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## Abstract

Large-eddy simulations of a free convective atmospheric boundary layer with an overlying capping inversion are considered. Attention is given to the dependence of the results upon the various factors influencing the simulation: the subgrid model, the domain size, and the mesh resolution. By providing artificial constraints upon the convection the results also provide extra insight into the underlying dynamics.

The gross features of the boundary layer, such as the overall energy budget, are not sensitive to the details of the simulations but a number of important factors are revealed. It has been found that near the surface the subgrid diffusivity must be larger than is usually supposed, in order for the vertical velocity skewness to have the correct sign. This region of the flow has a significant subgrid-scale heat flux and it seems that the subgrid model requires improvement in such cases. A revised model which under statically unstable conditions allows the mixing-length of the subgrid-scale turbulence to depend on the flow stability is found to give improved results. The domain size and mesh spacings have a significant influence upon the results and need a setting which allows resolution of the main, freely occurring scales of motion. The entrainment at the capping inversion is remarkable in its insensitivity to all factors. Finally, the higher resolution simulations provide a detailed view of the flow structure of the convective boundary layer. Downdrafts cover a large fraction of the surface area, and near the surface the flow converges into smaller areas comprising long narrow regions of updrafts. The plumes which penetrate through the depth of the boundary layer to the inversion mainly occur over the inter-sections of these long narrow regions of updrafts.

## Abstract

Large-eddy simulations of a free convective atmospheric boundary layer with an overlying capping inversion are considered. Attention is given to the dependence of the results upon the various factors influencing the simulation: the subgrid model, the domain size, and the mesh resolution. By providing artificial constraints upon the convection the results also provide extra insight into the underlying dynamics.

The gross features of the boundary layer, such as the overall energy budget, are not sensitive to the details of the simulations but a number of important factors are revealed. It has been found that near the surface the subgrid diffusivity must be larger than is usually supposed, in order for the vertical velocity skewness to have the correct sign. This region of the flow has a significant subgrid-scale heat flux and it seems that the subgrid model requires improvement in such cases. A revised model which under statically unstable conditions allows the mixing-length of the subgrid-scale turbulence to depend on the flow stability is found to give improved results. The domain size and mesh spacings have a significant influence upon the results and need a setting which allows resolution of the main, freely occurring scales of motion. The entrainment at the capping inversion is remarkable in its insensitivity to all factors. Finally, the higher resolution simulations provide a detailed view of the flow structure of the convective boundary layer. Downdrafts cover a large fraction of the surface area, and near the surface the flow converges into smaller areas comprising long narrow regions of updrafts. The plumes which penetrate through the depth of the boundary layer to the inversion mainly occur over the inter-sections of these long narrow regions of updrafts.

## Abstract

A new instrument package has been developed to measure turbulence at various heights in the boundary layer. The package or “turbulence probe” is designed to be attached to the tethering cable of a balloon. In contrast to previous balloon borne turbulence probes, it uses inclinometers and magnetometers rather than a damped pendulum to determine the probe sensor orientations. Comparisons with a mast mounted sonic anemometer have shown that all three components of turbulence energy can be measured with a good accuracy, even in strong wind conditions. A system of several such probes can also provide accurate corrections for errors introduced by the inevitable translational motions of the tether cable.

## Abstract

A new instrument package has been developed to measure turbulence at various heights in the boundary layer. The package or “turbulence probe” is designed to be attached to the tethering cable of a balloon. In contrast to previous balloon borne turbulence probes, it uses inclinometers and magnetometers rather than a damped pendulum to determine the probe sensor orientations. Comparisons with a mast mounted sonic anemometer have shown that all three components of turbulence energy can be measured with a good accuracy, even in strong wind conditions. A system of several such probes can also provide accurate corrections for errors introduced by the inevitable translational motions of the tether cable.

## Abstract

There are numerous reasons for calculating forecast verification scores, and considerable attention has been given to designing and analyzing the properties of scores that can be used for scientific purposes. Much less attention has been given to scores that may be useful for administrative reasons, such as communicating changes in forecast quality to bureaucrats and providing indications of forecast quality to the general public. The two-alternative forced choice (2AFC) test is proposed as a scoring procedure that is sufficiently generic to be usable on forecasts ranging from simple yes–no forecasts of dichotomous outcomes to forecasts of continuous variables, and can be used with deterministic or probabilistic forecasts without seriously reducing the more complex information when available. Although, as with any single verification score, the proposed test has limitations, it does have broad intuitive appeal in that the expected score of an unskilled set of forecasts (random guessing or perpetually identical forecasts) is 50%, and is interpretable as an indication of how often the forecasts are correct, even when the forecasts are expressed probabilistically and/or the observations are not discrete.

## Abstract

There are numerous reasons for calculating forecast verification scores, and considerable attention has been given to designing and analyzing the properties of scores that can be used for scientific purposes. Much less attention has been given to scores that may be useful for administrative reasons, such as communicating changes in forecast quality to bureaucrats and providing indications of forecast quality to the general public. The two-alternative forced choice (2AFC) test is proposed as a scoring procedure that is sufficiently generic to be usable on forecasts ranging from simple yes–no forecasts of dichotomous outcomes to forecasts of continuous variables, and can be used with deterministic or probabilistic forecasts without seriously reducing the more complex information when available. Although, as with any single verification score, the proposed test has limitations, it does have broad intuitive appeal in that the expected score of an unskilled set of forecasts (random guessing or perpetually identical forecasts) is 50%, and is interpretable as an indication of how often the forecasts are correct, even when the forecasts are expressed probabilistically and/or the observations are not discrete.

## Abstract

Previous experiments on thermal convection in a rotating fluid annulus (of depth *d*, inner radius *a* and outer radius *b* subject to an impressed horizontal temperature contrast Δ*T* have been extended to very small values of the aspect ratio &lambda=*D*(*b*-*a* by using in apparatus of large gap width *b*−*a* = 15.34 cm [and mean radius ½( *b*&plus*a* = 30.77 cm] and values of *d* as low as 1 cm. Particular attention was given to 1) the stable vertical temperature contrast (average value σ_{2}Δ*T* established by the fluid motions; and the dependence on λ and other parameters of (Θ_{c} the value of the dimensionless parameter Θ=gdΔ*p* at the transition, due to baroclinic instability, from axismmetric to non-axisymmetric flow. (Here *g* denotes acceleration due to gravity, Δ*p*/*p* is the fractional density contrast corresponding to Δ*T*, and &Omega is the angular speed of basic rotation.) At very low values of λ the area of the side walls, 2π(*b*+*a* is so very much less than that of the two rigid end walls,π(*b*
^{2}
*a*
^{2} that the experimental results can usefully he compared with the only available theoretical models in which frictional forces arise only in Ekman boundary layers on the end walls.

In agreement with theory, within the axisymmettic régimeσ_{2} is approximately equal to ½Π(Π=*g*δν^{½}λ^{2}/(8κρ¯ω^{2}) where ν denotes kinematic viscosity, and *k* thermal diffusivity] when II≪1 and approaches a value ≲1 when Π≫1; determinations of σ_{2} in the non-axisymmetric regime show that its value is comparatively unaffected by the presence of baroclinic waves.

Experimental values of Θ_{c} for values of λ<1 when the effects of viscosity are important, are compatible both quantitatively and qualitatively with theories of the effect of Ekman layer friction on Eady's theory of baroclinic instability. Furthermore, the values of the wavenumber close to the transition from axisymmetric to non-axisymmetric flow, for all values of λ considered, are in reasonable agreement with the linear theories.

## Abstract

Previous experiments on thermal convection in a rotating fluid annulus (of depth *d*, inner radius *a* and outer radius *b* subject to an impressed horizontal temperature contrast Δ*T* have been extended to very small values of the aspect ratio &lambda=*D*(*b*-*a* by using in apparatus of large gap width *b*−*a* = 15.34 cm [and mean radius ½( *b*&plus*a* = 30.77 cm] and values of *d* as low as 1 cm. Particular attention was given to 1) the stable vertical temperature contrast (average value σ_{2}Δ*T* established by the fluid motions; and the dependence on λ and other parameters of (Θ_{c} the value of the dimensionless parameter Θ=gdΔ*p* at the transition, due to baroclinic instability, from axismmetric to non-axisymmetric flow. (Here *g* denotes acceleration due to gravity, Δ*p*/*p* is the fractional density contrast corresponding to Δ*T*, and &Omega is the angular speed of basic rotation.) At very low values of λ the area of the side walls, 2π(*b*+*a* is so very much less than that of the two rigid end walls,π(*b*
^{2}
*a*
^{2} that the experimental results can usefully he compared with the only available theoretical models in which frictional forces arise only in Ekman boundary layers on the end walls.

In agreement with theory, within the axisymmettic régimeσ_{2} is approximately equal to ½Π(Π=*g*δν^{½}λ^{2}/(8κρ¯ω^{2}) where ν denotes kinematic viscosity, and *k* thermal diffusivity] when II≪1 and approaches a value ≲1 when Π≫1; determinations of σ_{2} in the non-axisymmetric regime show that its value is comparatively unaffected by the presence of baroclinic waves.

Experimental values of Θ_{c} for values of λ<1 when the effects of viscosity are important, are compatible both quantitatively and qualitatively with theories of the effect of Ekman layer friction on Eady's theory of baroclinic instability. Furthermore, the values of the wavenumber close to the transition from axisymmetric to non-axisymmetric flow, for all values of λ considered, are in reasonable agreement with the linear theories.

## Abstract

In a recent paper, Kuo and Schubert demonstrated the lack of observational support for the relevance of the criterion for cloud-top entrainment instability proposed by Randall and by Deardorff. Here we derive a new criterion, based on a model of the instability as resulting from the energy released close to cloud top, by Mixing between saturated boundary-layer air and unsaturated air from above the capping inversion. The condition is derived by considering the net conversion from potential to kinetic energy in a system consisting of two layers of fluid straddling cloud-top, when a small amount of mixing occurs between these layers. This contrasts with previous analyses, which only considered the change in buoyancy of the cloud layer when unsaturated air is mixed into it. In its most general form, this new criterion depends on the ratio of the depths of the layers involved in the mixing. It is argued that, for a self-sustaining instability, there must be a net release of kinetic energy on the same depth and time scales as the entrainment process itself. There are two plausible ways in which this requirement may be satisfied. Either one takes the depths of the layers involved in the mixing to each be comparable to the vertical scale of the entrainment process, which is typically of order tens of meters or less, or alternatively, one must allow for the efficiency with which energy released by mixing through a much deeper lower layer becomes available to initiate further entrainment. In both cases the same criterion for instability results. This criterion is much more restrictive than that proposed by Randall and by Deardorff; furthermore, the observational data is then consistent with the predictions of the current theory.

Further analysis provides estimates of the turbulent fluxes associated with cloud-top entrainment instability. This analysis effectively constitutes an energetically consistent turbulence closure for models of boundary layers with cloud. The implications for such numerical models are discussed. Comparisons are also made with other possible criteria for cloud-top entrainment instability which have recently been suggested.

## Abstract

In a recent paper, Kuo and Schubert demonstrated the lack of observational support for the relevance of the criterion for cloud-top entrainment instability proposed by Randall and by Deardorff. Here we derive a new criterion, based on a model of the instability as resulting from the energy released close to cloud top, by Mixing between saturated boundary-layer air and unsaturated air from above the capping inversion. The condition is derived by considering the net conversion from potential to kinetic energy in a system consisting of two layers of fluid straddling cloud-top, when a small amount of mixing occurs between these layers. This contrasts with previous analyses, which only considered the change in buoyancy of the cloud layer when unsaturated air is mixed into it. In its most general form, this new criterion depends on the ratio of the depths of the layers involved in the mixing. It is argued that, for a self-sustaining instability, there must be a net release of kinetic energy on the same depth and time scales as the entrainment process itself. There are two plausible ways in which this requirement may be satisfied. Either one takes the depths of the layers involved in the mixing to each be comparable to the vertical scale of the entrainment process, which is typically of order tens of meters or less, or alternatively, one must allow for the efficiency with which energy released by mixing through a much deeper lower layer becomes available to initiate further entrainment. In both cases the same criterion for instability results. This criterion is much more restrictive than that proposed by Randall and by Deardorff; furthermore, the observational data is then consistent with the predictions of the current theory.

Further analysis provides estimates of the turbulent fluxes associated with cloud-top entrainment instability. This analysis effectively constitutes an energetically consistent turbulence closure for models of boundary layers with cloud. The implications for such numerical models are discussed. Comparisons are also made with other possible criteria for cloud-top entrainment instability which have recently been suggested.

## Abstract

Large eddy simulations use a subgrid model, which is characterized by a length scale that is often related to the scale of the computational mesh by a numerical constant, *C*
_{
s
}. Mason and Callen argued that this subgrid model and its length scale define and impose the filter operation of the simulation. They saw *C*
_{
s
} as a measure of numerical accuracy. Others have sought to link the filter operation to the computational mesh and have viewed *C*
_{
s
} as needing determination for correct implementation. Here tests with a high resolution of 224 × 224 × 200 grid points are found to confirm Mason and Callen’s view. These simulations are also used together with lower-resolution simulations to illustrate the degree of convergence achieved. Some erroneous features of the simulations are identified through this test.

For the case of buoyant convection, the buoyancy dependence of the subgrid model is further examined. Most available subgrid models allow for buoyancy fluxes changing the level of the subgrid energy but only allow stable buoyancy gradients to modify the subgrid length scale—a reduction in this case. In contrast to most applications, it has been suggested that for a fixed filter operation, the subgrid length scale should always have a buoyancy dependence and should increase, in a finite way, with unstable buoyant transfer. Here an examination of spectral behavior in high-resolution simulations supports such an approach and shows that the model with the buoyancy-dependent length scale is indeed consistent with a fixed filter operation. The more conventional models are shown to have less satisfactory behavior.

## Abstract

Large eddy simulations use a subgrid model, which is characterized by a length scale that is often related to the scale of the computational mesh by a numerical constant, *C*
_{
s
}. Mason and Callen argued that this subgrid model and its length scale define and impose the filter operation of the simulation. They saw *C*
_{
s
} as a measure of numerical accuracy. Others have sought to link the filter operation to the computational mesh and have viewed *C*
_{
s
} as needing determination for correct implementation. Here tests with a high resolution of 224 × 224 × 200 grid points are found to confirm Mason and Callen’s view. These simulations are also used together with lower-resolution simulations to illustrate the degree of convergence achieved. Some erroneous features of the simulations are identified through this test.

For the case of buoyant convection, the buoyancy dependence of the subgrid model is further examined. Most available subgrid models allow for buoyancy fluxes changing the level of the subgrid energy but only allow stable buoyancy gradients to modify the subgrid length scale—a reduction in this case. In contrast to most applications, it has been suggested that for a fixed filter operation, the subgrid length scale should always have a buoyancy dependence and should increase, in a finite way, with unstable buoyant transfer. Here an examination of spectral behavior in high-resolution simulations supports such an approach and shows that the model with the buoyancy-dependent length scale is indeed consistent with a fixed filter operation. The more conventional models are shown to have less satisfactory behavior.

## Abstract

This article refers to the study of Mason and Weigel, where the generalized discrimination score *D* has been introduced. This score quantifies whether a set of observed outcomes can be correctly discriminated by the corresponding forecasts (i.e., it is a measure of the skill attribute of discrimination). Because of its generic definition, *D* can be adapted to essentially all relevant verification contexts, ranging from simple yes–no forecasts of binary outcomes to probabilistic forecasts of continuous variables. For most of these cases, Mason and Weigel have derived expressions for *D*, many of which have turned out to be equivalent to scores that are already known under different names. However, no guidance was provided on how to calculate *D* for ensemble forecasts. This gap is aggravated by the fact that there are currently very few measures of forecast quality that could be directly applied to ensemble forecasts without requiring that probabilities be derived from the ensemble members prior to verification. This study seeks to close this gap. A definition is proposed of how ensemble forecasts can be ranked; the ranks of the ensemble forecasts can then be used as a basis for attempting to discriminate between corresponding observations. Given this definition, formulations of *D* are derived that are directly applicable to ensemble forecasts.

## Abstract

This article refers to the study of Mason and Weigel, where the generalized discrimination score *D* has been introduced. This score quantifies whether a set of observed outcomes can be correctly discriminated by the corresponding forecasts (i.e., it is a measure of the skill attribute of discrimination). Because of its generic definition, *D* can be adapted to essentially all relevant verification contexts, ranging from simple yes–no forecasts of binary outcomes to probabilistic forecasts of continuous variables. For most of these cases, Mason and Weigel have derived expressions for *D*, many of which have turned out to be equivalent to scores that are already known under different names. However, no guidance was provided on how to calculate *D* for ensemble forecasts. This gap is aggravated by the fact that there are currently very few measures of forecast quality that could be directly applied to ensemble forecasts without requiring that probabilities be derived from the ensemble members prior to verification. This study seeks to close this gap. A definition is proposed of how ensemble forecasts can be ranked; the ranks of the ensemble forecasts can then be used as a basis for attempting to discriminate between corresponding observations. Given this definition, formulations of *D* are derived that are directly applicable to ensemble forecasts.

## Abstract

Large eddy simulations sometimes use monotone advection schemes. Such schemes are dissipative, and the effective subgrid model then becomes the combined effect of the intended model and of the numerical dissipation. The impacts on simulation reliability are examined for the cases of dry convective and neutral planetary boundary layers. In general it is found that the results in the well-resolved flow interior are insensitive to the details of the advection scheme. However, unsatisfactory results may be obtained if numerical dissipation dominates where the flow becomes less well resolved as the surface is approached.

## Abstract

Large eddy simulations sometimes use monotone advection schemes. Such schemes are dissipative, and the effective subgrid model then becomes the combined effect of the intended model and of the numerical dissipation. The impacts on simulation reliability are examined for the cases of dry convective and neutral planetary boundary layers. In general it is found that the results in the well-resolved flow interior are insensitive to the details of the advection scheme. However, unsatisfactory results may be obtained if numerical dissipation dominates where the flow becomes less well resolved as the surface is approached.

## Abstract

Detailed studies of the azimuthal structure of fully developed waves in a differentially heated rotating fluid annulus have been carried out with the aid of instrumentation capable of providing frequent determinations of the temperature variation around a circle concentric with the walls of the annulus. Owing to the cyclic nature of the data they are conveniently analyzed in terms of azimuthal Fourier modes. The time-averaged azimuthal spectra thus obtained show that in the regular regime, where the flow is dominated by a single mode of wavenumber *M*, say, significant “energy” is found not only in the harmonics required to describe the jet stream structure of the flow but also in the sideband modes of wavenumber *M*=1 which describe the observed azimuthal modulations in the amplitude and/or phase of the wave. At the high-wavenumber end of those spectra for which an inertial subrange can be resolved the “spectral energy” follows a (wavenumber)^{−3} law.

The time-dependent behavior of the phases of the sidebands and the main baroclinic mode, &phis_{
M−1}, ϕ_{
M+1} and ϕ_{
M
} respectively, is such that the value of ϕ≡2ϕ_{
M
}−ϕ_{
M−1}−ϕ_{
M+1}, remains nearly constant (and close to π), implying that a frame of reference can be found in which the average intrinsic frequencies of the main mode and its side bands are equal. This special frame is fixed relative to the rotating apparatus when the waves are only weakly dispersive, but it can be altered by sloping the endwalls of the apparatus so as to introduce dispersion and returned to the apparatus frame by introducing irregular topography. The theoretical implications of these results are explored with simple wave-interaction theory, which suggests that the sidebands interact strongly with baroclinically stable long waves, but in such a way that in equilibrium the *net* energy transfer into the long waves is small.

## Abstract

Detailed studies of the azimuthal structure of fully developed waves in a differentially heated rotating fluid annulus have been carried out with the aid of instrumentation capable of providing frequent determinations of the temperature variation around a circle concentric with the walls of the annulus. Owing to the cyclic nature of the data they are conveniently analyzed in terms of azimuthal Fourier modes. The time-averaged azimuthal spectra thus obtained show that in the regular regime, where the flow is dominated by a single mode of wavenumber *M*, say, significant “energy” is found not only in the harmonics required to describe the jet stream structure of the flow but also in the sideband modes of wavenumber *M*=1 which describe the observed azimuthal modulations in the amplitude and/or phase of the wave. At the high-wavenumber end of those spectra for which an inertial subrange can be resolved the “spectral energy” follows a (wavenumber)^{−3} law.

The time-dependent behavior of the phases of the sidebands and the main baroclinic mode, &phis_{
M−1}, ϕ_{
M+1} and ϕ_{
M
} respectively, is such that the value of ϕ≡2ϕ_{
M
}−ϕ_{
M−1}−ϕ_{
M+1}, remains nearly constant (and close to π), implying that a frame of reference can be found in which the average intrinsic frequencies of the main mode and its side bands are equal. This special frame is fixed relative to the rotating apparatus when the waves are only weakly dispersive, but it can be altered by sloping the endwalls of the apparatus so as to introduce dispersion and returned to the apparatus frame by introducing irregular topography. The theoretical implications of these results are explored with simple wave-interaction theory, which suggests that the sidebands interact strongly with baroclinically stable long waves, but in such a way that in equilibrium the *net* energy transfer into the long waves is small.