Chaotic mixing and the statistical properties of scalar turbulence
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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
∂tgt+ut⋅∇gt–κΔgt=st,
Figure 1: A numerical simulation of scalar turbulence on T2 advected by the stochastic Navier-Stokes equations
In his 1959 work (Batchelor (1959)) Batchelor made a significant step toward understanding these structures. He predicted that, on average, the L2 power spectral density of gt displays a |k|−1 power law (Batchelor’s law) for the L2 power spectral density of gt 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 ut evolves according to a randomly forced fluid model. The primary example is the incompressible stochastic Navier-Stokes equations on T2,
∂tut+ut⋅∇ut+∇pt–νΔut=ξt,divut=0,though other well-posed models (not restricted two dimensions) can be considered. Here the stochastic forcing ξt is assumed to be a non-degenerate, white-in-time, spatially Sobolev regular Gaussian forcing. The viscosity parameter ν>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 ν>0 is fixed.
Theorem 1 (Bedrossian, Blumenthal, and Punshon-Smith 2019a):
Let Π≤N denote the projection onto Fourier modes with |k|≤N. Let the source st in (AD) be a white-in-time Gaussian process and ut be given by (SNS) as described above. Then there exists a unique stationary probability measure μκ for the Markov process (ut,gt) and κ-independent constants C≥1 and ℓ0≤1 such that
1C0logN≤Eμκ‖Π≤Ng‖2L2≤C0logNforℓ−10≤|k|≤κ−1/2.Uniform-in-κ exponential mixing and Batchelor’s law
The key ingredient in obtaining Batchelor’s law is the mixing properties of the velocity field ut, and a quantitative understanding of how that mixing interacts with the diffusion. In the absence of a scalar source (st=0) and molecular diffusivity (κ=0), the velocity field ut filaments gt 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 gt (assuming it is mean zero) can be quantified using a negative Sobolev norm. Commonly chosen is the H−1 norm ‖gt‖H−1:=‖(−Δ)−1/2gt‖L2, 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 (SNS) cause the advection diffusion equation (without source but with diffusion) to mix exponentially fast with a rate that is uniform in the diffusivity parameter κ.
Theorem 2 (Uniform-in-diffusivity mixing, Bedrossian, Blumenthal, and Punshon-Smith 2021):
Let ut solve (SNS) with non-degenerate noise. There exists
a deterministic γ>0, independent of κ, such that for
all initial u0, and all κ∈[0,1] there is a random constant
Dκ=Dκ(u0,ω) so that for all zero-mean
g0∈H1 and all t>0 the following holds almost surely
‖gt‖H−1≤Dκe−γt‖g0‖H1.
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 (AD) 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 xt=ϕt(x) of the Lagrangian flow
map ϕt:Td→Td associated to the
velocity field ut, defined by ddtϕt(x)=ut(ϕt(x)),ϕ0(x)=x∈Td.
In the deterministic setting, proving positivity of Lyapunov exponents as in (PE) 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 ut solve (SNS) with non-degenerate noise as above, then there exists a deterministic constant λ1>0 (independent of u0 and ω) for which (PE) holds almost surely.
Decay of correlations and mixing
Let us now address how (EM) is obtained in the case
κ=0. In this case, the solution gt is given by
gt=g0∘(ϕt)−1. In view of this, (PE) suggests
that gt is `stretched out’ considerably as t increases, leading to
a rapid generation of high frequencies as oppositely signed values of
the concentration profile gt “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 |∫(f∘ϕt)gdx|≤Dκe−γt‖f‖H1‖g‖H1
Despite this simple picture, passing from (PE) to (1) 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 (ut,ϕt(x),ϕt(y)) for x≠y. Correlation decay for generic noise realizations is connected with the rate at which the probabilistic law of (ut,ϕt(x),ϕt(y)) relaxes to its equilibrium statistics (known as geometric ergodicity).
The proof of Theorem 2 with κ=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 (ut,ϕt(x),ϕt(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 (κ>0) into this scheme. This comes down again to the two-point process, now with Lagrangian flow ϕtκ augmented by an additional white noise term with variance √κ to account for molecular diffusivity. The primary step is to show that one can pass to the singular limit κ→0 in the dominant eigenvalue, eigenfunction pair for the Perron-Frobenius operator corresponding to (ut,ϕtκ(x),ϕtκ(y)); this is carried out in Bedrossian, Blumenthal, and Punshon-Smith (2021).
Bibliography
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