Harmonic Maa{\ss}-Jacobi forms of degree 1 with higher rank indices

We define and investigate real analytic weak Jacobi forms of degree 1 and arbitrary rank. En route we calculate the Casimir operator associated to the maximal central extension of the real Jacobi group, which for rank exceeding 1 is of order 4. In ranks exceeding 1, the notions of H-harmonicity and semi-holomorphicity are the same.Comment: 28 page

Vertical structure of near-bed cross-shore flow velocities in the swash zone of a dissipative beach

Cross-shore velocity profiles are measured at 0.001m vertical resolution and at 100Hz over the lower 0.02-0.07m of the water column in the mid swash zone on a dissipative, macrotidal beach. Swash motion is predominantly at infragravity frequencies and forced by significant wave heights exceeding 1.5m and peak wave periods over 15s. Observations of long duration (> 14s) swashes during two rising tides are used to quantify the vertical structure of cross-shore flow velocities and estimate corresponding bed shear stress and friction coefficients. Analysis is performed on an individual swash event to an elevation of 0.07 m and an ensemble event made up of 24 individual swash events to an elevation of 0.02m. Cross-shore velocities exceed 2 m s-1 and are of a similar magnitude during both the uprush and the backwash. Changes in velocity with elevation indicate that the swash zone boundary layer extends to 0.07m during the strongest flows and is well-represented by the logarithmic model applied to this elevation, except near flow reversal. Maximum bed shear stresses estimated using the logarithmic model are 22 N m-2 and 10 N m-2 for the individual event and ensemble event respectively and mean values are larger during the backwash than the uprush. Mean friction coefficients estimated from equating the logarithmic model and the quadratic drag law are 0.018 and 0.019 for the individual event and ensemble event respectively. Bed shear stress may be underestimated if the logarithmic model is fit to a velocity profile that is only part boundary layer, emphasising the need for high resolution velocity profiles close to the bed for accurate bed shear stress predictions in the swash zone