Search Results

You are looking at 1 - 10 of 17 items for

  • Author or Editor: Todd D. Ringler x
  • Refine by Access: All Content x
Clear All Modify Search
Todd D. Ringler and David A. Randall

Abstract

Shallow-water equations discretized on a perfect hexagonal grid are analyzed using both a momentum formulation and a vorticity-divergence formulation. The vorticity-divergence formulation uses the unstaggered Z grid that places mass, vorticity, and divergence at the centers of the hexagons. The momentum formulation uses the staggered ZM grid that places mass at the centers of the hexagons and velocity at the corners of the hexagons. It is found that the Z grid and the ZM grid are identical in their simulation of the physical modes relevant to geostrophic adjustment. Consistent with the continuous system, the simulated inertia–gravity wave phase speeds increase monotonically with increasing total wavenumber and, thus, all waves have nonzero group velocities.

Since a grid of hexagons has twice as many corners as it has centers, the ZM grid has twice as many velocity points as it has mass points. As a result, the ZM-grid velocity field is discretized at a higher resolution than the mass field and, therefore, resolves a larger region of wavenumber space than the mass field. We solve the ∇2 f = λf eigenvalue problem with periodic boundary conditions on both the Z grid and ZM grid to determine the modes that can exist on each grid. The mismatch between mass and momentum leads to computational modes in the velocity field. Two techniques that can be used to control these computational modes are discussed. One technique is to use a dissipation operator that captures or “sees” the smallest-scale variations in the velocity field. The other technique is to invert elliptic equations in order to filter the high wavenumber part of the momentum field.

Results presented here lead to the conclusion that the ZM grid is an attractive alternative to the Z grid, and might be particularly useful for ocean modeling.

Full access
Sara A. Rauscher and Todd D. Ringler

Abstract

The effects of a variable-resolution mesh on simulated midlatitude baroclinic eddies in idealized settings are examined. Both aquaplanet and Held–Suarez experiments are performed using the Model for Prediction Across Scales-Atmosphere (MPAS-A) hydrostatic dynamical core implemented within the National Science Foundation–Department of Energy (NSF–DOE) Community Atmosphere Model (CAM-MPAS-A). In the real world, midlatitude eddy activity is organized by orography, land–sea contrasts, and sea surface temperature anomalies. In these zonally symmetric idealized settings, transients should have an equal probability of occurring at any longitude. However, the use of a variable-resolution mesh with a circular high-resolution region centered at 30°N results in a maximum in eddy kinetic energy on the eastern side and downstream of this high-resolution region in both aquaplanet and Held–Suarez CAM-MPAS-A simulations. The presence of a geographically confined maximum in both simulations suggests this response is mainly attributable to CAM-MPAS-A’s ability to resolve eddies via the model dynamics as resolution increases. However, in the aquaplanet simulation, a secondary maximum in eddy kinetic energy is present, which is probably linked to the resolution dependencies of the CAM physics. These mesh responses must be considered when interpreting real-world variable-resolution CAM-MPAS-A simulations, particularly in climate change experiments.

Full access
Todd D. Ringler and Kerry H. Cook

Abstract

Idealized simulations of the atmosphere’s stationary response to the Rockies, Tibetan Plateau, and the Greenland Ice Sheet are made using a nonlinear, quasigeostrophic model and are compared to observations. Observational data indicate low-level heating (cooling) occurs above the Rockies and Tibet in the summer (winter). Low-level cooling is found above Greenland in both seasons. The atmosphere responds to both diabatic heating (termed thermal forcing) and low-level flow being obstructed by the mountain’s presence (termed mechanical forcing).

The response to thermal and mechanical forcing together can be very different from the response to either forcing individually. The presence of modest low-level heating or cooling (±1.5 K day−1) causes significant changes to the mechanical forcing and, thereby, to the stationary wave response. For example, while the nonlinear response to mechanical forcing and low-level heating is characterized by a cyclone over the orography, the response to mechanical forcing and low-level cooling consists of an anticyclone over the orography. These differences cannot be fully explained using linear theory. The presence of heating (cooling) tends to reduce (amplify) both the mechanical forcing and the far-field stationary wave response. In addition, the presence of low-level heating or cooling lowers the critical mountain height below which the response is essentially linear;including nonlinear temperature advection at the surface is especially important for obtaining an accurate response.

Full access
Todd D. Ringler and Kerry H. Cook

Abstract

The forcing of stationary waves by the earth’s large-scale orography is studied using a nonlinear stationary wave model based on the quasigeostrophic equations. The manner in which wind speed, meridional temperature gradient, Ekman pumping parameter, linear damping, orographic shape, and meridional wind structure affect the validity of the linearized equations is examined and the nonlinear response is investigated.

A critical mountain height that separates the linear from the nonlinear regime is defined based on the linear quasigeostrophic potential temperature equation applied at the surface. The largest critical heights (those responses in which nonlinearity is least important) are obtained when the surface damping is weak or nonexistent. Also, relative maximums in mountain critical heights are obtained when the ratio of surface wind to surface wind shear does not vary in the meridional direction. These critical height results are validated using the fully nonlinear stationary wave model.

The nonlinearly balanced response to imposed orography is diagnosed at the surface and aloft. The nonlinear effects of eddy wind/orography interaction and nonlinear advection are found to be important only in the vicinity of the orography. The structure of the nonlinear response at the surface is found to be robust and is characterized (in the Northern Hemisphere) by a high and low situated to the northwest and southeast, respectively, of the mountain center. This orientation of the surface response leads to a stationary wave train that propagates preferentially toward the equator.

The system is sensitive enough to both the surface wind and meridional temperature gradient that the observed seasonal variations in the zonal mean circulation will significantly alter the character of the response. As the meridional temperature gradient decreases, the relative importance of nonlinearity increases while the amplitude of the response at the upper levels decreases. Therefore, this model indicates that summertime mechanically forced stationary waves should be weaker, but more nonlinear, than their wintertime counterparts.

Full access
Phillip J. Wolfram and Todd D. Ringler

Abstract

Meridional diffusivity is assessed for a baroclinically unstable jet in a high-latitude idealized circumpolar current (ICC) using the Model for Prediction across Scales Ocean (MPAS-O) and the online Lagrangian in Situ Global High-Performance Particle Tracking (LIGHT) diagnostic via space–time dispersion of particle clusters over 120 monthly realizations of O(106) particles on 11 potential density surfaces. Diffusivity in the jet reaches values of O(6000) m2 s−1 and is largest near the critical layer supporting mixing suppression and critical layer theory. Values in the vicinity of the shelf break are suppressed to O(100) m2 s−1 because of the presence of westward slope front currents. Diffusivity attenuates less rapidly with depth in the jet than both eddy velocity and kinetic energy scalings would suggest. Removal of the mean flow via high-pass filtering shifts the nonlinear parameter (ratio of the eddy velocity to eddy phase speed) into the linear wave regime by increasing the eddy phase speed via the depth-mean flow. Low-pass filtering, in contrast, quantifies the effect of mean shear. Diffusivity is decomposed into mean flow shear, linear waves, and the residual nonhomogeneous turbulence components, where turbulence dominates and eddy-produced filamentation strained by background mean shear enhances mixing, accounting for ≥80% of the total diffusivity relative to mean shear [O(100) m2 s−1], linear waves [O(1000) m2 s−1], and undecomposed full diffusivity [O(6000) m2 s−1]. Diffusivity parameterizations accounting for both the nonhomogeneous turbulence residual and depth variability are needed.

Full access
Todd D. Ringler and David A. Randall

Abstract

Using the shallow water equations, a numerical framework on a spherical geodesic grid that conserves domain-integrated mass, potential vorticity, potential enstrophy, and total energy is developed. The numerical scheme is equally applicable to hexagonal grids on a plane and to spherical geodesic grids. This new numerical scheme is compared to its predecessor and it is shown that the new scheme does considerably better in conserving potential enstrophy and energy. Furthermore, in a simulation of geostrophic turbulence, the new numerical scheme produces energy and enstrophy spectra with slopes of approximately K −3 and K −1, respectively, where K is the total wavenumber. These slopes are in agreement with theoretical predictions. This work also exhibits a discrete momentum equation that is compatible with the Z-grid vorticity-divergence equation.

Full access
William H. Lipscomb and Todd D. Ringler

Abstract

Weather and climate models contain equations for transporting conserved quantities such as the mass of air, water, ice, and associated tracers. Ideally, the numerical schemes used to solve these equations should be conservative, spatially accurate, and monotonicity-preserving. One such scheme is incremental remapping, previously developed for transport on quadrilateral grids. Here the incremental remapping scheme is reformulated for a spherical geodesic grid whose cells are hexagons and pentagons. The scheme is tested in a shallow-water model with both uniform and varying velocity fields. Solutions for standard shallow-water test cases 1, 2, and 5 are obtained with a centered scheme, a flux-corrected transport (FCT) scheme, and the remapping scheme. The three schemes are about equally accurate for transport of the height field. For tracer transport, remapping is far superior to the centered scheme, which produces large overshoots, and is generally smoother and more accurate than FCT. Remapping has a high startup cost associated with geometry calculations but is nearly twice as fast as FCT for each added tracer. As a result, remapping is cheaper than FCT for transport of more than about seven tracers.

Full access
Todd D. Ringler, Ross P. Heikes, and David A. Randall

Abstract

This paper documents the development and testing of a new type of atmospheric dynamical core. The model solves the vorticity and divergence equations in place of the momentum equation. The model is discretized in the horizontal using a geodesic grid that is nearly uniform over the entire globe. The geodesic grid is formed by recursively bisecting the triangular faces of a regular icosahedron and projecting those new vertices onto the surface of the sphere. All of the analytic horizontal operators are reduced to line integrals, which are numerically evaluated with second-order accuracy. In the vertical direction the model can use a variety of coordinate systems, including a generalized sigma coordinate that is attached to the top of the boundary layer. Terms related to gravity wave propagation are isolated and an efficient semi-implicit time-stepping scheme is implemented. Since this model combines many of the positive attributes of both spectral models and conventional finite-difference models into a single dynamical core, it represents a distinctively new approach to modeling the atmosphere’s general circulation.

The model is tested using the idealized forcing proposed by Held and Suarez. Results are presented for simulations using 2562 polygons (approximately 4.5° × 4.5°) and using 10 242 polygons (approximately 2.25° × 2.25°). The results are compared to those obtained with spectral model simulations truncated at T30 and T63. In terms of first and second moments of state variables such as the zonal wind, meridional wind, and temperature, the geodesic grid model results using 2562 polygons are comparable to those of a spectral model truncated at slightly less than T30, while a simulation with 10 242 polygons is comparable to a spectral model simulation truncated at slightly less than T63.

In order to further demonstrate the viability of this modeling approach, preliminary results obtained from a full-physics general circulation model that uses this dynamical core are presented. The dominant features of the DJF climate are captured in the full-physics simulation.

In terms of computational efficiency, the geodesic grid model is somewhat slower than the spectral model used for comparison. Model timings completed on an SGI Origin 2000 indicate that the geodesic grid model with 10 242 polygons is 20% slower than the spectral model truncated at T63. The geodesic grid model is more competitive at higher resolution than at lower resolution, so further optimization and future trends toward higher resolution should benefit the geodesic grid model.

Full access
Todd D. Ringler, Doug Jacobsen, Max Gunzburger, Lili Ju, Michael Duda, and William Skamarock

Abstract

The ability to solve the global shallow-water equations with a conforming, variable-resolution mesh is evaluated using standard shallow-water test cases. While the long-term motivation for this study is the creation of a global climate modeling framework capable of resolving different spatial and temporal scales in different regions, the process begins with an analysis of the shallow-water system in order to better understand the strengths and weaknesses of the approach developed herein. The multiresolution meshes are spherical centroidal Voronoi tessellations where a single, user-supplied density function determines the region(s) of fine- and coarse-mesh resolution. The shallow-water system is explored with a suite of meshes ranging from quasi-uniform resolution meshes, where the grid spacing is globally uniform, to highly variable resolution meshes, where the grid spacing varies by a factor of 16 between the fine and coarse regions. The potential vorticity is found to be conserved to within machine precision and the total available energy is conserved to within a time-truncation error. This result holds for the full suite of meshes, ranging from quasi-uniform resolution and highly variable resolution meshes. Based on shallow-water test cases 2 and 5, the primary conclusion of this study is that solution error is controlled primarily by the grid resolution in the coarsest part of the model domain. This conclusion is consistent with results obtained by others. When these variable-resolution meshes are used for the simulation of an unstable zonal jet, the core features of the growing instability are found to be largely unchanged as the variation in the mesh resolution increases. The main differences between the simulations occur outside the region of mesh refinement and these differences are attributed to the additional truncation error that accompanies increases in grid spacing. Overall, the results demonstrate support for this approach as a path toward multiresolution climate system modeling.

Full access
Qing Yang, L. Ruby Leung, Sara A. Rauscher, Todd D. Ringler, and Mark A. Taylor

Abstract

This study investigates the moisture budgets and resolution dependency of precipitation extremes in an aquaplanet framework based on the Community Atmosphere Model, version 4 (CAM4). Moisture budgets from simulations using two different dynamical cores, the Model for Prediction Across Scales-Atmosphere (MPAS-A) and High Order Method Modeling Environment (HOMME), but the same physics parameterizations suggest that during precipitation extremes the intensity of precipitation is approximately balanced by the vertical advective moisture transport. The resolution dependency in extremes from simulations at their native grid resolution originates from that of vertical moisture transport, which is mainly explained by changes in dynamics (related to vertical velocity ω) with resolution. When assessed at the same grid scale by area-weighted averaging the fine-resolution simulations to the coarse grids, simulations with either dynamical core still demonstrate resolution dependency in extreme precipitation with no convergence over the tropics, but convergence occurs at a wide range of latitudes over the extratropics. The use of lower temporal frequency data (i.e., daily vs 6 hourly) reduces the resolution dependency. Although thermodynamic (moisture) changes become significant in offsetting the effect of dynamics when assessed at the same grid scale, especially over the extratropics, changes in dynamics with resolution are still large and explain most of the resolution dependency during extremes. This suggests that the effects of subgrid-scale variability of ω and vertical moisture transport during extremes are not adequately parameterized by the model at coarse resolution. The aquaplanet framework and analysis described in this study provide an important metric for assessing sensitivities of cloud parameterizations to spatial resolution and dynamical cores under extreme conditions.

Full access