Universal decay of scalar turbulence

M. Chaves, G. Eyink, U. Frisch, M. Vergassola

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M. Chaves, G. Eyink, U. Frisch, M. Vergassola. Universal decay of scalar turbulence. Physical Review Letters, 2001, pp.2305-2308. 10.1103/PhysRevLett.86.2305. hal-00139227

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Universal Decay of Scalar Turbulence

M. Chaves,¹ G. Eyink,² U. Frisch,¹ and M. Vergassola¹

¹CNRS, Observatoire de la Côte d'Azur, B.P. 4229, 06304 Nice Cedex 4, France

²Department of Mathematics, University of Arizona, Tucson, Arizona 85721

(Received 5 October 2000)

The asymptotic decay of passive scalar fields is solved analytically for the Kraichnan model, where the velocity has a short correlation time. At long times, two universality classes are found, both characterized by a distribution of the scalar—generally non-Gaussian—with global self-similar evolution in time. Analogous behavior is found numerically with a more realistic flow resulting from an inverse energy cascade.

DOI: 10.1103/PhysRevLett.86.2305

PACS numbers: 47.10.+g, 05.40.–a, 47.27.–i

Recent research on passive scalars in prescribed turbulent flows has mostly concentrated on universal small-scale properties in the presence of large-scale random pumping of the scalar (see Ref. [1] and references therein). Another interesting problem is the decay of a scalar field $\theta(\mathbf{r}, t)$ with prescribed random initial conditions. It is then possible to obtain a stronger form of universality at all scales, but only at long times, when (most) of the information in the initial conditions has been washed out. Specifically, we shall be interested in self-similar decay at large Péclet numbers. We ask the following questions: (i) Is it possible to find time-dependent rescaling factors of length and scalar amplitude such that in the rescaled variables the full single-time probability of the passive field becomes independent of time as $t \to \infty$? (ii) If the first question holds true, what are the universality classes (with respect to the initial conditions) of the asymptotically self-similar scalar field and what are their statistical properties? Such issues are, of course, not limited to passive scalars. For Burgers turbulence, a systematic theory of self-similar decay can be derived [2–4]. For 3D Navier-Stokes turbulence, we shall return to the corresponding issues in the conclusions. Here we first consider the Kraichnan model [5] of passive scalars in which the velocity has a short correlation time. This allows analytical and numerical work shedding light on the basic mechanisms of scalar decay. We show that there are two universality classes, both generally having self-similar non-Gaussian distributions. Spatially smooth and very rough velocity fields represent two limiting special cases: for the former self-similar decay is broken [6,7], while for the latter nearly Gaussian self-similar decay obtains. We then turn, using numerical simulations, to scalar decay in the presence of a more realistic 2D velocity field arising from an inverse energy cascade process [8]. This also leads to self-similar decay with exponential tails in the single-point scalar probability density function (pdf).

The advection-diffusion equation governing the evolution of a passive scalar field is

$$ \partial_t \theta(\mathbf{r}, t) + \mathbf{v}(\mathbf{r}, t) \cdot \nabla \theta(\mathbf{r}, t) = \kappa \nabla^2 \theta(\mathbf{r}, t), \quad (1) $$

where $\mathbf{v}(\mathbf{r}, t)$ is the incompressible advecting flow and $\kappa$ is the molecular diffusivity. Equation (1) governs the evolution

of the probability density at position $\mathbf{r}$ and time $t$ of tracer particles. Their Lagrangian trajectories $\rho(t)$ obey the stochastic equation

$$ d\rho(t) = \mathbf{v}(\rho(t), t) dt + \sqrt{2\kappa} d\beta(t), \quad (2) $$

where $\beta(t)$ is the $d$-dimensional isotropic Brownian motion. The scalar correlation functions at time $t$ can then be expressed in terms of the initial ones as

$$ \begin{aligned} C_N(\underline{\mathbf{r}}, t) &= \langle \theta(\mathbf{r}_1, t) \cdots \theta(\mathbf{r}_N, t) \rangle \\ &= \int \mathcal{P}_N(\underline{\boldsymbol{\rho}}, 0 | \underline{\mathbf{r}}, t) C_N(\underline{\boldsymbol{\rho}}, 0) d\underline{\boldsymbol{\rho}}. \end{aligned} \quad (3) $$

Here $\underline{\mathbf{r}}$ denotes the set $\mathbf{r}_1, \dots, \mathbf{r}_N$ and $\mathcal{P}_N$ is the probability density that $N$ Lagrangian particles, being at $\underline{\mathbf{r}}$ at time $t$, were previously at $\boldsymbol{\rho}$ at time $t=0$. The initial condition $\theta(\mathbf{r}, 0)$ is taken randomly and statistically homogeneously with correlation length $L$. The statistical homogeneity implies that the absolute position of the particles is irrelevant and only the separation variables $\mathbf{r}_{i,j} = \mathbf{r}_i - \mathbf{r}_j$ matter. In the sequel we shall always consider the propagator in the restricted homogeneous sector, i.e., in the $(N-1)d$ separation variables. Note that the evolution equation (1) conserves the so-called Corrsin integral [9]

$$ J_0 = \int C_2(\mathbf{r}, t) d\mathbf{r}. \quad (4) $$

That conservation law alone does not, however, allow one to predict the scalar energy decay: for this, we also need to know the growth of the scalar correlation length with time. This is where the specific properties of the advecting velocity come into play.

Let us begin with the Kraichnan model: the velocity is statistically homogeneous and stationary, isotropic, and Gaussian. The second-order covariance of its increments is

$$ \begin{aligned} D_{\alpha\beta}(\mathbf{r}, t) &\equiv \langle v_{\alpha}(\mathbf{r}, t) v_{\beta}(\mathbf{r}, 0) \rangle - \langle v_{\alpha}(\mathbf{r}, t) v_{\beta}(0, 0) \rangle \\ &= Dr^{-\gamma} [r^2(d - \gamma + 1)\delta_{\alpha\beta} \\ &\quad - (2 - \gamma)r_{\alpha}r_{\beta}] \delta(t), \end{aligned} \quad (5) $$

where $\gamma$ controls the roughness of the field. Smooth flows correspond to $\gamma = 0$, while for $\gamma = 2$ the opposite limit of very rough fields is achieved. Hereafter, we shall
concentrate on the generic nonsmooth case $\gamma \neq 0$. The growth of the scalar length scale is determined dimensionally to be $L(t) \sim t^{1/\gamma}$ and the units are chosen to set $L = D = 1$. Note that the velocity field in the Kraichnan model is statistically invariant under time inversion; i.e., $v(r, t)$ has the same statistical properties as $-v(r, -t)$. It follows that the N-particle propagator satisfies

$$ \mathcal{P}_N(\underline{\rho}, 0 | \underline{r}, t) = \mathcal{P}_N(\underline{\rho}, t | \underline{r}, 0) = \mathcal{P}_N(\underline{r}, t | \underline{\rho}, 0). \quad (6) $$

The major consequence of the short correlation time assumption is that the Lagrangian trajectories are a Markov process and that the N-particle propagators $\mathcal{P}_N$ obey Fokker-Planck equations (see, e.g., Ref. [1]):

$$ \partial_t \mathcal{P}_N + \sum_{i \neq j} \nabla_{\underline{\alpha}}^r \nabla_{\underline{\beta}}^{r_j} \left[ \kappa \delta_{\alpha\beta} + \frac{1}{2} D_{\alpha\beta}(\mathbf{r}_{i,j}) \right] \mathcal{P}_N = 0. \quad (7) $$

The advective term in (1) conserves all the moments $\langle \theta^n \rangle$. The scalar decay is therefore due to the effects of the molecular dissipation and, as usual in turbulence, we are interested in the singular limit $\kappa \to 0$ (infinite Péclet numbers). The scalar dissipation $\kappa \langle (\nabla\theta)^2 \rangle$ remains indeed finite in the limit. From the Lagrangian point of view, the molecular noise in (2) can be neglected and the singularity of the limit shows up in the fact that initially coinciding particles separate in finite time [10,11]. For $\kappa \to 0$, the fundamental solution of (7) is self-similar and the propagator possesses the scaling property

$$ \mathcal{P}_N(\lambda \underline{\rho}, 0 | \lambda \underline{r}, \lambda^\gamma t) = \lambda^{-(N-1)d} \mathcal{P}_N(\underline{\rho}, 0 | \underline{r}, t). \quad (8) $$

The simplest objects to investigate are the single-point moments $\langle \theta^{2n} \rangle (t)$, and we are interested in their long-time behavior $t \gg 1$. The first universality class is formed by the initial conditions where the Corrsin invariant $J_0$ defined in (4) does not vanish. We shall also make the natural assumption that the initial scalar correlations decay rapidly enough to ensure integrability of the cumulants of arbitrary order. The long-time behavior of the single-point moments is obtained by setting $r = 0$ in (3) and using (8) to rescale out the time dependence. The resulting object involves $t^{d/\gamma} C_2(\underline{\rho}, t^{1/\gamma}, 0)$ which tends to $\delta(\underline{\rho})J_0$. The final result is

$$ \langle \theta^{2n} \rangle (t) \approx \frac{J_0^n}{t^{nd/\gamma}} (2n - 1)!! \times \int \mathcal{P}_{2n}(\underline{\rho}_1, \underline{\rho}_2, \dots, \underline{\rho}_n, \underline{\rho}_n, 1 | \underline{\theta}, 0) \times d\underline{\rho}_1 \dots d\underline{\rho}_n, \quad (9) $$

as it is easily seen for the special case of initial Gaussian statistics. In fact, it is not hard to show that the same result holds for all initial data in the universality class $J_0 \neq 0$ and rapid correlation decay. The integrability condition on the cumulant of order $2n$ implies indeed that it must depend on all the $(2n - 1)$ independent interparticle separations. In the limit of long times, each dependence brings a function $\delta(\underline{\rho}_{ij})$ and a factor $t^{d/\gamma}$ in the denominator.

The total non-Gaussian contribution from the initial conditions will therefore decay faster than $t^{-nd/\gamma}$. Note that this argument implies a stronger form of universality than in forced steady states: the asymptotic pdf is here independent of the initial data, whereas in the steady states the scalar pdf depends on the details of the force. The expression (9) shows that the scalar statistics is self-similar in time. For example, the single-point pdf $\mathcal{P}(\theta, t)$ takes the form $t^{d/2\gamma} Q(\theta t^{d/2\gamma})$. The propagator in the limit of initially coinciding points that appears in (9) is a regular function of its arguments [10]. The smooth case $\gamma = 0$ is special in this respect since nearby particles separate exponentially and their separation is proportional to its initial value. This is the physical reason why (9) and the ensuing self-similarity of the scalar distribution do not hold for $\gamma = 0$ (see Refs. [6,7]). Note that the asymptotic pdf $\mathcal{P}(\theta, t)$, although universal and self-similar, is generally non-Gaussian. From (9), the hypothesis of Gaussianity is equivalent to the factorization of the probability for the $2n$ particles to collapse into pairs; i.e., the integral in (9) should be equal to $[\int \mathcal{P}_2(\underline{\rho}, \underline{\rho}, 1 | \underline{\theta}, 0) d\underline{\rho}]^n$. This is generally not the case, due to the correlations existing among the particle trajectories. The only exception is $\gamma = 2$ where the particles are independent. The degree of non-Gaussianity is thus expected to decrease when $\gamma$ increases, as confirmed in Fig. 1. The simulations were performed by the Lagrangian method presented in Refs. [11,12].

The second universality class is for initial conditions with Corrsin invariant $J_0 = 0$. The vanishing of $J_0$ constrains the scalar correlations in the infrared, i.e., at the very large scales, making them weaker than for $J_0 \neq 0$. It is therefore to be expected that the scalar decay will be asymptotically faster. Specifically, from the realizability of the field it follows that $J_1 = \int r^\gamma C_2(\underline{r}) d\underline{r} \neq 0$ [13]. It is also checked using (3) and (7) that the integral remains invariant in time. $J_0 = 0$ implies that the integral of the

<figure><img src="image_2.png" /><figcaption>FIG. 1. The normalized kurtosis $\langle \theta^4 \rangle / 3 \langle \theta^2 \rangle^2$ vs time in the Kraichnan model for the two values $\gamma = 5/4$ (dashed line) and $\gamma = 2/3$ (solid line). The error bars are comparable to the size of the symbols.</figcaption></figure>
correlation $C_N$ with respect to any of its arguments vanishes. Furthermore, the single-point moments can then be expressed as

$$ \langle \theta^{2n} \rangle (t) = \int \mathcal{Q}_{2n}(\underline{\rho}, t | \underline{r}, 0) C_{2n}(\underline{\rho}, 0) d\underline{\rho}. \quad (10) $$

Here

$$ \mathcal{Q}_{2n}(\underline{\rho}, t | \underline{r}, 0) = \left( \prod_{i=2}^{2n} \Delta_{\rho_{i,1},0} \right) S_{2n} \mathcal{P}_{2n}(\underline{\rho}, t | \underline{r}, 0), \quad (11) $$

$S_{2n}$ denotes the symmetrizer in the $\rho$ variables and the difference operator $\Delta$ acts on a function $f(\mathbf{x})$ as $\Delta_{x,y}f(\mathbf{x}) = f(\mathbf{x}) - f(\mathbf{y})$. To obtain (10), we set $\underline{r} = 0$ in (3), symmetrize in the $\rho$ variables, and use the condition $J_0 = 0$ to reexpress the result in terms of difference operators. For this, note that whenever any of the $\rho$ variables in the propagator is replaced by $\mathbf{0}$ the integral vanishes. The long-time behavior of the single-point moments is then derived from (10) exploiting again the scaling property (8). For Gaussian initial conditions, we obtain

$$ \begin{aligned} \langle \theta^{2n} \rangle (t) \approx & \frac{J_1^n}{t^{n(d/\gamma+1)}} (2n-1)!! \\ & \times \lim \int \frac{\mathcal{Q}_{2n}(\underline{\rho}_1, \underline{\rho}_2, \dots, \underline{\rho}_{2n-1}, \underline{\rho}_{2n}, 1 | \underline{0}, 0)}{[\rho_{1,2} \cdots \rho_{2n-1,2n}]^\gamma} \\ & \times \prod_{i=1}^n d\underline{\rho}_{2i-1}, \end{aligned} \quad (12) $$

where we used the convergence of $\lambda^{d/\gamma}(\lambda\rho)^\gamma C_2(\lambda\rho, 0)$ to $\delta(\rho)J_1$ for large $\lambda$'s. The limit of $\rho_{1,2}\cdots\rho_{2n-1,2n}$ tending to zero, which is understood in (12), has been shown to be well defined [14]. The strong universality of the scalar distribution is proved as for the $J_0 \neq 0$ case. The expression (12) indicates that the asymptotic scalar distribution is again self-similar although the rescaling factor is now $t^{(d/\gamma+1)/2}$, in agreement with the exact solution for the second-order case [13].

The long-time behavior of multipoint scalar correlations can be derived from (3) as for the single-point moments. Using the scaling property (8) we are generally led to analyze $\mathcal{P}_N(\underline{\rho}, 1 | \underline{r}/t^{1/\gamma}, 0)$, i.e., the limit where the $N$ particles are initially close. This limit is also crucial for the small-scale scaling properties in the presence of pumping and leads to the asymptotic expansion [10]

$$ \mathcal{P}_N(\underline{\rho}, t | \lambda \underline{r}, 0) \approx \sum_{a,k} \lambda^{\zeta_a + \gamma k} f_{a,k}(\underline{r}) g_{a,k}(\underline{\rho}, t), \quad (13) $$

valid for small $\lambda$. The first sum is over the zero modes $f_a = f_{a,0}$ with scaling dimensions $p_a$ (in increasing order). The functions $f_{a,k}$ for different $k$ are the so-called slow modes. The leading term in the expansion (13) comes from the constant mode $f_{0,0} = 1$ which corresponds to $g_{0,0}(\underline{\rho}, t) = \mathcal{P}_N(\underline{\rho}, t | \underline{0}, 0)$, i.e., the propagator for $N$ particles initially at the same position. Inserting (13) into (3) and using the dominance of $g_{0,0}$ it is checked that generic multipoint correlations asymptotically decay as the

corresponding single-point moment. Subdominant terms emerge by combining the correlation functions to cancel the dominant contribution. The classical example is given by the structure functions $S_{2n}(\underline{r}, t) = \langle [\theta(\underline{r}, t) - \theta(\underline{0}, t)]^{2n} \rangle$, whose expression in terms of the correlation functions reads $S_{2n} = (\prod_{i=1}^{2n} \Delta_{r_i,0}) C_{2n}(\underline{r}, t)$. The difference operator $\Delta$ was defined in (11). The point to note is that the reducible contributions to the correlations, i.e., not depending on all the coordinates, are annihilated by the combination of the structure functions. Inserting (13) into (3) we obtain

$$ \begin{aligned} S_{2n} \approx & \frac{(\prod_{i=1}^{2n} \Delta_{r_i,0}) f_{2n,0}(\underline{r})}{t^{p_{2n}/\gamma}} \int g_{2n,0}(\underline{\rho}, 1) \\ & \times C_{2n}(t^{1/\gamma} \underline{\rho}, 0) d\underline{\rho} \\ \propto & \left( \frac{r}{t^{1/\gamma}} \right)^{\zeta_{2n}} \langle \theta^{2n} \rangle (t), \end{aligned} \quad (14) $$

where $f_{2n,0}$ is the irreducible zero mode in (13) with the lowest dimension. The exponents $\zeta_{2n}$ coincide with those of the structure functions in forced steady states [10]. This justifies the classical notion of quasiequilibrium, i.e., that short-distance scaling laws are the same in forced and decaying situations provided the basic parameters are made appropriately time dependent. The analytical solution found for the Kraichnan model permits one to capture the basic features of the scalar decay. It is, in particular, quite clear that the key ingredient for the asymptotic self-similarity of the scalar pdf is the validity of the rescaling property (8). Simulations of transport by a more realistic field show that such property is not limited to short-correlated velocities and holds (at least for large enough times) for any self-similar velocity field. The specific 2D flow that we considered is generated by an inverse

<figure><img src="image_3.png" /><figcaption>FIG. 2. The scalar energy $\langle \theta^2 \rangle$ vs $t$ for the advection by a 2D flow generated by an inverse energy cascade. The solid and the long-dashed curves correspond to the two cases $J_0 \neq 0$ and $J_0 = 0$, respectively. The two straight dashed lines are the asymptotic power law behaviors $1/t^3$ and $1/t^4$.</figcaption></figure>
<figure><img src="image_4.png" /><figcaption>FIG. 3. The rescaled scalar pdf's vs $t^{3/2}\theta$ for the inverse energy cascade flow at the 3 times marked by a star in Fig. 2. The values on the abscissa are normalized by their rms values.</figcaption></figure>

cascade process [8], is scale invariant (no intermittency) of exponent 1/3, is isotropic, and, most important, has finite realistic correlation times [15–17]. The correlation length of the scalar initial distribution is taken comparable to the velocity injection scale (see Ref. [17] for more details). In Fig. 2 we present the curves of the energy decay showing that the exponents coincide with the values obtained from the Kraichnan model ($\gamma = 2/3$ for the inverse cascade flow). The rescaled scalar pdf's $P(\theta, t)/t^{3/2}$ for 3 times in the asymptotic power-law region are presented in Fig. 3. It is quite clear that the asymptotic distribution is again self-similar and its tails are exponential. The tails are indeed dominated by the rare events governed by Poisson statistics where the scalar is weakly mixed by diffusive effects [18].

For passive scalar decay in an intermittent velocity field, such as a 3D turbulent flow, the presence of various scaling exponents makes it unlikely that the propagator possesses a rescaling property like (8). The self-similarity of the scalar decay found here might then be destroyed, with the kurtosis and higher-order scalar moments possibly growing indefinitely with time. An even more challenging problem is that of the decay of 3D turbulent flow which has been frequently addressed, but usually in a phenomenological way and with the emphasis on the law of energy decay (see Refs. [19,20] for a review). Is there self-similar and universal decay as for the problem considered in this pa-

per or is the kurtosis of the velocity growing indefinitely, possibly leading to a highly intermittent spatial distribution of the kinetic energy? It is of great interest to address such issues experimentally and by advanced direct numerical simulations.

We are grateful to J. Bec, A. Celani, M. Chertkov, G. Falkovich, K. Gawędzki, B. Shraiman, K. R. Sreenivasan, and V. Yakhot for useful discussions. The support by the European Union under Contract No. HPRN-CT-2000-00162, by the National Science Foundation under Grant No. PHY94-07194, and by the Fundação para a Ciência e a Tecnologia under Grant No. PRAXIS XXI/BD/21413/99 is acknowledged. Simulations were performed at IDRIS (No. 991226).

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