Statistical field theory for a passive vector model with spatially linear advection

TL;DR

This work develops a statistically tractable Batchelor-regime model for a passive vector field advected by a spatially linear, Gaussian velocity. Using Hopf functional methods, the authors show that the full non-Gaussian statistics can be represented as a mixture of Gaussian sub-ensembles conditioned on the advecting field, reducing the problem to a stochastic PDE for the conditional spectrum and a Gaussian mixture for the full statistics. They derive a spectral energy budget, uncovering $E(k)\sim k^2$ at large scales and $E(k)\sim k^{-1}$ in the inertial-like range for $\gamma=0$, with anomalous dissipation and a Green’s-function solution linking forcing to the spectrum. Numerically, they solve the subensemble SPDEs using a method of characteristics on a dynamically remapped, logarithmic $k$-grid, and demonstrate intermittency through heavy-tailed small-scale statistics and a Gaussian-mixture description of longitudinal increments, while confirming ergodicity in the stationary state. They also analyze the effect of nonzero $\gamma$, finding stationary states only for $|\gamma|<\sqrt{3/8}$ and dynamo-like instabilities outside this range, linking to known results for passive vector models.

Abstract

One challenge in developing a statistical field theory of turbulence is the analysis of the functional equations that govern the complete statistics of the flow field. Simplified models of turbulence may help to develop such a statistical framework. Here, we consider the advection and stretching of an incompressible passive vector field by a spatially linear stochastic field as a model for small-scale turbulence. The model encompasses non-Gaussian statistics due to an intermittent energy flux from large scales to small scales, thereby displaying hallmark features of turbulence. We explore this model using the Hopf functional formalism, which naturally leads to a decomposition of the complex non-Gaussian statistics into Gaussian sub-ensembles based on different realizations of the advecting field. We then characterize intermittency of the model using a numerical implementation, which takes advantage of this statistical decomposition.

Figures (6)

• Figure 1: Slices through the trace of the spectrum tensor $\phi_{ii}^{\bm v}(\bm k, t)$ in the ($k_y=0$)-plane for one of the ensemble members. The first three panels are snapshots at three different times $t$. The last panel corresponds to an average over all snapshots in the interval $1 \leq t \leq 4$. The magnitude of $\bm k$ is spaced logarithmically. A forcing wavenumber $k_f = \mathrm{argmax}[k Q(k)]$ and a viscous wavenumber $k_\nu = \lambda^{-1}$ are indicated by dashed lines.
• Figure 2: Mean energy across scales. Top left: Mean energy spectrum of the ensemble simulations (red, solid) compared to the analytical solution \ref{['eq:complete_solution_spectral_energy_budget_gamma=0']} (black, dashed). The blue, shaded area indicates the distribution of the energy spectrum across snapshots of the ensemble of simulations in the stationary state, each level set corresponding to quantiles incremented by $5\%$. Bottom left: Dissipation, forcing, and transfer term (represented symbolically by $T(k)$) in the spectral energy budget equation \ref{['eq:spectral_energy_budget_gamma=0']}, computed for the analytical solution \ref{['eq:complete_solution_spectral_energy_budget_gamma=0']}. They are multiplied by $k$ so that the area below the curve corresponds to a rate of energy injection/dissipation in the semi-logarithmic plot. Right: Mean second-order longitudinal structure function. Colors and line styles are consistent with the top left panel.
• Figure 3: Energy and enstrophy of the model. Left panels: Energy and enstrophy in the ensemble simulations over time, starting at zero and then saturating at a statistically stationary state. The mean over the 10 ensemble simulations (solid, red) fluctuates around a numerical solution of the spectral energy budget equation \ref{['eq:spectral_energy_budget_gamma=0']} (black, dashed). The blue, shaded area indicates the distribution across the ensemble of simulations, each level set corresponding to quantiles incremented by $5\%$. A transient $T_\mathrm{tr}=1$ (grey, dotted) is chosen manually to demarcate the beginning of the stationary state. Right panels: Distribution of energy and enstrophy in the stationary state.
• Figure 4: How to construct longitudinal increment statistics. Left: Distribution of the second-order structure function $S^{\bm v}_\parallel(r)$ for different $r$ across realizations of $\phi^{\bm v}_{ij}(\bm k, t)$, computed by \ref{['eq:sub-ensemble_strucfunc_from_spectrum_tensor']}. Center: Gaussian PDFs (colored lines) with different variances, each weighted by their probability computed from the PDF of $S^{\bm v}_\parallel(r)$, for $r=0.21\lambda$. Their superposition (black line) forms the longitudinal increment PDF of the full model \ref{['eq:passive_vector_model']}. Right: Standardized longitudinal increment PDFs of the full model \ref{['eq:passive_vector_model']} for different values of $r$, compared to a Gaussian PDF (black, dashed), vertically shifted for clarity.
• Figure 5: Growth rate of the amplitude of power-law solutions, see \ref{['eq:amplitude_evolution']}. Red corresponds to positive growth rates, blue to negative ones. The dashed line indicates zero growth rates which corresponds to stationary solutions. For $|\gamma| > \sqrt{3/8}$ no stationary solutions exist. The black dots illustrate the $k^{-1}$ and $k^2$ solutions for $\gamma=0$ discussed in Sec. \ref{['sec:spectral_energy_budget']}. The black crosses demarcate the part of the line where both power-law exponents $\xi_{1,2}$ for a given $\gamma$ are negative, leading to a stationary solution that diverges for $k\to 0$.