Anisotropic spectra of acoustic type turbulence

We consider the problem of spectra for acoustic type of turbulence generated by shocks being randomly distributed in space. We show that for turbulence with a weak anisotropy such spectra have the same dependence in $k$-space as the Kadomtsev-Petviashvili (KP) spectrum: $E(k)\sim k^{-2}$. However, the frequency spectrum has always the falling $\sim \omega^{-2}$, independently on anisotropy. In the strong anisotropic case the energy distribution relative to wave vectors takes anisotropic dependence forming in the large $% k $ region the spectra of the jet type.

💡 Research Summary

The paper investigates the spectral properties of acoustic‑type turbulence that is generated by randomly distributed shock fronts (caustics). Starting from the well‑known Kadomtsev‑Petviashvili (KP) result for isotropic media—where the energy spectrum scales as E(k) ∝ k⁻² and the frequency spectrum as E(ω) ∝ ω⁻²—the authors extend the analysis to situations with anisotropic distributions of shocks, which are common in magnetised plasmas or when the turbulence source itself is directional.

In the weak‑turbulence regime, acoustic waves are essentially linear (ω ≈ cₛk) and the spectrum can be described by a superposition of independent plane waves. When the wave amplitude grows, nonlinear steepening leads to wave breaking and the formation of shock discontinuities. The authors model each shock as a thin density jump Δρ that, in one dimension, contributes a term proportional to δ(x‑x₀) in the spatial derivative of the density. By Fourier transforming this singular contribution and averaging over random shock positions and amplitudes, they recover the classic k⁻² spectrum for 1‑D, 2‑D, and 3‑D isotropic turbulence. This derivation reproduces the original KP result and also matches the Burgers‑type spectrum obtained by earlier works.

To treat anisotropy, the paper assumes that each shock is a circular disk (a “caustic”) whose normal vector n̂ and radius R are random but statistically independent. The density jump is taken as a cylindrically symmetric function Δρ(r⊥) that vanishes at the disk edge. The Fourier transform of a single disk yields an integral involving the Bessel function J₀(k⊥r⊥). Summing over many disks and averaging over their orientations leads to an expression for |ρₖ|² that contains the factor kₓ² ∫₀ᴿ Δρ(r⊥) r⊥ J₀(k⊥r⊥) dr⊥.

The authors focus on the short‑wavelength limit kR ≫ 1, where the Bessel function oscillates rapidly. Using the stationary‑phase method, they show that the integral is dominated by contributions near the disk boundary, producing an additional power‑law factor k⊥⁻³. Combined with the explicit kₓ² factor, the resulting three‑dimensional energy spectrum behaves as

 E(k‖, k⊥) ∝ k‖⁻² k⊥⁻⁵

for wave vectors whose transverse component satisfies k⊥R ≫ 1. In other words, along the direction parallel to the shock normal the spectrum retains the KP k⁻² scaling, while in the transverse direction it falls off much more steeply (k⁻⁵). This “jet‑type” anisotropic spectrum is a hallmark of strong anisotropy: energy is concentrated in narrow cones (jets) aligned with the preferred direction of the shocks.

The frequency spectrum is treated separately. Because a shock crossing a fixed point produces a time derivative of the density proportional to a Dirac delta, its Fourier transform yields |ρ(ω)|² ∝ ω⁻², irrespective of the spatial anisotropy. Hence the temporal spectrum always follows the universal ω⁻² law.

The paper also discusses practical implications. In space‑plasma observations (e.g., solar‑wind measurements) one often sees a mixture of a KP‑like k⁻² tail and a shallower inertial‑range scaling (such as k⁻⁵⁄³). By identifying the KP component, one can estimate the relative contribution of coherent shock structures to the overall turbulence. The authors note that two‑dimensional numerical simulations have already revealed jet‑type spectra, but limited spatial resolution prevented a detailed examination of the transverse k⁻⁵ scaling.

In conclusion, the study provides a comprehensive theoretical framework that (i) confirms the isotropic KP spectrum for shock‑dominated acoustic turbulence, (ii) predicts a universal ω⁻² frequency spectrum, and (iii) demonstrates that strong anisotropy transforms the spatial spectrum into a highly directional jet‑type form with k‖⁻² k⊥⁻⁵ scaling. These results bridge the gap between idealised isotropic theory and realistic anisotropic plasma environments, offering a quantitative tool for interpreting experimental and observational data where coherent shock structures coexist with weakly turbulent fluctuations.