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William C. Skamarock

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.

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William C. Skamarock

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.

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William C. Skamarock

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.

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William C. Skamarock and Maximo Menchaca

Abstract

The finite-volume transport scheme of Miura, for icosahedral–hexagonal meshes on the sphere, is extended by using higher-order reconstructions of the transported scalar within the formulation. The use of second- and fourth-order reconstructions, in contrast to the first-order reconstruction used in the original scheme, results in significantly more accurate solutions at a given mesh density, and better phase and amplitude error characteristics in standard transport tests. The schemes using the higher-order reconstructions also exhibit much less dependence of the solution error on the time step compared to the original formulation.

The original scheme of Miura was only tested using a nondeformational time-independent flow. The deformational time-dependent flow test used to examine 2D planar transport in Blossey and Durran is adapted to the sphere, and the schemes are subjected to this test. The results largely confirm those generated using the simpler tests. The results also indicate that the scheme using the second-order reconstruction is most efficient and its use is recommended over the scheme using the first-order reconstruction. The second-order reconstruction uses the same computational stencil as the first-order reconstruction and thus does not create any additional parallelization issues.

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Rashmi Mittal and William C. Skamarock

Abstract

An assessment of a recently developed (by Miura) second-order numerical advection scheme for icosahedral-hexagonal grids on the sphere is presented, and the effects of monotonic limiters that can be used with the scheme are examined. The cases address both deformational and nondeformational flow and continuous and discontinuous advected quantities; they include solid-body rotation of a cosine bell and slotted cylinder, and moving dynamic vortices. The limiters of Zalesak, Dukowicz and Kodis, and Thuburn are tested within this numerical scheme. The Zalesak and Thuburn limiters produce solutions with similar accuracy, and the Thuburn limiter, while computationally less expensive per time step, results in more stringent stability conditions for the overall scheme. The Dukowicz limiter is slightly more diffusive than the other two, but it costs the least.

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Hiroaki Miura and William C. Skamarock

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.

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William C. Skamarock and Almut Gassmann

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.

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May Wong and William C. Skamarock

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.

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Richard Rotunno, William C. Skamarock, and Chris Snyder

Abstract

Using a primitive equation (PE) model, we revisit two canonical flows that were previously studied using a semigeostrophic equation (SG) model. In a previous paper, the authors showed that the PE and the SG models can have significantly different versions of the large-scale dynamics—here they report on the implications of this difference for frontogenesis. The program for the study of frontogenesis developed by B. J. Hoskins and collaborators is followed to show how, in the PE version of the canonical cases, the surface warm front develops before the cold front, and why the upper-level front is a long, nearly continuous feature going from ridge to trough. The frontogenesis experienced by an air parcel is computed following the parcel to illustrate better the mechanisms involved. As the present calculations are carried out longer than most previous ones, the relation of the upper frontogenesis to the formation of the upper-level “cutoff” cyclone is also examined. Trajectory and three-dimensional graphical analyses show, with respect to the latter, the extreme distortions of the isentropic surfaces and mixing-induced variations in the potential vorticity field.

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Richard Rotunno, William C. Skamarock, and Chris Snyder

Abstract

A comparative analysis of simulations of baroclinic waves with and without surface drag is presented, with particular reference to surface features. As in recent studies, the present simulations show that, compared to simulations with no drag, those with surface drag are less inclined to develop a secluded warm sector, and that drag weakens the warm front while the cold front remains strong. The authors demonstrate that analogous effects occur when Ekman pumping is used in nonlinear quasigeostrophic numerical simulations of unstable baroclinic waves in a channel. However, since the quasigeostrophic model produces symmetric highs and lows in the unstable baroclinic wave, the cold and warm fronts are therefore also symmetric and hence equally affected by the Ekman pumping. The different effect that friction has on the warm front with respect to the cold front in the primitive-equation simulations is fundamentally related to the tendency for the lows to be strong and narrow and the highs weak and broad, and for the warm front to form just north of, and extend eastward from, the low, while the cold front extends between the high and the low. The authors’ thesis is that the Ekman pumping associated with the low, at the location where the warm front would form in the absence of surface friction, acts to resist the formation of the warm front, while the cold front, positioned between the high and the low where Ekman pumping associated with the baroclinic wave is weak, is therefore relatively unaffected.

Given the weakness of Ekman pumping associated with the baroclinic wave in the vicinity of the incipient cold front, the present simulations indicate that cold frontogenesis occurs in the drag case in much the same way as in the no-drag case. Present analysis shows that the horizontal advection creating the cold front is a combination of geostrophic and ageostrophic effects. A portion of the ageostrophic frontogenesis is a response to geostrophic frontogenesis, as in the case without surface drag; however with surface drag, a significant portion of the cross-front ageostrophic flow is due to the Ekman layer associated with the front itself.

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