Frictional effects in wind-driven ocean currents

Abstract

Surface ocean currents have a significant influence on the climate and their dynamics depend to a large extent on the behaviour of the vertical eddy viscosity. We present an analytic study of winddriven surface currents for general depth-dependent vertical eddy viscosities. A novel formulation for Ekman-type flows, that relies of a transformation to polar coordinates, enables us to show that in the Northern Hemisphere the horizontal current profile decays in magnitude and turns clockwise with increasing depth, irrespective of the vertical variations in diffusivity. Using a perturbation approach, we also derive a formula for the deflection angle of the current at the surface from the wind direction and discuss its implications. ARTICLE HISTORY

A Unifying Perspective on Transfer Function Solutions to the Unsteady Ekman Problem

“…In a simplifying work, Dritschel et al [16] find analytic solutions to the steady Ekman spiral for a piecewise linear eddy viscosity profile. Similarly, Ionescu-Kruse [41] finds steady solutions for power-law eddy viscosities proportional to z 2 or z 4/3 , while Bressan and Constantin [54] and Constantin [55] solve for the steady Ekman currents for an eddy viscosity profile that consists of a depth-dependent perturbation about a constant value. It is possible that results of those works could be fruitfully brought to bear on the unsteady problem considered here.…”

A Unifying Perspective on Transfer Function Solutions to the Unsteady Ekman Problem

“…Finally, several relevant recent studies have investigated solutions to the steady Ekman problem. In a simplifying work, Dritschel et al [16] find analytic solutions to the steady Ekman spiral for a piecewise linear eddy viscosity profile, while Bressan and Constantin [41] and Constantin [42] solve for the steady Ekman currents for a eddy viscosity profile that consists of a depth-dependent perturbation about a constant value. It is possible that the methods employed in those works could be extended to the unsteady problem considered here.…”

The Ekman spiral for piecewise-constant eddy viscosity

“…Properties (ii) and (iii) have been shown to hold in the general case of depth-dependent eddy viscosity (see [9]); however, property (i) is not valid in general and agrees only qualitatively (i.e. it predicts the correct direction of deflection) but not quantitatively with direct observations, which measure deflection angles in the range 10 • -75 • (cf.…”

“…It is reasonable to identify the simplified assumption of constant vertical eddy viscosity as the source for the inaccuracy in the prediction of the value of the deflection angle, but investigating how exactly a change in the eddy viscosity affects the deflection angle is very complicated and explicit solutions to other more general cases are scarce; most of the known ones can be found in [12][13][14]. The perturbative approach pursued in [9,15] in order to determine whether a given small perturbation of a constant eddy viscosity leads to a positive or negative increment of the deflection angle gives a good idea of how intricate the investigation of this problem is. Nevertheless, an analysis of situations that do not require the smallness assumptions required for a perturbative approach is of great interest.…”