Turbulent transport of angular momentum in magnetized fluids and torsional oscillations of the sun

Abstract

Fluxes of angular momentum produced by turbulence in rotating fluids are derived with the effects of a magnetic field included. It is assumed that the rotation is slow but that the magnetic field is of arbitrary strength. A mean magnetic field is shown to produce qualitative changes of the sources of the differential rotation rather than the quenching of differential rotation usually expected. A new equatorward flux of angular momentum arises through the influence of the toroidal magnetic field. The possibility of interpreting the torsional oscillations of the Sun as a consequence of the magnetic perturbations of the turbulent angular momentum fluxes is discussed.     

Nonlinear diffusive instabilities in differentially rotating stars

Abstract

Angular momentum driven instabilities in a stratified differentially rotating star are investigated. In the strong buoyancy limit axisymmetric instabilities of the Goldreich-Schubert type are the most important. A detailed discussion of the linear and small amplitude theories at an arbitrary latitude is given. The bifurcation to finite amplitude steady modes is typically transcritical, and occurs whenever the angular momentum or its gradient is neither parallel not perpendicular to local gravity. Such misalignments enhance the time scale for transport of angular momentum by the Goldreich-Schubert instability. Depending on the turbulent viscosity produced by secondary shear instabilities time scales as short as the Kelvin-Helmholtz time scale are possible.     

A laboratory demonstration of angular momentum mixing

Abstract

It is successfully demonstrated that substantial redistribution of the angular momentum within a completely liquid-filled cylinder in uniform rotation can be brought about by the induction of turbulent mixing through the resonant excitation of standing inertial waves. This means of mixing is accomplished without significant net circulation in the meridional plane, or strong boundary restraint.

Intense cyclonic vortices are created with an apparently high conversion of energy from the inertial wave excited. Visualizations and measurements of vortex strength and circulation distribution are presented and dimensional arguments are applied to interpret from the measurements the partition of the turbulence into relative velocity- and angular momentum-diffusing elements. This indicates tentatively the mechanism responsible; momentum advected by the inertial wave is irreversibly diffused by turbulence of smaller scale. Anisotropy with enhanced radial transport is an essential feature of the nett turbulence in such a mechanism. Similar combinations of large-scale waves and turbulence can be expected to occur in the geophysical situations to which the phenomenon of angular momentum mixing relates. The experiment does not, however, test the effectiveness of isotropic turbulence in the same rôle.     

An experimental study of global instabilities due to the tidal (elliptical) distortion of a rotating elastic cylinder

Abstract

Broad band secondary instability of elliptical vortex motion has been proposed as a principal source of shear-flow turbulence. Here experiments on such instability in an elliptical flow with no shear boundary layer are described. This is made possible by the mechanical distortion in the laboratory frame of a rotating fluid-filled elastic cylinder. One percent ellipticity of a 10 cm diameter cylinder rotating once each second can give rise to an exponentially-growing mode stationary in the laboratory frame. In first order this mode is a sub-harmonic parametric Faraday instability. The finite-amplitude equations represent angular momentum transfer on an inertial time scale due to Reynolds stresses. The growth of this mode is not limited by boundary friction but by detuning and centrifugal stabilization. On average, a generalized Richardson number achieves a marginal value through much of the evolved flow. However, the characteristic flow is intermittent with the cycle: rapid growth, stabilizing momentum transfer from the mean flow, interior re-spin up, and then again. Data is presented in which, at large Reynolds numbers, seven percent ellipticity causes a fifty percent reduction in the kinetic energy of the rotating fluid. In the geophysical setting, this tidal instability in the earth's interior could be inhibited by sub-adiabatic temperature gradients. A near adiabatic region greater than 10 km in height would permit the growth of tidally destabilized modes and the release of energy to three-dimensional disturbances. Such disturbances might play a central role in the geodynamo and add significantly to overall tidal dissipation.     

On the mean atmospheric circulation over Antarctica

Abstract

The south-easterly surface flow down the slopes of Antarctica induces a transfer of westerly angular momentum to the atmosphere, which must be removed from the Antarctic domain by atmospheric transports. It is suggested that synoptic eddies protruding from the northern baroclinic zone into the polar regions are modified by the topography such that they are able to perform these meridional transports. A simple linear two-layer model of the axisymmetric circulation of Antarctica is presented where the eddy effects are incorporated via a K-ansatz. It is shown that qualitatively realistic mean flow patterns can be obtained with this model. The limitations of this approach are exposed.     

The vortex created by mass transfer between layers of a rotating fluid

Abstract

A simple way to model stratification of the ocean or atmosphere is in terms of two superposed homogeneous layers of different density. Effects of cooling of the upper layer, such as that which occurs during bottom-water formation in the ocean, can be simulated by mass transfer from the upper layer to the lower layer. A model is constructed to see What effect such a mass transfer has on the flow when the mass transfer is confined to a limited region. The main effects are (i) doming of the interface, which maintains pressure gradients in balance with the velocity field, (ii) cyclonic rotation in the upper layer due to conservation of angular momentum of particles king drawn toward the sink, yet anticyclonic vorticity for those particles outside the mass transfer region due to shrinking of vortex lines drawn up over the dome. (iii) generally anticyclonic rotation in the lower layer due to particles tending to maintain their angular momentum while being pushed outwards, but some cyclonic rotation near the centre of mass transfer, due to momentum transfer from the upper layer. Similar effects to these are seen in the Greenland Sea where bottom water formation occurs. Results of the same sort are also found in a laboratory model of the process.     

A mechanism for angular momentum mixing

Abstract

If a contained homogeneous, rotating fluid is forced near to a resonance for elastoid-inertia waves, strong vortices are observed to form. Numerical experiments reported here lend support to the explanation that these are due to a redistribution of the angular momentum by the waves. If the waves grow until the Angular momentum gradient is overturned somewhere, turbulent mixing there make the redistribution irreversible, resulting in a vortex. The process is analogous to the formation of steps in a stratified fluid by breaking internal waves.     

On the excitation of short-term variations in the length of the day and polar motion

Variations in the distribution of mass within the atmosphere, and changes in the pattern of winds produce fluctuations in all three components of the angular momentum of the atmosphere on time-scales upwards of a few days. It, has been shown that variations in theaxial component of atmospheric angular momentum during the Special Observing Periods in the recent First GARP Global Experiment (FGGE, where GARP is the Global Atmospheric Research Programme) are well correlated with short-term changes in the length of the day. They are consistent with the total angular momentum of the atmosphere and solid Earth being conserved on short timescales (allowing for lunar and solar effects), without requiring significant angular momentum transfer between the Earth's liquid core and solid mantle on timescales of weeks or months. It has also been shown that fluctuations, in the equatorial components of atmospheric angular momentum make a major contribution to the observed wobble of the instantaneous pole of the Earth's rotation with respect to the Earth's crust. A necessary step in the investigation was a re-examination of the underlying theory of non-rigid body rotational dynamics and angular momentum exchange between the atmosphere and solid Earth. Since only viscous or topographic coupling between the atmosphere and solid Earth can transfer angular momentum, no atmospheric flow that everywhere satisfied inviscid equations (including, but not solely, geostrophic flow) could affect the rotation of a spherical solid Earth. New effective angular momentum functions were introduced in order to exploit the available data and allow for rotational and surface loading deformation of the Earth. A theoretical basis has now been established for future routine determinations of atmopheric, angular momentum fluctuations for the purpose of meteorological and geophysical research, including the assessment of the extent to which movements in the solid Earth associated with very large earthquakes contribute to the excitation of the Chandlerian wobble.     

Magnetic Rossby waves in a stably stratified layer near the surface of the Earth's outer core

Abstract

The stratification profile of the Earth's magnetofluid outer core is unknown, but there have been suggestions that its upper part may be stably stratified. Braginsky (1984) suggested that the magnetic analog of Rossby (planetary) waves in this stable layer (the ‘H’ layer) may be responsible for a portion of the short-period secular variation. In this study, we adopt a thin shell model to examine the dynamics of the H layer. The stable stratification justifies the thin-layer approximations, which greatly simplify the analysis. The governing equations are then the Laplace's tidal equations modified by the Lorentz force terms, and the magnetic induction equation. We linearize the Lorentz force in the Laplace's tidal equations and the advection term in the magnetic induction equation, assuming a zeroth order dipole field as representative of the magnetic field near the insulating core-mantle boundary. An analytical β-plane solution shows that a magnetic field can release the equatorial trapping that non-magnetic Rossby waves exhibit. A numerical solution to the full spherical equations confirms that a sufficiently strong magnetic field can break the equatorial waveguide. Both solutions are highly dissipative, which is a consequence of our necessary neglect of the induction term in comparison with the advection and diffusion terms in the magnetic induction equation in the thin-layer limit. However, were one to relax the thin-layer approximations and allow a radial dependence of the solutions, one would find magnetic Rossby waves less damped (through the inclusion of the induction term). For the magnetic field strength appropriate for the H layer, the real parts of the eigenfrequencies do not change appreciably from their non-magnetic values. We estimate a phase velocity of the lowest modes that is rather rapid compared with the core fluid speed typically presumed from the secular variation.     

Accelerating shear velocity in gravel-bed channels

Abstract

The behaviour of the shear velocity along a gravel-bed channel is investigated experimentally in the presence of a negative pressure gradient (accelerating flow). Different methods of estimation of the shear velocity, derived from vertical profiles of the mean longitudinal point velocity, are examined and a new method is proposed. Results show that the proposed method of estimation is comparable to the St Venant and Clauser's methods. At a specific cross section, for constant bottom slope and relative roughness, shear velocity increases with discharge.     

Nonlinear internal waves in ideal rotating basins

Abstract

Starting from Euler's equations of motion a nonlinear model for internal waves in fluids is developed by an appropriate scaling and a vertical integration over two layers of different but constant density. The model allows the barotropic and the first baroclinic mode to be calculated. In addition to the nonlinear advective terms dispersion and Coriolis force due to the Earth's rotation are taken into account. The model equations are solved numerically by an implicit finite difference scheme. In this paper we discuss the results for ideal basins: the effects of nonlinear terms, dispersion and Coriolis force, the mechanism of wind forcing, the evolution of Kelvin waves and the corresponding transport of particles and, finally, wave propagation over variable topography. First applications to Lake Constance are shown, but a detailed analysis is deferred to a second paper [Bauer et al. (1994)].     

Turbulent transport of angular momentum and differential rotation

Abstract

With the help of simplifying approximations, we have derived expressions for the non-diffusive fluxes of the angular momentum which are brought about by the action of Coriolis forces on the convective motion. The original turbulence, which is not perturbed by the Coriolis forces, is considered given and weakly anisotropic, the anisotropy having a preferred radial direction. The eddy viscosities are evaluated. Hence, a closed equation for the angular velocity is derived, and then solved for the case of slow rotation. It is shown that the differential rotation is generated most effectively in the case of moderate rotation when the Rossby number is of order unity. At small Rossby numbers, the rotation differentiality is inhibited. A negative eddy viscosity is suggested for the case of rapid rotation. Some implications for the Sun and other astrophysical objects are discussed.     

General circulation of the solar atmosphere from observational evidence

Summary One of the main results of the rotating cylinder experiments ofFultz andHide is that the general flow regime in them is essentially determined by the ratio of the angular velocity of the fluid motions (relative to the cylinder) to that of the cylinder itself. Extending these results to the atmosphere of the sun, leads to the hypothesis that the layer in which spots are imbedded should exhibit a non-axially symmetric pattern, of theRossby type.The fluid motions, characteristic of such a general circulation pattern, are mainly along spherical surfaces, and have a wavelike (eddy) appearance similar to the planetary waves in the upper troposphere of the terrestrial atmosphere. These eddies transport momentum along these spherical surfaces from regions of relatively lower angular velocity to regions of higher velocity. Tracers (e.g., sunspots) imbedded is such a flow would show a correlation between their proper motions in latitude and longitude, such that spots moving equatorward will tend to have larger longitudinal motions (toward the west limb), and vice versa.Analysis of ten years (1935 to 1944) of Greenwich spot data shows a consistent, and (statistically) very significant correlation of spot group proper motions, in the proper sense. These results provide strong support for the existence of large-scale waves which are some modest fraction of the solar circumference, but larger than the sunspot groups. Moreover, these waves transport angular momentum (up the gradient of angular velocity) toward the equatorial regions from higher latitudes across at least the entire sunspot zone. It is not known, however, whether these eddies are the primary (or only) source of momentum to maintain the equatorial acceleration of the sun. However, if this source were shut off, and all other processes continued unabated, this layer of the sun between latitudes ±20° would reach solid rotation in about 51/2 rotations.Because this eddy transport of momentum is counter to the gradient of angular velocity, there is an implied transformation of the kinetic energy of the eddies into the kinetic energy of the mean east-west flow. Of possibly even more interest, however, might be the possibility of transfers of kinetic energy between eddies of all different scale sizes extending down the entire spectrum to include sunspot groups and the spots themselves. Moreover, some eddy size(s) in this layer is likely to be primarily responsible for a conversion of potential to kinetic energy.A result of subsidiary interest is the systematically higher value of solar rotation (at all latitudes) derivable from this data, which includes all spots which survive for at least two days. In contrast to the work of previous authors who used only long-lived spots, the result obtained when many small spots are used, indicates perhaps a variation of the rotation rate with height in the solar atmosphere.The results provide no evidence to indicate the existence of significant meridional circulations (latitudinal driffs).     

On a general fluid dynamical theory of discrete unstable spiral modes in disk-shaped galaxies

Abstract

A general fluid dynamical theory of discrete unstable spiral modes in disk-shaped galaxies is described. This formulation of modes includes a radiation boundary condition and an exact numerical treatment of the Poisson equation. Thus, the modes are maintained by an outward transport of angular momentum, but they may be composed of both leading and trailing waves. A numerical scheme based on this formulation is described, and examples of modes obtained with this scheme are presented. These examples compare favorably with calculations based on the original asymptotic theory of Bertin, Lau, Lin and Mark. The implications of the present formulation of modes in galactic models support the hypothesis of a quasi-stationary spiral structure.     

Reynolds stresses and differential rotation. I. On recent calculations of zonal fluxes in slowly rotating stars

Abstract

In order to derive general zonal flux expressions this paper deals with the influence of slow (Ωτ<1) rotation on highly anisotropic stochastic motions. The radial and latitudinal fluxes of angular momentum are derived and depend on two spectral functions. The results are applied to the often used example of spatially periodic correlations, i.e., to one-mode spectra. The zonal fluxes in this case can be expressed as combinations of the intensities of the three velocity components and the wave number vector. Specializing these quantities to the cases discussed in the literature. we are able to reproduce previously published results. Moreover, after a replacement of the ensemble averages by those over the longitudinal coordinate, fluxes result which hare been derived for non-axisymmetric modes with l=m. Thus, a fairly complete account is given of results published in this field up to the present.     

Why ripples become sweiis,and swells in shallow water decay rapidly

Abstract

In an ocean with a horizontal bottom where no wind is blowing it is shown that the spin (angular momentum) of the ocean is conserved. Thus, when energy is dissipated, at least one of three things will happen: i) Wave spectra may move towards lower frequencies. ii) The directional distribution may be changed towards long-crested waves. iii) Shear currents may be generated. By neglecting ii) and iii), the frequency shift of a spectrum is calculated due to molecular dissipation. When all energy transforming phenomena as e.g. wave breaking and turbulence generation are taken into account, the conservation of spin seems to be able to explain the frequency shift of wave spectra. In shallow water it is shown that there is energy transfer from the waves to shear currents.     

The influence of global warming in Earth rotation speed

The tendency of the atmospheric angular momentum (AAM) is investigated using a 49-year set of monthly AAM data for the period January 1949–December 1997. This data set is constructed with zonal wind values from the reanalyses of NCEP/NCAR, used in conjunction with a variety of operationally produced AAM time series with different independent sources and lengths over 1976–1997. In all the analyzed AAM series the linear trend is found to be positive. Since the angular momentum of the atmosphere–earth system is conserved this corresponds to a net loss of angular momentum by the solid earth, therefore decreasing the Earth rotation speed and increasing the length of day (LOD). The AAM rise is significant to the budget of angular momentum of the global atmosphere–earth system; its value in milliseconds/century (ms/cy) is +0.56 ms/cy, corresponding to one-third of the estimated increase in LOD (+1.7 ms/cy). The major contribution to this secular trend in AAM comes from the equatorial Tropopause. This is consistent with results from a previous study using a simplified aqua-planet model to investigate the AAM variations due to near equatorial warming conditions. During the same time interval, 1949–1997, the global marine + land-surface tempera- ture increases by about 0.79 °C/cy, showing a linear correspondence between surface temperature increase and global AAM of about 0.07 ms per 0.1 °C. These results imply that atmospheric angular momentum may be used as an independent index of the global atmosphere’s dynamical response to the greenhouse forcing, and as such, the length of day may be used as an indirect indicator of global warming.     

Lateral friction in the oceans as an effect of potential vorticity mixing

Abstract

Applying a mixing-length calculation to potential vorticity rather than to momentum a new type of lateral friction appears in the oceanic mass transport equations. This friction is evaluated for the special case of horizontally homogeneous, quasi-geostrophic turbulence. The main effect is a westward force arising from the so-called β-term. This produces an additional southward interior transport and a strengthening of the western boundary current. A turbulent exchange coefficient K^H = 10⁸ em²s^?1 is sufficient to give a Gulf Stream transport twice that obtained by the classical Sverdrup model.     

Addressing Hurricane Intensity through Angular Momentum and Scale Energetics Approaches

This paper addresses two avenues for gaining insight into the hurricane intensity issue—the angular momentum approach and the scale interaction approach. In the angular momentum framework, the torques acting on a parcel's angular momentum are considered along an inflowing trajectory in order to construct the angular momentum budget. These torques are separable into three components: The pressure torque, the surface friction torque, and the cloud torque. All torques are found to diminish the angular momentum of an inflowing parcel, with the cloud torques having the most important role. In the scale interaction approach, energy exchanges among different scales within a hurricane are considered as a means of understanding hurricane intensity. It is found that the majority of kinetic energy contribution to the hurricane scales originates from potential-to-kinetic in-scale energy conversions. The contribution of mean-wave interactions in the kinetic energy varies with distance from the center and with the life stage of a storm. In the early stages, as the disorganized convection becomes organized on the hurricane scales, upscale energy transfers (i.e., from small to large scale) are found to take place in the outer radii of the storm. In a mature storm, the kinetic energy transfers are downscale, except for the inner radii.     

On the transport of moisture in polythermal glaciers

Abstract

We reconsider the problem of formulation of a model for polythermal glaciers, focussing attention in particular on the temperate zone where ice and water can coexist at the melting temperature. The energy equation for the ice-water mixture in this zone introduces a moisture flux, and a constitutive law for this flux is required. By analogy with the flow through a porous medium, we use Darcy's law (i.e. the second momentum equation of a two-phase flow model with “porous” geometry), and then require a mechanical constitutive relation relating the water pressure p _w to the average ice pressure p _i . Experience in two phase flows suggests that p _w =p _i may be problematical, and experience in soil mechanics suggests it is inaccurate. A constitutive relation is therefore presented based on work of Nye (1976), and its effect on the well-posedness of the model is examined. Considerations of the sort presented here have clear relevance in the formulation of similar problems in other geophysical situations, notably mantle convection.