We enumerate rational curves in toric surfaces passing through points and satisfying cross-ratio constraints using tropical and combinatorial methods. Our starting point is (Tyomkin in Adv Math 305:1356-1383, 2017), where a tropical-algebraic correspondence theorem was proved that relates counts of rational curves in toric varieties that satisfy point conditions and cross-ratio constraints to the analogous tropical counts. We proceed in two steps: based on tropical intersection theory we first study tropical cross-ratios and introduce degenerated cross-ratios. Second we provide a lattice path algorithm that produces all rational tropical curves satisfying such degenerated conditions explicitly. In a special case simpler combinatorial objects, so-called cross-ratio floor diagrams, are introduced which can be used to determine these enumerative numbers as well.

The present study focuses on understanding the bank undercut mechanism under controlled conditions in a laboratory flume, using cohesive sediments collected from the toe region of Ganges River (India). The velocity data were analyzed to determine the turbulence characteristics such as mean velocities, Reynolds stresses, intensities and other higher order turbulence statistics. Wavelet analysis of velocity time series delivered knowledge on the frequency of occurence of the tubulent eddy scales. Results showed that large eddies are distributed at a lower frequency band upstream and downstream of the undercut regions, which enhance the bank undercut succession. However, for different Reynolds numbers, the generation of eddies is different which plays a significant role for different erosion rates.