Geophysical flows with anisotropic turbulence and dispersive waves: flows with a β-effect

Geostrophic turbulence is a key paradigm in the current understanding of the large-scale planetary circulations. It implies that a flow is turbulent, rotating, stably stratified, and in near-geostrophic balance. When a small-scale forcing is present, geostrophic turbulence features an inverse energy cascade. When the meridional variation of the Coriolis parameter (or a β-effect) is included, the horizontal flow symmetry breaks down giving rise to the emergence of jet flows. The presence of a large-scale drag ensures that the flow attains a steady state. Dependent on the governing parameters, four steady-state flow regimes are possible, two of which are considered in this study. In one of these regimes, a flow is dominated by the drag while in the other one, the recently discovered regime of zonostrophic turbulence, a flow becomes strongly anisotropic and features slowly evolving systems of alternating zonal jets. Zonostrophic turbulence is distinguished by anisotropic inverse energy cascade and emergence of a new class of nonlinear waves known as zonons. In addition, meridional scalar diffusion is strongly modified in this regime. This paper provides an overview of various regimes of turbulence with a β-effect, elaborates main characteristics of friction-dominated and zonostrophic turbulence, elucidates the physical nature of the zonons, discusses the meridional diffusion processes in different regimes, and relates these results to oceanic observations.     

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.     

A model of a homogeneous isotropic turbulent flow and its application for the simulation of cloud drop tracks

Abstract

A model of a homogeneous isotropic turbulent flow is presented. The model provides different realizations of the random velocity field component with given correlation latitudinal and lateral functions and a spatial structure which obeys the Kolmogorov theory of homogeneous and isotropic turbulence. For the generation of the turbulent flow the structural function of the flow in the form suggested by Batchelor (Monin and Yaglom, 1975) was used. This function describes the spectrum of turbulence both in the viscous and inertial ranges. The isotropy and homogeneity of the velocity field of the model are demonstrated.

The model is aimed at simulating the ‘‘fine'’ features of drop's (aerosol particles') motion, such as the deviations of drops’ velocity from the velocity of the flow, detailed structures of drops’ tracks, related to drops’ (particles') inertia. The model is intended also for the purpose of studying cloud drops’ and aerosol particles’ motion and their diffusional spreading utilizing the Monte Carlo methods.

Some examples of drop tracks for drops of different size are presented. Drops’ tracks are very sophisticated, so that the relative position of drops falling initially from the same point can vary drastically. In some cases drops’ tracks diverge very quickly, in other cases all drops move within a turbulent eddy along nearly the same closed tracks, but with different speed. The concentration of drop tracks along isolated paths is found in spite of the existence of a large number of velocity harmonics. It is shown that drops (aerosol particles) tend to leave some areas of the turbulent flow apparently due to their inertia. These effects can possibly contribute to inhomogeneity of drops’ concentration in clouds at different spatial scales.     

Calculation of diffusivities of passive fields in turbulent media

Abstract

The exact numerical and approximate analytical solutions of the simplest nonlinear integral equation with second order nonlinearity for the averaged Green function are presented. It is assumed that the turbulence is stationary, homogeneous, isotropic and incompressible. Numerous examples of turbulent spectra are considered (peak-like spectrum, spectra of Kolmogorov's type with different forms of “pumping” regions, stepwise spectra etc.). Special emphasis is given to investigating the case of so called “frozen” turbulence when the parameter ξ =u ₀τ/R→∞ where uτ₀,R ₀ are characteristic velocity, lifetime and space scale of turbulent pulsations, respectively. It is shown that these solutions allow us to calculate the turbulent diffusivities accurately for arbitrary spectra with any values of the parameter ξ. The results take into account the possible helicity of turbulence concerned only with scalar passive fields (number density and temperature).     

A primitive equation,solar driven,perturbation model of the thermospheres of mars and venus

Abstract

A primitive equation, solar driven, thermospheric model is derived which has applications to the neutral gas components on Mars and Venus. The full effects of molecular viscosity and thermal conductivity are included, necessitating the development of a combined analytic and numerical solution technique. The model is applied to Venus in order to understand how thermospheric rotation, if present, would affect the dynamics. Results indicate that rotation periods of eight days or less should be observable. Application of the model to Mars indicates that the perturbation solar heating and the atmospheric response have primarily a diurnal component for which typical temperature and zonal wind maximum amplitudes are 20 K and 30 m/sec respectively. Because of uncertainty in the solar heating efficiency, calculations were made varying this parameter by an order of magnitude. The results imply that the response due to solar forcing alone is probably too small to account for observed concentrations of the minor constituents CO and O. An upper limit estimate is made of the upward propagation of wave energy from the lower atmosphere and the resulting response of the thermosphere.     

A vortex-tube model of eddies in the inertial range

Abstract

A vortex-tube geometry of the cascade of energy to small-scale eddies, in the inertial range of fully-developed turbulence, is proposed. The model is a special case of the beta model of Frisch, Sulem and Nelkin (1978). We require that the cascade conserve the principal invariants of inviscid, incompressible flow, namely volume, topological knottedness, circulation, and, at discrete times marking the termination of steps in the cascade, energy. The process terminates in a finite time, as in any beta model, leaving behind a self-similar network of “inactive” tubes. We associate a self-similar scaling dimension D with the structure, equal to the Hausdorff dimension of the set of “active” tubes at the termination of the cascade. Because circulation Λ plays a key role in the analysis of the cascade, we refer to these vortex-tube geometries as “gamma models”. The viewpoint throughout is entirely deterministic.

We describe two examples of gamma models. In the ring geometry, an eddy is a vortex ring, and the cascade produces “rings upon rings”, so we allow cutting and fusing of tubes while conserving total helicity. In the preferred helical model, no cutting is needed, and the cascade produces an infinite progression of braided “coils upon coils”. We suggest that latter geometry as a candidate for the topology of a singularity of the inviscid limit of a Navier-Stokes flow, when modeled by discrete vortex tubes.

A crucial ingredient of a gamma model, not explicitly present in a beta model, is the possibility of “splitting” a vortex tube into sub-tubes carrying smaller circulation. We suggest a dynamical basis for this process, as an instability of tubes whose cores violate the Rayleigh criterion.

The parameters describing a gamma model are not uniquely determined by our study, but there is a “simplest” helical gamma model, involving minimal splitting and distortion of tubes. The dimension D of the structure is 13/5, with a scale factor Λ = 2^?5/4. This value of D agrees with that suggested by Hentschel and Procaccia (1982), by analogy with established results for certain branched polymers.     

INTERNATIONALE GLAZIOLOGISCHE GRÖNLANDEXPEDITION, 1957–1959

Abstract

Equations are derived for vapor transport within a constant flux layer of the atmosphere. The physical model is based on the Reynolds analogy for fully turbulent flow, so that vapor and momentum flux are assumed similar. The shear stress is deduced from analysis of the wind profile by means of equations available for non-neutral conditions. Empirical expressions are proposed using the friction velocity rather than the wind velocity at a given level in order to take account of the stability of the air.     

Kinematic turbulent dynamo in a shear flow

Abstract

We consider the turbulent dynamo action in a differentially rotating flow by making use of a kinematic approach when the effect of a generated magnetic field on turbulent motions is neglected. The mean electromotive force is calculated in a quasilinear approximation. Differential rotation can stretch turbulent magnetic field lines and break the symmetry of turbulence in such a way that turbulent motions become suitable for the generation of a large scale magnetic field. The presence of shear changes the type of an equation governing the mean magnetic field. Due to shear stresses the mean magnetic field can be generated by a turbulent dynamo action even in a uniform turbulence. The growth rate depends on the length scale of the mean field being faster for the field with a smaller length scale.     

Turbulent viscosity optimized by data assimilation

As an alternative approach to classical turbulence modelling using a first or second order closure, the data assimilation method of optimal control is applied to estimate a time and space-dependent turbulent viscosity in a three-dimensional oceanic circulation model. The optimal control method, described for a 3-D primitive equation model, involves the minimization of a cost function that quantifies the discrepancies between the simulations and the observations. An iterative algorithm is obtained via the adjoint model resolution. In a first experiment, a k ± L model is used to simulate the one-dimensional development of inertial oscillations resulting from a wind stress at the sea surface and with the presence of a halocline. These results are used as synthetic observations to be assimilated. The turbulent viscosity is then recovered without the k + L closure, even with sparse and noisy observations. The problems of controllability and of the dimensions of the control are then discussed. A second experiment consists of a two-dimensional schematic simulation. A 2-D turbulent viscosity field is estimated from data on the initial and final states of a coastal upwelling event.     

Towards a realistic theory of the geodynamo

Abstract

The physics of the geodynamo is discussed. The main processes relevant for the buoyancy driven geodynamo are isolated. The successive stages of development of geodynamo theory are briefly described. The mechanism of local turbulence in the Earth's core is explained, and an estimate is presented of the turbulent transport of density inhomogeneities in the Earth's core. The significance of this turbulent transport to the geodynamo mechanism is stressed. The general scheme of the complete geodynamo theory of the future is outlined.     

Vortex evolution

Abstract

Friedmann's equation and the potential vorticity equation are generalised for turbulent motion. The generalised equations incorporate some new phenomena connected with turbulent transport of mass. It is proved that, if ?×[S×Ω+S(?·S)]≠0 where Ω is the absolute vorticity of the velocity and S is the turbulent density flux, then the Helmholtz-Kelvin theorem concerning the conservation of the velocity circulation around a closed path is violated and the potential vorticity is not a Lagrangian adiabatic invariant. The effects of this turbulent transport of mass on the creation or dissipation of vorticity discussed here is not equivalent to effects of baroclinicity or viscosity. Some possible implications of the new circulation theorem in geophysical and astrophysical fluid dynamics are discussed.     

Turbulence measurements in a submarine canyon

Profiles of velocity turbulence in Monterey Canyon, made with a recently developed expendable probe, show the existence of a very turbulent bottom boundary layer. The turbulent flow is up to 170 m thick and has peak microscale shears of 1 m s⁻¹ per meter. The rate of dissipation of kinetic energy, based on the observed shear variance, averaged over the depth of the turbulent boundary layer ranged from 70 to 500 × 10⁻⁶W m⁻³. Temperature measurements indicate that the flow was up canyon at a time of low tide. The upper bound for the vertical eddy viscosity is estimated to be17 × 10⁻⁴m²s⁻¹ and for the vertical eddy diffusivity is estimated to be 15 × 10⁻⁴m²s⁻¹. The large vertical scale and the intensity of the observed boundary layer suggest that the flow in Monterey Canyon may be important for the renewal and circulation of water over the continental shelf in the bay area.     

On the reynolds stress and lateral eddy viscosity due to transient oceanic Rossby waves

Summary From numerical solutions of a wind-driven homogeneous ocean model, anegative lateral eddy viscosity of the order 10⁴ cm² sec^–1 is inferred from the large-scale time-dependent currents in the interior of an enclosed shallow basin. The transient Rossby waves in this region produce a systematic convergence of eddy momentum at the latitude of the maximum average eastward current, and thus effect a transfer of zonal momentum from the large-scale eddies to the mean flow. In this sense they are analogous to the Rossby waves in the atmospheric general circulation, and it is speculated that such waves may help to maintain the mean zonal ocean currents. Although this negative viscosity induced by the large-scale transients is relatively small compared with the prescribed lateral viscosity of 10⁸ cm² sec^–1 and should be given a quite different physical interpretation, it is evidently an important viscous effect for the mean flow in the interior of the basin. The prescribed viscosity, on the other hand, is effective in controlling the model's simulated sub-grid scale dissipation, which occurs almost entirely in the nearby steady boundary currents.     

Convection and gravity waves in two layer models. I. Overstable modes driven in conducting boundary layers

Abstract

Stability analysis is formulated for a two-layer fluid model in which the upper and lower layers are convectively stable and unstable, respectively. With discontinuities in viscosity and conductivity at the interface, the exchange of stability does not generally hold and overstability is possible. A detailed analytical treatment is presented for the case of small viscosity and conductivity in which viscous and conducting boundary layers are formed at the interface.

The usual damping effect due to the energy dissipation by viscosity and thermal conductivity exists irrespective of whether the mode is the convection or the gravity wave, but, for larger horizontal wave lengths, the effect of the boundary layer can become more important. The jump in the thermal conductivity in the boundary layer can give rise to overstability of the gravity wave in agreement with Souffrin and Spiegel (1967). The jump in the viscosity provides a self-catalytic action for the unstable flow if the viscosity is assumed to be the nonlinear turbulent viscosity due to the motion itself. The effect, however, is not strong enough to overcome the usual viscous damping.     

On the interaction between a radiatively damped planetary wave and the zonally averaged circulation in the middle atmosphere

The interaction between a planetary wave damped by cooling to space and the zonally averaged circulation in the middle atmosphere is examined for a steady-state situation in middle latitudes. Quasi-geostrophic scaling of Type 2 is assumed (i.e. the space scales are planetary and the superrotation is small).A set of mean equations is derived for this scaling which is complementary to the set of perturbation equations previously studied. The mean equations show that a planetary wave induces a mean meridional circulation which is balanced by an eddy momentum forcing function and a mean diabatic heating which is balanced by an eddy heat flux forcing function. The vertical eddy fluxes enter the forcing at the same order as the horizontal eddy fluxes.An analytical wave solution is found for the case of an atmosphere in constant superrotation. The eddy fluxes and forcing functions are evaluated for this special case. It is found that they are very sensitive to the values of the radiative damping coefficient and the superrotation. Since the damping coefficient depends on the ozone concentration and the intensity of the solar ultraviolet flux, the results suggest that changes in these quantities can strongly modify the wave-mean flow interaction in the middle atmosphere. Possible implications for climate change are discussed.     

Mean and turbulent flow fields in a simulated ice‐covered channel with a gravel bed: some laboratory observations

Northern rivers experience freeze‐up over the winter, creating asymmetric under‐ice flows. Field and laboratory measurements of under‐ice flows typically exhibit flow asymmetry and its characteristics depend on the presence of roughness elements on the ice cover underside. In this study, flume experiments of flows under a simulated ice cover are presented. Open water conditions and simulated rough ice‐covered flows are discussed. Mean flow and turbulent flow statistics were obtained from an Acoustic Doppler Velocimeter (ADV) above a gravel‐bed surface. A central region of faster flow develops in the middle portion of the flow with the addition of a rough cover. The turbulent flow characteristics are unambiguously different when simulated ice covered conditions are used. Two distinct boundary layers (near the bed and in the vicinity of the ice cover, near the water surface) are clearly identified, each being characterized by high turbulent intensity levels. Detailed profile measurements of Reynolds stresses and turbulent kinetic energy indicate that the turbulence structure is strongly influenced by the presence of an ice cover and its roughness characteristics. In general, for y/d > 0·4 (where y is height above bed and d is local flow depth), the addition of cover and its roughening tends to generate higher turbulent kinetic energy values in comparison to open water flows and Reynolds stresses become increasingly negative due to increased turbulence levels in the vicinity of the rough ice cover. The high negative Reynolds stresses not only indicate high turbulence levels created by the rough ice cover but also coherent flow structures where quadrants one and three dominate. Copyright © 2012 John Wiley & Sons, Ltd.     

The impact of plant morphology on flow structure: comparative analysis of two types of submerged flexible macrophyte

ABSTRACT

The presence of aquatic vegetation in riverine and lacustrine environments alters the mean and turbulent flow structure and thus impacts the fate and transport of sediment and contaminants. Turbulent flows through Vallisneria natans (V. natans) and Potamogeton malaianus (P. malaianus) were investigated in a laboratory flume. The impact of plant morphology on mean velocity profile and turbulence distribution was analysed and discrepancies in flow alteration caused by different types of macrophyte were highlighted. Results show that a dense canopy of submerged macrophyte leads to a velocity profile featuring a counter velocity gradient in the lower part of the canopy. Negative Reynolds stress and its local maximum were observed there. Discrepancies in flow structure caused by different morphologies of both tested plants were further identified. With smaller frontal area in the lower part of the canopy, P. malaianus causes a much bigger gradient and local maximum in the velocity profile, and thus a larger local stress maximum than V. natans. The mean velocity gradient around the top of canopy, the Reynolds stress and the turbulence kinetic energy at the canopy interface are smaller than for the flow through the V. natans canopy. Larger reduction of the mean velocity within the V. natans canopy makes the suspended sediment of fine particles more easily deposited than in the P. malaianus canopy.     

Cross helicity and related dynamo

The turbulent cross helicity is directly related to the coupling coefficients for the mean vorticity in the electromotive force and for the mean magnetic-field strain in the Reynolds stress tensor. This suggests that the cross-helicity effects are important in the cases where global inhomogeneous flow and magnetic-field structures are present. Since such large-scale structures are ubiquitous in geo/astrophysical phenomena, the cross-helicity effect is expected to play an important role in geo/astrophysical flows. In the presence of turbulent cross helicity, the mean vortical motion contributes to the turbulent electromotive force. Magnetic-field generation due to this effect is called the cross-helicity dynamo. Several features of the cross-helicity dynamo are introduced. Alignment of the mean electric-current density J with the mean vorticity Ω , as well as the alignment between the mean magnetic field B and velocity U , is supposed to be one of the characteristic features of the dynamo. Unlike the case in the helicity or α effect, where J is aligned with B in the turbulent electromotive force, we in general have a finite mean-field Lorentz force J ?×? B in the cross-helicity dynamo. This gives a distinguished feature of the cross-helicity effect. By considering the effects of cross helicity in the momentum equation, we see several interesting consequences of the effect. Turbulent cross helicity coupled with the mean magnetic shear reduces the effect of turbulent or eddy viscosity. Flow induction is an important consequence of this effect. One key issue in the cross-helicity dynamo is to examine how and how much cross helicity can be present in turbulence. On the basis of the cross-helicity transport equation, its production mechanisms are discussed. Some recent developments in numerical validation of the basic notion of the cross-helicity dynamo are also presented.