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NASA NTRS1993

The Venus atmospheric response to solar cycle variations

Atmospheric drag measurements from the orbital decay of the Pioneer Venus Orbiter and Magellan spacecraft have recently been obtained of the Venus dayside and nightside atmosphere between 130 and 210 km during a period of low solar activity. These new measurements, combined with the earlier Pioneer Venus drag measurements (1978-80) obtained near the maximum of the 11-year solar cycle, have allowed the detection of the detailed response of temperature, atomic oxygen and carbon dioxide to solar variations. We have found a weak but detectable temperature response on the dayside which is in accord with the response predicted by Keating and Bougher when they assumed very strong CO2 radiative cooling resulting from atomic oxygen exciting CO2 into 15 micron emission. This same radiative process may cause strong cooling in the Earth's upper atmosphere with the doubling of CO2 in the future. With decreasing solar activity, the O/CO2 ratio in the lower thermosphere is found to decrease, apparently due to decreased photodissociation of CO2 and lower temperatures. The percent decrease in atomic oxygen with decreasing solar activity on the dayside is found to be approximately the same as the percent decreases of atomic oxygen transported to the nightside. A very weak response of nightside temperatures to solar activity variations has also been detected.

Keating, Gerald M.↗
NASA NTRS1994

The Balmer-dominated northeast limb of the Cygnus loop supernova remnant

We present a comprehensive investigation of the Balmar-dominated northeast limb of the Cygnus Loop supernova remnant. Data presented include H alpha (O III), and X-ray images, UV and visible spectrophotometry, and high-resolution spectroscophy. The two relatively bright Balmer-dominated filaments visible on the POSS prints are seen to be part of a very smooth and regular complex of filaments. These filaments mark the current location of the blast wave and are seen to bound the sharply limb-brightened X-ray emission, including the previously reported X-ray, 'halo.' The (O III)/h beta ratio throughout the region is approximately 0.1, except for regions in which the shock is undergoing a transition from nonradiative to incomplete radiative to incomplete radiative conditions. At these locations (O III) emission from the cooling region is quite strong, while collisionally excited Balmer-line emission can be weak because of photoionization of the preshock medium by UV from the nascent cooling region. As a result (O III)/H beta is greater than 100 in some locations. The nonradiative/radiative transition is best studied along the length of the northwestern of the two brightest filaments, where the shock velocity and swept-up column go from approximately 180 km/s and 10(exp 17)/sq cm at one end to approximately 140 km/s and 8 x 10(exp 17)/cm at the other. There are also a number of locations of such incomplete radiative emission where the shock has recently encountered denser regions with characteristic sizes of approximately 10(exp 18) cm. There is a considerable amount of evidence that the shock has decelerated from approximately 400 km/s to less than 200 km/s in the last 1000 yr. We interpret this as the result of the blast wave hitting the wall of a cavity which surround the supernova precursor and succeed in matching a wide range of data with a reflected shock model in which the density ofthe cavity wall is approximately 1.2/cu cm and the density in the interior of the cavity is about 0.09/cu cm. This analysis supports previous suggestions that the Cygnus Loop was a cavity explosion. We opredict that the X-ray shell will continue to brighten for the next 1000 years or so, during which time the shock front will become radiative throughout the northeast region. The similarity of the time scales for post-shock cooling, shock deceleration, and the 1ressure evolution of the region violates assumptions of steady flow, and greatly complicated interpretation of shock specptra in the region. This may be a common and important effect in many supernova remnants. High-resolutoin spectroscopy of the H alpha line profile shows that the broad component is surprisingly narrow (approximately 130 km/s). Consistency between this line width and the shock velocity inferred from spectrophotometry requires either rapid postshock equilibration of T(sub i) and T(sub e), or that the shock front has declerated from 180 km/s to 150 km/s in the last approximately 200 yr. While we cannot unambiguously distinquish between these models, the required deceleration is consistent with extrapolation of the shock decleration rate inferred over the last 1000 years. The width of the narrow component of the h alpha line profile is approximately 3 km/s, which indicates that the preshock medium has been heated to approximately 25,000 K by a shock precursor. The most promising types of precursor are heating either by fast neutrals which overtake the shock or by cosmic rays. In the latter case the observed properties of the precursor constrain the cosmic-ray diffusion coefficient to be between about 10(exp 23) sq cm/s. The characteristics of the precursor seem to be about right to explain the presence of weak (S II) and (N II) emission previously reported by Fesen & Itoh (1985).

Hester, J. Jeff↗
NASA NTRS1994

Parasitic instabilities in magnetized, differentially rotating disks

Velikhov, Chandrasekhar, Balbus & Hawley have discovered a MHD instability of differentially rotating fluids that may explain the effective viscosity of accretion disks. If the unperturbed magnetic energy density is much less than the gas pressure, this `magnetorotational' instability (MRI) arises on small scales and is approximately incompressible. If the unperturbed field is suffiently, strong, the MRI is suppressed. We therefore ask whether the MRI mechanism can amplify even very weak initial fields until they reach equipartition with the gas pressure. We show that in the total incompressible limit, the MRI modes are exact solutions of the nonlinear fluid equations, even if the perturbed magnetic field is much larger than the unperturbed field. Also, we present a new exact solution in the opposite limit that the magnetic pressure is much larger than that of the gas. On the other hand, we show that the incompressible MRI modes are themselves subject to parasitic instabilities with instantaneous growth rates proportional to the MRI amplitude. Some of the parasitic instabilities are of the Kelvin-Helmholtz type, while others are less familiar. The eigenfunctions of the latter group suggest that they may promote rapid reconnection of the MRI field. Thus, parasitic modes may stop MRI growth at subequipartition amplitudes if it developes from a sufficiently weak initital field.

Goodman, Jeremy↗
NASA NTRS1995

The physics of massive OB stars in different parent galaxies. 1: Ultraviolet and optical spectral morphology in the Magellanic Clouds

Hubble Space Telescope/Faint Object Spectrograph (HST/FOS) and European Space Observatory (ESO) 3.6-m/CASPEC observations have been made of 18 stars ranging in spectral type from O3 through B0.5 Ia, half of them in each of the Large and Small Magellanic Clouds, in order to investigate massive stellar winds and evolution as a function of metallicity. The spectroscopic data are initially presented and described here in an atlas format. The relative weakness of the stellar-wind features in the SMC early O V spectra, due to their metal deficiency, is remarkable. Because of their unsaturated profiles, discrete absorption components can be detected in many of them, which is generally not possible in LMC and Galactic counterparts at such early types, or even in SMC giants and supergiants. On the other hand, an O3 III spectrum in the SMC has a weak C IV but strong N V wind profile, possibly indicating the presence of processed material. Wind terminal velocities are also given and intercompared between similar spectral types in the two galaxies. In general, the terminal velocities of the SMC stars are smaller, in qualitative agreement with the predictions of radiation-driven wind theory. Further analyses in progress will provide atmospheric and wind parameters for these stars, which will be relevant to evolutionary models and the interpretation of composite starburst spectra.

Walborn, Nolan R.↗
NASA NTRS1994

Shock normal determination for multiple-ion shocks

We have adapted the single-ion Vinas and Scudder (1986) solution to the Rankine-Hugoniot (R-H) problem to a multiple-on solution. Using this technique, we can calculate a shock normal direction, shock speed, best estimate of the upstream and downstream magnetic field and plasma asymptotic states, and theta(sub Bn), the angle between the shock normal and the upstream magnetic field. We test the multi-ion solution with a theoretical case but are restricted to a perpendicular shock in order to close the multi-ion Rankine-Hugoniot equations. For this test case both single-ion and multi-ion solutions are equally valid. We examine parameter regimes to look for differences between single-ion and multi-ion solutions of the R-H equations, and we find that the largest differences occur for quasi-parallel shocks, small values of solar wind speed, large values of heavy ion density, and very strong and very weak shocks. For both the inbound and outbound crossing of comet Halley we have a slow solar wind speed, small values of water group ions and fairly weak shocks. We examine both the quasi-perpendicular inbound crossing and the quasi-parallel outbound crossing at comet Halley.

Kessel, R. L.↗
NASA NTRS1994

Latitudinal variability of large-scale coronal temperature and its association with the density and the global magnetic field

In this paper we utilize the latitiude distribution of the coronal temperature during the period 1984-1992 that was derived in a paper by Guhathakurta et al, 1993, utilizing ground-based intensity observations of the green (5303 A Fe XIV) and red (6374 A Fe X) coronal forbidden lines from the National Solar Observatory at Sacramento Peak, and establish it association with the global magnetic field and the density distributions in the corona. A determination of plasma temperature, T, was estimated from the intensity ratio Fe X/Fe XIV (where T is inversely proportional to the ratio), since both emission lines come from ionized states of Fe, and the ratio is only weakly dependent on density. We observe that there is a large-scale organization of the inferred coronal temperature distribution that is associated with the large-scale, weak magnetic field structures and bright coronal features; this organization tends to persist through most of the magnetic activity cycle. These high-temperature structures exhibit time-space characteristics which are similar to those of the polar crown filaments. This distribution differs in spatial and temporal characterization from the traditional picture of sunspot and active region evolution over the range of the sunspot cycle, which are manifestations of the small-scale, strong magnetic field regions.

Guhathakurta, M.↗
NASA NTRS1994

Arctic chlorine monoxide observations during spring 1993 over Thule, Greenland, and implications for ozone depletion

We have determined the vertical distribution of chlorine monoxide (ClO), from measurements of pressure-broadened molecular-emission spectra made over Thule, Greenland, during the 1993 Arctic spring. The measurements show a weak lower stratospheric layer of chlorine monoxide inside the vortex in late February, which was, however, significantly greater in mixing ratio than that seen in observations we made in the spring of 1992. ClO was also observed in much smaller quantities in early to mid-March 1993 when Thule was outside the vortex. The amount of ClO within the vortex was severely reduced by the time it returned over Thule in late March. This reduction occurred several weeks earlier relative to the winter solstice than the decline of ClO inside the Antarctic vortex in 1993. The enhanced Arctic lower stratospheric layer seen in late February 1993 at a nearly equivalent photochemical period, and beyond. We have calculated daily ozone loss rates, due primarily to the dimer chlorine catalytic cycle, from both sets of measurements. The vertical integral of the Arctic daily percentage ozone loss when the largest ClO levels were present, at the end of February, is found to be approximately one quarter of that in the Antarctic at a photochemical period only 1 week later. The relative weakness of daily ozone depletion, combined with the early disappearance of ClO in the Arctic, suggests that hemispheric dilution by ozone-poor air from within the Arctic vortex is unlikely to be sufficient to explain the historically extreme loss of midlatitude northern hemisphere ozone which began in 1992 and persisted throughout 1993.

Shindell, D. T.↗
NASA NTRS1995

Oscillation-convection coupling: Cause of supergranulation

A semianalytic method is derived for dealing simultaneously with large numbers of linear stellar oscillation modes trapped in a cavity (a shell) of fluid which is rotating and convecting. A simple generalization of mixing-length theory shows how convection is modulated by weak rotational effects and by the horizontal wind fields of linear r-mode oscillations. The modulated convection is then used to compute the energy lost to turbulent viscosity by a family of nondegenerate oscillations. Viscosity terms of fourth degree in the wind shear can be included if they are a perturbation affecting only a small portion of the r-mode. Viscous energy loss strenghthens convection in a narrow layer near the base of the H and He ionization zone. In the Sun, this layer is about 7 Mm thick and centered at 0.932 of a solar radius where convection cells have a typical size of about 20 Mm and a lifetime of 0.3 Ms, both similar to what is observed in supergranules. If the rms velocity of r-modes at the surface exceeds 5 m/s, then energy is deposited inside the Sun at a sufficient rate to power the supergranulation and impose on it a weak latitude dependence.

Wolff, C.↗
NASA NTRS1994

Ultraviolet observations of galaxies with the FAUST experiment

We have used the set of point sources detected by the Far Ultraviolet Space Telescope (FAUST) instrument to identify galaxies and study the total galaxy flux in a 250 A wide band peaking at 1650 A. A sample of 144 galaxies has been obtained after cross-reference with the RC3 catalog, elimination of objects confused with stars and various corrections for the photometry. The UV-B color dispersion is found to increase while the galaxies get redder from late to early types. The irregular galaxies appear on average redder and the Sbc galaxies bluer than indicated by the spectral energy distributions currently used for the calculations of K-corrections. Various arguments lead us to make the assumption of a constant dust extinction within each galaxy. The UV flux per unit area decreases on average from late to early type spirals. We find a weak correlation between the UV and far infra-red emission while the infra-red to UV flux ratio gets lower when galaxies get bluer (as measured by the UV to B flux ratio). The UV flux per unit area correlates with the HI gas surface density and the total gas surface density when this quantity is available. The correlation with the molecular gas alone is weak. In the Virgo cluster, the UV flux per unit area does not decrease in direct proportion to the HI deficiency. Galaxy counts per square degree and per magnitude interval have been obtained at high-galactic latitudes. Combined with data at fainter magnitudes, they show a variation as a function of magnitude with a near-euclidean slope over a range of 8 magnitudes.

Deharveng, J.-M.↗
NASA NTRS1995

Line-of-sight magnetic flux imbalances caused by electric currents

Several physical and observational effects contribute to the significant imbalances of magnetic flux that are often observed in active regions. We consider an effect not previously treated: the influence of electric currents in the photosphere. Electric currents can cause a line-of-sight flux imbalance because of the directionality of the magnetic field they produce. Currents associated with magnetic flux tubes produce larger imbalances than do smoothly-varying distributions of flux and current. We estimate the magnitude of this effect for current densities, total currents, and magnetic geometry consistent with observations. The expected imbalances lie approximately in the range 0-15%, depending on the character of the current-carying fields and the angle from which they are viewed. Observationally, current-induced flux imbalances could be indicated by a statistical dependence of the imbalance on angular distance from disk center. A general study of magnetic flux balance in active regions is needed to determine the relative importance of other- probably larger- effects such as dilute flux (too weak to measure or rendered invisible by radiative transfer effects), merging with weak background fields, and long-range connections between active regions.

Gary, G. Allen↗
NASA NTRS1995

Thermal emission from particulate surfaces: A comparison of scattering models with measured spectra

Emissivity spectra of particulate mineral samples are highly dependent on particle size when that size is comparable to the wavelength of light emitted (5-50 micrometers for the midinfrared). Proper geologic interpretation of data from planetary infrared spectrometers will require that these particle size effects be well understood. To address this issue, samples of quartz powders were produced with narrow, well-characterized particle size distributions. Mean particle diameters in these samples ranged from 15 to 227 micrometers. Emission spectra of these powders allow the first detailed comparison of the complex spectral variations with particle size observed in laboratory data with the predictions of radiative transfer models. Four such models are considered here. Hapke's relectance theory (converted to emissivity via Kirchoff's law) is the first model tested. Hapke's more recently published emission theory is also employed. The third model, the 'Mie/Conel' model, uses Mie single scattering with a two-stream approximation for multiple scattering. This model, like the first, is a converted reflec- tance model. Mie scattering assumes particles are both spherical and well separated, which is not true for the quartz powders, but includes diffraction effects. The fourth model uses the Mie solution for single scattering by spheres and inputs those results into the multiple scattering formalism of Hapke's emission theory. The results of the four models are considered in relation to the values of the optical constants n and k. We have grouped these as class 1 (k large), class 2 (k moderate, n is approximately 2), class 3 (k small, n is approximately 2), and class 4 (k small, n is approximately 1). In general, the Mie/Hapke hybrid model does best at predicting variations with grain size. In particular, it predicts changes of the correct pattern, although incorrect magnitude, for class 1 bands, where large increases in emissivity with decreasing grain size are observed. This model also does an excellent job on moderate (class 2) and very weak and intraband (class 3) regions, and correctly predicts the emission maximum and its invariance with grain size near the Christiansen frequency (class 4). The Mie/Hapke hybrid model also has the fewest free parameters of the four models examined, while maintaining the most physical treatment of the radiative transfer. The Mie/Conel model performs as well as the Mie/Hapke hybrid model in strong bands (class 1) but does not accurately model the behavior of moderate (class 2) and very weak (class 3) bands.

Moersch, J. E.↗
NASA NTRS1995

Bolometric temperature and young stars in the Taurus and Ophiuchus complexes

We calculated bolometric temperature (T(sub bol)) and luminosity (L(sub bol)) for 128 young stellar objects (YSOs) in Taurus, 74 in the Ophiuchus 'core', and 33 in the Ophiuchus 'off-core' region. We have constructed the bolometric luminosity-temperature (BLT) diagram, the log-log plot of L(sub bol) versus T(sub bol), for the three samples. T(sub bol) is defined as the temperature of a blackbody having the same frequency as the observed continuum spectrum. It measures the redness (or coldness) of an astronomical source. The BLT diagram is analogous to the H-R diagram and allows for a direct and quantitative comparison of YSOs at a wide variety of evolutionary states, ranging from the most deeply embedded stars to T Tauri stars nearly on the main sequence. We found (1) T(sub bol) increases monotonically from embedded sources (approximately 60-500 K) to classical T Tauri stars (approximately 1000-3000 K) to weak-line T Tauri stars (approximately 2000-5000 K); (2) T(sub bol) correlates reasonably well with the age inferred from the evolutionary models of pre-main-sequence stars and protostars for embedded 'protostars' and weak-line T Tauri stars. There is no significant correlation for the classical T Tauri stars. These results can be understood in terms of dissipation of circumstellar dust envelope and disk during the early stages of stellar evolution. Sources in the three regions have different distributions in the BLT diagram. The Ophiuchus core has the highest fraction of cold sources among the three regions. These cold sources are also more luminous than the YSOs in the other regions. The Ophiuchus off-core sample is dominated by the more evolved pre-main-sequence stars. The Taurus sources have distributions intermediate in L(sub bol), T(sub bol), and age between the Ophiuchus core and off-core distributions. These may suggest differences in the star formation history, and possibly in the stellar masses and mass accretion rates in these star-forming regions.

Chen, H.↗
NASA NTRS1995

Temporal variation of meandering intensity and domain-wide lateral oscillations of the Gulf Stream

The path of the Gulf Stream exhibits two modes of variability: wavelike spatial meanders associated with instability processes and large-sale lateral shifts of the path presumably due to atmospheric forcing. The objectives of this study are to examine the temporal variation of the intensity of spatial meandering in the stream, to characterize large-scale lateral oscillations in the stream's path, and to study the correlation betwen these two dynamically distinct modes of variability. The data used for this analysis are path displacemets ofthe Gulf Stream between 75 deg and 60 deg W obtained from AVHRR-derived (Advanced Very High Resolution Radiometer) infrared images for the period April 1982 through December 1989. Meandering intensity, measured by the spatial root-mean-sqaure displacement of the stream path, displays a 9-month dominant periodicity which is persistent through the study period. The 9-month fluctuation in meandering intensity may be related to the interaction of Rosseby waves with the stream. Interannual variation of meandering intensity is also found to be significant, with meandering being mich more intense during 1985 than it was in 1987. Annual variation, however,is weak and not well-defined.The spatially averaged position of the stream, which reflects nonmeandering large-scale lateral oscillations of the stream path, is dominated by an annual cycle. On average, the mean position is farthest north in November and farthest south in April. The first empirical orthogonal function mode of the space-time path displacements represents lateral oscillatins that are in-phase over the space-time domain. Interannual oscillations are also observed and are found to be weaker than the annual oscillation. The eigenvalue of the first mode indicates that about 21.5% of the total space-time variability of the stream path can be attibuted to domain-wide lateral oscillation. The correlation between meandering intensity and domain-wide lateral oscillations is very weak.

Lee, Tong↗
NASA NTRS1995

The nonlinear evolution of inviscid Goertler vortices in three-dimensional boundary layers

The nonlinear development of inviscid Gortler vortices in a three-dimensional boundary layer is considered. We do not follow the classical approach of weakly nonlinear stability problems and consider a mode which has just become unstable. Instead we extend the method of Blackaby, Dando, and Hall (1992), which considered the closely related nonlinear development of disturbances in stratified shear flows. The Gortler modes we consider are initially fast growing and we assume, following others, that boundary-layer spreading results in them evolving in a linear fashion until they reach a stage where their amplitudes are large enough and their growth rates have diminished sufficiently so that amplitude equations can be derived using weakly nonlinear and non-equilibrium critical-layer theories. From the work of Blackaby, Dando and Hall (1993) is apparent, given the range of parameters for the Gortler problem, that there are three possible nonlinear integro-differential evolution equations for the disturbance amplitude. These are a cubic due to viscous effects, a cubic which corresponds to the novel mechanism investigated in this previous paper, and a quintic. In this paper we shall concentrate on the two cubic integro-differential equations and in particular, on the one due to the novel mechanism as this will be the first to affect a disturbance. It is found that the consideration of a spatial evolution problem as opposed to temporal (as was considered in Blackaby, Dando, and Hall, 1992) causes a number of significant changes to the evolution equations.

Blackaby, Nicholas↗
NASA NTRS1995

Laminar soot processes

Soot processes within hydrocarbon fueled flames are important because they affect the durability and performance of propulsion systems, the hazards of unwanted fires, the pollutant and particulate emissions from combustion processes, and the potential for developing computational combustion. Motivated by these observations, the present investigation is studying soot processes in laminar diffusion and premixed flames in order to better understand the soot and thermal radiation emissions of luminous flames. Laminar flames are being studied due to their experimental and computational tractability, noting the relevance of such results to practical turbulent flames through the laminar flamelet concept. Weakly-buoyant and nonbuoyant laminar diffusion flames are being considered because buoyancy affects soot processes in flames while most practical flames involve negligible effects of buoyancy. Thus, low-pressure weakly-buoyant flames are being observed during ground-based experiments while near atmospheric pressure nonbuoyant flames will be observed during space flight experiments at microgravity. Finally, premixed laminar flames also are being considered in order to observe some aspects of soot formation for simpler flame conditions than diffusion flames. The main emphasis of current work has been on measurements of soot nucleation and growth in laminar diffusion and premixed flames.

Sunderland, P. B.↗
NASA NTRS1996

The Evolution of Finite Amplitude Wavetrains in Plane Channel Flow

We consider a viscous incompressible fluid flow driven between two parallel plates by a constant pressure gradient. The flow is at a finite Reynolds number, with an 0(l) disturbance in the form of a traveling wave. A phase equation approach is used to discuss the evolution of slowly varying fully nonlinear two dimensional wavetrains. We consider uniform wavetrains in detail, showing that the development of a wavenumber perturbation is governed by Burgers equation in most cases. The wavenumber perturbation theory, constructed using the phase equation approach for a uniform wavetrain, is shown to be distinct from an amplitude perturbation expansion about the periodic flow. In fact we show that the amplitude equation contains only linear terms and is simply the heat equation. We review, briefly, the well known dynamics of Burgers equation, which imply that both shock structures and finite time singularities of the wavenumber perturbation can occur with respect to the slow scales. Numerical computations have been performed to identify areas of the (wavenumber, Reynolds number, energy) neutral surface for which each of these possibilities can occur. We note that the evolution equations will breakdown under certain circumstances, in particular for a weakly nonlinear secondary flow. Finally we extend the theory to three dimensions and discuss the limit of a weak spanwise dependence for uniform wavetrains, showing that two functions are required to describe the evolution. These unknowns are a phase and a pressure function which satisfy a pair of linearly coupled partial differential equations. The results obtained from applying the same analysis to the fully three dimensional problem are included as an appendix.

Hewitt, R. E.↗
NASA NTRS1996

An Investigation of Cavity Vortex Generators in Supersonic Flow

The purpose of this report is to document the results of experiments performed at the University of Kansas and at the NASA Langley Research Center (LaRC) into the use of shaped cavities to generate vortices in supersonic flow, as well as the progress made in simulating the observed flow using the PAB3D flow solver. The investigation was performed on 18 different cavity configurations installed in a convergent-divergent nozzle at the Jet Exit Facility at the LaRC. Pressure sensitive paint, static-pressure ports, focusing Schliern, and water tunnel flow visualization techniques were used to study the nature of the flow created by these cavities. The results of these investigations revealed that a shaped cavity can generate a pair of counter-rotating streamwise vortices in supersonic flow by creating weak, compression Mach waves and weak shocks. The PAB3D computer program, developed at the LaRC, was used to attempt to reproduce the experimental results. Unfortunately, due to problems with matching the grid blocks, no converged results were obtained. However, intermediate results, as well as a complete definition of the grid matching problems and suggested courses of actions are presented.

Hazlewood, Richard↗
NASA NTRS1995

An extended structure-based model based on a stochastic eddy-axis evolution equation

We have proposed and implemented an extension of the structure-based model for weak deformations. It was shown that the extended model will correctly reduce to the form of standard k-e models for the case of equilibrium under weak mean strain. The realizability of the extended model is guaranteed by the method of its construction. The predictions of the proposed model were very good for rotating homogeneous shear flows and for irrotational axisymmetric contraction, but were seriously deficient in the case of plane strain and axisymmetric expansion. We have concluded that the problem behind these difficulties lies in the algebraic constitutive equation relating the Reynolds stresses to the structure parameters rather than in the slow model developed here. In its present form, this equation assumes that under irrotational strain the principal axes of the Reynolds stresses remain locked onto those of the eddy-axis tensor. This is correct in the RDT limit, but inappropriate under weaker mean strains, when the non-linear eddy-eddy interactions tend to misalign the two sets of principal axes and create some non-zero theta and gamma.

Kassinos, S. C.↗