Small Rotation Rates Research Articles

Mean zonal flows excited by critical thermal convection in rotating spherical shells

Mean zonal flows excited by critical modes of Boussinesq convection in rotating spherical shells are evaluated systematically by the weak nonlinear method. Calculations are executed in the ranges of the Prandtl number 0.01 ≤ P ≤ 102, the Taylor number 100 ≤ T ≤ 107, and the radius ratio 0.2 ≤ η ≤ 0.8. In the case of small rotation rates, two new states of zonal flow at the equatorial outer boundary are obtained; westerly flow emerges even when the Prandtl number is large, and easterly flow emerges even when the Prandtl number is small. The latter case is interesting since the induced equatorial easterly is in the opposite direction to that of the acceleration. In the case of large rotation rates, it is confirmed that the direction of mean zonal flow at the equatorial outer boundary does not depend on the thickness of the shell.

Electrochemical behaviour of sodium dithionite at a platinum electrode and determination of diffusion coefficient in alkaline solution

The electrochemical behaviour of sodium dithionite at a platinum electrode has been investigated in this paper. Two oxidation waves of sodium dithionite were observed with formation of sulfite as a stable intermediate species. It has been found that the oxidation of sodium dithionite to sulfite, according to the first wave, is controlled by the rate of mass transport, the oxidation of sulfite to sulfate, however, is not purely controlled by transport phenomena. Using the ratio of the limiting currents of both oxidation waves it was possible to determine the diffusion coefficient of sodium dithionite by extrapolation to the small rotation rate of the platinum electrode. From the obtained results it is also advisable to make use of the first oxidation reaction for analytical purposes.

Simulations of the Atmospheres of Synchronously Rotating Terrestrial Planets Orbiting M Dwarfs: Conditions for Atmospheric Collapse and the Implications for Habitability

Planets within the habitable zones of M dwarfs are likely to be synchronous rotators; in other words, one side is permanently illuminated while the other side is in perpetual darkness. We present results of three-dimensional simulations of the atmospheres of such planets, and comment on their possible habitability. Near the ground, a thermally direct longitudinal cell exists, transporting heat from the dayside to the nightside. The circulation is three-dimensional, with low-level winds returning mass to the dayside across the polar regions. Aloft, the zonally averaged winds display a pattern of strong superrotation due to these planets' finite (albeit small) rotation rate. With terrestrial values of insolation, a CO2/H2O atmosphere collapses, or condenses on the surface of the darkside, when surface pressure is approximately 30 mb, this value being much lower for a N2atmosphere. This temperature contrast is also sensitive to factors such as gravity, planetary radius, and IR optical depth τ. These results question the suitability of the concept of a habitable zone around M dwarfs that is independent of planetary parameters. If CO2partial pressure is controlled by the carbonate–silicate cycle, we find that these planets should have a minimum surface pressure of 1000–1500 mb of CO2, as this is the minimum pressure needed to support stable liquid water on the darkside at the inner edge of the habitable zone. We finally conclude that planets orbiting M stars can support atmospheres over a large range of conditions and, despite constraints such as stellar activity, are very likely to be habitable.

On convective turbulence and the influence of rotation

A series of experiments were performed in a rotating cylindrical tank over a wide range of rotation rates in which convective turbulence was generated by a bottom-mounted heated plate in both homogeneous and stratified fluids. Measurements were made of the turbulent velocities in all three axes over the full depth of the chamber, and of the temperatures at the mid-depth near the centre of the tank. For even small rotation rates, the measurements showed that the turbulent velocities were weakly affected by rotation at all depths, but as the rotation rate increased, the deviation from the non-rotational scaling slowly and progressively increased until eventually the turbulent velocities were fully rotationally controlled. The results indicated that there was no sudden transition of the turbulent field from the non-rotational state (a function only of the surface buoyancy flux B and the depth z ) to the rotational state (where the strength of the turbulent field is a function of only B and the Coriolis parameter f ). Rather the transition was a smooth asymptotic one from one state to the other. Nevertheless, it was possible to parametrize this transition by a single value of the turbulent or small scale Rossby number, defined by Ro = (B/f 3 z 2 ) 1 3 . Our measurements suggested a critical value of Ro c ≈ 0.1, below which the turbulence was fully rotationally controlled and which was equivalent to a critical depth z c = (35 ± 15)(B/f 3 ) 1 2 . Using typical oceanic values for B and f , the oceanic turbulence driven by surface cooling events becomes rotationally controlled only for depths greater than about 10 km, a depth which is greater than that of the bulk of the world's oceans. Thus, convective turbulence actively being generated by cooling of the ocean surface is best described by non-rotating turbulent velocity and length scales and is a function only of the surface buoyancy flux and the depth.

The effect of rotation on double-diffusive convection in a laterally heated vertical slot

The effect of rotation about a vertical axis on the linear stability of a salt-stratified fluid enclosed in a vertical slot when subjected to a temperature difference between the walls is investigated. It is found that for large salinity stratifications there are three distinct regimes of instability for different values of the rotation rate. For small rotation rates the convection cells resemble the thin, almost flat, convection cells predicted by non-rotating theory. The effect of the rotation is to marginally destabilize the fluid whilst inducing a small slope in the convection cells along the parallel walls. As the rotation rate is increased there is an abrupt change in the form of the most unstable convection cells as their aspect ratio and slope parallel to the walls both become of order one in magnitude. As the rotation rate is further increased there is a second less abrupt transition when the internal Rossby radius of deformation based on the vertical scale of the cells becomes of the same order of size as the slot width. After this point the slope of the cells increases in proportion to the rotation rate. The asymptotic nature of these three regimes is found.The effect of rotation on double-diffusive instabilities caused by more general horizontal temperature and salinity gradients in a salt-stratified fluid is also investigated, with particular reference to the case of heating the salinity gradient from a single sidewall. This analysis is restricted to the case where the rotation rate is low.

The effect of rotation on double-diffusive convection in a laterally heated vertical slot

The effect of rotation about a vertical axis on the linear stability of a salt-stratified fluid enclosed in a vertical slot when subjected to a temperature difference between the walls is investigated. It is found that for large salinity stratifications there are three distinct regimes of instability for different values of the rotation rate. For small rotation rates the convection cells resemble the thin, almost flat, convection cells predicted by non-rotating theory. The effect of the rotation is to marginally destabilize the fluid whilst inducing a small slope in the convection cells along the parallel walls. As the rotation rate is increased there is an abrupt change in the form of the most unstable convection cells as their aspect ratio and slope parallel to the walls both become of order one in magnitude. As the rotation rate is further increased there is a second less abrupt transition when the internal Rossby radius of deformation based on the vertical scale of the cells becomes of the same order of size as the slot width. After this point the slope of the cells increases in proportion to the rotation rate. The asymptotic nature of these three regimes is found.The effect of rotation on double-diffusive instabilities caused by more general horizontal temperature and salinity gradients in a salt-stratified fluid is also investigated, with particular reference to the case of heating the salinity gradient from a single sidewall. This analysis is restricted to the case where the rotation rate is low.

The configurations of a FENE bead-spring chain in transient rheological flows and in a turbulent flow

The changes in the configuration of a FENE bead-spring chain in a direct numerical simulation of turbulent channel flow and in some simple rheological flows are examined. Unraveling occurs both in uniaxial and shear flows, but the uniaxial flow is more effective. A vortex with a large rotation rate perpendicular to the principal strain of a uniaxial flow has only a minor retarding effect while a small rotation rate delays the unraveling substantially. In a turbulent flow, the chain unravels the most in the viscous sublayer, to about 90% of its fully extended length. It aligns at a 7° angle with the direction of mean flow. In the buffer zone, it unravels and coils up and takes different orientations at different times. Outside the wall region, the chain assumes a coiled configuration. The unraveling of the chain strongly depends on the relaxation time of the chain normalized with the wall shear rate, λ+. A value of λ+=10 exhibits strong unraveling while very weak unraveling is observed below λ+=1.

Rotating Rayleigh-Bénard convection: Küppers-Lortz transition

Rayleigh-Bénard convection with rotation about a vertical axis is investigated for small dimensionless rotation rates 0<Ω<50. The convection cell is cylindrical with aspect ratio Γ = 10 and the convecting fluid is water with a Prandtl number of 6.8 at T = 23.8° C. Comparisons are made between experimental data and linear stability theory for the onset Rayleigh number and for the wavenumber dependence of the convective pattern. The nonlinear Küppers-Lortz transition is found to occur significantly below the theoretically expected rotation rate Ω c and to be nucleated by defects created at the lateral cell walls.

Stability and chaotic advection in the eccentric Taylor problem

Theoretically, Lagrangian chaos or chaotic advection can occur by forcing a steady, two-dimensional velocity field with a time-periodic perturbation. This idea has recently been confirmed experimentally by Chaiken et al.1 and Chien et al.2 In principle, chaotic advection should also occur in a three-dimensional steady flow. To investigate this problem we constructed an eccentric Taylor–Couette apparatus with a rotating inner cylinder and a stationary outer cylinder. We obtain a 3-D steady flow in the following way. We first create a 2-D velocity field at small rotation rates of the inner cylinder by imposing a sufficiently large ecccentricity to obtain separated flow. (That is, an eddy in the region of largest gap is created by the separation of the fluid from the outer boundary and its reattachment downstream.) The inner cylinder rotation rate is then increased until Taylor vortices appear. The Taylor vortices modify the separated flow but do not destroy it. It is in this mixed Taylor vortex-separated flow regime that we carry out our studies of chaotic advection. On account of mathematical difficulties there exist few theoretical or experimental studies of stability of the flows in the eccentric geometry. We have therefore conducted numerical and laboratory experiments to identify the regions of parameter space where 3-D steady flow exists and where a transition of time dependence (i.e., 4-D flows) occurs. In our numerical work we used a commercial computational fluid dynamics program and we will report on our assessment of its accuracy in predicting the three-dimensional flow features observed in the experiments and its potential for investigating chaotic advection. In particular, we use it to find the ‘‘skeleton’’ of the flow and the fluxes between the associated regions of different flow type that it defines, and compare these with experimental observations.

Steady-state coating flows inside a rotating horizontal cylinder

Thin coating flows inside a rotating circular cylinder are investigated when the axis of rotation is perpendicular to the direction of gravity. Attention is restricted to flows of power-law fluids having negligible inertia. Four distinct steady-state liquid-film profiles are found to be possible. Two of the cases correspond to a continuous coating, i.e. films that cover the entire inner surface of the cylinder. The other two cases involve partial films covering a limited portion of the cylinder surface. Of the two continuous films, one is the expected configuration involving a coating that gradually changes in thickness as one moves around the cylinder, the film being thicker on the ascending portion of the cylinder and thinner on the descending portion. The second continuous-film configuration has regions on the rising side of cylinder where a rapid change in depth is possible. This case also has the potential to have recirculating zones where a portion of the fluid is trapped in either one or two eddies at fixed locations on the rising side of the cylinder. Of the two partial films, one corresponds to a weakly deformed puddle at the bottom of the cylinder and is the appropriate solution at small rotation rates. The second partial film is a film which coats a portion of the ascending side of the cylinder, the extent of which depends on the film volume.

Ellipsometric measurement of magnetooptical nonreciprocal effects

To detect small rotation rates with a ring laser the two counter-propagating beams must be uncoupled. This lock-off can be realized using nonreciprocal optical effects yielding different optical path lengths for the two waves. Usually magnetooptical effects are employed for nonreciprocity. To measure, instead of calculating from optical constants, the magnitude of this nonreciprocal effects an ellipsometric method is proposed. Lock-off elements using Faraday or Kerr effect (polar and transverse, respectively) are treated. First measurements were made on FeNi films.

Time-dependent viscous flow past an impulsively started rotating and translating circular cylinder

A numerical study is made of the development with time of the two-dimensional flow of a viscous, incompressible fluid around a circular cylinder which suddenly starts rotating about its axis with constant angular velocity and translating at right angles to this axis with constant speed. The governing partial differential equations in two space variables and time are reduced to sets of time-dependent equations in one space variable by means of Fourier analysis. By truncating the Fourier series to a finite number of terms, a finite set of differential equations is solved to give an approximation to the theoretical flow. The solutions are obtained by numerical methods. Results are given for the initial development with time of the asymmetrical wake at the rear of the cylinder at Reynolds numbers R [ges ] 200, based on the diameter of cylinder, and at small rotation rates. The detailed results show the formation of a Kármán vortex street. The time development of this separated flow is compared in detail at R = 200 with recent experimental results. The details of the formation and movement of the vortices behind the cylinder and the velocity profiles in several locations are virtually identical in the experimental and theoretical studies. The variations with time of the lift, drag and moment exerted by the fluid on the cylinder are determined both by calculations and by means of approximate analytical expressions. The agreement between these results at small times is excellent.

Beam width correction in high precision Sagnac interferometry

The effect of finite beam width on the Sagnac frequency shift is calculated using a general relativistic analysis. For small rotation rates, the beam width effect dominates over lowest order special relativistic corrections, and is a possible source of error in precision measurements.

Forced axial flow between rotating concentric cylinders

Forced axial flow in an annular gap of a cylindrical rotor is investigated analytically and experimentally. At small rotation rates and narrow gap widths, the axial flow is a simple Poiseuille flow over most of the rotor. The distance required for this Poiseuille flow to get established is estimated. An instability is observed at large rotation rates with certain input geometries.

Hydromagnetic planetary waves

A study is made of hydromagnetic oscillations in a rotating fluid sphere. The basic state is chosen as a uniform current parallel to the axis of rotation. It is found that the non-dissipative normal modes are described by a modified form of the Poincaré eigenvalue problem. For small rotation rates, the lowest non-axisymmetric modes are unstable. For rotation rates of geophysical interest all normal modes are stable. The introduction of ohmic dissipation leads to a hydro-magnetic boundary-layer problem. Solutions for the boundary layer are outlined indicating its role in altering the free periods, damping the oscillations and producing external magnetic fields. Dispersion relations are derived which establish that the zonal phase velocities of both ‘fast’ hydrodynamic and ‘slow’ hydro-magnetic waves can be of either sign. Observations of the secular variations of the earth's magnetic field indicate motion primarily towards the west. A mechanism for the selective excitation of the observed motion is discussed.