Formation and evolution of turbulence in convectively unstable internal solitary waves of depression shoaling over gentle slopes in the South China Sea

The shoaling of high-amplitude Internal Solitary Waves (ISWs) of depression in the South China Sea (SCS) is examined through large-scale parallel turbulence-resolving high-accuracy/resolution simulations. A select, near-isobath-normal, bathymetric transect of the gentle SCS continental slope is employed together with stratification and current profiles obtained by in-situ measurements. Three simulations of separate ISWs with initial deep-water amplitudes in the range [136m, 150m] leverage a novel wave-tracking capability for a propagation distance of 80km and accurately reproduce key features of in-situ-observed phenomena with significantly higher spatiotemporal resolution. The interplay between convective and shear instability and the associated turbulence formation and evolution, as a function of deep-water ISW amplitude are further studied in-part revealing processes previously not observed in the field. Across all three waves, the convective instability develops in a similar fashion. Heavier water entrained from the wave rear plunges into its interior, giving rise to transient, yet distinct, subsurface vortical structures. Ultimately, a gravity current is triggered which horizontally advances through the wave interior and mixes it down to pycnocline’s base. Although the waveform remains distinctly symmetric, Kelvin-Helmholtz billows emerge near the well-mixed ISW trough, disturb the wave’s trailing edge and give rise to an active wake. The evolution of the kinetic energy associated with finer-scale perturbations to the ISW-induced velocity field shows two different growth regimes, each dominated by either convective or shear instability. The wake’s perturbation kinetic energy is nonlinearly dependent on deep-water wave amplitude and can become a sizable fraction of the kinetic energy of the deep-water ISW.

💡 Research Summary

This paper presents the first large‑scale, three‑dimensional, turbulence‑resolving simulations of high‑amplitude, depression‑type internal solitary waves (ISWs) propagating over the gentle continental slope of the South China Sea (SCS). Using a spectral‑element method (SEM) with recently developed high‑order, non‑hydrostatic pressure solvers, the authors reproduce three separate ISWs whose deep‑water amplitudes range from 136 m to 150 m. The computational domain follows an 80 km transect that matches an in‑situ bathymetric track and incorporates measured background stratification and current profiles. A novel wave‑tracking algorithm allows the waves to be followed continuously over the full distance, providing unprecedented spatiotemporal resolution of the wave interior.

The study focuses on two sequential instability mechanisms that develop as the waves shoal: (1) convective instability and (2) shear (Kelvin‑Helmholtz) instability. When the wave‑induced horizontal velocity at the rear exceeds the phase speed, heavier water from the wave’s tail plunges into the interior, forming transient, subsurface vortical structures of order 100 m. These vortices evolve into a gravity‑current‑like front that propagates horizontally through the wave core, mixing the fluid down to the base of the pycnocline while preserving the overall symmetric shape of the ISW. The convective stage generates a rapid increase in perturbation kinetic energy (PKE), which the authors quantify as a distinct growth regime.

As the convective mixing weakens the stratification inside the wave, the Richardson number near the wave trough drops below the critical value (≈0.25). This triggers Kelvin‑Helmholtz (K‑H) billows that originate near the well‑mixed trough and travel downstream along the interface. The billows break, producing an active turbulent wake behind the wave. The wake’s PKE grows in a second, shear‑dominated regime and is found to depend non‑linearly on the deep‑water wave amplitude; for the largest wave (150 m) the wake’s PKE can reach more than 30 % of the total kinetic energy of the deep‑water ISW.

Energy spectra reveal that the convective stage concentrates energy at scales of 10–100 m, whereas the shear stage transfers energy to smaller scales (1–10 m), consistent with observed K‑H billow dimensions. The authors compare the simulated velocity and density fields with field observations from previous campaigns (Lien et al., Chang et al.) and find quantitative agreement in wave speed, shape, and the location of instability signatures.

Methodologically, the work demonstrates the capability of SEM to resolve multiscale processes ranging from basin‑scale propagation (∼100 km) down to meter‑scale turbulence, by locally refining the mesh within the wave core while coarsening elsewhere. The high‑order pressure solver overcomes long‑standing difficulties associated with high‑aspect‑ratio grids and non‑hydrostatic coupling, enabling efficient computation on modern supercomputers.

The findings have several implications. First, they confirm that convective and shear instabilities can coexist and act sequentially in ISWs propagating over gentle slopes, explaining why such waves can travel long distances without catastrophic breaking. Second, the gravity‑current‑like interior flow and the downstream turbulent wake represent significant mechanisms for vertical mixing and lateral transport of heat, salt, and nutrients, potentially influencing primary productivity on continental shelves. Third, the identified non‑linear relationship between deep‑water wave amplitude and wake PKE provides a quantitative basis for parameterizing wave‑induced turbulence in large‑scale ocean and climate models.

In summary, the paper delivers a comprehensive, high‑fidelity numerical investigation of ISW shoaling in the SCS, elucidating the detailed dynamics of convective overturning, gravity‑current propagation, Kelvin‑Helmholtz billow formation, and wake turbulence. It bridges the gap between sparse field observations and idealized laboratory or low‑resolution numerical studies, offering new insights into the role of internal solitary waves in oceanic mixing and energy budgets.