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- Author or Editor: J. W. Deardorff x
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Abstract
The criterion for instability of dry air mixing into a cloud top is determined both graphically and mathematically. It is found to be identical to the buoyancy-flux instability criterion of Randall (1976) based on stratocumulus mixed-layer jump-model equations which yield a cloud top jump in θ e , for marginal stability of order Δθ e =−1 to −3 K (θ e ), is equivalent potential temperature).
Numerical calculations from a three-dimensional boundary-layer turbulence model with stratocumulus are examined from the viewpoint of cloud top entrainment instability. The entrainment rate is found to increase decisively when Δθ e , drops below the critical value. The resulting rapid entrainment is found to dry out the cloud from the bottom up, leaving the cloud-base height intact; in conjunction with the negative Δθ e value, this causes ∂θ e /∂z in the cloud layer to become negative. This in turn sets the stage for the final phase of stratocumulus breakup, whereby conditional instability of the first kind begins to operate and the cloud fraction becomes small.
Abstract
The criterion for instability of dry air mixing into a cloud top is determined both graphically and mathematically. It is found to be identical to the buoyancy-flux instability criterion of Randall (1976) based on stratocumulus mixed-layer jump-model equations which yield a cloud top jump in θ e , for marginal stability of order Δθ e =−1 to −3 K (θ e ), is equivalent potential temperature).
Numerical calculations from a three-dimensional boundary-layer turbulence model with stratocumulus are examined from the viewpoint of cloud top entrainment instability. The entrainment rate is found to increase decisively when Δθ e , drops below the critical value. The resulting rapid entrainment is found to dry out the cloud from the bottom up, leaving the cloud-base height intact; in conjunction with the negative Δθ e value, this causes ∂θ e /∂z in the cloud layer to become negative. This in turn sets the stage for the final phase of stratocumulus breakup, whereby conditional instability of the first kind begins to operate and the cloud fraction becomes small.
Abstract
Convection between horizontal plates maintained at two different temperatures is studied numerically for a Rayleigh number of 6.75×105 and a Prandtl number of 0.71. The two-dimensional and Boussinesq approximations are applied. The computed motion eventually takes the form of large-scale and nearly steady-state vortices which induce plumes of warm (cool) air to impinge strongly upon the cool (warm) plate. When the width is restricted by lateral, insulated walls and is equal to the height, eddies which appear in corners suppress the single large-scale vortex which forms and cause the heat transport to be about 20 per cent smaller than is observed experimentally for a region of width much greater than height. With a width twice the height, a large-scale vortex pair develops. The effect of corner eddies is then diminished, and the heat flux becomes 18 per cent larger than the experimental value. When the lateral walls at this width are replaced by cyclic boundaries, no small-scale eddies appear, and the heat flux is 34 per cent too large. The tendency for air leaving the vicinity of either plate to be concentrated into narrow plumes, with temperature anomaly diminishing with distance from the plate, allows the convective heat flux to occur in the central region in the absence of an unstable lapse rate.
The absence of aperiodic, turbulent motions in the two-dimensional model of limited width is corroborated by measurements within a “two-dimensional” convection chamber of width equal to height.
A quasi-linear model of thermal convection is found to yield a heat transport at least 55 per cent too large, and a kinetic energy 270 per cent too large, in comparison with the non-linear model. The effect of replacing the rigid-surface boundary conditions by free-surface conditions is to increase the heat transport by 190 per cent.
Abstract
Convection between horizontal plates maintained at two different temperatures is studied numerically for a Rayleigh number of 6.75×105 and a Prandtl number of 0.71. The two-dimensional and Boussinesq approximations are applied. The computed motion eventually takes the form of large-scale and nearly steady-state vortices which induce plumes of warm (cool) air to impinge strongly upon the cool (warm) plate. When the width is restricted by lateral, insulated walls and is equal to the height, eddies which appear in corners suppress the single large-scale vortex which forms and cause the heat transport to be about 20 per cent smaller than is observed experimentally for a region of width much greater than height. With a width twice the height, a large-scale vortex pair develops. The effect of corner eddies is then diminished, and the heat flux becomes 18 per cent larger than the experimental value. When the lateral walls at this width are replaced by cyclic boundaries, no small-scale eddies appear, and the heat flux is 34 per cent too large. The tendency for air leaving the vicinity of either plate to be concentrated into narrow plumes, with temperature anomaly diminishing with distance from the plate, allows the convective heat flux to occur in the central region in the absence of an unstable lapse rate.
The absence of aperiodic, turbulent motions in the two-dimensional model of limited width is corroborated by measurements within a “two-dimensional” convection chamber of width equal to height.
A quasi-linear model of thermal convection is found to yield a heat transport at least 55 per cent too large, and a kinetic energy 270 per cent too large, in comparison with the non-linear model. The effect of replacing the rigid-surface boundary conditions by free-surface conditions is to increase the heat transport by 190 per cent.
Abstract
A numerical model for the study of three-dimensional, small-scale, horizontally homogeneous turbulence is presented for use with a two-dimensional grid in a vertical plane. The model is called pseudo three-dimensional because all quantities involving one of the horizontal dimensions are obtained by assumption during the calculations. It is applied to the problem of turbulent thermal convection between horizontal plates, with a Rayleigh number of 6.75×105, and Prandtl numbers of 0.71 and 10.
It is shown that this model allows turbulent motions to develop, whereas a two-dimensional model yields nearly steady convective rolls. The calculated heat flux is only about 5 per cent larger than experimental values, and the horizontal variances of temperature, vertical velocity and horizontal velocity are in rough agreement with observations at all levels. Terms in the vorticity and temperature variance equations are also evaluated, although experimental values are lacking for comparison.
Abstract
A numerical model for the study of three-dimensional, small-scale, horizontally homogeneous turbulence is presented for use with a two-dimensional grid in a vertical plane. The model is called pseudo three-dimensional because all quantities involving one of the horizontal dimensions are obtained by assumption during the calculations. It is applied to the problem of turbulent thermal convection between horizontal plates, with a Rayleigh number of 6.75×105, and Prandtl numbers of 0.71 and 10.
It is shown that this model allows turbulent motions to develop, whereas a two-dimensional model yields nearly steady convective rolls. The calculated heat flux is only about 5 per cent larger than experimental values, and the horizontal variances of temperature, vertical velocity and horizontal velocity are in rough agreement with observations at all levels. Terms in the vorticity and temperature variance equations are also evaluated, although experimental values are lacking for comparison.
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A review is presented on laboratory modeling of diffusion downwind of a continuous point source within a boundary layer of well defined height with turbulence driven by buoyant convection. Results of using mixed-layer scaling are summarized and comparisons with atmospheric field measurements discussed. Concentration probability distributions from the laboratory as a function of dimensionless downwind distance are presented in a manner which discloses the differences from a lognormal distribution.
Abstract
A review is presented on laboratory modeling of diffusion downwind of a continuous point source within a boundary layer of well defined height with turbulence driven by buoyant convection. Results of using mixed-layer scaling are summarized and comparisons with atmospheric field measurements discussed. Concentration probability distributions from the laboratory as a function of dimensionless downwind distance are presented in a manner which discloses the differences from a lognormal distribution.
Abstract
The first-order jump model for the potential temperature or buoyancy variable at the capping inversion atop a convectively mixed layer is reexamined and found to imply existence of an entrainment rate equation which is unreliable. The model is therefore extended here to allow all the negative buoyancy flux of entrainment to occur within the interfacial layer of thickness Δh and to allow realistic thermal structure within the layer. The new model yields a well behaved entrainment rate equation requiring scarcely any closure assumption in the cases of steady-state entrainment with large-scale subsidence, and pseudo-encroachment. For nonsteady entrainment the closure assumption need only be made on d(Δh)/dt in order to obtain the entrainment rates at both the outer and inner edges of the interfacial layer. A particular closure assumption for d(Δh)/dt is tested against five laboratory experiments and found to yield favorable results for both Δh and the mixed-layer thickness if the initial value of Δh is known. It is also compared against predictions from two zero-order jump models which do not attempt prediction of Δh and one first-order jump model.
Abstract
The first-order jump model for the potential temperature or buoyancy variable at the capping inversion atop a convectively mixed layer is reexamined and found to imply existence of an entrainment rate equation which is unreliable. The model is therefore extended here to allow all the negative buoyancy flux of entrainment to occur within the interfacial layer of thickness Δh and to allow realistic thermal structure within the layer. The new model yields a well behaved entrainment rate equation requiring scarcely any closure assumption in the cases of steady-state entrainment with large-scale subsidence, and pseudo-encroachment. For nonsteady entrainment the closure assumption need only be made on d(Δh)/dt in order to obtain the entrainment rates at both the outer and inner edges of the interfacial layer. A particular closure assumption for d(Δh)/dt is tested against five laboratory experiments and found to yield favorable results for both Δh and the mixed-layer thickness if the initial value of Δh is known. It is also compared against predictions from two zero-order jump models which do not attempt prediction of Δh and one first-order jump model.
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Abstract
The second-moment equations for buoyancy flux and density-fluctuation variance in the entrainment zone of a mixed layer are combined to yield an entrainment rate (we dependence having the nature of Turner's relation. That is, weq −1 is seen as a function solely of an interfacial Richardson number (Ri e ) involving q where q 2 is twice the interfacial turbulence kinetic energy (TKE). The mean thickness of the entrainment layer is found to be a satisfactory representation of the integral length scale near the mixed-layer interface which was used by Turner. The TKE equation in the entrainment layer is then utilized to evaluate Rie so that we can be obtained. This procedure allows the TKE equation to be solved past the point at which the interfacial-shear bulk Richardson number, Riv, becomes critical, and into the more unstable regime beyond where the experiment of Ellison and Turner can give guidance. Previous entrainment parameterizations which retain the critical-Ri v concept predict infinite entrainment rates at the critical point and negative ones beyond.
The present entrainment results are shown as a function of the various Richardson numbers: Ri v , Ri T based on the friction velocity, and Ri * based on the convective velocity scale, for values of each ranging from near zero to 1000. Comparison is made with previous results, which the present results resemble only in narrow regions of parameter space. For typical values of Ri T or Ri * , we is found to be enhanced by 10–20% if Ri v drops from ∞ to 5, and by a factor of from 3 to 8 if it drops to its critical value, (Ri v )crit near 0.7–1.0.
Abstract
The second-moment equations for buoyancy flux and density-fluctuation variance in the entrainment zone of a mixed layer are combined to yield an entrainment rate (we dependence having the nature of Turner's relation. That is, weq −1 is seen as a function solely of an interfacial Richardson number (Ri e ) involving q where q 2 is twice the interfacial turbulence kinetic energy (TKE). The mean thickness of the entrainment layer is found to be a satisfactory representation of the integral length scale near the mixed-layer interface which was used by Turner. The TKE equation in the entrainment layer is then utilized to evaluate Rie so that we can be obtained. This procedure allows the TKE equation to be solved past the point at which the interfacial-shear bulk Richardson number, Riv, becomes critical, and into the more unstable regime beyond where the experiment of Ellison and Turner can give guidance. Previous entrainment parameterizations which retain the critical-Ri v concept predict infinite entrainment rates at the critical point and negative ones beyond.
The present entrainment results are shown as a function of the various Richardson numbers: Ri v , Ri T based on the friction velocity, and Ri * based on the convective velocity scale, for values of each ranging from near zero to 1000. Comparison is made with previous results, which the present results resemble only in narrow regions of parameter space. For typical values of Ri T or Ri * , we is found to be enhanced by 10–20% if Ri v drops from ∞ to 5, and by a factor of from 3 to 8 if it drops to its critical value, (Ri v )crit near 0.7–1.0.
Abstract
Anomalously large values of the Reynolds stress (downstream component) obtained from measurements analysed by Angell are explained as being associated with changes in velocity at the top of a deepening planetary boundary layer. The proposed explanation is tested both by a simple interfacial velocity-jump model, and by a detailed three-dimensional numerical model. A method of parameterizing the complicated vertical profiles of momentum flux is proposed for use within large scale, multi-level circulation models.
Abstract
Anomalously large values of the Reynolds stress (downstream component) obtained from measurements analysed by Angell are explained as being associated with changes in velocity at the top of a deepening planetary boundary layer. The proposed explanation is tested both by a simple interfacial velocity-jump model, and by a detailed three-dimensional numerical model. A method of parameterizing the complicated vertical profiles of momentum flux is proposed for use within large scale, multi-level circulation models.
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