Schwarzschild Black Hole Turbulence: Scalar Probe

We explore how perturbations of a Schwarzschild black hole can redistribute energy among scalar modes and seed turbulent like cascades. We make use of the van der Pol-Krylov-Bogoliubov averaging method and derive coupled mode equations that describe near-resonant interactions between neighbouring multipoles. We compare two routes to instability, namely the difference-frequency mixing between adjacent modes and the diagonal (Mathieu) self-modulation channel. We show that, at high multipole number (eikonal limit), the difference-frequency route dominates and drives a one-way cascade from higher to lower frequencies. We chart the corresponding instability regions ("tongues") and quantify their detuning dependence. The framework provides a simple, quantitative mechanism for energy transfer in black hole ringdowns and clarifies when and how turbulent signatures can arise within linear probes on a weakly perturbed background.