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Abstract
An analytical model is derived for tropical relative humidity using only the Clausius–Clapeyron relation, hydrostatic balance, and a bulk-plume water budget. This theory is constructed for radiative–convective equilibrium and compared against a cloud-resolving model. With some reinterpretation of variables, it can be applied more generally to the entire tropics.
Given four variables—pressure, temperature, and the fractional entrainment and detrainment rates—the equations predict the relative humidity (RH) and the temperature lapse rate analytically. The RH is a simple ratio involving the fractional detrainment rate and the water-vapor lapse rate. When integrated upward in height, the equations give profiles of RH and temperature for a convecting atmosphere.
The theory explains the magnitude of RH and the “C” shape of the tropospheric RH profile. It also predicts that RH is an invariant function of temperature as the atmosphere warms, and this behavior matches what has been seen in global climate models and what is demonstrated here with cloud-resolving simulations. Extending the theory to include the evaporation of hydrometeors, a lower bound is derived for the precipitation efficiency (PE) at each height: PE > 1 − RH. In a cloud-resolving simulation, this constraint is obeyed with the PE profile taking the shape of an inverted C shape.
Abstract
An analytical model is derived for tropical relative humidity using only the Clausius–Clapeyron relation, hydrostatic balance, and a bulk-plume water budget. This theory is constructed for radiative–convective equilibrium and compared against a cloud-resolving model. With some reinterpretation of variables, it can be applied more generally to the entire tropics.
Given four variables—pressure, temperature, and the fractional entrainment and detrainment rates—the equations predict the relative humidity (RH) and the temperature lapse rate analytically. The RH is a simple ratio involving the fractional detrainment rate and the water-vapor lapse rate. When integrated upward in height, the equations give profiles of RH and temperature for a convecting atmosphere.
The theory explains the magnitude of RH and the “C” shape of the tropospheric RH profile. It also predicts that RH is an invariant function of temperature as the atmosphere warms, and this behavior matches what has been seen in global climate models and what is demonstrated here with cloud-resolving simulations. Extending the theory to include the evaporation of hydrometeors, a lower bound is derived for the precipitation efficiency (PE) at each height: PE > 1 − RH. In a cloud-resolving simulation, this constraint is obeyed with the PE profile taking the shape of an inverted C shape.
Abstract
The Gregory–Kershaw–Inness (GKI) parameterization of convective momentum transport, which has a tunable parameter C, is shown to be identical to a parameterization with no pressure gradient force and a mass flux smaller by a factor of 1 − C. Using cloud-resolving simulations, the transilient matrix for momentum is diagnosed for deep convection in radiative–convective equilibrium. Using this transilient matrix, it is shown that the GKI scheme underestimates the compensating subsidence of momentum by a factor of 1 − C, as predicted. This result is confirmed using a large-eddy simulation.
Abstract
The Gregory–Kershaw–Inness (GKI) parameterization of convective momentum transport, which has a tunable parameter C, is shown to be identical to a parameterization with no pressure gradient force and a mass flux smaller by a factor of 1 − C. Using cloud-resolving simulations, the transilient matrix for momentum is diagnosed for deep convection in radiative–convective equilibrium. Using this transilient matrix, it is shown that the GKI scheme underestimates the compensating subsidence of momentum by a factor of 1 − C, as predicted. This result is confirmed using a large-eddy simulation.
Abstract
A method is introduced for directly measuring convective entrainment and detrainment in a cloud-resolving simulation. This technique is used to quantify the errors in the entrainment and detrainment estimates obtained using the standard bulk-plume method. The bulk-plume method diagnoses these rates from the convective flux of some conserved tracer, such as total water in nonprecipitating convection. By not accounting for the variability of this tracer in clouds and in the environment, it is argued that the bulk-plume equations systematically underestimate entrainment. Using tracers with different vertical profiles, it is also shown that the bulk-plume estimates are tracer dependent and, in some cases, unphysical. The new direct-measurement technique diagnoses entrainment and detrainment at the gridcell level without any recourse to conserved tracers. Using this method in large-eddy simulations of shallow and deep convection, it is found that the bulk-plume method underestimates entrainment by roughly a factor of 2. The directly measured entrainment rates are then compared to cloud height and cloud buoyancy. Contrary to existing theories, fractional entrainment is not found to scale like the inverse of height, the cloud buoyancy, or the gradient of cloud buoyancy. On the other hand, fractional detrainment is found to scale linearly with cloud buoyancy. Finally, direct measurement is used to diagnose the spatial distribution of entrainment and detrainment during the evolution of an individual deep cumulonimbus.
Abstract
A method is introduced for directly measuring convective entrainment and detrainment in a cloud-resolving simulation. This technique is used to quantify the errors in the entrainment and detrainment estimates obtained using the standard bulk-plume method. The bulk-plume method diagnoses these rates from the convective flux of some conserved tracer, such as total water in nonprecipitating convection. By not accounting for the variability of this tracer in clouds and in the environment, it is argued that the bulk-plume equations systematically underestimate entrainment. Using tracers with different vertical profiles, it is also shown that the bulk-plume estimates are tracer dependent and, in some cases, unphysical. The new direct-measurement technique diagnoses entrainment and detrainment at the gridcell level without any recourse to conserved tracers. Using this method in large-eddy simulations of shallow and deep convection, it is found that the bulk-plume method underestimates entrainment by roughly a factor of 2. The directly measured entrainment rates are then compared to cloud height and cloud buoyancy. Contrary to existing theories, fractional entrainment is not found to scale like the inverse of height, the cloud buoyancy, or the gradient of cloud buoyancy. On the other hand, fractional detrainment is found to scale linearly with cloud buoyancy. Finally, direct measurement is used to diagnose the spatial distribution of entrainment and detrainment during the evolution of an individual deep cumulonimbus.
Abstract
The entropy budget has been a popular starting point for theories of the work, or dissipation, performed by moist atmospheres. For a dry atmosphere, the entropy budget provides a theory for the dissipation in terms of the imposed diabatic heat sources. For a moist atmosphere, the difficulties in quantifying irreversible moist processes or the value of the condensation temperature have so far frustrated efforts to construct a theory of dissipation. With this complication in mind, one of the goals here is to investigate the predictive power of the budget of dry entropy (i.e., the heat capacity times the logarithm of potential temperature).
Toward this end, the dry-entropy budget is derived for an atmosphere with realistic heat capacities and a solid-water phase, features that were absent from some previous studies of atmospheric entropy. It is shown that the dry-entropy budget may be interpreted as the sum of sources and sinks from six processes, which are, in order of decreasing magnitude, radiative cooling, condensation heating, sensible heating at the surface, wind-generated frictional dissipation, lifting of water, and transport of heat from the melting line to the upper troposphere. This picture leads to an alternative explanation for the low efficiency of the moist atmospheric engine.
Numerical simulations are presented from a new cloud-resolving model, Das Atmosphärische Modell, which was designed to conserve energy and close the dry-entropy budget. Simulations with and without subgrid diffusion of heat and water are compared to investigate the impact of subgrid parameterizations on the terms in the dry-entropy budget. The numerical results suggest a particularly simple parameterization of wind-generated dissipation that appears to be valid for changes in sea surface temperature and mean wind. The dry-entropy budget also points to various changes in forcings and parameterizations that could be expected to increase or decrease the wind-generated dissipation.
Abstract
The entropy budget has been a popular starting point for theories of the work, or dissipation, performed by moist atmospheres. For a dry atmosphere, the entropy budget provides a theory for the dissipation in terms of the imposed diabatic heat sources. For a moist atmosphere, the difficulties in quantifying irreversible moist processes or the value of the condensation temperature have so far frustrated efforts to construct a theory of dissipation. With this complication in mind, one of the goals here is to investigate the predictive power of the budget of dry entropy (i.e., the heat capacity times the logarithm of potential temperature).
Toward this end, the dry-entropy budget is derived for an atmosphere with realistic heat capacities and a solid-water phase, features that were absent from some previous studies of atmospheric entropy. It is shown that the dry-entropy budget may be interpreted as the sum of sources and sinks from six processes, which are, in order of decreasing magnitude, radiative cooling, condensation heating, sensible heating at the surface, wind-generated frictional dissipation, lifting of water, and transport of heat from the melting line to the upper troposphere. This picture leads to an alternative explanation for the low efficiency of the moist atmospheric engine.
Numerical simulations are presented from a new cloud-resolving model, Das Atmosphärische Modell, which was designed to conserve energy and close the dry-entropy budget. Simulations with and without subgrid diffusion of heat and water are compared to investigate the impact of subgrid parameterizations on the terms in the dry-entropy budget. The numerical results suggest a particularly simple parameterization of wind-generated dissipation that appears to be valid for changes in sea surface temperature and mean wind. The dry-entropy budget also points to various changes in forcings and parameterizations that could be expected to increase or decrease the wind-generated dissipation.
Abstract
Cloud-resolving simulations of convection over a surface temperature hot spot are used to evaluate the weak pressure gradient (WPG) and weak temperature gradient (WTG) approximations. The premise of the relaxed form of WTG—that vertical velocity is equal to buoyancy times a positive time scale—is found to be violated by thick layers of negative buoyancy in steady-state ascent. The premise of WPG—that horizontal divergence and pressure anomalies are collocated—is validated by these simulations. When implemented in a cloud-resolving model, WPG replicates buoyancy transients exceptionally well, including the adiabatic lifting of air below buoyancy anomalies. WTG captures neither this effect nor the associated triggering of moist convection. For steady states, WTG produces vertical velocity profiles that are too top heavy. On the other hand, WPG generates velocity profiles that closely match fully resolved hot-spot simulations. Taken together, the evidence suggests that WPG is a relatively accurate method for parameterizing supradomain-scale (SDS) dynamics.
Abstract
Cloud-resolving simulations of convection over a surface temperature hot spot are used to evaluate the weak pressure gradient (WPG) and weak temperature gradient (WTG) approximations. The premise of the relaxed form of WTG—that vertical velocity is equal to buoyancy times a positive time scale—is found to be violated by thick layers of negative buoyancy in steady-state ascent. The premise of WPG—that horizontal divergence and pressure anomalies are collocated—is validated by these simulations. When implemented in a cloud-resolving model, WPG replicates buoyancy transients exceptionally well, including the adiabatic lifting of air below buoyancy anomalies. WTG captures neither this effect nor the associated triggering of moist convection. For steady states, WTG produces vertical velocity profiles that are too top heavy. On the other hand, WPG generates velocity profiles that closely match fully resolved hot-spot simulations. Taken together, the evidence suggests that WPG is a relatively accurate method for parameterizing supradomain-scale (SDS) dynamics.
Abstract
By deriving analytical solutions to radiative–convective equilibrium (RCE), it is shown mathematically that convective available potential energy (CAPE) exhibits Clausius–Clapeyron (CC) scaling over a wide range of surface temperatures up to 310 K. Above 310 K, CAPE deviates from CC scaling and even decreases with warming at very high surface temperatures. At the surface temperature of the current tropics, the analytical solutions predict that CAPE increases at a rate of about 6%–7% per kelvin of surface warming. The analytical solutions also provide insight on how the tropopause height and stratospheric humidity change with warming. Changes in the tropopause height exhibit CC scaling, with the tropopause rising by about 400 m per kelvin of surface warming at current tropical temperatures and by about 1–2 km K−1 at surface temperatures in the range of 320–340 K. The specific humidity of the stratosphere exhibits super-CC scaling at temperatures moderately warmer than the current tropics. With a surface temperature of the current tropics, the stratospheric specific humidity increases by about 6% per kelvin of surface warming, but the rate of increase is as high as 30% K−1 at warmer surface temperatures.
Abstract
By deriving analytical solutions to radiative–convective equilibrium (RCE), it is shown mathematically that convective available potential energy (CAPE) exhibits Clausius–Clapeyron (CC) scaling over a wide range of surface temperatures up to 310 K. Above 310 K, CAPE deviates from CC scaling and even decreases with warming at very high surface temperatures. At the surface temperature of the current tropics, the analytical solutions predict that CAPE increases at a rate of about 6%–7% per kelvin of surface warming. The analytical solutions also provide insight on how the tropopause height and stratospheric humidity change with warming. Changes in the tropopause height exhibit CC scaling, with the tropopause rising by about 400 m per kelvin of surface warming at current tropical temperatures and by about 1–2 km K−1 at surface temperatures in the range of 320–340 K. The specific humidity of the stratosphere exhibits super-CC scaling at temperatures moderately warmer than the current tropics. With a surface temperature of the current tropics, the stratospheric specific humidity increases by about 6% per kelvin of surface warming, but the rate of increase is as high as 30% K−1 at warmer surface temperatures.
Abstract
Accurate, explicit, and analytic expressions are derived for the dewpoint and frost point as functions of temperature and relative humidity. These are derived theoretically in terms of physical constants using the Rankine–Kirchhoff approximations, which assume an ideal gas, fixed heat capacities, and zero specific volume of condensates. Compared to modern laboratory measurements, the expressions are accurate to within a few hundredths of a degree over the full range of Earth-relevant temperatures, from 180 to 273 K for the frost point and 230 to 330 K for the dewpoint.
Abstract
Accurate, explicit, and analytic expressions are derived for the dewpoint and frost point as functions of temperature and relative humidity. These are derived theoretically in terms of physical constants using the Rankine–Kirchhoff approximations, which assume an ideal gas, fixed heat capacities, and zero specific volume of condensates. Compared to modern laboratory measurements, the expressions are accurate to within a few hundredths of a degree over the full range of Earth-relevant temperatures, from 180 to 273 K for the frost point and 230 to 330 K for the dewpoint.
Abstract
This paper explores whether cumulus drag (i.e., the damping of winds by convective momentum transport) can be described by an effective Rayleigh drag (i.e., the damping of winds on a constant time scale). Analytical expressions are derived for the damping time scale and descent speed of wind profiles as caused by unorganized convection. Unlike Rayleigh drag, which has a constant damping time scale and zero descent speed, the theory predicts a damping time scale and a descent speed that both depend on the vertical wavelength of the wind profile. These results predict that short wavelengths damp faster and descend faster than long wavelengths, and these predictions are confirmed using large-eddy simulations. Both theory and simulations predict that the convective damping of large-scale circulations occurs on a time scale of O(1–10) days for vertical wavelengths in the range of 2–10 km.
Abstract
This paper explores whether cumulus drag (i.e., the damping of winds by convective momentum transport) can be described by an effective Rayleigh drag (i.e., the damping of winds on a constant time scale). Analytical expressions are derived for the damping time scale and descent speed of wind profiles as caused by unorganized convection. Unlike Rayleigh drag, which has a constant damping time scale and zero descent speed, the theory predicts a damping time scale and a descent speed that both depend on the vertical wavelength of the wind profile. These results predict that short wavelengths damp faster and descend faster than long wavelengths, and these predictions are confirmed using large-eddy simulations. Both theory and simulations predict that the convective damping of large-scale circulations occurs on a time scale of O(1–10) days for vertical wavelengths in the range of 2–10 km.
Abstract
A weak pressure gradient (WPG) approximation is introduced for parameterizing supradomain-scale (SDS) dynamics, and this method is compared to the relaxed form of the weak temperature gradient (WTG) approximation in the context of 3D, linearized, damped, Boussinesq equations. It is found that neither method is able to capture the two different time scales present in the full 3D equations. Nevertheless, WPG is argued to have several advantages over WTG. First, WPG correctly predicts the magnitude of the steady-state buoyancy anomalies generated by an applied heating, but WTG underestimates these buoyancy anomalies. It is conjectured that this underestimation may short-circuit the natural feedbacks between convective mass fluxes and local temperature anomalies. Second, WPG correctly predicts the adiabatic lifting of air below an initial buoyancy perturbation; WTG is unable to capture this nonlocal effect. It is hypothesized that this may be relevant to moist convection, where adiabatic lifting can reduce convective inhibition. Third, WPG agrees with the full 3D equations on the counterintuitive fact that an isolated heating applied to a column of Boussinesq fluid leads to a steady ascent with zero column-integrated buoyancy. This falsifies the premise of the relaxed form of WTG, which assumes that vertical velocity is proportional to buoyancy.
Abstract
A weak pressure gradient (WPG) approximation is introduced for parameterizing supradomain-scale (SDS) dynamics, and this method is compared to the relaxed form of the weak temperature gradient (WTG) approximation in the context of 3D, linearized, damped, Boussinesq equations. It is found that neither method is able to capture the two different time scales present in the full 3D equations. Nevertheless, WPG is argued to have several advantages over WTG. First, WPG correctly predicts the magnitude of the steady-state buoyancy anomalies generated by an applied heating, but WTG underestimates these buoyancy anomalies. It is conjectured that this underestimation may short-circuit the natural feedbacks between convective mass fluxes and local temperature anomalies. Second, WPG correctly predicts the adiabatic lifting of air below an initial buoyancy perturbation; WTG is unable to capture this nonlocal effect. It is hypothesized that this may be relevant to moist convection, where adiabatic lifting can reduce convective inhibition. Third, WPG agrees with the full 3D equations on the counterintuitive fact that an isolated heating applied to a column of Boussinesq fluid leads to a steady ascent with zero column-integrated buoyancy. This falsifies the premise of the relaxed form of WTG, which assumes that vertical velocity is proportional to buoyancy.
Abstract
A standard convention in moist thermodynamics, adopted by D. M. Romps and others, is to set the specific energy and entropy of dry air and liquid water to zero at the triple-point temperature and pressure. P. Marquet claims that this convention leads to physically incorrect results. To support this claim, Marquet presents numerical calculations of a lifted parcel. It is shown here that the claim is false and that the numerical calculations of Marquet are in error. In the context of a simple two-phase thermodynamic system, an analysis is presented here of the freedoms one has to choose additive constants in the definitions of energy and entropy. Many other misconceptions are corrected as well.
Abstract
A standard convention in moist thermodynamics, adopted by D. M. Romps and others, is to set the specific energy and entropy of dry air and liquid water to zero at the triple-point temperature and pressure. P. Marquet claims that this convention leads to physically incorrect results. To support this claim, Marquet presents numerical calculations of a lifted parcel. It is shown here that the claim is false and that the numerical calculations of Marquet are in error. In the context of a simple two-phase thermodynamic system, an analysis is presented here of the freedoms one has to choose additive constants in the definitions of energy and entropy. Many other misconceptions are corrected as well.
