Pure Appl. Geophys. 172 (2015), 2831–2857 Ó 2015 Springer Basel (outside the USA) DOI 10.1007/s00024-015-1063-4 Pure and Applied Geophysics Ekman Spiral in a Horizontally Inhomogeneous Ocean with Varying Eddy Viscosity 1 PETER C. CHU Abstract—The classical Ekman spiral is generated by surface wind and/or buoyancy forcing), (b) including ocean wind stress with constant eddy viscosity in a homogeneous ocean. wave effect, and (c) changing homogeneous to in- In real oceans, the eddy viscosity varies due to turbulent mixing caused by surface wind and buoyancy forcing. Horizontally inho- homogeneous density. mogeneous density produces vertical geostrophic shear which It was recognized that the eddy viscosity K^ is not contributes to current shear that also affects the Ekman spiral. a constant. After fitting observational ocean currents Based on similar theoretical framework as the classical Ekman spiral, the baroclinic components of the Ekman spiral caused by the to the Ekman spiral (e.g., HUNKINS 1966;STACEY et al. horizontally inhomogeneous density are obtained analytically with 1986;PRICE et al. 1987;RICHMAN et al. 1987; the varying eddy viscosity calculated from surface wind and CHERESKIN 1995;LENN and CHERESKIN 2009), the in- buoyancy forcing using the K-profile parameterization (KPP). ^ Along with the three existing types of eddy viscosity due to pure ferred K value varies more than an order of 2 -1 wind forcing (zero surface buoyancy flux), such an effect is magnitude, from 0.054 m s (PRICE et al. 1987) evaluated using the climatological monthly mean data of surface obtained from the field measurements acquired from wind stress, buoyancy flux, ocean temperature and salinity, and a surface mooring set in the western Sargasso Sea mixed layer depth. (34°N, 70°W) as part of the long-term upper ocean study phase 3 (LOTUS-3) during the summer of 2 -1 1982, to 0.006 m s (STACEY et al. 1986) obtained 1. Introduction from the low-frequency current measurements in the Strait of Georgia, British Columbia, Canada. The 2 -1 On the base of homogeneous density without smaller value (0.006 m s ) may be treated as a considering waves, EKMAN (1905) modeled turbulent lower bound of the eddy viscosity (PRICE et al. 1987). mixing in upper ocean as a diffusion process similar Recently, LENN and CHERESKIN (2009) obtained a to molecular diffusion with an eddy viscosity (tur- mean Ekman spiral from high-resolution repeat ob- bulent plus molecular), K^ (the symbol ‘^’ indicating servations of upper-ocean velocity in Drake Passage dimensional quantity), which was taken as a constant along with the constant temperature in the Ekman with many orders of magnitude larger than the layer (implying near neutral stratification). The eddy molecular viscosity. The turbulent mixing generates viscosities inferred from Ekman theory and the time- -2 an ageostrophic component of the upper ocean cur- averaged stress was directly estimated as O (10 – -1 2 -1 rents (called the Ekman spiral), decaying by an 10 )m s . e-folding over a depth as the current vector rotates to The turbulent mixing in upper-ocean is also the right (left) in the northern (southern) hemisphere viewed as being driven by the atmospheric fluxes of through one radian. Several approaches may advance momentum and buoyancy (heat and moisture), and the classical Ekman theory: (a) replacing constant the shear imposed by the ocean circulation, and is eddy viscosity by varying eddy viscosity, and relating characterized by the existence of a vertically quasi- the eddy viscosity to ocean mixing (under surface uniform layer of temperature and density (i.e., mixed layer). Underneath the mixed layer, there exists an- other layer with strong vertical gradients, such as the thermocline (in temperature) and pycnocline (in 1 Naval Ocean Analysis and Prediction Laboratory, Depart- ment of Oceanography, Naval Postgraduate School, Monterey, CA, density) (e.g., KRAUS and TURNER 1967;GARWOOD USA. E-mail: [email protected] 1977;CHU and GARWOOD 1991;STEGER et al. 1998; 2832 P. C. Chu Pure Appl. Geophys. CHU et al. 2002). Such vertical mixing generates random surface wave effects when the eddy viscosity varying upper-ocean eddy viscosity. The mixed layer is gradually varying with depth. Their solution was is a key component in studies of climate, and the link compared with observational data and with the results between the atmosphere and deep-ocean and directly from a large eddy simulation of the Ekman layer affects the air–sea exchange of heat, momentum, and (ZIKANOV et al. 2003). moisture (CHU 1993). However, effect of horizontally inhomogeneous Effect of vertical inhomogeneity of density on the density on the Ekman spiral with varying eddy vis- Ekman spiral (i.e., stratified Ekman layers) has been cosity due to vertical mixing under various surface identified by observational and modeling studies in forcing conditions has not yet been studied. Since the atmospheric boundary layer (LETTAU and DAB- horizontally inhomogeneous density leads to non-zero BERDT 1970;GRACHEV et al. 2008) and the oceanic vertical geostrophic shear and in turn contributes to the boundary layer (MCWILLIAMS et al. 2009;TAYLOR and current shear, the equations and surface boundary SARKAR 2008). Ocean observations from drifters/floats conditions for the classical Ekman model need to be show the role of horizontal density gradient in setting modified. Such modifications may lead to a new the stratification within the mixed layer. MCWILLIAMS structure of the Ekman spiral. The baroclinic compo- et al.(2009) computed vertical turbulent mixing nents of the Ekman spiral are identified analytically in within the boundary layer in a one-dimensional ver- this study using the KPP and three existing (due to pure tical column using the K-profile parameterization wind forcing) eddy viscosities without considering (KPP) scheme with surface mean wind stress, mean ocean waves. The rest of the paper is organized as heating, and solar absorption, and idealized repre- follows. Section 2 introduces the basic equations and sentations of the heat flux from the interior three- boundary conditions. Sections 3 and 4 describe the dimensional circulation. They found that there is not a Obukhov length scale (OBUKHOV 1946;MONIN and single, simple paradigm for the upper-ocean velocity OBUKHOV 1954), depth ratio, and KPP. Section 5 pre- profiles in stratified Ekman layers for the following sents the analytical solution of the Ekman spiral in reasons: (a) the Ekman layer is compressed by stable horizontally inhomogeneous ocean including analytical stratification and surface heating; (b) Ekman currents barotropic and baroclinic components due to KPP eddy penetrate down into the stratified layer; (c) penetrative viscosity. Sections 6 and 7 describe the baroclinic ef- solar absorption deepens the mean Ekman layer; fects with the KPP eddy viscosity under both surface (d) wind, and especially buoyancy rectification ef- wind and buoyancy forcing and with the three existing fects, yield a mean Ekman profile with a varying eddy eddy viscosities under pure wind forcing. Section 8 viscosity, where the mean turbulent stress and mean presents the conclusions. Appendices 1 and 2 list the shear are not aligned, whereas buoyancy rectification procedures for obtaining the analytical solutions of the induces profile flattening. These modeling results are Ekman spiral in horizontally inhomogeneous ocean for the one-dimensional ocean, i.e., no horizontal with depth-dependent eddy viscosity. gradients of any variables including the density. Effect of ocean surface gravity waves on the Ekman spiral has been identified through interacting 2. Ekman Layer Dynamics waves with ocean currents and wind stresses. As waves experience breaking and dissipation, momen- Let (x, y, z) be the zonal (positive eastward), tum passes from waves into ocean currents. Recent latitudinal (positive northward), and vertical (positive studies show that the influence of the surface wave upward with z = 0 at the ocean surface) coordinates motion via the Stokes drift and mixing is important to with (i, j, k) as the corresponding unit vectors, and u^ understanding the observed Ekman current profiles in be the velocity vector. Following the similar steady addition to wind stress, depth-varying eddy viscosity, dynamics of MCWILLIAMS and HUCKLE (2006) with and density inhomogeneity. SONG and HUANG (2011) modification from homogeneous to inhomogeneous used the WKB method to obtain the analytic solu- density, the steady-state horizontal momentum bal- tions for modified Ekman equations including ance with Boussinesq approximation is given by Vol. 172, (2015) Ekman Spiral in Horizontally Inhomogeneous Ocean 2833 1 1 o where j ¼ 0:41, is the von Karmen constant. Sub- f k  u^ ¼À rp À ðMÞ; ð1aÞ q h or stitution of (2) into (6a) and use of (6b) lead to w o o where q = 1,025 kg m-3, is the characteristic den- u^g u^E w MðzÞ¼ÀjuÃK þ : ð7Þ sity of seawater; h is the ocean surface mixed layer or or depth; r ¼Àz=h, is the non-dimensional vertical The velocity (u^E) is non-dimensionalized by coordinate; f is the Coriolis parameter (depending on M u^ u2 the latitude); is the vertical momentum flux due to u ¼ E ; V qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffià ; ð8Þ E V E turbulent mixing; p is the pressure. It is noted that the E 2jjf K^ð0Þ damping for currents due to vertical radiation of in- ertial waves into the oceanic interior is neglected. where K^ð0Þ is the eddy viscosity evaluated at the The mixed layer depth (h) can be determined from surface. Substitution of (7) into (5) and use of (4) and temperature and density profiles using subjective and (8) give objective methods (e.g., MONTEREY and LEVITUS 1997; juà o ouE juà o CHU et al. 2002;CHU and FAN 2010, 2011). The hy- f k  uE À K ¼ k  ½KS ; h or or fV or E ð9Þ drographic balance gives g S rq 1 op q qw o ¼ g À g ; ð1bÞ qw z qw Here, the vector, S = (sx, sy), is defined by where q is the density; g is the gravitational accel- g oq g oq -2 eration (9.81 m s ).