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Inertia-gravity waves in the atmosphere and ocean are transported and refracted by geostrophic turbulent currents. Provided that the wave group velocity is much greater than the speed of geostrophic turbulent currents, kinetic theory can be used to obtain a comprehensive statistical description of the resulting interaction (Savva et al. , J. Fluid Mech. , vol. 916, 2021, A6). The leading-order process is scattering of wave energy along a surface of constant frequency, $$\omega$$ , in wavenumber space. The constant- $$\omega$$ surface corresponding to the linear dispersion relation of inertia-gravity waves is a cone extending to arbitrarily high wavenumbers. Thus, wave scattering by geostrophic turbulence results in a cascade of wave energy to high wavenumbers on the surface of the constant- $$\omega$$ cone. Solution of the kinetic equations shows establishment of a wave kinetic energy spectrum $$\sim k_h^{-2}$$ , where $$k_h$$ is the horizontal wavenumber.
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On the Energy Cascade of 3-Wave Kinetic Equations: Beyond Kolmogorov–Zakharov Solutions
In weak turbulence theory, the Kolmogorov–Zakharov spectra is a class of time-independent solutions to the kinetic wave equations. In this paper, we construct a new class of time-dependent isotropic solutions to the decaying turbulence problems (whose solutions are energy conserved), with general initial conditions. These solutions exhibit the interesting property that the energy is cascaded from small wavenumbers to large wavenumbers. We can prove that starting with a regular initial condition whose energy at the infinity wave number |𝑝|=∞ is 0, as time evolves, the energy is gradually accumulated at {|𝑝|=∞}. Finally, all the energy of the system is concentrated at {|𝑝|=∞} and the energy function becomes a Dirac function at infinity 𝐸𝛿{|𝑝|=∞}, where E is the total energy. The existence of this class of solutions is, in some sense, the first complete rigorous mathematical proof based on the kinetic description for the energy cascade phenomenon for waves with quadratic nonlinearities. We only represent in this paper the analysis of the statistical description of acoustic waves (and equivalently capillary waves). However, our analysis works for other cases as well.
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- Award ID(s):
- 1854453
- PAR ID:
- 10137236
- Date Published:
- Journal Name:
- Communications in mathematical physics
- ISSN:
- 1432-0916
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/https://doi.org/10.1007/s00220-019-03651-w
