Clustering of Salt Fingers in Double-Diffusive Convection Leads to Staircaselike Stratification

We report on high-resolution, three-dimensional, high Rayleigh number, and low density ratio numerical simulations of fingering convection. We observe a previously unreported phenomenon of self-organization of fingers that cluster together to form larger-scale coherent structures. The flow ultimately forms density staircases, alternating well-mixed regions with fingering convective zones. We give evidence that the mechanical mixing induced by the clusters forms the staircases with a mechanism analogous to staircase formation in a stably stratified, nonconvective, stirred fluid.

Figure 1

Vertical sections at $y=L_y/2$ (lower panels) and horizontal sections at $z=0$ (upper panels) of salt anomaly, $S^{\prime }(x,y,z)=S(x,y,z)-\overline{S}(z)$, in the central portion of the domain $-0.3<z<0.3$, at three different times.

Figure 2

Evolution with time of total kinetic energy spectra as a function of the horizontal wavelength ($\lambda =2\pi /k$). The spectra have been averaged over the central portion of the domain (from $-0.3<z<0.3)$ and over three consecutive time steps spaced $\delta t={10}^{-7}$. The inset shows the time evolution of the integral scale $\Lambda$ defined in the text.

Figure 3

Enlargement of a vertical section of the flow at time $t=3.8\times {10}^{-6}$ and at $y=L_y/2$. The panels show salinity anomaly (upper left), temperature anomaly (upper right), vertical velocity (lower left), and buoyancy anomaly (lower right).

Figure 4

Time evolution of vertical profiles of horizontally averaged buoyancy, $\overline{B}(x)$, in the central portion of the domain. The frames after the first have been shifted to the right as a function of time.

Figure 5

Average buoyancy flux vs the vertical buoyancy gradient ${\tilde{B}}_z$. The statistics has been computed from $t=4\times {10}^{-6}$ to $t=1.5\times {10}^{-5}$, in the range $-0.3<z<0.3$. The gradient has been smoothed by convolution with a Hann window of height $h=0.01$. The inset shows the complete curve. The grey bands represent $95\%$ confidence bands obtained with a jackknife subsampling procedure.