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George C. Craig

Abstract

The three dimensional semigeostrophic equations are derived by a formal asymptotic analysis. The equations are scaled with typical values of the gradients across absolute momentum contours along isentropic surfaces and are accurate to first order in a small parameter that can be regarded as a modified Rossby number.

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John Thuburn and George C. Craig

Abstract

The sensitivity of the tropopause height to various external parameters has been investigated using a global circulation model (GCM). The tropopause height was found to be strongly sensitive to the temperature at the earth’s surface, less sensitive to the ozone distribution, and hardly sensitive at all to moderate changes in the earth’s rotation rate. The strong sensitivity to surface temperature occurs through changes in the atmospheric moisture distribution and its resulting radiative effects. The radiative and dynamical mechanisms thought to maintain the tropopause height have been investigated in some detail. The assumption that the lower stratosphere is close to radiative equilibrium leads to an easily computed relationship between tropospheric lapse rate and tropopause height. This relationship was found to hold well in the GCM in the extratropics away from the winter pole. Possible reasons for the breakdown of the relationship in the Tropics and near the winter pole are discussed. Simple relationships predicted by two different baroclinic adjustment theories, between parameters such as potential temperature gradients, the Coriolis parameter, and tropopause height, were examined. When some of these parameters were changed explicitly in GCM experiments, the remaining parameters, determined internally by the GCM, did not respond in the predicted way. These results cast doubt on the relevance of baroclinic adjustment to the height of the tropopause.

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John Thuburn and George C. Craig

Abstract

Earlier theoretical and modeling work introduced the concept of a radiative constraint relating tropopause height to tropospheric lapse rate and other factors such as surface temperature. Here a minimal quantitative model for the radiative constraint is presented and used to illustrate the essential physics underlying the radiative constraint, which involves the approximate balance between absorption and emission of thermal infrared (IR) radiation determining tropopause temperature.

The results of the minimal model are then extended in two ways. First, the effects of including a more realistic treatment of IR radiation are quantified. Second, the radiative constraint model is extended to take into account non-IR warming processes such as solar heating and dynamical warming near the tropopause. The sensitivity of tropopause height to non-IR warming is estimated to be a few kilometers per K day−1, with positive warming leading to a lower tropopause. Sensitivities comparable to this are found in GCM experiments in which imposed changes in the ozone distribution or in the driving of the stratospheric residual mean meridional circulation lead to changes in tropopause height. In the Tropics the influence of the stratospheric circulation is found to extend down at least as far as the main convective outflow level, some 5 km below the temperature minimum.

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George C. Craig and Andreas Dörnbrack

Abstract

Systematic numerical experiments were conducted to determine the spatial resolution required to resolve a moist thermal show convergence at a scale proportional to the smaller of the initial thermal diameter D 0 and a buoyancy length scale L buoy. The buoyancy length scale L buoy = ΔT 0/ΔΓ (ΔT 0 is the initial buoyancy excess of the thermal and ΔΓ is the ambient stratification) describes the maximum vertical displacement that can be induced against the stratification in the environment by buoyancy-driven pressure perturbations in the cloud and, thus, the maximum scale of eddies that cross the cloud boundary. For typical atmospheric conditions in which the cloud size D 0 is larger than L buoy, numerical simulations of the mixing processes in cumulus clouds must resolve L buoy.

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Kirstin Kober and George C. Craig

Abstract

Stochastic perturbations allow for the representation of small-scale variability due to unresolved physical processes. However, the properties of this variability depend on model resolution and weather regime. A physically based method is presented for introducing stochastic perturbations into kilometer-scale atmospheric models that explicitly account for these dependencies. The amplitude of the perturbations is based on information obtained from the model’s subgrid turbulence parameterization, while the spatial and temporal correlations are based on physical length and time scales of the turbulent motions. The stochastic perturbations lead to triggering of additional convective cells and improved precipitation amounts in simulations of two days with weak synoptic forcing of convection but different amounts of precipitation. The perturbations had little impact in a third case study, where precipitation was mainly associated with a cold front. In contrast, an unphysical version of the scheme with constant perturbation amplitude performed poorly since there was no perturbation amplitude that would give improved amounts of precipitation during the day without generating spurious convection at other times.

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Brenda G. Cohen and George C. Craig

Abstract

The theoretical predictions derived in Part I of this study for the equilibrium fluctuations of an idealized ensemble of noninteracting, pointlike cumulus clouds are tested against three-dimensional cloud resolving model (CRM) simulations of radiative–convective equilibrium. Simulations with different radiative cooling rates are used to give a range of cloud densities, while imposed vertical wind shear of different strengths is used to produce different degrees of convective organization. The distribution of mass flux of individual clouds is found to be exponential in all simulations, in agreement with the theory. The distribution of total mass flux over a finite region also agrees well (to within around 10%) with the theoretical prediction for all simulations, but only after a correction to the modeled variance to take account of the finite size of clouds has been made. In the absence of imposed vertical wind shear, some spatial clustering of convective cells is observed at lower forcings (−2 and −4 K day−1) on a scale of 10–20 km, while at higher forcings (−8, −12, and −16 K day−1), there is a tendency toward spatial regularity on the same scale. These localized cloud interactions, however, appear to have little effect on the magnitude of the mass flux variability. Surprisingly, the convective organization obtained in the simulations with vertical wind shear has only a small effect on the mass flux statistics, even though it shows clearly in the location of the clouds.

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George C. Craig and Han-Ru Cho

Abstract

Observational studies show that polar air stream cyclones form preferentially to the north of the polar front in regions of high vorticity and low static stability, although the baroclinicity may be stronger elsewhere. This phenomenon is investigated by considering a semigeostrophic linear stability analysis of a constant potential vorticity zonal jet in the presence of parameterized cumulus heating in the cold air mass. Three types of unstable modes are found for different amounts of heating. With relatively small heating rates, the usual baroclinic instability occurs, with the disturbance centered on the jet axis. With moderate heating, the fastest growing mode is again a mainly baroclinic system, but with significant amplitude only in the region of heating on the cold-air side of the jet. Finally, for sufficiently large heating rates, a small-scale disturbance, which is driven primarily by diabatic processes, forms in the cold air mass. There is a continuous transition between the three types of instability as the heating is varied. The short wavelength and meridionally confined structure of the second type of mode are characteristic of observed comma clouds. Sensitivity tests show that while it is necessary to have the release of latent heat confined to a part of the domain in order to produce a localized instability, confined modes only appear for physically reasonable parameter values in the high-vorticity, low-stability environment of the cyclonic shear region of the jet. It appears that this is due primarily to the effects of the reduced vertical stability in enhancing the feedback between convective heating and low-level convergence.

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Julia M. Windmiller and George C. Craig

Abstract

Self-aggregation in numerical simulations of tropical convection is described by an upscale growth and intensification of dry and moist regions. Previous work has focused on determining the relevant mechanism that induces moist regions to get moister and dry regions to get drier. Though different mechanisms have been identified, the spatial evolution of self-aggregation is remarkably universal. The first part of this study shows that different mechanisms can lead to a similar evolution of self-aggregation, if self-aggregation is described by a phase separation of moist and dry regions, through a process called coarsening. Though it was previously introduced based on a convection–humidity feedback, coarsening, importantly, is not tied to a specific feedback process but only requires an intensification of local humidity perturbations. Based on different feedback loops, three simple models of the evolution of the humidity field are introduced, all of which lead to coarsening. In each model, diffusive transport of humidity is assumed, which approximates a humidity increase due to convection, within a finite region around convective cores. In the second part, predictions made by coarsening are compared with atmospheric model simulations. Analyzing a set of radiative–convective equilibrium simulations shows that coarsening correctly predicts the upscale growth of the moist and dry regions in the early stages of self-aggregation. In addition, coarsening can explain why self-aggregation is not observed for small domains and why the shape of the final moist region changes with the shape of the domain.

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George C. Craig and Suzanne L. Gray

Abstract

Examination of conditional instability of the second kind (CISK) and wind-induced surface heat exchange (WISHE), two proposed mechanisms for tropical cyclone and polar low intensification, suggests that the sensitivity of the intensification rate of these disturbances to surface properties, such as surface friction and moisture supply, will be different for the two mechanisms. These sensitivities were examined by perturbing the surface characteristics in a numerical model with explicit convection. The intensification rate was found to have a strong positive dependence on the heat and moisture transfer coefficients, while remaining largely insensitive to the frictional drag coefficient. CISK does not predict the observed dependence of vortex intensification rate on the heat and moisture transfer coefficients, nor the insensitivity to the frictional drag coefficient since it anticipates that intensification rate is controlled by frictional convergence in the boundary layer. Since neither conditional instability nor boundary moisture content showed any significant sensitivity to the transfer coefficients, this is true of CISK using both the convective closures of Ooyama and of Charney and Eliassen. In comparison, the WISHE intensification mechanism does predict the observed increase in intensification rate with heat and moisture transfer coefficients, while not anticipating a direct influence from surface friction.

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George C. Craig and Brenda G. Cohen

Abstract

To provide a theoretical basis for stochastic parameterization of cumulus convection, the equilibrium fluctuations of a field of cumulus clouds under homogeneous large-scale forcing are derived statistically, using the Gibbs canonical ensemble from statistical mechanics. In the limit of noninteracting convective cells, the statistics of these convective fluctuations can be written in terms of the large-scale, externally constrained properties of the system. Using this framework, the probability density function of individual cloud mass fluxes is shown to be exponential. An analytical expression for the distribution function of total mass flux over a region of given size is also derived, and the variance of this distribution is found to be inversely related to the mean number of clouds in the ensemble. In a companion paper, these theoretical predictions are tested against cloud resolving model data.

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