Search Results
You are looking at 1 - 10 of 51 items for
- Author or Editor: William C. Skamarock x
- Refine by Access: All Content x
Abstract
General positive-definite and monotonic limiters are described for use with unrestricted-Courant-number flux-form transport schemes. These limiters are tested using a time-split multidimensional transport scheme. The importance of minimizing the splitting errors associated with the time-split operator and of the consistency between the transport scheme and the discrete continuity equation is demonstrated.
Abstract
General positive-definite and monotonic limiters are described for use with unrestricted-Courant-number flux-form transport schemes. These limiters are tested using a time-split multidimensional transport scheme. The importance of minimizing the splitting errors associated with the time-split operator and of the consistency between the transport scheme and the discrete continuity equation is demonstrated.
Abstract
Kinetic energy spectra derived from observations in the free atmosphere possess a wavenumber dependence of k −3 for large scales, characteristic of 2D turbulence, and transition to a k −5/3 dependence in the mesoscale. Kinetic energy spectra computed using mesoscale and experimental near-cloud-scale NWP forecasts from the Weather Research and Forecast (WRF) model are examined, and it is found that the model-derived spectra match the observational spectra well, including the transition. The model spectra decay at the highest resolved wavenumbers compared with observations, indicating energy removal by the model's dissipation mechanisms. This departure from the observed spectra is used to define the model's effective resolution. Various dissipation mechanisms used in NWP models are tested in WRF model simulations to examine the mechanisms' impact on a model's effective resolution. The spinup of the spectra in forecasts is also explored, along with spectra variability in the free atmosphere and in forecasts under different synoptic regimes.
Abstract
Kinetic energy spectra derived from observations in the free atmosphere possess a wavenumber dependence of k −3 for large scales, characteristic of 2D turbulence, and transition to a k −5/3 dependence in the mesoscale. Kinetic energy spectra computed using mesoscale and experimental near-cloud-scale NWP forecasts from the Weather Research and Forecast (WRF) model are examined, and it is found that the model-derived spectra match the observational spectra well, including the transition. The model spectra decay at the highest resolved wavenumbers compared with observations, indicating energy removal by the model's dissipation mechanisms. This departure from the observed spectra is used to define the model's effective resolution. Various dissipation mechanisms used in NWP models are tested in WRF model simulations to examine the mechanisms' impact on a model's effective resolution. The spinup of the spectra in forecasts is also explored, along with spectra variability in the free atmosphere and in forecasts under different synoptic regimes.
Abstract
The NCAR Community Climate System Model (CCSM) finite-volume atmospheric core uses a C–D-grid discretization to solve the equations of motion. A linear analysis of this discretization shows that it behaves as a D grid to leading order; it possesses the poor response of the D grid for short-wavelength divergent modes, the poor response of the C and D grids for short-wavelength rotational modes, and is only first-order accurate in time and damping. The scheme combines a modified forward–backward time integration for gravity waves with forward-in-time upwind-biased advection schemes, and the solver uses a vector-invariant form of the momentum equations. Other approaches using these equations are considered that circumvent some of the problems inherent in the current approach.
Abstract
The NCAR Community Climate System Model (CCSM) finite-volume atmospheric core uses a C–D-grid discretization to solve the equations of motion. A linear analysis of this discretization shows that it behaves as a D grid to leading order; it possesses the poor response of the D grid for short-wavelength divergent modes, the poor response of the C and D grids for short-wavelength rotational modes, and is only first-order accurate in time and damping. The scheme combines a modified forward–backward time integration for gravity waves with forward-in-time upwind-biased advection schemes, and the solver uses a vector-invariant form of the momentum equations. Other approaches using these equations are considered that circumvent some of the problems inherent in the current approach.
Abstract
An adaptive implicit–explicit vertical transport method is implemented in the Advanced Research version of the Weather Research and Forecasting Model (WRF-ARW), and improved integration efficiency is demonstrated for configurations employing convective-allowing horizontal and vertical resolutions. During the warm season over the continental United States, stable forecasts at convective-allowing resolutions are more challenging because localized regions of extreme thermodynamic instability generate large vertical velocities within thunderstorms that cause the integrations to become unstable because of violations of the Courant–Friedrichs–Lewy (CFL) condition for the explicit advection scheme used in WRF-ARW. The implicit–explicit vertical transport scheme removes the CFL instability but maintains accuracy for typical vertical velocities. Tests using this scheme show that the new scheme permits a time step that is 20%–25% percent larger, and it reduces the wall clock time by 10%–13% percent relative to a configuration similar to a current operational convection-allowing model while also producing more realistic updraft intensities within the most intense storms. Other approaches to maintain stability are either less efficient (e.g., reducing the time step) or significantly impact the solution accuracy (e.g., increasing the damping and/or reducing the latent heating, which severely limits the updraft magnitudes during the forecasts).
Abstract
An adaptive implicit–explicit vertical transport method is implemented in the Advanced Research version of the Weather Research and Forecasting Model (WRF-ARW), and improved integration efficiency is demonstrated for configurations employing convective-allowing horizontal and vertical resolutions. During the warm season over the continental United States, stable forecasts at convective-allowing resolutions are more challenging because localized regions of extreme thermodynamic instability generate large vertical velocities within thunderstorms that cause the integrations to become unstable because of violations of the Courant–Friedrichs–Lewy (CFL) condition for the explicit advection scheme used in WRF-ARW. The implicit–explicit vertical transport scheme removes the CFL instability but maintains accuracy for typical vertical velocities. Tests using this scheme show that the new scheme permits a time step that is 20%–25% percent larger, and it reduces the wall clock time by 10%–13% percent relative to a configuration similar to a current operational convection-allowing model while also producing more realistic updraft intensities within the most intense storms. Other approaches to maintain stability are either less efficient (e.g., reducing the time step) or significantly impact the solution accuracy (e.g., increasing the damping and/or reducing the latent heating, which severely limits the updraft magnitudes during the forecasts).
Abstract
As an alternative to traditional precipitation analysis and forecast verification, 1D and 2D spectral decompositions of NCEP/Stage IV and Multi-Radar Multi-Sensor (MRMS) precipitation products and convective-scale model forecasts are examined. Both the stage IV and MRMS analyses and the model forecasts show a similar weak power-law behavior in 1D spectral decompositions, although the MRMS analysis does not drop off in power at wavelengths less than approximately 20 km as found in the stage IV analysis. The convective-scale forecasts produce similar behavior to the MRMS when the forecast model’s effective resolution is sufficient. Neither the MRMS analyses nor the forecasts suggest the existence of a break in the spectral slope at the scales for which the analyses and forecasts are valid. The 2D spectra of both observations and forecasts, expressed in terms of an absolute wavenumber and azimuthal angle, show power varying significantly as a function of azimuthal angle for a given wavenumber. This azimuthal anisotropy is significant, and is dominated by the second mode (wavenumber 2). The phase of the mode is the result of the orientation of precipitation features and, hence, convective system orientations and propagation. Observations show a shift in orientation (phase) over May–June–July. The convective forecasts reproduce this shift in phase, although with a consistent but small phase error.
Abstract
As an alternative to traditional precipitation analysis and forecast verification, 1D and 2D spectral decompositions of NCEP/Stage IV and Multi-Radar Multi-Sensor (MRMS) precipitation products and convective-scale model forecasts are examined. Both the stage IV and MRMS analyses and the model forecasts show a similar weak power-law behavior in 1D spectral decompositions, although the MRMS analysis does not drop off in power at wavelengths less than approximately 20 km as found in the stage IV analysis. The convective-scale forecasts produce similar behavior to the MRMS when the forecast model’s effective resolution is sufficient. Neither the MRMS analyses nor the forecasts suggest the existence of a break in the spectral slope at the scales for which the analyses and forecasts are valid. The 2D spectra of both observations and forecasts, expressed in terms of an absolute wavenumber and azimuthal angle, show power varying significantly as a function of azimuthal angle for a given wavenumber. This azimuthal anisotropy is significant, and is dominated by the second mode (wavenumber 2). The phase of the mode is the result of the orientation of precipitation features and, hence, convective system orientations and propagation. Observations show a shift in orientation (phase) over May–June–July. The convective forecasts reproduce this shift in phase, although with a consistent but small phase error.
Abstract
A forward-in-time splitting method for integrating the elastic equations is presented. A second-order Runge–Kutta time integrator (RK2) for the large-time-step integration is combined with the forward–backward scheme in a manner similar to the Klemp and Wilhelmson method. The new scheme produces fully second-order-accurate integrations for advection and gravity wave propagation. The RK2 scheme uses upwind discretizations for the advection terms and is easily combined with standard vertically semi-implicit techniques so as to improve computational efficiency when the grid aspect ratio becomes large. A stability analysis of the RK2 split-explicit scheme shows that it is stable for a wide range of advective and acoustic wave Courant numbers.
The RK2 time-split scheme is used in a full-physics nonhydrostatic compressible cloud model. The implicit damping properties associated with the RK2’s third-order horizontal differencing allows for a significant reduction in the value of horizontal filtering applied to the momentum and pressure fields, while qualitatively the solutions appear to be better resolved than solutions from a leapfrog model.
Abstract
A forward-in-time splitting method for integrating the elastic equations is presented. A second-order Runge–Kutta time integrator (RK2) for the large-time-step integration is combined with the forward–backward scheme in a manner similar to the Klemp and Wilhelmson method. The new scheme produces fully second-order-accurate integrations for advection and gravity wave propagation. The RK2 scheme uses upwind discretizations for the advection terms and is easily combined with standard vertically semi-implicit techniques so as to improve computational efficiency when the grid aspect ratio becomes large. A stability analysis of the RK2 split-explicit scheme shows that it is stable for a wide range of advective and acoustic wave Courant numbers.
The RK2 time-split scheme is used in a full-physics nonhydrostatic compressible cloud model. The implicit damping properties associated with the RK2’s third-order horizontal differencing allows for a significant reduction in the value of horizontal filtering applied to the momentum and pressure fields, while qualitatively the solutions appear to be better resolved than solutions from a leapfrog model.
Abstract
Higher-order finite-volume flux operators for transport algorithms used within Runge–Kutta time integration schemes on irregular Voronoi (hexagonal) meshes are proposed and tested. These operators are generalizations of third- and fourth-order operators currently used in atmospheric models employing regular, orthogonal rectangular meshes. Two-dimensional least squares fit polynomials are used to evaluate the higher-order spatial derivatives needed to cancel the leading-order truncation error terms of the standard second-order centered formulation. Positive definite or monotonic behavior is achieved by applying an appropriate limiter during the final Runge–Kutta stage within a given time step.
The third- and fourth-order formulations are evaluated using standard transport tests on the sphere. The new schemes are more accurate and significantly more efficient than the standard second-order scheme and other schemes in the literature examined by the authors. The third-order formulation is equivalent to the fourth-order formulation plus an additional diffusion term that is proportional to the Courant number. An optimal value for the coefficient scaling this diffusion term is chosen based on qualitative evaluation of the test results. Improvements using the higher-order scheme in place of the traditional second-order centered approach are illustrated within 3D unstable baroclinic wave simulations produced using two global nonhydrostatic models employing spherical Voronoi meshes.
Abstract
Higher-order finite-volume flux operators for transport algorithms used within Runge–Kutta time integration schemes on irregular Voronoi (hexagonal) meshes are proposed and tested. These operators are generalizations of third- and fourth-order operators currently used in atmospheric models employing regular, orthogonal rectangular meshes. Two-dimensional least squares fit polynomials are used to evaluate the higher-order spatial derivatives needed to cancel the leading-order truncation error terms of the standard second-order centered formulation. Positive definite or monotonic behavior is achieved by applying an appropriate limiter during the final Runge–Kutta stage within a given time step.
The third- and fourth-order formulations are evaluated using standard transport tests on the sphere. The new schemes are more accurate and significantly more efficient than the standard second-order scheme and other schemes in the literature examined by the authors. The third-order formulation is equivalent to the fourth-order formulation plus an additional diffusion term that is proportional to the Courant number. An optimal value for the coefficient scaling this diffusion term is chosen based on qualitative evaluation of the test results. Improvements using the higher-order scheme in place of the traditional second-order centered approach are illustrated within 3D unstable baroclinic wave simulations produced using two global nonhydrostatic models employing spherical Voronoi meshes.
Abstract
Several transport schemes developed for spherical icosahedral grids are based on the piecewise linear approximation. The simplest one among them uses an algorithm where the tracer distribution in the upwind side of a cell face is reconstructed using a linear surface. Recently, it was demonstrated that using second- or fourth-order reconstructions instead of the linear one produces better results. The computational cost of the second-order reconstruction method was not much larger than the linear one, while that of the fourth-order one was significantly larger. In this work, the authors propose another second-order reconstruction scheme on the spherical icosahedral grids, motivated by some ideas from the piecewise parabolic method. The second-order profile of a tracer is reconstructed under two constraints: (i) the area integral of the profile is equal to the cell-averaged value times the cell area and (ii) the profile is the least squares fit to the cell-vertex values. The new scheme [the second upwind-biased quadratic approximation (UQA-2)] is more accurate than the preceding second-order reconstruction scheme [the first upwind-biased quadratic approximation (UQA-1)] in most of the tests in this work. Solutions of UQA-2 are sharper than those of UQA-1, although with slightly larger phase errors. The computational cost of UQA-2 is comparable to UQA-1.
Abstract
Several transport schemes developed for spherical icosahedral grids are based on the piecewise linear approximation. The simplest one among them uses an algorithm where the tracer distribution in the upwind side of a cell face is reconstructed using a linear surface. Recently, it was demonstrated that using second- or fourth-order reconstructions instead of the linear one produces better results. The computational cost of the second-order reconstruction method was not much larger than the linear one, while that of the fourth-order one was significantly larger. In this work, the authors propose another second-order reconstruction scheme on the spherical icosahedral grids, motivated by some ideas from the piecewise parabolic method. The second-order profile of a tracer is reconstructed under two constraints: (i) the area integral of the profile is equal to the cell-averaged value times the cell area and (ii) the profile is the least squares fit to the cell-vertex values. The new scheme [the second upwind-biased quadratic approximation (UQA-2)] is more accurate than the preceding second-order reconstruction scheme [the first upwind-biased quadratic approximation (UQA-1)] in most of the tests in this work. Solutions of UQA-2 are sharper than those of UQA-1, although with slightly larger phase errors. The computational cost of UQA-2 is comparable to UQA-1.
Abstract
Two time-splitting methods for integrating the elastic equations are presented. The methods are based on a third-order Runge–Kutta time scheme and the Crowley advection schemes. The schemes are combined with a forward–backward scheme for integrating high-frequency acoustic and gravity modes to create stable split-explicit schemes for integrating the compressible Navier–Stokes equations. The time-split methods facilitate the use of both centered and upwind-biased discretizations for the advection terms, allow for larger time steps, and produce more accurate solutions than existing approaches. The time-split Crowley scheme illustrates a methodology for combining any pure forward-in-time advection schemes with an explicit time-splitting method. Based on both linear and nonlinear tests, the third-order Runge–Kutta-based time-splitting scheme appears to offer the best combination of efficiency and simplicity for integrating compressible nonhydrostatic atmospheric models.
Abstract
Two time-splitting methods for integrating the elastic equations are presented. The methods are based on a third-order Runge–Kutta time scheme and the Crowley advection schemes. The schemes are combined with a forward–backward scheme for integrating high-frequency acoustic and gravity modes to create stable split-explicit schemes for integrating the compressible Navier–Stokes equations. The time-split methods facilitate the use of both centered and upwind-biased discretizations for the advection terms, allow for larger time steps, and produce more accurate solutions than existing approaches. The time-split Crowley scheme illustrates a methodology for combining any pure forward-in-time advection schemes with an explicit time-splitting method. Based on both linear and nonlinear tests, the third-order Runge–Kutta-based time-splitting scheme appears to offer the best combination of efficiency and simplicity for integrating compressible nonhydrostatic atmospheric models.
Abstract
A positive-definite transport scheme for moisture is tested in a nonhydrostatic forecast model using convection-permitting resolutions. Use of the positive-definite scheme is found to significantly reduce the large positive bias in surface precipitation forecasts found in the non-positive-definite model forecasts, in particular at high precipitation thresholds. The positive-definite scheme eliminates spurious sources of water arising from the clipping of negative moisture values in the non-positive-definite model formulation, leading to the bias reduction.
Abstract
A positive-definite transport scheme for moisture is tested in a nonhydrostatic forecast model using convection-permitting resolutions. Use of the positive-definite scheme is found to significantly reduce the large positive bias in surface precipitation forecasts found in the non-positive-definite model forecasts, in particular at high precipitation thresholds. The positive-definite scheme eliminates spurious sources of water arising from the clipping of negative moisture values in the non-positive-definite model formulation, leading to the bias reduction.
