Ekman Spiral in a Horizontally Inhomogeneous Ocean with Varying Eddy Viscosity

Home , Ekman spiral, Ekman velocity

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

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 Â 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 uÊ 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 (uÊ) 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 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiÃ ; 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; juÃ o ouE juÃ o CHU et al. 2002;CHU and FAN 2010, 2011). The hy- f k Â uE À K ¼ k Â 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 ). The horizontal velocity consists sx ¼ o ; sy ¼ o ; ð10Þ qw x qw y of two parts: geostrophic current, [u^g ¼ðu^g; v^gÞ], and ageostrophic current [uÊ ¼ðuÊ; vÊÞ] (Ekman flow), which represents the baroclinicity (i.e., sx 6¼ 0; sy 6¼ 0). The ocean is barotropic if u^ ¼ u^g þ uÊ: ð2Þ sx ¼ sy ¼ 0: ð11Þ where the geostrophic current is given by 1 The second-order differential Eq. (9) needs two f k Â u^g ¼À rp; ð3Þ boundary conditions. At the surface (r = 0) we have qw s^ qa and computed solely from the density field (CHU Mð0Þ¼ ¼ C jju^ u^ u2h ð12aÞ q D q a a Ã Ã 1995, 2000, 2006). Differentiation of (3) with respect w w to z and use of (1b) lead to the thermal wind relation where CD is the drag coefficient; s^ is the surface wind -3 ou^ g stress; qa = 1.29 kg m , is the characteristic atmo- g k q : 4 o ¼À Â r ð Þ spheric density; uâ is the wind near the ocean surface; z f qw hÃ ¼½cos h; sin h is the unit vector of the wind di- Substitution of (2) - (3) into (1a) leads to rection; h is the angle of the wind from the east; and 1 o u is the ocean friction velocity, f k Â u^ ¼À ðMÞ: ð5Þ * E h or "# 2 1=2 M CDqajjuâ The vertical momentum flux (i.e., turbulent Rey- uÃ ¼ : ð12bÞ nolds stress) is modeled by qw ^ o^ Evaluation of (7) and (4) at the surface leads to M K u ðzÞ¼À o ; ð6aÞ h r o^ o^ M ug uE ^ ð0Þ¼ÀjuÃK o þ o ; ð13aÞ where K is the eddy viscosity that is non-dimen- r r r¼0 sionalized by ou^g gh K^ o ¼ k Â rq : ð13bÞ K ¼ ; ð6bÞ r r¼0 f qw r¼0 hjuÃ 2834 P. C. Chu Pure Appl. Geophys.

Substitution of (10), (12a), and (13b) into (13a) leads Here, L is the depth where the wind-generated tur- to the surface boundary condition for the non-di- bulence is balanced by the downward buoyancy flux mensional Ekman flow uE, (B \ 0) due to surface warming (Q \ 0) and/or freshening (P [ E) and is comparable to the con- ouE uÃ h ¼À h À k Â Sð0Þ: ð14Þ vection-generated turbulence by the upward or jKð0ÞV Ã fV r¼0 E E buoyancy flux (B [ 0) due to surface cooling (Q [ 0) where K (0) and S (0) represent the values of (K, S) and/or salinisation (P \ E); and k is the depth ratio. evaluated at the surface (r = 0). Moreover, the gen- Monthly depth ratio (k) (Fig. 1) are calculated from eral solution of (9) contains exponentially increasing the monthly mean global ocean (10 m)/L and ocean and decreasing parts with the non-dimensional depth friction velocity (u*) data (1° 9 1°) downloaded from r. The exponentially increasing part is unphysical and http://iridl.ldeo.columbia.edu/SOURCES/.DASILVA/. needs to be eliminated. Therefore, the lower boundary SMD94/.climatology/ (DASILVA et al. 1994), and the condition of Eq. (9) is used monthly mixed layer depth (h) data downloaded from http://www.nodc.noaa.gov/OC5/WOA94/mix.html uE finite as r !1 ð15Þ (MONTEREY and LEVITUS 1997). to filter out the unphysical solution. In fact, the lower The depth ratio (k) is used to determine the forcing boundary condition (15) is also used in the classical regimes (LOMBARDO and GREGG 1989): convective Ekman spiral. Generally, Eq. (9) is not closed. One regime (k\ À 10), wind-forcing regime (k [ À 1), more equation for the density q is needed. If q is given, and combined forcing regime (À10 k À1). The the second-order differential Eq. (9) with the bound- depth ratio (k) also serves as a stability parameter (see ary conditions (14) and (15) are well-posed. For depth- next section). The calculated monthly depth ratio (k) dependent eddy viscosity, (9) is an inhomogeneous (Fig. 1) shows strong seasonal variability with only linear differential equation with variable coefficient K. two regimes evident: wind-forcing and combined forcing regimes since almost no data with k \ -10. In January, the combined forcing (À10 k À1) pre- 3. Obukhov Length (L) and Depth Ratio (k) vails most of the Northern Hemisphere including North Atlantic, North Pacific, Arabian Sea, Mediter- The eddy viscosity is to characterize vertical ranean Sea, and eastern tropical South Pacific; the mixing, which is generated by surface wind stress (s) wind forcing (k [ À 1) prevails most of the Southern and surface buoyancy flux (B in m2 s-3, upward Hemisphere. In July, the combined forcing prevails in positive), the southern hemisphere, and the wind forcing pre- gaQ vails in the Northern Hemisphere B ¼ þ gbðE À PÞS; ð16Þ qð0Þcp

-2 where Q is the net heat flux (upward positive, W m ); 4. KPP cp is the specific heat for the sea water; S is the surface salinity [in practical salinity units (psu)]; a is the co- With the surface wind and buoyancy forcing, the efficient of thermal expansion; b is the coefficient of KPP rules for the non-dimensional eddy viscosity (K) haline contraction; and (E, P) are evaporation and are given by the product of a depth-dependent non- -1 precipitation (m s ). Ocean mixed layer is generally dimensional turbulent velocity wx (r) (scaled by ju*) developed by wind stirring and convection (upward and a dimensionless vertical shape function G (r) surface buoyancy flux B). To examine dominant (LARGE et al. 1994) mixing mechanisms, the Obukhov length scale (L) and w ðr; kÞGðrÞ; if 1 r 0 the depth ratio (k) are calculated by Kðr; kÞ¼ x ; ð18Þ wxð1; kÞGð1Þ; if r [ 1 u3 h ð10 mÞ h L ¼À Ã ; k ¼ ¼ : ð17Þ to represent the capability of deeper mixed layers to jB L L ð10 m) contain larger more effect turbulent eddies. It is noted Vol. 172, (2015) Ekman Spiral in Horizontally Inhomogeneous Ocean 2835

Figure 1 Monthly depth ratio (k): (a) January, and (b) July. It is noted that only two regimes are evident with the monthly mean data: wind-forcing and combined forcing regimes. Here, the black contours are referred as k =-1 that the extension of eddy viscosity K (r, k) from in the right-hand side of (19) is the mathematical 1 r 0tor 0 is because h was deﬁned as the aesthetics and computational regularity. The depth boundary layer depth in the original KPP model, dependent non-dimensional turbulent velocity scale which is usually deeper than the mixed layer depth. wx (r) is given by (LARGE et al. 1994) The shape function G (r) is assumed to be a cubic ( 1 polynomial (O’BRIEN 1970) and given by (MCWIL- /ðekÞ ; e\r\1 k\0; e¼ 0:1 wxðr; kÞ¼ LIAMS and HUCKLE 2006; Fig. 2) 1 /ðrkÞ ; otherwise ÂÃðr À rÞ2 ð20Þ GðrÞ¼r 1 À r2 þ 0 Hðr À rÞ; ð19Þ 2r 0 0 Here, the function / is deﬁned by the Monin– where r0 ¼ 0:05; H(a) is the Heaviside step function Obukhov similarity theory (MONIN and OBUKHOV (equal to 1 for a [ 0 and 0 otherwise). As pointed out 1954) such that the dimensional turbulent velocity by MCWILLIAMS and HUCKLE (2006), the second term scales equal ju* with neutral forcing (k = 0) and are 2836 P. C. Chu Pure Appl. Geophys. enhanced and reduced in unstable (k \ 0) and stable given depth r, / increases with k (Fig. 3a); and K (r, (k [ 0) conditions. It is given by (LARGE et al. 1994) k) decreases with k (Fig. 3b). Such k-dependence of / 8 and K (r, k) is quite smooth for k [ 0 and k \ 0, but < 1 þ 5rk; 0 k 1 16rk À1=4; k\0; r a =k very abrupt at k = 0. The /-values are small for k \ 0 /ðrkÞ¼: ð À Þ m am À1=3 (e.g., / ¼ 0:05 for r = 0.5, k =-1) and very large ð1:26 À 8:38rkÞ ; k\0;r [ am=k ¼À0:2 for k [ 0 (e.g., / 30; 000 for r = 0.5, k = 1). The K-values are large for k \ 0 (e.g., K ’ 7 for r = 0.5, ð21Þ k =-1) and very small for k [ 0 (e.g., K 10À5 for For neutral forcing (k = 0), /ðrkÞ¼1. Substitution r = 0.5, k = 1). However, the dependence of / and K of (19), (20), and (21) into (18) leads to an analytical (r, k)onr is quite mild. Substitution of the KPP eddy non-dimensional KPP eddy viscosity K (r, k). For a viscosity at the surface K (0, k) into (8) leads to

3=2 u Ã VE ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2jjf hjKð0; kÞ

Monthly Ekman velocity scale (VE), calculated from the same data sets for the computation of k, has strong seasonal variability (Fig. 4). In January, larger -1 VE-values ([0.5 m s ) occur in the Northern Hemisphere such as in the Gulf Stream, Kuroshio, equatorial regions (especially in the eastern Paciﬁc), -1 and smaller VE-values (\0.2 m s ) occur in the Southern Hemisphere. In July, smaller VE-values (\0.2 m s-1) occur in the Northern Hemisphere except some northern tropical regions such as near the northern African coast and west Arabian Sea, and -1 Figure 2 larger VE-values ([0.5 m s ) occur in the Southern The shape function G (r) Hemisphere.

Figure 3 Dependence of (a) / and (b) K on r and k for 1 r 0 Vol. 172, (2015) Ekman Spiral in Horizontally Inhomogeneous Ocean 2837

Figure 4 Monthly Ekman velocity scale (VE): a January, and b July

5. Ekman Spiral approximate analytical solutions uE [= (uE, vE)] (see Appendix 1), Substitution of (18) into (9) leads to uE ¼ uE þ DuE; vE ¼ vE þ DvE; ð23Þ where juÃ o ouE f k Â uE À Kðr; kÞ þ À h or or uE ¼ exp½Fðr; kÞ fga cos½þFðr; kÞ a sin½Fðr; kÞ ; ju o ¼ Ã k Â ½Kðr; kÞS ; ð22Þ ð24aÞ fVE or v ¼ exp½Fðr; kÞ fgaþ sin½ÀFðr; kÞ aÀ cos½Fðr; kÞ which is an ordinary differential equation with depth- E 24b varying K (r, k). The WKB method was used in this ð Þ study to solve the differential Eq. (22) with the are the barotropic components of the Ekman velocity boundary conditions (14) and (15) to get the (i.e., sx = 0, sy = 0); and 2838 P. C. Chu Pure Appl. Geophys.

ÈÉ þ À DuE ¼ c exp½Fðr; kÞ b cos½þFðr; kÞ b sin½Fðr; kÞ 2 8 "#ÂÃpffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 9 3 > o Kðf; kÞðs À s Þ o K f; k > 6 > x y ð Þ > 7 6 > o þðsx À syÞ o cos½Fðr; kÞÀFðf; kÞ > 7 6 < f f = 7 Zr 6 7 csgnðf Þ 6 > "#ÂÃpffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi > 7 À 6 > > 7df 2 6 > o Kðf; kÞðsx þ syÞ o Kðf; kÞ > 7 2f 6 > À þðs þ s Þ sin½Fðr; kÞÀFðf; kÞ > 7 0 6 : o x y o ; 7 4 f f 5 Â expðÞ½Fðr; kÞÀFðf; kÞ ð25Þ 2 8 "#ÂÃpffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 9 3 > o Kðf; kÞðs À s Þ o K f; k > 6 > x y ð Þ > 7 6 > o þðsx À syÞ o cos½Fðr; kÞÀFðf; kÞ > 7 6 < f f = 7 Z1 6 7 csgnðfÞ 6 > "#ÂÃpffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi > 7 À 6 > > 7df 2 6 > o Kðf; kÞðsx þ syÞ o Kðf; kÞ > 7 2f 6 > þ þðs þ s Þ sin½Fðr; kÞÀFðf; kÞ > 7 r 6 : o x y o ; 7 4 f f 5 Â expðÞÀ½Fðr; kÞÀFðf; kÞ ÈÉ þ À DvE ¼ c exp½Fðr; kÞ b sin½ÀFðr; kÞ b cos½Fðr; kÞ 2 8 "#ÂÃpffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 9 3 > o Kðf; kÞðs þ s Þ o K f; k > 6 > x y ð Þ > 7 6 > o þðsx þ syÞ o cos½Fðr; kÞÀFðf; kÞ > 7 6 < f f = 7 Zr 6 7 csgnðfÞ 6 > "#ÂÃpffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi > 7 À 6 > > 7df 2 6 > o Kðf; kÞðsx À syÞ o Kðf; kÞ > 7 2f 6 > þ þðs À s Þ sin½Fðr; kÞÀFðf; kÞ > 7 0 6 : of x y of ; 7 4 5 Â expðÞ½Fðr; kÞÀFðf; kÞ ð26Þ 2 8 "#ÂÃpffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 9 3 > o Kðf; kÞðs þ s Þ o K f; k > 6 > x y ð Þ > 7 6 > o þðsx þ syÞ o cos½Fðr; kÞÀFðf; kÞ > 7 6 < f f = 7 Z1 6 7 csgnðfÞ 6 > "#ÂÃpffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi > 7 À 6 > > 7df 2 6 > o Kðf; kÞðsx À syÞ o Kðf; kÞ > 7 2f 6 > À þðs À s Þ sin½Fðr; kÞÀFðf; kÞ > 7 r 6 : o x y o ; 7 4 f f 5 Â expðÞÀ½Fðr; kÞÀFðf; kÞ Vol. 172, (2015) Ekman Spiral in Horizontally Inhomogeneous Ocean 2839 are the baroclinic components of the Ekman velocity mix.html (MONTEREY and LEVITUS 1997), and the (i.e., nonzero if sx 6¼ 0; sy 6¼ 0). The parameters are angle of surface wind (h) data, which is computed defined as follows: from the monthly zonal wind data downloaded pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi from the website: http://iridl.ldeo.columbia.edu/ jhfjj Kð0; kÞ c ; aÆ ðcos h Æ sin hÞsgnðf Þ; SOURCES/.DASILVA/.SMD94/.climatology/.u3/ u Ã and the monthly meridional wind data downloaded 1; if f 0 sgnðf Þ¼ ð27Þ from the website: http://iridl.ldeo.columbia.edu/ À1; if f \0 SOURCES/.DASILVA/.SMD94/.climatology/.v3/.

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ 2 b À2sgnðf Þ Kð0; kÞsyð0Þ=f 8 "#ÂÃpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 9 > o Kðr; kÞsy o Kðr; kÞ > > þ s cos½Fðr; kÞ > Z1 <> or y or => sgnðfÞ ð28aÞ þ "#ÂÃpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp½Fðr; kÞ dr f 2 > o > > Kðr; kÞsx o Kðr; kÞ > 0 :> þ þ s sin½Fðr; kÞ ;> or x or pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi À 2 b 2sgnðf Þ Kð0; kÞsxð0Þ=f 8 "#ÂÃpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 9 > o Kðr; kÞsy o Kðr; kÞ > > þ s sin½Fðr; kÞ > Z1 <> or y or => sgnðfÞ ð28bÞ þ "#ÂÃpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp½Fðr; kÞ dr f 2 > o > > Kðr; kÞsx o Kðr; kÞ > 0 :> À þ s cos½Fðr; kÞ ;> or x or

sffiffiffiffiffiffiffiffiffiffi Z r Figure 5 shows examples of dimensional Ekman jjf h pffiffiffiffiffiffiffiffiffiffiffiffiffiffid1 FðÞ¼Àr; k : ð29Þ spirals (uÊ ¼ VEuE) with (solid curve) and without 2juÃ 0 KðÞ1; k (dashed curve) baroclinic components. The upper left Here, sgn (f) is the sign function. The non-dimen- panels (a) and (b) show the Ekman spirals at Loca- sional barotropic (uE; vE), and baroclinic (DuE; DvE) tion-1 (11°N, 159°W) (i.e., north equatorial Pacific) components of the Ekman spiral are calculated for the with large baroclinic components. The upper right global oceans except the regions near the equator panels (c) and (d) indicate the Ekman spirals at Lo- (5°S–5°N) using (24a), (24b), (25a), (25b) with the cation-2 (43°N, 169°E) (i.e., northwestern Pacific monthly mean global ocean density q (kg m-3) cal- mid-latitude) with small baroclinic components. culated from the World Ocean Atlas 2009 Profiles of the horizontal density gradient temperature and salinity data (1° 9 1° resolution) (oq=ox,oq=oy) are much larger at Location-1 (lower (http://www.nodc.noaa.gov/OC5/WOA09/pubwoa09. left panels) than at Location-2 (lower right panels). html) using the International Thermodynamic Equa- tion of Seawater (http://www.teos-10.org/pubs/ TEOS-10_Manual.pdf), the computed KPP eddy 6. Baroclinic Effect viscosity data K (r, k), the ocean friction velocity (u*) data (1° 9 1°) from http://iridl.ldeo.columbia.edu/ The baroclinic portion of the Ekman spiral can be SOURCES/.DASILVA/.SMD94/.climatology/ (DASILVA effectively determined by the ratio of the vertical et al. 1994), the monthly mixed layer depth (h)data integration of baroclinic Ekman component over the from http://www.nodc.noaa.gov/OC5/WOA94/ Ekman velocity, 2840 P. C. Chu Pure Appl. Geophys.

Figure 5 Examples of Ekman spirals (a) uÊ,(b) vÊ at Location-1 (11°N, 159°W), (c) uÊ; and (d) vÊ at Location-2 (43°N, 169°E) (upper panels)as well as corresponding horizontal density gradients (lower panels). It is noted that the horizontal density gradients are much stronger at Location-1 than Location-2

R1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k and M:large(small)k corresponds to small Du2 Dv2 dr E þ E (large) M. Such negative correlation is found in 0 M ¼ R1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : ð30Þ the scatter diagrams of (k, M) for the global 2 2 uE þ vEdr oceans in January (Fig. 7a) and July (Fig. 7b) with 0 linear regression equations, Horizontal distribution of M has strong seasonal 2 and spatial variability with large M-values (>0.2) M ¼ 0:0239 À 0:0195k ðR ¼ 0:561Þ for January; occurring in the tropical North Pacific Ocean, ð31Þ tropical Atlantic Ocean (10°N–25°N), and eastern M ¼ 0:0305 À 0:0171k ðR2 ¼ 0:463Þ for July: ð32Þ Arabian Sea with the largest value of 0.9 in the central tropical North Pacific Ocean near the The two regression equations are significant on the dateline, and with small M-values (\0.2) occur- level of 0.0005 with the numbers of paired data are ring in the Southern Hemisphere in January, and 6945 in January (Fig. 7a) and 6940 in July (Fig. 7b). vice versa in July (Fig. 6). Comparison between The negative correlation between k and M may be Figs. 6 and 1 shows negative correlation between related to the increase of the KPP eddy viscosity with Vol. 172, (2015) Ekman Spiral in Horizontally Inhomogeneous Ocean 2841

Figure 6 Monthly horizontal distribution of M:(a) January and (b) July the decrease of k especially for k \ 0. The baro- inversely proportional to f2 [see (25), (26), (28a), and clinicity parameter is identified by the vertical (28b)].The baroclinic effect on the Ekman spiral is integration of the magnitude of horizontal s-gradient evaluated by the correlation coefficient (R) between (crossing the mixed layer) scaled by f2, the two parameters, M and b, under the wind Z1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffi (k À1 and combined (k\ À 1Þ forcing regimes in 1 2 2 the Northern and Southern hemispheres (Table 1). It b ¼ sx þ sy dr; ð33Þ f 2 is found that M and b are positively correlated for all 0 the situations with large R ([0.73) for both hemi- which shows evident spatial variability and weak spheres and months (January and July) under the seasonal variability (Fig. 8). Since B is inversely surface wind forcing regime (k À1, with large R in proportional to f2, the B-value is usually large (B [ 5) the Northern Hemisphere in January (0.77) and in the in low latitudes (20°S–20°N), and small (B 5) in Southern Hemisphere in July (0.62) under the com- middle and high latitudes. It is noted that the scale bined forcing regime (k\ À 1Þ, and with small R in factor of f2 for the baroclinicity parameter (B) [see the Northern Hemisphere in July (0.49) and in the (33)] (scaled by f2) is only used for searching for Southern Hemisphere in January (0.36). The scatter baroclinic Ekman components since (DuE; DvE) are diagrams of (b, M) for the Northern Hemisphere 2842 P. C. Chu Pure Appl. Geophys.

and weakened in the hemisphere with prevailing wind forcing regime.

7. Eddy Viscosity due to Pure Wind Forcing

7.1. General Description

Earlier studies such as in McWILLIAMS and HUCKLE (2006) and SONG and HUANG (2011) assume no surface buoyancy ﬂux (B = 0), i.e., the depth ratio k = 0 [see (17)], the depth-dependent, non-dimen-

sional turbulent velocity scale wx (r) equals 1 [see (20) and (21)]. Also, the dimensional form of the Ekman equation is used o ou^ 1 o ÂÃ f k Â u^ À K^ðzÞ E ¼À k Â K^ðzÞS ; E oz oz f oz ð35Þ with the surface (dimensional) boundary condition, ou^ K^ð0ÞSð0Þ K^ð0Þ E ¼ u2h^ þ k Â ; ð36aÞ Figure 7 oz Ã Ã f Scatter diagrams of (k, M) with linear regression: (a) January and z¼0 (b) July. It is noted that the negative correlation between (k, M)is signiﬁcant on the level of 0.0005 and the lower boundary condition,

uÊ finite as z !À1; ð36bÞ (Fig. 9) and Southern Hemisphere (Fig. 10) also where show similar statistical relationships between M and b K^ : u2= f : (M increase as B increases). Since M vanishes as b ð0Þ¼0 004 Ã jj ð36cÞ vanishes, (i.e., no baroclinic Ekman components With the monthly mean surface wind stress data, the when the horizontal density gradient equals zero), the ocean friction velocity uÃ [using (12b)] and in turn linear regression equation between M and b is written the surface eddy viscosity K^ð0Þ [using (36c)] is cal- by culated except the equatorial region 5°S–5°N. Figure 11 clearly shows strong horizontal and sea- M ¼ cb; ð34Þ sonal variations of K^ð0Þ. In January, it has large where the regression coefficient c is obtained using values ([0.02 m2 s-1) in the western/central tropical the least square error method. The regression coeffi- North Pacific and Atlantic oceans (6°N–25°N), cient c is always positive (Table 1). It has the largest medium values (0.01–0.02 m2 s-1) in the mid-lati- value under the combined forcing regime in January tudes associated with the Kuroshio and Gulf Stream, for the Northern Hemisphere (0.0498), and in July for and small values (\0.01 m2 s-1) in rest of the global the Southern Hemisphere (0.0370). It has smallest oceans. However, in July, it has very large values value under the combined forcing regime in July for ([0.03 m2 s-1) in the western Arabian Sea (related to the Northern Hemisphere (0.00551) (prevailing wind the southwest monsoon), large values (0.02– forcing regime), and in January for the Southern 0.03 m2 s-1) in the western Bay of Bengal and the Hemisphere (0.00932) (prevailing wind forcing southern tropical Indian and Atlantic oceans (6°S– regime). Thus, the baroclinic effect is enhanced in the 25°S), medium values (0.01–0.02 m2 s-1) in the hemisphere with prevailing combined forcing regime southern tropical Pacific Ocean (6°S–25°S), mid- Vol. 172, (2015) Ekman Spiral in Horizontally Inhomogeneous Ocean 2843

Figure 8 Monthly horizontal distribution and zonal mean of B:(a) January and (b) July

Table 1 January and July correlation coefﬁcients and number of paired data between (M, B) for Northern and Southern hemispheres under wind and combined forcing regimes

Location Statistics Januaryðk À1Þ January (k \ -1) July ðk À1Þ July (k \ -1)

Northern Hemisphere Number of paired data 1,181 1,225 2,120 143 R 0.753 0.773 0.797 0.487 c 0.00939 0.0498 0.0101 0.00551 Southern Hemisphere Number of paired data 4,015 220 2,573 1,660 R 0.794 0.358 0.733 0.624 c 0.0142 0.00932 0.0134 0.037 latitude (35°S–45°S) Indian Ocean, and small values taking surface value K^ð0Þ for the whole water col- (\0.01 m2 s-1) in the rest of the global oceans in umn, and (c) wind- and depth-independent, i.e., July. The eddy viscosity K^ðzÞ has three different assigning a constant value. Correspondingly, the so- types: (a) wind and depth dependent using the KPP, lutions for the three types of eddy viscosity are ð1Þ ð2Þ ð3Þ (b) wind-dependent and depth-independent, i.e., represented by uÊ ; uÊ ; uÊ , 2844 P. C. Chu Pure Appl. Geophys.

Figure 9 Scatter diagrams of (B, M) with linear regression for the Northern Hemisphere: (a) January, wind forcing regime (k À1, (b) January, combined forcing regime (k\ À 1), (c) July, wind forcing regime (k À1, and (d) July, combined forcing regime (k\ À 1). It is noted that the positive correlation between (k, M) is signiﬁcant on the level of 0.0005

ðiÞ ^ðiÞ ðiÞ ðiÞ ^ðiÞ ðiÞ 7.2. Wind- and Depth-Dependent Eddy Viscosity uÊ ¼ uE þ DuÊ ; vÊ ¼ vE þ DvÊ ; i ¼ 1; 2; 3 ð37Þ The vertically varying eddy viscosity due to the surface wind stress is given by (SONG and HUANG where ðu^ðiÞ; v^ðiÞÞ are the components of the Ekman E E 2011) velocity in barotropic ocean [i.e., the Ekman solu- ðiÞ ðiÞ ^ ^ tions when (11) is satisfied], and ðDuÊ ; DvÊ Þ are the KðzÞ¼Kð0Þð1 À a1zÞ expða2zÞ; ð39Þ baroclinic components of the Ekman spiral. The where ða ; a Þ are positive constants. Fitting (37) baroclinic effect is identified by the root-mean square 1 2 with the flow in the f-plane using the large-eddy (RMS) within the ocean mixed layer of the baroclinic ð1Þ ð1Þ ð2Þ ð2Þ ð3Þ simulations (ZIKANOV et al. 2003) gives the following components [(DuÊ ; DvÊ ), (DuÊ ; DvÊ ), (DuÊ ; ð3Þ semi-empirical formula (SONG and HUANG 2011) DvÊ )], vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ^ ^ jjf z u hihi KðzÞ¼Kð0ÞGð Þ; u XJ 2 2 uÃ ð40Þ t1 ðiÞ ðiÞ R ¼ Du^ ðjÞ þ Dv^ ðjÞ ; i ¼ 1; 2; 3 i J E E GðtÞ¼ð1 À 64:0327tÞ expð4:0073tÞ; j¼1 ð38Þ to calculate the depth-dependent eddy viscosity due to the surface wind stress. Here, GðtÞ is the shape where j denotes the vertical level; and J is the total function. Figure 12 shows the dependence of GðtÞ number of the vertical levels from the surface to the versus t, where t is the non-dimensional depth, t ¼ mixed layer depth. jjf z=uÃ: It is noted that K^ð0Þ is inversely proportional Vol. 172, (2015) Ekman Spiral in Horizontally Inhomogeneous Ocean 2845

ÈÉ to the magnitude of the Coriolis parameter |f|. For the ^ð1Þ ^þ ^À vE ¼ exp½FðzÞ V sin½ÀFðzÞ V cos½FðzÞ ; same ocean friction velocity u , the lower the lati- * ð41bÞ tude, the higher the value ofK^ð0Þ. ð1Þ ð1Þ ð1Þ The Ekman velocity, uÊ ¼½uÊ ; vÊ are the where approximate analytical solutions of (35) by the WKB rffiffiffiffiffiffi Z0 method (see Appendices 1 and 2) with the eddy jjf df V^Æ ¼ V aÆsgnðf Þ; FðzÞ¼À qffiffiffiffiffiffiffiffiffiffi K^ z E 2 viscosity ð Þ given by (40). The barotropic compo- K^ðfÞ nents are given by z ÈÉ ð42Þ u^ð1Þ ¼ exp½FðzÞ V^þ cos½þFðzÞ V^À sin½FðzÞ ; E The baroclinic components are given by ð41aÞ

ÈÉ ð1Þ ^þ ^À DuÊ ¼ exp½FðzÞ D1V cos½þFðzÞ D1V sin½FðzÞ Z0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½s ð0ÞÀs ð0Þ 2 oK^ðfÞ þ x y cos½FðzÞÀFðfÞ expðÞ½FðzÞÀFðfÞ df 4f jjf K^ðfÞ of z Z0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½sxð0Þþsyð0Þ 2 oK^ðfÞ À o sin½FðzÞÀFðfÞ expðÞ½FðzÞÀFðfÞ df 4f jjf K^ðfÞ f ð43aÞ z Zz sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½s ð0ÞÀs ð0Þ 2 oK^ðfÞ þ x y cos½FðzÞÀFðfÞ expðÞÀ½FðzÞÀFðfÞ df 4f jjf K^ðfÞ of À1 Zz sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½s ð0Þþs ð0Þ 2 oK^ðfÞ þ x y sin½FðzÞÀFðfÞ expðÞÀ½FðzÞÀFðfÞ df 4f jjf K^ðfÞ of À1 ÈÉ ð1Þ ^þ ^À DvÊ ¼ exp½FðzÞ D1V sin½ÀFðzÞ D1V cos½FðzÞ Z0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½s ð0ÞÀs ð0Þ 2 oK^ðfÞ þ x y sin½FðzÞÀFðfÞ expðÞ½FðzÞÀFðfÞ df 4f jjf K^ðfÞ of z Z0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½sxð0Þþsyð0Þ 2 oK^ðfÞ þ o cos½FðzÞÀFðfÞ expðÞ½FðzÞÀFðfÞ df 4f jjf K^ðfÞ f ð43bÞ z Zz sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½s ð0Þþs ð0Þ 2 oK^ðfÞ þ x y cos½FðzÞÀFðfÞ expðÞÀ½FðzÞÀFðfÞ df 4f jjf K^ðfÞ of À1 Zz sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½s ð0ÞÀs ð0Þ 2 oK^ðfÞ À x y sin½FðzÞÀFðfÞ expðÞÀ½FðzÞÀFðfÞ df 4f jjf K^ðfÞ of À1 2846 P. C. Chu Pure Appl. Geophys.

Figure 10 Scatter diagrams of (B, M) with linear regression for the Southern Hemisphere: (a) January, wind forcing regime (k À1, (b) January, combined forcing regime (k\ À 1), (c) July, wind forcing regime (k À1, and (d) July, combined forcing regime (k\ À 1). It is noted that the positive correlation between (k, M) is signiﬁcant on the level of 0.0005

Here sffiffiffiffiffiffiffiffiffiffi Figure 13 shows the global horizontal distribution -1 K^ s s and zonal mean R1 (m s ) in January and July. In ^þ ð0Þ yð0Þ ffiffiffiyðp0Þffiffiffiffiffiffi D1V ¼À À p January, it has large values (>0.2 m s-1) in the tro- 2jjf f 2 2f jjf Z0 pical North Pacific and Atlantic oceans (6°N–20°N), exp½FðzÞ oK^ðzÞ Â qffiffiffiffiffiffiffiffiffiffi cos½FðzÞ dz the tropical North Indian Ocean (6°N–10°N), the oz K^ðzÞ tropical South Pacific and Atlantic oceans (6°S– À1 -1 Z0 10°S), medium values (0.02–0.2 m s ) in the mid- sxð0Þ exp½FðzÞ oK^ðzÞ latitudes associated with the Kuroshio and Gulf À pffiffiffi pffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffi sin½FðzÞ dz; oz 2 2f jjf K^ðzÞ Stream, south tropical Indian Ocean (6°S–30°S) and À1 small values (<0.01 m s-1) in rest of the global ð44aÞ oceans. In July, it has large values (>0.2 m s-1) in the sffiffiffiffiffiffiffiffiffiffi tropical oceans (15°S–15°N) with a maximum value ^ À Kð0Þ sxð0Þ sxð0Þ -1 D1V^ ¼ þ pffiffiffi pffiffiffiffiffiffi of 0.62 m s in the Arabian Sea. It generally de- 2jjf f 2 2f jjf creases with the increasing latitude. In high latitudes, Z0 -1 o ^ it is very small (<0.01 m s ). Zonal mean R1 de- expq½ffiffiffiffiffiffiffiffiffiffiFðzÞ KðzÞ Â o cos½FðzÞ dz creases with latitude near-exponentially in both ^ z À1 KðzÞ Northern and Southern hemispheres) from 0.6 m s-1 Z0 -1 -3 -1 o ^ (0.4 m s )at6°N(6°S) to less than 10 ms at sffiffiffiyðp0Þffiffiffiffiffiffi expq½ffiffiffiffiffiffiffiffiffiffiFðzÞ KðzÞ À p o sin½FðzÞ dz: high latitudes near 60°N (60°S). The histograms and 2 2f jjf ^ z À1 KðzÞ associated probability density functions (PDFs) for ð44bÞ January and July (Fig. 14) show near-exponentially Vol. 172, (2015) Ekman Spiral in Horizontally Inhomogeneous Ocean 2847

Figure 11 Horizontal distribution of eddy viscosity at the ocean surface K^ð0Þ calculated from monthly mean surface wind stress data (downloaded from the websites: http://iridl.ldeo.columbia.edu/SOURCES/.DASILVA/.SMD94/.climatology/.u3/ and http://iridl.ldeo.columbia.edu/SOURCES/. DASILVA/.SMD94/.climatology/.v3/ using (20) and (21): (a) January, and (b) July

decreasing probability with R1 (log scale used in the 7.3. Wind-Dependent and Depth-Independent Eddy vertical axis). The probability of R1 larger than Viscosity 0.2 m s-1 is 2.6 % [= 1 - P (R 0.2 m s-1) = The eddy viscosity is given by 1 - 0.974)] in January and 4.6 % (= 1 - 0.954) in July. The probability of R larger than 0.1 m s-1 is 1 ^ ^ 10.0 % in January and 11.5 % in July. The 95th KðzÞ¼Kð0Þ: ð45Þ -1 -1 percentile is 0.173 m s in January and 0.212 m s Substitution of (45) into (42) leads to, in July (also see Tables 2 and 3). 2848 P. C. Chu Pure Appl. Geophys.

Hemisphere middle latitudes, zonal belts with

medium values of Dw (20–50 m) appear at 35°S– 50°S in the three southern oceans (Atlantic, Indian, and Pacific) in January, but does not appear in the Southern Pacific in July. Substitution of (46) into (41a), (41b), (43a), (43b), (44a), and (44b) leads to the barotropic components z z z ^ð2Þ ^þ ^À uE ¼ expð Þ V cosð ÞþV sinð Þ ; DW DW DW ð47aÞ z z z ^ð2Þ ^þ ^À vE ¼ expð Þ V sinð ÞÀV cosð Þ ; DW DW DW ð47bÞ and the baroclinic components ð2Þ z ^þ z ^À z DuÊ ¼ expð Þ D2V cosð ÞþD2V sinð Þ ; DW DW DW ð48aÞ z z z Figure 12 ð2Þ ^þ ^À DvÊ ¼ expð Þ D2V sinð ÞÀD2V cosð Þ Vertical structure function GðtÞ versus t DW DW DW ð48bÞ sffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ À z 2K^ð0Þ 0:008u2 Here (V^ ; V^ ) are given by (42) and F z ; D Ã; 46 ð Þ¼ W ¼ ¼ 3=2 ð Þ DW jjf jjf D s ð0Þ D s ð0Þ D V^þ ¼À W y ; D V^À ¼ W x : ð49Þ 2 f 2 2f where (40) is used; Dw is the e-folding decay scale -1 of the Ekman depth, which varies with the surface The global horizontal distribution of R2 (m s ) wind stress through K^ð0Þ, and latitude. For the same (Fig. 16) is similar to that of R1 (Fig. 13) in both ocean friction velocity (uÃ), the lower the latitude, January and July with smaller values in the Gulf the higher the value of DW. The e-folding scale Dw Stream and Kuroshio regions. Latitudinal decrease of is computed from the global data of K^ð0Þ using zonal mean RMS with the eddy viscosity is also (46). It also has evident horizontal and weak sea- comparable in both Northern and Southern hemi- sonal variations (Fig. 15, equatorial region 5°S–5°N spheres. The histograms and associated PDFs for not computed): large values ([100 m) generally January and July (Fig. 17) also show near-exponen- occur in subtropical regions (10°N–20°N, 10°S– tially decreasing probability with R2. For large R2 -1 20°S) in January and July due to small Coriolis values (>0.2 m s ), the probability is zero in Jan- parameter. In the Northern Hemisphere middle and uary and 0.6 % (= 1–0.994) in July. The probability -1 high latitudes, Dw is larger in January (50–100 m in of R2 larger than 0.1 m s is 0.28 % in January and -1 the Atlantic Ocean and western Pacific Ocean) than 4.7 % in July. The 95th percentile is 0.089 m s in -1 in July (mostly less than 20 m). In the Southern January and 0.11 m s in July. Vol. 172, (2015) Ekman Spiral in Horizontally Inhomogeneous Ocean 2849

Figure 13 Horizontal distribution and zonal mean of vertical root-mean square of baroclinic components of the Ekman spiral in neutral ocean, R1 (m s-1), inside the mixed layer with wind-dependent and depth-dependent eddy viscosity K (z): a January and b July

^ð3Þ ^ð3Þ 7.4. Wind- and Depth-Independent Eddy Viscosity components (uE ; vE ) and (48a) and (48b) lead to the baroclinic components (Du^ð3Þ; Dmv^ð3Þ) of the Ekman For the wind- and depth-independent eddy E E velocity. Here, the constant eddy viscosity K^ is taken viscosity, 0 as 0.054 m2 s-1 (Price et al. 1987). -1 K^ðzÞ¼K^0 ¼ const: ð50Þ The global horizontal distribution of R3 (m s ) (Fig. 18) is similar to that of R (Fig. 13) in January Substitution of (50) into (46) gives 1 sffiffiffiffiffiffiffiffi and July with latitudinal decrease of zonal mean RMS in the Northern and Southern hemispheres. The 2K^0 DW ¼ D ¼ ; ð51Þ histograms and associated PDFs for January and July jjf (Fig. 19) also show near-exponentially decreasing -1 which is the classical Ekman depth. After replacing probability with R3. For large R3 values (>0.2 m s ), DW by D,(47a) and (47b) lead to the barotropic the probability is zero in January and July. The 2850 P. C. Chu Pure Appl. Geophys.

Figure 14 -1 Histogram and probability density function of R1 (m s ) with wind-dependent and depth-dependent eddy viscosity K^ðzÞ:(a) January, and (b) July

Table 2 Statistical characteristics of the VRMA (within the ocean mixed layer) of the baroclinic components over the global oceans for the three types of eddy viscosity under zero surface buoyancy ﬂux in January

RMS Q0.5 (m/s) Q0.95 (m/s) P (R 0.02 m/s) P (R 0.1 m/s) P (R 0.2 m/s)

R1 0.0113 0.173 0.669 0.900 0.974 R2 0.00401 0.0888 0.800 0.972 1.000 R3 0.00107 0.0399 0.901 0.999 1.000

Table 3 Statistical characteristics of the VRMA (within the ocean mixed layer) of the baroclinic components over the global oceans for the three types of eddy viscosity under zero surface buoyancy ﬂux in July

RMS Q0.5 (m/s) Q0.95 (m/s) P (R 0.02 m/s) P (R 0.1 m/s) P (R 0.2 m/s)

R1 0.00929 0.212 0.694 0.875 0.954 R2 0.00306 0.110 0.795 0.953 0.994 R3 0.000769 0.0575 0.883 0.988 1.000 Vol. 172, (2015) Ekman Spiral in Horizontally Inhomogeneous Ocean 2851

Figure 15 Horizontal distribution of e-folding depth (Dw) of the Ekman layer with wind-dependent and depth-independent eddy viscosity: (a) January and (b) July

-1 probability of R3 larger than 0.1 m s is 0.1 % in ageostrophic shears linking to turbulent stress in up- January and 1.2 % in July. The 95th percentile is per oceans, under surface wind and buoyancy forcing 0.04 m s-1 in January and 0.058 m s-1 in July. using the KPP eddy viscosity. The Ekman spiral contains barotropic and baroclinic components. The barotropic component is similar to the classical Ek- 8. Conclusions man spiral. The baroclinic component is caused by horizontally inhomogeneous density. The baroclinic Analytical solution of the Ekman spiral in real component vanishes as the horizontal density gradi- oceans is obtained with vertical geostrophic and ent vanishes. 2852 P. C. Chu Pure Appl. Geophys.

Figure 16 Horizontal distribution and zonal mean of vertical root-mean square of baroclinic components of the Ekman spiral in neutral ocean, R2 (m s-1), inside the mixed layer with wind-dependent and depth-independent eddy viscosity K^ð0Þ:(a) January and (b) July

Climatological monthly data of global ocean Statistical analysis on the calculated global (k, M, mixed layer depth, Monin–Obukhov length scale, B) values shows significant negative correlation be- friction velocity, surface winds, and density profiles tween k and M: large (small) k corresponds to small are used to calculate the depth ratio (k), KPP eddy (large) M, and significant positive correlation be- viscosity, the barotropic and baroclinic Ekman ve- tween B and M: large (small) B corresponds to large locities, the baroclinicity parameter (B), and the (small) M. The negative correlation between k and proportion of the baroclinic Ekman component (M). M may be related to the increase of the KPP eddy Large baroclinic proportion is usually associated viscosity with the decrease of k especially for k \ 0. with the prevailing combined forcing regime such as The positive correlation coefficient between B and in the Northern (Southern) Hemisphere in January M varies with the prevailing wind and combined (July). forcing regimes. The baroclinic effect is enhanced in Vol. 172, (2015) Ekman Spiral in Horizontally Inhomogeneous Ocean 2853

Figure 17 -1 Histogram and probability density function of R2 (m s ) inside the mixed layer with wind-dependent and depth-independent eddy viscosity K^ð0Þ:(a) January and (b) July the hemisphere with prevailing combined forcing for all the cases (three types of eddy viscosity). In regime and weakened in the hemisphere with pre- middle and high latitudes (especially in the Southern vailing wind forcing regime. Hemisphere), it is generally very small (i.e., the For pure wind forcing (i.e., zero surface buoyancy classical Ekman spiral applies) except in the Gulf ﬂux), three types of eddy viscosity from existing Stream and Kuroshio regions in January. These re- parameterization [wind- and depth-dependent, wind- sults are consistent with the earlier observational dependent and depth-independent, and wind- and studies such as conducted in the Drake Passage (LENN depth-independent (i.e., constant eddy viscosity)] as and CHERESKIN 2009). well as the vertical root-mean square of the baroclinic The near-exponentially decreasing probability component within the ocean mixed layer (R) of the with the vertical root-mean square of the baroclinic analytical Ekman spiral are used to investigate the component is obtained from the histograms. The baroclinic effect. It enhances with the decreasing statistical characteristics show that the baroclinic latitude and usually very evident ([0.2 m s-1) in the components for the three types of eddy viscosity tropical oceans in January and July, and extremely under pure wind forcing are all comparable in Jan- large value of 0.62 m s-1 in the Arabian Sea in July uary and July. Near-exponentially decreasing 2854 P. C. Chu Pure Appl. Geophys.

Figure 18 Horizontal distribution and zonal mean of vertical root-mean square of baroclinic components of the Ekman spiral in neutral ocean, R3 (m s-1), inside the mixed layer with a constant eddy viscosity (0.054 m2 s-1): (a) January and (b) July

probability with R1, R2,orR3 is found. The Finally, it is noted that the monthly mean density -1 probability of R1 larger than 0.2 m s is 2.6 % ﬁelds from the WOA-2009 temperature and salinity [=1 - P (R 0.2 m s-1) = 1 – 0.974)] in January data cannot represent density fronts associated with and 4.6 % (= 1 – 0.954) in July. The probability of submesoscale processes. The computation here is -1 R1 larger than 0.1 m s is 10.0 % in January and only to show the importance of horizontally inho- 11.5 % in July. The 95th percentile is 0.173 m s-1 in mogeneous density on the Ekman spiral. Further January and 0.212 m s-1 in July (also see Tables 2 computation is needed to verify the good ap- and 3). proximate/analytical solutions again high horizontal- Vol. 172, (2015) Ekman Spiral in Horizontally Inhomogeneous Ocean 2855

Figure 19 -1 2 -1 Histogram and probability density function of R3 (m s ) inside the mixed layer with a constant eddy viscosity (0.054 m s ): (a) January and (b) July resolution wind and density data if they will be where available. fh h2 o ÂÃ f ¼ ; / ¼ Ks 0 ju f ju V or y Ã 0 Ã E ð54Þ h2 o Acknowledgments À i ½Ksx : f0juÃVE or The author would like to thank Mr. Chenwu Fan for The function / represents the baroclinic effect. The his outstanding efforts on the computation and second-order differential equation (53) needs two graphics. boundary conditions. The surface boundary condition (14) becomes

dw uÃ Appendix 1: General Solutions of Eq. (22) jr¼0 ¼À ðÞcos h þ i sin h dr jKð0ÞVE h2 Let the Ekman currents (u , v ) be represented by þ ½s ð0ÞÀis ð0Þ: ð55Þ E E f ju V y x a complex variable w, 0 Ã E pffiffiffiffiffiffiffi The lower boundary condition of equation (53)is w ¼ u þ iv ; i ¼ À1: ð52Þ E E given by Substitution of (52) into (22) leads to jjw finite as r !1: ð56Þ 2 d w dK dw to guarantee a physically meaningful solution, i.e., K 2 þ À if0w ¼ /; ð53Þ dr dr dr jjw cannot be infinity as r !1. Eq. (53) is a linear 2856 P. C. Chu Pure Appl. Geophys.

inhomogeneous ordinary differential equation with Solving for the first two terms S0 and S1 yields the depth-varying coefficient K (z). Following BERGER rffiffiffiffiffiffi Zr and GRISOGONO (1998), studying the Ekman atmo- jjf0 df S0 ¼Æð1 þ iÞ pffiffiffiffiffiffiffiffiffiffi; ð63Þ spheric boundary layer, an approximate solution to 2 KðfÞ 0 the inhomogeneous problem (53) can be found with the variation of parameters technique, provided that 1 Kð0Þ S1 ¼ ln : ð64Þ an approximate solution of the homogeneous problem 4 KðrÞ of (53), Thus, the two approximate solutions of the homoge- d2w dK dw neous equation (55) are K þ À if w ¼ 0; ð57Þ dr2 dr dr 0 w1ðrÞ¼AðrÞ exp½ð1 þ iÞFðrÞ; exists. If two independent approximate solutions to w2ðrÞ¼AðrÞ exp½Àð1 þ iÞFðrÞ; ð65Þ homogeneous problem of Eq. (57) are given by w1ðrÞ where and w2ðrÞ, the general solution of Eq. (53) is given rffiffiffiffiffiffi r by 1=4 Z Kð0Þ jjf0 pdffiffiffiffiffiffiffiffiffiffif w ¼ c w ðrÞþc w ðrÞþc^ ðrÞw ðrÞþc^ ðrÞw ðrÞ; AðrÞ¼ ; FðrÞ¼À : 1 1 2 2 1 1 2 2 KðrÞ 2 KðfÞ ð58Þ 0 ð66Þ where Substitution of (55) and (66) into (59) and (60) gives Z0 w2/ Zr c^1ðrÞ¼À df; ð1 À iÞ KðfÞ½w1ðfÞdw2ðfÞ=df À w2ðfÞdw1ðfÞ=df ^ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r c1ðrÞ¼À AðfÞ/ðfÞ exp½Àð1 þ iÞFðfÞ df; 2 2jjf0 Kð0Þ ð59Þ 0 ð67Þ Z1 w1/ Z1 c^2ðrÞ¼ df: ð1 À iÞ KðfÞ½w1ðfÞdw2ðfÞ=df À w2ðfÞdw1ðfÞ=df pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r c^2ðrÞ¼À AðfÞ/ðfÞ exp½ð1 þ iÞFðfÞ df: 2 2jjf0 Kð0Þ ð60Þ r ð68Þ Substitution of (65) into (58) gives Appendix 2: The WKB Method for Solving Eq. (57) w ¼½c1 þ c^1ðrÞAðrÞ exp½ð1 þ iÞFðrÞ þ ½c2 þ c^ ðrÞAðrÞ exp½Àð1 þ iÞFðrÞ: ð69Þ The WKB method can be used to obtain a good 2 approximate solution of Eq. (57) if the vertical var- It is noted that F (r) \ 0 leads to iation of K (r) is slower than that of wðrÞ (GRISOGONO w !1as r !1: ð70Þ 1995), 2 This leads to ðS þ S e þ S e2 þ ...Þ w / exp 0 1 2 ; ð61Þ e c2 ¼ 0 ð71Þ where e is a presumably small parameter. Substitution Substitution of (69) into the surface boundary con- of (61) into (57) leads to a set of equations in terms of dition (55) gives powers of e.IfK (r) does not vary too quickly with c1 ¼ c^2ð0Þþð1 À iÞ depth, we have ()sffiffiffiffiffiffiffiffiffiffi u ðcos h þ i sin hÞ Kð0Þ h Â Ã pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi À ½s ð0ÞÀis ð0Þ : jjSnþ1ðrÞ y x 1; n ¼ 0; 1; 2; ... ð62Þ j 2jjf0 Kð0Þ jj2f0 f jjSnðrÞ Vol. 172, (2015) Ekman Spiral in Horizontally Inhomogeneous Ocean 2857

þ À KRAUS, E. B., TURNER, J.S. (1967). A one-dimensional model of the ¼ c^2ð0ÞþV À iV sffiffiffiffiffiffiffiffiffiffi seasonal thermocline, Part II, Tellus, 19, 98–105. ÈÉÂÃÂÃ Kð0Þ h LARGE, W.G., MCWILLIAMS, J. C., DONEY, S. C. (1994). Oceanic þ sxð0ÞÀsyð0Þ þ isxð0Þþsyð0Þ : vertical mixing: A review and a model with a non-local K-profile jj2f0 f boundary layer parameterization. Rev. Geophys., 32, 363–403. ð72Þ LENN, Y., CHERESKIN, T.K. (2009). Observation of Ekman Currents in the Southern Ocean. J. Phys. Oceanogr., 39, 768–779. LETTAU, H. H., DABBERDT, W. F. (1970). Variangular Wind Spirals. Boundary-Layer Meteorol.1, 64–79. EFERENCES R LOMBARDO, C. P., M. C. GREGG (1989). Similarity scaling of viscous and thermal dissipation in a convecting surface boundary layer, BERGER, B.W., GRISOGONO, B. (1998). The baroclinic, variable eddy J. Geophys. Res., 94, 6273–6284. viscosity Ekman layer. Bound.-Lay. Meteorol., 87, 363–380. MCWILLIAMS, J. C., HUCKLE, E. (2006). Ekman layer rectification. CHERESKIN, T.K. (1995). Direct evidence for an Ekman balance in J. Phys. Oceanogr., 36, 1646–1659. the California Current. J. Geophys. Res., 100, 18261–18269. MCWILLIAMS, J. C., HUCKLE,E.&SHCHEPETKIN, A. F. (2009). CHU, P.C. (1993). Generation of low frequency unstable modes in a Buoyancy effects in a stratified Ekman layer. J. Phys. Oceanogr., coupled equatorial troposphere and ocean mixed layer. J. Atmos. 39, 2581–2599. Sci., 50, 731–749. MONIN, A. S., OBUKHOV, A. M. (1954). Basic laws of turbulent CHU, P.C. (1995). P-vector method for determining absolute ve- mixing in the surface layer of the atmosphere. Trudy Geofiz. Inst. locity from hydrographic data. Marine Tech. Soc. J., 29, 2, 3–14. Acad. Nauk SSSR., 24, 163–187. CHU, P.C. (2000). P-vector spirals and determination of absolute MONTEREY, G., LEVITUS, S. (1997). Seasonal Variability of Mixed velocities. J. Oceanogr., 56, 591–599. Layer Depth for the World Ocean. NOAA Atlas NESDIS 14, CHU, P. C. (2006). P-Vector Inverse Method. Springer, 605 pp. U.S. Gov. Printing Office, Wash., D.C., 96 pp. CHU, P.C., GARWOOD, R.W. Jr. (1991). On the two-phase thermo- O’BRIEN, J. J. (1970). A note on the vertical structure of the eddy dynamics of the coupled cloud-ocean mixed layer. J. Geophys. exchange coefficient in the planetary boundary layer. J. Atmos. Res., 96, 3425–3436. Sci., 27, 1213–1215. CHU, P.C., LIU, Q.-Y., JIA, Y.-L., FAN, C.W. (2002). Evidence of a OBUKHOV, A.M. (1946). Turbulence in an atmosphere with a non- barrier layer in the Sulu and Celebes Seas. J. Phys. Oceanogr., uniform temperature. Trudy Inst. Teoret. Geophys. Akad. Nauk 32, 3299–3309. SSSR. 1: 95–115 (translation in: Boundary-Layer Meteorol. CHU, P.C., FAN, C.W. (2010). Optimal linear fitting for objective 1971. 2: 7–29). determination of ocean mixed layer depth from glider profiles. PRICE, J. F., WELLER, R. A., Schudlich, R. R. (1987). Wind-driven J. Atmos. Oceanic Technol., 27, 1893–1898. ocean currents and Ekman transport. Science, 238: 1534–1538. CHU, P.C. and FAN, C.W. (2011). Maximum angle method for de- RICHMAN, J.F., DESZOEKE, R., DAVIS, R.E. (1987). Measurements of termining mixed layer depth from seaglider data. J. Oceanogr., near-surface shear in the Ocean. J. Geophys. Res., 92, 67, 219–230. 2851–2858. DASILVA, A., YOUNG, A. C., LEVITUS, S. (1994). Atlas of Surface SONG, J.-B., HUANG, Y.-S. (2011). An approximate solution of Marine Data, Volume 1: Algorithms and Procedures, Number 6, wave-modified Ekman current for gradually varying eddy vis- NOAA/NODC. cosity. Deep-Sea Res. I, 58, 668–676. EKMAN, V. W. (1905). On the influence of the earth’s rotation on STACEY, M.W., POND, S., LEBLOND, P. H. (1986). A wind-forced ocean-currents. Ark. Mat. Astron. Fys., 2, 1–52. Ekman spiral as a good statistical fit to low-frequency currents in GARWOOD, R.W. Jr. (1977). An oceanic mixed layer capable of a coastal strait. Science, 233, 470–472. simulating cyclic states. J. Phys. Oceanogr., 7, 455–468. STEGER, J., COLLINS, C.A., and CHU, P.C. (1998). Circulation in the GRACHEV A.A., ANDREAS E.L, FAIRALL C.W., GUEST P.S., PERSSON Archipelago de Colon (Galapagos Islands), November 1993. P.O.G. (2008). Turbulent measurements in the stable atmo- Deep-Sea Res. II, 45, 1093–1114. spheric boundary layer during SHEBA: ten years after. Acta TAYLOR, J. R., SARKAR, S. (2008). Stratification effect in a bottom Geophysica.56, 1, 142–166. Ekman layer. J. Phys. Oceanogr., 38, 2535–2553. GRISOGONO, B. (1995). A generalized Ekman layer profile with ZIKANOV, O., SLINN, D.N., DHANAK, M.R. (2003). Large-eddy gradually varying eddy diffusivities. Quart. J. Roy. Meteorol. simulations of the wind-induced turbulent Ekman layer. J. Fluid Soc. 121, 445–453. Mech., 495, 343–368. HUNKINS, K. (1966). Ekman drift currents in the Arctic Ocean. Deep-Sea Res.13, 607–620.

(Received December 11, 2014, revised February 24, 2015, accepted February 26, 2015, Published online March 21, 2015)