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## Abstract

An analytical solution is presented for the baroclinic stability problem of a Boussinesq fluid in a β-plane channel with Ekman suction boundary conditions. All of the modes, stable and unstable, belonging to this problem are identified. It is found that an unstable mode exists for only a certain range of values of the Burger number. The value of the Burger number at the upper limit of this range increases as the Ekman number decreases. Beyond this upper limit only a damped mode exists. It is also found that this transition in parameter space from the unstable to the stable mode occurs in a discontinuous manner.

## Abstract

An analytical solution is presented for the baroclinic stability problem of a Boussinesq fluid in a β-plane channel with Ekman suction boundary conditions. All of the modes, stable and unstable, belonging to this problem are identified. It is found that an unstable mode exists for only a certain range of values of the Burger number. The value of the Burger number at the upper limit of this range increases as the Ekman number decreases. Beyond this upper limit only a damped mode exists. It is also found that this transition in parameter space from the unstable to the stable mode occurs in a discontinuous manner.

## Abstract

When a vertical rotating annulus of liquid is subject to a horizontal temperature gradient, provided that the coefficient of kinematical viscosity, ν¯, is not too great and the angular velocity of rotation,Ω is sufficiently high, four distinct regimes of hydrodynamical flow are possible, as shown in previous work by Hide. Only one of these regimes is characterized by symmetry about the axis of rotation.

_{2}≡

*d*/(

*b*−

*a*), Π

_{4}

*gd*Δρ/ρ¯Ω

^{2}(

*b*−

*a*)

^{2}, Π

_{5}≡4Ω

^{2}(

*b*−

*a*)

^{5}/ν¯

^{2}

*d*and Π

_{6}≡ν¯/κ¯, where

*d*is the depth of the fluid,

*b*and

*a*are the radi of curvature of the surfaces of the annulus,

*g*is the acceleration of gravity, ρ¯ is the mean density of the fluid, Δρ is the density contrast associated with the impressed horizontal temperature gradient and κ¯ is the thermal diffusivity of the fluid. In a diagram with log

_{10}Π

_{5}as abscissa and log

_{10}Π

_{4}as ordinate, axisymmetric flow is found outside an anvil-shaped region whose upper boundary lies below the line 11,&=2.0, the flow being symmetric for an 114 when 115 where lis. The lower boundary of the anvil-shaped region is given by the equation.

_{10}

_{4}

_{6}

_{10}

_{2}

_{5}

Theory suggests that the non-axisymmetric flow is a manifestation of “baroclinic instability,” a process first studied in the investigation of planetary-scale atmospheric motions. The experimental results are discussed in terms of Eady' theoretical baroclinic instability model, as extended by others to include effects due to viscous boundary layers. Qualitatively, agreement between theory and experiment is satisfactory, and at the highest values of II attained in the experiments the quantitative agreement is remarkably good. The poor quantitative agreement found at the lowest values of II_{5} attained indicates that a major source of frictional dissipation has not yet been properly accounted for in the theory.

## Abstract

When a vertical rotating annulus of liquid is subject to a horizontal temperature gradient, provided that the coefficient of kinematical viscosity, ν¯, is not too great and the angular velocity of rotation,Ω is sufficiently high, four distinct regimes of hydrodynamical flow are possible, as shown in previous work by Hide. Only one of these regimes is characterized by symmetry about the axis of rotation.

_{2}≡

*d*/(

*b*−

*a*), Π

_{4}

*gd*Δρ/ρ¯Ω

^{2}(

*b*−

*a*)

^{2}, Π

_{5}≡4Ω

^{2}(

*b*−

*a*)

^{5}/ν¯

^{2}

*d*and Π

_{6}≡ν¯/κ¯, where

*d*is the depth of the fluid,

*b*and

*a*are the radi of curvature of the surfaces of the annulus,

*g*is the acceleration of gravity, ρ¯ is the mean density of the fluid, Δρ is the density contrast associated with the impressed horizontal temperature gradient and κ¯ is the thermal diffusivity of the fluid. In a diagram with log

_{10}Π

_{5}as abscissa and log

_{10}Π

_{4}as ordinate, axisymmetric flow is found outside an anvil-shaped region whose upper boundary lies below the line 11,&=2.0, the flow being symmetric for an 114 when 115 where lis. The lower boundary of the anvil-shaped region is given by the equation.

_{10}

_{4}

_{6}

_{10}

_{2}

_{5}

Theory suggests that the non-axisymmetric flow is a manifestation of “baroclinic instability,” a process first studied in the investigation of planetary-scale atmospheric motions. The experimental results are discussed in terms of Eady' theoretical baroclinic instability model, as extended by others to include effects due to viscous boundary layers. Qualitatively, agreement between theory and experiment is satisfactory, and at the highest values of II attained in the experiments the quantitative agreement is remarkably good. The poor quantitative agreement found at the lowest values of II_{5} attained indicates that a major source of frictional dissipation has not yet been properly accounted for in the theory.

## Abstract

Thermally driven flows in rotating laboratory containers with cylindrical geometry can he axially symmetric or they can be wavelike, depending on experimental parameters. In the traditional *regime diagram* of thermal Rossby number versus Taylor number the region of axially symmetric motion is separated from the regime of wavelike motion by a knee-shaped boundary. The simplest theoretical model that predicts the shape of this curve is due to Barcilon (1964) and consists of the Eady model of baroclinic instability applied to a rotating channel with Ekman layers at the top and bottom. Anticipating that rotating fluid experiments might soon be done in spherical shell geometry, we have extended Barcilon's model to a beta-plane channel. The purpose of our study is to predict with a simple model the changes which the beta-effect should produce in the shape and position of the boundary separating the regions of axially symmetric and wavelike motion.

## Abstract

Thermally driven flows in rotating laboratory containers with cylindrical geometry can he axially symmetric or they can be wavelike, depending on experimental parameters. In the traditional *regime diagram* of thermal Rossby number versus Taylor number the region of axially symmetric motion is separated from the regime of wavelike motion by a knee-shaped boundary. The simplest theoretical model that predicts the shape of this curve is due to Barcilon (1964) and consists of the Eady model of baroclinic instability applied to a rotating channel with Ekman layers at the top and bottom. Anticipating that rotating fluid experiments might soon be done in spherical shell geometry, we have extended Barcilon's model to a beta-plane channel. The purpose of our study is to predict with a simple model the changes which the beta-effect should produce in the shape and position of the boundary separating the regions of axially symmetric and wavelike motion.

## Abstract

Wave-amplitude vacillation in a thermally driven, rotating cylindrical annulus of fluid is studied using temperature and speed data from a 98-probe network of tiny bead thermistors suspended in the fluid. Mid-depth synoptic charts and estimates of heat and momentum fluxes, spectral characteristics and energetics are presented as a function of time. Relationships to time-dependent atmospheric phenomena are discussed. A section is included which deals with the effects of probes.

## Abstract

Wave-amplitude vacillation in a thermally driven, rotating cylindrical annulus of fluid is studied using temperature and speed data from a 98-probe network of tiny bead thermistors suspended in the fluid. Mid-depth synoptic charts and estimates of heat and momentum fluxes, spectral characteristics and energetics are presented as a function of time. Relationships to time-dependent atmospheric phenomena are discussed. A section is included which deals with the effects of probes.

## Abstract

A Stability analysis of a thin horizontal rotating fluid layer which is subjected to arbitrary horizontal and vertical temperature gradients is presented. The basic state is a nonlinear Hadley cell which contains both Ekman and thermal boundary layers; it is given in closed form. The stability analysis is based on the linearized Navier-Stokes equations, and zonally symmetric perturbations in the form of waves propagating in the meridional direction are considered. Numerical methods were used for the stability problem. The objective of this investigation was to extend previous work on symmetric baroclinic instability with a more realistic model. Hence, the study deals with flows for which the Richardson number (based on temperature and flow gradients at mid-depth) is of order unity and less. The computations cover ranges of Prandtl number 0.2 ≤ σ ≤ 5, Rossby number 10^{−2} ≤ ε ≤ 10^{2} and Ekman number 10^{−4} ≤ E ≤ 10^{−1}. It was found, in agreement with previous work, that the instability sets in when the Richardson number is close to unity and that the critical Richardson number is a non-monotonic function of the Prandtl number. Further, it was found that the critical Richardson number decreases with increasing Ekman number until a critical value of the Ekman number is reached beyond which the fluid is stable. The principal of exchange of stability was not assumed and growth rates wore calculated. A wavelength of maximum growth rate was found. For our model overstability was not found. Some computations were performed for Richardson numbers less than zero. No discontinuities in growth rates are noticeable when the Richardson number changes sign. This result indicates a smooth transition from symmetric baroclinic instability to a convective instability.

## Abstract

A Stability analysis of a thin horizontal rotating fluid layer which is subjected to arbitrary horizontal and vertical temperature gradients is presented. The basic state is a nonlinear Hadley cell which contains both Ekman and thermal boundary layers; it is given in closed form. The stability analysis is based on the linearized Navier-Stokes equations, and zonally symmetric perturbations in the form of waves propagating in the meridional direction are considered. Numerical methods were used for the stability problem. The objective of this investigation was to extend previous work on symmetric baroclinic instability with a more realistic model. Hence, the study deals with flows for which the Richardson number (based on temperature and flow gradients at mid-depth) is of order unity and less. The computations cover ranges of Prandtl number 0.2 ≤ σ ≤ 5, Rossby number 10^{−2} ≤ ε ≤ 10^{2} and Ekman number 10^{−4} ≤ E ≤ 10^{−1}. It was found, in agreement with previous work, that the instability sets in when the Richardson number is close to unity and that the critical Richardson number is a non-monotonic function of the Prandtl number. Further, it was found that the critical Richardson number decreases with increasing Ekman number until a critical value of the Ekman number is reached beyond which the fluid is stable. The principal of exchange of stability was not assumed and growth rates wore calculated. A wavelength of maximum growth rate was found. For our model overstability was not found. Some computations were performed for Richardson numbers less than zero. No discontinuities in growth rates are noticeable when the Richardson number changes sign. This result indicates a smooth transition from symmetric baroclinic instability to a convective instability.

## Abstract

The stability of a thin fluid layer between two rotating plates which are subjected to a horizontal temperature gradient is studied. First, the solution for the stationary basic state is obtained in a closed form. This solution identifies Ekman and thermal layers adjacent to the plates and interior temperature and velocity fields which are almost linear functions of height. Then the stability of that basic state with respect to infinitesimal zonal waves is analyzed via the solution of the complete viscous linear equations for the perturbations. The character of the growth rates is found to be similar to the growth rates of the classical baroclinic waves. The neutral stability curves for these waves possessed a knee in the Rossby-Taylor number plane to the left of which all perturbations are stable. The region of instability is found to depend on the Prandtl number, the vertical stratification parameter, and both the meridional and zonal wavenumbers. It is found in general that the flow is unstable for small enough Ekman numbers and for Rossby numbers less than 10. It is also found that increased vertical stable stratification and increased Prandtl number stabilize the flow.

## Abstract

The stability of a thin fluid layer between two rotating plates which are subjected to a horizontal temperature gradient is studied. First, the solution for the stationary basic state is obtained in a closed form. This solution identifies Ekman and thermal layers adjacent to the plates and interior temperature and velocity fields which are almost linear functions of height. Then the stability of that basic state with respect to infinitesimal zonal waves is analyzed via the solution of the complete viscous linear equations for the perturbations. The character of the growth rates is found to be similar to the growth rates of the classical baroclinic waves. The neutral stability curves for these waves possessed a knee in the Rossby-Taylor number plane to the left of which all perturbations are stable. The region of instability is found to depend on the Prandtl number, the vertical stratification parameter, and both the meridional and zonal wavenumbers. It is found in general that the flow is unstable for small enough Ekman numbers and for Rossby numbers less than 10. It is also found that increased vertical stable stratification and increased Prandtl number stabilize the flow.

## Abstract

During a traverse of the regular wave regime in a rotating, differentially heated annulus of viscous liquid (Prandtl number = 86.0), an asymmetrical wave pattern was observed in the transition region between 4 and 5 regular waves. The shape and time development of this pattern is interpreted as being primarily the result of the simultaneous presence of a 4-wave and a 5-wave pattern, and of the dispersion resulting from the relative motion of these two waves.

## Abstract

During a traverse of the regular wave regime in a rotating, differentially heated annulus of viscous liquid (Prandtl number = 86.0), an asymmetrical wave pattern was observed in the transition region between 4 and 5 regular waves. The shape and time development of this pattern is interpreted as being primarily the result of the simultaneous presence of a 4-wave and a 5-wave pattern, and of the dispersion resulting from the relative motion of these two waves.

## Abstract

The radial distribution of the azimuthally averaged temperature at one depth in a rotating, differentially heated annulus of fluid undergoing “amplitude vacillation” is measured with an array of 48 thermocouples suspended on fine wires in the fluid. The properties of amplitude vacillation and the feasibility of making measurements with multi-probe arrays (properly staggered in the fluid to avoid disrupting the organized flow) are discussed.

## Abstract

The radial distribution of the azimuthally averaged temperature at one depth in a rotating, differentially heated annulus of fluid undergoing “amplitude vacillation” is measured with an array of 48 thermocouples suspended on fine wires in the fluid. The properties of amplitude vacillation and the feasibility of making measurements with multi-probe arrays (properly staggered in the fluid to avoid disrupting the organized flow) are discussed.