Transport across a bathymetric interface in quasi-two-dimensional flow

Geophysical flows are often approximated as being two-dimensional on large scales due to their high aspect ratios. Two-dimensional flows in turn are well known to be prone to producing coherent structures that impact their mixing and transport. However, real geophysical flows are not exactly two-dimensional, in that they typically have nonuniform bathymetry that may also affect mixing. We study the interplay of quasi-two-dimensionality, nonuniform bathymetry, and lateral transport via laboratory experiments in an electromagnetically stirred thin-layer flow that is weakly turbulent. We find that spatial variations in bathymetry lead to laterally coexistent zones with different levels of turbulence, and we investigate the transport across the interface dividing these zones using an approach based on transfer operators. We find that this transport is asymmetric and that fluid elements cross from low turbulence to high turbulence via directed advection but from high to low turbulence via random eddying motion. Our results may have implications for understanding geophysical flows where lateral transport is suppressed, such as ocean dead zones.

Figure 1

(a) Instantaneous forward finite-time Lyapunov exponent (FTLE) field computed for an integration time of $2.5T_L$, where $T_L=L_m/u^{\prime }$ is the eddy turnover time associated with the forcing. The dashed line shows the location of the bathymetric step; the low Reynolds number side is on the left of the figure, and the high Reynolds number side is on the right. (b) Backward FTLE field, computed for the same integration time. (c) Temporally averaged forward FTLE field. (d) Temporally averaged backward FTLE field. All FTLE values have been nondimensionalized by $T_L$, and the size of the domain shown in all cases is $8L_m\times 8L_m$ (or $21\times 21{\mathrm{cm}}^2$).

Figure 2

Evolution of the transfer operator singular values as a function of the integration time $\tau$ over which the transfer operator is computed, in units of $T_L=L_m/u^{\prime }$, for (a) experiments with a bathymetric step and (b) the control case of featureless bathymetry. The largest singular value is fixed at 1 by construction, but the other singular values decay with increasing $\tau$, indicating loss of coherence. The second and third singular values ${\sigma }_2$ and ${\sigma }_3$ are plotted with thicker lines. As discussed in the text, the value of ${\sigma }_2$ indicates the degree of coherence of our partition, and the gap between ${\sigma }_2$ and ${\sigma }_3$ indicates our confidence in this partition. The vertical dashed lines in panel (a) show the values of $\tau$ for which partitions are shown in Fig. 3. In the control case, ${\sigma }_2$ decays very quickly as $\tau$ increases, indicating a rapid drop in coherence. Additionally, the spectral gap between ${\sigma }_2$ and ${\sigma }_3$ remains small for all $\tau$, indicating low confidence.

Figure 3

Flow partitions for (a) $\tau =8T_L$, (b) $10T_L$, and (c) $12T_L$ for the case with variable bathymetry. For each value of $\tau$, the top two panels show the singular vectors $X$ (at the initial time) and $Y$ (at the final time), while the bottom two panels show the same data but binarized following the work of Froyland et al. [19]. The dashed lines show the location of the bathymetric step; the low Reynolds number side is on the left, and the high Reynolds number side is on the right. The size of the domain shown in all cases is $8L_m\times 8L_m$ (or $21\times 21{\mathrm{cm}}^2$). The values of the second singular value ${\sigma }_2$ and the ratio ${\sigma }_2/{\sigma }_3$ are (a) 0.94 and 1.119, (b) 0.90 and 1.198, and (c) 0.84 and 1.304.

Figure 4

Probability density functions (PDFs) of the kinetic energy $k$ of fluid elements located at fixed distances (ranging from $0.2L_m$ to $1.8L_m$) to the bathymetric step on (a) the low Reynolds number side and (b) the high Reynolds number side. Solid lines show the statistics of fluid elements that eventually crossed the porous transport barrier, while dashed lines show the statistics of fluid elements that did not.

Figure 5

PDFs of the magnitude of the vorticity $\left|\omega \right|$ of fluid elements located at fixed distances (ranging from $0.2L_m$ to $1.8L_m$) to the bathymetric step on (a) the low Reynolds number side and (b) the high Reynolds number side. Solid lines show the statistics of fluid elements that eventually crossed the porous transport barrier, while dashed lines show the statistics of fluid elements that did not.

Figure 6

PDFs of the angle between the velocity vector of fluid elements located at fixed distances to the bathymetric step and the direction normal to the step for (a) fluid elements initially on the low Reynolds number side and (b) fluid elements initially on the high Reynolds number side.

Figure 7

PDFs of the initial distance to the barrier for fluid elements that eventually cross it (in a time less than or equal to $2T_L$) from low to high Reynolds number (dashed line) and from high to low Reynolds number (solid line).