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- Author or Editor: J. R. Herring x

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## Abstract

The thermal convection equations for a thin layer of fluid are solved numerically as an initial value problem. The calculations include only those nonlinear terms which have the form of an interaction of a fluctuation in the velocity and temperature with the mean temperature field. In the present calculations, the velocity and temperature fluctuations have one horizontal wave number, and satisfy free boundary conditions on two conducting horizontal surfaces.

The computed steady state velocity and temperature amplitudes show many of the observed qualitative features. In particular, the experimentally observed boundary layering of the mean temperature field is correctly reproduced, and, at large Rayleigh number, the total heat transport is found to be proportional to the cube root of the Rayleigh number, provided the fluctuating temperature and velocity amplitudes have that horizontal wave number which maximizes the total heat transport. However, the heat transport found here for free boundaries is about three times the experimental value for rigid boundaries. The mean temperature gradient can become negative near the boundaries for large Rayleigh numbers and large horizontal scale motions.

The linear stability of the system is also investigated, and it is concluded that the stable solutions for all Rayleigh numbers investigated (R < 10^{8}) have horizontal wave numbers which very nearly maximize the total heat transport. The stability study also indicates regions in which two or more horizontal wave numbers are required to support convection.

## Abstract

The thermal convection equations for a thin layer of fluid are solved numerically as an initial value problem. The calculations include only those nonlinear terms which have the form of an interaction of a fluctuation in the velocity and temperature with the mean temperature field. In the present calculations, the velocity and temperature fluctuations have one horizontal wave number, and satisfy free boundary conditions on two conducting horizontal surfaces.

The computed steady state velocity and temperature amplitudes show many of the observed qualitative features. In particular, the experimentally observed boundary layering of the mean temperature field is correctly reproduced, and, at large Rayleigh number, the total heat transport is found to be proportional to the cube root of the Rayleigh number, provided the fluctuating temperature and velocity amplitudes have that horizontal wave number which maximizes the total heat transport. However, the heat transport found here for free boundaries is about three times the experimental value for rigid boundaries. The mean temperature gradient can become negative near the boundaries for large Rayleigh numbers and large horizontal scale motions.

The linear stability of the system is also investigated, and it is concluded that the stable solutions for all Rayleigh numbers investigated (R < 10^{8}) have horizontal wave numbers which very nearly maximize the total heat transport. The stability study also indicates regions in which two or more horizontal wave numbers are required to support convection.

## Abstract

An analytic approximation method is developed to treat the problem of thermal convection at large Rayleigh numbers R. The method is applied to a convection model in which the fluctuating self-interactions are omitted. The results of the method compare satisfactorily to the exact solutions at large Rayleigh numbers. The results include a derivation of the R^{½} law for the Nusselt number, and closed form estimates for the shape of the mean temperature field and temperature and velocity fluctuation fields.

## Abstract

An analytic approximation method is developed to treat the problem of thermal convection at large Rayleigh numbers R. The method is applied to a convection model in which the fluctuating self-interactions are omitted. The results of the method compare satisfactorily to the exact solutions at large Rayleigh numbers. The results include a derivation of the R^{½} law for the Nusselt number, and closed form estimates for the shape of the mean temperature field and temperature and velocity fluctuation fields.

## Abstract

Two-dimensional rotating turbulent flow above a random topography is investigated using the direct interaction approximation and an extension of the test field model, which includes equations for the lagged covariance spectra. For topographic dominated flows (at large scales) the flow predicted is strongly locked to topography. If inertial effects dominate (at smaller scales), three enstrophy-inertial subranges of progressively smaller scales are suggested: a *k*
^{−1} energy range, followed by two physically distinguishable *k*
^{−3} ranges. We discuss these inertial ranges by a heuristic theory based on the test field model similar to that proposed by Leith (1968). The origins of these inertial subranges are explained by considering the dominant vorticity distortion (or transfer) process at a given scale, and the coherence time (the length of time the distorting process lasts) at that scale. If topography determines both distortion and the time scale, a *k*
^{−1} range results; the first *k*
^{−3} range is an inertial distortion and topographic Rossby wave time-scale regime, and the second *k*
^{−3} range is the usual two-dimensional inertial range. We examine in some detail the predictions of the theory for stationary turbulence maintained by random stirring at large scales. The theory predicts that the lagged covariance of the vorticity field has a static component which is strongly correlated with topography. The relative magnitude of this static component is determined in terms of a nondimensional measure of topography.

## Abstract

Two-dimensional rotating turbulent flow above a random topography is investigated using the direct interaction approximation and an extension of the test field model, which includes equations for the lagged covariance spectra. For topographic dominated flows (at large scales) the flow predicted is strongly locked to topography. If inertial effects dominate (at smaller scales), three enstrophy-inertial subranges of progressively smaller scales are suggested: a *k*
^{−1} energy range, followed by two physically distinguishable *k*
^{−3} ranges. We discuss these inertial ranges by a heuristic theory based on the test field model similar to that proposed by Leith (1968). The origins of these inertial subranges are explained by considering the dominant vorticity distortion (or transfer) process at a given scale, and the coherence time (the length of time the distorting process lasts) at that scale. If topography determines both distortion and the time scale, a *k*
^{−1} range results; the first *k*
^{−3} range is an inertial distortion and topographic Rossby wave time-scale regime, and the second *k*
^{−3} range is the usual two-dimensional inertial range. We examine in some detail the predictions of the theory for stationary turbulence maintained by random stirring at large scales. The theory predicts that the lagged covariance of the vorticity field has a static component which is strongly correlated with topography. The relative magnitude of this static component is determined in terms of a nondimensional measure of topography.

## Abstract

Utilizing an abridged version of the test field model, we examine the relaxation of two-dimensional homogeneous turbulence back to its isotropic state. Our procedure is to represent the departure from isotropy in terms of an angular Fourier series and to derive equations governing the temporal relaxation of higher angular harmonics from the test field model. The resulting equations for the anisotropic part of the Reynolds stress tensor are linearized, and examined in some detail both analytically, and for a simple atmospheric spectrum with an enstrophy inertial-range, numerically. It is found that the relaxation back to isotropy is very non-local in wavenumber space, a result seemingly in counter-distinction to three-dimensional turbulence for which the relaxation is supposedly local. The difference is explained by the importance in two-dimensional flows of direct straining of small scales by large scales. Some preliminary direct spectral numerical simulation data in support of these ideas are also presented. Utilizing the linearized version of the theory, we give an estimate of the relaxation rate of the anisotropic part of the total Reynolds stress, similar to that originally given by Rotta for three-dimensional turbulence If the anisotropy is centered in the energy-containing range, we obtain a value for the rate coefficient of ∼0.25(*E*
^{½}/*L*), where *E* is the total kinetic energy, and *L* the turbulence integral scale. The implications of these findings for subgrid-scale parameterization are discussed, and a formalism for describing the evolution of the large scales with parameterized treatment of the small scale is sketched. Two new effects beyond those customarily represented in three-dimensional turbulence theory appear to require attention: a production of subgrid-scale turbulence energy which depends on a certain measure of the excess of (large scale) strain rate over the (large scale) vorticity, and a production of subgrid-scale anisotropy by means of the direct straining by the large scales. Formulas estimating these effects are presented.

## Abstract

Utilizing an abridged version of the test field model, we examine the relaxation of two-dimensional homogeneous turbulence back to its isotropic state. Our procedure is to represent the departure from isotropy in terms of an angular Fourier series and to derive equations governing the temporal relaxation of higher angular harmonics from the test field model. The resulting equations for the anisotropic part of the Reynolds stress tensor are linearized, and examined in some detail both analytically, and for a simple atmospheric spectrum with an enstrophy inertial-range, numerically. It is found that the relaxation back to isotropy is very non-local in wavenumber space, a result seemingly in counter-distinction to three-dimensional turbulence for which the relaxation is supposedly local. The difference is explained by the importance in two-dimensional flows of direct straining of small scales by large scales. Some preliminary direct spectral numerical simulation data in support of these ideas are also presented. Utilizing the linearized version of the theory, we give an estimate of the relaxation rate of the anisotropic part of the total Reynolds stress, similar to that originally given by Rotta for three-dimensional turbulence If the anisotropy is centered in the energy-containing range, we obtain a value for the rate coefficient of ∼0.25(*E*
^{½}/*L*), where *E* is the total kinetic energy, and *L* the turbulence integral scale. The implications of these findings for subgrid-scale parameterization are discussed, and a formalism for describing the evolution of the large scales with parameterized treatment of the small scale is sketched. Two new effects beyond those customarily represented in three-dimensional turbulence theory appear to require attention: a production of subgrid-scale turbulence energy which depends on a certain measure of the excess of (large scale) strain rate over the (large scale) vorticity, and a production of subgrid-scale anisotropy by means of the direct straining by the large scales. Formulas estimating these effects are presented.

## Abstract

An investigation of thermal convection in a thin layer of fluid has recently been reported (Herring, 1963). The calculation included only those nonlinearities having the form of an interaction of a fluctuating quantity with the mean temperature field. In addition, free boundary conditions were employed and the fluctuating velocity and temperature fields were composed of one horizontal wave number, α. In the present paper, the calculation is extended to include the effects associated with rigid boundaries and many horizontal wave numbers.

The results of the multi-α study indicate that the stable steady state solution contains only one α, provided the Rayleigh number, R, is less than ≅10^{6}. Above R≅10^{6}, the stable solution contains at least two &alpha's. The stable single-α solutions have a somewhat different value of α than either that predicted by the maximum heat flux principle of Malkus (1954) or that predicted by the relative stability criterion of Malkus and Veronis (1958). At present, we are not able to characterize the stability of the system by postulating an extremal for some simple property of the flow.

The value of the Nusselt number found here for rigid boundaries is N=0.115R½, for large R. This value for N is within ∼20 per cent of the experimental value for large Prandtl number fluids. The rms values of the velocity and temperature fluctuation fields computed here appear to have the form expected for large Prandtl number fluids. The lack of accurate experimental data prevents us from drawing definite conclusions as to the numerical accuracy for these quantities. The computed mean temperature profile is qualitatively correct, but develops a thin stabilizing region with a stable temperature gradient just exterior to the thermal boundary layer. It is concluded that the stabilizing region represents a self-adjustment in the flow which compensates for the omission of the effects of eddy processes on the equations of motion.

## Abstract

An investigation of thermal convection in a thin layer of fluid has recently been reported (Herring, 1963). The calculation included only those nonlinearities having the form of an interaction of a fluctuating quantity with the mean temperature field. In addition, free boundary conditions were employed and the fluctuating velocity and temperature fields were composed of one horizontal wave number, α. In the present paper, the calculation is extended to include the effects associated with rigid boundaries and many horizontal wave numbers.

The results of the multi-α study indicate that the stable steady state solution contains only one α, provided the Rayleigh number, R, is less than ≅10^{6}. Above R≅10^{6}, the stable solution contains at least two &alpha's. The stable single-α solutions have a somewhat different value of α than either that predicted by the maximum heat flux principle of Malkus (1954) or that predicted by the relative stability criterion of Malkus and Veronis (1958). At present, we are not able to characterize the stability of the system by postulating an extremal for some simple property of the flow.

The value of the Nusselt number found here for rigid boundaries is N=0.115R½, for large R. This value for N is within ∼20 per cent of the experimental value for large Prandtl number fluids. The rms values of the velocity and temperature fluctuation fields computed here appear to have the form expected for large Prandtl number fluids. The lack of accurate experimental data prevents us from drawing definite conclusions as to the numerical accuracy for these quantities. The computed mean temperature profile is qualitatively correct, but develops a thin stabilizing region with a stable temperature gradient just exterior to the thermal boundary layer. It is concluded that the stabilizing region represents a self-adjustment in the flow which compensates for the omission of the effects of eddy processes on the equations of motion.

## Abstract

The evolution of Ertel's potential vorticity (PV) is examined in direct numerical simulations (DNS) of decaying turbulence advecting passive scalars and in a generalized Taylor-Green vortex (TGV). It is noted that although PV itself is advected as a passive scalar, its dissipation occurs over all scales and is not concentrated in the velocity or scalar dissipations range. Thus, attempts to invoke cascade arguments to infer an inertial range for PV variance are vitiated. Moreover, for the TGV it is noted that molecular dissipation can create PV from an initial state for which it is everywhere zero. For the random initial value problem, the DNS results suggest a simple characterization of PV dissipation, which implies that for isotropic turbulence (and small Prandtl numbers) PV decays roughly exponentially on a lime scale ∼ (*L*/*U*
_{rms})*R*
_{λ}
^{½}, *L* being the integral scale, *u*
_{rms} the large rms velocity, and *R*
_{rms}, the microscale Reynolds number. The statistics of PV are also examined, and it is noted that it is far from Gaussian, even at modest values of Reynolds number *R*
_{λ}.

## Abstract

The evolution of Ertel's potential vorticity (PV) is examined in direct numerical simulations (DNS) of decaying turbulence advecting passive scalars and in a generalized Taylor-Green vortex (TGV). It is noted that although PV itself is advected as a passive scalar, its dissipation occurs over all scales and is not concentrated in the velocity or scalar dissipations range. Thus, attempts to invoke cascade arguments to infer an inertial range for PV variance are vitiated. Moreover, for the TGV it is noted that molecular dissipation can create PV from an initial state for which it is everywhere zero. For the random initial value problem, the DNS results suggest a simple characterization of PV dissipation, which implies that for isotropic turbulence (and small Prandtl numbers) PV decays roughly exponentially on a lime scale ∼ (*L*/*U*
_{rms})*R*
_{λ}
^{½}, *L* being the integral scale, *u*
_{rms} the large rms velocity, and *R*
_{rms}, the microscale Reynolds number. The statistics of PV are also examined, and it is noted that it is far from Gaussian, even at modest values of Reynolds number *R*
_{λ}.

## Abstract

Computer simulators are made of the growth of the difference-velocity field for pairs of realizations of isotropic, three-dimensional turbulence at Reynolds number R_{&lambda}≈40. The simulations involve full-scale integration of the Navier-Stokes equation in the Fourier representation. It is found that the difference-velocity variance (error energy) grows with time even when the initial difference-velocity is confined to wave numbers strongly damped by viscosity. The numerical integrations are compared with results of the direct-interaction approximation (DIA). It is found that the DIA gives reasonably satisfactory quantitative agreement for the evolution of the error energy and the error. energy spectrum. What discrepancies there are represent an underestimate of error energy growth by the DIA. This is explained by theoretical analysis of the approximation.

## Abstract

Computer simulators are made of the growth of the difference-velocity field for pairs of realizations of isotropic, three-dimensional turbulence at Reynolds number R_{&lambda}≈40. The simulations involve full-scale integration of the Navier-Stokes equation in the Fourier representation. It is found that the difference-velocity variance (error energy) grows with time even when the initial difference-velocity is confined to wave numbers strongly damped by viscosity. The numerical integrations are compared with results of the direct-interaction approximation (DIA). It is found that the DIA gives reasonably satisfactory quantitative agreement for the evolution of the error energy and the error. energy spectrum. What discrepancies there are represent an underestimate of error energy growth by the DIA. This is explained by theoretical analysis of the approximation.

## Abstract

The goal of the U.S. Climate Resilience Toolkit’s (CRT) Climate Explorer (CE) is to provide information at appropriate spatial and temporal scales to help practitioners gain insights into the risks posed by climate change. Ultimately, these insights can lead to groups of local stakeholders taking action to build their resilience to a changing climate. Using CE, decision-makers can visualize decade-by-decade changes in climate conditions in their county and the magnitude of changes projected for the end of this century under two plausible emissions pathways. They can also check how projected changes relate to user-defined thresholds that represent points at which valued assets may become stressed, damaged, or destroyed. By providing easy access to authoritative information in an elegant interface, the Climate Explorer can help communities recognize—and prepare to avoid or respond to—emerging climate hazards. Another important step in the evolution of CE builds on the purposeful alignment of the CRT with the U.S. Global Change Research Program’s (USGCRP) National Climate Assessment (NCA). By closely linking these two authoritative resources, we envision that users can easily transition from static maps and graphs within NCA reports to dynamic, interactive versions of the same data within CE and other resources within the CRT, which they can explore at higher spatial scales or customize for their own purposes. The provision of consistent climate data and information—a result of collaboration among USGCRP’s federal agencies—will assist decision-making by other governmental entities, nongovernmental organizations, businesses, and individuals.

## Abstract

The goal of the U.S. Climate Resilience Toolkit’s (CRT) Climate Explorer (CE) is to provide information at appropriate spatial and temporal scales to help practitioners gain insights into the risks posed by climate change. Ultimately, these insights can lead to groups of local stakeholders taking action to build their resilience to a changing climate. Using CE, decision-makers can visualize decade-by-decade changes in climate conditions in their county and the magnitude of changes projected for the end of this century under two plausible emissions pathways. They can also check how projected changes relate to user-defined thresholds that represent points at which valued assets may become stressed, damaged, or destroyed. By providing easy access to authoritative information in an elegant interface, the Climate Explorer can help communities recognize—and prepare to avoid or respond to—emerging climate hazards. Another important step in the evolution of CE builds on the purposeful alignment of the CRT with the U.S. Global Change Research Program’s (USGCRP) National Climate Assessment (NCA). By closely linking these two authoritative resources, we envision that users can easily transition from static maps and graphs within NCA reports to dynamic, interactive versions of the same data within CE and other resources within the CRT, which they can explore at higher spatial scales or customize for their own purposes. The provision of consistent climate data and information—a result of collaboration among USGCRP’s federal agencies—will assist decision-making by other governmental entities, nongovernmental organizations, businesses, and individuals.