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Hiroaki Miura

Abstract

Fourth-order-centered operators on regular hexagonal grids with the ZM-grid arrangement are described. The finite-volume method is used and operators are defined at hexagonal cell centers. The gradient operator is calculated from 12 surrounding cell center scalars. The divergence operator is defined from 12 surrounding cell corner vectors. A linear combination of local or interpolated values generates cell corner values used to calculate the operators. The flux-divergence operator applies the same cell corner values as those used in the gradient and divergence operators. The fourth-order convergence of the gradient and divergence operators is obtained in numerical tests using sufficiently smooth and differentiable test functions. The flux-divergence operator is formally second-order accurate. However, the results from a cone advection test show that the flux-divergence operator performs better than a commonly used second-order flux-divergence operator. Numerical dispersion and phase error are small because mean wind advection is computed with fourth-order accuracy.

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Hiroaki Miura

Abstract

A discrete form of the flux-divergence operator is developed to compute advection of tracers on spherical hexagonal–pentagonal grids. An upwind-biased advection scheme based on a piecewise linear approximation for one-dimensional regular grids is extended simply for spherical hexagonal–pentagonal grids. The distribution of a tracer over the upwind side of a cell face is linearly approximated using a nodal value and a gradient at a computational node on the upwind side. A piecewise linear approximation is relaxed to a local linear approximation, and the relaxation precludes the complicated conditional branching present in remapping schemes. Results from a cosine bell advection test show that the new scheme compares favorably with other upwind-biased schemes for spherical hexagonal–pentagonal grids.

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Hiroaki Miura

Abstract

A standard nominally third-order upwind-biased spatial discretization of the flux-divergence operator was extended to a spherical icosahedral grid. The method can be used with multistage time-stepping schemes such as the Runge–Kutta method to compute the transport of variables on both hexagonal–pentagonal and triangular meshes. Two algorithms can be used to determine mesh cell face values: 1) interpolation using a quadratic function reconstructed subject to an integral constraint, or 2) calculation of the weighted mean of two linearly interpolated and extrapolated values. The first approach was adopted for a triangular mesh because the second approach depends on the mesh having a hexagonal or pentagonal shape. Both approaches were tested on the hexagonal–pentagonal mesh.

These schemes were subjected to standard transport tests on a spherical icosahedral grid. A three-stage Runge–Kutta time stepping method was used, and if necessary a flux limiter was applied to maintain monotonicity. The two different methods produced very similar solutions on a hexagonal–pentagonal mesh. Their accuracy was very close to the accuracy of a preexisting method designed for a Voronoi mesh only. When compared to another method that uses a quadratic polynomial interpolation, the phase error of the solutions was reduced, and their accuracy was much improved. The accuracies of the solutions were comparable on triangular and hexagonal–pentagonal meshes.

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Hiroaki Miura

Abstract

A shallow-water model using the hexagonal synchronized B grid (SB grid) is developed on the spherical icosahedral grid. The SB grid adopts the same variable arrangement as the ZM grid, but does not suffer from a computational mode problem of the ZM grid since interactions in the extra degrees of freedom of velocity fields through the nonlinear terms are excluded. For better representations of the geostrophic balance, a quadratic reconstruction of fluid height inside hexagonal/pentagonal cells is used to configure the gradient with the second-order accuracy. When nongeostrophic motions are more dominant than geostrophic ones, smaller-scale noises arise. To prevent a decoupling of the velocity fields, a hyperviscosity is added to force velocities adjacent to each other to evolve synchronously. Some standard tests are performed to examine the SB-grid shallow-water model. The model is almost second-order accurate if both the initial conditions and the surface topography are smooth and if the influence of the hyperviscosity is small. The SB-grid model is superior to a C-grid model regarding the convergence of error norms in a steady-state geostrophically balanced flow test, while it is inferior to that concerning conservation of total energy in a case of flow over an isolated mountain. An advantage of the SB-grid model is that both accuracy and stability are weakly sensitive to whether a grid optimization is applied or not. The SB grid is an attractive alternative to the conventional A grid and is competitive with the C grid on the spherical icosahedral grid.

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Hiroaki Miura and Masahide Kimoto

Abstract

Construction and optimization methods of spherical hexagonal–pentagonal geodesic grids are investigated. The objective is to compare grid structures on common ground.

The distinction between two types of hexagonal–pentagonal grids is made. Three conventional grid optimization methods are summarized. In addition, three new optimization methods are proposed. Six desirable conditions for an ideal grid are described, and the grid optimization methods are organized in view of such conditions.

Interval uniformity, area uniformity, isotropy, and bisection of cell faces are systematically investigated for optimized grids. There are compensations of preferable grid features in each optimization method, and an optimal method cannot be decided based only on the research of grid features. It is suggested that grid optimization methods should be selected based on research of numerical schemes.

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Tamaki Suematsu and Hiroaki Miura

Abstract

An environment favorable for the development of the Madden–Julian oscillation (MJO) was investigated by classifying MJO-like atmospheric patterns as MJO and regionally confined convective (RCC) events. Comparison of MJO and RCC events showed that even when preceded by a major convective suppression event, convective events did not develop into an MJO when large-scale buildup of moist static energy (MSE) was inhibited. The difference in the MSE accumulation between MJO and RCC is related to the contrasting low-frequency basic-state sea surface temperature (SST) pattern; the MJO and RCC events were associated with anomalously warm and cold low-frequency SSTs prevailing over the western to central Pacific, respectively. Differences in the SST anomaly field were absent from the intraseasonal frequency range of 20–60 days. The basic-state SST pattern associated with the MJO was characterized by a positive zonal SST gradient from the Indian Ocean to the western Pacific, which provided a long-standing condition that allowed for sufficient buildup of MSE across the Indian Ocean to the western Pacific via large-scale low-level convergence over intraseasonal and longer time scales. The results of this study suggest the importance of such a basic-state SST, with a long-lasting positive zonal SST gradient, for enhancing convection over a longer than intraseasonal time scale in realizing a complete MJO life cycle.

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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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Chia Rui Ong, Hiroaki Miura, and Makoto Koike

Abstract

The terminal velocity of cloud drops and raindrops used in numerical model calculations can significantly affect weather predictions. Current formulations rely on laboratory experiments made in the 1940s and 1960s. Because these experiments were performed only at typical environmental conditions of 20°C and 1013 hPa, parameterizations have been introduced to deduce the terminal velocity aloft without rigorous evaluation. In this study, an incompressible two-phase flow direct numerical simulation model is used to calculate the free-falling motion of axisymmetric drops with diameters between 0.025 and 0.5 mm to study the terminal fall velocity. Simulated terminal fall velocities of free-falling drops at 20°C and 1013 hPa agree within 3.2% with the previous empirical parameterization (Beard formula), and 4.5% with existing laboratory data in the diameter range between 0.3 and 0.5 mm. The velocities converge to the analytic Hadamard–Rybczynski solution within 2% for small Reynolds numbers, demonstrating the robustness of our simulations. Simulations under various atmospheric conditions show that existing empirical parameterizations that account for the air density dependence of the terminal velocity have errors up to 11.8% under the conditions examined in this study. We propose a new empirical formula that describes the air density dependence of the terminal velocity. It is also shown that the falling speed of a small drop is not sensitive to shape oscillation, and the terminal velocity decreases by only less than 1.3% when the axis ratio increases by 12% with reduced surface tension. Internal circulation within falling drops is also presented and compared with previous studies.

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Tomoe Nasuno, Hirofumi Tomita, Shinichi Iga, Hiroaki Miura, and Masaki Satoh

Abstract

This study investigated the multiscale organization of tropical convection on an aquaplanet in a model experiment with a horizontal mesh size of 3.5 km (for a 10-day simulation) and 7 km (for a 40-day simulation). The numerical experiment used the nonhydrostatic icosahedral atmospheric model (NICAM) with explicit cloud physics.

The simulation realistically reproduced multiscale cloud systems: eastward-propagating super cloud clusters (SCCs) contained westward-propagating cloud clusters (CCs). SCCs (CCs) had zonal sizes of several thousand (hundred) kilometers; typical propagation speed was 17 (10) m s−1. Smaller convective structures such as mesoscale cloud systems (MCs) of O(10 km) and cloud-scale elements (<10 km) were reproduced. A squall-type cluster with high cloud top (z > 16 km) of O(100 km) area was also reproduced.

Planetary-scale equatorial waves (with wavelengths of 10 000 and 40 000 km) had a major influence on the eastward propagation of the simulated SCC; destabilization east of the SCC facilitated generation of new CCs at the eastern end of the SCC. Large-scale divergence fields associated with the waves enhanced the growth of deep clouds in the CCs. A case study of a typical SCC showed that the primary mechanism forcing westward propagation varies with the life stages of the CCs or with vertical profiles of zonal wind. Cold pools and synoptic-scale waves both affected CC organization. Cloud-scale elements systematically formed along the edges of cold pools to sustain simulated MCs. The location, movement, and duration of the MCs varied with the large-scale conditions.

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Tomoe Nasuno, Hirofumi Tomita, Shinichi Iga, Hiroaki Miura, and Masaki Satoh

Abstract

Large-scale tropical convective disturbances simulated in a 7-km-mesh aquaplanet experiment are investigated. A 40-day simulation was executed using the Nonhydrostatic Icosahedral Atmospheric Model (NICAM). Two scales of eastward-propagating disturbances were analyzed. One was tightly coupled to a convective system resembling super–cloud clusters (SCCs) with a zonal scale of several thousand kilometers (SCC mode), whereas the other was characterized by a planetary-scale dynamical structure (40 000-km mode). The typical phase velocity was 17 (23) m s−1 for the SCC (40 000 km) mode. The SCC mode resembled convectively coupled Kelvin waves in the real atmosphere around the equator, but was accompanied by a pair of off-equatorial gyres. The 40 000-km mode maintained a Kelvin wave–like zonal structure, even poleward of the equatorial Rossby deformation radius. The equatorial structures in both modes matched neutral eastward-propagating gravity waves in the lower troposphere and unstable (growing) waves in the upper troposphere. In both modes, the meridional mass divergence exceeded the zonal component, not only in the boundary layer, but also in the free atmosphere. The forcing terms indicated that the meridional flow was primarily driven by convection via deformation in pressure fields and vertical circulations. Moisture convergence was one order of magnitude greater than the moisture flux from the sea surface. In the boundary layer, frictional convergence in the (anomalous) low-level easterly phase accounted for the buildup of low-level moisture leading to the active convective phase. The moisture distribution in the free atmosphere suggested that the moisture–convection feedback operated efficiently, especially in the SCC mode.

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