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- Author or Editor: H. Tennekes x

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## Abstract

This paper deals with the problem of fully developed free convection in the atmospheric boundary layer. In free convection, the height of the Ekman layer is much larger than the absolute value of the Monin-Oboukhov length. The kinetic energy budget of the turbulence above the surface layer shows that the standard deviations of vertical velocity and of temperature are related to *h*/*L* by σ_{
w
}/*u*
_{*}∝(−*h*/*L*)^{⅓} and σ_{θ}/θ_{*}∝(−*h*/*L*)^{⅓}. Because convection has no natural length scale, the height of the neutral Ekman layer (*h*∝*u*
_{*}/*f*) is used to explore the consequences of the proposed expressions for σ_{
w
} and σ_{θ}. The dissimilarity between the heat flux and the momentum flux is studied in terms of time- and length-scale ratios and in terms of a flux Richardson number. A definitive solution of the problem, however, cannot be formulated until an expression for the height of unstable Ekman layers, as a function of the time of day and the stability conditions at the top of the boundary layer, can he found.

## Abstract

This paper deals with the problem of fully developed free convection in the atmospheric boundary layer. In free convection, the height of the Ekman layer is much larger than the absolute value of the Monin-Oboukhov length. The kinetic energy budget of the turbulence above the surface layer shows that the standard deviations of vertical velocity and of temperature are related to *h*/*L* by σ_{
w
}/*u*
_{*}∝(−*h*/*L*)^{⅓} and σ_{θ}/θ_{*}∝(−*h*/*L*)^{⅓}. Because convection has no natural length scale, the height of the neutral Ekman layer (*h*∝*u*
_{*}/*f*) is used to explore the consequences of the proposed expressions for σ_{
w
} and σ_{θ}. The dissimilarity between the heat flux and the momentum flux is studied in terms of time- and length-scale ratios and in terms of a flux Richardson number. A definitive solution of the problem, however, cannot be formulated until an expression for the height of unstable Ekman layers, as a function of the time of day and the stability conditions at the top of the boundary layer, can he found.

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## Abstract

The differential equations governing the strength Δ (a potential temperature difference) and the height *h* of inversions associated with dry penetrative convection are considered. No assumptions on the magnitude of the downward heat flux at the inversion base are needed to obtain an algebraic equation that relates *h* and Δ to the heating history of the boundary layer and to the initial conditions. After the nocturnal inversion has been filled in by heating, the inversion base generally grows linearly with time in the morning, but is proportional to the square root of time in the afternoon. The variation of Δ with time differs greatly from case to case.

## Abstract

The differential equations governing the strength Δ (a potential temperature difference) and the height *h* of inversions associated with dry penetrative convection are considered. No assumptions on the magnitude of the downward heat flux at the inversion base are needed to obtain an algebraic equation that relates *h* and Δ to the heating history of the boundary layer and to the initial conditions. After the nocturnal inversion has been filled in by heating, the inversion base generally grows linearly with time in the morning, but is proportional to the square root of time in the afternoon. The variation of Δ with time differs greatly from case to case.

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## Abstract

This paper explores the practical consequences of the asymptotic nature of the logarithmic wind profile in neutral, barotropic, planetary boundary layers. Recent developments in boundary-layer theory have shown that the von Kármán constant is a universal constant only in a very specific asymptotic sense; in typical atmospheric conditions its value is probably about 10% larger than the asymptotic one. Pending the development of a second-order theory, the value κ = 0.35 ± 0.02 is recommended for micrometeorological applications over smooth terrain. It is shown that *K* theory cannot be used in attempts to detect any trends of deviations from the logarithmic law.

## Abstract

This paper explores the practical consequences of the asymptotic nature of the logarithmic wind profile in neutral, barotropic, planetary boundary layers. Recent developments in boundary-layer theory have shown that the von Kármán constant is a universal constant only in a very specific asymptotic sense; in typical atmospheric conditions its value is probably about 10% larger than the asymptotic one. Pending the development of a second-order theory, the value κ = 0.35 ± 0.02 is recommended for micrometeorological applications over smooth terrain. It is shown that *K* theory cannot be used in attempts to detect any trends of deviations from the logarithmic law.

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## Abstract

It is likely that several features of the mid-latitude circulation in the earth's atmosphere wig also be observed in two-dimensional, nondivergent flow with buoyant forcing and surface friction. Properly scalled, buoyancy effects are surprisingly similar to baroclinic effects. A linear stability analysis shows that the growth rate of unstable disturbances depends on zonal wavenumber in much the same way as that of baroclinic waves, except for the absence of a high-wavenumber cutoff related to the Rossby radius of deformation. The energy conversion mechanisms in buoyancy-driven two-dimensional flow closely resemble those in the atmosphere: eddy kinetic energy is maintained primarily by conversion of eddy potential energy, the kinetic energy of the mean zonal flow is maintained primarily by a reverse energy cascade, and the flow owes its existence and dynamics to the mean temperature contrast between latitude circles. The equations studied in this paper include these for enstrophy and temperature variance; the spectral fluxes of these quantities are taken into account. The maintenance of the general circulation in two-dimensional flow is described in part by a system of flux-maintenance equations. These shed light on such issues as the magnitude of the poleward eddy heat flux in developing storms and the countergradient eddy momentum flux in middle latitudes.

## Abstract

It is likely that several features of the mid-latitude circulation in the earth's atmosphere wig also be observed in two-dimensional, nondivergent flow with buoyant forcing and surface friction. Properly scalled, buoyancy effects are surprisingly similar to baroclinic effects. A linear stability analysis shows that the growth rate of unstable disturbances depends on zonal wavenumber in much the same way as that of baroclinic waves, except for the absence of a high-wavenumber cutoff related to the Rossby radius of deformation. The energy conversion mechanisms in buoyancy-driven two-dimensional flow closely resemble those in the atmosphere: eddy kinetic energy is maintained primarily by conversion of eddy potential energy, the kinetic energy of the mean zonal flow is maintained primarily by a reverse energy cascade, and the flow owes its existence and dynamics to the mean temperature contrast between latitude circles. The equations studied in this paper include these for enstrophy and temperature variance; the spectral fluxes of these quantities are taken into account. The maintenance of the general circulation in two-dimensional flow is described in part by a system of flux-maintenance equations. These shed light on such issues as the magnitude of the poleward eddy heat flux in developing storms and the countergradient eddy momentum flux in middle latitudes.

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## Abstract

The budget of turbulent kinetic energy at the base of the inversion which caps the daytime atmospheric boundary layer depends on the lapse rate of potential temperature in the air aloft. The principal gain term in the energy budget is turbulent transport of kinetic energy, the principal loss term is buoyant conversion of kinetic energy into potential energy. The contributions made by these and other terms in the energy budget need to be parameterized for applications to inversion-rise prediction schemes. This paper contains a detailed analysis of the effects of dissipation near the inversion base, which leads to reduced entrainment if the air aloft is very stable. The parameterized energy budget also includes the Zilitinkevich correction, the influence of mechanical energy production near the inversion base, and modifications needed to incorporate cases in which the surface heat flux is negligible. Extensive comparisons of the theoretical model with experimental data indicate that a simplified treatment of the energy budget is adequate for forecasts of the development of convective mixed layers. The parameterization scheme is also applicable to thermocline erosion in the ocean; in that case, however, some of the minor terms in the energy budget often play a major role.

## Abstract

The budget of turbulent kinetic energy at the base of the inversion which caps the daytime atmospheric boundary layer depends on the lapse rate of potential temperature in the air aloft. The principal gain term in the energy budget is turbulent transport of kinetic energy, the principal loss term is buoyant conversion of kinetic energy into potential energy. The contributions made by these and other terms in the energy budget need to be parameterized for applications to inversion-rise prediction schemes. This paper contains a detailed analysis of the effects of dissipation near the inversion base, which leads to reduced entrainment if the air aloft is very stable. The parameterized energy budget also includes the Zilitinkevich correction, the influence of mechanical energy production near the inversion base, and modifications needed to incorporate cases in which the surface heat flux is negligible. Extensive comparisons of the theoretical model with experimental data indicate that a simplified treatment of the energy budget is adequate for forecasts of the development of convective mixed layers. The parameterization scheme is also applicable to thermocline erosion in the ocean; in that case, however, some of the minor terms in the energy budget often play a major role.

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## Abstract

In this paper we develop an abbreviated model for the pressure-gradient velocity correlation terms in the equations for the Reynolds-stress components in the neutral boundary layer. The model contains three terms: a nonlinear return-to-isotropy term, a mean strain-rate term, and a mean vorticity term. There are three free constants in the model, which are determined with the aid of experimental results on the ratios between the Reynolds-stress components in the neutral surface layer. Since three independent equations are involved, the model is self-contained. Through its mean vorticity term, the model incorporates the effects of a rotating coordinate system. The application of the model to a neutral Ekman layer gives realistic results.

## Abstract

In this paper we develop an abbreviated model for the pressure-gradient velocity correlation terms in the equations for the Reynolds-stress components in the neutral boundary layer. The model contains three terms: a nonlinear return-to-isotropy term, a mean strain-rate term, and a mean vorticity term. There are three free constants in the model, which are determined with the aid of experimental results on the ratios between the Reynolds-stress components in the neutral surface layer. Since three independent equations are involved, the model is self-contained. Through its mean vorticity term, the model incorporates the effects of a rotating coordinate system. The application of the model to a neutral Ekman layer gives realistic results.

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## Abstract

New similarity expressions for turbulent Ekman layers have recently appeared, the theoretical foundation of which is analyzed in this paper. It is shown that the flow in Ekman layers has to be solved by singular perturbation methods. The similarity laws given by Gill and Csanady are, in first approximation, independent of the surface Rossby number if it is sufficiently large.

## Abstract

New similarity expressions for turbulent Ekman layers have recently appeared, the theoretical foundation of which is analyzed in this paper. It is shown that the flow in Ekman layers has to be solved by singular perturbation methods. The similarity laws given by Gill and Csanady are, in first approximation, independent of the surface Rossby number if it is sufficiently large.

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## Abstract

A rate equation is derived which describes the development of the boundary-layer height under stable conditions as a function of time.

It takes the form of a linear relaxation equation; its solution is forced toward an equilibrium value. The equilibrium height is connected to the work done by the ageostrophic wind in the boundary layer. The time scale of the relaxation process increases monotonically from a few hours shortly after sunset to a value of the order of 10 h later on. This means that the boundary-layer height evolves very slowly, which may lead to the unwarranted impression that stationary conditions have been reached. The main features of the rate equation are confirmed by comparison with the results of computer simulations and with field observations of the boundary-layer height during clear nights.

## Abstract

A rate equation is derived which describes the development of the boundary-layer height under stable conditions as a function of time.

It takes the form of a linear relaxation equation; its solution is forced toward an equilibrium value. The equilibrium height is connected to the work done by the ageostrophic wind in the boundary layer. The time scale of the relaxation process increases monotonically from a few hours shortly after sunset to a value of the order of 10 h later on. This means that the boundary-layer height evolves very slowly, which may lead to the unwarranted impression that stationary conditions have been reached. The main features of the rate equation are confirmed by comparison with the results of computer simulations and with field observations of the boundary-layer height during clear nights.

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## Abstract

The observed wintertime distribution of the poleward flux of westerly momentum by transient eddies at the jet stream (250 mb) level is characterized by 1) a strong convergence of momentum flux into “storm tracks” over the western and central oceans near 40°N and 2) strong poleward fluxes over the western parts of the continents. The former fluxes are strongly countergradient. This observed distribution is discussed in light of the flux maintenance equation, which is analogous to the equations for the local time rate of change of eddy kinetic energy. The terms of major interest in this equation are a so-called “mixing term,” which is always acting to produce a down-gradient flux, and a pair of terms which involve temporal correlations between the eddy velocity components and their respective ageostrophic departures. It is shown that the observed countergradient fluxes over the oceans must be maintained by these ageostrophic correlations terms.

The geographical distributions of the terms described above are estimated on the basis of ten winters' hemispheric synoptic charts. The ageostrophic correlation terms tend to produce eddy fluxes of westerly momentum into the storm tracks, as observed. It is proposed that the convergence of westerly momentum into the storm tracks is a consequence of the fact that there is a strong tendency for air with high westerly momentum to be accelerated in the direction of the center of the storm track by the imbalance between pressure gradient and Coriolis forces.

## Abstract

The observed wintertime distribution of the poleward flux of westerly momentum by transient eddies at the jet stream (250 mb) level is characterized by 1) a strong convergence of momentum flux into “storm tracks” over the western and central oceans near 40°N and 2) strong poleward fluxes over the western parts of the continents. The former fluxes are strongly countergradient. This observed distribution is discussed in light of the flux maintenance equation, which is analogous to the equations for the local time rate of change of eddy kinetic energy. The terms of major interest in this equation are a so-called “mixing term,” which is always acting to produce a down-gradient flux, and a pair of terms which involve temporal correlations between the eddy velocity components and their respective ageostrophic departures. It is shown that the observed countergradient fluxes over the oceans must be maintained by these ageostrophic correlations terms.

The geographical distributions of the terms described above are estimated on the basis of ten winters' hemispheric synoptic charts. The ageostrophic correlation terms tend to produce eddy fluxes of westerly momentum into the storm tracks, as observed. It is proposed that the convergence of westerly momentum into the storm tracks is a consequence of the fact that there is a strong tendency for air with high westerly momentum to be accelerated in the direction of the center of the storm track by the imbalance between pressure gradient and Coriolis forces.