A Multimoment Constrained Finite-Volume Model for Nonhydrostatic Atmospheric Dynamics

Abstract

The two-dimensional nonhydrostatic compressible dynamical core for the atmosphere has been developed by using a new nodal-type high-order conservative method, the so-called multimoment constrained finitevolume (MCV) method. Different from the conventional finite-volume method, the predicted variables (unknowns) in an MCV scheme are the values at the solution points distributed within each mesh cell. The time evolution equations to update the unknown point values are derived from a set of constraint conditions based on the multimoment concept, where the constraint on the volume-integrated average (VIA) for each mesh cell is cast into a flux form and thus guarantees rigorously the numerical conservation. Two important features make the MCV method particularly attractive as an accurate and practical numerical framework for atmospheric and oceanic modeling. 1) The predicted variables are the nodal values at the solution points that can be flexibly located within a mesh cell (equidistant solution points are used in the present model). It is computationally efficient and provides great convenience in dealing with complex geometry and source terms. 2) High-order and physically consistent formulations can be built by choosing proper constraints in view of not only numerical accuracy and efficiency but also underlying physics. In this paper the authors present a dynamical core that uses the third-and the fourth-order MCV schemes. They have verified the numerical outputs of both schemes by widely used standard benchmark tests and obtained competitive results. The present numerical core provides a promising and practical framework for further development of nonhydrostatic compressible atmospheric models.

Verification of a non-hydrostatic dynamical core using the horizontal spectral element method and vertical finite difference method: 2-D aspects

“…For a quantitative comparison, we present the root-mean-square errors for u , w , and θ in Table 2. These values are very comparable with those of other numerical models (Giraldo and Restelli, 2008;Li et al, 2013). Table 2.…”

“…Figure 5 shows the simulated results of the perturbed horizontal and vertical wind speeds after 10 h. In comparison with the analytic solution, the numerical solutions match quite well. The results of the present model are also very similar to the results of other numerical models (Giraldo and Restelli, 2008;Li et al, 2013). For a quantitative comparison, we present the root-mean-square errors for u , w , and θ in Table 2.…”

“…where θ c = −15 K and r = In this study, the potential temperature perturbation of θ c = −15 K was adopted for comparison with GR08 and Li et al (2013). Straka et al (1993) originally used a −15 K temperature perturbation.…”

“…For example, GR08 gave the ranges of θ ∈ −1.51 × 10 −3 , 2.78 × 10 −3 from the model based on the spectral element and discontinuous Galerkin methods. Additionally, Li et al (2013), using the high-order conservative finite volume model, showed θ ∈ −1.53 × 10 −3 , 2.80 × 10 −3 .…”

Simulation of all-scale atmospheric dynamics on unstructured meshes

“…In the quest for unified all-scale atmospheric models (recall the last paragraph of section 1.1), the top-down and bottom-up paths of extending the spectral range of simulated scales were advancing their preferred integration methods for stiff PDEs, with marked examples including semi-Lagrangian semi-implicit time integrators originated in NWP [187,28,74,10,117,173] on the one hand, and Eulerian split-explicit time stepping methods [114,118,6,119], a heritage of small-scale limited-area models [153,64], on the other. The collection of schemes was further enhanced with various forms of spatial discretisation, including finite differences [187,28,118,6], spectral transforms [187,10,173], finite volumes [114,119], and more recently high-order element based methods such as spectral element [154,42,155,29] and discontinuous Galerkin [102,43,7,44], or the multi-moment finite-volume approach [17,79,80,18], or combinations thereof [158]. The technical literature devoted to the advancement of nonhydrostatic atmospheric models is extensive, and the references provided merely illustrate its diversity.…”