Inertial Waves Research Articles

Internal gravity waves versus inertial waves in the laboratory

Internal gravity waves versus inertial waves in the laboratory

Instability and warping in vertically oscillating accretion discs

Abstract Many accretion discs have been found to be distorted: either warped due a misalignment in the system, or non-circular as a result of orbital eccentricity or tidal deformation by a binary companion. Warped, eccentric, and tidally distorted discs are not in vertical hydrostatic equilibrium, and thus exhibit vertical oscillations in the direction perpendicular to the disc, a phenomenon that is absent in circular and flat discs. In extreme cases, this vertical motion is manifested as a vertical ‘bouncing’ of the gas, potentially leading to shocks and heating, as observed in recent global numerical simulations. In this paper we isolate the mechanics of vertical disc oscillations by means of quasi-2D and fully 3D hydrodynamic local (shearing-box) models. To determine the numerical and physical dissipation mechanisms at work during an oscillation we start by investigating unforced oscillations, examining the effect of initial oscillation amplitude, as well as resolution, boundary conditions, and vertical box size on the dissipation and energetics of the oscillations. We then drive the oscillations by introducing a time-dependent gravitational potential. A key result is that even a purely vertically oscillating disc is (parametrically) unstable to developing inertial waves, as we confirm through a linear stability analysis. The most important of these has the character of a bending wave, whose radial wavelength depends on the frequency of the vertical oscillation. The nonlinear phase of the instability exhibits shocks, which dampen the oscillations, although energy can also flow from the bending wave back to the vertical oscillation.

Synergistic Impact of Diurnal Warm Layers and Inertial Wave Mixing on Sea Surface Temperature Warming and Upper Ocean Stratification

AbstractWe study two sea surface temperature (SST) warming events and upper ocean stratification changes in the northern South China Sea in 2022 using data from an EM‐APEX float and satellite observations. The diurnal warm layers (DWLs) and the increasing buoyancy frequency N2 above the top of the thermocline can restrict the penetration depth of nighttime convection and wind‐driven mixing, which prevents cooler water from mixing upward, allowing solar heating to increase the SST by more than 1°C in a few days. The stratification budget approach is used to reproduce observations below 40 m despite some uncertainties in estimating variables such as horizontal gradient. After the first SST warming event, the stratification changes in the subsurface layers constituted by an increase in N2 above 70 m and a decrease below this depth can be attributed to the combined effects of turbulent diffusion and vertical advection rather than to horizontal advection or penetrative solar radiation. This ocean interior mixing is likely caused by the shear of near‐inertial waves at ∼50 m, when the nighttime convection could not penetrate through the DWL's base around 20 m. The stratification budget approach fails to simulate the changes above 40 m after the second SST warming event partly due to the presence of a near‐surface freshwater layer. Our observations offer insights into the effect of inertial wave‐induced mixing in the ocean interior when near‐surface stratified layers are present, which can lead to changes in upper ocean stratification and SST.

Investigation of the characteristics of low-level jets over North America in a convection-permitting Weather Research and Forecasting simulation

Abstract. In this study, we utilized a high-resolution (4 km) convection-permitting Weather Research and Forecasting (WRF) simulation spanning a 13-year period (2000–2013) to investigate the climatological features of low-level jets (LLJs) over North America. The 4 km simulation enabled us to represent the effects of orography and the underlying surface on the boundary layer winds better. Focusing on the continental US and the adjacent border regions of Canada and Mexico, this study not only identified several well-known large-scale LLJs, such as the southerly Great Plains LLJ and the summer northerly California coastal LLJ, but also the winter Quebec northerly LLJ which received less focus before. All these LLJs reach their peak in the nighttime in the diurnal cycle. Thus, the different thermal and dynamic mechanisms forming these three significant LLJs are investigated in this paper. Inertial oscillation theory dominates in the Great Plain LLJ, and the California coastal LLJ is formed by the baroclinic theory, whereas the Quebec LLJ is associated with both theories. Moreover, the high-resolution simulation revealed climatic characteristics of weaker and smaller-scale LLJs or low-level wind maxima in regions with complex terrains, such as the northerly LLJs in the foothill regions of the Rocky Mountains and the Appalachians during the winter. This study provides valuable insights into the climatological features of LLJs in North America, and the high-resolution simulation offers a more detailed understanding of LLJ behavior near complex terrains and other smaller-scale features.

A One-Dimensional Model for Investigating Scale-separated Approaches to the Interaction of Oceanic Internal Waves

Abstract High-frequency wave propagation in near-inertial wave shear has, for four decades, been considered fundamental in setting the spectral character of the oceanic internal wave continuum and for transporting energy to wave-breaking. We compare idealized ray tracing numerical results with metrics derived using a wave turbulence derivation for the kinetic equation and a path integral to study this specific process. Statistical metrics include the changing ensemble mean vertical wavenumber, referred to as a mean drift, dispersion about the mean drift, time lagged correlation estimates of wavenumber and phase locking of the wave packets with the background. The path integral permits us to identify the mean drift as a resonant process and dispersion about that mean drift as non-resonant. At small inertial wave amplitudes; ray tracing, wave turbulence and the path integral provide consistent descriptions for the mean drift of wavepackets in the spectral domain and dispersion about the mean drift. Extrapolating these results to the background internal wavefield over-predicts downscale energy transports by an order of magnitude. At oceanic amplitudes, however, the numerics support diminished transport and dispersion that coincide with the mean drift time scale becoming similar to the lagged correlation time scale. We parse this as the transition to a non-Markovian process. Despite this decrease, numerical estimates of downscale energy transfer are still too large. We argue that residual differences result from an unwarranted discard of Bragg scattering resonances. Our results support replacing the long standing interpretive paradigm of extreme scale-separated interactions with a more nuanced slate of ‘local’ interactions in the kinetic equation.

Multi-Objective Optimization of an Inertial Wave Energy Converter for Multi-Directional Wave Scatter

To advance wave energy devices towards commercialization, it is essential to optimize their design to enhance system performance. Additionally, a thorough economic evaluation is crucial for making these technologies competitive with other renewable energy sources. This study focuses on the techno-economic optimization of an innovative inertial system, the so-called SWINGO system, which is based on gyropendulum technology. SWINGO stands out due to its high energy efficiency in multi-directional installation sites, where wave directions vary significantly throughout the year. The study introduces the application of a multi-objective Evolutionary Algorithm (EA), specifically the Non-dominated Sorting Genetic Algorithm II (NSGA-II), to optimize the techno-economic performance of the SWINGO system. This approach aims to identify optimal design parameters that maximize energy extraction while considering economic viability. By deriving a Pareto frontier, a set of optimal devices is selected for further analysis. The performance of the SWINGO system is also compared to an alternative (mono-directional) inertial wave energy converter, the Inertial Sea Wave Energy Converter (ISWEC), to highlight the differences in techno-economic outcomes. Both systems are evaluated at two different installation sites: Pantelleria island and the North Sea in Denmark, with a focus on the directional wave scatter at each location.

Weakening the effect of boundaries: “diffusion-free” boundary conditions as a “do least harm” alternative to Neumann

In this note, we discuss a poorly known alternative boundary condition to the usual Neumann or “stress-free” boundary condition typically used to weaken boundary layers when diffusion is present but very small. These “diffusion-free” boundary conditions were first developed (as far as the authors know) in 1995 (Sureshkumar and Beris J. Non-Newtonian Fluid Mech. 60, 53–80, 1995) in viscoelastic flow modelling but are worthy of general consideration in other research areas. To illustrate their use, we solve two simple ODE problems and then treat a PDE problem – the inertial wave eigenvalue problem in a rotating cylinder, sphere and spherical shell for small but non-zero Ekman number E. Where inviscid inertial waves exist (cylinder and sphere), the viscous flows in the Ekman boundary layer are O ( E 1 / 2 ) weaker than for the corresponding stress-free layer and fully O ( E ) weaker than in a non-slip layer. These diffusion-free boundary conditions can also be used with hyperdiffusion and provide a systematic way to generate as many further boundary conditions as required. The weakening effect of this boundary condition could allow precious numerical resources to focus on other areas of the flow and thereby make smaller, more realistic values of diffusion accessible to simulations.

The Paris low-level jet during PANAME 2022 and its impact on the summertime urban heat island

Abstract. The low-level jet (LLJ) and the urban heat island (UHI) are common nocturnal phenomena. While the UHI has been studied extensively, interactions of the LLJ and the urban atmosphere in general (and the UHI in particular) have received less attention. In the framework of the PANAME (PAris region urbaN Atmospheric observations and models for Multidisciplinary rEsearch) initiative in the Paris region, continuous profiles of horizontal wind speed and vertical velocity were recorded with two Doppler wind lidars (DWLs) – for the first time allowing for a detailed investigation of the summertime LLJ characteristics in the region. Jets are detected for 70 % of the examined nights, often simultaneously at an urban and a suburban site, highlighting the LLJ regional spatial extent. Emerging at around sunset, the mean LLJ duration is ∼ 10 h, the mean wind speed is 9 m s−1, and the average core height is 400 m above the city. The temporal evolution of many events shows signatures that indicate that the inertial oscillation mechanism plays a role in the jet development: a clockwise veering of the wind direction and a rapid acceleration followed by a slower deceleration. The horizontal wind shear below the LLJ core induces variance in the vertical velocity (σw2) above the urban canopy layer. It is shown that σw2 is a powerful predictor for regional contrast in air temperature, as the UHI intensity decreases exponentially with increasing σw2 and strong UHI values only occur when σw2 is very weak. This study demonstrates how DWL observations in cities provide valuable insights into near-surface processes relevant to human and environmental health.

Low‐Level Jet and Its Effect on the Onset of Summertime Nocturnal Rainfall in Beijing

AbstractLow‐level Jets (LLJs) play a critical role in triggering nocturnal rainfall in many areas on our planet. Nevertheless, little is known regarding the vertically resolved evolution of LLJs preceding rainfall. Here, the high‐resolution wind measurements from a radar wind profiler network in Beijing are used to monitor the continuous evolution of boundary‐layer jets (BLJs) and synoptic‐system‐related LLJs (SLLJs) before nocturnal rainfall under the southwesterlies synoptic pattern for the summers of 2020–2023. The results show that inertial oscillations precede LLJ‐induced nocturnal rainfall. The typical northward propagation of LLJ is driven primarily by horizontal advection. Starting 48 min before rainfall, the BLJ nose rapidly strengthens and moves downward from 0.8 km above ground level (AGL), while the SLLJ nose moves upward to 2.4 km AGL. This leads to enhanced moisture transport and convergence, which triggers nocturnal rainfall. The findings are of great significance for improving the forecast skill of nocturnal rainfall.

Tayler Instability Revisited

Tayler instability of toroidal magnetic fields B ϕ is broadly invoked as a trigger for turbulence and angular momentum transport in stars. This paper presents a systematic revision of the linear stability analysis for a rotating, magnetized, and stably stratified star. For plausible configurations of B ϕ , instability requires diffusive processes: viscosity, magnetic diffusivity, or thermal/compositional diffusion. Our results reveal a new physical picture, demonstrating how different diffusive effects independently trigger instability of two types of waves in the rotating star: magnetostrophic waves and inertial waves. It develops via overstability of the waves, whose growth rate sharply peaks at some characteristic wavenumbers. We determine instability conditions for each wave branch and find the characteristic wavenumbers. The results are qualitatively different for stars with magnetic Prandtl number Pm ≪ 1 (e.g., the Sun) and Pm ≫ 1 (e.g., protoneutron stars). The parameter dependence of unstable modes suggests a nonuniversal scaling of the possible Tayler–Spruit dynamo.

Electron-only Reconnection and Ion Heating in 3D3V Hybrid-Vlasov Plasma Turbulence

We perform 3D3V hybrid-Vlasov simulations of turbulence with quasi-isotropic, compressible injection near ion scales to mimic the Earth’s magnetosheath plasma, and investigate the novel electron-only reconnection, recently observed by NASA’s Magnetospheric Multiscale mission, and its impact on ion heating. Retaining electron inertia in the generalized Ohm's law enables collisionless magnetic reconnection. Spectral analysis shows a shift from kinetic Alfvén waves to inertial kinetic Alfvén and inertial whistler waves near electron scales. To distinguish the roles of inertial scale and gyroradius (d i and ρ i), three ion beta (β i = 0.25, 1, 4) values are studied. Ion-electron decoupling increases with β i, as ions become less mobile when the injection scale is closer to ρ i than d i, highlighting the role of ρ i in achieving an electron magnetohydrodynamic regime at sub-ion scales. This regime promotes electron-only reconnection in turbulence with small-scale injection at β i ≳ 1. We observe significant ion heating even at large β i, with Q i/ϵ ≈ 69%, 91%, and 96% at β i = 0.25, 1, and 4, respectively. While ion heating is anisotropic at β i ≤ 1 (T i,⊥ > T i,∥), it is marginally anisotropic at β i > 1 (T i,⊥ ≳ T i,∥). Our results show ion turbulent heating in collisionless plasmas is sensitive to the separation between injection scales (λ inj) and ρ i, β i, and finite-k ∥ effects, necessitating further investigation for accurate modeling. These findings have implications for other collisionless astrophysical environments, like high-β plasmas in intracluster medium, where processes such as microinstabilities or shocks may inject energy near ion-kinetic scales.

Mechanical interactions modeling of inertial wave energy converters

Numerous technological solutions for wave energy converters (WECs), referred as inertial reaction mass (IRM) systems, incorporate a reacting mass within the floater, coupled with a power take-off (PTO) system, to shelter all electronic components from the hostile sea environment. While the overall complexity of the system increases, the current modeling procedures persist in considering only a limited number of modes of motion, neglecting relevant dynamical effects. In this context, this paper proposes a systematic procedure for defining the kinematic characteristics and overall analytical model for the dynamics of IRM WECs. The significance of the proposed procedure lies in the statement of the reaction mass-related dynamic equation, considering the floater’s parametric excitation in six degrees of freedom (DoF). Additionally, it introduces the procedure for defining the reaction forces that the inertial mass exerts on the floater, which are often neglected in the literature for the full simulation of such systems. Furthermore, the proposed analytical modeling procedure allows the definition of approximated models in more simplified nonlinear forms for dynamic analysis and ultimately in fully linear approximations. This enables the application of methodologies and techniques commonly used in the literature for linear systems. The development of the framework is kept generic, in order to introduce a versatile mathematical procedure, that can be easily adjusted, with minor modifications, to accurately capture and represent the mechanical interaction for a wide family of IRM WEC devices. Subsequently, a case study on a vertical-hinged pendulum WEC is analyzed, to showcase the effectiveness of the proposed methodology. Moreover, to test the reliability of the analytical framework, a comparison with the output of a commercial software is conducted.

Maximization of inertial waves focusing in linear and nonlinear regimes

Maximization of inertial waves focusing in linear and nonlinear regimes

COMPARING SCIENTIFIC COMPUTING ENVIRONMENTS FOR SIMULATING 2D NON-BUOYANT FLUID PARCEL TRAJECTORY UNDER INERTIAL OSCILLATION: A PRELIMINARY EDUCATIONAL STUDY

This study presents a preliminary numerical investigation of the two-dimensional trajectory of a non-buoyant fluid parcel subjected to inertial oscillations and abrupt external forcing events. The simulations were implemented using Python, GNU Octave, R, Julia, and Fortran open-source scientific computing environments. By running 1,000 iterations in each environment, we evaluated the computational performance of these languages in tackling this idealized problem. The results, visualized through static plots and animation, validate the numerical model's ability to represent the fundamental physics governing fluid motion. Statistical analysis using the Kruskal-Wallis test and Dunn's post-hoc test with Bonferroni correction revealed that Fortran exhibits significantly faster execution times than other environments. However, the choice of programming language should also consider factors such as coding expertise, library availability, and scalability requirements. This study focuses on the performance of scientific computing environments within each language rather than the languages themselves. The observed execution times should be interpreted in the context of the specific environments used, as they often leverage optimized libraries written in lower-level languages. Despite the limitations of this work, such as the simplified 2D model and the use of a single hardware configuration, this study provides valuable insights into selecting appropriate computational tools. It contributes to educational resources for teaching idealized fluid dynamics models. Future studies could explore more complex scenarios, a more comprehensive range of programming environments, and the impact of different numerical schemes and physical parameterizations.

The physical mechanism of the streaming instability

Abstract The main hurdle of planet formation theory is the metre-scale barrier. One of the most promising ways to overcome it is via the streaming instability (SI). Unfortunately, the mechanism responsible for the onset of this instability remains mysterious. It has recently been shown that the SI is a Resonant Drag Instability (RDI) involving inertial waves. We build on this insight and clarify the physical picture of how the SI develops, while bolstering this picture with transparent mathematics. Like all RDIs, the SI is built on a feedback loop: in the ‘forward action’, an inertial wave concentrates dust into clumps; in the ‘backward reaction’, those drifting dust clumps excite an inertial wave. Each process breaks into two mechanisms, a fast one and a slow one. At resonance, each forward mechanism can couple with a backward mechanism to close a feedback loop. Unfortunately, the fast-fast loop is stable, so the SI uses the fast-slow and slow-fast loops. Despite this last layer of complexity, we hope that our explanation will help understand how the SI works, in which conditions it can grow, how it manifests itself, and how it saturates.

Linear analysis of a swirling jet with a realistic swirler model

The dynamics of an axisymmetrical swirling jet is studied via global linear stability and resolvent analyses. The modeled flow represents a combustor-like swirling jet, that is turbulent, compressible, non-parallel, and enclosed. In particular, the computational domain embeds a realistic axisymmetrical swirler model to resolve the mode conversion process. Swirl fluctuations are non-negligible on this configuration representative of a swirl burner, and match the analytical mode shapes of inertial waves of an inviscid uniform flow as obtained from global stability analysis. The stability map presents two eigenvalues driving a modal amplification. These eigenmodes couple a standing acoustic wave sustained in the mixing duct and the combustion chamber with the Kelvin-Helmholtz mechanism at the mixing duct exit and the acoustic-vorticity mode conversion process at the swirler, and act as a frequency selection criterion. Finally, the most amplified forcing from the resolvent analysis is similar to an unsteady heat source in the combustion chamber, and the identified optimal amplification mechanism is likely to be triggered in reacting flow with unsteady heat release rate.

Submesoscale Eddies Detected by SWOT and Moored Observations in the Northwestern Pacific

AbstractThe Surface Water and Ocean Topography (SWOT) mission provides a good opportunity to study fine‐scale processes in the global ocean but whether it can detect balanced submesoscale eddies is uncertain due to the “contamination” by unbalanced inertial gravity waves. Here, based on concurrent observations from SWOT and a mooring array in the northwestern Pacific, we successfully captured two submesoscale cyclonic eddies with negative sea level anomalies (SLAs) in spring 2023. We find that the SLA amplitude and equivalent radius of the first (second) eddy are 2.5 cm and 16.0 km (2.0 cm and 18.8 km), respectively. For both eddies, their vertical scales are around 150 m and their horizontal velocities and Rossby numbers exceed 15.0 cm/s and 0.4, respectively. Further analysis suggests that similar submesoscale eddies can commonly occur in the northwestern Pacific and that SWOT is capable to detect larger submesoscale eddies with scales greater than ∼10 km.

A large deformation viscoelasticity theory for elastomeric materials and its numerical implementation in the open-source finite element program FEniCSx

Elastomeric solid materials typically exhibit a pronounced viscoelastic response. In this paper we consider a large deformation viscoelasticity theory for isotropic elastomeric materials which uses a multi-branch multiplicative decomposition of the deformation gradient. We then describe the numerical implementation of the theory in the open-source finite element program FEniCSx. Several example simulations which demonstrate the capability of the theory and its numerical implementation to model stress-relaxation, creep, stretch-rate sensitivity, hysteresis, damped inertial oscillations, and dynamic column buckling are shown. The source codes for these simulations are provided. The theory and the codes presented in this paper lay the foundation for future extensions of the theory and its numerical implementation to include the effects of coupling with thermal, electrical, and magnetic fields — extensions which are of central importance in modeling the response of soft-active materials.

Existence of small-scale Rossby waves points to low convective velocity amplitudes in the Sun

Inertial waves occur naturally in rotating fluids such as the Sun and the Earth's atmosphere. Rossby waves in the Sun have the potential to shed fresh light on interior turbulence and convection that prior seismic methods, reliant on sound waves, have been unable to accomplish. Here, we utilize ∼13 years of observational products taken by the space-based helioseismic and magnetic imager, onboard the solar dynamics observatory, to characterize solar equatorial Rossby waves. By examining maps of motions at the surface using two different methods, we are able to identify Rossby modes up to azimuthal order m = 30, approximately up to twice the spatial wavenumber limit of previous studies. The dispersion relation of these modes departs significantly from the classical two-dimensional Rossby-Haurwitz description. A parameter study of the effect of superadiabaticity and viscous diffusion on these inertial modes indicates that each parameter plays a role in influencing both the frequencies and linewidths of high m modes. Using the Rhines-scale relation, we constrain the root mean square amplitude of turbulent convection more tightly to ∼2 m/s, adding more evidence to the paradigm of weakly convective amplitudes at large scales.

Inertial oscillation modelling of low-level jets: an application to the complex terrain and double-nosed wind profiles

This study investigates the role of inertial oscillations in the evolution of a nocturnal Low-Level Jet (LLJ) in complex terrain and explores the impacts of local perturbations on wind dynamics. Specifically, a conceptual model based on inertial oscillations (Van de Wiel et al. J Atmos Sci 67(8):2679–2689 (2010)) is used to replicate the evolution of an LLJ in a gentle-sloping valley ensuring to capture its long-period dynamics under weak synoptic forcing. The analysis is performed on an already-analysed case study from the Mountain Terrain Atmospheric Modeling and Observations (MATERHORN) data set, taking advantage of the known local flow characteristics and the existence of a temporary anomaly in the LLJ shape called double-nosed LLJ. In an attempt to capture this last flow feature, a model modification is introduced, revealing appropriate to capture the double-nosed shape of the LLJ. Further observational studies will be needed to corroborate the operational use of this model and explore its application potential in different wind and energy sectors.