Passive scalar turbulence is the study of how a scalar quantity, such as
temperature or salinity, is transported by an incompressible fluid. This
process 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 an
incompressible velocity field, \(\kappa>0\) is the diffusivity parameter
and \(s_t\) is a replenishing source. As \(g_t\) evolves, it often settles
into a statistical steady state and complex self-similar structures
arise 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 (all rights to the video content belong to Sam Punshon-Smith).

In his 1959 work (Batchelor (1959)) Batchelor made a significant step toward
understanding 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 \(g_t\) along
frequencies \(k\) in the so-called *viscous convective range*, i.e.,
length-scales sufficiently small such that the fluid motion is
viscosity-dominated but large enough so as not to be dissipated by
molecular 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 the
distribution of pollutants and biological matter in the ocean and
atmosphere. Despite this success, Batchelor’s law has evaded rigorous
mathematical proof.

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

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

For this model and a host of other fluid models, in Bedrossian, Blumenthal, and Punshon-Smith (2019a) we prove 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-time Gaussian 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 properties
of the velocity field \(u_t\), and a quantitative understanding of how
that mixing interacts with the diffusion. In the absence of a scalar
source (\(s_t =0\)) and molecular diffusivity (\(\kappa = 0\)), the velocity
field \(u_t\) filaments \(g_t\) and forms small scales as it homogenizes, a
process known as *mixing* (see Figure 2).

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

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

In Bedrossian, Blumenthal, and Punshon-Smith (2021) we show that solutions to \(\eqref{eqSNS}\) cause the advection
diffusion equation (without source but with diffusion) to mix
exponentially 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 exists
a **deterministic** \(\gamma > 0\), independent of \(\kappa\), such that for
all 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 second
moment **uniformly bounded in** \(\kappa\).

Theorem 2 can be seen as a direct consequence of Theorem 1 and follows from a fairly straight forward argument using the mild form of \(\eqref{eqAD}\) and estimates on the stochastic convolution. This argument is 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 predominant
mechanism for mixing in spatially regular fluids is the chaotic motion
of the particle trajectories \(x_t = \phi^t(x)\) of the **Lagrangian flow
map** \(\phi^t : \mathbb{T}^d \to \mathbb{T}^d\) associated to the
velocity 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 Lyapunov
exponent* \[\begin{equation}
0< \lambda_1 := \lim_{t\to \infty} \frac{1}{t}\log|D_x\phi^t| \, .\label{eqPE}\tag{PE}
\end{equation}\] This property is typically referred to as **Lagrangian
chaos** in the fluid mechanics literature.

In the deterministic setting, proving positivity of Lyapunov exponents
as in \(\eqref{eqPE}\) is currently hopelessly out of reach due to the
possible formation of coherent structures and lack of ergodicity.
However, starting with the seminal work of Furstenberg
(Furstenberg (1963)), significant success has been achieved in
proving 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 are
what enabled us to prove the following Lagrangian chaos result, the
first 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, then
there exists a **deterministic** constant \(\lambda_1 > 0\) (independent
of \(u_0\) 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, \(\eqref{eqPE}\) suggests
that \(g_t\) is `stretched out’ considerably as \(t\) increases, leading to
a rapid generation of high frequencies as oppositely signed values of
the concentration profile \(g_t\) ``pile up’’ against each other almost
everywhere in the domain. Indeed, this local-to-global mechanism is
widely used in dynamics. It is known as *decay of correlations* and
takes 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 is
equivalent 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-point
process** 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 tracks
two particles subjected to the same velocity field
\((u_t,\phi^t(x),\phi^t(y))\) for \(x \neq y\). Correlation decay for
generic noise realizations is connected with the rate at which the
probabilistic law of \((u_t, \phi^t(x), \phi^t(y))\) relaxes to its
equilibrium statistics (known as *geometric ergodicity*).

The proof of Theorem 2 with \(\kappa = 0\) utilizes this connection using tools from the theory of Markov chains, particularly the Harris theorem Meyn and Tweedie (2012). The main difficulty to overcome here is the degeneracy in 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, they separate again exponentially fast. This effectively amounts to a large deviation estimate on the convergence of finite-time Lyapunov exponents to the asymptotic Lyapunov exponent deduced in Theorem 3, and is carried out in Bedrossian, Blumenthal, and Punshon-Smith (2019b).

It remains to incorporate molecular diffusion (\(\kappa > 0\)) into this scheme. This comes down again to the two-point process, now with Lagrangian flow \(\phi^t_\kappa\) augmented by an additional white noise term 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 the Perron-Frobenius operator corresponding to \((u_t, \phi^t_\kappa(x), \phi^t_\kappa(y))\); this is carried out in Bedrossian, Blumenthal, and Punshon-Smith (2021).

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