Multi-scale waves in sound-proof global simulations with EULAG

EULAG is a computational model for simulating flows across a wide range of scales and physical scenarios. A standard option employs an anelastic approximation to capture nonhydrostatic effects and simultaneously filter sound waves from the solution. In this study, we examine a localized gravity wave packet generated by instabilities in Held-Suarez climates. Although still simplified versus the Earth’s atmosphere, a rich set of planetary wave instabilities and ensuing radiated gravity waves can arise. Wave packets are observed that have lifetimes ≤ 2 days, are negligibly impacted by Coriolis force, and do not show the rotational effects of differential jet advection typical of inertia-gravity waves. Linear modal analysis shows that wavelength, period, and phase speed fit the dispersion equation to within a mean difference of ∼ 4%, suggesting an excellent fit. However, the group velocities match poorly even though a propagation of uncertainty analysis indicates that they should be predicted as well as the phase velocities. Theoretical arguments suggest the discrepancy is due to nonlinearity — a strong southerly flow leads to a critical surface forming to the southwest of the wave packet that prevents the expected propagation

Forward-in-time differencing for fluids: Nonhydrostatic modeling of fluid motions on a sphere

Traditionally, numerical models for simulating planetary scale weather and climate employ the hydrostatic primitive equations--an abbreviated form of Navier-Stokes` equations that neglect vertical accelerations and use simplified Coriolis forces. Although there is no evidence so far that including nonhydrostatic effects in global models has any physical significance for large scale solutions, there is an emerging trend in the community toward restoring Navier-Stokes` equations (or at least their less constrained forms) in global models of atmospheres and oceans. The primary motivation is that state-of-the-art computers already admit resolutions where local nonhydrostatic effects become noticeable. much of this present research aims to improve the design of a high-performance numerical model for simulating the flows of moist (and precipitating), rotating, stratified fluids past a specified time-dependent irregular lower boundary. This model is representative of a class of nonhydrostatic atmospheric codes that employs the anelastic equations of motion in a terrain-following curvilinear framework, and contains parallel implementations of semi-Lagrangian and Eulerian approximations selectable by the user. The model has been employed in a variety of application; the quality of results suggest that modern nonoscillatory forward-in-time (NFT) methods are superior to the more traditional centered-in-time-and-space schemes, in terms of accuracy, computational efficiency, flexibility and robustness. The authors have extended the Cartesian NFT model to a mountainous sphere and, consequently, have dispensed with the traditional geophysical simplifications of hydrostaticity, gentle terrain slopes, and weak rotation. In this paper, they discuss the algorithmic design, relative efficiency and accuracy of several different variants (hydrostatic, nonhydrostatic, implicit, explicit, etc.) of the NFT global model. They substantiate their theoretical discussions with the results of simulations of idealized global orographic flows and climates

Strong and auxiliary forms of the semi-Lagrangian method for incompressible flows

We present a review of the semi-Lagrangian method for advection-diusion and incompressible Navier-Stokes equations discretized with high-order methods. In particular, we compare the strong form where the departure points are computed directly via backwards integration with the auxiliary form where an auxiliary advection equation is solved instead; the latter is also referred to as Operator Integration Factor Splitting (OIFS) scheme. For intermediate size of time steps the auxiliary form is preferrable but for large time steps only the strong form is stable

MPDATA: A positive definite solver for geophysical flows

This article is a review of MPDATA, a class of methods for the numerical simulation of advection based on the sign-preserving properties of upstream differencing. MPDATA was designed originally as an inexpensive alternative to flux-limited schemes for evaluating the transport of nonnegative thermodynamic variables (such as liquid water or water vapour) in atmospheric models. During the last decade, MPDATA has evolved from a simple advection scheme to a general approach for integrating the conservation laws of geophysical fluids on micro-to-planetary scales. The purpose of this paper is to summarize the basic concepts leading to a family of MPDATA schemes, review the existing MPDATA options, as well as to demonstrate the efficacy of the approach using diverse examples of complex geophysical flows

Forward in time methods for global climate research. Final report

Purpose is to demonstrate feasibility and utility of nonoscillatory forward-in-time (NFT) methods formodeling the global dynamics of the atmosphere and oceans. This includes development of new algorithms, construction of numerical models, and testing these models. One aspect of the research is to compare two variants of NFT methods, one based on Eulerian approximations and the other based on semi-Lagrangian approximations

Forward-in-Time Differencing for Fluids: Nonhydrostatic Modeling of Rotating Stratified Flow on a Mountainous Sphere

Traditionally, numerical models for simulating planetary scale weather and climate employ the hydrostatic primitive equations-an abbreviated form of Navier-Stokes equations that neglect vertical accelerations and use simplified inertial forces. 1 Although there is no evidence so far that including nonhydrostatic effects in global models has any physical significance for large scale solutions, there is an apparent trend in the community toward restoring Navier-Stokes equations (or at least their less constrained forms) in global models of atmospheres and oceans. The primary motivation for this is that the state-of-the-art computers already admit resolutions where local nonhydrostatic effects become noticeable. Other advantages include: the convenience of local mesh refinement; better overall accuracy; insubstantial computational overhead relative to hydrostatic models; universality and therefore convenience of maintaining a single large code; as well as conceptual simplicity and mathematical elegancy--features important for education. The few existing nonhydrostatic global models differ in analytic formulation and numerical design, reflecting their different purposes and origins. Much of our present research improves the design of a high-performance numerical model for simulating the flows of moist (and precipitating), rotating, stratified fluids past a specified time-dependent irregular lower boundary. This model is representative of a class of nonhydrostatic atmospheric codes employing the an elastic equations of motion in a terrain-following curvilinear framework, and contains parallel implementations of semi-Lagrangian and Eulerian approximations selectable by the user. The model has been employed in a variety of applications; the quality of results suggest that modern nonoscillatory forward-in-time (NFT) methods are superior to the more traditional centered-in-time-and-space schemes, in terms of accuracy, computational efficiency, flexibility and robustness