330. Some properties of the Burgers dynamics with Brownian or white-noise initial velocity

Posted by Patrick Valageas

March 31, 2009

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Patrick Valageas is a permanent researcher at the IPhT (Theoretical Physics department) of CEA, Saclay. His interests include turbulence, observational cosmology (LSS formation in particular) and astrophysics. Dmitry.

I would like to thank Dmitry for giving me the opportunity to present two recent papers of mine (arXiv:0810.4332 and arXiv:0903.0956), on the Burgers equation, from the point of view of a cosmologist. They consider the one-dimensional Burgers dynamics for Brownian and white-noise initial velocity, and expand some previous results on the probability distributions of velocity and Lagrangian increments, as well as on the distribution of the density and the shock mass function.

First, let me recall that the Burgers equation,

$\partial_t {\bf v} + ({\bf v}.\nabla){\bf v}=\nu \Delta {\bf v}$ ,

was originally introduced as a simplified model for fluid turbulence, as it shares the same hydrodynamical (advective) nonlinearity and several conservation laws with the Navier-Stokes equation. However, it also appears in many physical problems, such as the propagation of nonlinear acoustic waves or the formation of large-scale structures (filaments, clusters of galaxies) in cosmology. In the latter context, where one considers the inviscid limit ( $\nu\rightarrow 0^+$ ), it is known as the “adhesion model” and it provides a good description of the large-scale filamentary structure of the cosmic web (this involves a rescaling of time and velocities, so that is actually the linear growing mode of density perturbations, proportional to the scale factor in a critical-density universe, ${\bf x}$ is a comoving coordinate, and ${\bf v}$ is – up to a time-dependent factor – the peculiar velocity, i.e. the mean Hubble flow associated with the expansion of the universe has been subtracted).

In this context, one is interested in the statistical properties of the dynamics, as described by the density and velocity fields, starting with a random Gaussian initial velocity at t=0 and a uniform density (the density evolving through the usual continuity equation). These initial conditions are the signature of quantum fluctuations generated in the primordial Universe and agree with the small Gaussian fluctuations observed on the cosmic microwave background. In the hydrodynamical context, this setup corresponds to “decaying Burgers turbulence” (whereas stationary “Burgulence” would be obtained by adding an external stochastic forcing). Thus, the evolution is deterministic and the stochasticity arises from the random initial conditions.

In the inviscid limit, the one-dimensional Burgers dynamics can be understood in very simple terms through a discrete model (as the Burgers dynamics also appears as the continuum limit of a ballistic aggregation process). Indeed, if we set $\nu=0$ , we obtain the equation of motion of free particles, ${\rm d} v/{\rm d}t=0$ , that keep forever their initial velocity and can cross each other. This is actually the well-known Zeldovich approximation in the cosmological context (which is exact at linear order).

Parabolic construction of the solution

Fig. 1. The parabolic construction of the solution. At time t₁ , x₁ is a regular point, while there is a shock at time t₂ at position x₂ ( t₂>t₁ ).

Then, adding an infinitesimal viscosity prevents such particle crossings, so that when particles collide they stick together with conservation of momentum (but with a loss of energy). In the hydrodynamical continuum case, this corresponds to the formation of shocks, that lead to negative jumps of the velocity field (as faster particles overtake slower ones). This is actually the reason why the Burgers equation was introduced in cosmology, as an improvement over the Zeldovich approximation to prevent particles from crossing each other and escaping to infinity. The sticking is intended to mimic the trapping of particles in the gravitational potential wells built by the dynamics.

Most theoretical studies of the Burgers dynamics focus on the one-dimensional case, with power-law initial energy spectra, $E_0(k)\propto k^n$ , where E_0(k) is the Fourier transform of the initial velocity correlation. Then, for -3<n<1 , the evolution is self-similar and the characteristic scale grows as $L(t)\propto t^{2/(n+3)}$ .

Hopf-Cole solution and geometrical construction

In my papers I mostly focussed on the cases n=-2 (Brownian initial velocity) and n=0 (white-noise initial velocity), which are two standard cases where many exact results can be obtained. I now describe the general method used in these works. They rely on the well-known Hopf-Cole transformation, that provides the explicit solution of the Burgers equation. Indeed, introducing the velocity potential, $v=\partial\psi/\partial x$ , and making the change of variable $\psi(x,t)=-2\nu\ln\theta(x,t)$ , transforms the nonlinear Burgers equation into the linear heat equation. This gives the explicit solution

$\psi(x,t)=-2\nu \ln \int_{-\infty}^{\infty} \frac{{\rm d} q}{\sqrt{4\pi\nu t}} \; \exp\left[-\frac{(x-q)^2}{4\nu t}-\frac{\psi_0(q)}{2\nu}\right]$ ,

where $\psi_0(q)$ is the initial potential. Then, in the inviscid limit, $\nu\rightarrow 0^+$ , a steepest-descent method gives

$\psi(x,t)=\min_q \left[ \psi_0(q) + \frac{(x-q)^2}{2t} \right] \hspace{0.5cm} \mbox{and} \hspace{0.5cm} v(x,t)=\frac{x-q(x,t)}{t}$ ,

where q(x,t) is the point where the minimum in the first expression is reached. Outside of shocks, this is the initial Lagrangian position of the particle that is located at the Eulerian position at time . If there are two solutions, $q_-{}<q_+$ , there is a shock at position that contains all the mass coming from $\left[{}q_-,q_+{}\right]{}$

This explicit solution has several well-known geometrical interpretations. For instance, let us consider the downward parabola ${\cal P}_{x,c}(q)$ , centered at and of maximum ,

${\cal P}_{x,c}(q)=- \frac{(q-x)^2}{2 t} + c$ .

Then, starting from below with a large negative value of , such that the parabola is everywhere well below $\psi_0(q)$ , we increase until the two curves touch one another. Then, the abscissa of the point of first-contact is the Lagrangian coordinate q(x,t) and the potential is given by $\psi(x,t)=c$ . At early times, the parabolas have a large curvature so that the contact point is close to , while at late times the parabolas are almost flat, so that many shocks have formed and particles have travelled over a large distance. For the non-smooth, power-law initial conditions, $E_0(k)\propto k^n$ with -3<n<1 , shocks actually form as soon as t>0 . Moreover, shocks are dense in Eulerian space for -3<n<-1 and isolated for -1<n<1 .

Let us first consider the case of white-noise initial velocity ( n=0 ), that is, the initial potential $\psi_0(q)$ is a Brownian motion. Then, the process $q\mapsto\psi_0$ is Markovian, and from the previous geometrical construction we can see that a key quantity is the conditional probability density $K_{x,c}(q_1,\psi_1;q_2,\psi_2)$ for the Markov process $\psi_0(q)$ , starting from $\psi_1$ at q_1 , to end at $\psi_2$ at $q_2 \geq q_1$ , while staying above the parabolic barrier, $\psi_0(q)>{\cal P}_{x,c}(q)$ , for $q_1\leq q\leq q_2$ . Since $\psi_0(q)$ is a Brownian motion, this kernel $K_{x,c}$ obeys the usual diffusion equation but with an absorbing parabolic boundary at ${\cal P}_{x,c}$ , and it is possible to obtain its explicit expression in terms of Airy functions. In the case n=-2 , $\psi_0(q)$ is the integral of a Brownian motion, and the process $q\mapsto\{\psi_0,v_0\}$ is Markovian. Thus, we are now led to consider the conditional probability density $K_{c,x}(q_1,\psi_1,v_1;q_2,\psi_2,v_2)$ . It obeys an advective-diffusion equation with parabolic absorbing barrier and we can again derive its explicit expression.

Then, from the geometrical construction in terms of parabolas, the properties of the system can be expressed in terms of these kernels $K_{x,c}$ . For instance, in the case n=0 (white-noise), restricted to the range [q_-,q_+] , the probability $p_x(q_-\leq q'\leq q)$ that the Lagrangian coordinate q'(x,t) is in the range [0,q] , factorizes into 2 terms, i) $\psi_0$ stays above ${\cal P}_{x,c}$ but crosses ${\cal P}_{x,c+{\rm d} c}$ over $q_-\leq q'\leq q$ , ii) $\psi_0$ stays above ${\cal P}_{x,c}$ for $q{}<{}q'{}<{}q_+$ , which can both be expressed in terms of $K_{x,c}$ , and we eventually integrate over the height and take the limit $q_\pm\to\pm\infty$

Thus, the reason why the two cases n=-2 and n=0 can be explicitly solved is that one obtains Markovian processes, so that the geometrical construction for point distributions can be broken into such pieces, each involving a simple kernel $K_{x,c}$ , which are joined by matching at the boundaries. This is no longer possible for generic initial conditions, where the behavior of the initial potential $\psi_0(q)$ over some range [q_1,q_2] does not only depend on a few values at the boundaries but on the full curve $\psi_0(q)$ on both sides.

Brownian initial velocity

In the case n=-2 , one obtains the peculiar property that Lagrangian increments, $\Delta q=q(x_2)-q(x_1)$ , are exactly independent and homogeneous for non-overlapping cells (far from any boundaries). Then, point distributions factorize. Next, these increments show for small Eulerian distance, $\Delta x=|x_2-x_1|\rightarrow 0$ , the bifractal behavior

$\nu>\frac{1}{2}: \langle (\Delta q)^{\nu}\rangle \sim \Delta x , \;\; \nu<\frac{1}{2}: \langle (\Delta q)^{\nu}\rangle \sim (\Delta x)^{2\nu}$ .

As is well-known, the scaling obtained for large $\nu$ is due to shocks, that are associated with finite jumps for $\Delta q \sim 1$ (the factor \Delta x arises from the probability to encounter a massive shock in the interval of size $\Delta x$ ). This also leads to the small-scale scaling $\langle [v(x+\ell)-v(x)]^p\rangle \propto \ell$ ( $p\geq 2$ ) for the velocity structure functions. These exponents are universal, in the sense that they appear as soon as shocks have formed, independently of the initial conditions. They are different from those obtained in Navier-Stokes turbulence, where the relevant structures are more varied and less singular (because of pressure effects).

Probability distribution of the overdensity for the Brownian case

Fig. 2. The probability distribution $P_X(\eta)$ of the overdensity, $\eta=m/({\overline \rho} \Delta x)$ , for Brownian initial velocity. Smaller corresponds to more nonlinear scales or times.

A quantity of interest, within the cosmological context, is the distribution, $P_X(\eta)$ , of the mean overdensity, $\eta=\rho/{\overline \rho}$ , within a cell of size $\Delta x$ , where $X\propto \Delta x/t^2$ is the rescaled length that expresses the self-similar evolution. As shown in the figure, at large scales (or early times) it goes to a sharp Gaussian peaked around the mean, $\langle \eta\rangle=1$ , as we recover the Gaussian initial conditions. At smaller nonlinear scales (or late times), where shocks govern the dynamics, an intermediate power-law regime develops. This is quite similar to the behavior observed in numerical simulations of cosmological gravitational clustering. In fact, it is possible to obtain its exact expression as

$P_X(\eta)=\sqrt{\frac{X}{\pi}} \, \eta^{-3/2} \, e^{-X(\sqrt{\eta}-1/\sqrt{\eta})^2}$ .

Moreover, the ratios of the density cumulants are scale-invariant,

$\frac{\langle\eta^p\rangle_c}{\langle\eta^2\rangle_c^{p-1}}=(2p-3)!!$ .

In the cosmological context, this property is known as the “stable-clustering ansatz”, which is a reasonable approximation in the nonlinear regime. Thus, the 1d Burgers dynamics with Brownian initial velocity provides a dynamical model where this property is exactly fulfilled. Finally, it is interesting to note that the mass function of shocks obtained in this system, $n(M)=1/\sqrt{\pi} M^{-3/2} e^{-M}$ , with $M \propto m/t^2$ , happens to be identical to the prediction of the Press-Schechter ansatz, that is also widely used in cosmology.

White-noise initial velocity

In the case n=0 , a qualitative difference from the previous case is that socks are no longer dense but isolated. Then, at small scales most cells are empty. One recovers the scaling $\langle(\Delta q)^{\nu}\rangle \propto \Delta x$ at small scales, now for all $\nu>0$ , that is associated with shocks, and the similar scaling for the velocity structure functions.

Probability distribution of the overdensity for white-noise

Fig. 3. The probability distribution $P_X(\eta)$ of the overdensity for white-noise initial velocity.

The system is now dominated by shocks, even at large scales, so that the distribution of the overdensity no longer goes to a Gaussian in the limit of large scales (or early times). Thus, at large densities it always shows a cubic exponential cutoff, $P_X(\eta) \sim e^{-X^3\eta^3/12}$ , with $X \propto x/t^{2/3}$ . At small nonlinear scales, apart from the Dirac contribution, $\delta(\eta)$ , associated with empty cells, the regular contribution again shows a power-law regime for low and moderate densities (but there is no longer a cutoff at very low densities).

The “stable-clustering ansatz” is no longer satisfied, but the ratios $\langle\eta^p\rangle_c/\langle\eta^2\rangle_c^{p-1}$ still have finite limits for $\Delta x\rightarrow 0$ . This is again an universal property due to shocks. As with the comparison with Navier-Stokes turbulence, the cosmological gravitational clustering dynamics appears more complex, as the relevant structures are extended halos rather than point-like singularities (which could explain why the “stable-clustering ansatz” is not exactly realized in cosmology).

Conclusion

I have given here a very brief introduction to studies of the Burgers equation, focussing on analogies with gravitational clustering. Although such works do not provide quantitative predictions for actual turbulence or gravitational dynamics, the Burgers dynamics still shows a great interest per se, as it appears in many other contexts. Moreover, it can serve as a useful benchmark to test approximation schemes devised for turbulence or gravitational dynamics, as the nonlinearities are similar, and one can compare with exact results. Readers who are interested in such topics can have a look at the recent review arXiv:0704.1611, which contains all references I have not given above.

Some literature

1. S. Gurbatov, A. Malakhov and A. Saichev, “Nonlinear random waves and turbulence in nondispersive media: waves, rays and particles”, Manchester University Press, 1991.
2. M. Vergassola and B. Dubrulle and U. Frisch and A. Noullez, “Burgers’equation, Devil’s staircases and the mass distribution for large-scale structures”, Astron. Astrophys. 289 (1994) 325 – one nice paper which also makes the link with cosmology.
3. S. N. Gurbatov, S. I. Simdyankin, E. Aurell, U. Frisch and G. Toth, “On the decay of Burgers turbulence”, J. Fluid Mech. 344 (1997) 339 – and one paper for the turbulence point of view.

Cosmology, Hydrodynamics, Journal club, Master Postdoc, Turbulence

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330. Some properties of the Burgers dynamics with Brownian or white-noise initial velocity

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