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NASA NTRS1984

An analysis of the stability of the compressible Ekman boundary layer

The linear stability problem for the compressible Ekman boundary layer common to rotating fluids is formulated and the stability properties determined numerically. Three classes of unstable waves are identified (called class A, B, and C), their properties are described. The class C waves have only recently been reported in the literature and are present only in compressible Ekman boundary layers. Most of the calculations presented here are for uranium hexafluoride gas; however, critical Reynolds numbers are also computed for air and ammonia gas. Compressibility is generally found to decrease the critical Reynolds number for each class of wave. A comparison of results for the three different gases shows the stability to be largely unaffected by changes in the gas properties. Maximum growth rate calculations for each wave show the class A and B waves to be the dominant instabilities.

Spall, J. R.↗

NASA NTRS1985

Baroclinic adjustment and midlatitude temperature profiles

A baroclinic adjustment that alters the vertical temperature structure and shows how features of the observed midlatitude temperature structure can be understood in terms of such an adjustment is presented. The formulation of the adjustment and its properties are described, giving a comparison between observed and adjusted profiles. Results from a simple model for the midlatitude atmosphere's vertical temperature structure that display such an adjustment are presented, and a comparison is made between the adjustment and two-layer model stability. The results suggest that vertical heat fluxes from baroclinically unstable waves can produce significant variations of midlatitude lapse rates in both height and time. Midlatitude models that omit these variations may be missing an important aspect of wave-mean flow interaction.

Gutowski, W. J., Jr.↗

NASA NTRS1985

The utilization of satellite data and dynamics in understanding and predicting global weather phenomena

A two layer spectral quasi-geostrophic model is used to simulate the effects of topography on the equilibria, the stability, and the long term evaluation of incipient unstable waves. The flow is forced by latitudinally dependent radiational heating. The nature of the form drag instability of high index equilibria is investigated. The proximity of the equilibrium shear to a resonant value is essential for the instability, provided the equilibrium occurs at a slightly stronger shear than resonance. The properties of the steady Hadley and Rossby required for a thermally forced rotating fluid on a sphere are further explained. An objective parameterization technique is developed for general nonlinear hydrodynamical systems. The typical structure is one in which the rates of change of the dependent variables depend on homogeneous quadratic and linear forms, as well as on inhomogeneous forcing terms. Also documented is a steady, axisymmetric model of the general circulation developed as a basis for climate stability studies. The model includes the effects of heating, rotation, and internal friction, but neglects topography. Included is further research on cloud street phenomena. Orientation angles and horizontal wavelengths of boundary layer rolls and cloud streets are determined from an analysis of a truncated spectral model of three dimensional shallow moist Boussinesq convection in a shearing environment is further explained. Relatively broadly spaced roll clouds have orientations for which the Fourier component of the roll perpendicular shear is nearly zero, but the second corresponds to narrowly spaced rolls having orientations for which the Fourier coefficients of both the perpendicular and the parallel components of the shear are nearly equal.

Shirer, H. N.↗

NASA NTRS1985

The effect of topography on the evolution of unstable disturbances in a baroclinic atmosphere

A two layer spectral quasi-geostrophic model is used to simulate the effects of topography on the equilibria, their stability, and the long term evolution of incipient unstable waves. The flow is forced by latitudinally dependent radiative heating. Dissipation is in the form of Rayleigh friction. An analytical solution is found for the propagating finite amplitude waves which result from baroclinic instability of the zonal winds when topography is absent. The appearance of this solution for wavelengths just longer than the Rossby radius of deformation and disappearance of ultra-long wavelengths is interpreted in terms of the Hopf bifurcation theory. Simple dynamic and thermodynamic criteria for the existence of periodic Rossby solutions are presented. A Floquet stability analysis shows that the waves are neutral. The nature of the form drag instability of high index equilibria is investigated. The proximity of the equilibrium shear to a resonant value is essential for the instability, provided the equilibrium occurs at a slightly stronger shear than resonance.

Clark, J. H. E.↗

NASA NTRS1985

Active control of compressible flows on a curved surface

The effect of localized, time periodic surface heating and cooling over a curved surface is studied. This is a mechanism for the active control of unstable disturbances by phase cancellation and reinforcement. It is shown that the pressure gradient induced by the curvature significantly enhances the effectiveness of this form of active control. In particular, by appropriate choice of phase, active surface heating can completely stabilize and unstable wave.

Maestrello, L.↗

NASA NTRS1985

Active control of compressible flows on a curved surface

Maestrello, L.↗

NASA NTRS1986

Parametric instabilities of the circularly polarized Alfven waves including dispersion

A class of parametric instabilities of large-amplitude, circularly polarized Alfven waves is considered in which finite frequency (dispersive) effects are included. The dispersion equation governing the instabilities is a sixth-order polynomial which is solved numerically. As a function of K identically equal to k/k-sub-0 (where k-sub-0 and k are the wave number of the 'pump' wave and unstable sound wave, respectively), there are three regionals of instability: a modulation instability at K less than 1, a decay instability at K greater than 1, and a relatively weak and narrow instability at K close to squared divided by v-sub-A squared (where c-sub-s and v-sub-A are the sound and Alfven speeds respectively), the modulational instability occurs when beta is less than 1 (more than 1) for left-hand (right-hand) pump waves, in agreement with the previous results of Sakai and Sonnerup (1983). The growth rate of the decay instability of left-hand waves is greater than the modulational instability at all values of beta. Applications to large-amplitude wave observed in the solar wind, in computer simulations, and in the vicinity of planetary and interplanetary collisionless shocks are discussed.

Wong, H. K.↗

NASA NTRS1987

An analytic study of the Charney problem on the sphere

An analytic study of the Charney (1947) problem is presented, in which a shortwave approximation and a two-scale assumption are adopted to derive approximate solutions for the problem. The solutions contain a second-order turning point whose location is important in determining the properties of baroclinic unstable waves. This turning point is located at the maximum of the meridional temperature gradient. The solutions indicate a close relation between baroclinic instability on the sphere and on a Beta-plane. The solutions have some limitations and are not valid for long waves. It is suggested that the solutions have implications for parameterizations of eddy fluxes in climate modeling.

Lee, Yung-An↗

NASA NTRS1989

Disturbances along the Arizona monsoon boundary

The possibility of the occurrence of disturbances along the 'Arizona monsoon boundary' (the boundary between two air masses in the southwestern U.S. and northwestern Mexico) during the summer is evaluated. Linear calculations demonstrate that the monsoon boundary is indeed unstable, with maximum growth rates of 0.4-0.6/d at a wavelength near 1200 km. It is found that the energy source for the most unstable waves is baroclinic and that the waves are barotropically damped.

Moore, Thomas J.↗

NASA NTRS1990

Second mode interactions in supersonic boundary layers

The nonlinear evolution of a two-dimensional second mode unstable wave in a Mach 4.5 wall-bounded flow is computed by solving the full time-dependent compressible Navier-Stokes equations. A highly accurate solution is obtained using spectral collocation methods. It is shown that departure from linearity first occurs in the critical layer due to the cubic nonlinearities in the momentum equation. This is a direct result of the large density perturbations in this regime. Time evolution studies of the growth rate as a function of normal distance from the plate suggests that the mode is evolving toward a nonlinear saturated state, and that this problem is possibly amenable to standard weakly nonlinear perturbation methods.

Erlebacher, Gordon↗

NASA NTRS1991

Nonlinear dynamics near the stability margin in rotating pipe flow

The nonlinear evolution of marginally unstable wave packets in rotating pipe flow is studied. These flows depend on two control parameters, which may be taken to be the axial Reynolds number R and a Rossby number, q. Marginal stability is realized on a curve in the (R, q)-plane, and the entire marginal stability boundary is explored. As the flow passes through any point on the marginal stability curve, it undergoes a supercritical Hopf bifurcation and the steady base flow is replaced by a traveling wave. The envelope of the wave system is governed by a complex Ginzburg-Landau equation. The Ginzburg-Landau equation admits Stokes waves, which correspond to standing modulations of the linear traveling wavetrain, as well as traveling wave modulations of the linear wavetrain. Bands of wavenumbers are identified in which the nonlinear modulated waves are subject to a sideband instability.

Yang, Z.↗

NASA NTRS1992

Coarsening following a morphological instability in the one-sided model

We study the coarsening of the interfacial pattern separating two coexisting phases after the pattern becomes morphologically unstable. Shortly after the instability, the scale of the structure is given by the most unstable wave number in the linear regime. At later times, nonlinear interactions cause the structure to coarsen. Coarsening is studied by monitoring the time dependence of the two-dimensional power spectrum of the pattern, especially for wave vectors transverse to the direction of growth. Characteristic length scales of the pattern obtained from moments of the power spectrum are asymptotically linear in time. Furthermore, the power spectrum is seen to satisfy a scaling relation, in agreement with previous studies. A normal-velocity autocorrelation function is calculated and found to decay substantially over length scales that are of the order of the scale of the pattern. The issue of spatial anisotropy in the correlation functions is also discussed.

Vinals, Jorge↗

NASA NTRS1994

Physical description of boundary-layer transition: Experimental evidence

The problems of understanding the origins of turbulent flow and transition to turbulent flow are the most important unsolved problems of fluid mechanics and aerodynamics. It is well known that the stability, transition, and turbulent characteristics of bounded shear layers are fundamentally different from those of free shear layers. Likewise, the stability, transition, and turbulent characteristics of open systems are fundamentally different from those of closed systems. Because of the influence of indigenous disturbances, surface geometry and roughness, sound, heat transfer, and ablation, it is not possible to develop general prediction schemes for transition location and the nature of turbulent structures in boundary-layer flows. At the present time no mathematical model exists that can predict the transition Reynolds number on a flat plate. The recent progress in this area is encouraging, in that a number of distinct transition mechanisms have been found experimentally. The theoretical work finds them to be amplitude and Reynolds-number dependent. The theory remains rather incomplete with regard to predicting transition. Amplitude and spectral characteristics of the disturbances inside the laminar viscous layer strongly influence which type of transition occurs. The major need in this area is to understand how freestream disturbances are entrained into the boundary layer, i.e., to answer the question of receptivity. We refer receptivity to the mechanism(s) that cause freestream disturbances to enter the boundary layer and create the initial amplitudes for unstable waves.

Saric, William S.↗

NASA NTRS1999

Swept-Wing Receptivity Studies Using Distributed Roughness

Transition to turbulence in swept-wing flows has resisted correlation with linear theory because of its sensitivity to freestream conditions and 3-D roughness and because one of the principal instability modes quickly 'becomes nonlinear. In the face of such a formidable problem, two rather long-term fundamental efforts have been underway at DLR Gottinberg and Arizona State University that address swept-wing transition. These efforts have been recently reviewed by Bippes (1997) and Reibert and Saric (1997). Thus, the present work is a continuation of a series of studies on swept-wing boundary layers which have led to a better understanding of the transition process. In particular, we have taken advantage of the sensitivity to 3-D roughness and the modal nature of the instability in order to propose a particular control strategy. Complementing the two aforementioned reviews, general reviews of the swept-wing transition problem are found in Arnal (1997) and Kachanov (1996). Other recent reviews include Reshotko (t997), Crouch (1997), and Herbert (1997a,b). The failure of linear theory is discussed in Reed et al. (1996). The historical work is found in Reed and Sar-ic (1989). The basic idea is that the combination of sweep and chordwise pressure gradient within the boundary layer creates a velocity component perpendicular to the inviscid streamline. This crossflow profile is inflectional and exhibits both traveling and stationary unstable waves called crossflow vortices that are (approximately) aligned along the inviscid streamlines. Under conditions of low freestream turbulence levels, the dominant crossflow wave is stationary (Reibert and Saric t997) while moderate to high turbulence levels initiate dominant traveling waves (Dehle and Bippes 1996; Bippes 1997). 'Me mechanism is relatively insensitive to sound and 2-D surface roughness (Radeztsky et al. 1993) but very sensitive to 3-D roughness near the attachment line. We concentrate our work on low-turbulence freestream flows and stationary crossflow waves. Although the v' and w' components of the disturbances are very small, by convecting streamwise momentum in the wall-normal direction, they produce 0(l) changes inu'.Thus the mean flow is highly distorted with localized inflection points. Transition is then triggered by a high-frequency secondary instability of the distorted mean profile.

Saric, William S.↗

NASA NTRS1994

The Nonlinear Evolution of the Inviscid Secondary Instability of Streamwise Vortex Structures

The weakly nonlinear evolution of an inviscid marginally unstable wave growing on a boundary layer supporting a streamwise vortex structure is investigated. The nonlinear growth of the wave is found to be controlled by the diffusion layer located at the edge of the critical layer associated with the wave. The evolution equation is found to depend on the upstream history of the wave and the solution of the equation suggests that the wave either restructures the mean state so as to make it stable or develops a singularity at a finite distance downstream of the point of neutral stability.

Blackaby, Nicholas D.↗

NASA NTRS2014

A Vortical Dawn Flank Boundary Layer for Near-Radial IMF: Wind Observations on 24 October 2001

We present an example of a boundary layer tailward of the dawn terminator which is entirely populated by rolled-up flow vortices. Observations were made by Wind on 24 October 2001 as the spacecraft moved across the region at the X plane approximately equal to −13 Earth radii. Interplanetary conditions were steady with a near-radial interplanetary magnetic field (IMF). Approximately 15 vortices were observed over the 1.5 hours duration of Wind's crossing, each lasting approximately 5 min. The rolling up is inferred from the presence of a hot tenuous plasma being accelerated to speeds higher than in the adjoining magnetosheath, a circumstance which has been shown to be a reliable signature of this in single-spacecraft observations. A blob of cold dense plasma was entrained in each vortex, at whose leading edge abrupt polarity changes of field and velocity components at current sheets were regularly observed. In the frame of the average boundary layer velocity, the dense blobs were moving predominantly sunward and their scale size along the X plane was approximately 7.4 Earth radii. Inquiring into the generation mechanism of the vortices, we analyze the stability of the boundary layer to sheared flows using compressible magnetohydrodynamic Kelvin-Helmholtz theory with continuous profiles for the physical quantities. We input parameters from (i) the exact theory of magnetosheath flow under aligned solar wind field and flow vectors near the terminator and (ii) the Wind data. It is shown that the configuration is indeed Kelvin-Helmholtz (KH) unstable. This is the first reported example of KH-unstable waves at the magnetopause under a radial IMF.

Magnetosheath↗

NASA NTRS1985

Nonlinear forcing of planetary scale waves by amplifying unstable baroclinic eddies generated in the troposphere

The present investigation is mainly concerned with isolating the effects of wave-wave coupling on the growth of planetary-scale baroclinic modes of zonal wavenumber 1-3. It is shown that a global distribution of amplifying intermediate-scale baroclinic modes can cause rapid growth of planetary-scale modes. The growth rates of the planetary modes are found to be comparable to the growth rates of the most unstable baroclinic modes during the first 5-10 days. Thereafter the planetary waves continue to grow (albeit at a slower rate and not necessarily monotonically), and do not exhibit a decay phase as pronounced as that which is characteristic of the intermediate-scale modes.

Young, R. E.↗

NASA NTRS1981

Chaotic /strange/ and periodic behavior in instability saturation by the oscillating two-stream instability

The nonlinear Schroedinger equation with linear growth and damping is truncated to three waves. The resulting system of nonlinear ordinary differential equations describes the excitation of linearly damped waves by the oscillating two-stream instability driven by a linearly unstable pump wave. This system represents a simple model for the nonlinear saturation of a linearly unstable wave. The model is examined analytically and numerically as a function of the dimensionless parameters of the system. It is found that the model can exhibit a wealth of characteristic dynamical behavior including stationary equilibria, Hopf bifurcations to periodic orbits, period doubling bifurcations, chaotic solutions characteristic of a strange attractor, tangent bifurcations from chaotic to periodic solutions, transient chaos, and hysteresis. Many of these features are shown to be explainable on the basis of one-dimensional maps. In the case of chaotic solutions, evidence for the presence of a strange attractor is provided by demonstrating Cantor set-like structure (i.e., scale invariance) in the surface of section.

Russell, D. A.↗