Stably-stratified Layer Research Articles

Tidal dissipation in rotating and evolving giant planets with application to exoplanet systems

ABSTRACT We study tidal dissipation in models of rotating giant planets with masses in the range 0.1–10MJ throughout their evolution. Our models incorporate a frequency-dependent turbulent effective viscosity acting on equilibrium tides (including its modification by rapid rotation consistent with hydrodynamical simulations) and inertial waves in convection zones, and internal gravity waves in the thin radiative atmospheres. We consider a range of planetary evolutionary models for various masses and strengths of stellar instellation. Dissipation of inertial waves is computed using a frequency-averaged formalism fully accounting for planetary structures. Dissipation of gravity waves in the radiation zone is computed assuming these waves are launched adiabatically and are subsequently fully damped (by wave breaking/radiative damping). We compute modified tidal quality factors Q′ and evolutionary time-scales for these planets as a function of their ages. We find inertial waves to be the dominant mechanism of tidal dissipation in giant planets whenever they are excited. Their excitation requires the tidal period (Ptide) to be longer than half the planetary rotation (Prot/2), and we predict inertial waves to provide a typical Q′ ∼ 103(Prot/1d)2, with values between 105 and 106 for a 10-d period. We show correlations of observed exoplanet eccentricities with tidal circularization time-scale predictions, highlighting the key role of planetary tides. A major uncertainty in planetary models is the role of stably-stratified layers resulting from compositional gradients, which we do not account for here, but which could modify predictions for tidal dissipation rates.

Retracted: Self-consistent models of Earth’s mantle and core from long-period seismic and tidal constraints

SUMMARYIn this study, we inverted a large set of normal-mode centre-frequencies and quality (attenuation) factors, including astronomic-geodetic data (mass, moment of inertia and tidal response), using self-consistently built models of the radial elastic and anelastic seismic structure of the Earth. The mantle models are constructed using petrologic phase equilibria in combination with a laboratory-based viscoelastic model that connects dissipation from seismic to tidal periods, whereas seismic properties for a well-mixed and homogeneous core are computed using equations-of-state. Relative to the preliminary seismic reference model (PREM), we find that for the models to fit the observations, mantle P- and S-wave velocities have to be slightly faster and slower, respectively, while outer-core P-wave velocity is slower on account of a different velocity gradient, whereas inner-core velocity structure is similar, within the uncertainties of the inferred model parameters. In terms of density, we find that the lower mantle is less dense and the outer core more dense than PREM, while the inner core is similar to PREM. To study the impact of the inferred mantle seismic velocity structure, we computed P- and S-wave traveltimes and compared these to the observations of globally-averaged P- and S-wave traveltimes from the reprocessed ISC catalogue that resulted in an excellent match. In an attempt to further refine the seismic P-wave velocity structure of the outer core, we also considered multiple core–mantle-boundary underside-reflected body wave traveltime data. Although the match to the underside reflections clearly improves as a result of a steeper velocity gradient in the outer core relative to the normal-modes- and astronomic-geodetic-data-only case, subtle differences nevertheless persist that appear to support a change in velocity gradient in the outermost core, evocative of a stably-stratified layer. The laboratory-based viscoelastic model considered here resolves the anelastic response of Earth’s mantle from long-period seismic (∼100 s) to tidal (18.6 yr) periods, accounting for both normal-mode and tidal dissipation measurements. Finally, as a potential means of refining core composition, we considered the density contrast across the inner-core boundary (ICB) based on our inverted models. The most probable ICB density difference found here is 0.3–0.45 g cm−3, which is in the lower range of earlier body-wave- and normal-mode-based predictions. This suggests that the compositional heterogeneity associated with light-element partitioning, which is considered the principal driving mechanism for the compositional convection that powers the geodynamo, may be less effective than previously thought, calling for exsolution of solids from the liquid outer core as a possible additional source of energy. This would also help address the problem of a young inner core.

A Global Simulation of the Dynamo, Zonal Jets, and Vortices on Saturn

Abstract The fluid dynamics planet Saturn gives rise to alternating east–west jet streams, large cyclonic and anticyclonic vortices, and a dipole-dominant magnetic field that is highly axisymmetric about the planetary rotation axis. Modeling these features in a self-consistent manner is crucial for understanding the dynamics of Saturn’s interior and atmosphere. Here we report a turbulent high-resolution dynamo simulation in a spherical shell that produces these features simultaneously for the first time. A crucial model ingredient is a long-hypothesized stably stratified layer (SSL), sandwiched between a deep metallic hydrogen layer and an outer low-conductivity molecular layer, born out of the limited solubility of helium inside metallic hydrogen at certain depths. The model spontaneously produces polar cyclones and significant low-latitude and midlatitude jet stream activity in the molecular layer. The off-equatorial low-latitude jet streams partially penetrate into the SSL and interact with the magnetic field. This helps to axisymmetrize the magnetic field about the rotation axis and convert some of the poloidal magnetic field to a toroidal field, which appears as two global magnetic energy rings surrounding the deeper dynamo region. The simulation also mimics a distinctive dip in the fifth spherical harmonic in Saturn’s magnetic energy spectrum as inferred from the Cassini Grand Finale measurements. Our model highlights the role of an SSL in shaping the fluid dynamical and magnetic features of giant planets, as exemplified at Saturn.

Nocturnal jets over wind farms in complex terrain

Recent wind-energy field experiments have enhanced our understanding of stratified flows over topography by observing the flow of nocturnal jets over complex terrains in the natural environment. There is a research gap on how the intricacies of such flows could impact wind-energy production. In this study, we investigated how nocturnal jets influence the performance of two operational wind farms built over complex terrain, by combining operational data and numerical mesoscale simulations. The wind farms are similarly designed as two rows of turbines roughly aligned in the crosswind direction and separated by microscale distances. Front and back rows are located, respectively, near the leading edge and the lee of a microscale plateau with a downstream valley. Nocturnal jets occur close to midnight as the cold fast-moving air of a gravity current rushes inland after the evening transition. They produce a deeper stably-stratified layer and strong downslope winds and in some cases can cause turbines in the back row to produce twice as much power as those in the front row.

A Dynamo Simulation Generating Saturn‐Like Small Magnetic Dipole Tilts

AbstractAmong planetary dynamos, the magnetic field of Saturn stands out in its exceptional level of axisymmetry. One of its peculiar features is that the magnetic dipole mode is tilted with respect to the planetary rotation axis by only ≈0.007° or less. Numerical dynamo simulations performed in this context have had great difficulty in producing such small dipole tilt angles without introducing ad hoc ingredients such as a latitudinally varying heat flux pattern in the outer layers or stably stratified layers (SSLs). Here we present a numerical dynamo simulation that generates a highly axisymmetric dynamo with a dipole tilt of about ≈0.0008° on average. The model consists of a deep dynamo layer and an overlying low‐conductivity layer but without any SSLs. We highlight a novel mechanism where strong differential rotation generated in the atmospheric layer penetrates into the dynamo region, helping to maintain a very small magnetic dipole tilt.

Stable stratification promotes multiple zonal jets in a turbulent jovian dynamo model

The ongoing NASA's Juno mission puts new constraints on the internal dynamics of Jupiter. Data gathered by its onboard magnetometer reveal a dipole-dominated surface magnetic field accompanied by strong localised magnetic flux patches. The gravity measurements indicate that the fierce surface zonal jets extend several thousands of kilometers below the cloud level before rapidly decaying below 0.94 − 0.96RJ, RJ being the mean Jovian radius at the one bar level. Jupiter's internal structure can be grossly decomposed in two parts: (i) an outer layer filled with a mixture of molecular hydrogen and helium where the zonal flows are thought to be driven; (ii) an inner region where hydrogen becomes metallic and dynamo action is expected to sustain the magnetic field. Several internal models however suggest a more intricate structure with a thin intermediate region in which helium would segregate from hydrogen, forming a compositionally-stratified layer. Here, we develop the first global Jovian dynamo which incorporates an intermediate stably-stratified layer between 0.82RJ and 0.86RJ. Using much lower diffusivities than previous models enables us to more clearly separate the dynamics of the metallic core and the molecular envelope. Analysing the energy balance reveals that magnetic energy is almost one order of magnitude larger than kinetic energy in the metallic region, while most of the kinetic energy is pumped into zonal motions in the molecular envelope. Those result from the different underlying force hierarchy with a triple balance between Lorentz, Archimedean and ageostrophic Coriolis forces in the metallic core and inertia, buoyancy and ageostrophic Coriolis forces controlling the external layers. The simulation presented here is the first to demonstrate that multiple zonal jets and dipole-dominated dynamo action can be consolidated in a global simulation. The inclusion of a stable layer is a necessary ingredient that allows zonal jets to develop in the outer envelope without contributing to the dynamo action in the deeper metallic region. Stable stratification however also smooths out the small-scale features of the magnetic field by skin effect, yielding a too-dipolar surface field as compared to the observations. These constraints suggest that possible stable layers in Jupiter should be located much closer to the surface (0.9 − 0.95RJ).

Turbulent convection in subglacial lakes

Abstract

A regime diagram for the slurry F-layer at the base of Earth's outer core

Seismic observations of a slowdown in P wave velocity at the base of Earth's outer core suggest the presence of a stably-stratified region known as the F-layer. This raises an important question: how can light elements that drive the geodynamo pass through the stably-stratified layer without disturbing it? We consider the F-layer as a slurry containing solid particles dispersed within the liquid iron alloy that snow under gravity towards the inner core. We present a regime diagram showing how the dynamics of the slurry F-layer change upon varying the key parameters: Péclet number (Pe), the ratio between advection and chemical diffusion; Stefan number (St), the ratio between sensible and latent heat; and Lewis number (Le), the ratio between thermal and chemical diffusivity. We obtain four regimes corresponding to stable, partially stable, unstable and no slurries. No slurry is found when the heat flow at the base of the layer exceeds the heat flow at the top, while a stably-stratified slurry arises when advection overcomes thermal diffusion (Pe≳Le) that exists over a wide range of parameters relevant to the Earth's core. Our results estimate that a stably-stratified F-layer gives a maximum inner-core boundary (ICB) body wave density jump of Δρbod≤534kgm−3 which is compatible with the lower end of the seismic observations where 280≤Δρbod≤1,100kgm−3 is reported in the literature. With high thermal conductivity the model predicts an inner core age between 0.6 and 1.2Ga, which is consistent with other core evolution models. Our results suggest that a slurry model with high core conductivity predicts geophysical properties of the F-layer and core that are consistent with independent seismic and geodynamic calculations.

Coupled convection and internal gravity waves excited in water around its density maximum at 4°C

Coupled mixed convective and stratified systems are common in natural flows. To study experimentally the associated dynamics, we use a singular property of water: its non-linear equation of state is characterised by a maximum density close to $4^{\circ}$C. By heating the top of a tank at $35^{\circ}$C and cooling the bottom at $0^{\circ}$C, a two-layer configuration spontaneously appears. The convective motion in the bottom layer consists mostly of a large-scale circulation and rising cold plumes. This turbulent flow generates propagating internal gravity waves in the upper stably-stratified layer. PIV measurements are performed and spectral characteristics of the convection and internal gravity waves are presented. An horizontal large-scale reversing flow in the stratified layer is observed which is viscously driven by a third, intermediate layer. This buffer layer is located between the convective and stratified layers and is thermally coupled with the convective one, hence sustaining a strong horizontal shear. Three dimensional direct numerical simulations with geometry and physical parameters close to the experimental ones corroborate our experimental results.

Penetration of Alfvén waves into an upper stably-stratified layer excited by magnetoconvection in rotating spherical shells

The penetration of magneto-hydrodynamic (MHD) disturbances into an upper strongly stratified stable layer excited by magnetoconvection in rotating spherical shells is investigated. An analytic expression for the penetration distance is derived by considering perturbations of a stably stratified rotating MHD Boussinesq fluid in a semi-infinite region, with the rotation axis and a uniform magnetic field tilted relative to the gravity axis. Solutions for the response to MHD disturbances applied at the bottom boundary show that the disturbances propagate as Alfvén waves in the stable layer. Their propagation distance is proportional to the Alfvén wave speed and inversely proportion to both the arithmetic average of viscosity and magnetic diffusion and the total wavenumber of the disturbance. The derived expression for penetration distance is in good agreement with the numerical results for neutral convection in a rotating spherical shell with an upper stably stratified layer embedded in an axially uniform basic magnetic field.

An Evaluation Of Local Similarity At The Top Of The Mixed Layer Based On Large-Eddy Simulations

Local similarity, referred to as type II similarity,in the interfacial, stably-stratified layer at thetop of the atmospheric (or oceanic) mixed layer isdiscussed. Type II scales for scalars are based onthe local values of scalar gradients. Similaritypredictions are derived from the second-orderclosure model of Yamada and Mellor, and also fromsimilarity arguments. The obtainedformulation is verified for active and passive scalarsbased on the large-eddy simulation model.

Intermittency in the two-dimensional inverse cascade of energy: Experimental observations

An extensive experimental study of the two-dimensional inverse energy cascade is presented. The experiments are performed in electromagnetically driven flows, using thin, stably-stratified layers. Complete instantaneous velocity fields are measured using particle imaging velocimetry techniques. Depending on the bottom-wall friction, two different regimes are observed: when the friction is low, the energy transferred from the forcing scale towards large scales accumulates in the lowest accessible mode, leading to a mean rotation of the flow and to an energy spectrum displaying a sharp peak at the minimum wave-number. This condensation is accompanied by the emergence of a very strong vortex around which the rotation is organized. At higher frictions, the inverse energy cascade conjectured by Kraichnan [Phys. Fluids 10, 1417 (1967)] is observed and is found to be stationary, homogeneous and isotropic, with a Kolmogorov constant consistent with numerical estimates. This inverse cascade does not appear to be characterized by the presence of strong coherent vortices. The characteristic size of the latter is of the order of the injection scale. Their statistical properties tend to show that the cascade is rather driven by a clustering mechanism involving same sign vortices rather than a sequence of merging events producing larger and larger vortices. Intermittency effects are also investigated for the inverse cascade range. It is found that, within experimental errors, there is no intermittency in the inverse cascade range of two-dimensional turbulence and that the statistics of velocity increments, either longitudinal or transverse, are close to Gaussian. These results constitute the first experimental study of intermittency in two-dimensional turbulence as well as the first observation of normal scaling in a field of research which has been increasingly concerned with anomalous exponents.

Some experiments on stably-stratified shear flows

This note summarises the results of an experimental investigation of the structure of turbulent shear flows with stable density stratification. The experiments were carried out in a wind tunnel with a fully-developed rectangular channel flow of air into which carbon dioxide was introduced through a line source at floor level where it formed a stably-stratified layer. Measurements made in three such layers included the vertical distributions of mean velocity and mean concentration at a number of positions downstream of the source. Measurements of the intensities of the three turbulence components were also made for one of the layers.