Inertial Waves Research Articles

Modal Decomposition and Linear Modeling of Swirl Fluctuations in the Mixing Section of a Model Combustor Based on Particle Image Velocimetry Data

Abstract In order to determine the flame transfer function of a combustion system, different mechanisms have been identified that need to be modeled. This study focuses on the generation and propagation of one of these mechanisms, namely, the swirl fluctuations downstream of a radial swirl combustor under isothermal conditions. Swirl fluctuations are generated experimentally by imposing acoustic perturbations. Time-resolved longitudinal and crosswise particle image velocimetry (PIV) measurements are conducted inside the mixing tube and combustion chamber to quantify the evolution of the swirl fluctuations. The measured flow response is decomposed using spectral proper orthogonal decomposition to unravel the contributions of different dynamical modes. In addition a resolvent analysis is conducted based on the linearized Navier–Stokes equations to reveal the intrinsically most amplified flow structures. Both, the data-driven and analytic approach, show that inertial waves are indeed present in the flow response and an inherent flow instability downstream of the swirler, which confirms recent theoretical works on inertial waves. However, the contribution of the identified inertial waves to the total swirl fluctuations turns out to be very small. This is suggested to be due to the very structured forcing at the swirler and the additional amplification of shear-driven modes. Overall, this work confirms the presence of inertial waves in highly turbulent swirl combustors and evaluates its relevance for industry-related configurations. It further outlines a methodology to analyze and predict their characteristics based on mean fields only, which is applicable for complex geometries of industrial relevance.

Vertical coordinate and resolution dependence of the second moment turbulent closure models and their limitations

Coordinate and resolution dependence of three second moment turbulent closure models are studied using one-dimensional Navy Coastal Ocean Model (NCOM) experiments and large eddy simulations at Ocean Station Papa. Our results suggest that finer resolution near the base of the mixed layer is critical for better model performance. A mixed layer enhanced vertical grid is proposed that outperforms both the uniform and the stretched grids with significantly fewer vertical layers used. For the new grid, the model accuracy is strongly dependent on the resolution near the base of the mixed layer, and not affected much by the total number of vertical layers used. However, given the success of the new grid, the lack of representation for the near inertial gravity waves below the mixed layer has hampered the ability of second moment turbulent closure models on accurate representation of turbulent mixing in the water column. While both the Langmuir circulation and the variation of surface heat fluxes are shown to be able to significantly change the strength of the near inertial waves, they have negligible effect on the eddy viscosity in the transition layer.

Response of the Bay of Bengal to super cyclone Amphan examined using synergistic satellite and in-situ observations

Tropical cyclone Amphan is the first super cyclone that happened in the north Indian Ocean in the last 20 years. In this work, multi-platform datasets were used to investigate the responses of the upper ocean to cyclone Amphan. The most striking response was the cold wake left by the cyclone spanning the entire Bay of Bengal with an amplitude up to ∼4°C. Satellite salinity observations revealed that the maximum increase in surface salinity was ∼1.5 PSU on the right side of the track of Amphan. Surface circulation was also observed to be modulated with the passage of a cyclone with a rightward bias in the change in its speed and direction. The currents observed from a moored buoy showed strong inertial oscillations. Argo observations showed that changes induced by the cyclone occurred up to 150 m depth of the cyclone and ocean heat content in the upper 150 m depth decreased due to the passage of the cyclone. There was an enhancement of surface chlorophyll concentration (∼1.5 mg/m3) after the passage of the cyclone, which was centred along the track of the cyclone where the winds were the highest. Mixed layer heat and salinity budget analysis showed that the sea surface cooling and increase in salinity was primarily driven by vertical mixing processes, though horizontal advection contributed meagrely. This study also brings forward the fact that regional differences exist in the responses of the ocean to the forcing of cyclones.

Diurnal Variations of Coastal Boundary Layer Jets over the Northern South China Sea and Their Impacts on Diurnal Cycle of Rainfall over Southern China during the Early-Summer Rainy Season

ABSTRACTThe diurnal variations of coastal boundary layer jets (CBLJs) over the northern South China Sea (SCS) and their effects on the diurnal cycle of rainfall over southern China during the early-summer rainy season of 2010–16 are examined based on the merged hourly gauge and satellite rainfall dataset, ERA5 data, and numerical simulations. CBLJs over the northern SCS exhibit two high-frequency regions with different diurnal cycles: one shows a nocturnal peak, located at the Beibu Gulf (WCBLJ); the other shows a double peak in the early morning and late afternoon, located over the coastal area of Guangdong (ECBLJ). Both CBLJs are occurred more frequently near Hainan Island. The inertial oscillations triggered by the land–sea thermal contrast, dynamic effects of Hainan Island, and effects of mountains over Vietnam are mainly attributed to the nocturnal WCBLJs, whereas the inertial oscillations induced by the large-scale sea breezes are responsible for nocturnal ECBLJs. The formation of afternoon ECBLJs mainly result from the thermal effects of Hainan Island. Nocturnal and morning rainfall over southern China are pronounced near the coast and the windward slope, while the afternoon rainfall peaks over the inland region in Guangdong Province. Due to dynamically forced lifting and moisture transport, WCBLJs and ECBLJs are essential for the formation of nocturnal and morning rainfall over Guangxi and Guangdong Province, respectively. In addition to the effects of solar heating, the afternoon rainfall over inland Guangdong is mainly affected by the ECBLJs and eastward-propagating rainfall belts from Guangxi Province, which are highly related to the occurrence of WCBLJs.

Effects of Coriolis Force on the Nonlinear Interactions of Acoustic-Gravity Waves in the Atmosphere

The nonlinear theory of acoustic-gravity waves ( AGWs ) in the atmosphere is revisited with the effects of the Coriolis force. Previous theory in the literature [Phys. Scr . 90 (2015) 055001] is advanced. Starting from a set of fluid equations modified by the Coriolis force, a general linear dispersion relation is derived which manifests the coupling of the high-frequency acoustic-gravity waves ( AGWs ) and the low-frequency inertial gravity waves ( IGWs ). The frequency of IGWs is enhanced by the Earth's angular velocity. The latter also significantly modifies the nonlinear coupling of AGWs and IGWs whose evolutions are described by the Zakharov approach as well as the Wigner- Moyal formalism. The consequences of the two equivalent evolution equations modified by the Coriolis force are briefly discussed.

Tidal migration of hot Jupiters: introducing the impact of gravity wave dissipation

ABSTRACT We study the migration of hot Jupiters orbiting solar-type pre-main sequence and main-sequence stars under the effect of tidal dissipation. The explored range of stellar mass extends from 0.6 to 1.3 M⊙. We apply recently developed prescriptions that allow us to explore the orbital evolution over the wide parameter space. Three types of tides are considered: equilibrium tide, inertial waves, and gravity waves. We combine the results of our simulations with the observed distribution of stellar and planetary parameters to evaluate the infall rate of hot Jupiters in the Milky Way Galaxy. In particular, we find that, for 11–21 per cent of the initial hot Jupiter population, coalescence occurs before the host star’s main sequence termination. If the planet is massive enough, such an event can potentially be accompanied by a powerful transient detectable with new facilities. Orbital decay by itself can be observed through transit-timing variation. However, the obtained coalescence rate in the Galaxy is too low (340–650 events per million years) to make positive predictions about the observational possibility. Potentially identifiable decaying systems formed by a star corresponding to a given mass interval might be too rare to be detected with the modern space telescopes, like TESS, within a 10-yr baseline. At the same time, the forthcoming missions, like PLATO, look more promising in this regard.

The influence of front strength on the development and equilibration of symmetric instability. Part 2. Nonlinear evolution

In Part 1 (Wienkers, Thomas & Taylor, J. Fluid Mech., vol. 926, 2021, A6), we described the theory for linear growth and weakly nonlinear saturation of symmetric instability (SI) in the Eady model representing a broad frontal zone. There, we found that both the fraction of the balanced thermal wind mixed down by SI and the primary source of energy are strongly dependent on the front strength, defined as the ratio of the horizontal buoyancy gradient to the square of the Coriolis frequency. Strong fronts with steep isopycnals develop a flavour of SI we call ‘slantwise inertial instability’ by extracting kinetic energy from the background flow and rapidly mixing down the thermal wind profile. In contrast, weak fronts extract more potential energy from the background density profile, which results in ‘slantwise convection.’ Here, we extend the theory from Part 1 using nonlinear numerical simulations to focus on the adjustment of the front following saturation of SI. We find that the details of adjustment and amplitude of the induced inertial oscillations depend on the front strength. While weak fronts develop narrow frontlets and excite small-amplitude vertically sheared inertial oscillations, stronger fronts generate large inertial oscillations and produce bore-like gravity currents that propagate along the top and bottom boundaries. The turbulent dissipation rate in these strong fronts is large, highly intermittent and intensifies during periods of weak stratification. We describe each of these mechanisms and energy pathways as the front evolves towards the final adjusted state, and in particular focus on the effect of varying the dimensionless front strength.

Vortex cluster arising from an axisymmetric inertial wave attractor

We present an experimental and numerical study of the nonlinear dynamics of an inertial wave attractor in an axisymmetric geometrical setting. The rotating ring-shaped fluid domain is delimited by two vertical coaxial cylinders, a conical bottom and a horizontal wave generator at the top: the vertical cross-section is a trapezium, while the horizontal cross-section is a ring. Forcing is introduced via axisymmetric low-amplitude volume-conserving oscillatory motion of the upper lid. The experiment shows an important result: at sufficiently strong forcing and long time scale, a saturated fully nonlinear regime develops as a consequence of an energy transfer draining energy towards a slow two-dimensional manifold represented by a regular polygonal system of axially oriented cyclonic vortices undergoing a slow prograde motion around the inner cylinder. We explore the long-term nonlinear behaviour of the system by performing a series of numerical simulations for a set of fixed forcing amplitudes. This study shows a rich variety of dynamical regimes, including a linear behaviour, a triadic resonance instability, a progressive frequency enrichment reminiscent of weak inertial wave turbulence and the generation of a slow manifold in the form of a polygonal vortex cluster confirming the experimental observation. This vortex cluster is discussed in detail, and we show that it stems from the summation and merging of wave-like components of the vorticity field. The nature of these wave components, the possibility of their detection under general conditions and the ultimate fate of the vortex clusters at even longer time scale remain to be explored.

The influence of front strength on the development and equilibration of symmetric instability. Part 1. Growth and saturation

Submesoscale fronts with large horizontal buoyancy gradients and $O(1)$ Rossby numbers are common in the upper ocean. These fronts are associated with large vertical transport and are hotspots for biological activity. Submesoscale fronts are susceptible to symmetric instability (SI) – a form of stratified inertial instability which can occur when the potential vorticity is of the opposite sign to the Coriolis parameter. Here, we use a weakly nonlinear stability analysis to study SI in an idealised frontal zone with a uniform horizontal buoyancy gradient in thermal wind balance. We find that the structure and energetics of SI strongly depend on the front strength, defined as the ratio of the horizontal buoyancy gradient to the square of the Coriolis frequency. Vertically bounded non-hydrostatic SI modes can grow by extracting potential or kinetic energy from the balanced front and the relative importance of these energy reservoirs depends on the front strength and vertical stratification. We describe two limiting behaviours as ‘slantwise convection’ and ‘slantwise inertial instability’ where the largest energy source is the buoyancy flux and geostrophic shear production, respectively. The growing linear SI modes eventually break down through a secondary shear instability, and in the process transport considerable geostrophic momentum. The resulting breakdown of thermal wind balance generates vertically sheared inertial oscillations and we estimate the amplitude of these oscillations from the stability analysis. We finally discuss broader implications of these results in the context of current parameterisations of SI.

Resonant Tidal Responses in Rotating Fluid Bodies: Global Modes Hidden beneath Localized Wave Beams

Abstract In rotating stars and planets, excitation of inertial waves in convective envelopes provides an important channel for tidal dissipation, but the dissipation rate due to inertial waves depends erratically on the tidal frequency. Tidal dissipation is significantly enhanced at some frequencies, suggesting possible resonances between the tidal forcing and some eigenmodes. However, the nature of these resonances remains enigmatic owing to the singularity of the eigenvalue problem of inertial waves, and the resonances are often mistakenly attributed to wave attractors in the literature. In this Letter, we reveal that resonant tidal responses correspond to inertial modes with large-scale flows hidden beneath localized wave beams. Strong couplings between the tidal forcing and the hidden large-scale flows intensify the localized wave beams emanating from the critical latitudes, leading to enhanced tidal dissipation. This study resolves a longstanding puzzle regarding the frequency dependence of tidal dissipation due to inertial waves in convective envelopes.

Shear Flow in Cylindrical Open Channel Under Precession

The study of forced oscillations in open cylindrical channel under precession is extended to include the shear effect, that is induced by inertial waves in such systems. The linear part of the problem led to two equations for stability one for the viscous part similar to Orr-Sommerfeld equation and one for the inviscid part similar to Rayleigh equation, the second was solved and discussed depending on the stream function observation. The linear part also led to relationship that connects between the stream velocity and the disturbance one is derived in a form similar to Burns conditions for open flows under normal conditions. Experimentally measuring the horizontal velocity distribution with depth showed that this distribution is sinusoidal one. Burns condition was solved based on this assumption. The nonlinear part of the problem led to a new version of Koteweg De-Vries (KdV) equation that is solved numerically by applying the leapfrog method for time discretization, Fourier transformation for the space one, and the trapezoidal rule for solving the integrals with depth, the results showed that the shear has no specific impact on the wave form which is similar to the classical results obtained by the theories under normal conditions.

Three-dimensionality of the triadic resonance instability of a plane inertial wave

In rotating flows dominated by inertial waves, triadic resonant interactions, and among them the triadic resonance instability, are the elementary bricks of the energy transfers between spatial scales. We demonstrate theoretically and experimentally that the triadic resonance instability of a plane inertial wave is a three-dimensional process, very efficient at redistributing the energy in the plane normal to rotation. A remarkable consequence of this result is that a turbulent sea of weakly nonlinear inertial waves will have a strong tendency to reach statistical axisymmetry.

Convection‐Permitting Hindcasting of Diurnal Variation of Mei‐yu Rainfall Over East China With a Global Variable‐Resolution Model

AbstractTwo sets of 48‐h hindcast experiments for Mei‐yu season using non‐hydrostatic global variable‐resolution model (MPAS‐Atmosphere) with regional refinement at 4 km resolution (V4km) and uniform 60 km resolution (U60km) are first‐time conducted to attempt at simulating the Mei‐yu rainfall in 2015 over East China. The analysis focuses on the seasonal average of diurnal variation of precipitation amount (PA), intensity (PI), and frequency (PF). Both simulations reasonably reproduce the spatial distribution of PA. V4km is more skillful in simulating the spatial distributions and magnitudes of PF and PI. The diurnal cycle of Mei‐yu rainfall shows a major early morning peak and a minor afternoon peak, contributed by precipitation during two sub‐periods with distinct synoptic circulations. With strong Mei‐yu in the first sub‐period, the diurnal variation of PA is controlled by nocturnal southwesterly jet. V4km overestimates the morning peak mainly due to its bias in simulating boundary layer inertial oscillation. Although differences in PA between the simulations are small, U60km overestimates PF and underestimates PI. With weak Mei‐yu in the second sub‐period, the diurnal variation of PA is controlled by both synoptic circulation and local convection. At both resolutions, deviations in the large‐scale circulation modulated by a few typhoons lead to positive biases in the morning peak of PA. After removing the typhoon impacts, V4km captures the observed diurnal cycle of PA well, while U60km significantly underestimates PA and PI particularly in the afternoon. Future studies focusing on advancing modeling of southwesterly jet and typhoons may further improve convection permitting simulation of Mei‐yu rainfall.

Evolution of forced magnetohydrodynamic waves in a stratified fluid

The evolution of a buoyancy disturbance in a stratified incompressible fluid permeated by a uniform vertical magnetic field is investigated. Two regimes are considered in the absence of background rotation – that of strong stratification, where the internal gravity wave frequency $\omega _A$ is much higher in magnitude than the magnetic (Alfvén) wave frequency $\omega _M$ , and that of strong magnetic field, where $\omega _M$ is dominant. For small but finite magnetic diffusion, perturbations that initially lie in the strong-field regime are shown to cross over to the regime of strong stratification, so that small-scale motions may exist as damped internal gravity waves at large times. The induced magnetic field propagates as damped Alfvén waves for a much longer time than the velocity before undergoing the above transition. With strong rotation, the unstably stratified system that satisfies the inequality $|\omega _C| > |\omega _M| \gg |\omega _A| \gg |\omega _\eta |$ , where $\omega _C$ is the inertial wave frequency and $\omega _\eta$ is the diffusion frequency, is of relevance to convection-driven dynamos. Here, a parameter space with $|\omega _M/\omega _C| \sim 0.1$ is found wherein the flow intensity of the slow magnetic-Archimedean-Coriolis (MAC) waves is of the same order of magnitude as that of the fast MAC waves. Slow wave motions at horizontal length scales much smaller than the width of the fluid layer can therefore generate substantial helicity in rapidly rotating dynamos. The excitation of slow MAC waves at scales of $\sim$ 10 km in the Earth's core may play a crucial role in the generation of the axial dipole field.

Diurnal variability of frontal dynamics, instability, and turbulence in a submesoscale upwelling filament

AbstractRecent high-resolution numerical simulations have shown that the diurnal variability in the atmospheric forcing strongly affects the dynamics, stability, and turbulence of submesoscale structures in the surface boundary layer (SBL). Field observations supporting the real-ocean relevance of these studies are, however, largely lacking at the moment. Here, the impact of large diurnal variations in the surface heat flux on a dense submesoscale upwelling filament in the Benguela upwelling system is investigated, based on a combination of densely-spaced turbulence microstructure observations and surface drifter data. Our data show that during nighttime and early-morning conditions, when solar radiation is still weak, frontal turbulence is generated by a mix of symmetric and shear instability. In this situation, turbulent diapycnal mixing is approximately balanced by frontal restratification associated with the cross-front secondary circulation. During daytime, when solar radiation is close to its peak value, the SBL quickly restratifies, the conditions for frontal instability are no longer fulfilled, and SBL turbulence collapses except for a thin wind-driven layer near the surface. The drifter data suggest that inertial oscillations periodically modulate the stability characteristics and energetics of the submesoscale fronts bounding the filament.

Geostrophic adjustment on the f-plane: Symmetric versus anti-symmetric initial height distributions

We study the fundamental process of geostrophic adjustment in infinitely long zonal and meridional channels of widths Ly and Lx, respectively, by deriving analytic solutions and simulating the linearized rotating shallow water equations (LRSWE). All LRSWE's variables are divided into time-independent (geostrophic) and time-dependent components. The latter includes Kelvin waves, Poincaré waves, and inertial oscillations. Explicit expressions are derived for both components, which are confirmed numerically. Anti-symmetric and symmetric initial height distributions, η0(x), are considered, both of which introduce a length scale, D, into the problem. We show that for an anti-symmetric η0(x), (i) the rate of approach to geostrophy is D-independent; (ii) the decay rate of inertial oscillations is ∝t−1/2; and (iii) for DLy≫1, the energy of the final state in any finite sub-domain of the channel exceeds that of the initial state, while for DLy≪1 the energy in the final state is smaller than in the initial state. In contrast, for a symmetric η0(x): (i) the rate of approach to geostrophy increases with D; (ii) the decay rate of inertial oscillations is ∝t−3/2, and (iii) the energy of the final state is always smaller than that of the initial state. In meridional channels, the effect of boundaries is to (i) block the waves, propagation to infinity; (ii) alter the spatial structure of the geostrophic flow; and (iii) discretize the frequency spectrum, thus eliminating the inertial oscillations. The ratio of wave energy to initial energy decreases with Lx for anti-symmetric η0(x) and increases with Lx for symmetric η0(x).

Nonlinear pulsations of horizontal jets

A shallow-water model with horizontally nonuniform density is used to study the dynamics of jet flows that arise under the influence of buoyancy and the Coriolis force. Within this approach, the jet is described by a self-similar compactly-localized solution and interpreted as a band of shear flow having a temperature contrast with the ambient fluid. In addition to stationary states, the dynamics of such jets admit cyclonic rotation with a constant angular velocity and transverse nonlinear pulsations. The phase portrait corresponding to this model shows that regimes with pulsating jets develop along closed trajectories bounded by the separatrix loop. The theory predicts that the period for warm jet pulsations is longer than the inertial oscillation period caused by the Earth’s rotation, while for cold jet pulsations, it is shorter. Thus, only warm jets can have a noticeable effect on the atmospheric dynamics in the synoptic range. In particular, they may well be responsible for additional spectral peaks that appear in this range of wind speed fluctuations.

On the Origin of Geostrophic Liquid Cylinders in Rotating Sphere with Oscillating Inner Core

The effect of non-axisymmetric inertial waves on the zonal flow structure in a rotating spherical shell is studied experimentally. The wave source is the inner core, which performs circular oscillations in the equatorial plane. The case of positive frequencies that corresponds to the advanced core motion is considered. As a result of non-linear effects, a quasi-two-dimensional axisymmetric steady flow with a complex distribution of angular velocity is generated. The extrema in the velocity profile are equivalent to the nested liquid geostrophic cylinders, which appearance is due to the interaction of the inertial waves within a viscous boundary layer. One of the extrema is associated with the critical latitude at the core boundary, where the inertial waves are excited. With an increase in the oscillation frequency, the position of this extremum gradually shifts to the rotation axis. Additional geostrophic circulation occurs close to the points where the inertial wave reflects from the cavity boundary.

On the linear and non-linear fluid response to the circular forcing in a rotating spherical shell

Fluid flow excited by a core oscillating in a rotating spherical cavity is experimentally investigated. Oscillations are set by an external inertial field so that in the reference frame of the cavity, the core moves along a circular trajectory around the rotation axis. Two situations are considered: when the core oscillations are co-directed or counter-directed with respect to the rotation of the cavity. The oscillating core is a source of non-axisymmetric inertial waves that form a system of cone-shaped shear layers in fluid bulk. Depending on the oscillation frequency, various inertial flow regimes arise, the spatial structure of which strongly depends on the sign of the oscillations. It is found that a strong non-linear response in the form of a steady zonal flow corresponds to each flow regime. The flow structure is a system of nested liquid geostrophic cylinders, one of which is associated with the critical latitude at the core boundary, where inertial waves are generated. The next one is associated with the wave reflection from the cavity boundary and is clearly manifested when they are focused on the wave attractor. The most intense zonal flow occurs when inertial waves are superposed and global vortex structures are resonantly excited.

Statistical Characteristics of Inertial Gravity Waves Over a Tropical Station in the Western Pacific Based on High‐Resolution GPS Radiosonde Soundings

AbstractThe characteristics of gravity wave activity in tropical areas have been extensively explored, but at present, less attention has been paid to the activity of inertial gravity waves (IGWs) in the Western Pacific based on in‐situ detection data. Based on the radiosonde data over Guam (13.3°N, 144.4°E) for 6 years (2013–2018), the hodograph method and spectrum analysis are used to analyze the statistical characteristics of IGWs in the troposphere (2–14 km) and stratosphere (18–28 km). The gravity wave (GW) energy in the troposphere has clear annual cycle, with the strongest activity in winter and the weakest in monsoon. In the stratosphere, the periodic variation of wave energy is interrupted due to the 2016 quasi‐biennial oscillation (QBO) disruption reflected in the zonal wind field. In the troposphere, the quantity of up‐propagating GWs is close to that of the down‐propagating GWs, while in the stratosphere, they are almost up‐propagating GWs. The horizontal propagation direction of GWs is mainly northeast and southwest in both the troposphere and the stratosphere, since the weak background wind field does not produce significant filtering effect. The spectral amplitude based on the normalized temperature perturbation has clear uniform annual variations in the troposphere and stratosphere, which are the largest in winter and the smallest in monsoon. Due to the “saturation” process in the up‐propagating process of GWs, the spectral slope is more negative in the stratosphere.