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- Author or Editor: J. B. Klemp x
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Abstract
An alternative form for a height-based terrain-following coordinate is presented here that progressively smoothes the coordinate surfaces with height to remove smaller scale (steeper) terrain structure from the surfaces. Testing this approach in comparison with traditional and hybrid terrain-following formulations in resting-atmosphere simulations demonstrates that it can significantly reduce artificial circulations caused by inaccuracies in the horizontal pressure gradient term. The simulations also suggest that some further improvement in the accuracy of the horizontal pressure gradient terms can be achieved using a simplified version of Mahrer’s approach, which can be implemented with little increase in computational cost or complexity.
Abstract
An alternative form for a height-based terrain-following coordinate is presented here that progressively smoothes the coordinate surfaces with height to remove smaller scale (steeper) terrain structure from the surfaces. Testing this approach in comparison with traditional and hybrid terrain-following formulations in resting-atmosphere simulations demonstrates that it can significantly reduce artificial circulations caused by inaccuracies in the horizontal pressure gradient term. The simulations also suggest that some further improvement in the accuracy of the horizontal pressure gradient terms can be achieved using a simplified version of Mahrer’s approach, which can be implemented with little increase in computational cost or complexity.
Abstract
Cut cells use regular or nearly regular polygonal cells to describe fields. For a given orography, some cells may be completely under the mountain, some completely above the mountain, and some are partially filled with air. While there are reports indicating considerably improved simulations with cut cells, inaccuracies may arise with some approximations, producing noise in fields near the surface. This behavior may depend strongly on the approximations made for the advection terms near the surface. This paper investigates the accuracy of advection for numerical schemes for a nondivergent flow near a mountain surface. The schemes use C-grid staggering with densities located at cell centers or on the corners of cells. Also, a nonconserving scheme is considered, which was used in the past with real-data cut-cell simulations. Since the cut cells near the surface create an irregular resolution, the accuracy and order of some approximations may break down near the surface. The objective of this paper is to find schemes having the same accuracy for advection near the surface as in the interior of the domain. As a test problem, uniform advection by a nondivergent velocity field is used with a 45° slope mountain (represented as a straight line) on a rectangular grid. Along the surface a sequence of triangular and pentagonal cells of quite different sizes are generated. Some schemes being discussed for cut cells lead to inaccurate and noisy solutions for this perfectly smooth mountain. A scheme using piecewise linear basis functions in a C grid with density points at the cell corners avoids these inaccuracies.
Abstract
Cut cells use regular or nearly regular polygonal cells to describe fields. For a given orography, some cells may be completely under the mountain, some completely above the mountain, and some are partially filled with air. While there are reports indicating considerably improved simulations with cut cells, inaccuracies may arise with some approximations, producing noise in fields near the surface. This behavior may depend strongly on the approximations made for the advection terms near the surface. This paper investigates the accuracy of advection for numerical schemes for a nondivergent flow near a mountain surface. The schemes use C-grid staggering with densities located at cell centers or on the corners of cells. Also, a nonconserving scheme is considered, which was used in the past with real-data cut-cell simulations. Since the cut cells near the surface create an irregular resolution, the accuracy and order of some approximations may break down near the surface. The objective of this paper is to find schemes having the same accuracy for advection near the surface as in the interior of the domain. As a test problem, uniform advection by a nondivergent velocity field is used with a 45° slope mountain (represented as a straight line) on a rectangular grid. Along the surface a sequence of triangular and pentagonal cells of quite different sizes are generated. Some schemes being discussed for cut cells lead to inaccurate and noisy solutions for this perfectly smooth mountain. A scheme using piecewise linear basis functions in a C grid with density points at the cell corners avoids these inaccuracies.
Abstract
A theory for the dynamics of strong surface winds on the lee side of a large mountain range is derived and compared with observations. The strong winds observed near Boulder, Colo., are found to be surface manifestations of standing gravity waves whose wavelength is long compared to typical resonant lee wavelengths. The theory indicates that such waves can become very intense if an inversion is present neat mountain-top level in the upstream environment and if the stability and wind profiles are such that the waves approximately reverse phase between the surface and the tropopause. The theory is extended to the development of a numerical model for estimation of maximum surface winds from upstream sounding data. Comparisons of the model predictions with observations are sufficiently encouraging to suggest the future utilization of such a model for operational forecasting. The differences between the predictions of this theory and those of hydraulic jump models are explored.
Abstract
A theory for the dynamics of strong surface winds on the lee side of a large mountain range is derived and compared with observations. The strong winds observed near Boulder, Colo., are found to be surface manifestations of standing gravity waves whose wavelength is long compared to typical resonant lee wavelengths. The theory indicates that such waves can become very intense if an inversion is present neat mountain-top level in the upstream environment and if the stability and wind profiles are such that the waves approximately reverse phase between the surface and the tropopause. The theory is extended to the development of a numerical model for estimation of maximum surface winds from upstream sounding data. Comparisons of the model predictions with observations are sufficiently encouraging to suggest the future utilization of such a model for operational forecasting. The differences between the predictions of this theory and those of hydraulic jump models are explored.
Abstract
A numerical model is developed for simulating the flow of stably stratified nonrotating air over finite-amplitude, two-dimensional mountain ranges. Special attention is paid to accurate treatment of internal dissipation and to formulation of an upper boundary region and lateral boundary conditions which allow upward and lateral propagation of wave energy out of the model. The model is hydrostatic and uses potential temperature for the vertical coordinate. A local adjustment procedure is derived to parameterize low Richardson number instability. The model behavior is tested against analytic theory and then applied to a variety of idealized and real flow situations, leading to some new insights and new questions on the nature of large-amplitude mountain waves. The model proves to be effective in simulating the structure of two observed cases of strong mountain waves with very different characteristics.
Abstract
A numerical model is developed for simulating the flow of stably stratified nonrotating air over finite-amplitude, two-dimensional mountain ranges. Special attention is paid to accurate treatment of internal dissipation and to formulation of an upper boundary region and lateral boundary conditions which allow upward and lateral propagation of wave energy out of the model. The model is hydrostatic and uses potential temperature for the vertical coordinate. A local adjustment procedure is derived to parameterize low Richardson number instability. The model behavior is tested against analytic theory and then applied to a variety of idealized and real flow situations, leading to some new insights and new questions on the nature of large-amplitude mountain waves. The model proves to be effective in simulating the structure of two observed cases of strong mountain waves with very different characteristics.
Abstract
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Abstract
The effects of vertical wind shear and buoyancy on convective storm structure and evolution are investigated with the use of a three-dimensional numerical cloud model. By varying the magnitude of buoyant energy and one-directional vertical shear over a wide range of environmental conditions associated with severe storms, the model is able to produce a spectrum of storm types qualitatively similar to those observed in nature. These include short-lived single cells, certain types of multicells and rotating supercells. The relationship between wind shear and buoyancy is expressed in terms of a nondimensional convective parameter which delineates various regimes of storm structure and, in particular, suggests optimal conditions for the development of supercell type storms. Applications of this parameter to well-documented severe storm cases agree favorably with the model results, suggesting both the value of the model in studying these modes of convection as well as the value of this representation in identifying the proper environment for the development of various storm types.
Abstract
The effects of vertical wind shear and buoyancy on convective storm structure and evolution are investigated with the use of a three-dimensional numerical cloud model. By varying the magnitude of buoyant energy and one-directional vertical shear over a wide range of environmental conditions associated with severe storms, the model is able to produce a spectrum of storm types qualitatively similar to those observed in nature. These include short-lived single cells, certain types of multicells and rotating supercells. The relationship between wind shear and buoyancy is expressed in terms of a nondimensional convective parameter which delineates various regimes of storm structure and, in particular, suggests optimal conditions for the development of supercell type storms. Applications of this parameter to well-documented severe storm cases agree favorably with the model results, suggesting both the value of the model in studying these modes of convection as well as the value of this representation in identifying the proper environment for the development of various storm types.
Abstract
For the numerical simulation of atmospheric flows that extend as high as the thermosphere, it is more appropriate to represent the upper boundary of the model domain as a material surface at constant pressure rather than one characterized by a rigid lid. Consequently, in adapting the Model for Prediction Across Scales (MPAS) for geospace applications, a modification of the height-based vertical coordinate is presented that permits the coordinate surfaces at upper levels to transition toward a constant pressure surface at the model’s upper boundary. This modification is conceptually similar to a terrain-following coordinate at low levels, but now modifies the coordinate surfaces at upper levels to conform to a constant pressure surface at the model top. Since this surface is evolving in time, the height of the upper boundary is adaptively adjusted to follow a designated constant pressure upper surface. This is accomplished by applying the hydrostatic equation to estimate the change in height along the boundary that is consistent with the vertical pressure gradient at the model top. This alteration in the vertical coordinate requires only minor modifications and little additional computational expense to the original height-based time-invariant terrain-following vertical coordinate employed in MPAS. The viability of this modified vertical coordinate formulation has been verified in a 2D prototype of MPAS for an idealized case of upper-level diurnal heating.
Significance Statement
Most atmospheric numerical models that use a height-based vertical coordinate employ a rigid lid at the top of the model domain. While a rigid lid works well for applications in the troposphere and stratosphere, it is not well suited for applications extending into the thermosphere where significant vertical expansion/contraction occurs due to deep heating/cooling of the atmosphere. This paper develops and tests a simple modification to the height-based coordinate formulation that allows the height of the upper boundary to adaptively follow a constant pressure surface. This added flexibility in the treatment of the upper domain boundary for height-based models may be particularly beneficial in facilitating their transition to a deep atmosphere configuration without significant retooling of the model numerics.
Abstract
For the numerical simulation of atmospheric flows that extend as high as the thermosphere, it is more appropriate to represent the upper boundary of the model domain as a material surface at constant pressure rather than one characterized by a rigid lid. Consequently, in adapting the Model for Prediction Across Scales (MPAS) for geospace applications, a modification of the height-based vertical coordinate is presented that permits the coordinate surfaces at upper levels to transition toward a constant pressure surface at the model’s upper boundary. This modification is conceptually similar to a terrain-following coordinate at low levels, but now modifies the coordinate surfaces at upper levels to conform to a constant pressure surface at the model top. Since this surface is evolving in time, the height of the upper boundary is adaptively adjusted to follow a designated constant pressure upper surface. This is accomplished by applying the hydrostatic equation to estimate the change in height along the boundary that is consistent with the vertical pressure gradient at the model top. This alteration in the vertical coordinate requires only minor modifications and little additional computational expense to the original height-based time-invariant terrain-following vertical coordinate employed in MPAS. The viability of this modified vertical coordinate formulation has been verified in a 2D prototype of MPAS for an idealized case of upper-level diurnal heating.
Significance Statement
Most atmospheric numerical models that use a height-based vertical coordinate employ a rigid lid at the top of the model domain. While a rigid lid works well for applications in the troposphere and stratosphere, it is not well suited for applications extending into the thermosphere where significant vertical expansion/contraction occurs due to deep heating/cooling of the atmosphere. This paper develops and tests a simple modification to the height-based coordinate formulation that allows the height of the upper boundary to adaptively follow a constant pressure surface. This added flexibility in the treatment of the upper domain boundary for height-based models may be particularly beneficial in facilitating their transition to a deep atmosphere configuration without significant retooling of the model numerics.
Abstract
Through the interactive use of Doppler-radar analyses and a three-dimensional numerical storm simulation the detailed structure of a supercell, tornadic storm is analyzed. This storm, named the Del City storm, occurred in central Oklahoma on 20 May 1977. The storm exhibits certain important features which are essential to maintaining its longevity and which promote the storm's transition to its tornadic phase. These features are strongly influenced by the rotational character of the storm separates the precipitation from the updraft and which orients the resulting downdrafts to which reinforce low-level convergence along the gust front and sustain the storm. Analyses of air parcel and rain trajectories within the storm provide a detailed visualization of this internal structure. These trajectories reveal that. air parcels rising through the cyclonically rotating updraft actually turn anticyclonically with height owing to the influence of the storm relative environmental wind field. Downdraft trajectories suggest that the cold outflow air behind the gust front originates in the environment at heights below 2 km. The distribution of vorticity is also investigated within the mature storm. At low levels the strong cyclonic vorticity is found to be located downwind of the convergence line, along the strong gradient between the updraft and down-draft. The similarities in structure between the observed and simulated storm suggest that the larger scale environment plays a dominant role in structuring many of the detailed features of the storm.
Abstract
Through the interactive use of Doppler-radar analyses and a three-dimensional numerical storm simulation the detailed structure of a supercell, tornadic storm is analyzed. This storm, named the Del City storm, occurred in central Oklahoma on 20 May 1977. The storm exhibits certain important features which are essential to maintaining its longevity and which promote the storm's transition to its tornadic phase. These features are strongly influenced by the rotational character of the storm separates the precipitation from the updraft and which orients the resulting downdrafts to which reinforce low-level convergence along the gust front and sustain the storm. Analyses of air parcel and rain trajectories within the storm provide a detailed visualization of this internal structure. These trajectories reveal that. air parcels rising through the cyclonically rotating updraft actually turn anticyclonically with height owing to the influence of the storm relative environmental wind field. Downdraft trajectories suggest that the cold outflow air behind the gust front originates in the environment at heights below 2 km. The distribution of vorticity is also investigated within the mature storm. At low levels the strong cyclonic vorticity is found to be located downwind of the convergence line, along the strong gradient between the updraft and down-draft. The similarities in structure between the observed and simulated storm suggest that the larger scale environment plays a dominant role in structuring many of the detailed features of the storm.
Abstract
Although the use of a damping layer near the top of a computational model domain has proven effective in absorbing upward-propagating gravity-wave energy in idealized simulations, this technique has been less successful in real atmospheric applications. Here, a new technique is proposed for nonhydrostatic model equations that are solved using split-explicit time-integration techniques. In this method, an implicit Rayleigh damping term is applied only to the vertical velocity, as a final adjustment at the end of each small (acoustic) time step. The adjustment is equivalent to including an implicit Rayleigh damping term in the vertical momentum equation together with an implicit vertical diffusion of w, and could be applied in this manner in other time-integration schemes. This implicit damping for the vertical velocity is unconditionally stable and remains effective even for hydrostatic gravity waves. The good absorption characteristics of this layer across a wide range of horizontal scales are confirmed through analysis of the linear wave equation and numerical mountain-wave simulations, and through simulations of an idealized squall line and of mountain waves over the Colorado Rocky Mountains.
Abstract
Although the use of a damping layer near the top of a computational model domain has proven effective in absorbing upward-propagating gravity-wave energy in idealized simulations, this technique has been less successful in real atmospheric applications. Here, a new technique is proposed for nonhydrostatic model equations that are solved using split-explicit time-integration techniques. In this method, an implicit Rayleigh damping term is applied only to the vertical velocity, as a final adjustment at the end of each small (acoustic) time step. The adjustment is equivalent to including an implicit Rayleigh damping term in the vertical momentum equation together with an implicit vertical diffusion of w, and could be applied in this manner in other time-integration schemes. This implicit damping for the vertical velocity is unconditionally stable and remains effective even for hydrostatic gravity waves. The good absorption characteristics of this layer across a wide range of horizontal scales are confirmed through analysis of the linear wave equation and numerical mountain-wave simulations, and through simulations of an idealized squall line and of mountain waves over the Colorado Rocky Mountains.
Abstract
Historically, time-split schemes for numerically integrating the nonhydrostatic compressible equations of motion have not formally conserved mass and other first-order flux quantities. In this paper, split-explicit integration techniques are developed that numerically conserve these properties by integrating prognostic equations for conserved quantities represented in flux form. These procedures are presented for both terrain-following height and hydrostatic pressure (mass) vertical coordinates, two potentially attractive frameworks for which the equation sets and integration techniques differ significantly. For each set of equations, the linear dispersion equation for acoustic/gravity waves is derived and analyzed to determine which terms must be solved in the small (acoustic) time steps and how these terms are represented in the time integration to achieve stability. Efficient techniques for including numerical filters for acoustic and external modes are also presented. Simulations for several idealized test cases in both the height and mass coordinates are presented to demonstrate that these integration techniques appear robust over a wide range of scales, from subcloud to synoptic.
Abstract
Historically, time-split schemes for numerically integrating the nonhydrostatic compressible equations of motion have not formally conserved mass and other first-order flux quantities. In this paper, split-explicit integration techniques are developed that numerically conserve these properties by integrating prognostic equations for conserved quantities represented in flux form. These procedures are presented for both terrain-following height and hydrostatic pressure (mass) vertical coordinates, two potentially attractive frameworks for which the equation sets and integration techniques differ significantly. For each set of equations, the linear dispersion equation for acoustic/gravity waves is derived and analyzed to determine which terms must be solved in the small (acoustic) time steps and how these terms are represented in the time integration to achieve stability. Efficient techniques for including numerical filters for acoustic and external modes are also presented. Simulations for several idealized test cases in both the height and mass coordinates are presented to demonstrate that these integration techniques appear robust over a wide range of scales, from subcloud to synoptic.
