General Temporal Instability Criteria For Stably Stratified Inviscid Flow

The temporal instability of stably stratified flow was investigated by analyzing the Taylor-Goldstein equation theoretically. According to this analysis, the stable stratification $N^2\geq0$ has a destabilization mechanism, and the flow instability is due to the competition of the kinetic energy with the potential energy, which is dominated by the total Froude number $Fr_t^2$. Globally, $Fr_t^2 \leq 1$ implies that the total kinetic energy is smaller than the total potential energy. So the potential energy might transfer to the kinetic energy after being disturbed, and the flow becomes unstable. On the other hand, when the potential energy is smaller than the kinetic energy ($Fr_t^2>1$), the flow is stable because no potential energy could transfer to the kinetic energy. The flow is more stable with the velocity profile $U’/U’’’>0$ than that with $U’/U’’’<0$. Besides, the unstable perturbation must be long-wave scale. Locally, the flow is unstable as the gradient Richardson number $Ri>1/4$. These results extend the Rayleigh’s, Fj{\o}rtoft’s, Sun’s and Arnol’d’s criteria for the inviscid homogenous fluid, but they contradict the well-known Miles-Howard theorem. It is argued here that the transform $F=\phi/(U-c)^n$ is not suitable for temporal stability problem, and that it will lead to contradictions with the results derived from the Taylor-Goldstein equation. However, such transform might be useful for the study of the Orr-Sommerfeld equation in viscous flows.

💡 Research Summary

The paper revisits the temporal stability of stably stratified inviscid shear flows by a rigorous analysis of the Taylor‑Goldstein equation (TGE). The authors introduce a global energy‑based parameter, the total Froude number (Fr_t^2 = K/P), where (K) is the integrated kinetic energy of the mean flow and (P) is the integrated potential energy associated with the stratification. By manipulating the TGE, they demonstrate that when (Fr_t^2 \le 1) the potential energy exceeds the kinetic energy, allowing a small disturbance to draw energy from the stratification and grow. Conversely, if (Fr_t^2 > 1) the kinetic reservoir dominates, suppressing any transfer of potential energy and rendering the flow temporally stable. This global criterion extends classical inviscid stability results (Rayleigh, Fjørtoft, Sun, Arnold) to stratified environments.

A second key finding concerns the sign of the ratio (U’/U’’’ ) of the first to third derivative of the base‑flow velocity profile. When (U’/U’’’ > 0) the velocity profile is “convex” in a sense that limits the phase‑speed variation of disturbances, thereby inhibiting the energy exchange that drives instability. In contrast, (U’/U’’’ < 0) corresponds to a “concave” profile that amplifies phase‑speed gradients, facilitating energy transfer and promoting instability. Hence the curvature of the shear profile at higher order plays a decisive role in the stability landscape.

The authors also establish a wavelength selection rule: only long‑wave (small‑k) disturbances can possess a positive growth rate. In the short‑wave limit (k → ∞) the terms in the TGE cancel, yielding a neutral or damped response. This long‑wave requirement distinguishes the identified instability from classic Kelvin‑Helmholtz modes, which can be short‑wave unstable.

On a local scale the paper proposes a novel Richardson‑number condition. Contrary to the well‑known Miles‑Howard theorem (which states that instability requires (Ri < 1/4)), the analysis here predicts that instability occurs when the gradient Richardson number exceeds the critical value, i.e., (Ri > 1/4). The authors attribute the discrepancy to the use of the transformation (F = \phi/(U-c)^n) in traditional proofs. They argue that this transformation, while converting the TGE into a Sturm‑Liouville form, improperly treats the complex eigenvalue (c) and therefore misrepresents the temporal growth rate. Consequently, the Miles‑Howard result is not applicable to the temporal problem considered.

Finally, the paper remarks that the same transformation may still be valuable for the Orr‑Sommerfeld equation governing viscous flows, where the eigenvalue problem is fundamentally different. However, for inviscid, temporally growing disturbances, the transformation leads to contradictions with the direct TGE analysis.

In summary, the study provides a comprehensive set of instability criteria for stably stratified inviscid shear flows: (1) a global energy balance expressed through (Fr_t^2), (2) the sign of (U’/U’’’ ) as a higher‑order curvature indicator, (3) a long‑wave requirement for unstable modes, and (4) a reversed Richardson‑number threshold. These results broaden the theoretical framework of shear‑flow stability, challenge the universality of the Miles‑Howard theorem, and suggest new avenues for interpreting internal wave dynamics and energy exchange processes in geophysical and engineering contexts.