Uncertainty Growth in Stably Stratified Turbulence

TL;DR

This study quantifies uncertainty growth in statistically steady stably stratified turbulence by employing twin DNS runs and decorrelator diagnostics within the Boussinesq framework. It reveals a monotonic decrease of the largest Lyapunov exponent $\lambda$ with increasing stratification (via $N$ and ${\rm Fr}$), while preserving a universal sequence of decorrelation: initial decay, exponential growth with self-similar spectra, and saturation. The growth is highly anisotropic, with vertical spread strongly suppressed by buoyancy, and horizontal spread dominating as stratification strengthens. The analysis shows that chaos suppression arises primarily from strain-mediated alignment with the compressive eigen-direction of the rate-of-strain tensor, rather than direct buoyancy effects, highlighting the utility of decorrelator-based methods for anisotropic geophysical flows.

Abstract

We investigate uncertainty growth and chaotic dynamics in statistically steady, stably stratified three-dimensional turbulence. Using direct numerical simulations of the Boussinesq equations, we quantify the divergence of initially infinitesimal perturbations via twin simulations and decorrelator diagnostics. At short times, perturbations exhibit exponential growth, allowing us to define a (largest) Lyapunov exponent. We systematically examine how this exponent depends on stratification strength, quantified by the Brunt--Väisälä frequency and the Froude number, in a parameter regime relevant to oceanic flows. We find that increasing stratification leads to a monotonic reduction of the Lyapunov exponent, indicating suppressed chaoticity. Despite this reduction, uncertainty growth retains the universal temporal sequence observed in homogeneous isotropic turbulence -- initial decay, exponential growth, and saturation. The growth phase is characterized by self-similar decorrelator spectra, but exhibits strong anisotropy: uncertainty spreads much more slowly along the stratification direction than horizontally, with the disparity increasing with stratification strength. An analysis of the decorrelator evolution equation reveals that the suppression of chaos arises primarily from strain-mediated alignment dynamics rather than direct buoyancy coupling. Our results provide a quantitative characterization of predictability and uncertainty growth in stratified turbulence and highlight the utility of decorrelator-based methods for anisotropic geophysical flows.

Figures (5)

• Figure 1: Pseudocolor plots of $\phi_u(\mathbf{x},t)=\tfrac{1}{2}|\delta\mathbf{u}(\mathbf{x},t)|^2$ for a representative stratified run with $N = 12$ during the early stages of decorrelation for slices in the (a) XY and (b) XZ planes. Insets correspond to an earlier time before exponential growth begins. The inner colorbar corresponds to the inset, while the outer colorbar corresponds to the main figure. The perturbation field develops pronounced anisotropy, forming extended structures perpendicular to the direction of stratification. The reduced vertical variability highlights the suppression of uncertainty propagation along the stratification axis due to buoyancy effects.
• Figure 2: (a) A semilog plot of the temporal evolution of the spatially averaged velocity decorrelator $\Phi^u(t)=\langle \tfrac{1}{2}|\delta\mathbf{u}(\mathbf{x},t)|^2\rangle$ for different values of the non-dimensional Brunt--Väisälä frequency $\tilde{N}$ (see legend). After an initial decay, $\Phi^u(t)$ exhibits a clear exponential growth regime followed by saturation, indicating complete decorrelation of the two flow realizations. (b) The Lyapunov exponent $\lambda$ extracted from the exponential growth phase of $\Phi^u(t)$ as a function of $\tilde{N}$ (lower axis). The monotonic decrease of $\lambda$ with increasing stratification strength shows that stable stratification systematically suppresses chaoticity. The same data is also plotted against the Froude number ${\rm Fr}$ (upper axis), confirming that increased buoyancy effects lead to reduced sensitivity to infinitesimal perturbations.
• Figure 3: (a) Time evolution of the individual contributions to the decorrelator growth rate in Eq. \ref{['rate_equation']} for $N = 4$: the strain-induced term $\beta_S$, the viscous term $\beta_\eta$, the buoyancy–velocity cross-correlation $N\langle\delta u_z\,\delta b\rangle$, and the forcing contribution, each normalized by $\Phi^u(t)$. The initial decay of $\Phi^u(t)$ is driven by large, negative contributions from $\beta_S$ and $\beta_\eta$, while the buoyancy term remains subdominant throughout. (b) Temporal evolution of the spherically averaged decorrelator $\phi^u(r,t)$ with respect to the centre at $\mathbf{x}_0$ for different radii $r$ (see legend), illustrating spatially homogeneous exponential growth of uncertainty. The inset shows $\phi^u(z,t)$ at fixed $r$ for seven different values of $z$ in the range $0.53 \pi \leq z \leq 1.47 \pi$ , confirming uniform growth even along the direction of stratification. The dashed curve represents the spectra for the reference state.
• Figure 4: Evolution of the uncertainty spectrum in spectral space: (a)--(c) Horizontal decorrelator spectra $\tilde{\phi}^u(k_\perp,t)$ at representative times during the decay, exponential growth, and saturation phases. The inset shows the rescaled spectra $\tilde{\phi}^u(k_\perp,t)/[\ell^\Delta_\perp(t)\Phi^u(t)]$ plotted against $k_\perp \ell^\Delta_\perp(t)$. As can be seen from inset of (b), the rescaled spectra collapses demonstrating self-similar evolution in the growth phase. (d)--(f) The corresponding vertical decorrelator spectra $\tilde{\phi}^u(k_z,t)$, exhibiting analogous behaviour along the direction of stratification.
• Figure 5: Temporal evolution of the uncertainty integral length scales in the (a) vertical ($\ell^\Delta_z$) and (b) horizontal ($\ell^\Delta_\perp$) directions. The evolution is shown till the time the two systems A and B decorrelate completely. Throughout the exponential growth phase, the vertical uncertainty scale remains significantly smaller than the horizontal one, reflecting the anisotropic suppression of vertical motions by stable stratification.