Piotr K. Smolarkiewicz

A Fully Multidimensional Positive Definite Advection Transport Algorithm with Small Implicit Diffusion

In numerical modeling of physical phenomena it is often necessary to solve the advective transport equation for positive definite scalar functions. Numerical schemes of second- or higher-order accuracy can produce negative values in the solution due to the dispersive ripples. Lower-order schemes, such as the donor cell or Lax-Friedrichs, or higher-order schemes with zeroth-order diffusion added produce no ripples but suffer from excessive implicit diffusion. In the last ten years a possible resolution of this dilemma has been developed in the form of hybrid schemes, in which the advective fluxes are given as a weighted average of a first-order positive definite scheme’s fluxes and a higher-order scheme’s fluxes. The difference in determination of the weights in the calculation of the average advective fluxes has led to different hybrid schemes. Two main hybrid-type schemes have been developed. One, the so called flux-corrected transport (FCT) method, was originated by Boris and Book [ I-31 and generalized by Zalesak [ 141; the other was developed by Harten and Zwas [ 6,9] in the form of the self-adjusting hybrid schemes (SAHS) method. Both methods were constructed to deal effectively with shocks and contact discontinuities. Solutions of the advection transport equation obtained by using FCT or SAHS maintain positive definiteness of the initial condition and, as can be seen from presented tests (Zalesak [ 141, Harten [6]), be very accurate. Unfortunately, application of these methods to the modeling of complex multidimensional hydrodynamical systems like atmospheric phenomena is rather limited due to the excessive computer time required. Furthermore, in many hydrodynamical systems,

A Simple Positive Definite Advection Scheme with Small Implicit Diffusion

Abstract The development of negative values for positive definite scalars in the solution of the advection equation is an important difficulty in numerical modeling. This paper proposes a new positively definite advection scheme which has a simple form, small implicit diffusion and low computational cost. Comparisons of the present scheme with some other known positive definite schemes are also presented.

Low Froude Number Flow Past Three-Dimensional Obstacles. Part I: Baroclinically Generated Lee Vortices

Abstract We study the flow of a density-stratified fluid past a three-dimensional obstacle, using a numerical model. Our special concern is the response of the fluid when the Froude number is near or less than unity. Linear theory is inapplicable in this range of Froude number, and the present numerical solutions show the rich variety of phenomena that emerge in this essentially nonlinear flow regime. Two such phenomena, which occupy Parts I and II of this study, are the formation of a pair of vertically oriented vortices on the lee side and a zone of flow reversal on the windward side of the obstacle. The Ice vortices have been explained as a consequence of the separation of the viscous boundary layer from the obstacle however, this boundary layer is absent (by design) in the present experiments and lee vortices still occur. We argue that a vertical component of vorticity develops on the lee side owing to the tilting of horizontally oriented vorticity produced baroclinically as the isentropes deform in r...

The multidimensional positive definite advection transport algorithm: nonoscillatory option

Abstract This paper presents a nonoscillatory option (i.e., free of dispersive ripples) of the advection algorithm described previously in J. Comput. Phys. ( 54 (1984), 325; 67 (1986), 396). The approach adopted merges the flux-corrected transport methodology with the iterative formalism of the algorithm. Further discussion of the algorithms accuracy is included. Theoretical considerations are illustrated through numerical tests and examples of applications to atmospheric fluid dynamics problems.

Gravity Waves, Compensating Subsidence and Detrainment around Cumulus Clouds

Abstract Gravity waves play an important role in the redistribution of heat and moisture in a deep convecting cloud field. We explore this role in a two-dimensional numerical experiment on a simple moist convecting system consisting of an isolated long-lasting nonprecipitating cloud in a calm atmosphere with no surface forcing. The cloud develops a horizontally averaged density variation with height which is neutrally buoyant with respect to a moist adiabatic. The buoyancy difference between the cloud and the undisturbed sounding produces circulations that can be understood as spreading gravity waves which adjust the environmental buoyancy to be equal to the cloud buoyancy by compensating subsidence. Unlike the circulations inside clouds, this adjustment takes place without turbulent mixing. Hence, the “buoyancy adjustment time” T1 during which the environment comes into rough buoyant equilibrium with the clouds is much shorter than the “mixing time” T2 which it takes a tracer, initially concentrated at s...

CRCP: a cloud resolving convection parameterization for modeling the tropical convecting atmosphere

Abstract A new computational approach, CRCP, is proposed in which both the large-scale (LS) tropical dynamics and cloud-scale (CS) dynamics are captured explicitly. The leading idea is to represent subgrid scales of the LS model by imbedding a 2D CS model in each column of the 3D LS model – the approach tailored for distributed memory architectures. The overall philosophy underlying CRCP is the reinvestment of efforts from large-eddy simulation to elaborate yet ‘embarrassingly parallel’ turbulence models. Similar as in the traditional ‘convection parameterization’, the LS model provides ‘ambient forcings’ for the CS model imbedded inside each LS column, and the CS model feeds back a ‘convective response’ for every column of the LS model. Furthermore, availability of the cloud-scale data allows for explicit coupling of moist convection with radiative and surface processes. Following our experience with cloud-resolving modeling of the tropical convection, the CS model is oriented along the E–W direction inside each LS model column. A simple strategy for the coupling the LS and CS models derives from physical understanding of interactions between LS flow and moist tropical convection. Theoretical considerations are illustrated with an example of application to observational data from the Phase III of the Global Atmospheric Research Programme Atlantic Tropical Experiment (GATE).

The multidimensional positive definite advection transport algorithm: further development and applications

Abstract This paper presents further generalizations of the advection algorithm described in J. Comput. Phys. (54 (1984), 325). Time-dependent velocity fields and a generalized form of the continuity equation are considered. Applicability of the algorithm to the diffusion equation and to the transport of nonpositive scalars is also discussed. Theoretical considerations are illustrated through numerical tests and applications to some particular geophysical fluid dynamics problems.

MAGNETIC CYCLES IN GLOBAL LARGE-EDDY SIMULATIONS OF SOLAR CONVECTION

We report on a global magnetohydrodynamical simulation of the solar convection zone, which succeeds in generating a large-scale axisymmetric magnetic component, antisymmetric about the equatorial plane and undergoing regular polarity reversals on decadal timescales. We focus on a specific simulation run covering 255 years, during which 8 polarity reversals are observed, with a mean period of 30 years. Time-latitude slices of the zonally averaged toroidal magnetic component at the base of the convecting envelope show a well-organized toroidal flux system building up in each solar hemisphere, peaking at mid-latitudes and migrating toward the equator in the course of each cycle, in remarkable agreement with inferences based on the sunspot butterfly diagram. The simulation also produces a large-scale dipole moment, varying in phase with the internal toroidal component, suggesting that the simulation may be operating as what is known in mean-field theory as an αΩ dynamo.

On the Dynamics of Hawaiian Cloud Bands: Island Forcing

Abstract This study focuses on basic island scale forcing mechanisms for the formation and evolution of a band cloud typically present upwind of the island of Hawaii. By means of numerical experiments and verification of our results against observations and laboratory experiments reported in the literature, we show that the band cloud is a complex three-dimensional phenomenon which is inseparable from the airflow around the island. In particular, we demonstrate that the event needs to be analyzed in terms of the basic fluid dynamics of strongly stratified flow past a three-dimensional obstacle. The band cloud is found to arise primarily from the dynamic interaction of the trade winds with the island. The upwind surface flow forms a separation line with an associated stagnation point. A low-level convergence zone forms along this line, resulting in an updraft line. If the updrafts are strong enough, a band cloud forms. Formation and characteristics of such a system are mostly controlled by the environmenta...

On Forward-in-Time Differencing for Fluids: an Eulerian/Semi-Lagrangian Non-Hydrostatic Model for Stratified Flows

ABSTRACT In this paper, we describe a non-hydrostatic anelastic model for simulating stratified flows in terrain-following coordinates. The model is based solely on non-oscillatory forward-in-rime integration schemes, and our primary goal is to demonstrate the utility of such methods for modelling small-scale atmospheric dynamics. We use the formal similarity of the Eulerian and semi-Lagrangian equations of two-time-level approximations to construct a unified model that readily allows selection of either formulation. We apply the model to two test problems of stratified flows past isolated obstacles. We use these tests to validate the forward-in-time approach against a traditional centred-in-time-and-space Eulerian model, and to discuss the relative accuracy and efficiency of the two formulations of our model. One problem illustrates the efficacy of flux-form Eulerian methods, while the other demonstrates strengths of the semi-Lagrangian approach.