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Abstract
The University of Wisconsin hybrid isentropic–sigma (θ–σ) coordinate channel model and the nominally identical sigma (σ) model are used to test the relative capabilities of nine trace constituent transport algorithms. The nine are “standard” second-order finite differencing, the standard with two local “borrow and fill” fixers, the standard with a global fixer, four conservative flux-integrated approaches, and the conservation of second-order moments (CSOM). Transport of two analytically specified initial trace constituent distributions is simulated within a common initial atmosphere, which includes a baroclinically amplifying synoptic-scale wave. Two different vertical resolution θ–σ models and four vertical resolution σ models provide excellent test beds for comparison of the transport algorithms because their 48-h predictions of standard synoptic fields are virtually identical.
Although no analytic solution exists against which detailed comparisons can be made, the constraint of adiabatic conditions for a continuum provides that the maximum of a trace constituent within explicit or implicit isentropic layers of a model should be conserved throughout the simulations, and that the area between any two trace constituent contours on an isentropic surface should remain constant. With these conditions as bases of comparison, several results are unambiguous. First, in the σ models the standard with fixers is better than the other schemes except for the CSOM at the highest resolution. Second, in the θ–σ models, the piecewise parabolic and CSOM schemes produce results approximately as accurate as the standard with fixers. Third, when comparing all algorithms, models, resolutions, and distributions, the CSOM scheme produces the most consistent results. Finally, for a large majority of the cases, the θ–σ models perform more accurately than the σ models with respect to the conservation of constituent extrema.
Abstract
The University of Wisconsin hybrid isentropic–sigma (θ–σ) coordinate channel model and the nominally identical sigma (σ) model are used to test the relative capabilities of nine trace constituent transport algorithms. The nine are “standard” second-order finite differencing, the standard with two local “borrow and fill” fixers, the standard with a global fixer, four conservative flux-integrated approaches, and the conservation of second-order moments (CSOM). Transport of two analytically specified initial trace constituent distributions is simulated within a common initial atmosphere, which includes a baroclinically amplifying synoptic-scale wave. Two different vertical resolution θ–σ models and four vertical resolution σ models provide excellent test beds for comparison of the transport algorithms because their 48-h predictions of standard synoptic fields are virtually identical.
Although no analytic solution exists against which detailed comparisons can be made, the constraint of adiabatic conditions for a continuum provides that the maximum of a trace constituent within explicit or implicit isentropic layers of a model should be conserved throughout the simulations, and that the area between any two trace constituent contours on an isentropic surface should remain constant. With these conditions as bases of comparison, several results are unambiguous. First, in the σ models the standard with fixers is better than the other schemes except for the CSOM at the highest resolution. Second, in the θ–σ models, the piecewise parabolic and CSOM schemes produce results approximately as accurate as the standard with fixers. Third, when comparing all algorithms, models, resolutions, and distributions, the CSOM scheme produces the most consistent results. Finally, for a large majority of the cases, the θ–σ models perform more accurately than the σ models with respect to the conservation of constituent extrema.
Abstract
Part I of this paper examined nine trace constituent advection algorithms as applied in channel versions of the University of Wisconsin hybrid isentropic–sigma (θ–σ) and sigma (σ) models. This paper examines the performance of 12 semi-Lagrangian transport (SLT) algorithms in the same models. The interpolants of trace constituent include second- through twelfth-order Lagrangian, an overlapping polynomial, a quasi-monotonic scheme, and two sixth-order schemes that employ derivative estimates. Additional experiments are performed that emulate the SLT algorithm in the NCAR Community Climate Model 2. As in Part I, these three-dimensional simulations are under adiabatic conditions so that conservation of the initial trace constituent maximum on an isentropic surface, and conservation of the areas between any two constituent contours, can be used as objective measures of SLT algorithm accuracy. Further, the experiments provide comparisons not only between the different SLT formulations but also between their performance in models using θ and σ coordinates.
Similar to Part I, an important result of the experiments is that comparison of algorithms is most revealing under three-dimensional transport within a baroclinic wave in which vertical transport is important. The experiments also show 1) that the “cascade” interpolation scheme is a reasonable method of greatly reducing computation time in SLT without affecting accuracy; 2) that “shape-preserving” interpolation schemes reduce accuracy in both the θ–σ and σ models; and 3) that Lagrangian interpolants of tenth and twelfth order do not significantly improve results. Comparisons to results in Part I suggest that the conservation of second-order moments advection scheme is the most consistent of all options tested.
Abstract
Part I of this paper examined nine trace constituent advection algorithms as applied in channel versions of the University of Wisconsin hybrid isentropic–sigma (θ–σ) and sigma (σ) models. This paper examines the performance of 12 semi-Lagrangian transport (SLT) algorithms in the same models. The interpolants of trace constituent include second- through twelfth-order Lagrangian, an overlapping polynomial, a quasi-monotonic scheme, and two sixth-order schemes that employ derivative estimates. Additional experiments are performed that emulate the SLT algorithm in the NCAR Community Climate Model 2. As in Part I, these three-dimensional simulations are under adiabatic conditions so that conservation of the initial trace constituent maximum on an isentropic surface, and conservation of the areas between any two constituent contours, can be used as objective measures of SLT algorithm accuracy. Further, the experiments provide comparisons not only between the different SLT formulations but also between their performance in models using θ and σ coordinates.
Similar to Part I, an important result of the experiments is that comparison of algorithms is most revealing under three-dimensional transport within a baroclinic wave in which vertical transport is important. The experiments also show 1) that the “cascade” interpolation scheme is a reasonable method of greatly reducing computation time in SLT without affecting accuracy; 2) that “shape-preserving” interpolation schemes reduce accuracy in both the θ–σ and σ models; and 3) that Lagrangian interpolants of tenth and twelfth order do not significantly improve results. Comparisons to results in Part I suggest that the conservation of second-order moments advection scheme is the most consistent of all options tested.
Abstract
The description of a global version of the University of Wisconsin (UW) hybrid isentropic-sigma (θ − σ) model and the results from an initial numerical weather prediction experiment are presented in this paper. The main objectives of this initial test are to 1) discuss θ − σ model development and computer requirements, 2) demonstrate the ability of the UW θ − σ model for global numerical weather prediction using realistic orography and parameterized physical processes, and 3) compare the transport of an inert trace constituent against a nominally “identical” sigma (σ) coordinate model. Initial and verifying data for the 5-day simulations presented in this work were supplied by the Goddaird Earth Observing System (GEOS-1) data assimilation system. The time period studied is 1–6 February 1985.
This validation experiment demonstrates that the global UW θ − σ model produces a realistic 5-day simulation of the mass and momentum distributions when compared to both the identical σ model and GEOS-1 verification. Root-mean-square errors demonstrate that the θ − σ model is slightly more accurate than the nominally identical σ model with respect to standard synoptic variables. Of particular importance, the UW θ − σ model displays a distinct advantage over the conventional σ model with respect to the prognostic simulation of inert trace constituent transport in amplifying baroclinic waves of the extratropics. This is especially true in the upper troposphere and stratosphere where the spatial integrity and conservation of an inert trace constituent is severely compromised in the a model compared to the θ − σ model.
Abstract
The description of a global version of the University of Wisconsin (UW) hybrid isentropic-sigma (θ − σ) model and the results from an initial numerical weather prediction experiment are presented in this paper. The main objectives of this initial test are to 1) discuss θ − σ model development and computer requirements, 2) demonstrate the ability of the UW θ − σ model for global numerical weather prediction using realistic orography and parameterized physical processes, and 3) compare the transport of an inert trace constituent against a nominally “identical” sigma (σ) coordinate model. Initial and verifying data for the 5-day simulations presented in this work were supplied by the Goddaird Earth Observing System (GEOS-1) data assimilation system. The time period studied is 1–6 February 1985.
This validation experiment demonstrates that the global UW θ − σ model produces a realistic 5-day simulation of the mass and momentum distributions when compared to both the identical σ model and GEOS-1 verification. Root-mean-square errors demonstrate that the θ − σ model is slightly more accurate than the nominally identical σ model with respect to standard synoptic variables. Of particular importance, the UW θ − σ model displays a distinct advantage over the conventional σ model with respect to the prognostic simulation of inert trace constituent transport in amplifying baroclinic waves of the extratropics. This is especially true in the upper troposphere and stratosphere where the spatial integrity and conservation of an inert trace constituent is severely compromised in the a model compared to the θ − σ model.
Abstract
In an initial effort in regional numerical weather prediction, results from the University of Wisconsin isentropic-sigma (UW θ−σ) hybrid model and an “identical” sigma model are compared. The two main objectives are to demonstrate the capability of the UW θ−σ model for regional numerical weather prediction and to identify advantages of the hybrid model in simulating atmospheric water vapor transport and precipitation relative to the sigma model.
The 72-h simulations produced by the two models extend over a region covering the western Pacific Ocean, North America, and the western Atlantic Ocean. The simulations begin at 0000 UTC 13 January 1979, a period during which an intense Chicago blizzard (sometimes called the Mayor Jane Byrne storm) develops over the central United States. This period also includes the rapid development of a cyclone in the western Pacific Ocean.
Results using the Global Weather Experiment (GWE) ECMWF level IIIB data as initial and verification data indicate that both models produce reasonable and similar 72-b simulations, with the UW θ−σ model mass and momentum distributions being slightly more accurate than the sigma model. Of particular importance for the Chicago blizzard is that the UW θ−σ model more accurately simulates water vapor transport northward from the Gulf of Mexico and westward from the Atlantic Ocean. As a result, the hybrid model more accurately simulates observed precipitation, especially over the northeastern United States and southeastern Canada.
Abstract
In an initial effort in regional numerical weather prediction, results from the University of Wisconsin isentropic-sigma (UW θ−σ) hybrid model and an “identical” sigma model are compared. The two main objectives are to demonstrate the capability of the UW θ−σ model for regional numerical weather prediction and to identify advantages of the hybrid model in simulating atmospheric water vapor transport and precipitation relative to the sigma model.
The 72-h simulations produced by the two models extend over a region covering the western Pacific Ocean, North America, and the western Atlantic Ocean. The simulations begin at 0000 UTC 13 January 1979, a period during which an intense Chicago blizzard (sometimes called the Mayor Jane Byrne storm) develops over the central United States. This period also includes the rapid development of a cyclone in the western Pacific Ocean.
Results using the Global Weather Experiment (GWE) ECMWF level IIIB data as initial and verification data indicate that both models produce reasonable and similar 72-b simulations, with the UW θ−σ model mass and momentum distributions being slightly more accurate than the sigma model. Of particular importance for the Chicago blizzard is that the UW θ−σ model more accurately simulates water vapor transport northward from the Gulf of Mexico and westward from the Atlantic Ocean. As a result, the hybrid model more accurately simulates observed precipitation, especially over the northeastern United States and southeastern Canada.
Abstract
The main goals of this paper are 1) to demonstrate the feasibility of incorporating a prognostic equation for water vapor and diabatic processes in the University of Wisconsin θ−σ model discussed in Part I, 2) to document methods applied to overcome difficulties stemming from the inclusion of moist processes and 3) to present results illustrating the effects of latent heat release on baroclinic development. The results confirm earlier studies that a prognostic equation for water vapor and the diabatic component of latent heat release may be included in a hybrid model. However, the modifications made in this study were important for eliminating spurious supersaturation and release of latent heat within grid volumes emerging and submerging through the interface between sigma and isentropic model domains. The results demonstrate the hybrid model's robust nature and potential for use in prediction.
For this demonstration, model simulations of an analytically specified synoptic-scale wave that amplified baroclinically under dry and moist conditions are compared. Simulations with and without a hydrological component show that the overall effect of latent heat release is to markedly enhance cyclo- and frontogenesis. The resultant pattern of precipitation is coherent, and the structure of the developing wave is consistent with the concepts of self-development. No detrimental effects are evident in either the structure or processes resulting from the inclusion of moist processes and the presence of an interface between sigma and isentropic model domains.
Abstract
The main goals of this paper are 1) to demonstrate the feasibility of incorporating a prognostic equation for water vapor and diabatic processes in the University of Wisconsin θ−σ model discussed in Part I, 2) to document methods applied to overcome difficulties stemming from the inclusion of moist processes and 3) to present results illustrating the effects of latent heat release on baroclinic development. The results confirm earlier studies that a prognostic equation for water vapor and the diabatic component of latent heat release may be included in a hybrid model. However, the modifications made in this study were important for eliminating spurious supersaturation and release of latent heat within grid volumes emerging and submerging through the interface between sigma and isentropic model domains. The results demonstrate the hybrid model's robust nature and potential for use in prediction.
For this demonstration, model simulations of an analytically specified synoptic-scale wave that amplified baroclinically under dry and moist conditions are compared. Simulations with and without a hydrological component show that the overall effect of latent heat release is to markedly enhance cyclo- and frontogenesis. The resultant pattern of precipitation is coherent, and the structure of the developing wave is consistent with the concepts of self-development. No detrimental effects are evident in either the structure or processes resulting from the inclusion of moist processes and the presence of an interface between sigma and isentropic model domains.
Abstract
The primary objectives of this study are threefold: 1) to compare simulators of dry and moist baroclinic development from 10-and 22-layer hybrid isentropic-sigma coordinate models with those from 11-, 27-, and 35-layer sigma coordinate models; 2) to examine the ability of the models to transport water vapor and simulate equivalent potential temperature θ e ; and 3) to compare predictions of the timing, location, and amount of precipitation. A model's capability to predict precipitation sterns from the accuracy of its simulation of the joint distribution of mass, potential temperature, and water vapor throughout the model domain. In a series of experiments to compare simulations of precipitation, several analytic distributions of water vapor are specified initially. The water vapor distributions include a “cylinder”extending vertically throughout the atmosphere and “lenses” within isentropic, sigma, and isobaric layers. The effect of increased horizontal resolution are also studied.
Results indicate that when the relative humidity is vertically uniform through a substantial extent of the atmosphere, all the models produce very similar precipitation distributions. However, when water vapor is confined to relatively shallow layers, the ability of the sigma coordinate models to simulate the timing, location, and amount of precipitation is severely compromised. Furthermore, the 10-layer hybrid model conserves θ e to a higher degree of accuracy and simulates a more realistic evolution of precipitation even when compared to results from sigma models with increased vertical and horizontal resolution. In all instances, the experiments demonstrate that advantages reside in prediction of precipitation with the hybrid model. Both theoretical and conceptual bases for thew differences are provided.
Abstract
The primary objectives of this study are threefold: 1) to compare simulators of dry and moist baroclinic development from 10-and 22-layer hybrid isentropic-sigma coordinate models with those from 11-, 27-, and 35-layer sigma coordinate models; 2) to examine the ability of the models to transport water vapor and simulate equivalent potential temperature θ e ; and 3) to compare predictions of the timing, location, and amount of precipitation. A model's capability to predict precipitation sterns from the accuracy of its simulation of the joint distribution of mass, potential temperature, and water vapor throughout the model domain. In a series of experiments to compare simulations of precipitation, several analytic distributions of water vapor are specified initially. The water vapor distributions include a “cylinder”extending vertically throughout the atmosphere and “lenses” within isentropic, sigma, and isobaric layers. The effect of increased horizontal resolution are also studied.
Results indicate that when the relative humidity is vertically uniform through a substantial extent of the atmosphere, all the models produce very similar precipitation distributions. However, when water vapor is confined to relatively shallow layers, the ability of the sigma coordinate models to simulate the timing, location, and amount of precipitation is severely compromised. Furthermore, the 10-layer hybrid model conserves θ e to a higher degree of accuracy and simulates a more realistic evolution of precipitation even when compared to results from sigma models with increased vertical and horizontal resolution. In all instances, the experiments demonstrate that advantages reside in prediction of precipitation with the hybrid model. Both theoretical and conceptual bases for thew differences are provided.
Abstract
Five- and 10-day inert trace constituent distributions prognostically simulated with the University of Wisconsin (UW) hybrid isentropic–sigma (θ–σ) model, the nominally identical UW sigma (σ) model, and the National Center for Atmospheric Research Community Climate Model 2 (CCM2) are analyzed and compared in this study. The UW θ–σ and σ gridpoint models utilize the flux form of the primitive equations, while CCM2 is based on the spectral representation and uses semi-Lagrangian transport (SLT) for trace constituents. Results are also compared against a version of the CCM that uses spectral transport for the trace constituent. These comparisons 1) contrast the spatial and temporal evolution of the filamentary transport of inert trace constituents simulated with the UW θ–σ and σ models against a “state of the art” GCM under both isentropic and nonisentropic conditions and 2) examine the ability of the models to conserve the initial trace constituent maximum value during 10-day integrations.
Results show that the spatial distributions of trace constituent evolve in a similar manner, regardless of the transport scheme or model type. However, when compared to the UW θ–σ model’s ability to simulate filamentary structure and conserve the initial trace constituent maximum value, results from the other models in this study indicate substantial spurious dispersion. The more accurate conservation demonstrated with the UW θ–σ model is especially noticeable within extratropical amplifying baroclinic waves, and it stems from the dominance of two-dimensional, quasi-horizontal isentropic exchange processes in a stratified baroclinic atmosphere. This condition, which largely precludes spurious numerical dispersion associated with vertical advection, is unique to isentropic coordinates. Conservation of trace constituent maxima in sigma coordinates suffers from the complexity of, and inherent need for, resolving three-dimensional transport in the presence of vertical wind shear during baroclinic amplification, a condition leading to spurious vertical dispersion. The experiments of this study also indicate that the shape-preserving SLT scheme used in CCM2 further reduces conservation of the initial maximum value when compared to the spectral transport of trace constituents, although the patterns are more coherent and the Gibbs phenomenon is eliminated.
Abstract
Five- and 10-day inert trace constituent distributions prognostically simulated with the University of Wisconsin (UW) hybrid isentropic–sigma (θ–σ) model, the nominally identical UW sigma (σ) model, and the National Center for Atmospheric Research Community Climate Model 2 (CCM2) are analyzed and compared in this study. The UW θ–σ and σ gridpoint models utilize the flux form of the primitive equations, while CCM2 is based on the spectral representation and uses semi-Lagrangian transport (SLT) for trace constituents. Results are also compared against a version of the CCM that uses spectral transport for the trace constituent. These comparisons 1) contrast the spatial and temporal evolution of the filamentary transport of inert trace constituents simulated with the UW θ–σ and σ models against a “state of the art” GCM under both isentropic and nonisentropic conditions and 2) examine the ability of the models to conserve the initial trace constituent maximum value during 10-day integrations.
Results show that the spatial distributions of trace constituent evolve in a similar manner, regardless of the transport scheme or model type. However, when compared to the UW θ–σ model’s ability to simulate filamentary structure and conserve the initial trace constituent maximum value, results from the other models in this study indicate substantial spurious dispersion. The more accurate conservation demonstrated with the UW θ–σ model is especially noticeable within extratropical amplifying baroclinic waves, and it stems from the dominance of two-dimensional, quasi-horizontal isentropic exchange processes in a stratified baroclinic atmosphere. This condition, which largely precludes spurious numerical dispersion associated with vertical advection, is unique to isentropic coordinates. Conservation of trace constituent maxima in sigma coordinates suffers from the complexity of, and inherent need for, resolving three-dimensional transport in the presence of vertical wind shear during baroclinic amplification, a condition leading to spurious vertical dispersion. The experiments of this study also indicate that the shape-preserving SLT scheme used in CCM2 further reduces conservation of the initial maximum value when compared to the spectral transport of trace constituents, although the patterns are more coherent and the Gibbs phenomenon is eliminated.
Abstract
In a validation experiment of a hybrid isentropic–sigma coordinate primitive equation model developed at the University of Wisconsin (the UW θ−σ model), an initial value technique is used to investigate numerically the normal-mode characteristics of baroclinically amplifying disturbances over a spectrum of meteorologically significant wavelength. The experiments are designed to determine the impact of coupling an isentropic-coordinate free atmospheric domain to a sigma-coordinate planetary boundary layer (PBL) on the normal-mode characteristics. The growth rate and phase speed spectra of the most unstable normal modes are obtained for an analytically prescribed zonal flow field. The evolution and vertical structure of the kinetic energy, energy conversions, growth rates, and geopotential fields are investigated.
Several modifications have been made to earlier versions of the UW θ−σ model to overcome noise introduced by adjustments associated with emerging and submerging grid volumes at the sigma–isentropic interface. With these modifications, the hybrid model accurately simulates the evolution and structure of normal-mode baroclinic disturbances for all wavenumbers considered except for wavenumber two. The normal-mode growth rate and phase speed spectra compare well with previous studies using standard sigma coordinate models. There is no evidence of aliasing the baroclinic normal-mode characteristics due to the coupling of isentropic and sigma domains.
Abstract
In a validation experiment of a hybrid isentropic–sigma coordinate primitive equation model developed at the University of Wisconsin (the UW θ−σ model), an initial value technique is used to investigate numerically the normal-mode characteristics of baroclinically amplifying disturbances over a spectrum of meteorologically significant wavelength. The experiments are designed to determine the impact of coupling an isentropic-coordinate free atmospheric domain to a sigma-coordinate planetary boundary layer (PBL) on the normal-mode characteristics. The growth rate and phase speed spectra of the most unstable normal modes are obtained for an analytically prescribed zonal flow field. The evolution and vertical structure of the kinetic energy, energy conversions, growth rates, and geopotential fields are investigated.
Several modifications have been made to earlier versions of the UW θ−σ model to overcome noise introduced by adjustments associated with emerging and submerging grid volumes at the sigma–isentropic interface. With these modifications, the hybrid model accurately simulates the evolution and structure of normal-mode baroclinic disturbances for all wavenumbers considered except for wavenumber two. The normal-mode growth rate and phase speed spectra compare well with previous studies using standard sigma coordinate models. There is no evidence of aliasing the baroclinic normal-mode characteristics due to the coupling of isentropic and sigma domains.
