The Application of Flow-Dependent Ekman Transport to a Two-Layer Shallow Water Model Yanxu Chen1, David Straub1, Louis-Philippe Nadeau1,2 1Department of Atmospheric and Oceanic Sciences, McGill University, Canada 2Institut des Sciences de la Mer de Rimouski, Université du Québec à Rimouski, Canada

Background Model Formulation Results Results

Ø The modified Ekman layer ØAnalysis 1: How is the Ekman transport of a balanced vortex Ø Ekman transport and pumping are known to be modified by different from intuition? surface currents. Ekman transport equations: Here, we use a simple vortex to represent one eddy in the ! ⃑ 3 ocean and the vortex can be either cyclonic or anticyclonic. Wenegrat & Thomas #$ + &' ( ∇ #$ + #$ ( ∇ &' + *+̂×#$ = " − 12∇ #$ Ekman (1905) Stern (1965) !" (2017) Transport Allows for shear in unsteady term vorticity term diffusion term depends on Extends Stern’s results the surface velocity C A the stress and to better account for Content field to affect the the Coriolis curvature in the surface advection term wind forcing term transport: “nonlinear” parameter flow path. Ekman theory. only. Notice: &' represents the background flow.

Ekman Ø The two-layer shallow water model transport

Typically, wind stress is applied as a body force over the Scan here for Valid for plane The new forcing shows more enstrophy input at high-frequencies, parallel flows (e.g., ocean upper layer. We instead assume a thin Ekman-like the video of Curvilinear flows, with Homogeneous straight jets); this simulation. whereas the standard forcing shows a peak at intermediate-to- Ekman Rossby number layer embedded in the upper layer. Divergent Ekman Assumptions deep ocean at however, validity for small scales. The latter appears related to coherent eddies with <<1 and the balanced transports then enter into the upper layer mass equation. We rest. curvilinear flows has Rossby number <1. large interface height displacements. been questioned by compare different formulations using a two-layer shallow water Wenegrat & Thomas. model.

Ø Note that W&T formulation has been carried out in curvilinear Standard method new method Our model produces transients (see videos), however, the coordinates, thus, it would be difficult to apply their Ekman Ekman layer time-average is consistent with Wenegrat and Thomas. Upper layer Upper layer H h1 equations to complicated background flow fields, e.g., jets with 1 H1 h1 a random shape, turbulent eddies, etc. The zonal transport develops a quadrupole pattern,

H2 Lower H2 Lower emphasizing that the nonlinear Ekman transport is not strictly Ø We extend the W&T Ekman formulation by adding a time- layer layer perpendicular to the wind stress. The meridional transport dependent term. This step removes the need for integrating converges (diverges) on the north (south) side of the cyclonic over streamlines, and also introduces a near-inertial vortex, with the pattern reversed for the vortex with anticyclonic component to the Ekman pumping. Simulation Standard method flow. Process Ekman ØAnalysis 2: How does the interior respond to this modified Research Objective transport Ekman layer? Upper-layer Equations Next, let’s consider the upper-layer response. In contrast to momentum Ø The questions are: notable differences in PV forcing, upper-layer kinetics of different Upper-layer Wind stress Scan here for simulations act similarly. Our future work will continue this Whether this flow-dependent Ekman layer produces a different mass structure videos of my structure for Ekman transport; simulations! analysis by adding a high-frequency component to the wind and Whether the interior flow responds to this Ekman layer compare upper-layer responses. differently from the one with usual boundary layer setup. Simulation New method Process First, we analyze the RHS of the upper-layer PV equations, References Ekman which can be called PV forcings. If PV forcings are different for wind transport standard and new Ekman representations, then we expect [1] Wenegrat and Thomas. Ekman transport in balanced currents Upper-layer Equations different interior responses. with curvature. momentum [2] Niiler. On the Ekman divergence in an oceanic jet. Upper-layer Simulations Standard method New method [3] Stern. Interaction of a uniform wind stress with a geostrophic mass Upper-layer PV vortex. equations