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- Author or Editor: Nadir Jeevanjee x
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Abstract
Clear-sky CO2 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 CO2 forcing, which depends only on surface temperature T s , stratospheric temperature, and column relative humidity (RH). This model shows that CO2 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 CO2 forcing is thus largely due to the strong meridional gradient in T s . In the tropics and midlatitudes, however, the presence of H2O 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 CO2 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 CO2 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 CO2 forcing, which depends only on surface temperature T s , stratospheric temperature, and column relative humidity (RH). This model shows that CO2 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 CO2 forcing is thus largely due to the strong meridional gradient in T s . In the tropics and midlatitudes, however, the presence of H2O 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 CO2 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 CO2, H2O, and the H2O 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 CO2 and H2O 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 CO2, H2O, and the H2O 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 CO2 and H2O 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 Ar , defined as height divided by width of the initial buoyancy perturbation. An analytic equation for thermal ascent rate wt that depends on α is derived from the thermal-volume-averaged momentum budget equation. The thermal top height when wt is maximum, defining a critical height zc , is inversely proportional to α. The height zc also corresponds to the thermal top height when buoyant fluid along the thermal’s vertical axis is fully replaced by entrained nonbuoyant environmental fluid rising from below the thermal. The time scale for this process is controlled by the vertical velocity of parcels rising upward through the thermal’s core. This parcel vertical velocity is approximated from Hill’s analytic spherical vortex, yielding an analytic inverse relation between α and Ar . Physically, this α–Ar relation is connected to changes in circulation as Ar is modified. Numerical simulations of thermals with Ar varied from 0.5 to 2 give α values close to the analytic theoretical relation, with a factor of ∼3 decrease in α as Ar is increased from 0.5 to 2. The theory also explains why α of initially spherical thermals from past laboratory and modeling studies is about 0.15. Overall, this study provides a theoretical underpinning for understanding the entrainment behavior of thermals, relevant to buoyantly driven atmospheric flows.
Significance Statement
Thermals, which are coherent, quasi-spherical regions of upward-moving buoyant fluid, are a key feature of many convective atmospheric flows. The purpose of this study is to characterize how thermals entrain surrounding fluid and spread out as they rise. We use theory and numerical modeling to explain why entrainment rate decreases with an increase in the initial thermal aspect ratio—the ratio of height to width. This work also explains why the nondimensional spreading rate (change in thermal radius with height) of initially spherical thermals from past laboratory and numerical modeling studies is about 0.15. Overall, this work provides a framework for conceptualizing the entrainment behavior of thermals and thus improved understanding of vertical transport in convective atmospheric flows.
Abstract
This study uses theory and numerical simulations to analyze the nondimensional spreading rate α (change in radius with height) of buoyant thermals as they rise and entrain surrounding environmental fluid. A focus is on how α varies with initial thermal aspect ratio Ar , defined as height divided by width of the initial buoyancy perturbation. An analytic equation for thermal ascent rate wt that depends on α is derived from the thermal-volume-averaged momentum budget equation. The thermal top height when wt is maximum, defining a critical height zc , is inversely proportional to α. The height zc also corresponds to the thermal top height when buoyant fluid along the thermal’s vertical axis is fully replaced by entrained nonbuoyant environmental fluid rising from below the thermal. The time scale for this process is controlled by the vertical velocity of parcels rising upward through the thermal’s core. This parcel vertical velocity is approximated from Hill’s analytic spherical vortex, yielding an analytic inverse relation between α and Ar . Physically, this α–Ar relation is connected to changes in circulation as Ar is modified. Numerical simulations of thermals with Ar varied from 0.5 to 2 give α values close to the analytic theoretical relation, with a factor of ∼3 decrease in α as Ar is increased from 0.5 to 2. The theory also explains why α of initially spherical thermals from past laboratory and modeling studies is about 0.15. Overall, this study provides a theoretical underpinning for understanding the entrainment behavior of thermals, relevant to buoyantly driven atmospheric flows.
Significance Statement
Thermals, which are coherent, quasi-spherical regions of upward-moving buoyant fluid, are a key feature of many convective atmospheric flows. The purpose of this study is to characterize how thermals entrain surrounding fluid and spread out as they rise. We use theory and numerical modeling to explain why entrainment rate decreases with an increase in the initial thermal aspect ratio—the ratio of height to width. This work also explains why the nondimensional spreading rate (change in thermal radius with height) of initially spherical thermals from past laboratory and numerical modeling studies is about 0.15. Overall, this work provides a framework for conceptualizing the entrainment behavior of thermals and thus improved understanding of vertical transport in convective atmospheric flows.
