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- Author or Editor: Nadir Jeevanjee x

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## Abstract

Syukoro (Suki) Manabe’s Nobel Prize in Physics was awarded largely for his early work on one-dimensional models of “radiative–convective equilibrium” (RCE), which produced the first credible estimates of Earth’s climate sensitivity. This article reviews that work and tries to identify those aspects that make it so distinctive. We argue that Manabe’s model of RCE contained three crucial ingredients. These are (i) a tight convective coupling of the surface to the troposphere, (ii) an assumption of fixed relative humidity rather than fixed absolute humidity, and (iii) a sufficiently realistic representation of greenhouse gas radiative transfer. Previous studies had separately identified these key ingredients, but none had properly combined them. We then discuss each of these ingredients in turn, highlighting how subsequent research in the intervening decades has only cemented their importance for understanding global climate change. We close by reflecting on the elegance of Manabe’s approach and its lasting value.

## Abstract

Syukoro (Suki) Manabe’s Nobel Prize in Physics was awarded largely for his early work on one-dimensional models of “radiative–convective equilibrium” (RCE), which produced the first credible estimates of Earth’s climate sensitivity. This article reviews that work and tries to identify those aspects that make it so distinctive. We argue that Manabe’s model of RCE contained three crucial ingredients. These are (i) a tight convective coupling of the surface to the troposphere, (ii) an assumption of fixed relative humidity rather than fixed absolute humidity, and (iii) a sufficiently realistic representation of greenhouse gas radiative transfer. Previous studies had separately identified these key ingredients, but none had properly combined them. We then discuss each of these ingredients in turn, highlighting how subsequent research in the intervening decades has only cemented their importance for understanding global climate change. We close by reflecting on the elegance of Manabe’s approach and its lasting value.

## 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

Tropical cyclone (TC) potential intensity (PI) theory has a well-known form, consistent with a Carnot cycle interpretation of TC energetics, which relates PI to mean environmental conditions: the difference between surface and TC outflow temperatures and the air–sea enthalpy disequilibrium. PI has also been defined as a difference in convective available potential energy (CAPE) between two parcels, and quantitative assessments of future changes make use of a numerical algorithm based on this definition. Here, an analysis shows the conditions under which these Carnot and CAPE-based PI definitions are equivalent. There are multiple conditions, not previously enumerated, which in particular reveal a role for irreversible entropy production from surface evaporation. This mathematical analysis is verified by numerical calculations of PI’s sensitivity to large changes in surface-air relative humidity. To gain physical insight into the connection between the CAPE and Carnot formulations of PI, we use a recently developed analytic theory for CAPE to derive, starting from the CAPE-based definition, a new approximate formula for PI that nearly recovers the previous Carnot PI formula. The derivation shows that the difference in undilute buoyancies of saturated and environmental parcels that determines CAPE PI can in fact be expressed as a difference in the parcels’ surface moist static energy, providing a physical link between the Carnot and CAPE formulations of PI. This combination of analysis and physical interpretation builds confidence in previous numerical CAPE-based PI calculations that use climate model projections of the future tropical environment.

## Abstract

Tropical cyclone (TC) potential intensity (PI) theory has a well-known form, consistent with a Carnot cycle interpretation of TC energetics, which relates PI to mean environmental conditions: the difference between surface and TC outflow temperatures and the air–sea enthalpy disequilibrium. PI has also been defined as a difference in convective available potential energy (CAPE) between two parcels, and quantitative assessments of future changes make use of a numerical algorithm based on this definition. Here, an analysis shows the conditions under which these Carnot and CAPE-based PI definitions are equivalent. There are multiple conditions, not previously enumerated, which in particular reveal a role for irreversible entropy production from surface evaporation. This mathematical analysis is verified by numerical calculations of PI’s sensitivity to large changes in surface-air relative humidity. To gain physical insight into the connection between the CAPE and Carnot formulations of PI, we use a recently developed analytic theory for CAPE to derive, starting from the CAPE-based definition, a new approximate formula for PI that nearly recovers the previous Carnot PI formula. The derivation shows that the difference in undilute buoyancies of saturated and environmental parcels that determines CAPE PI can in fact be expressed as a difference in the parcels’ surface moist static energy, providing a physical link between the Carnot and CAPE formulations of PI. This combination of analysis and physical interpretation builds confidence in previous numerical CAPE-based PI calculations that use climate model projections of the future tropical environment.

## 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}