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DOE OSTI2022

High-order multirate explicit time-stepping schemes for the baroclinic-barotropic split dynamics in primitive equations

In order to treat the multiple time scales of ocean dynamics in an efficient manner, the baroclinic-barotropic splitting technique has been widely used for solving the primitive equations for ocean modeling. Based on the framework of strong stability-preserving Runge-Kutta approach, we propose two high-order multirate explicit time-stepping schemes (SSPRK2-SE and SSPRK3-SE) for the resulting split system in this paper. The proposed schemes allow for a large time step to be used for the three-dimensional baroclinic (slow) mode and a small time step for the two-dimensional barotropic (fast) mode, in which each of the two mode solves just need to satisfy their respective CFL conditions for numerical stability. Specifically, at each time step, the baroclinic velocity is first computed by advancing the baroclinic mode and fluid thickness of the system with the large time-step and the assistance of some intermediate approximations of the barotropicmode obtained by substepping with the small time step; then the barotropic velocity is corrected by using the small time step to re-advance the barotropic mode un-der an improved barotropic forcing produced by interpolation of the forcing terms from the preceding baroclinic mode solves; lastly, the fluid thickness is updated by coupling the baroclinic and barotropic velocities. Additionally, numerical inconsistencies on the discretized sea surface height caused by the mode splitting are relieved via a reconciliation process with carefully calculated flux deficits. Here, two benchmark tests from the “MPAS-Ocean” platform are carried out to numerically demonstrate the performance and parallel scalability of the proposed SSPRK-SE schemes.

54 ENVIRONMENTAL SCIENCES↗

NASA NTRS1994

Numerical simulation of baroclinic Jovian vortices

We examine the evolution of baroclinic vortices in a time-dependent, nonlinear numerical model of a Jovian atmosphere. The model uses a normal-mode expansion in the vertical, using the barotropic and first two baroclinic modes. Results for the stability of baroclinic vortices on an f plane in the absence of a mean zonal flow are similar to results of Earth vortex models, although the presence of a fluid interior on the Jovian planets shifts the stability boundaries to smaller length scales. The presence of a barotropic mean zonal flow in the interior stabilizes vortices against instability and significantly modifies the finite amplitude form of baroclinic instabilities. The effect of a zonal flow on a form of barotropic instability produces periodic oscillations in the latitude and longitude of the vortex as observed at the level of the cloud tops. This instability may explain some, but not all, observations of longitudinal oscillations of vortices on the outer planets. Oscillations in aspect ratio and orientation of stable vortices in a zonal shear flow are observed in this baroclinic model, as in simpler two-dimensional models. Such oscillations are also observed in the atmospheres of Jupiter and Neptune. The meridional propagation and decay of vortices on a beta plane is inhibited by the presence of a mean zonal flow. The direction of propagation of a vortex relative to the mean zonal flow depends upon the sign of the meridional potential vorticity gradient; combined with observations of vortex drift rates, this may provide a constraint on model assumption for the flow in the deep interior of the Jovian planets.

Achterberg, Richard K.↗

DOE OSTI2021

Improved Time-Stepping Methods in Global to Regional Ocean Modeling (Annual Status Report 2020)

Time stepping algorithms are an important part of ocean models, and strongly influence both the accuracy of solution and performance. There have been a number of projects investigating various improvements for ocean time-stepping schemes in the Model for Prediction Across Scales-Ocean (MPAS-Ocean), a component of the DOE Energy Exascale Earth System Model. Ocean dynamics include fast surface gravity waves, which are two-dimensional, and slower internal waves, which are three-dimensional, so ocean models use a split time-stepping scheme that separates these barotropic and baroclinic modes for efficiency. MPAS-Ocean runs on variable-resolution horizontal meshes, and must scale to tens of thousands of cores and millions of horizontal gridcells. Ocean models require time stepping algorithms that are customized to these needs, and which are tuned for performance on various resolutions and architectures.

58 GEOSCIENCES↗

DOE OSTI2022

Improved Time-Stepping Methods in Global to Regional Ocean Modeling (Annual Status Report)

58 GEOSCIENCES↗

DOE OSTI2023

The r -modes of slowly rotating, stratified neutron stars

ABSTRACT The only r-modes that exist in a globally barotropic, rotating, Newtonian star are the fundamental l = |m| solutions, where l and m are the indices of the spherical harmonic $Y_l^m$ that describe the mode’s angular dependence. This is in stark contrast to a stellar model that is non-barotropic throughout its interior, which hosts all the l ≥ |m| perturbations including radial overtones. In reality, neutron stars are stratified with locally barotropic regions. Therefore, we explore how stratification alters a star’s ability to support r-modes. We consider the globally stratified case and examine the behaviour of the modes as the star gets close to barotropicity. In this limit, we find that all but the fundamental l = |m| perturbations change character and become generic inertial modes. Restricting the analysis to l = |m| perturbations, we develop the r-mode equations in order to consider stellar models that exhibit local barotropicity. Our results for such models show that the r-mode overtones diverge and join the inertial modes. In order to see which r-modes persist and retain their character in realistic neutron stars, these calculations will need to be brought into full general relativity.

79 ASTRONOMY AND ASTROPHYSICS↗

NASA NTRS1979

A theoretical study of three-dimensional barotropic instability with applications to the upper stratosphere

A numerical method for tropospheric baroclinic instability is used to establish the structures, phase speeds and growth rates of the normal modes of a number of idealized barotropically unstable mean zonal wind fields in a quasi-geostrophic Boussinesq framework. The objective was to gain insight into the effects of vertical mean flow variation on barotropically unstable stratosphere. Both horizontally symmetric and asymmetric flows are considered, but in all cases the flows are vertically symmetric. Major findings show that (1) the wave growth rates associated with the strongest expected horizontal shears decrease with increasing vertical shear and are at least 28% smaller for the weakest expected vertical shear than with no shear; (2) differences in the form of either the vertical or the horizontal shear flow can affect this growth rate reduction by about 25%; and (3) the vertical scale of the barotropic unstable waves is of the order of the geometric mean of the vertical mean flow scale and the vertical penetrat

Pfister, L.↗

DOE OSTI2022

Universal relations for neutron star 𝑓-mode and 𝑔-mode oscillations

Among the various oscillation modes of neutron stars, 𝑓- and 𝑔- modes are the most likely to be ultimately observed in binary neutron star mergers due to their relatively large coupling and shared frequencies with tidal excitations. The 𝑓-mode frequency and damping time are known to correlate in normal neutron stars with their compactness, and previous fits to hadronic stars are extended and shown to be valid for an extremely broad sampling of equations of state using a piecewise polytropic parametrization scheme for hadrons and a constant sound-speed parametrization for quark matter. Separate fits applicable to quark (self-bound) stars are improved. Much more significant correlations exist with tidal deformability, and therefore with moment of inertia and quadrupole moment. It is conclusively demonstrated that these correlations are the same for all types of stars, whether hadronic, hybrid, or pure quark, and its accuracy is quantified. A novel 1-node branch of the 𝑓-mode that occurs in low-mass hybrid stars in a narrow mass range just beyond the critical mass necessary for a phase transition to appear is identified. This 1-node branch shows the largest, but still small, deviations from the universal correlation for any configuration. It is characterized by a nonmonotonic relation between neutron star mass and 𝑓-mode frequency, in contrast to the behavior otherwise observed in normal, quark and hybrid stars. The 𝑔-mode only exists in matter with a nonbarotropic equation of state involving temperature, chemical potential or composition (such as being out of beta equilibrium), or a phase transition in barotropic matter. Here, the 𝑔-mode therefore could serve as a probe for studying phase transitions in hybrid stars. In contrast with the 𝑓-mode, 𝑔-mode frequencies do not correlate well with tidal deformability, but depend strongly on properties of the transition (the density and the magnitude of the discontinuity) at the transition. Imposing causality and maximum mass constraints, a fit involving neutron star and phase transition properties is found and the 𝑔-mode frequency is determined to have an upper bound of about 1.25 kHz. However, if the sound speed 𝑐𝑠 in the inner core at densities above the phase transition density is restricted to 𝑐$^{2}_{𝑠}$ ≤1/3, 𝑔-mode frequencies can only reach about 0.8 kHz, which are significantly lower than 𝑓-mode frequencies (1.3–2.8 kHz). 𝑔-mode gravitational wave damping times are found to be extremely long, >10 4 s (102 s) in the inner core with 𝑐$^{2}_{𝑠}$ ≤1/3 (1), in comparison with 𝑓-mode damping times (0.1–1 s).

Composition of astronomical objects↗

NASA NTRS1986

A comparison of the bounded derivative and the normal-mode initialization methods using real data

Application of the bounded-derivative and normal-mode methods to a simple linear barotropic model at a typical middle latitude shows that the two methods lead to identical constraints up to a certain degree of approximation. Beyond this accuracy the two methods may differ from each other. When applied to a global nonlinear barotropic model using real data, again the two methods lead to similar balanced initial states. The gravity oscillations in the unbalanced height field, which have amplitudes of up to 60 m with a dominant periodicity of about 5 to 6 h, are practically eliminated by both initialization methods. The rotational wind component is smooth even for the unbalanced initial state. The small-scale spatial features of the irrotational wind component are drastically reduced by initialization. Both the nonlinear normal-mode and the bounded-derivative initialization methods yield similar divergence fields centered around the areas of highest orography. The comparison shows that there is no significant loss of information in the mass and momentum fields, despite the fact that the bounded-derivative method employs only the original, rotational wind component to construct a balanced initial state compared to the normal-mode method, which, in addition, makes use of the unbalanced divergent wind and height fields.

Semazzi, F. H. M.↗

NASA NTRS1984

Ageostrophic instabilities in a two fluid system with rotation

The stability of synoptic scale waves formed on a frontal surface is studied including nongeostrophic effects with the basic flow subjected to both vertical and horizontal shear. Spectral method is used to obtain the desired solutions. The stability characteristics of the developed unstable modes are presented as a function of shears of the basic flow. With the inclusion of barotropic shear the spectrum of instabilities increase. The lower speeded member of the mixed mode (gravitational-rotational) pair is influenced by the barotropic shear in the basic current and it appears at lower vertical shears. The structure of the height perturbations are utilized to distinguish the various unstable modes developed in the system together with their stability characteristics. This investigation has shown that the ageostrophic effects can be a significant factor in the development of synoptic scale waves on a frontal surface.

Ramanathan, N.↗

NASA NTRS1983

Absolute barotropic instability and monsoon depressions

Monsoon depressions over the Bay of Bengal develop almost entirely in July and August. After studies conducted by Lindzen et al. (1980) and Stevens and Lindzen (1978), only barotropic instability remains as a mechanism for the development of the wave disturbances associated with monsoon depressions. The present investigation has the objective to show that barotropic instability is able to explain the wave aspects of monsoon depressions, but that normal mode analysis is inadequate. It is found that the local barotropically unstable response to regional perturbations in the Bay of Bengal during July and August will be dominated by the lower troposphere. The analysis clearly identifies the features of the mean flow which lead to monsoon depressions in July. The features include the development of an easterly jet as opposed to semijet structure in the mean flow, and the development of a modest easterly flow at the jet center as opposed to westerly flow.

Lindzen, R. S.↗

NASA NTRS1984

The Influence of Synoptic Scales on Low-frequency Modes of Variability

Model analysis and predictablity studies using an equivalent barotropic empirical orthogonal function (EOF) model described in Schubert (1984b) are dicussed. An extensive set of 10-day forecasts was produced for the 12 component version of the model. Comparisons with observed data show apparently useful forecasts for individual winters. However, when averaged over 10 winters, the improvement over persistence was marginal and only one of the EOF's (resembling North Atlantic blocking) showed a statistically significant improvement. These results suggest the need for a more realistic model which includes shorter time scale baroclinic processes and spatial modes capable of efficiently representing the characteristic structures associated with baroclinic instability.

Schubert, S. D.↗

DOE OSTI2023

Nonequilibrium effects on stability of hybrid stars with first-order phase transitions

The stability of hybrid stars with first-order phase transitions as determined by calculating fundamental radial oscillation modes is known to differ from the predictions of the widely used Bardeen-Thorne-Meltzer criterion. We consider the effects of out-of-chemical-equilibrium physics on the radial modes and hence stability of these objects. For a barotropic equation of state, this is done by allowing the adiabatic sound speed to differ from the equilibrium sound speed. We show that doing so extends the stable branches of stellar models, allowing stars with rapid phase transitions to support stable higher-order stellar multiplets similarly to stars with multiple slow phase transitions. We also derive a new junction condition to impose on the oscillation modes at the phase transition. Termed the reactive condition, it is physically motivated, consistent with the generalized junction conditions between two phases, and has the common rapid and slow conditions as limiting cases. Unlike the two common cases, it can only be applied to nonbarotropic stars. Here, we apply this junction condition to hybrid stellar models generated using a two-phase equation of state consisting of nuclear matter with unpaired quark matter at high densities joined by a first-order phase transition and show that, like in the slow limiting case, stars that are classically unstable are stabilized by a finite chemical reaction speed.

79 ASTRONOMY AND ASTROPHYSICS↗

NASA NTRS1981

Barotropic instability of zonal flows

The barotropic instability of zonal flows due to infinitesimal normal-mode perturbations is considered, with the zonal flow assumed to be continuous and either monotonic or nonmonotonic and allowed to have one or more inflection points. A sufficient condition for instability is derived for this general flow profile that complements that of Arnol'd (1965), which restricted itself to the prediction of stability only.

Tung, K. K.↗

NASA NTRS1985

A statistical-dynamical study of empirically determined modes of atmospheric variability

The observed wintertime intraseasonal variability of the northern hemisphere midtropospheric circulation is analyzed within the framework of an equivalent barotropic model. The data set and the form of the eigenvalue/vector problem in the wave domain are described, and the statistical significance of the empirical orthogonal functions (EOFs) is discussed, and some selected modes are briefly described. The anomaly model and the transformation to the EOF domain are presented together with a discussion of the model coefficients and forcing term. The fitting procedure for the parameterized terms are described and the level of correspondence between the modelled and observed streamfunction tendencies as a function of the EOFs is examined. The stability of the time mean flow and other base states is investigated.

Schubert, S. D.↗

NASA NTRS1986

Vertical structure of the wind field during the Special Observing Period I of the Global Weather Experiment

The vertical structure of the global atmosphere is analyzed for selected periods of the Special Observing Period I (SOP-I) for the Global Weather Experiment (GWE). The analysis consists of projection of the stream-function and velocity potential at 200 and 850 mb on spherical harmonics and of the wind and height fields on the normal modes of a linearized form of the primitive equations for a basic state at rest. The kinematic vertical structure is discussed in terms of correlation coefficients of the 200 mb and 850 mb winds and analysis of the internal and external normal modes of the primitive equations. The reliability of the results is checked by applying the same analysis methods to data sets obtained from three different institutions: Geophysical Fluid Dynamics Laboratory (GFDL), European Center for Medium Range Weather Forecasting (ECMWF), and Goddard Laboratory for the Atmospheres (GLA). It is found that, on a global basis, vertically reversing circulations are as important as the equivalent barotropic structures. For the verticaly reversing components, the gravity and mixed Rossby-gravity modes have contributions of the same order of magnitude as those of the Rossby modes in tropical latitudes.

Paegle, J. N.↗

NASA NTRS1994

Low-frequency modes and nonbarotropic effects in pseudo-Newtonian accretion disks

A recently developed formalism is used to reexamine the question of the existence of hydrodynamical modes that pulsate with very low frequencies in the inner regions of accretion disks. The formalism is valid in an exact sense for the adiabatic pulsations of rotating Newtonian fluids that are generally nonbarotropic (such as those with 'nonadiabatic temperature gradients,' for example), and hence its application in the present context represents an improvement over previous analyses that are more approximate. The formalism is applied to thin non-self-gravitating disks, with the gravitational potential of the central source modified in the usual way in order to simulate relativistic effects. In the barotropic limit, the analyses indicate that in many cases nearly Keplerian disks exhibit nonaxisymmetric modes of pulsation that are trapped in the inner disk regions, with pulsation periods much longer than the dynamical timescale. These results are similar to those of earlier calculations that assume disks pulsate without changing the temperature distribution. A method is developed for including lowest order nonbarotropic effects. Previous analyses have been incapable of accurately treating the nonbarotropic regime. The application of the present method to the low-frequency modes reveals that, due to unexpected cancellations among terms, the nonbarotropic correction to the pusation frequency omega is only of order tilde-omega(sub BV exp 2) omega, where tilde-omega(sub BV) is the appropriate dimensionless Brunt-Vaisala frequency. This correction is much smaller than the expected correction of order tilde-omega(sub BV) Omega, where Omega is the rotation angular velocity. The important conclusion drawn from this is that nonbarotropic corrections are generally small and hence that low-frequency modes persist into the nonbarotropic regime. For disk temperatures appropriate to X-ray emission, the adiabatic frequencies of trapped modes are of the same order as the frequencies associated with quasi-periodic oscillations (QPOs) of certain low-mass X-ray binaries and other X-ray sources. It is suggested that further consideration be given to the possiblity that modulation of emission processes by these modes can explain at least some of the QPO phenomena.

Ipser, James R.↗

NASA NTRS1978

Solitary Rossby waves in zonal shear flows and their interactions

Interactions of long-wave solitons propagating in shear flows are described by a coupled pair of Korteweg-de Vries equations. The basic equation of motion for the analysis is the quasi-geostrophic forecast equation, and the interaction of two wave modes is studied. The solution for mode 1-mode 2 interaction of solitary waves in an asymmetric shear flow of a barotropic atmosphere with divergence is constructed. Streamline patterns for certain flows are obtained. An unsteady solitary wave solution for a modified Korteweg-de Vries equation is derived.

Redekopp, L. G.↗

NASA NTRS1982

Wave instability in the polar region of Venus

Infrared and radio observations of the upper cloud region of Venus indicate that the north polar region contains features of large thermal contrast. A cold collar, encompassing a region of temperature inversions, lies between latitudes of about 65 and 75 deg, and a pair of warm features, separated by about 180 deg of longitude and centered near 80 deg latitude, rotate about the pole with a period of about 2.9 days. It is shown that the cold temperatures associated with the inversions lead to an enhancement in the mean zonal wind in a localized area near the pole, and that this enhancement makes the mean flow barotropically unstable. Since data for this region are limited, a model for the thermal structure has been used for calculating growth times and phase periods of the unstable modes. Choosing model parameters to agree as closely as possible with available data, it has been determined that the rotating warm features are likely to be manifestations of barotropically unstable waves.

Elson, L. S.↗