May 12, 1999 Strati ed Ekman Layers James F. Price DepartmentofPhysical Oceanography,Wo o ds Hole Oceanographic Institution, Wo o ds Hole, Massachusetts 1 Miles A. Sundermeyer Joint Program in Physical Oceanography, Massachusetts Institute of Technology, and Wo o ds Hole Oceanographic Institution, Wo o ds Hole, Massachusetts 1 Now at the Center for Marine Science and Technology, University of Massachusetts, New Bedford. Abstract: Under fair weather conditions the time-averaged, wind-driven current forms a spiral in which the currentvector decays and turns cum sole with increasing depth. These spirals resemble classical Ekman spirals. They di er in that their rotation depth scale exceeds the e-folding depth of the sp eed by a factor of two to four and they are compressed in the downwind direction. A related prop erty is that the time-averaged stress is evidently not parallel to the time-averaged vertical shear. We develop the hyp othesis that the at spiral structure may b e a consequence of the temp oral variability of strati cation. Within the upp er 10-20 m of the water column this variability is asso ciated primarily with the diurnal cycle and can b e treated bya time-dep endent di usion mo del or a mixed-layer mo del. The latter can b e simpli ed to yield a closed solution that gives an explicit account of strati ed spirals and reasonable hindcasts of midlatitude cases. At mid and higher latitudes the wind-driven transp ort is found to b e trapp ed mainly within the upp er part of the Ekman layer, the diurnal warm layer. At tropical latitudes the e ects of diurnal cycling are in some ways less imp ortant, and Ekman layer currents are likely to b e signi canttomuch greater depths. In that event, the lower part of the Ekman layer is likely to b e a ected by strati cation variability that may b e nonlo cal. 2 1 Observing and Mo deling Wind-Driven Currents The upp er o cean Ekman layer problem was de ned in a complete and almost mo dern form in Walfrid Ekman's landmark analysis of 1905. From very limited observations Ekman inferred that the momentum balance for steady wind-driven currents must b e primarily between the Coriolis acceleration acting on the current and the divergence of a turbulent stress imp osed by the wind. He understo o d that a mo del of wind-driven currents must b e built around a mo del of the turbulent stress, and wentontoshowhow eld observations and di usion theory might b e used to develop such mo dels. Despite this promising b eginning and a long history of research, there are fundamental asp ects of the Ekman layer problem that remain unsettled, including even the list of imp ortant external variables. The great and enduring diculty is that turbulent stress is imp ortantatlowest order in the Ekman layer, and yet is almost imp ossible to measure in situ against a background of surface gravity waves. Thus the observational basis needed for a full understanding of the Ekman layer is incomplete. Measurement of Ekman layer currents presents similarchallenges, but mo dern measurement to ols and techniques Weller and Davis, 1980; Weller, 1981 havenow given what app ears to b e a detailed and reliable view of wind-driven currents under fair weather conditions Price et al., 1986; Wij els et al., 1994; Chereskin, 1995. 1.1 Goals, Scop e and Outlie Our goals are to describ e the vertical structure of wind-driven currents using these historical eld observations, and then identify the simplest mo dels that can serve to explain and predict this structure. Toward these goals we take up four questions in turn: Q1 What is the structure of the fair weather Ekman layer? By structure we mean the shap e and thickness of the current pro le, including the current direction. This question is addressed rst by a review and analysis of the eld observations noted ab ove in Section 2. The scop e of this study is limited to op en o cean and fair weather conditions wind stress less than ab out 0.2 Pa and signi cant solar heating primarily b ecause those are the conditions that held in the present data sets. Other imp ortant cases, e.g., the winter Ekman layer Krauss, 1993; Schudlich and Price, 1998 and the Ekman layer under ice 3 McPhee, 1990 are omitted from consideration, though we will try to indicate where the transition to other regimes may o ccur. Q2 Do es the classical di usion theory lead to a useful mo del of the fair weather Ekman layer? Ekman's 1905 classical di usion theory was one of the rst attempts to apply ideas of turbulent transfer to a natural system, and it remains valuable to dayasa reference or starting p oint for b oundary layer mo dels. The present analysis shows that a very simple, optimized classical di usion mo del can give a fairly go o d account of the observed Ekman layer. There are small but systematic errors however, and a more realistic mo del evidently requires a complex di usion co ecient Sections 3 and 6. Q3 What physical pro cesses have to b e represented in a minimum, realistic mo del of the fair weather Ekman layer? The central hyp othesis of this analysis is that time-dep endentvariations of strati cation are crucially imp ortant for the upp er o cean Ekman layer. One imp ortant mo de of fair weather variability is the diurnal cycle, evidence of which is reviewed in Section 4.1. A time-dep endent di usion mo del that incorp orates a diurnal cycle is shown to give a qualitatively realistic Ekman layer structure Section 4.2. A layered mo del of this pro cess can b e integrated to yield an approximate, closed solution for a strati ed Ekman layer Sections 5.1 and 5.2. This solution is a successor to the Price et al. 1986 numerical mo del. Comparison of this mo del with mid-latitude cases is generally favorable, while comparison with a tropical case is less so, p erhaps b ecause of neglected strati cation variability in the lower part of the Ekman layer Section 7. Q4 How do es the Ekman layer structure vary with wind stress, latitude, and other external variables? The observations alone provide a very limited view of parameter dep endence, and a full assessment requires a mo del Section 5.4. A fundamental and conventional assumption is that the Ekman layer structure is determined mainly by the lo cal surface uxes wind stress and heat uxes and by Earth's rotation. A secondary assumption, made in the hop e that simple mo dels and explicit solutions will suce for prediction, is that the surface uxes can b e represented well enough by suitable time-averages. Neither of these is strictly true, and part of the analysis will seek to evaluate the resulting errors and suggest remedies. 4 Finally, our attempt to answer the questions p osed ab ove is summarized in Section 8, where we also p oint out some of the many missing pieces to a complete understanding of the Ekman layer problem. The remainder of this section de nes the terms of a familiar momentum balance. 1.2 An Upp er Ocean Momentum Balance This analysis is meant to treat the lo cal resp onse of currents, aside from all consequences of top ography and spatially varying wind stress. Thus it will apply only to deep, op en o cean regions away from the equator. Consideration is also limited to regimes in which the momentum balance of an observed horizontal current v averaged over tens of minutes is o nearly linear and can b e written in the usual form, 1 @ 1 @ v o + if v = rP + HOT 1 o @t @z where f is the Coriolis parameter, is the nominal densityofseawater, rP is the horizontal pressure gradient, and HOT are higher order terms, for example a horizontal eddy stress, that are presumed small. Bold symb ols are complex with comp onents real, imaginary = 0 0 crosswind, downwind. The stress, , is a turbulent momentum ux, = <v w >, o where < > is a time average over tens of minutes and w is the vertical comp onentof velo city. As noted at the outset, turbulence measurements able to resolve this momentum ux are generally not p ossible over the op en o cean, though some imp ortant qualitative features of upp er o cean turbulence are known from observations, and referred to in Section 4.1. For contrast, see McPhee and Martinson, 1994 for detailed turbulent stress measurements b elow an ice cover. Thus the present analysis deals solely with current phenomena rather than with turbulence per se. This is a signi cant limitation in so far as quite di erent parameterizations of turbulent stress may give rather similar current pro les, and of course turbulence prop erties are imp ortant in their own right. The imp ortant pressure gradient term could result from tides, eddies, or p erhaps the basin-scale circulation but not, presumably, directly from the lo cal wind. That b eing so the observed current can b e decomp osed into a sum of wind and pressure-driven comp onents McPhee, 1990, v = v + v where the momentum balance of the wind-driven currentis o p 5 then 1 @ @ v + if v = : 2 @t @z How this decomp osition might b e accomplished with eld observations is reviewed in Section 2.1.