Continental Self Researc ] (]]]]) ]]] ]]] www.elsevier.com/locate/csr ynamic caracteristics and orizontal transports of internal solitons generated at te Columbia River plume front Jiayi Pan, avid A. Jay epartment of Civil and Environmental Engineering, Portland State University, P.O. Box 751, Portland, OR 97201, USA Received 15 January 2007; received in revised form 17 November 2007; accepted 2 January 2008 Abstract River plume front-generated internal solitons play an important role in te interaction between te plume and coastal waters. Te internal solitons drive a non-armonic velocity field, resulting in a orizontal transport tat carries plume water seaward and redistributes nutrients and sediments. In tis study, we present observations of internal solitons generated at te Columbia River plume front tat separates te new, tidal plume, older plume and coastal waters. Scale analyses suggest tat te plume front-generated internal solitons are igly non-linear waves, and teir dynamic properties do not conform to any weakly non-linear teory. Tus, a ig-order Korteweg de Vries (KdV) teory is used to analyze te internal solitons. Te comparison between teoretical values and cruise data sows tat te ig-order KdV model is muc better tan te weakly non-linear teories for prediction of te soliton dynamic parameters. Based on te model, we develop teoretical and numerical solutions of te soliton-induced upper layer orizontal transport and Lagrangian water parcel transport distance, wic sows tat te water particle drift, during te internal soliton passage, is as far as 1 km, and demonstrates te role of te internal solitons on te excange between te plume and ambient coastal water. Energy fluxes caused by te internal solitons are estimated using te ig-order KdV teory. Te leading soliton fluxes 2.0 10 3 Wm 1 per unit crest lengt, and carries energy of 4.2 10 5 Jm 1. Te total energy carried by te eigt internal solitons is 1.6 10 6 Jm 1, about 70% of te total frontal energy. r 2008 Elsevier Ltd. All rigts reserved. Keywords: Columbia River plume; Internal solitons; Horizontal transports 1. Introduction Te discarge of te Columbia River forms a buoyant coastal plume, wic transports dissolved and particulate load, pyto- and zooplankton, larvae and pollutants into nortwest US coastal waters (Barnes et al., 1972; Grimes and Kingsford, 1996; Hickey et al, 1998). Te evolution and dynamics of te plume are affected by te local wind stress, ambient coastal current, and Eart rotation (Strub et al., 1987; Muncow and Garvine, 1993; Cao and Boicourt, 1986; Cao, 1988; Hickey et al., 1998; Fong and Geyer, 2001). Horner-evine et al. (2008) sowed tat te Columbia plume consists of four distinct water masses: source water at te lift-off point, te tidal, re-circulating and far-field plumes. Small-scale penomena in te Corresponding autor. Tel.: +1 503 725 2960. E-mail address: panj@cecs.pdx.edu (J. Pan). strongly stratified plume can also greatly influence mixing and excanges between te plume and ocean waters (Garcı a-berdeal et al., 2002; Yankovsky et al., 2001; Kay and Jay, 2003a, b; Luketina and Imberger, 1989; Orton and Jay, 2005; Peters and Jons, 2005; Pritcard and Huntley, 2006; Jay et al., 2008). Among tese, internal waves are a major dynamic feature. Previous investigators ave sown tat te internal waves in te plume area ave diverse origins. One category is generated by interaction between soreward propagating tides and te sarp topograpy of te continental slope. Moum et al. (2003) made detailed observations of an internal solitary wave train propagating soreward over te Oregon s continental self. In tis region, igly nonlinear and soreward traveling internal waves were also reported by Stanton and Ostrovsky (1998). Anoter important kind of internal wave, wic we study ere, is generated at te tidal plume front, and propagates 0278-4343/$ - see front matter r 2008 Elsevier Ltd. All rigts reserved. doi:10.1016/j.csr.2008.01.002
2 ARTICLE IN PRESS J. Pan,.A. Jay / Continental Self Researc ] (]]]]) ]]] ]]] Fig. 1. A SAR image taken on July 5, 2004 at 14:37 UTC sowing multiple internal wave packets generated at and traveling seaward from te Columbia River plume front. offsore. Fig. 1 sows a typical Syntetic Aperture Radar (SAR) image of plume frontal internal waves taken on July 5, 2004 at 14:37 UTC. Multiple internal wave packets are seen in tis image. Pan et al. (2007) analyzed a group of internal waves generated at and traveling a distance from te Columbia River front using a SAR image, and extracted te maximum amplitude, pase speed and soliton-induced velocities. Nas and Moum (2005) explored te generation mecanism of internal waves at te Columbia River plume front. uring ebb, te Columbia River plume is pused out of te river mout, and its front forms a gravitational current in te new, tidal plume (Horner-evine et al., 2008). At first, te tidal front propagates faster tan te intrinsic wave and te frontal internal Froude number is 41; a depression forms and is trapped in te front. However, as te front speed decreases below te intrinsic wave speed and te Froude number becomes o1, te depression separates from te front and develops into internal waves troug a fission process. Tese waves ten propagate troug te plume near-field into te far-field. Te frontal Froude number is a critical parameter for generation of internal waves at plume front. Jay et al. (2008) reported tat tere are obvious differences in internal wave generation at upstream fronts between upwelling and downwelling conditions. Under typical downwelling conditions, te tidal plume front is usually broad (up to 5 km) and diffuse on its upstream soutern side. However, under summer upwelling conditions, te upstream front remains sarp and narrow (only 50 200 m wide on its upwind or nortern side) and marks a transition from supercritical to subcritical flow for 6 12 after ig water. Internal wave generation occurs regularly under upwelling conditions. Generation is first seen on te soutern side, and te front proceeds to unzip from sout to nort as internal waves are generated for several ours. Jay et al. suggested tat potential vorticity conservation causes norterly fronts to tin and explains, terefore, te asymmetry of frontal beavior. Internal waves generated at te river plume front are non-linear solitary waves, and are referred to as internal solitons. Tese internal solitons ave considerable impact on frontal processes. Te plume frontal kinetic and potential energies are dramatically canged by te sedding of internal solitons. Te internal solitons also form associated non-linear velocity fields, causing orizontal volume transport and mixing plume water wit te ambient environment. In addition, te induced current can redistribute nutrients and suspended particulate matters. Te study of plume front-generated internal solitons illuminates te interaction of te tidal plume wit te plume near-field, te far-field and ambient. However, te dynamic features and impact of te internal solitons traveling beyond te plume region remain to be investigated. In tis study, we present detailed observations of internal solitons generated at te Columbia River plume front and a dynamic analysis. Te observation data were collected by te River Influences on Self Ecosystems (RISE) project, wic ypotesizes tat waters influenced by te Columbia River plume are more productive tan adjacent coastal waters, especially off Wasington. Section 2 describes vessel measurements. Te relevant internal soliton teory is given in Section 3. Section 4 displays te analysis results. Te volume transport and energy flux caused by te internal solitons are analyzed in Sections 5 and 6, respectively. Section 7 presents te summary and discussion. 2. Observations Te data were collected in te RISE cruise in June 2006 by te R/V Pt Sur. Te R/V Pt Sur carried out rapid surveys using a towed body (TRIAXUS, steerable in 3), in wic was mounted a 911 Seabird conductivity temperature dept (CT) profiler equipped wit sensors for nitrate (N), C, T, pressure, transmissivity and fluorescence and two Ocean Sensors fast-response OS-200 CTs to measure density fine structure. Te ig mobility of te TRIAXUS was used to sample surface waters (from 60 m up to witin 0.5 2 m from te surface, depending on sea state) outside of te sip wake. Te vessel s nearsurface underway data acquisition system (UAS) acquired position, meteorological data, salinity (S), temperature (T), and fluorescence at 3-m dept. Te R/V Pt Sur also carried a pole-mounted 1200 khz acoustic oppler current profiler (ACP), wic measured te current velocity at te dept from about 3 to 25 m. X-band sipboard radar images were stored every minute, and tey are useful in tracking plume fronts and internal solitons. An internal soliton packet was observed on June 8, 2006 around 14:00 UTC, 8.5 after iger ig water at Ft. Canby, WA. Fig. 2a c sow tree sipboard radar images, taken on June 8, 2006, at 13:55, 14:14, and 14:30 UTC,
J. Pan,.A. Jay / Continental Self Researc ] (]]]]) ]]] ]]] 3 respectively. Te images reveal a group of more tan 10 internal solitons traveling seaward, off te Columbia River front. Along te sip transect, te internal solitons are propagating nortwestward at 277.01, and te sip cruised at 90.71 wit te speed of 3.4 m s 1. Tese parameters are listed in Table 1. To determine te soliton propagation speed, we generate a geo referenced internal soliton interpretation map (Fig. 3), in wic te solid and das lines correspond to te internal solitons and plume front at te time 13:55 and 14:30 UTC, respectively. From tese internal soliton locations, we measure te soliton pase speeds, wic are listed in Table 1. Te leading internal soliton is propagating at 1.1 m s 1. Te front is traveling at 0.5 m s 1, muc slower tan te internal solitons. Fig. 4 illustrates te cruise transect and te internal solitons observed at 14:14 UTC by te sip radar. Te density profiles and te ACP observations are sown in Fig. 5. Te density profiles (Fig. 5a), measured by te Seabird CT onboard te TRIAXUS towfis, suggest te existence of, but do not fully resolve te solitons. Figs. 5b sows te orizontal velocity normal to te wave crests wit te positive values in te internal wave propagation direction of 277.01. Te positive value Table 1 Soliton propagation and sip navigation parameters Soliton direction (1) Sip direction (1) Sip speed (m s 1 ) 277.0 90.7 3.4 Fig. 2. Sipboard X-band radar images taken on June 8, 2006 at 13:55 UTC (a), 14:14 UTC (b) and 14:30 UTC (c) around 8.5 after iger ig water at Ft. Canby, WA. Fig. 3. Georeferenced internal soliton interpretations. Te lines sow te internal waves and fronts at 13:55 UTC (A1-8) and 14:30 UTC (C1-8) in solid and das lines, respectively. Te circle and asterisk represent te sip locations at 13:55 and 14:30 UTC.
4 ARTICLE IN PRESS J. Pan,.A. Jay / Continental Self Researc ] (]]]]) ]]] ]]] indicates tat wave-induced particle orizontal movements are in te direction of te wave propagation. Fig. 5c gives a velocity plot at te 5-m dept, and velocity is lowpass filtered using five-point running mean (30 m) to suppress te noise. Te velocity peaks sow te internal solitons. Te ACP beam eco intensity (Fig. 5d) reveals te igly non-linear soliton structure and a total of about 17 solitons traveling away from te front. However, only te leading eigt solitons are clear in te radar images, density profiles and velocity sections, so only tese solitons are analyzed. Fig. 4. Te cruise transect (tick line) and te internal wave crests at 14:14 UTC. Because te solitons were observed from a moving vessel, te observed soliton scales must be corrected for te vessel navigation speed. Te vessel and te solitons were traveling in almost opposite directions wic makes te measured soliton scales less tan te actual values. Te correction formula is given by c L ¼ L 0 cos j 1, (1) V cos j were L represents te corrected soliton scale, L 0 te soliton scale measured by te vessel, c te soliton speed determined from sip radar, V te sip speed and j is te angle between te vessel navigation and te soliton traveling (ere j ¼ 186.31). After te soliton scale correction, we measure te soliton widts, and maximum amplitude and velocity using ACP eco and velocity sections and CT density profiles. Te procedure is as follows. (1) Using ACP eco and velocity sections troug an internal soliton, we can detect te interface dept between upper and lower layers at te soliton center. Te undisturbed interface dept is obtained from te ambient density profile. Te difference of interface depts between te soliton center and te undisturbed ambient water gives te soliton maximum amplitude. Alternatively, soliton maximum amplitude is also deduced from density profiles by using te interfacial dept variations witin te soliton. Te final maximum Fig. 5. TRIAXUS CT and sip-mounted ACP measurements. (a) ensity profiles by CT, (b) ACP crest-normal orizontal velocity, (c) ACP crest-normal orizontal velocity at 5-m dept and (d) ACP beam eco intensity.
J. Pan,.A. Jay / Continental Self Researc ] (]]]]) ]]] ]]] 5 Table 2 Sip cruise data and teoretical predictions Z 0 (m) (m) c (m s 1 ) U (m s 1 ) Q t (m 3 s 1 m 1 ) (m) E (10 5 Jm 1 ) F (10 3 Wm 1 ) n Soliton 1 Observed 8.5 100.0 1.1 0.73 5.1 145.4 4.2 2.0 0.843 Predicted 99.4 0.97 0.71 4.6 187.1 Soliton 2 Observed 7.3 104.9 1.0 0.55 4.7 138.8 3.2 1.5 0.798 Predicted 102.2 0.95 0.61 3.7 142.5 Soliton 3 Observed 6.6 106.4 1.0 0.52 4.8 158.3 2.7 1.2 0.767 Predicted 104.7 0.93 0.55 3.2 123.3 Soliton 4 Observed 3.3 136.6 1.0 0.29 2.2 85.5 0.8 0.3 0.572 Predicted 132.9 0.85 0.27 1.4 62.9 Soliton 5 Observed 5.2 111.0 1.0 0.47 3.4 116.4 1.8 0.7 0.697 Predicted 112.3 0.90 0.43 2.4 93.4 Soliton 6 Observed 4.8 119.0 0.9 0.42 2.9 124.7 1.5 0.6 0.674 Predicted 115.4 0.89 0.40 2.2 86.3 Soliton 7 Observed 4.3 127.4 0.9 0.39 2.5 101.1 1.3 0.5 0.643 Predicted 120.0 0.88 0.36 1.9 78.0 Soliton 8 Observed 3.4 135.6 0.9 0.29 2.2 94.3 0.9 0.3 0.579 Predicted 131.3 0.85 0.28 1.4 64.3 amplitude is te mean of te amplitudes from ACP eco and velocity sections and te density profiles. (2) Wit te derived soliton maximum amplitude, we can determine te alf-amplitude dept. Te ACP eco and velocity sections give te outlines of te soliton sapes, and based on tese, we measure te soliton widts at te alf-amplitude dept from te two sections. Te widts are referred to as te soliton alf-widt. Te obtained alf-widt is te average of te widts from te two sections. (3) Using te ACP velocity observations, we obtain te velocity profile in te upper layer at te soliton center, and te maximum velocity is derived by averaging te velocity profile. Table 2 lists te maximum amplitude, alf-widt and maximum velocity of te internal solitons derived from te measurements using te above procedure. 3. Internal soliton teory 3.1. Scale analysis Normally, tere are tree teoretical regimes tat describe te beavior of internal solitons: sallow water (Benjamin, 1966; Benney, 1966), finite-dept (Josep, 1977; Kubota et al., 1978) and deep water (Benjamin, 1967; Ono, 1975). Te criteria for discriminating te tree categories depend on te ratio of te water dept to te soliton orizontal scale (Liu et al., 1985; Zeng et al., 1993, 1995). (1) For sallow water teory, scale parameters satisfy L H 1; H Oð1Þ; Z 0 L 2 Oð1Þ. (2) 3 H (2) In finite-dept case, te parameters are L 1; H 1; L H Oð1Þ; Z 0 L Oð1Þ. (3) 2 (3) For deep water teory, te parameters are governed by L H! 0; L 1; Z 0 L ¼ Oð1Þ, (4) 2 were L is te soliton caracteristic lengt, Z 0 te maximum amplitude, te upper layer dept and H is te water dept. For tis soliton case, te environmental and soliton parameters are given by =O(7 m), Z 0 =O(8 m), H=O(140 m), and L=O(120 m). Terefore, we ave ¼ 0:05, (5) H Z 0 L 2 ¼ 0:04, (6) 3 H Z 0 L ¼ 19:6, (7) 2 and L ¼ 0:9. (8) H Eqs. (5) and (6) suggest tat te solitons are not sallow water waves (Eq. (2)). By Eq. (7), te finite-dept criterion does not old (Eq. (3)). Eqs. (7) and (8) sow tat criterion 3 is not satisfied (Eq. (4)). Furtermore, criterion 1 (Eq. (2)) requires tat H. Criteria 2 and 3 (Eqs. (3) and (4)) suggest ðz 0 L= 2 ÞOð1Þ and ðl=þ 1, and lead to Z 0. Tis means tat all te tree criteria require tat te
6 ARTICLE IN PRESS J. Pan,.A. Jay / Continental Self Researc ] (]]]]) ]]] ]]] upper layer is deep and te layer dept sould be greater tan te soliton amplitude. However, te internal solitons generated by a river plume front ave large amplitudes relative to te surface layer dept. Te scale analysis sows tat te solitons in tis case do not belong to any of te above tree weakly non-linear regimes. For large-amplitude internal solitons in a sallow upper layer dept, non-linear effects are very strong, and, terefore, te weakly non-linear assumption cannot fully describe te actual situation. Tus, we use a igly non-linear approac to te soliton analysis. 3.2. Internal soliton teory Two-dimensional internal waves may be described by a stream function c. Te stream function is a product of linear wave speed and te vertical displacement: c=(c 0 U)I(x, z), were x=x ct, c 0 and c are te linear wave and soliton speeds, respectively, and U is te background velocity. Te vertical displacement as a separable form I(x, z, t)= Z(x)f(z). Te background velocity sear is very small, so tat we can neglect ambient sear (Fig. 5b); terefore, te vertical structure function (f) is a solution of a linear Sturm Liouville equation (Apel et al., 1985; Fu and Holt, 1984) wit te normalization f(z 0 ) ¼ 1, were z 0 is te dept corresponding to te maximum value of f. d 2 f dz 2 þ N 2 f ¼ 0 and fð0þ ¼fð HÞ ¼0. (9) 2 ðc 0 UÞ In te orizontal dimension, a ig-order Korteweg de Vries (KdV) equation is used (Stanton and Ostrovsky, 1998; Ostrovsky and Stepanyants, 1989; Grimsaw et al., 2002). It is given by qz qt þðc 0 þ az þ a 1 Z 2 Þ qz qx þ b q3 Z ¼ 0, (10) qx3 were a ¼ 3 ðc 0 UÞ R 0 H ðqf=qzþ3 dz R 2 0, (11) H ðqf=qzþ2 dz b ¼ 1 2 ðc 0 UÞ R 0 H f2 dz R 0 H ðqf=qzþ2 dz, (12) wit a 1 3ðc 0 UÞ 7 2 ðh Þ 2 8 ðh 2Þ2 3 þðh Þ 3, (13) H were is te upper-layer dept, defined, ere, as te dept at wic appears te maximum buoyancy frequency. Te ig-order KdV Eq. (10) as a soliton solution Z ¼ a a 1 n 2 x ct tan þ d tan were n and d are parameters satisfying dðnþ ¼ 1 4 ln 1 þ n 1 n x ct i d, (14) (15) and rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 24a 1 b ¼ a 2 n 2. (16) Te soliton pase speed is given by c ¼ c 0 a2 n 2. (17) 6a 1 Te soliton-induced velocity is derived from te soliton stream function. Te orizontal velocity u is obtained by u ¼ qc qz ¼ðc 0 UÞZ qf qz. (18) Te soliton alf-widt ( ) is derived from Eq. (14) r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! 1 ¼ 2atan 2 tan 2. (19) d 4. Analyses 4.1. Vertical structure function Te undisturbed density profile is sown in Fig. 6a, wic is derived from te data p displayed in Fig. 5a. Te buoyancy frequency (N ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðg=rþðqr=qzþ) is calculated and sown in Fig. 6b (solid line). Wit te N profile, we solve numerically te eigenvalue Eq. (9) using te MATLAB partial differential equation toolbox. We only consider te first mode because te pase speed of iger order modes is muc slower tan tat of te first mode and less tan te frontal speed. Tus, only te first mode can escape from te front. A tree-parameter buoyancy model is also sown in Fig. 6b (das line, Vlasenko, 1994) and was used for te simulation of te stratification in te Columbia River plume region by Pan et al. (2007). Fig. 6c displays te numerical solution to f for te first mode in solid line. Te das line sows tat analytic solution to te eigenvalue equation using te tree-parameter buoyancy frequency model. Fig. 6c suggests tat te two solutions are very close, and te tree-parameter model is useful for description of te buoyancy profile in te Columbia River plume region. From te first mode eigenvalue, c 0 U is solved as 0.65 m s 1. Te vertically averaged background velocity (U) is 0.09 m s 1, and, terefore, te first mode linear wave speeds (c 0 ) is 0.74 m s 1. 4.2. Soliton alf-widt, maximum amplitude, pase speed and maximum velocity Te parameters a and b are calculated as 0.105 s 1 and 61.93 m 3 s 1, respectively, from te soliton structure function (f) using Eqs. (11) and (12). Te value of a 1 is 0.0045 m 1 s 1 from Eq. (13) wit ¼ 7.5 m and water dept H ¼ 137.5 m. Tese values are used to calculate te soliton alf-widt, maximum amplitude and velocity and pase speed. From Eq. (14), te soliton maximum
J. Pan,.A. Jay / Continental Self Researc ] (]]]]) ]]] ]]] 7 Fig. 6. Te background density profile (a), te observed buoyancy frequency (solid line) and te tree-parameter buoyancy model (das line) (b) and numerical solution of te vertical structure function f (solid line) and analytic solution of f from te tree-parameter buoyancy model (das line) (c). amplitude is Z 0 ¼ a a 1 n tan s. (20) From Eq. (21), we obtain te soliton-induced maximum velocity, u 0 qf u 0 ¼ ðc 0 UÞZ 0 qz. (21) In te upper layer, qf/qz is almost a constant as 0.1281 m 1. Fig. 7a c sow te teoretical relationsips between te soliton maximum amplitude (maximum velocity, pase speed) and alf-widt for te ig-order KdV, KdV, finitedept Josep and deep-water Benjamin and Ono equations. Te observed soliton parameters are displayed in Fig. 7 as asterisks. Clearly, te ig-order KdV teory is closest to te measured soliton maximum amplitude and velocity, pase speed, and alf-widt data. Compared wit te ig-order KdV teory, te oter teories suggest lower maximum amplitude, maximum velocity and soliton pase speed for te same soliton widt. Only te igorder KdV teory can describe large-amplitude internal solitons (relative to upper layer dept) in te plume region wit a sallow surface layer. For te leading soliton, te maximum amplitude Z 0 ¼ 8.5 m, from wic and Eq. (20), n ¼ 0.843. Terefore, te alf-widt of te leading soliton is obtained from Eq. (19) as 99.4 m, te soliton-induced maximum velocity in te upper layer is calculated using Eq. (21) as 0.71 m s 1, and its pase speed is derived from Eq. (17) as 0.97 m s 1, wereas te measured soliton alf-widt, maximum velocity and pase speed are 100.0 m, 0.73 m s 1 and 1.1 m s 1, respectively. For te leading soliton, te difference between te predictions and te measurements for alf-widt, maximum velocity and pase speed are 0.6%, 3% and 12% of teir values, respectively. Te results for all te solitons are listed in Table 2. Water dept variability can affect te teoretical interpretation on te soliton sape, since we use te constant water dept. However, in tis case water dept variations are small for tese soliton positions, ranging from 136.3 to 138.6 m. Te relationsip between maximum amplitude and soliton widt canges slowly wit dept (Fig. 8), wic suggests te constant water dept would not cange te interpretation results. Fig. 8 also sows tat, as te water dept increases, te soliton maximum amplitude increases for any given soliton alf-widt. 5. Soliton-induced volume transport Te sallow surface layer of plume fronts generates depression solitons. Associated wit te depression solitons, te near-surface velocity is in te same direction as tat of te soliton travel, wile te lower level velocity is in te opposing direction. Tus, te orizontal transport in te upper layer carries plume water away from te area, resulting in te orizontal mixing between te plume and te adjacent water. As te solitons travel away from te front, te upper layer orizontal transport spreads te tidal
8 ARTICLE IN PRESS J. Pan,.A. Jay / Continental Self Researc ] (]]]]) ]]] ]]] Fig. 7. Te teoretical relationsips of soliton amplitude (a), velocity (b) and pase speed (c) wit te alf-widt for te ig-order KdV, KdV, finitedept Josep equation, and deep water Benjamin and Ono equation. Te vessel measurements are in asterisks. Fig. 8. Te relationsip between internal soliton amplitude and widt at water depts of 120.0, 136.3, 138.6 and 160.0 m. plume water into te plume near-field or far-field, or even into ambient waters, depending on circumstances. Tis transport is analyzed as follows. Te cross-frontal velocity induced by an internal soliton generated at te plume front is given by u ¼ u 0 2 tan d x ct tan þ d tan x ct i d. (22) Te upper layer velocity average is written as Z 0 Ū ¼ 1 udz ¼ ðc UÞZ 0 z 0 z 0 2z 0 tan d x ct tan þ d x ct i tan d. (23) Let te time origin be suc tat a soliton peak is at x ¼ 0, and (c U)Z 0 /(2 tan d) be represented by U 0. Te average
J. Pan,.A. Jay / Continental Self Researc ] (]]]]) ]]] ]]] 9 transport rate (Q t ) in te upper layer (from z 0 to 0) over te period from /c to /c caused by a single soliton is derived as Q t ¼ 1 Z =c Ūðz 0 þ ZÞ dt 2 =c =c Z =c ¼ U n 0 tan ct 2 =c =c þ d tan ct i d Z 0 ct tan 2z 0 tan d þ d tan ct i 2 d dt ¼ U 0 ln cosð = þ dþ U 0 Z þ 0 c 2 cosð = dþ 4 jz 0 jtan d Z =c tan ct þ d tan ct i 2dt. d (24) =c Tis net orizontal transport results from te non-linearity of te internal soliton. For linear internal waves, in contrast, te velocity field is armonic, so tat te integral of te velocity wit respect to time over te wave period is zero (Pillips, 1977). Suc linear waves cannot generate net orizontal transport in te upper and lower layers. Using Eq. (24), we calculate te transport rate caused by te internal solitons. For te leading soliton, te predicted Q t is 4.6 m 3 s 1 m 1 (Table 2). Tis is comparable wit te Q t (over te period from /c to /c) deduced from sip ACP data, 5.1 m 3 s 1 m 1. For all solitons, te observed transports are 25 60% greater tan te teoretical ones, even toug te maximum measured velocities are less tan te predictions. Tis migt result from te uneven background velocities, or te differences between te measurement and te predictions of oter parameters. In te upper layer, an internal soliton forces a water parcel to move forward during te soliton passage due to te associated velocity field. Because te soliton travels faster tan te water particle, eventually, te soliton will leave te water parcel beind, but te water parcel in te upper layer will be transported a distance seaward. Tus, te wole upper layer is moved forward by te passage of a soliton packet. In order to analyze te transport distances of water parcels in te upper layer caused by multiple solitons, a Lagrangian metod is employed. From Eq. (22), te differential form of te velocity equation is given by x nþ1 ¼ x n 1 þ u 0t tan d tan x n ct n þ d tan x n ct i n d, (25) were x n 1, x n and x n+1 are successive water parcel positions. Considering a water parcel initially at x ¼ 2 wit te initial time t 0 ¼ 0, we calculate te transport distance for te water parcel caused by te solitons. For te leading soliton, te transport distance is 187.1 m, and te total transport distance is 837.8 m for te eigt solitons, wereas te total transport distance deduced from ACP measurements is 964.5 m. In tis soliton packet, tere are 17 solitons. Tus, te actual water parcel transport distance is greater tan 1 km. Also, more tan one packet of internal solitons is sometimes spawned and released from te front (cf. Fig. 1). Tis will cause even greater transport of frontal waters. Normally, te tidal plume is about 10 30 km. Tus, te transport caused by te internal solitons can expand plume radius by te 3 10%, and increases te plume area by 6 20%. Te spreading of plume waters intensifies te excange between tidal plume and oter coastal waters and enances te frontal mixing. 6. Energy flux caused by te internal solitons Te internal solitons transport energy, gained from te front, beyond river plume waters. Te energy flux caused by te internal waves may provide energy for mixing processes. In te previous sections, te internal solitons reported in tis study are well described by te ig-order KdV equation. Te teory is furter used ere to derive te soliton energy flux. Moum et al. (2007) indicated tat te potential and kinetic energies are generally equal for internal solitary waves. From te kinetic energy expression, we ave te total energy per unit crest lengt over te distance from +ct to +ct associated wit an internal soliton Z Z 0 E ¼ Z 2 dx rðc 0 UÞ 2 qf 2 dz, (26) H qx were r is te undisturbed density. For an internal soliton, te group velocity is same as te pase speed (Zeng et al., 1995; Moum et al., 2007). Tus, te energy flux caused by te internal soliton is written as F ¼ cei ¼ c Z Z 0 Z 2 dx rðc 0 UÞ 2 qf 2 dz. (27) 2 H qx Using Eqs. (26) and (27), we can estimate te total energy and energy flux of te internal solitons. Te estimated energy flux for te leading soliton is 2.0 10 3 Wm 1. Te total energy per unit crest lengt for te soliton is 4.2 10 5 Jm 1, and te total energy for te eigt solitons is 1.6 10 6 Jm 1. Table 2 lists total energy and energy fluxes for all te eigt solitons. Compared wit a packet of typical internal solitons generated at and traveling off te Columbia River plume front in te study by Pan et al. (2007), in wic te energy flux of te leading soliton was 417.4 W m 1, tis group is more energetic. To understand te role of te internal soliton energy flux in plume frontal energetics, we estimate te total frontal energy (E f ), wic is a summation of te frontal kinetic (KE f ) and available potential (APE f ) energies (E f ¼ KE f +APE f ). Te frontal region is defined in Fig. 5a by te two parallel lines. Te frontal available potential energy is estimated using te metod suggested by Huang (2005), in wic te reference density profile is selected as te orizontal average. Using tis metod, te KE f and APE f in a unit frontal lengt are calculated as 6.9 10 5 and 1.4 10 6 Jm 1, respectively.
10 ARTICLE IN PRESS J. Pan,.A. Jay / Continental Self Researc ] (]]]]) ]]] ]]] Tus, te total frontal energy E f ¼ 2.1 10 6 Jm 1. Te energy of te internal solitons originates from te front, and it is seen tat, te internal solitons fluxed away 70% of te total frontal energy. Tus, internal wave generation plays an important role in te frontal energy dissipation processes tat begin wen te plume left te estuary. 7. Summary and discussion River plume water forms a region of strong stratification favorable for te internal soliton generation and propagation. An internal soliton packet generated at te Columbia River plume front was observed during te RISE cruise on June 8, 2006. Using te observation data, we derive te internal soliton alf-widt, te maximum amplitude and velocity, and pase speed. Scale analyses suggest tat tese internal solitons do not conform to te weakly non-linear teories suc as te sallow water KdV equation, finitedept Josep and Kubota equation, and deep-water Benjamin and Ono equation, because te plume frontgenerated internal solitons ave large amplitude compared wit te upper layer dept and are propagating in te sallow pycnocline. Te ig-order KdV equation is used to analyze te soliton observations. It is considerably better in predicting te soliton properties of alf-widt, maximum velocity and pase speed tan te weakly non-linear teories. For te leading soliton, te difference of between te teoretical predictions and te measurements for alfwidt, maximum velocity and pase speed are 0.6%, 3% and 12% of teir values, respectively. Tese igly non-linear internal solitons induce nonarmonic velocity fields, resulting in orizontal transports in te upper layer and causing te redistribution of low salinity waters, suspended sediments and biological properties in te plume area. Employing te ig-order KdV teory, we derive te teoretical orizontal transport in te upper layer. For te leading soliton, te transport rate per unit crest lengt is 4.6 m 3 s 1 m 1, close to te vessel ACP data deduced value, 5.1 m 3 s 1 m 1. Te water parcel transport distance is solved by using te Lagrangian metod, and tat te total transport distance caused by te observed solitons is more tan 1 km, suggesting plume radius and area were expended by 3 10% and 6 20%, respectively. Tis process will enance te interaction between te plume and ambient coastal water. Te expression of te energy flux per unit crest lengt is derived in terms of te ig-order KdV teory; it gives 2.0 10 3 Wm 1 for te leading soliton. Te total energy carried by te eigt soliton is as ig as 1.6 10 6 Jm 1. Compared wit te total energy in te frontal region sown in Fig. 5a, te internal soliton energy flux is an important factor dissipating te frontal energy. Front-generated internal solitons appear on about 25% of available SAR images for upwelling and neutral conditions and a few SAR images for downwelling conditions. Considering tat only about alf of te SAR images portray te plume during te appropriate tidal pase for soliton propagation to be visible, we conclude tat internal soliton generation is a frequent plume penomenon. Because nutrient-ric ig salinity waters are close to te surface under upwelling conditions wen soliton generation is most prevalent, it is likely tat mixing induced by tese solitons plays a role in te ig zooplankton becomes observed around te margins of te plume (Jay et al., 2008). Acknowledgments Te study is supported by te National Science Foundation (project RISE-River Influences on Ecosystems, OCE 0239072), and te Bonneville Power Administration and National Oceanic and Atmosperic Administration (NOAA) Fiseries (Project: Ocean Survival of Salmonids). We tank Captain Ron L. Sort of te R/V Pt Sur and Marine Tecnicians Stewart Lamberdin and Ben Jokinen for teir superb support of in situ data collection. Te SAR images were provided by Compreensive Large Array-data Stewardsip System (CLASS) of NOAA. Te autors are grateful to te anonymous reviewers for teir valuable suggestions and comments. References Apel, J.R., Holbrook, J.R., Liu, A.K., Tsai, J.J., 1985. Te Sulu Sea internal soliton experiment. Journal of Pysical Oceanograpy 15, 1625 1651. Barnes, C.A., uxbury, A.C., Morse, B.A., 1972. Circulation and selected properties of te Columbia River effluent at sea. In: Pruter, A.T., Alverson,.L. (Eds.), Te Columbia River Estuary and Adjacent Ocean Waters. 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