Spontaneous generation of helical flows by salt fingers

TL;DR

Three-dimensional DNS of salt-finger turbulence are performed in regimes with $\\tau \\ll 1$ and large $R_\\rho$ to reveal multiscale, anisotropic finger dynamics and spontaneous generation of a helical horizontal mean flow. A multiscale reduced model, the Modified-IFSC (MIFSC), is derived to retain Reynolds stresses while filtering inertia and internal gravity waves, and it reproduces full-DNS fluxes up to $\\varepsilon \\approx 1$. The key finding is that helicity in the mean flow emerges from strictly nonhelical fluctuations, enabling a spontaneous symmetry breaking that twists fingers into corkscrew structures. The results provide a practical reduced framework for predicting fluxes in oceanic double-diffusive systems and may have implications for stellar interiors where similar parameter regimes occur.

Abstract

We study the dynamics of salt fingers in the regime of slow salinity diffusion (small inverse Lewis number) and strong stratification (large density ratio), focusing on regimes relevant to Earth's oceans. Using three-dimensional direct numerical simulations in periodic domains, we show that salt fingers exhibit rich, multiscale dynamics in this regime, with vertically elongated fingers that are twisted into helical shapes at large scales by mean flows and disrupted at small scales by isotropic eddies. We use a multiscale asymptotic analysis to motivate a reduced set of partial differential equations that filters internal gravity waves and removes inertia from all parts of the momentum equation except for the Reynolds stress that drives the helical mean flow. When simulated numerically, the reduced equations capture the same dynamics and fluxes as the full equations in the appropriate regime. The reduced equations enforce zero helicity in all fluctuations about the mean flow, implying that the symmetry-breaking helical flow is spontaneously generated by strictly non-helical fluctuations.

Figures (6)

• Figure 1: Flow velocity snapshots at $y=0$ in the saturated state from simulations of equations \ref{['eq:full_mom']}-\ref{['eq:full_S']} with varying supercriticality: $\varepsilon = 1/80$ (a-c), $\varepsilon = 1/10$ (d-f) and $\varepsilon = 1$ (g-i), with timetraces of the corresponding salinity flux $|F_S|$(blue solid lines) shown in panels j, k, and l, respectively, alongside fluxes from different reduced models (orange dashed and red dash-dotted lines, see Sec. \ref{['sec:models']}). All cases exhibit a multiscale and anisotropic flow where fingers with large vertical extent and vertical velocity (compared to horizontal width and velocity) coexist with small-scale, isotropic disturbances. Magenta curves (panels e, f, h, and i) show the time-average (over the saturated state) of the horizontal, helical mean flow $\hat{\mathbf{U}}_\perp$ that becomes a significant feature for $\varepsilon \gtrsim 0.1$.
• Figure 2: Relative helicity (see text) of the mean flow (blue) and of the fluctuations about the mean flow (orange; multiplied by $10^4$ to ease comparison) for two values of $\varepsilon$. At small $\varepsilon$, the flow is almost maximally helical, and in both cases the fluctuations are highly non-helical, with $H_\mathrm{rel}[\hat{\mathbf{u}}'] \sim 10^{-5}$.
• Figure 3: Horizontal (blue) and vertical (orange) kinetic energy spectra (time-averaged over the statistically stationary state) versus $\hat{k}_z$ at $\hat{k}_y = 0$ and $\hat{k}_x = \hat{k}_\mathrm{opt}$. Black lines show $\hat{k}_z = \hat{k}_\mathrm{opt}$ to highlight the small-scale isotropic flow component while the red vertical lines correspond to the secondary peak in the horizontal spectrum to highlight the anisotropic, small $\hat{k}_z$ flow component. The ratio between these two wavenumbers provides one measure of anisotropy shown in Fig. \ref{['fig:RMS_trends']}.
• Figure 4: Scalings with respect to $\varepsilon$ of several quantities (indicated in the caption for each panel) for the full system, Eqs. \ref{['eq:full_mom']}-\ref{['eq:full_S']} (blue dots), the IFSC model, with Eqs. \ref{['eq:full_T']} and \ref{['eq:full_mom']} replaced by Eqs. \ref{['eq:LPN_T']} and \ref{['eq:IFSC_mom']} (green diamonds), and the modified IFSC (MIFSC) model, where Eq. \ref{['eq:full_mom']} is replaced instead by Eqs. \ref{['eq:MIFSC_mean_mom']}-\ref{['eq:MIFSC_fluct_mom']} (orange crosses). Black dashed lines show scalings predicted by the multiscale asymptotic analysis described in the text. The green dashed lines and the two measures of anisotropy are described in the text.
• Figure 5: Volume rendering of the vertical velocity for the same $\mathcal{R} = 2$ simulation as shown in Fig. \ref{['fig:dynamics']}(g-i).