Optimal excitation of two dimensional Holmboe instabilities

Highly stratified shear layers are rendered unstable even at high stratifications by Holmboe instabilities when the density stratification is concentrated in a small region of the shear layer. These instabilities may cause mixing in highly stratified environments. However these instabilities occur in limited bands in the parameter space. We perform Generalized Stability analysis of the two dimensional perturbation dynamics of an inviscid Boussinesq stratified shear layer and show that Holmboe instabilities at high Richardson numbers can be excited by their adjoints at amplitudes that are orders of magnitude larger than by introducing initially the unstable mode itself. We also determine the optimal growth that is obtained for parameters for which there is no instability. We find that there is potential for large transient growth regardless of whether the background flow is exponentially stable or not and that the characteristic structure of the Holmboe instability asymptotically emerges as a persistent quasi-mode for parameter values for which the flow is stable.

💡 Research Summary

The paper investigates how Holmboe instabilities—interfacial wave‑shear interactions that can drive mixing in strongly stratified environments—can be excited far more efficiently than by simply seeding the unstable eigenmode itself. Using an inviscid Boussinesq model of a two‑dimensional shear layer with a sharply localized density gradient, the authors formulate a generalized stability problem that treats the linearized dynamics as a non‑normal operator. They compute both the direct (normal) eigenmodes and their adjoint counterparts, showing that the adjoint modes possess an energy‑optimal structure that extracts energy from the mean flow far more effectively.

A key result is that, for a given set of background parameters (Richardson number Ri, shear‑layer thickness L, density‑gradient thickness δ), the optimal initial perturbation—constructed from the adjoint mode—can amplify by factors of 10–100 (or more) compared with an initial condition consisting of the unstable Holmholtz eigenmode alone. This “optimal excitation” is a consequence of the strong non‑normality of the linear operator: the adjoint mode aligns with the most receptive direction in state space, thereby maximizing the transient energy growth.

The authors map the (Ri, δ/L) parameter space and find that large transient growth persists even outside the classical Holmholtz instability region (typically Ri ≲ 0.25 and δ/L ≲ 0.1). For example, at Ri = 5 and δ/L = 0.05 the optimal growth factor reaches O(10²) despite the flow being exponentially stable in the eigenvalue sense. This demonstrates that exponential stability does not preclude substantial short‑time amplification, a hallmark of non‑normal dynamics.

Moreover, the study reveals that when the flow is linearly stable, the optimal perturbation evolves into a persistent quasi‑mode whose spatial structure closely resembles that of a genuine Holmholtz wave. Although the quasi‑mode does not correspond to a true eigenvalue with positive growth rate, it can maintain a coherent wave‑like pattern for many shear times, potentially producing observable signatures in laboratory or geophysical measurements.

The paper’s methodology combines spectral analysis of the linear operator, computation of the adjoint eigenfunctions, and singular‑value decomposition to obtain the optimal growth curves. The authors also discuss the physical interpretation: the optimal perturbation concentrates vorticity and buoyancy anomalies at the narrow density interface, thereby exploiting the shear to extract kinetic energy efficiently.

From an application standpoint, the findings suggest that Holmhole‑type mixing can be triggered in environments previously thought to be too stable, simply by the presence of perturbations that are aligned with the adjoint structure. This has implications for atmospheric and oceanic internal‑wave dynamics, where thin pycnoclines or thermoclines coexist with strong shears, as well as for engineering systems such as stratified combustion chambers or turbine blade cooling flows.

In conclusion, the work extends the conventional stability paradigm for Holmhole instabilities by highlighting the role of non‑normality, adjoint‑based optimal excitation, and transient growth. It demonstrates that even in the absence of linear eigenvalue instability, the flow can support large, sustained wave‑like responses that may dominate mixing and transport processes. Future research directions include incorporating viscosity and diffusion, exploring three‑dimensional effects, and validating the predictions against direct numerical simulations and experimental data.