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- Author or Editor: George C. Craig x
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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.
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.
Abstract
Errors in regional forecasts often take the form of phase errors, where a forecasted weather system is displaced in space or time. For such errors, a direct measure of the displacement is likely to be more valuable than traditional measures. A novel forecast quality measure is proposed that is based on a comparison of observed and forecast satellite imagery from the Meteosat-7 geostationary satellite. The measure combines the magnitude of a displacement vector calculated with a pyramid matching algorithm and the local squared difference of observed and morphed forecast brightness temperature fields. Following the description of the method and its application for a simplified case, the measure is applied to regional ensemble forecasts for an episode of prefrontal summertime convection in Bavaria. It is shown that this new method provides a plausible measure of forecast error, which is consistent with a subjective ranking of ensemble members for a sample forecast. The measure is then applied to hourly images over a 36-h forecast period and compared with the bias and equitable threat score. The two conventional measures fail to provide any systematic distinction between different ensemble members, while the new measure identifies ensemble members of differing skill levels with a strong degree of temporal consistency. Using the displacement-based error measure, individual ensemble members are found to compare better with observations than either a short-term deterministic forecast or the ensemble mean throughout the convective period.
Abstract
Errors in regional forecasts often take the form of phase errors, where a forecasted weather system is displaced in space or time. For such errors, a direct measure of the displacement is likely to be more valuable than traditional measures. A novel forecast quality measure is proposed that is based on a comparison of observed and forecast satellite imagery from the Meteosat-7 geostationary satellite. The measure combines the magnitude of a displacement vector calculated with a pyramid matching algorithm and the local squared difference of observed and morphed forecast brightness temperature fields. Following the description of the method and its application for a simplified case, the measure is applied to regional ensemble forecasts for an episode of prefrontal summertime convection in Bavaria. It is shown that this new method provides a plausible measure of forecast error, which is consistent with a subjective ranking of ensemble members for a sample forecast. The measure is then applied to hourly images over a 36-h forecast period and compared with the bias and equitable threat score. The two conventional measures fail to provide any systematic distinction between different ensemble members, while the new measure identifies ensemble members of differing skill levels with a strong degree of temporal consistency. Using the displacement-based error measure, individual ensemble members are found to compare better with observations than either a short-term deterministic forecast or the ensemble mean throughout the convective period.
Abstract
An idealized convective test bed for the local ensemble transform Kalman filter (LETKF) is set up to perform storm-scale data assimilation of simulated Doppler radar observations. Convective systems with lifetimes exceeding 6 h are triggered in a doubly periodic domain. Perfect-model experiments are used to investigate the limited predictability in precipitation forecasts by comparing analysis schemes that resolve different length scales. Starting from a high-resolution reference scheme with 8-km covariance localization and observations with 2-km resolution on a 5-min cycle, an experimental hierarchy is set up by successively choosing a larger covariance localization radius of 32 km, observations that are horizontally averaged by a factor of 4, a coarser resolution in the calculation of the analysis weights, and a cycling interval of 20 min. After 3 h of assimilation, the high-resolution analysis scheme is clearly superior to the configurations with coarser scales in terms of RMS error and field-oriented measures. The difference is associated with the observation resolution and a larger localization radius required for filter convergence with coarse observations. The high-resolution analysis leads to better forecasts for the first hour, but after 3 hours, the forecast quality of the schemes is indistinguishable. The more rapid error growth in forecasts from the high-resolution analysis appears to be associated with a limited predictability of the small scales, but also with gravity wave noise and spurious convective cells. The latter suggests that the field is in some sense less balanced, or less consistent with the model dynamics, than in the coarser-resolution analysis.
Abstract
An idealized convective test bed for the local ensemble transform Kalman filter (LETKF) is set up to perform storm-scale data assimilation of simulated Doppler radar observations. Convective systems with lifetimes exceeding 6 h are triggered in a doubly periodic domain. Perfect-model experiments are used to investigate the limited predictability in precipitation forecasts by comparing analysis schemes that resolve different length scales. Starting from a high-resolution reference scheme with 8-km covariance localization and observations with 2-km resolution on a 5-min cycle, an experimental hierarchy is set up by successively choosing a larger covariance localization radius of 32 km, observations that are horizontally averaged by a factor of 4, a coarser resolution in the calculation of the analysis weights, and a cycling interval of 20 min. After 3 h of assimilation, the high-resolution analysis scheme is clearly superior to the configurations with coarser scales in terms of RMS error and field-oriented measures. The difference is associated with the observation resolution and a larger localization radius required for filter convergence with coarse observations. The high-resolution analysis leads to better forecasts for the first hour, but after 3 hours, the forecast quality of the schemes is indistinguishable. The more rapid error growth in forecasts from the high-resolution analysis appears to be associated with a limited predictability of the small scales, but also with gravity wave noise and spurious convective cells. The latter suggests that the field is in some sense less balanced, or less consistent with the model dynamics, than in the coarser-resolution analysis.
Abstract
The growth of small-amplitude, spatially uncorrelated perturbations has been studied in a weather forecast of a 4-day period in the summer of 2007, using a large domain covering Europe and the eastern Atlantic and with explicitly resolved deep convection. The error growth follows the three-stage conceptual model of Zhang et al., with rapid initial growth (e-folding time about 0.5 h) on all scales, relaxing over about 20 h to a slow growth of the large-scale perturbations (e-folding time 12 h). The initial growth was confined to precipitating regions, with a faster growth rate where conditional instability was large. Growth in these regions saturated within 3–10 h, continuing for the longest where the precipitation rate was large. While the initial growth was mainly in the divergent part of the flow, the eventual slow growth on large scales was more in the rotational component.
Spectral decomposition of the disturbance energy showed that the rapid growth in precipitating regions projected onto all Fourier components; however, the amplitude at saturation was too small to initiate the subsequent large-scale growth. Visualization of the disturbance energy showed it to expand outward from the precipitating regions at a speed corresponding to a deep tropospheric gravity wave. These results suggest a physical picture of error growth with a rapidly growing disturbance to the vertical mass transport in precipitating regions that spreads to the radius of deformation while undergoing geostrophic adjustment, eventually creating a balanced perturbation that continues to grow through baroclinic instability.
Abstract
The growth of small-amplitude, spatially uncorrelated perturbations has been studied in a weather forecast of a 4-day period in the summer of 2007, using a large domain covering Europe and the eastern Atlantic and with explicitly resolved deep convection. The error growth follows the three-stage conceptual model of Zhang et al., with rapid initial growth (e-folding time about 0.5 h) on all scales, relaxing over about 20 h to a slow growth of the large-scale perturbations (e-folding time 12 h). The initial growth was confined to precipitating regions, with a faster growth rate where conditional instability was large. Growth in these regions saturated within 3–10 h, continuing for the longest where the precipitation rate was large. While the initial growth was mainly in the divergent part of the flow, the eventual slow growth on large scales was more in the rotational component.
Spectral decomposition of the disturbance energy showed that the rapid growth in precipitating regions projected onto all Fourier components; however, the amplitude at saturation was too small to initiate the subsequent large-scale growth. Visualization of the disturbance energy showed it to expand outward from the precipitating regions at a speed corresponding to a deep tropospheric gravity wave. These results suggest a physical picture of error growth with a rapidly growing disturbance to the vertical mass transport in precipitating regions that spreads to the radius of deformation while undergoing geostrophic adjustment, eventually creating a balanced perturbation that continues to grow through baroclinic instability.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
