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NASA NTRS1986

A parameterization technique for nonlinear spectral models

An objective parameterization technique is developed for general nonlinear hydrodynamical systems. The typical structure of hydrodynamical systems, regardless of their complexity, 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. As a prototype of the generic problem containing this typical structure, the parameterization technique is applied to various three component subsets of a five component nonlinear spectral model of forced, dissipative quasi-geostrophic flow in a channel. The results obtained lead to specification of the necessary data coverage requirements for applying the technique in general.

Gelaro, R.↗

NASA NTRS1986

On cloud street development in three dimensions - Parallel and Rayleigh instabilities

Expected orientation angles and horizontal wavelengths of boundary layer rolls or cloud streets are determined from an analysis of a truncated spectral model of three dimensional shallow moist Boussinesq convection in a shearing environment. The nonlinear secondary circulations are organized into two dimensional forms by the height dependent wind field, and these rolls may develop from the combined effects of thermal stratification and mean wind shear. The associated thermal and parallel instability mechanisms are shown to be special cases of a single one. Only one mode is found when the stratification is unstable or neutral, but a second one is possible when the stratification is weakly stable. The first corresponds to relatively broadly spaced rolls having orientations for which the Fourier component of the roll perpendicular shear is nearly zero, but the second corresponds to relatively 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 NTRS1986

Cloud streets during KonTur - A comparison of parallel/thermal instability modes with observations

Estimates of cloud street geometry produced by a model of the parallel/thermal instability modes of shallow convection are compared with observations obtained during the 1981 KonTur experiment. Good agreement between the modeled and observed orientation angles, wavelengths and Reynolds numbers are found when the streets are assumed to derive their energy from the average shear and the lowest order sine terms of a Fourier expansion of the mean wind profile (or equivalently from the lowest order cosine terms of the mean shear profile). The modes associated with the cosine terms of the wind profile (or the sine terms of the wind shear profile) do not agree well with the observations. These results suggest that the boundary layer rolls observed during KonTur might have developed owing to a combined parallel/thermal instability originating primarily from the cosine terms of the ambient roll parallel wind shear.

Shirer, H. N.↗

NASA NTRS1985

The use of satellite data in understanding and predicting convective and large-scale dynamical processes

A two-layer truncated baroclinic spectral model was developed to study the long-term evolution of disturbances to a baroclinically unstable mean flow. Topography and crudely-parameterized radiative processes were accounted for. As a result of Robert Schlaak's discovery of the underlying barotropic nature of the index oscillation as well as reviewers suggestions about the original manuscript, the model has been revised to allow for barotropic as well as baroclinic wave-mean flow interactions. The form-drag exerted by the topography on the barotropic part of the mean flow is larger than on the baroclinic part and thus researchers anticipate significant changes from the original calculations on the index oscillation when it is strongly modulated by topography. Researchers believe that since the index oscillation accounts for a significant portion of atmospheric temporal variance, the long term predictability could be improved if reliable forecasts of the index oscillation were available. Two spectral models of the index oscillation, one barotropic and the other baroclinic, have been developed. The latter allows for moisture, radiation, land-sea temperature countrasts, and energy exchanges with the underlying surface.

Dutton, J. A.↗

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

A parameterization technique for nonlinear spectral models

An objective parameterization technique is developed for general nonlinear hydrodynamical systems. The typical structure of hydrodynamical systems, regardless of their complexity, 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. As a prototype of the generic problem containing this typical structure, the parameterization technique is applied to various three component subsets of a five component nonlinear spectral model of forced, dissipative quasi-geostrophic flow in a channel. The results obtained lead to specification of the necessary data coverage requirements for applying the technique in general.

Gelaro, R.↗

NASA NTRS1985

On cloud street development in three dimensional parallel and Rayleigh instabilities

Expected orientation angles and horizontal wavelengths of boundary layer rolls or cloud streets are determined from an analysis of a truncated spectral model of three dimensional shallow moist Boussinesq convection in a shearing environment. The nonlinear secondary circulations are organized into two dimensional forms by the height dependent wind field, and these rolls may develop from the combined effects of thermal stratification and mean wind shear. The associated thermal and parallel instability mechanisms are shown to be special cases of a single one. Only one mode is found when the stratification is unstable or neutral, but a second one is possible when the stratification is weakly stable. The first corresponds to relatively broadly spaced rolls having orientations for which the Fourier component of the roll perpendicular shear is nearly zero, but the second corresponds to relatively 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

Cloud streets during KonTur: A comparison of parallel/thermal instability modes with observations

Estimates of cloud street geometry produced by a model of the parallel/thermal instability modes of shallow convection are compared with observations obtained during the 1981 KonTur experiment. Good agreement between the modeled and observed orientation angles, wavelengths and Reynolds numbers are found when the streets are assumed to derive their energy from the average shear and the lowest order sine terms of a Fourier expansion of the mean wind profile (or equivalently from the lowest order cosine terms of the mean shear profile). The modes associated with the cosine terms of the wind profile (or the sine terms of the wind shear profile) do not agree well with the observations. These results suggest that the boundary layer rolls observed during KonTur might have developed owing to a combined parallel/thermal instability originating primarily from the cosine terms of the ambient roll parallel wind shear.

Shirer, H. N.↗

NASA NTRS1985

On the nonlinear characteristics of the axisymmetric flow regime: Cylindrical and spherical systems

The physical relationship between steady axisymmetric flows that might be observed in the atmosphere and in laboratory vessels is investigated theoretically. This is accomplished by comparing both the nonlinear structure and the thermal forcing mechanisms in two truncated spectral models of flow in the atmosphere and the rotating laboratory cylinder, respectively. Under statically stable conditions, the response of the internally forced spherical model (which is developed here from a set of new orthonormal basis functions) exhibits steady behavior different from that in the externally forced cylindrical model. Two regions of multiple steady solutions occur in the cylindrical model, under stable conditions, that are not found in the spherical one. The possible physical relevance of these multiple solutions is investigated by determining their location in parameter space with respect to the classical Hadley-Rossby transition curve. The results suggest that the wave flow regime, in an annulus, might develop catastrophically when an upper symmetric flow ceases to exist.

Higgins, R. W.↗

NASA NTRS1984

The use of satellite data in understanding and predicting convective and large-scale dynamical processes

Mesoscale convective processes and how they affect and interact with mid-latitude cyclones were studied. The ageostrophic and associated vertical motion field was calculated using a highly accurate iterative method of solving the semigeostrophic omega equation. The tendencies for convective destabilization in the 850-750 mb layer due to differential geostrophic and ageostrophic advection and differential moist adiabatic ascent, were found. The spectral models of the index oscillation, one barotropic and the other baroclinic, were developed. Theoretical and observational studies of cloud streets were conducted.

Dutton, J. A.↗

NASA NTRS1984

Transitions in shallow convection - An explanation for lateral cell expansion

The generalized, seven-coefficient model presented for two-dimensional Rayleigh-Benard convection successfully simulates one way in which lateral cell expansion can occur as the value of the imposed vertical temperature difference is changed. The triad of interacting wavenumbers is varied as the vertical heating rate is varied and only the energetically active components are retained, so that transitional behavior within two-dimensional convective flow can be both simulated and physically interpreted as the representation of the cell expansion process via successive secondary branching. It is found that while a two-dimensional cell broadening mechanism is likely to operate for small Prandtl number values, a three-dimensional mechanism is expected at values greater than about 0.7.

Chang, H.-R.↗

NASA NTRS1982

Bifurcation and stability of low-order steady flows in horizontally and vertically forced convection

A nonlinear spectral model of two-dimensional, shallow Boussinesq convection which responds to heating in both the horizontal and vertical directions is examined. The governing partial differential system is converted to an infinite set of ordinary differential equations and truncated to a small set to permit detailed study of the number and types of transitions from one flow configuration to another. The Hadley number and the Rayleigh number are defined as the horizontal and vertical thermal forcing mechanisms, respectively, for inclusion in the nonlinear spectral model, which is composed of three equations. The model is then used to describe steady states, linearly stable solutions, and balancing factors in unstable stratification. The number and the distribution of the steady states are found to be qualitatively independent of the aspect ratio and the Prandtl number.

Yost, D. A.↗

NASA NTRS1982

Toward a unified theory of atmospheric convective instability

A nonlinear three-dimensional truncated spectral model of shallow and moist Boussinesq convection indicates that parallel instability and thermal forcing are linked, in view of the fact that only one convective mode exists in which either or both mechanisms are operating to generate convection in the planetary boundary layer. It is also established that the wind field causes two-dimensional roll convection formation, an alignment of the convection with the wind in a preferred manner, and a propagation speed that is related to the wind component perpendicular to the roll axis. Latent heating is responsible for the decrease of the critical value of the environmental lapse rate in accordance with the slice method stability criterion. When only the upper part of the upward branch is moist and all of the downward branch is dry, latent heating also causes a finite-amplitude convective solution for Rayleigh number values lower than the critical value of linear analysis.

Shirer, H. N.↗

NASA NTRS1980

Bifurcation and stability in a model of moist convection in a shearing environment

The truncated spectral system (model I) of shallow moist two-dimensional convection discussed by Shirer and Dutton (1979) is expanded to eleven coefficients (model II) in order to include a basic wind. Cloud streets, the atmospheric analog of the solutions to model II, are typically observed in an environment containing a shearing basic motion field. Analysis of the branching behavior of solutions to mode II shows that, if the basic wind direction varies with height, very complex temporal behavior is possible as the modified Rayleigh number HR is increased sufficiently. The first convective solution is periodic, corresponding to a cloud band that propagates downwind; but secondary branching to a two-dimensional torus can occur for larger values of HR. Orientation band formulas are derived whose predictions generally agree with the results of previous studies.

Shirer, H. N.↗