Taylor Goldstein Equation Research Articles

Stratified inclined duct: two-layer hydraulics and instabilities

The stratified inclined duct (SID) sustains an exchange flow in a long, gently sloping duct as a model for continuously forced density-stratified flows such as those found in estuaries. Experiments have shown that the emergence of interfacial waves and their transition to turbulence as the tilt angle is increased appears to be linked to a threshold in the exchange flow rate given by inviscid two-layer hydraulics. We uncover these hydraulic mechanisms by (i) using recent direct numerical simulations (DNS) providing full flow data in the key flow regimes (Zhu et al., J. Fluid Mech., vol. 969, 2023, A20), (ii) averaging these DNS into two layers, and (iii) using an inviscid two-layer shallow-water and instability theory to diagnose interfacial wave behaviour and provide physical insight. The laminar flow is subcritical and stable throughout the duct and hydraulically controlled at the ends of the duct. As the tilt is increased, the flow becomes supercritical everywhere and unstable to long waves. An internal jump featuring stationary waves first appears near the centre of the duct, then leads to larger-amplitude travelling waves, and to stronger jumps, wave breaking and intermittent turbulence at the largest tilt angle. Long waves described by the (nonlinear) shallow-water equation are interpreted locally as linear waves on a two-layer parallel base flow described by the Taylor–Goldstein equation. This link helps us to interpret long-wave instability and contrast it with short-wave (e.g. Kelvin–Helmholtz) instability. Our results suggest a transition to turbulence in SID through long-wave instability relying on vertical confinement by the top and bottom walls.

Spatial and temporal variability in mode-1 and mode-2 internal solitary waves from MODIS-Terra sun glint off the Amazon shelf

Abstract. The Amazon shelf is a key region for intense internal tides (ITs) and nonlinear internal solitary wave (ISWs) generation associated with them. The region shows well-marked seasonal variability (from March to July, MAMJJ, and from August to December, ASOND) of the circulation and stratification, which can both induce changes in the ISW physical characteristics. The description of the seasonal and neap–spring tidal variability in the ISWs off the Amazon shelf is performed for the first time using a meaningful data set composed of 140 MODIS-Terra imagery from 2005 to 2021, where about 500 ISW signatures were identified in the sun glint region. Previous studies have documented the existence of mode-1 ISWs, but the region appears as a newly described hotspot for mode-2 ISWs. ISW packets separated by typical mode-1 (95–170 km; 2.1–3.8 m s−1) and mode-2 (46–85 km; 1.0–1.9 m s−1) IT wavelengths have been identified and mapped coming from different IT generation sites. For each ISW, a group of waves (3 to 10) is generally follows the largest crest. The intra-packet distance between each wave in the group is about 10 to 20 km. Regions of higher occurrence of ISWs are spaced by a IT mode-1 wavelength. We make the assumption that it might correspond to the IT reflection beams at the surface, which may generate newer ISWs. The mean mode-1 and mode-2 inter-packet distances do not show significant differences according to their IT generation sites. The ISW activity is higher (more than 60 % of signatures) during spring tides than neap tides. In the region under the influence of the North Equatorial Counter Current (NECC), ISWs are separated by a mean mode-1 IT wavelength which is 14.3 % higher during ASOND than during MAMJJ due to a deeper thermocline and the reinforcement of the NECC. These ISWs are also characterized by a wider inter-packet distance distribution (higher standard deviation) that may be related to the stronger eddy kinetic energy (EKE) during ASOND compared to MAMJJ. The mean inter-packet distance of mode-2 ISWs remains almost unchanged during the two seasons, but the inter-packet distance distribution is wider in ASOND than in MAMJJ as for mode 1. Note that these results need to be treated with caution, as only few occurrences of mode-2 waves were found during MAMJJ. In the region of the NECC, the direction of propagation for all modes is very similar in MAMJJ (about 30∘ clockwise from the north), whereas, for ASOND, the ISWs propagate in a wider pathway (from 0 to 60∘ clockwise from the north), due to a much larger eddy activity. During ASOND, as the background flux goes further east, the inter-packet distances become larger (4 % for mode 1 and 7.8 % for mode 2). These results show that the reinforcement of the NECC in ASOND appears to play a role in diverting the waves towards the east, increasing their phase velocities and their eastern traveling direction component when compared to MAMJJ. Calculations of the IT velocities using the Taylor–Goldstein equation supported our results regarding the presence of ISWs associated with mode-2 ITs and additionally the IT seasonal variability.

Disk wakes in nonlinear stratification

Nonlinearity of density stratification modulates buoyancy effects. We report results from a body-inclusive large eddy simulation of a wake in nonlinear stratification, specifically for a circular disk at diameter-based Reynolds number ( ${\it Re} $ ) of $5000$ . Five density profiles are considered; the benchmark has linear stratification and the other four have hyperbolic tangent profiles of the same thickness to model a pycnocline. The disk moves inside the central core of the pycnocline in two of those four cases and, in the other two cases with a shifted density profile, the disk moves partially/completely outside the pycnocline. The maximum buoyancy frequency ( $N_{max}$ ) for all the profiles is the same. The first part of the study investigates the centred cases. Non-uniform stratification results in increasing wake turbulence relative to the benchmark owing to reduced suppression of turbulence production as well as wave trapping in the pycnocline. Steady lee waves are also quantified to understand the limitations of linear theory. The second part pays attention to the effect of a relative shift between the pycnocline and the disk. The wake defect velocity decays substantially faster in the cases with a shift and the wake has higher turbulence level. The effect of disk location on the Kelvin wake waves (a family of steady waves within the pycnocline) and its modal form is obtained and explained by solving the Taylor–Goldstein equation. The family of unsteady internal gravity waves that are generated by the wake is also studied and the effect of disk shift is quantified.

The Role of Ambient Turbulence in Canopy Wave Generation by Kelvin–Helmholtz Instability

We test the hypothesis that internal waves observed in flow over forest canopies are generated by Kelvin–Helmholtz instability. The waves were observed at night, under stably stratified and weak wind conditions, with a horizontally scanning aerosol lidar and an instrumented tower. The lidar images are used to determine the salient wavelength and phase propagation velocity of each episode. Time series data measured at the tower are then used to form vertical profiles of background velocity and buoyancy just before each observed wave event. The profiles are input to the Taylor–Goldstein equation to predict the phase velocity, wavelength and period of the fastest-growing linear instability, and the results compared with the lidar observations. The observed wavelengths tend to be longer than predicted by the Taylor–Goldstein theory, typically by a factor of two. That discrepancy is removed when the theory is extended to account for the effects of ambient, small-scale turbulence.

Internal Hydraulic Transition and Turbulent Mixing Observed in the Kuroshio over the I-Lan Ridge off Northeastern Taiwan

Abstract Complex small-scale processes and energetic turbulence are observed at a sill located on the I-Lan Ridge that spans across the strong Kuroshio off Taiwan. The current speed above the sill is strong (1.5 m s−1) and unsteady (±0.5 m s−1) due to the Kuroshio being modulated by the semidiurnal tide. Above the sill crest, isothermal domes, with vertical scales of ∼20 and ∼50 m during the low and high tides, respectively, are generated by turbulent mixing as a result of shear instability in the bottom boundary layer. Tidally modulated hydraulic character modifies the small-scale processes occurring on the leeward side of the sill. Criticality analysis, performed by solving the Taylor–Goldstein equation, suggests that the observed lee waves and intermediate layer sandwiched by two free shear layers are related to the mode-1 and mode-2 critical control between the sill crest and immediate lee, respectively. Around high tide, lee waves are advected further downstream, and only mode-1 critical control can occur, leading to a warm water depression. The shear instabilities ensuing from the hydraulic transition processes continuously mediate flow kinetic energy to turbulence such that the status of marginal instability where the Richardson number converges at approximately 0.25 is reached. The resultant eddy diffusivity Kρ is concentrated from O(10−4) to O(10−3) m2 s−1 and has a maximum value of 10 m2 s−1. The sill on the western flank of the Kuroshio is a hotspot for energetic mixing of Kuroshio waters and South China Sea waters.

Linear temporal stability analysis on the inviscid sheared convective boundary layer flow

A linear temporal stability analysis is conducted for inviscid sheared convective boundary layer flow, in which the sheared instability with stable stratification coexists with and caps over the thermal instability with unstable stratification. The classic Taylor–Goldstein equation is applied with different stratification factors Js and Jb in the Brunt–Väisälä frequency, respectively. Two shear-thermal hybrid instabilities, the hybrid shear stratified (HSS) and hybrid Rayleigh–Bénard (HRB) modes, are obtained by solving the eigenvalue problems. It is found that the temporal growth rates of the HSS and HRB modes vary differently with increased Jb in two distinct wavenumber (α̃) regions defined by the intersection point between the stability boundaries of the HSS and HRB modes. Based on Jb,cr where the temporal growth rate of the HSS and HRB are equal, a map of the unique critical boundary, which separates the effective regions of the HSS and HRB modes, is constructed and found to be dependent on Js, Jb, and α̃. The examinations of the subordinate eigenfunctions indicate that the shear instability is well developed in the HSS mode, in which the large vortex structures may prevail and suppress the formation of convective rolls; the shear instability in the HRB mode is either “partly developed” when Jb<Jb,cr or “undeveloped” when Jb>Jb,cr, thus only plays a secondary role to modify the dominant convective rolls, and as Jb increases, the eigenfunctions of the HSS mode exhibit different transitional behaviors in the two regions, signifying the “shear enhancement” and “shear sheltering” of the entrainment of buoyancy flux.

Some aspects in Kelvin-Helmholtz instability with and without Boussinesq approximation

The topic of this paper is the Kelvin-Helmholtz instability, a phenomenon which occurs on the interface of a stratified fluid, in the presence of a parallel shear flow, when there is a velocity and density difference across the interface of two adjacent layers. This paper focuses on a numerical simulation modelled by the Taylor-Goldstein equation, which represents a more realistic case compared to the basic Kelvin-Helmholtz shear flow. The Euler system is solved with new modelled smooth velocity and density profiles at the interface. The flux at cell boundaries is reconstructed by implementing a third order WENO (Weighted Essentially Non-Oscillatory) method. Next, a Riemann solver builds the fluxes at cell interfaces. The use of both Rusanov and HLLC solvers is investigated. Temporal discretization is done by applying the second order TVD (total variation diminishing) Runge-Kutta method on a uniform grid. Numerical simulations are performed comparatively for both Kelvin-Helmholtz and Taylor-Goldstein instabilities, on the same simulation domains. We find that increasing the number of grid points leads to a better accuracy in shear layer vortices visualization. Thus, we can conclude that applying the Taylor-Goldstein equation improves the realism in the general fluid instability modelling.

Impacts of subtidal motions and the earth rotation on modal characteristics of the semidiurnal internal tide

Subtidal motions, such as large-scale circulations and mesoscale eddies, frequently occupy on the propagation path of internal waves in the ocean. Through solving a modified Taylor–Goldstein equation with subtidal currents, density stratification, and the earth rotation, this study investigates the impacts of subtidal motions and the earth rotation on the modal characteristics of the semidiurnal internal tide (SIT) based on long-term mooring measurements in the northern South China Sea (SCS). It is shown that the modal characteristics of the SIT are significantly influenced by the time-varying subtidal current and density stratification associated with energetic subtidal motions in the northern SCS. The earth rotation plays a minor role in modulating the modal characteristics of the SIT in the northern SCS, but it becomes effective at middle and high latitudes when strong subtidal current shear associated with subtidal motions is present. Moreover, the strong subtidal current associated with subtidal motions may significantly affect the critical latitude of the SIT. The observational results reported here will help to improve our understanding of the modal characteristics of internal tides under the influence of subtidal motions in global oceans.

Initiation of ray tracing models: evolution of small-amplitude gravity wave packets in non-uniform background

This study introduces a new computational scheme for the linear evolution of internal gravity wave packets passing over strongly non-uniform stratifications and background flows as found, e.g., near the tropopause. Focusing on linear dispersion, which is dominant at small wave amplitudes, the scheme describes general wave superpositions arising from wave reflections near strong variations of the background stratification. Thus, it complements WKB theory, which is restricted to nearly monochromatic waves but covers weakly nonlinear effects in turn. One envisaged application of the method is the formulation of bottom-of-the-stratosphere starting conditions for ray tracing parameterizations that follow nonlinear gravity wave packets into the upper atmosphere. A key feature in this context is the method’s separation of wave packets into up- and downward-propagating components. The paper first summarizes a multilayer method for the numerical solution of the Taylor–Goldstein equation. Borrowing ideas from Eliassen and Palm (Geophys Publ 22:1–23, 1961), the scheme is based on partitioning the atmosphere into several uniformly stratified layers. This allows for analytical plane wave solutions in each layer, which are matched carefully to obtain continuously differentiable global eigenmode solutions. This scheme enables rapid evaluations of reflection and transmission coefficients for internal waves impinging on the tropopause from below as functions of frequency and horizontal wavenumber. The study then deals with a spectral method for propagating wave packets passing over non-uniform backgrounds. Such non-stationary solutions are approximated by superposition of Taylor–Goldstein eigenmodes. Particular attention is paid to an algorithm that translates wave packet initial data in the form of modulated sinusoidal signals into amplitude distributions for the system’s eigenmodes. With this initialization in place, the state of the perturbations at any given subsequent time is obtained by a single superposition of suitably phase-shifted eigenmodes, i.e., without any time-stepping iterations. Comparisons of solutions for wave packet evolution with those obtained from a nonlinear atmospheric flow solver reveal that apparently nonlinear effects can be the result of subtle linear wave packet dispersion.

The Miles Theorem and the First Boundary Value Problem for the Taylor-Goldstein Equation

The Miles Theorem and the First Boundary Value Problem for the Taylor-Goldstein Equation

The Miles Theorem and the first boundary value problem for the Taylor–Goldstein equation

The Miles Theorem and the first boundary value problem for the Taylor–Goldstein equation

HIMAWARI-8 GEOSTATIONARY SATELLITE OBSERVATION OF THE INTERNAL SOLITARY WAVES IN THE SOUTH CHINA SEA

Abstract. The new generation geostationary meteorological satellite, Himawari-8 (H-8), was launched in 2015. Its main payload, the Advanced Himawari Imager (AHI), can observe the earth with 10-minute interval and as high as 500–m spatial resolution. This makes the H-8 satellite an ideal data source for marine and atmospheric phenomena monitoring. In this study, the propagation of internal solitary waves (ISWs) in the South China Sea is investigated using AHI imagery time series for the first time. Three ISWs cases were studied at 3:30–8:00 UTC on 30 May, 2016. In all, 28 ISWs were detected and tracked between the time series image pairs. The propagation direction and phase speeds of these ISWs are calculated and analyzed. The observation results show that the properties of ISW propagation not stable and maintains nonlinear during its lifetime. The resultant ISW speeds agree well with the theoretical values estimated from the Taylor-Goldstein equation using Argo dataset. This study has demonstrated that the new generation geostationary satellite can be a useful tool to monitor and investigate the oceanic internal waves.

Effect of free surface on submerged stratified shear instabilities

In this paper, we have considered the effects of the shallowness of the domain as well as the air–water free surface on the stratified shear instabilities of the fluid underneath. First, we numerically solve the non-Boussinesq Taylor–Goldstein equation for smooth velocity and density profiles of a model shear layer with a free surface. When the depth of the fluid is relatively shallow compared to the shear layer thickness, the surface gravity waves existing at the free surface come closer to the waves existing in the shear layer. This can lead to resonant wave interactions, making the flow unstable to more varieties of modal instabilities. In order to obtain a deeper understanding of the instability mechanisms, we have performed analytical studies with broken-line profiles (profiles for which vorticity and density are piecewise constant). Furthermore, reduced-order broken-line profiles have also been developed, based on which dispersion diagrams are constructed. Through these diagrams we have underpinned the resonantly interacting waves leading to each type of instability. Two new instabilities have been found; one of them, referred to as the ‘surface gravity – interfacial gravity (SG-IG) mode’, arises due to the interaction between a surface gravity wave and an interfacial gravity wave, and would therefore be absent if there is no internal density stratification. The other one – the ‘surface gravity – lower vorticity (SG-LV) mode’, which arises due to the interaction between a surface gravity wave and the lower vorticity wave, surpasses Kelvin–Helmholtz (KH) instability to become the most unstable mode, provided the system is significantly shallow. Stability boundary of the SG-LV mode is found to be quite different from that of KH. In fact, KH becomes negligible for relatively shallow flows, while SG-LV’s growth rate is significant – comparable to the growth rate of KH for unbounded domains (${\approx}0.18$). Moreover, the SG-LV mode is found to be analogous to the barotropic mode observed in two-layer quiescent flows. We have found that the effect of a free surface on the Holmboe instability is not appreciable. Holmboe in the presence of a free surface is found to be analogous to the baroclinic mode observed in two-layer quiescent flows. Except for the Holmboe instability, remarkable differences are observed in all other instabilities occurring in shallow domains when the air–water interface is replaced by a rigid lid. We infer that the rigid-lid approximation is valid for large vertical domains and should be applied with caution otherwise. Furthermore, we have also shown that if shear is absent at the free surface, our problem can be modelled using Boussinesq approximation, that is, $O(1)$ density variations in the inertial terms can still be neglected.

The miles theorem and new particular solutions to the Taylor–Goldstein equation

The direct Lyapunov method is used to prove the absolute linear instability of steadystate plane-parallel shear flows of an inviscid stratified incompressible fluid in the gravity field with respect to plane perturbations both in the Boussinesq and non-Boussinesq approximations. A strict description is given for the applicability of the known necessary condition for linear instability of steady-state plane-parallel shear flows of an ideal nonuniform (by density) incompressible fluid in the gravity field both in the Boussinesq and non-Boussinesq approximations (the Miles theorem). Analytical examples of illustrative character are constructed.

Formation and evolution of flapping and ballooning waves in magnetospheric plasma sheet

By adopting Lembege & Pellat’s 2D plasma-sheet model, we investigate the flankward flapping motion and Sunward ballooning propagation driven by an external source (e.g., magnetic reconnection) produced initially at the sheet center. Within the ideal MHD framework, we adopt the WKB approximation to obtain the Taylor–Goldstein equation of magnetic perturbations. Fourier spectral method and Runge–Kutta method are employed in numerical simulations, respectively, under the flapping and ballooning conditions. Studies expose that the magnetic shears in the sheet are responsible for the flapping waves, while the magnetic curvature and the plasma gradient are responsible for the ballooning waves. In addition, the flapping motion has three phases in its temporal development: fast damping phase, slow recovery phase, and quasi-stabilized phase; it is also characterized by two patterns in space: propagating wave pattern and standing wave pattern. Moreover, the ballooning modes are gradually damped toward the Earth, with a wavelength in a scale size of magnetic curvature or plasma inhomogeneity, only 1–7% of the flapping one; the envelops of the ballooning waves are similar to that of the observed bursty bulk flows moving toward the Earth.

Stability boundaries and sufficient stability conditions for stably stratified, monotonic shear flows

Linear stability of inviscid, parallel, and stably stratified shear flow is studied under the assumption of smooth strictly monotonic profiles of shear flow and density, so that the local Richardson number is positive everywhere. The marginally unstable modes are systematically found by solving a one-parameter family of regular Sturm–Liouville problems, which can determine the stability boundaries more efficiently than solving the Taylor–Goldstein equation directly. By arguing for the non-existence of a marginally unstable mode, we derive new sufficient conditions for stability, which generalize the Rayleigh–Fjørtoft criterion for unstratified shear flows.

On instability and mixing on the UK Continental Shelf

The stability of stratified flows at locations in the Clyde, Irish and Celtic Seas on the UK Continental Shelf is examined. Flows are averaged over periods of 12–30min in each hour, corresponding to the times taken to obtain reliable estimates of the rate of dissipation of turbulent kinetic energy per unit mass, ε. The Taylor–Goldstein equation is solved to find the maximum growth rate of small disturbances to these averaged flows, and the critical gradient Richardson number, Ric. The proportion of unstable periods where the minimum gradient Richardson number, Rimin, is less than Ric is about 35%. Cases are found in which Ric < 0.25; 37% of the flows with Rimin < 0.25 are stable, and Ric < 0.24 in 68% of the periods where Rimin < 0.25. Marginal conditions with 0.8 < Rimin/Ric < 1.2 occur in 30% of the periods examined. The mean dissipation rate at the level where the fastest growing disturbance has its maximum amplitude is examined to assess whether the turbulence there is isotropic and how it relates to the wave–turbulence boundary. It is concluded that there is a background level of dissipation that is augmented by instability; instability of the averaged flow does not account for all the turbulence observed in mid-water. The effects of a horizontal separation of the measurements of shear and buoyancy are considered. The available data do not support the hypothesis that the turbulent flows observed on the UK shelf adjust rapidly to conditions that are close to being marginal, or that flows in a particular location and period of time in one sea have stability characteristics that are very similar to those in another.

Laboratory experiments and simulations for solitary internal waves with trapped cores

AbstractWe perform simultaneous coplanar measurements of velocity and density in solitary internal waves with trapped cores, as well as viscous numerical simulations. Our set-up comprises a thin stratified layer (approximately 15 % of the overall fluid depth) overlaying a deep homogeneous layer. We consider waves propagating near a free surface, as well as near a rigid no-slip lid. In the free-surface case, all trapped-core waves exhibit a strong shear instability. We propose that Marangoni effects are responsible for this instability, and use our velocity measurements to perform quantitative calculations supporting this hypothesis. These surface-tension effects appear to be difficult to avoid at the experimental scale. By contrast, our experiments with a no-slip lid yield robust waves with large cores. In order to consider larger-amplitude waves, we complement our experiments with viscous numerical simulations, employing a longer virtual tank. Where overlap exists, our experiments and simulations are in good agreement. In order to provide a robust definition of the trapped core, we propose bounding it as a Lagrangian coherent structure (instead of using a closed streamline, as has been done traditionally). This construction is less sensitive to small errors in the velocity field, and to small three-dimensional effects. In order to retain only flows near equilibrium, we introduce a steadiness criterion, based on the rate of change of the density in the core. We use this criterion to successfully select within our experiments and simulations a family of quasi-steady robust flows that exhibit good collapse in their properties. The core circulation is small (at most, around 10 % of the baroclinic wave circulation). The core density is essentially uniform; the standard deviation of the density, in the core region, is less than 4 % of the full density range. We also calculate the circulation, kinetic energy and available potential energy of these waves. We find that these results are consistent with predictions from Dubreil-Jacotin–Long theory for waves with a uniform-density irrotational core, except for an offset, which we suggest is associated with viscous effects. Finally, by computing Richardson-number fields, and performing a temporal stability analysis based on the Taylor–Goldstein equation, we show that our results are consistent with empirical stability criteria in the literature.

Dynamic responses of a riser under combined excitation of internal waves and background currents

In this study, the dynamic responses of a riser under the combined excitation of internal waves and background currents are studied. A modified Taylor-Goldstein equation is used to calculate the internal waves vertical structures when background currents exist. By imposing rigid-lid boundary condition, the equation is solved by Thompson-Haskell method. Based on the principle of virtual work, a nonlinear differential equation for riser motions is established combined with the modified Morison formula. Using Newmark-β method, the motion equation is solved in time domain. It is observed that the internal waves without currents exhibit dominated effect on dynamic response of a riser in the first two modes. With the effects of the background currents, the motion displacements of the riser will increase significantly in both cases that wave goes along and against the currents. This phenomenon is most obviously observed at the motions in the first mode.

Topographic control of stratified flows: upstream jets, blocking and isolating layers

AbstractOptimal solutions to the nonlinear, hydrostatic, Boussinesq equations are developed for steady, density-stratified, topographically controlled flows characterized by blocking and upstream influence. These flows are jet-like upstream of an isolated obstacle and are contained within an asymmetric, thinning stream tube that is accelerated as it passes over the crest. A stagnant, nearly uniform-density isolating layer, surrounded by a bifurcated uppermost streamline, separates the accelerated flow from an uncoupled flow above. The flows are optimal in the sense that, for a given stratification, the solutions maximize the topographic rise above the blocking level required for hydraulic control while minimizing the total energy of the flow. Hydraulic control is defined mathematically by the asymmetry of the accelerated flow as it passes the crest. A subsequent analysis of the Taylor–Goldstein equation shows that these sheared, non-uniformly stratified flows are indeed subcritical upstream, critical at the crest, and supercritical downstream with respect to gravest-mode, long internal waves. The flows obtained are relevant to arrested wedge flows, selective withdrawal, stratified towing experiments, tidal flow over topography and atmospheric flows over mountains.