Thermodynamics of driven systems via the Kuramoto-Sivashinsky equation

TL;DR

The paper investigates whether driven KS turbulence can be embedded within a metriplectic thermodynamic framework. It shows that the unmodified KS dynamics cannot support monotone entropy due to the positive spectrum of $T_{KS} = -\partial_{xx}-\nu\partial_{xxxx}$ under forcing, and then constructs a thermodynamically consistent KS via spectral filtering to obtain $\partial_t v + v v_x = T_{diss} v$ with entropy production $\frac{dS}{dt} = \frac{1}{T}\int_0^L (\mathcal{L}^v v)^2 dx \ge 0$. Numerical experiments compare the KS and the thermodynamic KS, revealing that forcing can increase entropy even in the metriplectic setting and that selective injection of positive spectra can yield equilibria, relative equilibria, or chaos depending on which modes are excited. The work links pattern formation in driven PDEs to the second law and shows how forcing spectra shape long-time dynamics, with implications for thermodynamic modeling of driven fluids and plasmas. Equilibria in the metriplectic KS systems are shown to be spatial constants, and the approach highlights the trade-offs between thermodynamic consistency and the rich dynamics of KS-type systems.

Abstract

We examine the differences between the driven turbulence described by the Kuramoto-Sivashinsky (KS) equation and the second law of thermodynamics. A general velocity and entropy density system is analyzed with the unified thermodynamic algorithm of metriplectic dynamics, and we show that the positive spectra of the KS equation due to an external energy source prevent its metriplectic description. A variant of the KS equation is produced that monotonically generates an entropy, but the only equilibria of this variant system are spatially constant. Numerical experiments are performed comparing the evolution of the KS equation and its thermodynamic variant. The entropy of this thermodynamic system is increased further by the driving effects of the KS equation, reconciling the generation of entropy with the energy source of the KS equation. Further numerical experiments restrict the positive spectra in the KS equation to determine the effect on the system time evolution. While rescaling the growth rates of instabilities reproduces similar behavior on a slower time scale, introduction of individual positive spectra reproduces the formation of equilibria, relative equilibria, and a transition to chaos.

Figures (9)

• Figure 1: Solutions of the KS equation (left) are dissipated under metriplectic dynamics (right). Each row corresponds to a different initial condition (see text for details).
• Figure 2: Entropy generation associated with metriplectic dynamics of the Kassam Trefethen benchmarking conditions, traveling wave solutions, a strange hyperbolic point, and a hyperbolic secant profile.
• Figure 3: Interpolated profile of TW1 used for initialization.
• Figure 4: Positive spectra of the Kassam Trefethen benchmarking and Cvitanovic et al TW1 condition parameters.
• Figure 5: Spatiotemporal variation of $v(x,t)$ as the growth rate of positive spectra of $T_{KS}$ is increased linearly. As the parameter $\epsilon$ decreases, the transition to chaos remains, but occurs later in the dynamics and the amplitude of the oscillations decreases.