Dubreil Jacotin Long Research Articles

Graph theoretic methods for assessing the effects of a background shear current on internal solitary waves

Internal solitary waves are a widely observed phenomenon in natural waters. Mathematically, they are fundamentally a nonlinear phenomenon that differs from the paradigm of turbulence, in that energy does not move across scales. Internal solitary waves may be computed from the Dubreil–Jacotin Long equation, which is a scalar partial differential equation that is equivalent to the stratified Euler equations. When a background shear current is present the algebraic complexity of the problem increases substantially. We present an alternative point of view for characterizing the situation with a shear current using Lagrangian (particle-like) models analysed with graph theoretic methods. We find that this yields a novel, data-centric framework for analysis that could prove useful well beyond the study of internal solitary waves.

Experimental study of the vertical structure of internal solitary waves in the continuous pycnocline

In order to reveal the complex structural characteristics of internal solitary waves (ISWs) in the actual ocean, an experimental study of the vertical structure of ISWs in the continuous pycnocline (a transition layer with sharp density changes) was conducted in a stratified fluid flume. The gravity collapse method was used to generate ISWs, and their wave-flow fields were measured using a coupled wave-flow measurement technique. The vertical structure of wave-flow fields was investigated as was the applicability of the Dubreil–Jacotin–Long (DJL) equation. The results show that the waveform of ISWs contains multiple isodensity lines that varied with fluid depth. The wave amplitude and wavelength of ISWs exhibited depth-dependent changes, which were negatively correlated. The vertical structure of the flow fields exhibited an approximate circular wave packet, with stronger horizontal flow than vertical flow. The larger the characteristic amplitude, the stronger the intensity of the flow field, and the faster the intensity of the vertical flow field increased. The applicability of the DJL equation was closely related to the stratified environment, with better agreement when the upper layer fluid constituted a larger ratio of the total fluid thickness.

Non-Boussinesq effect on the internal solitary wave propagation under a Lagrangian-like description for various pycnocline thickness conditions

The Dubreil–Jacotin–Long (DJL) equation is an exact model for the steady propagation of internal solitary waves (ISWs) under Boussinesq approximation. The aim of this study is to discuss the non-Boussinesq effect on ISWs initialized by DJL equation for various pycnocline thickness conditions. The simulated results show that the non-Boussinesq effect can indeed lead to instability of DJL equation, and the pycnocline thickness will affect the amplitude of re-stable ISWs and the separation process of tail waves. Moreover, in order to extract features of the non-Boussinesq instability and the location of the tail wave separation, we propose an effective analytical approach for the quasi-traveling wave problem inspired by the definition of classical Lagrangian coherent structure (LCS). The improved finite-time Lyapunov exponent with a moving window (FTLEmw) is a Lagrangian-like description with considering the prior information of physical phenomenon. For a steady ISW propagation in the Boussinesq system, the repelling and attracting FTLEmv fields are symmetrical and the field ridge behaves as a continuous spiral topology, which suggests that particles on different streamlines in the flow structure have different rotation periods, while the streamlines are not actually completely closed due to the forward propagation of the wave. During the main unstable stage of non-Boussinesq effect, the topology of separated tail wave which is disconnected from the ISW main part can be clearly distinguished. The tail wave separation locations identified by the improved FTLEmv field and the classical FTLE field are not the same, which illustrates the difference between the two perspectives of flow structure change and fluid particle motion. In addition, the leading edge, trailing edge and main part of ISWs will gradually be divided into three-part FTLEmv topologies in the thicker pycnocline conditions.

Anatomy of Mode-1 Internal Solitary Waves Derived From Seaglider Observations in the Northern South China Sea

Abstract Observations from a Seaglider, two pressure-sensor-equipped inverted echo sounders (PIESs), and a thermistor chain (T-chain) mooring were used to determine the waveform and timing of internal solitary waves (ISWs) over the continental slope east of Dongsha Atoll. The Korteweg–de Vries (KdV) and Dubreil–Jacotin–Long (DJL) equations supplemented the data from repeated profiling by the glider at a fixed position (depth ∼1017 m) during 19–24 May 2019. The glider-recorded pressure perturbations were used to compute the rarely measured vertical velocity (w) with a static glider flight model. After removing the internal tide–caused vertical velocity, the w of the eight mode-1 ISWs ranged from −0.35 to 0.36 m s−1 with an uncertainty of ±0.005 m s−1 due to turbulent oscillations and measurement error. The horizontal velocity profiles, wave speeds, and amplitudes of the eight ISWs were further derived from the KdV and DJL equations using the glider-observed w and potential density profiles. The mean speed of the corresponding ISW from the PIES deployed at ∼2000 m depth to the T-chain moored at 500 m depth and the 19°C isotherm displacement computed from the T-chain were used to validate the waveform derived from KdV and DJL. The validation suggests that the DJL equation provides reasonably representative wave speed and amplitude for the eight ISWs compared to the KdV equation. Stand-alone glider data provide near-real-time hydrography and vertical velocities for mode-1 ISWs and are useful for characterizing the anatomy of ISWs and validating numerical simulations of these waves. Significance Statement Internal solitary waves (ISWs), which vertically displace isotherms by approximately 100 m, considerably affect nutrient pumping, turbulent mixing, acoustic propagation, underwater navigation, bedform generation, and engineering structures in the ocean. A complete understanding of their anatomy and dynamics has many applications, such as predicting the timing and position of mode-1 ISWs and evaluating their environmental impacts. To improve our understanding of these waves and validate the two major theories based on the Korteweg–de Vries (KdV) and Dubreil–Jacotin–Long (DJL) equations, the hydrography data collected from stand-alone, real-time profiling of an autonomous underwater vehicle (Seaglider) have proven to be useful in determining the waveform of these transbasin ISWs in deep water. The solutions to the DJL equation show good agreement with the properties of mode-1 ISWs obtained from the rare in situ data, whereas the solutions to the KdV equation underestimate these properties. Seaglider observations also provide in situ data to evaluate the performance of numerical simulations and forecasting of ISWs in the northern South China Sea.

Effects of the inhomogeneous vertical structure of an internal solitary wave on the force exerted on a horizontal transverse cylinder

The vertical inhomogeneous structure of an internal solitary wave (ISW) in a continuous density pycnocline and its force exerted on a horizontal transverse cylinder are investigated theoretically and experimentally. The Dubreil–Jacotin–Long equation is used to describe the inhomogeneous vertical structure of the ISW, and a formula for calculating the vertical force on the cylinder in the inhomogeneous vertical structure is proposed. The inhomogeneous vertical structure of the ISW and its vertical force on a horizontal transverse cylinder are experimentally measured in a large stratified fluid flume. It is shown that the inhomogeneous vertical structure of the ISW is characterized by both inhomogeneous vertical distributions of ISW envelopes and amplitudes. The inhomogeneous vertical structure of the flow field is characterized by the shear distribution of the horizontal velocity above and below the continuous density pycnocline, as well as the reversed distribution of the vertical velocity on the windward and leeward sides of the ISW. The vertical force characteristics on the cylinder in the continuous density pycnocline, as well as the physical mechanism of the influence of the ISW inhomogeneous vertical structure on the vertical force, are obtained. The peak value of the vertical force on the cylinder situated at the pycnocline increases with the increase in ISW amplitude, and also, it increases and then decreases with the increase in submerged depth. Considering the inhomogeneous vertical structure characteristics of the ISW in an actual ocean environment, the average relative error in the vertical force calculation can be more effectively reduced by using the continuous density pycnocline force model than the strict two-layer fluid force model. The actual characteristics of ISWs in a stratified ocean environment can be objectively described, and the estimation accuracy of the vertical force on underwater objects can be greatly improved.

ISWFoam: a numerical model for internal solitary wave simulation in continuously stratified fluids

Abstract. A numerical model, ISWFoam, for simulating internal solitary waves (ISWs) in continuously stratified, incompressible, viscous fluids is developed based on a fully three-dimensional (3D) Navier–Stokes equation using the open-source code OpenFOAM®. This model combines the density transport equation with the Reynolds-averaged Navier–Stokes equation with the Coriolis force, and the model discrete equation adopts the finite-volume method. The k–ω SST turbulence model has also been modified according to the variable density field. ISWFoam provides two initial wave generation methods to generate an ISW in continuously stratified fluids, including solving the weakly nonlinear models of the extended Korteweg–de Vries (eKdV) equation and the fully nonlinear models of the Dubreil–Jacotin–Long (DJL) equation. Grid independence tests for ISWFoam are performed, and considering the accuracy and computing efficiency, the appropriate grid size of the ISW simulation is recommended to be 1/150th of the characteristic length and 1/25th of the ISW amplitude. Model verifications are conducted through comparisons between the simulated and experimental data for ISW propagation examples over a flat bottom section, including laboratory scale and actual ocean scale, a submerged triangular ridge, a Gaussian ridge, and slope. The laboratory test results, including the ISW profile, wave breaking location, ISW arrival time, and the spatial and temporal changes in the mixture region, are well reproduced by ISWFoam. The ISWFoam model with unstructured grids and local mesh refinement can effectively simulate the evolution of ISWs, the ISW breaking phenomenon, waveform inversion of ISWs, and the interaction between ISWs and complex topography.

The effect of strong shear on internal solitary-like waves

Abstract. Large-amplitude internal waves in the ocean propagate in a dynamic, highly variable environment with changes in background current, local depth, and stratification. The Dubreil–Jacotin–Long, or DJL, theory of exact internal solitary waves can account for a background shear, doing so at the cost of algebraic complexity and a lack of a mathematical proof of algorithm convergence. Waves in the presence of shear that is strong enough to preclude theoretical calculations have been reported in observations. We report on high-resolution simulations of stratified adjustment in the presence of strong shear currents. We find instances of large-amplitude solitary-like waves with recirculating cores in parameter regimes for which DJL theory fails and of wave types that are completely different in shape from classical internal solitary waves. Both are spontaneously generated from general initial conditions. Some of the waves observed are associated with critical layers, but others exhibit a propagation speed that is very near the background current maximum. As such they are not freely propagating solitary waves, and a DJL theory would not apply. We thus provide a partial reconciliation between observations and theory.

Long‐Term Observations of Shoaling Internal Solitary Waves in the Northern South China Sea

AbstractShoaling internal solitary waves (ISWs) were observed at three mooring sites on the upper continental slope in the northern South China Sea over a period of 5–11 months at water depths of 600, 430, and 350 m. Their properties exhibit a fortnightly variation because of their origination from internal tides. ISW amplitudes, current speeds, and propagation speeds are greater and wave widths narrower in summer than in winter, consistent with the effect of increased stratification in summer, as confirmed by Dubreil‐Jacotin‐Long (DJL) solutions. As ISWs propagate up the slope, the differential response of current and propagation speeds to bottom topography provides an opportunity for convective breaking of ISWs. Convective breaking occurs mostly between 430 and 600‐m depths and exhibits a marginal convective instability status such that (a) the maximum current speed remains nearly equal to the propagation speed and (b) for large‐amplitude waves the current speed and propagation speed decrease at nearly the same rate between 600 and 430‐m depths. The marginal convective instability occurs because ISWs adjust gradually to the gently sloping bottom and preserve their structural integrity after the onset of breaking. Vertical velocity variances behind the leading ISWs, which serve as a surrogate for the number of trailing waves, increase when ISWs reach the convective breaking limit, suggesting that convective breaking may accelerate the fission process in leading ISWs or that convective breaking is accompanied by an enhanced nonlinear dispersion of waves trailing ISWs generated by internal tides.

Internal solitary waves with subsurface cores

Large internal solitary waves with subsurface cores have recently been observed in the South China Sea. Here fully nonlinear solutions of the Dubreil–Jacotin–Long equation are used to study the conditions under which such cores exist. We find that the location of the cores, either at the surface or below the surface, is largely determined by the sign of the vorticity of the near-surface background current. The results of a numerical simulation of a two-dimensional shoaling internal solitary wave are presented which illustrate the formation of a subsurface core.

Optimal transient growth in thin-interface internal solitary waves

The dynamics of perturbations to large-amplitude internal solitary waves (ISWs) in two-layered flows with thin interfaces is analysed by means of linear optimal transient growth methods. Optimal perturbations are computed through direct–adjoint iterations of the Navier–Stokes equations linearized around inviscid, steady ISWs obtained from the Dubreil-Jacotin–Long (DJL) equation. Optimal perturbations are found as a function of the ISW phase velocity $c$ (alternatively amplitude) for one representative stratification. These disturbances are found to be localized wave-like packets that originate just upstream of the ISW self-induced zone (for large enough $c$) of potentially unstable Richardson number, $Ri<0.25$. They propagate through the base wave as coherent packets whose total energy gain increases rapidly with $c$. The optimal disturbances are also shown to be relevant to DJL solitary waves that have been modified by viscosity representative of laboratory experiments. The optimal disturbances are compared to the local Wentzel–Kramers–Brillouin (WKB) approximation for spatially growing Kelvin–Helmholtz (K–H) waves through the $Ri<0.25$ zone. The WKB approach is able to capture properties (e.g. carrier frequency, wavenumber and energy gain) of the optimal disturbances except for an initial phase of non-normal growth due to the Orr mechanism. The non-normal growth can be a substantial portion of the total gain, especially for ISWs that are weakly unstable to K–H waves. The linear evolution of Gaussian packets of linear free waves with the same carrier frequency as the optimal disturbances is shown to result in less energy gain than found for either the optimal perturbations or the WKB approximation due to non-normal effects that cause absorption of disturbance energy into the leading face of the wave. Two-dimensional numerical calculations of the nonlinear evolution of optimal disturbance packets leads to the generation of large-amplitude K–H billows that can emerge on the leading face of the wave and that break down into turbulence in the lee of the wave. The nonlinear calculations are used to derive a slowly varying model of ISW decay due to repeated encounters with optimal or free wave packets. Field observations of unstable ISW by Moum et al. (J. Phys. Oceanogr., vol. 33 (10), 2003, pp. 2093–2112) are consistent with excitation by optimal disturbances.

Observations of nonlinear internal waves at a persistent coastal upwelling front

We collected high-resolution observations of nonlinear internal waves (NLIWs) at a persistent upwelling front in the shallow coastal environment (~20m) of northern Monterey Bay, CA. The coastal upwelling front forms between recently upwelled waters and warmer stratified waters that are trapped in the bay (upwelling shadow). The front propagates up and down the coast in the along-shore direction as a buoyant plume front due to modulation by strong diurnal wind forcing. The evolution of the coastal upwelling front, and the subsequent modulation of background environmental conditions, is examined using both individual events and composite day averages. We demonstrate that regional-scale upwelling and local diurnal wind forcing are key components controlling local stratification and the formation of internal wave guides that allow for high-frequency internal wave activity. Finally, we discuss the ability of theoretical models to describe particularly large-amplitude internal waves that exist in the presence of a strong background shear and test a fully nonlinear model (i.e., the Dubreil–Jacotin–Long equation).

Strong mode-mode interactions in internal solitary-like waves

Classical linear theory presents vertically trapped internal waves of different modes as completely uncoupled. This description carries over to the simplest weakly nonlinear theory for internal solitary waves, the Korteweg-de Vries theory. The balance between weakly nonlinear and dispersive effects in this theory allows for soliton solutions, meaning that waves emerge from collisions without changing form. However, exact mode-1 internal solitary waves have been shown to exhibit departures from soliton behaviour during overtaking collisions. We present a numerical investigation of the strong modal coupling between mode-1 and mode-2 internal solitary-like waves during head-on and overtaking collisions. We begin by presenting a “clean” theoretical setup using an exact theory (the Dubreil Jacotin Long equation) for the mode-1 wave and weakly nonlinear theory for the mode-2 wave to initialize the numerical model. During the collision, the mode-2 wave is significantly deformed by the mode-1 wave-induced currents, and indeed, by the end of the collision, the mode-2 wave has lost coherence almost entirely. We discuss how the collisions change as the amplitude of the mode-1 wave decreases, as the mode-1 wave becomes broad crested, and when multiple pycnoclines preclude mode-2 wave breaking and the formation of quasi-trapped cores in the mode-2 waves. We demonstrate where viscous dissipation occurs during the collisions, finding it slightly enhanced in the near pycnocline region, but not to the point where it can explain the loss of coherence. Subsequently, we use linear theory to demonstrate that it is a combination of the pycnocline deformation and the shear across the pycnocline centre due to the mode-1 waves, which alters the structure of the mode-2 waves and leads to the loss of coherence. In fact, the shear is vital, and with only a deformed pycnocline, mode-2 wave structure is only slightly altered. We present the results of a direct numerical simulation on experimental scales in which both mode-1 and mode-2 waves are generated by stratified adjustment. This simulation confirms that the numerical results should be readily observable in the laboratory. We conclude by revisiting existing weakly nonlinear theory for collisions, finding a surprising twist on the well established notions of “weak” and “strong” collisions.

Trapped internal waves over topography: Non-Boussinesq effects, symmetry breaking and downstream recovery jumps

It is well-known that in certain parameter regimes, the steady flow of a density stratified fluid over topography can yield large amplitude internal waves. We discuss an embedded boundary method to solve the Dubreil-Jacotin-Long (DJL) equation for steady-state, supercritical flows over topography in an inviscid, stratified fluid. The DJL equation is equivalent to the full set of stratified steady Euler equations and thus the waves we compute are exact nonlinear solutions. The numerical method presented yields far better scaling with increase in grid size than other iterative methods that have been used to solve this equation, and this in turn allows for a more thorough exploration of parameter space. For waves under the Boussinesq approximation, we contrast the properties of trapped waves over hill-like and valley-like topography, finding that the symmetry of freely propagating solitary waves when the stratification is reflected across the middepth is not present for trapped waves. We extend the derivation of the DJL equation to the non-Boussinesq case and discuss the effect of the new, non-Boussinesq terms on the structure of the trapped waves, finding that the sharp transition between large and small amplitude waves observed under the Boussinesq approximation is much more gradual when the Boussinesq approximation is relaxed. Finally, we demonstrate the existence of asymmetric steady states over hill-like topography where the flow is subcritical upstream of the topography but transitions to supercritical somewhere over the hill. Waves in this new class of exact solutions are related to so-called downstream recovery jumps predicted on the basis of hydrostatic (shallow water) theories, but when breaking does not occur the recovery jump does not stop propagating downstream and an asymmetric state across the topography maximum is reached for long times.

Shear instability of internal solitary waves in Euler fluids with thin pycnoclines

AbstractThe stability with respect to initial condition perturbations of solitary travelling-wave solutions of the Euler equations for continuously, stably stratified, near two-layer fluids is examined numerically and analytically for a set of parameters of relevance for laboratory experiments. Numerical travelling-wave solutions of the Dubreil–Jacotin–Long equation are first obtained with a variant of Turkington, Eyland and Wang’s iterative code by testing convergence on the equation’s residual. In this way, stationary solutions with very thin pycnoclines (and small Richardson numbers) approaching the near two-layer configurations used in experiments can be obtained, allowing for a stability study free of non-stationary effects, introduced by lack of numerical resolution, which develop when these solutions are used as initial conditions in a time-dependent evolution code. The thin pycnoclines in this study permit analytical results to be derived from strongly nonlinear models and their predictions compared with carefully controlled numerical simulations. This brings forth shortcomings of simple criteria for shear instability manifestations based on parallel shear approximations due to subtle higher-order effects. In particular, evidence is provided that the fore–aft asymmetric growth observed in all simulations requires non-parallel shear analysis. Collectively, the results of this study reveal that while the wave-induced shear can locally reach unstable configurations and give rise to local convective instability, the global wave/self-generated shear system is in fact stable, even for extreme cases of thin pycnoclines and near-maximum-amplitude waves.

Intrinsic Breaking of Internal Solitary Waves in a Deep Lake

Based on simulations with the Dubreil-Jacotin-Long (DJL) equation, the limiting amplitude and the breaking mechanisms of internal solitary waves of depression (ISWs) are predicted for different background stratifications. These theoretical predictions are compared to the amplitude and the stability of the leading internal solitary waves of more than 200 trains of ISWs observed in the centre of a sub-basin of Lake Constance. The comparison of the model results with the field observations indicates that the simulated limiting amplitude of the ISWs provides an excellent prediction of the critical wave height above which ISWs break in the field. Shear instabilities and convective instabilities are each responsible for about half of the predicted wave breaking events. The data suggest the presence of core-like structures within the convectively unstable waves, but fully developed and stable cores were not observed. The lack of stable trapped cores in the field can be explained by the results from dynamic simulations of ISWs with trapped cores which demonstrate that even slight disturbances of the background stratification cause trapped cores to become unstable.

Seasonal variation of solitary wave properties in Lake Constance

The properties of internal solitary waves (ISWs) depend on the stratification of the water body. In most climatic regions the stratification in lakes and oceans varies during the year, and hence the properties of the ISWs can also be expected to change over the seasons. On the basis of a long‐term temperature time series recorded over 6 years, this paper investigates seasonal changes in the characteristic properties of ISWs in Lake Überlingen, a subbasin of Lake Constance. A large number of ISWs with amplitudes ranging from 3 m to 30 m were identified. More than 15% of the leading ISWs of a wave train were associated with density inversions, often indicating shear instabilities or trapped cores. For all waves the propagation depth and the value of a nonlinearity index nlp providing the degree of nonlinearity were determined, propagation depth being the rest height of the isotherm undergoing maximum displacement and nlp the ratio between wave amplitude and propagation depth. The index nlp was found to be a good parameter for predicting the occurrence of inversions. The statistical analysis of the wave properties derived from the observations revealed that the degree of nonlinearity of the ISWs changes with season. Complementary to the statistical analysis, the seasonally averaged ISW properties were compared with wave prototypes obtained numerically from the Dubreil‐Jacotin‐Long (DJL) and the stratified Korteweg‐deVries (KdV) models. The simulations indicate that the typical stratification and its seasonal variation are responsible for the degree and the seasonality of nonlinearity of the ISWs.

Spectral methods for internal waves: indistinguishable density profiles and double-humped solitary waves

Abstract. Internal solitary waves are widely observed in both the oceans and large lakes. They can be described by a variety of mathematical theories, covering the full spectrum from first order asymptotic theory (i.e. Korteweg-de Vries, or KdV, theory), through higher order extensions of weakly nonlinear-weakly nonhydrostatic theory, to fully nonlinear-weakly nonhydrostatic theories and finally exact theory based on the Dubreil-Jacotin-Long (DJL) equation that is formally equivalent to the full set of Euler equations. We discuss how spectral and pseudospectral methods allow for the computation of novel phenomena in both approximate and exact theories. In particular we construct markedly different density profiles for which the coefficients in the KdV theory are very nearly identical. These two density profiles yield qualitatively different behaviour for both exact, or fully nonlinear, waves computed using the DJL equation and in dynamic simulations of the time dependent Euler equations. For exact, DJL, theory we compute exact solitary waves with two-scales, or so-called double-humped waves.

Observations of a Large-Amplitude Internal Wave Train and Its Reflection off a Steep Slope

Abstract Remote and in situ field observations documenting the reflection of a normally incident, short, and large-amplitude internal wave train off a steep slope are presented and interpreted with the help of the Dubreil–Jacotin–Long theory. Of the seven remotely observed waves that composed the incoming wave train, five were observed to reflect. It is estimated that the incoming wave train carried Ei = (24 ± 4) × 104 J m−1 to the boundary. The reflection coefficient, defined as the ratio of reflected to incoming wave train energies, is estimated to be R = 0.5 ± 0.2. This is about 0.4 lower than parameterizations in the literature, which are based on reflections of single solitary waves, would suggest. It is also shown that the characteristics of the wave-boundary situation observed in the field are outside the parameter space examined in previous laboratory and numerical experiments on internal solitary wave reflectance. This casts doubts on extrapolating current laboratory-based knowledge to fjord-like systems and calls for more research on internal solitary wave reflectance.

Numerical simulation of supercritical trapped internal waves over topography

We consider the steady flow of a stratified fluid over topography in a fluid of finite vertical extent, as typified by experimental flumes with a rigid lid or the ocean under the rigid lid approximation. We do not specify a functional form of the upstream stratification or background current and derive a general version of the Dubreil–Jacotin–Long equation appropriate for the problem. This elliptic equation is strongly nonlinear and we develop an efficient, pseudospectral, iterative method for its numerical solution. The method allows us to compute laminar, trapped waves with amplitudes more than 50% of the depth of the fluid. We find that when either a background shear current is present or the topography is narrow enough, multiple steady states are possible and we confirm this finding by using integrations of the full time-dependent Euler equations. We discuss instances of waves with closed streamlines, finding that the presence of shear allows for waves with vortex cores that persist for long times in time-dependent simulations and match well with solutions of the steady theory. In contrast, streamline overturning in the absence of upstream shear only occurs for flows that are stratified near the surface and in this instance, time-dependent simulations yield unsteady cores that do not match steady results very well.

A model for large-amplitude internal solitary waves with trapped cores

Abstract. Large-amplitude internal solitary waves in continuously stratified systems can be found by solution of the Dubreil-Jacotin-Long (DJL) equation. For finite ambient density gradients at the surface (bottom) for waves of depression (elevation) these solutions may develop recirculating cores for wave speeds above a critical value. As typically modeled, these recirculating cores contain densities outside the ambient range, may be statically unstable, and thus are physically questionable. To address these issues the problem for trapped-core solitary waves is reformulated. A finite core of homogeneous density and velocity, but unknown shape, is assumed. The core density is arbitrary, but generally set equal to the ambient density on the streamline bounding the core. The flow outside the core satisfies the DJL equation. The flow in the core is given by a vorticity-streamfunction relation that may be arbitrarily specified. For simplicity, the simplest choice of a stagnant, zero vorticity core in the frame of the wave is assumed. A pressure matching condition is imposed along the core boundary. Simultaneous numerical solution of the DJL equation and the core condition gives the exterior flow and the core shape. Numerical solutions of time-dependent non-hydrostatic equations initiated with the new stagnant-core DJL solutions show that for the ambient stratification considered, the waves are stable up to a critical amplitude above which shear instability destroys the initial wave. Steadily propagating trapped-core waves formed by lock-release initial conditions also agree well with the theoretical wave properties despite the presence of a "leaky" core region that contains vorticity of opposite sign from the ambient flow.