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Introduction

Wind-driven upwelling is important for the water quality and ecology in coastal zone. The traditional spin-up theory tells that wind stress curl determines the sign of vertical velocity at the base of the surface boundary layer, or Ekman layer. However, during their study of horizontal and residual circulation in Tokyo Bay, my collaborators, Prof. Keisuke Nakayama at Kobe Uni. and Dr. Tetsuya Shintani at Tokyo Metropolitan Uni., came up with an idea that the centrifugal force can induces upwelling regardless of the sign of wind stress curl (Fig. 1). I was asked to investigate the effects of strong nonlinearity in the surface Ekman layer from a theoretical perspective. Previous studies in oceanography considered only the traditional Ekman downwelling under weak^2),4),6-7) and strong^1),3) anti-cyclonic forcing (i.e., negative, or clockwise, wind stress curl in Northern Hemisphere).

Fully-nonlinear Ekman layer solution

Following previous studies on the von Karman swirling flow problem^5),8), we numerically solved a set of ordinary differential equations for fully-nonlinear (steady) Ekman layer. We imposed wind stress curl at the surface and a slip condition at the base, which approximates reduced mixing (eddy viscosity) due to stratification.

There are two important branches of the solution, one corresponding to the traditional Ekman layer, and the other corresponding to upwelling under anti-cyclonic forcing. Numerical solutions in an idealized cylindrical basin under spatially uniform wind stress curl suggests bifurcation from the first to second branch (Fig. 2). The second branch appears to be physically realisable only under strong anti-cyclonic forcing, when the centrifugal force exceeds the Coriolis force. This suggests the following three Regimes in nonlinear surface Ekman layer:

1. Ekman and nonlinear upwelling under cyclonic forcing (positive wind stress curl in Northern Hemisphere)
2. Ekman downwelling under weak anti-cyclonic forcing
3. Nonlinear upwelling under strong anti-cyclonic forcing

The idealized numerical simulations suggest that the threshold between Regimes 2 and 3 is $(\nabla \times \vec{\tau})/(\rho f) \approx -7 \times 10^{-6} \text{(m/s)}$, where $\vec{\tau}$ is the wind stress vector, $\rho$ is the water density, and $f$ is the Coriolis parameter.

Application to Tokyo Bay

Tokyo Bay is an semi-enclosed bay with a length and width of about 60 and 15 km, respectively. The depth changes sharply from a few hundreds of meters at the bay mouth to 15 m at the bay head. Fig. 3 shows an example of observed downwelling under weak anti-cyclonic forcing.

Statistics of wind stress curl over Tokyo Bay shows that anti-cyclonic forcing is strong enough to induce nonlinear upwelling in winter (Fig. 4). However, time-dependent numerical simulations show that it takes approximately an inertial period (~18 h in Tokyo Bay) for an Ekman layer to induce nonlinear upwelling under anti-cyclonic forcing. Since typical duration of anti-cyclonic forcing over Tokyo Bay is less than ~6 h, Regimes 3 would occur only under exceptional conditions.

Implications to other sites

The magnitude of wind stress curl over Tokyo bay, O(10^-6 N/m³), is an order of magnitude larger than over Great Lakes or open ocean, O(10^-7 N/m³), which is probably due to orographic effects. It implies that strongly nonlinear Ekman layer is relevant to medium-sized bays and lakes surrounded by complex terrain.

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Related Publications

Details are available in

Nakayama, K., T. Shintani, K. Shimizu, T. Okada, H. Hinata, and K. Komai. 2014. Horizontal and residual circulation driven by wind stress curl in Tokyo Bay. Journal of Geophysical Research – Oceans, 119: 1977-1992, doi:10.1002/2013JC009396.

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