# Search Results

## You are looking at 1 - 10 of 13 items for

- Author or Editor: Alan J. Faller x

- Refine by Access: All Content x

## Abstract

It is proposed that when a dependent variate *y* is classified into *N* sets of values with different predictive statistical relations for each classification, the associated correlation coefficients *r _{n}
* (

*n*= 1 −

*N*) can be usefully combined into a single correlation coefficient by a weighted average of the

*r*

_{n}^{2}. The weighting factor is the variance of

*y*for each

*n*. In assessing the combined predictability of

*y*by the set of

*N*relations an additional weighting factor is

*J*, the number of predictions to be made for each

_{n}*n*.

## Abstract

It is proposed that when a dependent variate *y* is classified into *N* sets of values with different predictive statistical relations for each classification, the associated correlation coefficients *r _{n}
* (

*n*= 1 −

*N*) can be usefully combined into a single correlation coefficient by a weighted average of the

*r*

_{n}^{2}. The weighting factor is the variance of

*y*for each

*n*. In assessing the combined predictability of

*y*by the set of

*N*relations an additional weighting factor is

*J*, the number of predictions to be made for each

_{n}*n*.

## Abstract

Turbulent shear flow generally contains large eddies which appear to be due to a shear instability of the profile of the mean flow. By analogy with laboratory experimental results, it is inferred that large eddies in the planetary boundary layer of the atmosphere should take the form of horizontal roll vortices with an orientation between that of the surface wind and that of the geostrophic flow. The vertical extent and intensity of the vertical motions associated with these roll vortices often may be sufficient to give rise to bands of clouds in the lower atmosphere. For an adiabatic boundary layer the spacing of cloud bands formed by this mechanism is tentatively predicted to be given by the relation *L*=200*U*/sinϕ, where *L* is measured in meters, *U* is the geostrophic speed near the ground in meters per second and ϕ is latitude.

## Abstract

Turbulent shear flow generally contains large eddies which appear to be due to a shear instability of the profile of the mean flow. By analogy with laboratory experimental results, it is inferred that large eddies in the planetary boundary layer of the atmosphere should take the form of horizontal roll vortices with an orientation between that of the surface wind and that of the geostrophic flow. The vertical extent and intensity of the vertical motions associated with these roll vortices often may be sufficient to give rise to bands of clouds in the lower atmosphere. For an adiabatic boundary layer the spacing of cloud bands formed by this mechanism is tentatively predicted to be given by the relation *L*=200*U*/sinϕ, where *L* is measured in meters, *U* is the geostrophic speed near the ground in meters per second and ϕ is latitude.

## Abstract

The remarkable similarity between the motions in a heated rotating tank of water (dishpan experiments) and the large-scale motions of the atmosphere has raised the question of the similarity of the details of the motion. It has been found possible to demonstrate visually the existence of frontal systems and associated wave cyclones. Analyses and photographs showing the relationships of these systems to the large-scale circulations are presented. It is concluded that similarity in laboratory models is not confined to that scale of motion commonly referred to as the general circulation.

## Abstract

The remarkable similarity between the motions in a heated rotating tank of water (dishpan experiments) and the large-scale motions of the atmosphere has raised the question of the similarity of the details of the motion. It has been found possible to demonstrate visually the existence of frontal systems and associated wave cyclones. Analyses and photographs showing the relationships of these systems to the large-scale circulations are presented. It is concluded that similarity in laboratory models is not confined to that scale of motion commonly referred to as the general circulation.

## Abstract

A hierarchy of theoretical and numerical models for the dispersion of discrete floating tracers on lakes and oceans is presented. Central to these models is the role of Langmuir circulations, which concentrate tracers into narrow windrows this inhibiting tracer dispersion. But time-dependent Langmuir circulations cause the rows of tracers to wander and so split, by local time dependence and by downwind advection, thus promoting dispersion. Accordingly, the Langmuir circulations generally render the smaller-scale background turbulence irrelevant for direct estimates of surface dispersion.

Analytical models includes: 1) a theory of tracers in a linear mean-flow convergence plus homogeneous turbulence, this theory being applicable to the width of windrows; and 2) a model with a spatially periodic mean flow and a periodic small-scale eddy diffusion coefficient that allows an estimate of the Langmuir-scale dispersivity for steady parallel cells.

Random-flight calculations for a model of complex time-dependent and downwind dependent Langmuir circulations have led to the explicit prediction *K* = 0.5*T*
_{C}
^{*−½} where *K*
^{ast;} and *T*
_{C}
^{*} are the nondimensional dispersivity and cellular time scale, respectively.

## Abstract

A hierarchy of theoretical and numerical models for the dispersion of discrete floating tracers on lakes and oceans is presented. Central to these models is the role of Langmuir circulations, which concentrate tracers into narrow windrows this inhibiting tracer dispersion. But time-dependent Langmuir circulations cause the rows of tracers to wander and so split, by local time dependence and by downwind advection, thus promoting dispersion. Accordingly, the Langmuir circulations generally render the smaller-scale background turbulence irrelevant for direct estimates of surface dispersion.

Analytical models includes: 1) a theory of tracers in a linear mean-flow convergence plus homogeneous turbulence, this theory being applicable to the width of windrows; and 2) a model with a spatially periodic mean flow and a periodic small-scale eddy diffusion coefficient that allows an estimate of the Langmuir-scale dispersivity for steady parallel cells.

Random-flight calculations for a model of complex time-dependent and downwind dependent Langmuir circulations have led to the explicit prediction *K* = 0.5*T*
_{C}
^{*−½} where *K*
^{ast;} and *T*
_{C}
^{*} are the nondimensional dispersivity and cellular time scale, respectively.

## Abstract

A procedure for statistical correction of numerical prediction equations at the end of each predictive time step is described and tested with a two-dimensional prediction model. The model equations are modified Burgers' equations which contain space and velocity dependent sources of energy to maintain the flow against dissipation. A detailed flow is calculated from a fine-grid numerical integration on a rectangular region with periodic boundary conditions. Coarse-grid values, for initiating and testing various coarse-grid prediction equations, are obtained by space-time mesh-box averages.

Since the coarse-grid equations cannot represent subgrid-scale motions, statistical corrections are added in the form of parametric terms as a pragmatic substitute for the missing subgrid-scale effects. Tests with different forms of the equations show that substantial improvements can be obtained when the coefficients of the parametric terms are determined by multiple regression. Still further improvements are found when separate regression equations are calculated for each grid point and when the coefficients are adjusted in time to stabilize the calculations.

The model equations are found to give periodic behavior in time despite somewhat complex sources of energy dependent upon the model geography and upon the flow itself. In addition, the predictions are stable to small perturbations of the initial conditions. It is concluded, therefore, that the coarse-grid prediction errors, which grow rapidly with time in the manner of real atmospheric prediction errors, are due entirely to the truncation errors of the coarse-grid equations.

## Abstract

A procedure for statistical correction of numerical prediction equations at the end of each predictive time step is described and tested with a two-dimensional prediction model. The model equations are modified Burgers' equations which contain space and velocity dependent sources of energy to maintain the flow against dissipation. A detailed flow is calculated from a fine-grid numerical integration on a rectangular region with periodic boundary conditions. Coarse-grid values, for initiating and testing various coarse-grid prediction equations, are obtained by space-time mesh-box averages.

Since the coarse-grid equations cannot represent subgrid-scale motions, statistical corrections are added in the form of parametric terms as a pragmatic substitute for the missing subgrid-scale effects. Tests with different forms of the equations show that substantial improvements can be obtained when the coefficients of the parametric terms are determined by multiple regression. Still further improvements are found when separate regression equations are calculated for each grid point and when the coefficients are adjusted in time to stabilize the calculations.

The model equations are found to give periodic behavior in time despite somewhat complex sources of energy dependent upon the model geography and upon the flow itself. In addition, the predictions are stable to small perturbations of the initial conditions. It is concluded, therefore, that the coarse-grid prediction errors, which grow rapidly with time in the manner of real atmospheric prediction errors, are due entirely to the truncation errors of the coarse-grid equations.

## Abstract

Two modifications of the procedure for applying statistical corrections to the prediction equations discussed in Faller and Schemm (1977) are considered. In the first, the constant terms which normally appear in the regression equations are suppressed, leading to improved extended range STAT predictions. In the second, a noncentered advection term which gave exceptionally high correlations with one time step prediction errors is tested; however, we have been unable to use this type of term successfully in extended predictions. These results indicate the need for careful selection and testing of parametric terms for the proposed statistical corrections.

## Abstract

Two modifications of the procedure for applying statistical corrections to the prediction equations discussed in Faller and Schemm (1977) are considered. In the first, the constant terms which normally appear in the regression equations are suppressed, leading to improved extended range STAT predictions. In the second, a noncentered advection term which gave exceptionally high correlations with one time step prediction errors is tested; however, we have been unable to use this type of term successfully in extended predictions. These results indicate the need for careful selection and testing of parametric terms for the proposed statistical corrections.

## Abstract

A procedure for statistical correction of numerical prediction equations at the end of each predictive time step is described and tested with a one-dimensional prediction model. The model equation is a modified Burgers equation that allows the formation of shocks, analogous to atmospheric fronts, and which contains a space and velocity-dependent source of energy to maintain the flow against dissipation. The detailed flow is calculated from a fine-grid numerical integration. Coarse-grid values, for testing a coarse-grid prediction scheme, are obtained by space-time averages.

Since the coarse-grid prediction equation cannot represent the sub-grid-scale motions, statistical corrections are added in the form of parametric terms as a pragmatic substitute for the missing sub-grid-scale effects. Tests with different versions of the model show that substantial improvement over the straight- forward coarse-grid prediction can he obtained when the coefficients of appropriate parametric terms are determined by a multiple regression procedure.

## Abstract

A procedure for statistical correction of numerical prediction equations at the end of each predictive time step is described and tested with a one-dimensional prediction model. The model equation is a modified Burgers equation that allows the formation of shocks, analogous to atmospheric fronts, and which contains a space and velocity-dependent source of energy to maintain the flow against dissipation. The detailed flow is calculated from a fine-grid numerical integration. Coarse-grid values, for testing a coarse-grid prediction scheme, are obtained by space-time averages.

Since the coarse-grid prediction equation cannot represent the sub-grid-scale motions, statistical corrections are added in the form of parametric terms as a pragmatic substitute for the missing sub-grid-scale effects. Tests with different versions of the model show that substantial improvement over the straight- forward coarse-grid prediction can he obtained when the coefficients of appropriate parametric terms are determined by a multiple regression procedure.

## Abstract

This study is concerned with the instability of laminar Ekman flow which is characteristic of boundary-layer flows in rotating systems of many kinds. By numerical integration of the equations of motion we have obtained critical values of a Reynolds number for this flow, growth rates of the unstable perturbations and equilibrium finite-amplitude solutions. The results are in good agreement with experimental data, and the numerical solutions extend the results beyond what could be obtained readily in laboratory experiments.

## Abstract

This study is concerned with the instability of laminar Ekman flow which is characteristic of boundary-layer flows in rotating systems of many kinds. By numerical integration of the equations of motion we have obtained critical values of a Reynolds number for this flow, growth rates of the unstable perturbations and equilibrium finite-amplitude solutions. The results are in good agreement with experimental data, and the numerical solutions extend the results beyond what could be obtained readily in laboratory experiments.

## Abstract

With stable density stratification the shear-flow instability of the Ekman boundary layer exhibits two distinct regimes. At low values of a Richardson number the growth rate of instability, at specified Reynolds number, wavelength and angle, decreases linearly with Ri. At higher values of Ri the growth rate may decrease more slowly or may increase with Ri. The peculiar effects at the large values of Ri are interpreted as a resonance of the shear-flow instability with internal gravity waves. This resonance occurs when the speed of the shear-flow instability relative to the basic flow lies within the range of speeds of internal gravity waves relative to the basic flow, as determined by the Brunt-Väisälä frequency. Under these conditions the growth of waves appears to be dominated by the Type II mechanism of energy exchange for Ekman layer instability. Internal gravity waves generated by the shear-flow instability have their crests nearly parallel to the geostrophic flow above the boundary layer and move to the left of the geostrophic flow with speeds between approximately 0. 15 and 0.7 times the geostrophic speed. The type II energy exchange mechanism with the apparent resonance is permitted by the Coriolis forces.

## Abstract

With stable density stratification the shear-flow instability of the Ekman boundary layer exhibits two distinct regimes. At low values of a Richardson number the growth rate of instability, at specified Reynolds number, wavelength and angle, decreases linearly with Ri. At higher values of Ri the growth rate may decrease more slowly or may increase with Ri. The peculiar effects at the large values of Ri are interpreted as a resonance of the shear-flow instability with internal gravity waves. This resonance occurs when the speed of the shear-flow instability relative to the basic flow lies within the range of speeds of internal gravity waves relative to the basic flow, as determined by the Brunt-Väisälä frequency. Under these conditions the growth of waves appears to be dominated by the Type II mechanism of energy exchange for Ekman layer instability. Internal gravity waves generated by the shear-flow instability have their crests nearly parallel to the geostrophic flow above the boundary layer and move to the left of the geostrophic flow with speeds between approximately 0. 15 and 0.7 times the geostrophic speed. The type II energy exchange mechanism with the apparent resonance is permitted by the Coriolis forces.