Large-amplitude internal waves, often referred to as internal solitary-like waves or ISWs, are a well-studied coherent, nonlinear phenomenon accessible via field measurements, laboratory experiments, and simulations of density stratified fluids. Historically, ISWs were described by approximate perturbation theories that lead to an equation of Korteweg–de Vries (KdV) type for the temporal–horizontal portion of the object . The vertical component is described by solving a linear ordinary differential eigenvalue problem. This is generally referred to as the weakly nonlinear description, or WNL. While aspects of WNL continue to be actively developed in the recent literature e.g., with an excellent survey available in , it has been known for some time that WNL is lacking as a quantitative descriptor of ISWs. The KdV theory itself predicts that wave amplitude is bounded above by the onset of breaking, in contrast to observations of wave broadening and the formation of waves with flat crests i.e. tabletop waves;. Exact ISWs are solutions of the nonlinear elliptic eigenvalue problem referred to as the Dubreil–Jacotin–Long (DJL) equation. This equation is formally equivalent to the full stratified Euler equations (in a frame moving with the wave), and thus this theory is referred to as fully nonlinear or exact.

Comparisons of WNL with exact ISWs typically yield poor results for the vertical structure, especially for complex stratifications. Higher-order WNL theory, with the Gardner or mKdV equations as prominent examples, does a better job of qualitatively matching the type of upper bound on wave amplitude , but there is a significant gap between what is observed in simulations and what WNL can describe . In the context of ISWs with strong shear, this gap is well illustrated by the study of , which develops a WNL theory for weakly stratified critical layers. The theory is algebraically quite complex, with significant ties to work on Rossby wave critical layers. However, the theory is not compared to simulations or experiments, and thus it is unclear to what extent it applies to a situation with a dynamically evolving wave field. Interestingly, it is very similar in style to , which considered a closely linked topic some 30 years earlier.

While field measurements are often taken in a dynamic, complex environment with changes in the stratification structure and a non-trivial background current, much of the understanding of ISWs has been developed based on relatively simple stratifications (typically quasi-two-layer or exponential stratification). The presence of a background shear current complicates the algebra needed to derive theory, whether WNL or the DJL equation. In the case of WNL, the numerical methods are unchanged, but for the DJL equation the iterative algorithm used to obtain solutions requires an ad hoc modification seeand the Methods section below. In practice, publicly available software for the DJL equation can handle many situations, but there is no a priori proof of convergence. It has been shown that the presence of a background shear current can modify the type of upper bound on wave amplitude and can also change the polarity of exact ISWs again seefor details. The detail-oriented reader is cautioned that Fig. 3 in is based on ISWs of elevation which have an opposite sign of wave-induced vorticity to ISWs of depression. For ISWs of depression a background current with positive vorticity tends to decrease the limiting wave amplitude. In his study of ISW energetics in the presence of a background shear current, thus employed a negative background current.

The DJL equation has also been extended to model ISWs past breaking or waves with trapped cores. Such waves are known to form during shoaling , and the questions of whether the core is completely trapped , quiescent , or found to be immediately adjacent to the boundary or at depth have all been the subject of recent studies. Any ISW can be described as a propagating baroclinic vortex, but trapped cores provide a more complex wave–vortex coupling that will be revisited below.

In the classical theory of hydrodynamic stability the gradient Richardson number (Ri) is the standard necessary condition; when Ri>0.25 the flow is linearly stable. However, Ri<0.25 is not a sufficient condition for instability . Large-amplitude ISWs, both with and without a background shear current, can induce strong shear currents near the wave crest. This situation has drawn interest over several decades , first as a pure conjecture and then with two-dimensional and finally three-dimensional situations. It has been found that shear instability within ISWs is quite complex, with onset dependent on the strength of upstream perturbations, the Reynolds number, and duration of time spent below Ri=0.25 as a perturbation passes through the wave. Three-dimensionalization, at least on scales for which direct numerical simulation (DNS) has been accessible, occurs preferentially at the rear of the wave, and in some parameter regimes instability is episodic, with bursts and quiescent periods.

measured large-amplitude ISWs in northern Monterey Bay, California (USA), as part of a complex interplay between a coastal upwelling front and local wind forcing. These waves were large amplitude (up to a 10 m maximum isopycnal displacement in water approximately 20 m deep) with strong background shear currents. Indeed, the background shear currents were so strong that solutions of the DJL equation were impossible to compute. Some analysis of large ISW-like waves in the presence of strong background shear currents is provided in . These authors considered the resonant generation of ISWs by flow of a background current with shear over isolated topography. They found large-amplitude wave trains as well as waves trapped behind the topography with vortex-rich cores that were quasi-trapped near the surface (i.e. both the upstream-propagating wave trains and the trapped waves exhibited vortex-rich cores). However, the connection between simulations and field measurements is incomplete since it is unclear whether the waves observed in were resonantly generated.

In recent work, considered the effect of a background current on mode-2 waves. The waves were generated via stratified adjustment, similar to the methodology we follow below. The stratification used was centred at the mid-depth, so that in the absence of a shear current initial perturbations have the dominant part of their energy deposited into the mode-2 wave field. The authors document the manner in which the adjustment process and the resulting leading mode-2 wave and trailing mode-1 tail are modified by the presence of shear. They also compare their results to the observations of , though the fact that both the density profile and shear layer in are centred at the mid-depth makes a detailed comparison impossible.

The interaction of ISWs with background currents has also been explicitly demonstrated in the context of frontal generation. discussed observations and simulations of a case in the Saguenay fjord, Quebec, Canada. The authors found that convergence near the front was important in numerical generation experiments, though the parameter space they explored was fairly limited. In all cases discussed by the authors, rightward- and leftward-propagating wave trains propagating away from the front differ in amplitude, with larger waves observed in stronger shear.

In a related non-ISW context, considered the generation of internal waves by tidal flow in the presence of shear. They concentrated on characterizing energy flux and found profound asymmetries between the waves propagating with the shear and against the shear. Interestingly, whether the downstream or upstream side of the ridge that generated the internal waves dominated the energy flux was found to depend on the slope of the flanks of the ridge. It is difficult to ascertain implications of this study for the problem of ISWs with shear, since these authors only considered a linear stratification. Nevertheless, the study clearly suggests that shear can profoundly influence wave energetics.

In this paper, we report on high-resolution simulations of stratified adjustment and the resulting internal wave field in the presence of strong shear currents. The structure of the paper is organized as follows. The Methods section presents the governing equations, the DJL theory, and the design of and parameter values for all numerical experiments we present. The Results section presents the major findings, divided into three parts: (i) waves modified by shear but without critical layers, (ii) waves with critical layers, and (iii) situations in which the stratification is in the opposite half of the water column from the shear layer. The paper concludes with a Discussion section and a brief set of Conclusions.

We consider an incompressible fluid in the absence of rotation that obeys the Boussinesq and rigid lid approximations. The stratified Navier–Stokes equations in the absence of rotation read 1∂u∂t+u⋅∇u=-1ρ0∇P+ν∇2u-ρgρ0k^,2∇⋅u=0,3∂ρ∂t+u⋅∇ρ=κ∇2ρ, where u is the velocity, P is the pressure, ρ is the density, and ρ0 is some reference density of the fluid. The physical parameters are the molecular kinematic viscosity ν (set to equal 1×10-6 ms-2) and scalar diffusivity κ (set to equal 2×10-7 ms-2). The unit vector in the vertical direction is denoted by k^. The computational domain is rectangular, with the x axis running left to right along the bottom of the domain, so that 0<x<Lx and 0<z<Lz.

Simulations were carried out with the pseudospectral code SPINS . The code has been thoroughly validated in a number of different configurations (e.g. shear instabilities, internal wave generation, internal solitary wave propagation) and is available for download through its online manual: https://wiki.math.uwaterloo.ca/fluidswiki/index.php?title=SPINS_User_Guide (last access: 28 September 2021). All simulations reported employed regularly spaced grids. Approximately 30 exploratory simulations were carried out prior to identifying a parameter space of interest. Table  provides a list of cases discussed in this paper along with their key physical and numerical parameters. As a general comment, simulations were much higher resolution than many comparable studies in the literature, and the pseudospectral aspect of SPINS means that the model has very little numerical dissipation. This proved vital for some of the results reported below and is discussed further in the Discussion section.

All simulations were initialized with a background shear current 4U(z)=Umaxexp⁡(z-Lz)/dU. The total depth was fixed as Lz=20 m for all simulations, but the horizontal extent of simulations was varied on a case-by-case basis. The thickness of the shear layer was fixed at dU=3 m or 0.15 of the total depth. Umax was varied as described in Table . The form of the background current is such that, in the absence of stratification, it is linearly stable (i.e. Fjortoft's criterion is not satisfied).

The initial density field was specified as ρ(x,z)=ρ‾(z-η) with 5η=η0exp⁡-(x-0.5Lx)2/wd2. While other values were used in preliminary experiments, all numerical experiments reported below set η0=-5 m (0.25 of the total depth) and wd=100 m.

The background density field was specified as 6ρ‾(z)=1-Δρ2tanh⁡(z-z0)/d. The dimensionless density difference was set as Δρ=0.001 for all simulations, while the pycnocline centre and thickness, (z0,d), were varied as shown in Table .

Figure  shows a sample representation of the initial conditions, while Fig.  shows the 1-D background profiles for the various simulations discussed below. Figure a shows the N2(z) profile, Fig. b shows the background shear current U(z), and Fig. c shows the gradient Richardson number (Ri).

Exact internal solitary waves are solutions of the DJL equation. In the presence of a background shear current this equation has a rather complex form: 7∇2η+Uz(z-η)[U(z-η)-c]1-ηx2+1-ηz2+N2(z-η)[U(z-η)-c]2η=0. For no background current the DJL equation takes the more familiar form 8∇2η+N2(z-η)c2η=0, with the strong non-linearity encapsulated in the evaluation of the buoyancy frequency squared profile at its upstream height (i.e. at z-η). While several different techniques for the solution of the DJL equation without a background current are described in the literature, the only publicly available approach to the DJL equation with a background current that the present authors are aware of employs an ad hoc extension of the direct variational method of , as described in . This consists of an iterative algorithm, which while generally stable for weak currents leads to “wandering” when shear is strong. The simulations reported below specifically concentrate on parameter regimes in which the DJL was found not to converge (unless otherwise noted in the discussion).

The numerical experiments reported above illustrate that when a strong background shear current is present, the set of possible wave–vortex phenomena is considerably larger than the tidy ISWs described by the DJL equation. Indeed, essentially none of the phenomena we have simulated could be termed truly steady. The main coherent feature from our simulations one could expect to see in the field, for a strong background shear current, are ISWs with a strong vortex core in the near-surface, high-shear region. Classical theory of ISWs with cores generally considers nearly quiescent cores , while recent theory and simulations that include background shear currents yield so-called “subsurface cores” . Neither of these matches the strong wave–vortex coupling that was observed above (e.g. Fig. ). In cases in which a critical layer is possible, the vortex cores coexist with smaller disturbances that extend ahead of the leading ISW.

The simulations we performed required resolution that was much higher than that typically used in coastal ocean simulations (i.e. multi-kilometre domains with a horizontal resolution of less than a metre and a vertical resolution of less than 5 cm). Even recent process studies on mode-2 waves in the presence of shear using the finite-volume MITgcm used a resolution of 5×0.5 m (which was more than sufficient for the phenomena these authors set out to model). Moreover, the pseudospectral model used has very low inherent numerical dissipation, so that even phenomena with relatively few grid points across them are only weakly damped. The high resolution necessarily makes extension to three dimensions a complicated task. However, 3-D simulations initialized from a 2-D simulation, i.e. using the early portion of the simulations described above, should be possible with present computational resources. It is likely that performing a detailed energy analysis, as in for ISWs propagating against the background shear current (hence ones with a DJL theory), would require 3-D simulations.

However, a more serious issue when interpreting the above results is in the large gap between theory-inspired simulations and those on the regional scale in coastal oceans. To the authors' knowledge there has been no systematic study of ISW behaviour in marginally resolved situations, especially when parameterizations for mixed layers (e.g. the KPP scheme) are employed. As noted above, the key feature of ISWs is their ability to outrun local instabilities or mixing events without a loss of coherence. A globally high value of eddy viscosity cannot be “outrun”, and thus care should be taken in extrapolating simulation results to particular field measurements.

Even with the above caveats, the simulations we have conducted pose a very clear question for the theorist: “In deep water in which a pycnocline may be far removed from local shear layers can an ISW be destabilized by a shear that would be thought of as so far from the wave that it would be irrelevant to its evolution?”

We have provided one possible explanation for why the measurements in yielded large-amplitude waves with no DJL-based description. That is, that for a portion of parameter space, mode-1 ISWs generated via stratified adjustment in the presence of a background shear current are not free waves, since they have a propagation speed very close to the maximum value of the background current. When some stratification is observed near the surface, these waves take the form of ISWs with strongly recirculating cores.

The simulations above were 2-D, while the observations of were influenced by rotation and likely included some large-scale 3-D structures (comments on small-scale three-dimensionalization were made in the Discussion above). Moreover, the observations take a different form from the wave trains that are spontaneously generated in our stratified adjustment simulations. This is not surprising, and a set of initial conditions more closely tied to the field observations remains a clear goal for future work.

A different avenue for future work would be to more closely link our results to the work of on mode-2 waves. Mode-2 waves lack a DJL theory due to the possible presence of a trailing tail of mode-1 waves, and the question of whether a particular shear current more strongly affects the leading mode-2 wave or its mode-1 tail remains largely open.

Finally, we provided well-resolved examples of stratified adjustment with critical layers (cases labelled “CL”). When a significant stratification was present near the surface, the waves formed consisted of strong vortex cores, with some evidence of fine-scale structure on the leading side of the wave. In the absence of stratification near the surface, a train of vortices spontaneously formed near the surface, and this led to small-amplitude deformations of the underlying pycnocline. Our simulations did not yield evidence of the common theoretical assumption that ISWs are the dominant object, with critical layers providing a slow drain of energy . This observation should be re-examined on scales for which 3-D DNS is possible.