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Passive scalar turbulence is the study of how a scalar quantity, such astemperature or salinity, is transported by an incompressible fluid. Thisprocess is modeled by the *advection diffusion equation*\[\begin{equation}\partial_tg_t + u_t\cdot\nabla g_t – \kappa \Delta g_t = s_t,\label{eqAD}\tag{AD}\end{equation}\] where \(g_t\) is the scalar quantity, \(u_t\) is anincompressible velocity field, \(\kappa>0\) is the diffusivity parameterand \(s_t\) is a replenishing source. As \(g_t\) evolves, it often settlesinto a statistical steady state and complex self-similar structuresarise due to repeated stretching and folding by the velocity field.

Figure 1: A numerical simulation of scalar turbulence on \(\mathbb{T}^2\)advected by the stochastic Navier-Stokes equations

In his 1959 work (Batchelor (1959)) Batchelor made a significant step towardunderstanding these structures. He predicted that, on average, the \(L^2\)power spectral density of \(g_t\) displays a \(|k|^{-1}\) power law(Batchelor’s law) for the \(L^2\) power spectral density of \(g_t\) alongfrequencies \(k\) in the so-called *viscous convective range*, i.e.,length-scales sufficiently small such that the fluid motion isviscosity-dominated but large enough so as not to be dissipated bymolecular diffusion. This law has since been verified in physical,numerical, and experimental settings (e.g. Grant et al. (1968), Antonia and Orlandi (2003),Gibson and Schwarz (1963)) and is frequently used by scientists to predict thedistribution of pollutants and biological matter in the ocean andatmosphere. Despite this success, Batchelor’s law has evaded rigorousmathematical proof.

The purpose of this post is to report progress with Jacob Bedrossian andAlex Blumenthal on the development of rigorous mathematical tools forstudying Batchelor’s law when \(u_t\) evolves according to a randomlyforced fluid model. The primary example is the *incompressiblestochastic Navier-Stokes equations* on \(\mathbb{T}^2\),

\[\begin{equation}\partial_t u_t + u_t\cdot\nabla u_t + \nabla p_t – \nu \Delta u_t = \xi_t, \\\mathbf{div} u_t=0,\label{eqSNS}\tag{SNS}\end{equation}\]

though other well-posed models (not restricted two dimensions) can beconsidered. Here the stochastic forcing \(\xi_t\) is assumed to be anon-degenerate, white-in-time, spatially Sobolev regular Gaussianforcing. The viscosity parameter \(\nu > 0\) can be considered the inverseReynolds number.

For this model and a host of other fluid models, in Bedrossian, Blumenthal, and Punshon-Smith (2019a) weprove a version Batchelor’s prediction on the cumulative power spectrum,when the viscosity parameter \(\nu>0\) is fixed.

Theorem 1 (Bedrossian, Blumenthal, and Punshon-Smith 2019a):

Let \(\Pi_{\leq N}\) denote the projection onto Fourier modes with\(|k|\leq N\). Let the source \(s_t\) in \(\eqref{eqAD}\) be a white-in-timeGaussian process and \(u_t\) be given by \(\eqref{eqSNS}\) as described above.Then there exists a unique stationary probability measure \(\mu^\kappa\)for the Markov process \((u_t,g_t)\) and \(\kappa\)-independent constants\(C \geq 1\) and \(\ell_0\leq 1\) such that

\[\begin{equation}\frac{1}{C_0}\log N \leq \mathbb{E}_{\mu^\kappa} \|\Pi_{\leq N}g\|^{2}_{L^2} \leq C_0\log N \quad \text{for}\quad \ell_{0}^{-1} \leq |k|\leq \kappa^{-1/2}.\label{eqBL}\tag{BL}\end{equation}\]

## Uniform-in-\(\kappa\) exponential mixing and Batchelor’s law

The key ingredient in obtaining Batchelor’s law is the mixing propertiesof the velocity field \(u_t\), and a quantitative understanding of howthat mixing interacts with the diffusion. In the absence of a scalarsource (\(s_t =0\)) and molecular diffusivity (\(\kappa = 0\)), the velocityfield \(u_t\) filaments \(g_t\) and forms small scales as it homogenizes, aprocess known as *mixing* (see Figure 2).

Figure 2: Mixing of a circular blob, showing filamentation and formationof small scales.

Mixing of the scalar \(g_t\) (assuming it is mean zero) can be quantifiedusing a negative Sobolev norm. Commonly chosen is the \(H^{-1}\) norm\(\|g_t\|_{H^{-1}} := \|(-\Delta)^{-1/2}g_t\|_{L^2}\), which essentiallymeasures the average filamentation width, though there are many otherexpedient choices Thiffeault (2012).

In Bedrossian, Blumenthal, and Punshon-Smith (2021) we show that solutions to \(\eqref{eqSNS}\) cause the advectiondiffusion equation (without source but with diffusion) to mixexponentially fast with a rate that is *uniform in the diffusivity*parameter \(\kappa\).

Theorem 2 (Uniform-in-diffusivity mixing, Bedrossian, Blumenthal, and Punshon-Smith 2021):

*Let* \(u_t\) solve \(\eqref{eqSNS}\) with non-degenerate noise. There existsa **deterministic** \(\gamma > 0\), independent of \(\kappa\), such that forall initial \(u_0\), and all \(\kappa \in [0,1]\) there is a random constant\(D_\kappa = D_\kappa(u_0,\omega)\) so that for all zero-mean\(g_0 \in H^1\) and all \(t>0\) the following holds almost surely\[\begin{equation}\|g_t\|_{H^{-1}} \leq D_\kappa e^{-\gamma t}\|g_0\|_{H^1}.\label{eqEM}\tag{EM}\end{equation}\] *The random constant* \(D_\kappa\), has finite secondmoment **uniformly bounded in** \(\kappa\).

Theorem 2 can be seen as a direct consequence of Theorem 1 and followsfrom a fairly straight forward argument using the mild form of\(\eqref{eqAD}\) and estimates on the stochastic convolution. This argumentis carried out in Bedrossian, Blumenthal, and Punshon-Smith (2019a).

## Lagrangian chaos

It has long been understood in the physics community (e.g. Bohr et al. (2005),Antonsen Jr and Ott (1991), Ott (1999), Shraiman and Siggia (2000)) that the predominantmechanism for mixing in spatially regular fluids is the chaotic motionof the particle trajectories \(x_t = \phi^t(x)\) of the **Lagrangian flowmap** \(\phi^t : \mathbb{T}^d \to \mathbb{T}^d\) associated to thevelocity field \(u_t\), defined by \[ \frac{d}{dt} \phi^t(x) = u_t(\phi^t(x)), \quad \phi^0(x) = x\in \mathbb{T}d.\] Here we characterize chaos through having a *positive Lyapunovexponent* \[\begin{equation}0< \lambda_1 := \lim_{t\to \infty} \frac{1}{t}\log|D_x\phi^t| \, .\label{eqPE}\tag{PE}\end{equation}\] This property is typically referred to as **Lagrangianchaos** in the fluid mechanics literature.

In the deterministic setting, proving positivity of Lyapunov exponentsas in \(\eqref{eqPE}\) is currently hopelessly out of reach due to thepossible formation of coherent structures and lack of ergodicity.However, starting with the seminal work of Furstenberg(Furstenberg (1963)), significant success has been achieved inproving existence and positivity of Lyapunov exponents in the context of*random dynamical systems* (Arnold (2013), Kifer (2012),P. H. Baxendale (1989), Ledrappier and Young (1985)). Ideas in this vein arewhat enabled us to prove the following Lagrangian chaos result, thefirst step in the proof of Theorem 3.

Theorem 3 (Lagrangian chaos, Bedrossian, Blumenthal, and Punshon-Smith 2018):

Let \(u_t\) solve \(\eqref{eqSNS}\) with non-degenerate noise as above, thenthere exists a **deterministic** constant \(\lambda_1 > 0\) (independentof \(u_0\) and \(\omega\)) for which \(\eqref{eqPE}\) holds almost surely.

## Decay of correlations and mixing

Let us now address how \(\eqref{eqEM}\) is obtained in the case\(\kappa = 0\). In this case, the solution \(g_t\) is given by\(g_t = g_0 \circ (\phi^t)^{-1}\). In view of this, \(\eqref{eqPE}\) suggeststhat \(g_t\) is `stretched out’ considerably as \(t\) increases, leading toa rapid generation of high frequencies as oppositely signed values ofthe concentration profile \(g_t\) “pile up’’ against each other almosteverywhere in the domain. Indeed, this local-to-global mechanism iswidely used in dynamics. It is known as *decay of correlations* andtakes the form \[\begin{equation}\left|\int (f\circ \phi^t)\, g dx \right| \leq D_\kappa e^{-\gamma t}\|f\|_{H^1}\|g\|_{H^1}\label{eq1}\tag{1}\end{equation}\] for each mean zero \(f,g\in H^1\) and \(t>0\). This isequivalent to exponential mixing \(\eqref{eqEM}\).

Despite this simple picture, passing from \(\eqref{eqPE}\) to \(\eqref{eq1}\)requires serious work. When random driving is present, the **two-pointprocess** is a powerful tool for proving exponential correlation decay(P. Baxendale and Stroock (1988), Dolgopyat et al. (2004)).

In our context, this is the Markov process that simultaneously trackstwo particles subjected to the same velocity field\((u_t,\phi^t(x),\phi^t(y))\) for \(x \neq y\). Correlation decay forgeneric noise realizations is connected with the rate at which theprobabilistic law of \((u_t, \phi^t(x), \phi^t(y))\) relaxes to itsequilibrium statistics (known as *geometric ergodicity*).

The proof of Theorem 2 with \(\kappa = 0\) utilizes this connection usingtools from the theory of Markov chains, particularly the Harris theoremMeyn and Tweedie (2012). The main difficulty to overcome here is the degeneracyin the \((u_t, \phi^t(x), \phi^t(y))\) process near the diagonal\(\{x=y\}\). One needs to show that any time two particles are close, theyseparate again exponentially fast. This effectively amounts to a largedeviation estimate on the convergence of finite-time Lyapunov exponentsto the asymptotic Lyapunov exponent deduced in Theorem 3, and is carriedout in Bedrossian, Blumenthal, and Punshon-Smith (2019b).

It remains to incorporate molecular diffusion (\(\kappa > 0\)) into thisscheme. This comes down again to the two-point process, now withLagrangian flow \(\phi^t_\kappa\) augmented by an additional white noiseterm with variance \(\sqrt{\kappa}\) to account for molecular diffusivity.The primary step is to show that one can pass to the singular limit\(\kappa \to 0\) in the dominant eigenvalue, eigenfunction pair for thePerron-Frobenius operator corresponding to\((u_t, \phi^t_\kappa(x), \phi^t_\kappa(y))\); this is carried out inBedrossian, Blumenthal, and Punshon-Smith (2021).

## Bibliography

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