Acoustic2D solver

Description

In this case, the collisional terms simpilfies to

\[St_{k}=\frac{2 \pi}{\sqrt{6}a k}\int \left(\mathcal{N}^k_{12}-\mathcal{N}^1_{2k}-\mathcal{N}^2_{k1}\right)\mathrm{d}k_1\mathrm{d}k_2\]

With $\mathcal{N}^k_{12}=|V_{k12}|^2(n_1n_2-n_1n_k-n_kn_2)\delta(k-k_1-k_2)$ and $V_{k12}=V_0\sqrt{kk_1k_2}$.

This is WKE results from assuming a disperse regularisation in the dispersion relation $\omega_k=ck(1+a^2k^2)$, with $ak\ll1$. See Griffin et al., Phys. Rev. Lett. 128, 224501 (2022).

Solver

The computation of the collision integral is relatively simple. We use the resonance condition $k - k_1 - k_2 = 0$ and the symmetry $k_1 \leftrightarrow k_2$ to reduce the collision integral to

\[St_{k} = I_{k} - 2 J_{k}\]

where

\[I_{k} = \frac{2 \pi}{\sqrt{6}a k} \int\limits_{0}^{k} ~ |V_{k12}|^2(n_1n_2-n_1n_k-n_kn_2) ~ \mathrm{d}k_1\]

and

\[J_{k} = \frac{2 \pi}{\sqrt{6}a k} \int\limits_{k}^{k_{\rm max}} ~ |V_{k12}|^2(n_kn_2-n_kn_1-n_1n_2) ~ \mathrm{d}k_1\]

with the integrands being computed using $k_2 = |k - k_1|$ and $k_{\rm max}$ the maximal wave vector modulus.

The wave frequency is $\omega_{k} = ck$ and we use logarithmic grid wave_spectrum. The different options are described in the Acoustic2D documentation.

Diagnostics

The standard diagnostics should work for this system.

Theoretical predictions

The Kolmogorov-Zakharov energy spectrum is

\[e_k = 2\pi k \omega_k n_{\bf k} = \frac{4^{3/2} \sqrt{a c}}{3 \pi} ~ k^{-1}.\]

Acoustic3Dsolver

Description

In this case, the collisional terms simpilfies to

\[St_{k}=\frac{4 \pi^2}{c k^2}\int \left(\mathcal{N}^k_{12}-\mathcal{N}^1_{2k}-\mathcal{N}^2_{k1}\right)k_1 k_2 \mathrm{d}k_1\mathrm{d}k_2\]

With $\mathcal{N}^k_{12}=|V_{k12}|^2(n_1n_2-n_1n_k-n_kn_2)\delta(k-k_1-k_2)$ and $V_{k12}=V_0\sqrt{kk_1k_2}$.

This WKE describes 3D acoustic waves with frequency $\omega_k=ck$. See Zakharov and Sagdeev, Dokl. Akad. Nauk SSSR 192, 297–300 (1970) and Kochurin and Kuznetsov, arXiv, 2407.08352, (2024).

Solver

The computation of the collision integral is the same than for Acoustic2D solver, up to the prefactor. We use logarithmic grid wave_spectrum. The different options are described in the Acoustic3D documentation.

Diagnostics

The standard diagnostics should work for this system.

Theoretical predictions

The Kolmogorov-Zakharov energy spectrum is the Zakharov-Sagdeev (ZS) spectrum.

\[e_k = 4\pi k^2 \omega_k n_{\bf k} = \frac{\sqrt{c P} k^{-3/2}}{\sqrt{3\pi(\pi + 4 \ln 2 − 1)}}\]

with $P$ the energy injection rate.

List of structures and methods for Acoustic solvers

WavKinS.Acoustic2D — Type
Acoustic2D

Simulation structure for Acoustic 2D wave turbulence. It contains

name::String #name of the simulation type
Nk_arguments::Int # Number of arguments of ``n_k``. 1: (fully symetric) , 2: (cylindrical average in 3D or mirror symmetric in 2D), 3: Only mirror symmetric in 3D
ω # Dispersion relation. This is a function of ``k``. It takes `Nk_argument` arguments

Nk::wave_spectrum #wave action
Sk::wave_spectrum #collisional integral
F1::wave_spectrum #working field
FSt::Vector{wave_spectrum} #Array of working fields for multithreading
partition::Vector{UnitRange{Int64}} #partition for multithreading

# Type of interpolation and time stepping scheemes
interp_scheeme::Interp_Scheeme
time_stepping::Time_Stepping

# Outputs and diagnostics
diags::diagnostic_container

t::Float64 #current time
a::Float64 # dispersive length
c::Float64 # speed of sound
dimension::Int # physical dimension of the system (or of the isotropic sector)
dΩ::Float64 # surface of the unit sphere (or of the isotropic sector)

FD::force_dissipation # Contains all the terms about force and dissipation.
WavKinS.Acoustic3D — Type
Acoustic3D

Simulation structure for Acoustic 3D wave turbulence. It contains

name::string #name of the simulation type
Nk_arguments::Int # Number of arguments of ``n_k``. 1: (fully symetric) , 2: (cylindrical average in 3D or mirror symmetric in 2D), 3: Only mirror symmetric in 3D
ω # Dispersion relation. This is a function of ``k``. It takes `Nk_argument` arguments

Nk::wave_spectrum #wave action
Sk::wave_spectrum #collisional integral
F1::wave_spectrum #working field
FSt::Vector{wave_spectrum} #Array of working fields for multithreading
partition::Vector{UnitRange{Int64}} #partition for multithreading

# Type of interpolation and time stepping scheemes
interp_scheeme::Interp_Scheeme
time_stepping::Time_Stepping

# Outputs and diagnostics
diags::diagnostic_container

t::Float64 #current time
c::Float64 # speed of sound
dimension::Int # physical dimension of the system (or of the isotropic sector)
dΩ::Float64 # surface of the unit sphere (or of the isotropic sector)

FD::force_dissipation # Contains all the terms about force and dissipation.