Sparsity of Fourier mass of passively advected scalars in the Batchelor regime
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| Format: | Preprint |
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2024
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| author | Blumenthal, Alex Huynh, Manh Khang |
| author_facet | Blumenthal, Alex Huynh, Manh Khang |
| contents | In 1959, Batchelor gave a prediction for the power spectral density of a passive scalar advected by an incompressible fluid exhibiting shear-straining, a mechanism for the creation of small scales in the scalar [Bat59]. Recently, a `cumulative' version of this law, summing over Fourier modes below a given wavenumber $N$, was given for a broad class of passive scalars under incompressible advection, including by solutions to the stochastic Navier-Stokes equations [BBPS22c].
This paper addresses to what extent Fourier mass of such passive scalars truly saturates the predicted power law scaling due to Batchelor. Via discrete-time pulsed-diffusion models of the advection-reaction equations, we exhibit situations compatible with the cumulative law but for which the distribution of Fourier mass among wavenumbers $|k| \leq N$ is relatively \emph{sparse} and much smaller than a `mode-wise' version of Batchelor's original prediction. In the same situations we also establish an `exponential radial shell' version of Batchelor's laws via a novel application of the method of spectral distributions. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_05473 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Sparsity of Fourier mass of passively advected scalars in the Batchelor regime Blumenthal, Alex Huynh, Manh Khang Dynamical Systems Analysis of PDEs 37A25, 76F20, 37D20 In 1959, Batchelor gave a prediction for the power spectral density of a passive scalar advected by an incompressible fluid exhibiting shear-straining, a mechanism for the creation of small scales in the scalar [Bat59]. Recently, a `cumulative' version of this law, summing over Fourier modes below a given wavenumber $N$, was given for a broad class of passive scalars under incompressible advection, including by solutions to the stochastic Navier-Stokes equations [BBPS22c]. This paper addresses to what extent Fourier mass of such passive scalars truly saturates the predicted power law scaling due to Batchelor. Via discrete-time pulsed-diffusion models of the advection-reaction equations, we exhibit situations compatible with the cumulative law but for which the distribution of Fourier mass among wavenumbers $|k| \leq N$ is relatively \emph{sparse} and much smaller than a `mode-wise' version of Batchelor's original prediction. In the same situations we also establish an `exponential radial shell' version of Batchelor's laws via a novel application of the method of spectral distributions. |
| title | Sparsity of Fourier mass of passively advected scalars in the Batchelor regime |
| topic | Dynamical Systems Analysis of PDEs 37A25, 76F20, 37D20 |
| url | https://arxiv.org/abs/2410.05473 |