Onset of intermittency in stochastic Burgers hydrodynamics

We study the onset of intermittency in stochastic Burgers hydrodynamics, as characterized by the statistical behavior of negative velocity gradient fluctuations. The analysis is based on the response functional formalism, where specific velocity configurations—the viscous instantons—are assumed to play a dominant role in modeling the left tails of velocity gradient probability distribution functions. We find, as expected on general grounds, that the field-theoretical approach becomes meaningful in practice only if the effects of fluctuations around instantons are taken into account. Working with a systematic cumulant expansion, it turns out that the integration of fluctuations yields, in leading perturbative order, to an effective description of the Burgers stochastic dynamics given by the renormalization of its associated heat kernel propagator and the external force-force correlation function.

Figure 1

Feynman diagrammatic representation of (a) the heat kernel propagator ${\langle u(x,t)p(x^{\prime },t^{\prime })\rangle }_0$ and (b) the velocity-velocity correlator ${\langle u(x,t)u(x^{\prime },t^{\prime })\rangle }_0$.

Figure 2

One loop contributions for the renormalization of (a) the noise kernel and (b) the heat kernel propagator in the effective MSRJD action. Incoming and outgoing dashed lines are associated to the instanton fields $p^c(x,t)$ and $u^c(x,t)$, respectively, as they appear in Eqs. (2.28) and (2.29).

Figure 3

The Lagrange multiplier $\lambda$ is given as a function of the velocity gradient $\xi$. Open circles represent values obtained from the numerical solutions of Eqs. (2.9)–(2.11) (the black solid line is just a polynomial interpolation of the numerical data); red solid line: approximated instanton relation, Eq. (3.13); dashed line: $\lambda =2\xi$, which holds for asymptotically small velocity gradients.

Figure 4

Comparison between the surrogate saddle-point action, as prescribed by Grafke et al. [20] for the case of noise strength parameter $g=1.7$ (black solid line), and a four-parameter fitting function (red line) which provides distinct power-law asymptotics for domains of small and large velocity gradients.

Figure 5

Modeled (red lines) and empirical (black lines) vgPDFs are compared for noise strengths $g=1.0,1.2,1.5,1.7,1.8,1.9$, and 2.0. They have been shifted along the vertical axis to ease visualization, and their associated values of $g$ grow from the bottom to the top in each one of the PDF sets. Panels (a) and (b) give the modeled vgPDFs that include and neglect, respectively, the effects of fluctuations around instantons.

Figure 6

Solid lines, labeled by values of $g$, represent relative corrections to the MSRJD surrogate saddle-point action due to fluctuations around instantons. The intersection points of each one of the solid lines with the vertical and horizontal dashed lines define the range of normalized velocity gradients $\xi /g$ where the perturbative cumulant expansion is assumed to work (highlighted region in the plot).