THE PARTITION OF ENERGY INTO WAVES AND CURRENTS W. Perrie, C. Tang, Y. Hu and B.M. DeTracy Fisheries & Oceans Canada, Bedfrd Institute f Oceangraphy, Dartmuth, Nva Sctia, Canada 1. INTRODUCTION Ocean mdels usually estimate surface currents withut explicit mdelling f the cean waves. T cnsider the impact f waves n surface currents, we use a wave mdel in a mdified Ekman layer mdel, which is imbedded in a diagnstic cean mdel. Thus, we explicitly cnsider wave effects, fr example Stkes drift and wave-breaking dissipatin, in cnjunctin with the Ekman current, mean currents and wind-driven pressure gradient currents. This is an explicit implementatin f the equatins fr wave-induced currents, as derived by Jenkins (1987ab). WAMtype mdel terms are used t estimate energy input t waves by wind and remved by wave-breaking dissipatin. The cean mdel fllws that described by Tang et al. (1999). Previusly these equatins by Jenkins were used t estimate wave-induced frcing n ice fles (Perrie and Hu, 1997). This cupled wave-cean mdel is cmpared t measurements frm the Labradr Sea Deep Cnvectin Experiment f 1997. We shw that the wave effect is largest in rapidly develping intense strms, when wave-mdified currents can briefly exceed the usual Ekman currents by as much as 40%. A large part f the increase in velcity can be attributed t the Stkes drift. Reductins in mmentum transfer t the cean due t wind input t waves, and enhancements due t wave breaking dissipatin are each f the rder 20-30%. 2. NUMERICAL MODEL In the absence f waves, the near-surface currents can be btained by slving the gverning equatins f the cean mdel at the surface. If the mdel has sufficient vertical reslutin, these equatins include an Ekman layer. The presence f waves mdifies the Ekman layer and the assciated surface currents, resulting frm wind inputs int waves, wave evlutin and wave-breaking. T accunt fr wave-induced currents, it is necessary t replace the cean mdel s riginal Ekman layer with a wave-mdified Ekman layer. 2.1 Wave Mdel Ocean wave spectra usually cnsiders three dminant prcesses: input f energy t the waves by wind, S in, nnlinear transfer between spectral cmpnents due t wave-wave interactins, S nl, and dissipatin due t whitecapping and wave-breaking,, which transfers energy t currents. Three prcesses are described fr the WAM S ds peratinal wave mdel by Hasselmann et al. (1988) and Kmen et al. (1994). The crrespnding balance equatin describing wave grwth and develpment is E( t, x, f, θ ) + C g E( t, x, f, θ ) = S in + S ds + S nl (1) E is the tw-dimensinal wave spectrum in terms f frequency f, wave prpagatin directin, θ, time, t, and psitin, x, and where C g is the grup velcity. In this study, we extend the apprach f Jenkins (1987ab, 1989) and Perrie and Hu (1997) by cupling the wave mdel t a simple realistic three-dimensinal cean where ( t, x, f,θ )
mdel (Tang and Gui, 1996). This allws us t estimate the impact f surface waves n surface currents. We d nt cnsider ther cupling mechanisms here, such as the impact f currents n waves. 3. Ocean Mdel We assume that the density current in the Ekman layer is much smaller than the Ekman current. This is generally a valid assumptin in the Labradr Sea and the Grand Banks, because wind mixing and winter cling create a surface mixed layer f unifrm density in the upper tens t hundreds f meters f the water clumn. We use a diagnstic cean mdel in which the water density des nt change with time. The gverning equatins are given by 1 + f u = p + ( w u ) (2) w u + = 0 ρ 0 p = gρ ζ + g ρdz (4) z where f is the Crilis parameter and the (hrizntal, vertical) cmpnents f velcity are ( u, w), respectively. We assume c-rdinates where z is upward and ζ is the adjusted sea level. Water density is ρ and ρ is a reference density. The last term f Equatin (2) is the vertical gradient f the Reynlds stress, which is created by windmixing and wave effects. This term is assumed significant nly in the Ekman layer. The hrizntal current is represented by a decmpsitin int an interir velcity, u, and an Ekman velcity, u : u = u i + u w. (5) The Ekman current u w is large nly in the Ekman layer. The interir velcity includes the density current as well as the current assciated with hrizntal pressure gradients. Expanding Equatins (2)-(4) in pwers f the Ekman number, ε = d / H, where d is the Ekman depth and H is water depth, we get gverning equatins fr ui and u w, 0 i g + f u i = g ζ ρdz (6) t ρ z w i + f u w = ( wu w ) (7) where the Reynlds stress is parameterized by a vertical eddy cefficient, w wu w = AV. (8) The surface bundary cnditin fr the Ekman current is: w Ta AV = (9) ρ z=0 where Ta is wind stress, cmputed frm surface wind fields, U, at 10m height, Ta = ρ a C d U U, (10) ρ a is the air density, and A V, C d, the air-sea drag cefficient. Further details are presented in Perrie et al. (2002). w (3)
2.3 Wave-Current Cupling Frm Jenkins (1987ab, 1989) and Perrie and Hu (1997), wave mdified surface currents are determined by, E 2 u E kz + f u E = AV f U S 2π df f Sds ke dθ k 2. (11) The assciated surface bundary cnditins, incrprating the effects f the energy input t waves by the wind S in, A V E c c=0 = T ρ a 2 π df fks dθ (12) in where c is the vertical crdinate z at time t = 0. These equatins are in terms f the quasi-eulerian current, u E, which is the Lagrangian mean current minus the Stkes drift, u L U S. We can identify the quasi-eulerian current with the Eulerian-mean current, u w as lng as particle displacements are nt s large as t mve frm ne space grid t anther, during a given simulatin. The secnd term n the right f (12) is a reductin in the wind stress fr current generatin, because sme ges int wave generatin. Stkes drift U S is the Lagrangian velcity at time x, y, c, t = 0, fllwing Huang (1971), t f a particle whse initial psitin was ( ) U 2kc S ) (x, t) = 4π fke E( f, θ dfdθ. (13) 2.4 Mdel Implementatin In the presence f waves the ttal hrizntal current at the surface, u surf, is given by the sum, u = u + u + U. (14) surf i E S The cean mdel dmain is a rectangle measuring 1200 km X 2240 km with the left side apprximately parallel t the Labradr cast, as shwn in Figure 1a. Within this dmain, the mdel equatins were implemented n a 20 km X 20 km staggered spatial grid (C grid). Seasnal climatlgical temperature and salinity data frm an bjective analysis were used t cmpute the density field (Tang et al., 1999). Six-hurly wind data frm Canadian Meterlgical Service were used fr frcing. A finite difference methd is used fr u E and Stkes drift U S, with an implicit time-stepping t satisfy the stability cnditins. A grid f 200 pints was used in the vertical directin [0, 100] m, varying frm 5 mm at the surface t ~1 m at the bttm. The fineness f the grid near the upper surface necessitates a time-step f abut 70sec t satisfy the stability cnditin. The wave mdel time-step and grid spacing are 1200 sec and 50 km. We used a E f,θ as in Perrie and Hu (1997). well-tuned secnd-generatin peratinal wave mdel t estimate ( ) 4. CASE STUDY The strm ccurring n 26-27 January is an example f develping synptic strms that pass acrss Newfundland and the Labradr Sea t the Nrtheast. This strm develped high winds ver an extensive regin, as it intensified and prpagated acrss the Grand Banks and the Labradr Sea. Winds reached a maximum f almst 25 m/s n 27
January at 00 UTC. Shrtly thereafter, winds in the suthern Labradr Sea began t significantly weaken while a strng new system pushed acrss the nrthern Labradr Sea with assciated f high winds. Figure (1a). The cean mdel dmain (1200 km X 2240 km) with left side parallel t the Labradr cast. Grid spacing is 20 km. The wave mdel is implemented n the entire map n a 50 km grid. The psitin f buy #23549 is indicated. The lcatin f ice cver fr the perid 26-31 January 1997 is shwn ( ). (1b) Surface currents frm the cean mdel, excluding its Ekman layer, fr 27 January. This is u i, with n ue r U S. In Figure 1b, we give the surface current field, u i, frm the cean mdel fr 27 January 00 UTC, resulting frm the density current as well as the current assciated with hrizntal pressure gradients. This shws the Labradr Current and its extensin alng the shelf edge f the Labradr Shelf, the N.E. Newfundland Shelf and the Grand Banks. The crrespnding Ekman currents u w (withut waves) fr 26-29 January fllw the intensificatin and weakening f the strms as they prpagate acrss the Labradr Sea, driven by winds that increase in magnitude as strms becme strnger, and decrease as the strms weaken. These are shwn in Figure 2, fr the usual Ekman relatins Equatins (7)-(9). These currents have n explicit cnsideratin f waves. Mmentum is passed directly t currents. As mst surface current data are cllected with drifters, we will present a cmparisn with buy drift data frm the LSDCE (Labradr Sea Deep Cnvectin Experiment 1997) campaign, at the Wrkshp. The buy (#23549) is shwn in Figure 1a. The cmparisn is fr 16-21 December 1996. During the perid f the test, wind speeds were reasnably high, ften in excess f 10m/s. When winds are lw, waves are als lw, Ekman currents are small and waves have little effect. We will shw (a) the cean mdel nly, withut Ekman layer r wave-induced currents, (b) with Ekman layer currents, and (c) with mdified Ekman layer currents and wave-induced currents. Thus, it will be shwn that the cean mdel s estimate fr displacement is abut 40% f the buy trajectry, in magnitude. The difference in directin is abut 10. The Ekman currents gives a cmbined cean + Ekman mdel that is abut 80% f the buy trajectry, and the cean + wave-mdified Ekman currents is almst the same displacement as the buy
drift in this five-day perid. Thus, we can shw that wave-current interactins have a majr effect n surface currents. T crss-link the traditinal Ekman layer mdel which has current u w t the wave-mdified Ekman layer with. We will shw the latter as a per centage enhancement n the frmer fr the case study. u E Figure (2). Ekman currents fr 27 January at 00, 06, 12, 18 UTC, withut wave effects. 5. CONCLUSIONS The effects f waves n surface currents are cnsidered using a diagnstic cean mdel and an peratinal wave frecasting mdel. During wave grwth, the part f the wind mmentum transferred t surface waves thrugh Sin exceeds that remved thrugh, and as a result wind stress is less efficient in driving the surface current. When S ds winds and waves are high, wave-induced currents are imprtant even when cean currents are strng. In rapidly develping strms, wave-mdified currents can exceed 40% cmpared t the ttal current field, fr large cean regins. Wave-enhanced surface currents evlve as the strms evlve, fllwing the high wind and wave areas. During quiescent perids, wave-induced currents are cmparatively weak. The wave effect causes a Stkes drift that can be cmparable t the surface currents estimated by the usual Ekman frmulatin. In terms f ttal mmentum atmsphere-cean transfers, parameterizatin f wave-generatin causes reductins in the estimated mmentum transfer t the cean due t the wind input t waves, and enhancements due t wave dissipatin. ACKNOWLEDGEMENTS This study was funded by the Canadian Panel n Energy Research and Develpment (PERD) f Canada under Prjects #534201 n Advanced Wave Mdels and #21758 n Ocean Surface Currents and Waves. High quality wind
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