict that the CTS approximation should hold well for H_{2}O but less so for CO_{2}, a conclusion we verify with line-by-line radiative transfer calculations. Along the way we also discuss the well-known “*τ* = 1 law,” and its dependence on the choice of vertical coordinate.

## Abstract

This study examines dynamic pressure drag on rising dry buoyant thermals. A theoretical expression for drag coefficient *C _{d}
* as a function of several other nondimensional parameters governing thermal dynamics is derived based on combining the thermal momentum budget with the similarity theory of Scorer. Using values for these nondimensional parameters from previous studies, the theory suggests drag on thermals is small relative to that on solid spheres in laminar or turbulent flow. Two sets of numerical simulations of thermals in an unstratified, neutrally stable environment using an LES configuration of the Cloud Model 1 (CM1) are analyzed. One set has a relatively low effective Reynolds number

*R*and the other has an order-of-magnitude-higher

_{e}*R*; these produce laminar and turbulent resolved-scale flows, respectively. Consistent with the theoretical

_{e}*C*, the magnitude of drag is small in all simulations. However, whereas the laminar thermals have

_{d}*C*≈ 0.01, the turbulent thermals have weakly

_{d}*negative*drag (

*C*≈ −0.1). This difference is explained by the laminar thermals having near vertical symmetry but the turbulent thermals exhibiting considerable vertical asymmetry of their azimuthally averaged flows. In the laminar thermals, buoyancy rapidly becomes concentrated around the main centers of rotation located along the horizontal central axis, leading to expansion of thermals via baroclinic vorticity generation but doing little to break vertical symmetry of the flow. Vertical asymmetry of the azimuthally averaged flow of turbulent thermals is attributed mainly to small-scale resolved eddies that are concentrated in the upper part of the thermals.

_{d}## Abstract

This study examines dynamic pressure drag on rising dry buoyant thermals. A theoretical expression for drag coefficient *C _{d}
* as a function of several other nondimensional parameters governing thermal dynamics is derived based on combining the thermal momentum budget with the similarity theory of Scorer. Using values for these nondimensional parameters from previous studies, the theory suggests drag on thermals is small relative to that on solid spheres in laminar or turbulent flow. Two sets of numerical simulations of thermals in an unstratified, neutrally stable environment using an LES configuration of the Cloud Model 1 (CM1) are analyzed. One set has a relatively low effective Reynolds number

*R*and the other has an order-of-magnitude-higher

_{e}*R*; these produce laminar and turbulent resolved-scale flows, respectively. Consistent with the theoretical

_{e}*C*, the magnitude of drag is small in all simulations. However, whereas the laminar thermals have

_{d}*C*≈ 0.01, the turbulent thermals have weakly

_{d}*negative*drag (

*C*≈ −0.1). This difference is explained by the laminar thermals having near vertical symmetry but the turbulent thermals exhibiting considerable vertical asymmetry of their azimuthally averaged flows. In the laminar thermals, buoyancy rapidly becomes concentrated around the main centers of rotation located along the horizontal central axis, leading to expansion of thermals via baroclinic vorticity generation but doing little to break vertical symmetry of the flow. Vertical asymmetry of the azimuthally averaged flow of turbulent thermals is attributed mainly to small-scale resolved eddies that are concentrated in the upper part of the thermals.

_{d}## Abstract

Clear-sky CO_{2} forcing is known to vary significantly over the globe, but the state dependence that controls this is not well understood. Here we extend the formalism of Wilson and Gea-Banacloche to obtain a quantitatively accurate analytical model for spatially varying instantaneous CO_{2} forcing, which depends only on surface temperature *T*
_{
s
}, stratospheric temperature, and column relative humidity (RH). This model shows that CO_{2} forcing can be considered a swap of surface emission for stratospheric emission, and thus depends primarily on surface–stratosphere temperature contrast. The strong meridional gradient in CO_{2} forcing is thus largely due to the strong meridional gradient in *T*
_{
s
}. In the tropics and midlatitudes, however, the presence of H_{2}O modulates the forcing by replacing surface emission with RH-dependent atmospheric emission. This substantially reduces the forcing in the tropics, introduces forcing variations due to spatially varying RH, and sets an upper limit (with respect to *T*
_{
s
} variations) on CO_{2} forcing that is reached in the present-day tropics. In addition, we extend our analytical model to the instantaneous tropopause forcing, and find that this forcing depends on *T*
_{
s
}
*only*, with no dependence on stratospheric temperature. We also analyze the *τ* = 1 approximation for the emission level and derive an exact formula for the emission level, which yields values closer to *τ* = 1/2 than to *τ* = 1.

## Abstract

Clear-sky CO_{2} forcing is known to vary significantly over the globe, but the state dependence that controls this is not well understood. Here we extend the formalism of Wilson and Gea-Banacloche to obtain a quantitatively accurate analytical model for spatially varying instantaneous CO_{2} forcing, which depends only on surface temperature *T*
_{
s
}, stratospheric temperature, and column relative humidity (RH). This model shows that CO_{2} forcing can be considered a swap of surface emission for stratospheric emission, and thus depends primarily on surface–stratosphere temperature contrast. The strong meridional gradient in CO_{2} forcing is thus largely due to the strong meridional gradient in *T*
_{
s
}. In the tropics and midlatitudes, however, the presence of H_{2}O modulates the forcing by replacing surface emission with RH-dependent atmospheric emission. This substantially reduces the forcing in the tropics, introduces forcing variations due to spatially varying RH, and sets an upper limit (with respect to *T*
_{
s
} variations) on CO_{2} forcing that is reached in the present-day tropics. In addition, we extend our analytical model to the instantaneous tropopause forcing, and find that this forcing depends on *T*
_{
s
}
*only*, with no dependence on stratospheric temperature. We also analyze the *τ* = 1 approximation for the emission level and derive an exact formula for the emission level, which yields values closer to *τ* = 1/2 than to *τ* = 1.

## Abstract

Climate models and observations robustly agree that Earth’s clear-sky longwave feedback has a value of about −2 W m^{−2} K^{−1}, suggesting that this feedback can be estimated from first principles. In this study, we derive an analytic model for Earth’s clear-sky longwave feedback. Our approach uses a novel spectral decomposition that splits the feedback into four components: a surface Planck feedback and three atmospheric feedbacks from CO_{2}, H_{2}O, and the H_{2}O continuum. We obtain analytic expressions for each of these terms, and the model can also be framed in terms of Simpson’s law and deviations therefrom. We validate the model by comparing it against line-by-line radiative transfer calculations across a wide range of climates. Additionally, the model qualitatively matches the spatial feedback maps of a comprehensive climate model. For present-day Earth, our analysis shows that the clear-sky longwave feedback is dominated by the surface in the global mean and in the dry subtropics; meanwhile, atmospheric feedbacks from CO_{2} and H_{2}O become important in the inner tropics. Together, these results show that a spectral view of Earth’s clear-sky longwave feedback elucidates not only its global-mean magnitude, but also its spatial pattern and its state dependence across past and future climates.

### Significance Statement

The climate feedback determines how much our planet warms due to changes in radiative forcing. For more than 50 years scientists have been predicting this feedback using complex numerical models. Except for cloud effects the numerical models largely agree, lending confidence to global warming predictions, but nobody has yet derived the feedback from simpler considerations. We show that Earth’s clear-sky longwave feedback can be estimated using only pen and paper. Our results confirm that numerical climate models get the right number for the right reasons, and allow us to explain regional and state variations of Earth’s climate feedback. These variations are difficult to understand solely from numerical models but are crucial for past and future climates.

## Abstract

Climate models and observations robustly agree that Earth’s clear-sky longwave feedback has a value of about −2 W m^{−2} K^{−1}, suggesting that this feedback can be estimated from first principles. In this study, we derive an analytic model for Earth’s clear-sky longwave feedback. Our approach uses a novel spectral decomposition that splits the feedback into four components: a surface Planck feedback and three atmospheric feedbacks from CO_{2}, H_{2}O, and the H_{2}O continuum. We obtain analytic expressions for each of these terms, and the model can also be framed in terms of Simpson’s law and deviations therefrom. We validate the model by comparing it against line-by-line radiative transfer calculations across a wide range of climates. Additionally, the model qualitatively matches the spatial feedback maps of a comprehensive climate model. For present-day Earth, our analysis shows that the clear-sky longwave feedback is dominated by the surface in the global mean and in the dry subtropics; meanwhile, atmospheric feedbacks from CO_{2} and H_{2}O become important in the inner tropics. Together, these results show that a spectral view of Earth’s clear-sky longwave feedback elucidates not only its global-mean magnitude, but also its spatial pattern and its state dependence across past and future climates.

### Significance Statement

The climate feedback determines how much our planet warms due to changes in radiative forcing. For more than 50 years scientists have been predicting this feedback using complex numerical models. Except for cloud effects the numerical models largely agree, lending confidence to global warming predictions, but nobody has yet derived the feedback from simpler considerations. We show that Earth’s clear-sky longwave feedback can be estimated using only pen and paper. Our results confirm that numerical climate models get the right number for the right reasons, and allow us to explain regional and state variations of Earth’s climate feedback. These variations are difficult to understand solely from numerical models but are crucial for past and future climates.

## 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 *A _{r}
*, defined as height divided by width of the initial buoyancy perturbation. An analytic equation for thermal ascent rate

*w*that depends on

_{t}*α*is derived from the thermal-volume-averaged momentum budget equation. The thermal top height when

*w*is maximum, defining a critical height

_{t}*z*, is inversely proportional to

_{c}*α*. The height

*z*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

_{c}*α*and

*A*. Physically, this

_{r}*α*–

*A*relation is connected to changes in circulation as

_{r}*A*is modified. Numerical simulations of thermals with

_{r}*A*varied from 0.5 to 2 give

_{r}*α*values close to the analytic theoretical relation, with a factor of ∼3 decrease in

*α*as

*A*is increased from 0.5 to 2. The theory also explains why

_{r}*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.

## 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 *A _{r}
*, defined as height divided by width of the initial buoyancy perturbation. An analytic equation for thermal ascent rate

*w*that depends on

_{t}*α*is derived from the thermal-volume-averaged momentum budget equation. The thermal top height when

*w*is maximum, defining a critical height

_{t}*z*, is inversely proportional to

_{c}*α*. The height

*z*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

_{c}*α*and

*A*. Physically, this

_{r}*α*–

*A*relation is connected to changes in circulation as

_{r}*A*is modified. Numerical simulations of thermals with

_{r}*A*varied from 0.5 to 2 give

_{r}*α*values close to the analytic theoretical relation, with a factor of ∼3 decrease in

*α*as

*A*is increased from 0.5 to 2. The theory also explains why

_{r}*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.