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How does the El Niño-generated coastal current propagate past the Mendocino escarpment? Allen, Susan E.; Hsieh, William W. May 29, 1997

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JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 102, NO. Cll, PAGES 24,977-24,985, NOVEMBER 15, 1997 How does the E1 Nifio-generated coastal current propagate past the Mendocino escarpment? Susan E. Allen and William W. Hsieh Department of Earth and Ocean Sciences, University of British Columbia, Vancouver, Canada Abstract. During an E1 Nifio, an internal coastal Kelvin wave bore propagates poleward along the west coast of North America, leaving behind a steady anomalous coastal jet. A nonlinear, two-layer, f plane, shallow-water model is used to determine the amplitude change of the steady coastal current over an escarpment. As the E1 Nifio-generated warm coastal current passes the Mendocino escarpment off northern California, its amplitude is marginally enhanced. In contrast, for a cold coastal current, the amplitude will be reduced north of the Mendocino escarpment. When the Kelvin wave bore travels over a depth increase, the amplitude change is predicted to be much larger than over a depth decrease (as in the case of the Mendocino escarpment). This model is also applicable to bottom water flowing equatorward along a western boundary. In this case, much larger amplitude changes are found. Introduction E1 Nifio, a prominent atmosphere-ocean interaction centered around the tropical Pacific, is characterized by the appearance of anomalously warm waters in the eastern and central tropical Pacific on an interannual timescale [Philander, 1990]. These tropical E1 Nifio ef- fects spread to the extratropical northern hemisphere by two main mechanisms: (1) The warm water prop- agates poleward along the west coast of North Amer- ica by coastally trapped Kelvin waves [Pares-Sierra and O'Brien, 1989], and (2) the atmosphere responds to the heating in the central equatorial Pacific by developing a series of alternating high- and low-pressure cells in the northern hemisphere known as the atmospheric telecon- nection [Wallace and Gutzler, 1981]. The coastal signal is clearly seen in both the tem- perature and the sea level data [Enfield and Allen, 1980] and can be explained dynamically as a first-mode baroclinic Kelvin wave. The strength of the anoma- lies, for example, a change in the surface temperature of approximately 5øC and associated alongshore cur- rents of 40 cm/s in October 1982 off Oregon [Huyer and Smith, 1985], indicates a strongly nonlinear phe- nomenon. With the observed sharp temperature rise occurring over days, the signal can be dynamically mod- eled as an internal Kelvin wave bore. In the alongshore direction the signal has a sharp drop followed by a long flat region of depressed pycnocline height (Figure 1). Copyright 1997 by the American Geophysical Union. Paper number 97J C01583. 0148-0227/97/97JC-01583509.00 This region of small spatial gradients corresponds to the steady warm coastal current which is left behind after the passage of the initial bore. Offshore the expected shape is the exponential decrease over the Rossby ra- dius, typical of Kelvin waves. This model is consis- tent with a nonlinear wave balance alongshore and a geostrophic balance cross shore. The northward traveling warm coastal current asso- ciated with the Kelvin wave encounters the steep Men- docino escarpment off California (about 41øN), where the ocean depths abruptly changes from about 4000 to 3000 m further north. There has been a series of in- triguing theoretical studies on Kelvin wave transmis- sion or the adjustment to steady flow over escarpments and ridges [Johnson, 1985; Gill et al., 1986; Allen, 1988; Killworth, 1989a, b; Johnson and Davey, 1990; Johnson 1990; Willmort and Grimshaw, 1991; Johnson, 1993; Willmort and Johnson, 1995; Allen, 1996]. These stud- ies and the numerical study of Wajsowicz [1991] have shown that low-frequency flow over a sharp depth de- crease in the alongstream direction generates double Kelvin waves [Rhines, 1969; Longuet-Higgins, 1968a, b; Willmort, 1984] that propagate along the depth de- crease. An incoming Kelvin bore, in the case of a sharp depth decrease in the northern hemisphere, will generate an offshore propagating, long double Kelvin mode as shown in Figure 2a, with possible significant energy loss for the incident Kelvin bore. Since the warm coastal current was observed during an E1 Nifio north of the Mendo- cino escarpment [Huyer and Smith, 1985], the coastal current is clearly capable of propagating past the es- carpment. Our first objective is to theoretically esti- mate how the steady coastal current formed after the 24,977 24,.978 ALLEN AND HSIEH: ESCARPMENT EFFECT ON EL NIfO COASTAL CURRENT North Sea surface .•• Pycnocline Bore Figure 1. Sketch of the alongshore structure of the coastal Kelvin wave bore. passage of the Kelvin wave bore is affected by the Men- docino escarpment. No attempt is made to analyze the initial interaction between the bore and the escarpment. Furthermore, the studies concerning escarpments list- ed above showed clear asymmetry; the situation for a Kelvin wave encountering a steep depth increase in the alongshore direction (Figure 2b) is totally different from the steep decrease shown in Figure 2a. In Figure 2b the long double Kelvin wave creates a singularity in the lin- ear solution at the coastline (because the wave propa- gates shoreward) [Johnson, 1985], so frictional and non- linear effects become important [Allen, 1988]. Hence the main energy loss for the incident Kelvin bore in Figure 2b is from friction rather than from the offshore propagating long double Kelvin mode in Figure 2a. Our second objective is to study the difference between an alongshore depth increase and an alongshore depth de- crease in influencing the steady coastal current left after the passage of the Kelvin bore. The large body of research on scattering over ridges and escarpments cannot directly answer these ques- tions. Most of the previous research has considered the linear homogeneous problem [Johnson, 1985; Gill et al., 1986; Johnson, 1990; Willmort and Grimshaw, 1991]. Stratified flows were considered in the linear regime [Killworth, 1989a, b; Johnson, 1993; Willmort and Johnson, 1995]. Only Allen [1988] considered non- linear flow, but all her theoretical results were for ho- mogeneous flow. Wajsowicz [1991] used the Bryan- Cox model including the advection terms but consid- ered cases well within the linear regime. As the Kelvin wave generated by an E1 Nifio can have a large pycno- cline displacements compared to the average pycnocline depth [ttuyer and Smith, 1985], we consider a nonlinear bore. In the linear limit, our results agree with those of Wajsowicz [1991]. We will proceed in the spirit of Killworth [1989a, b], who considered Kelvin wave transmission over a ridge, and use a "black box" approach. The details in the interaction region near the escarpment will not be con- sidered. Our problem is somewhat simpler than the ridge case. We either have flow over a depth decrease (step up, case 1) for which frictional effects hould re- main small and a momentum balance can be used, or we have flow over a depth increase (step down, case 2) for which the flow is confined near the coast and a mass balance can be used. The ridge, of course, includes a step up and a step down, hence combining the com- plexities. The simpler geometry allows the inclusion of the nonlinear advection terms, important for the large observed pycnocline displacements. Flow over a Steep Depth Decrease The problem of long, nonlinear Kelvin waves was considered by Bennett [1973]. The equations indicate that such waves must steepen, and he postulates that they would eventually form a bore. This result is con- sistent with experiments on similar nonrotating flows such as those by Wood and Simpson [1984]. For long waves, Bennett [1973] shows that the cross-shore bal- ance is geostrophic, and providing potential vorticity is conserved, the cross-shore shape of the long-shore ve- locity field and the sea surface elevation is the classic Kelvin wave exponential. As the wavelength becomes much larger than the (bareclinic) Rossby radius, the cross-shore velocity introduced by the nonlinear terms vanishes. Consider a Kelvin wave bore (Figure 1) propagat- ing northward past an escarpment into shallower water (Figure 2a) in a two-layer stratified fluid in the north- ern hemisphere (Figure 3). After the passage of the bore, the flow tends toward a steady state. As there is little alongshore variation and no cross-shore flow, all the advection terms in the momentum equations of the shallow water model are 0. In this steady state region south of the escarpment, assuming potential vorticity is conserved, the disturbance has an offshore interface displacement with the classic Kelvin wave shape, where y increases offshore and the bareclinic Rossby ra- dius is R - {g'hH/[(H + h)f•']} •/•', with H the upper a) Shallow Deep Shallow b) Figure 2. Sketch showing (a) case 1, the incoming Kelvin wave bore, the outgoing bore and the offshore propagating, long double Kelvin mode and (b) case 2, just the bores, as no offshore propagating long double Kelvin mode is excited. The shaded region represents the "black box" around the escarpment where evanes- cent waves, nonlinear and frictional effects occur. Our analysis i  for the near-steady currents formed after the passage of the Kelvin wave bore. ALLEN AND HSIEH: ESCARPMENT EFFECT ON EL NI•IO COASTAL CURRENT 24,979 Figure 3. Vertical section showing the variables in the two-layer model. layer depth, h the lower layer depth, •/• the interface displacement at the coast, g' the reduced gravity and f the Coriolis parameter. The shape is consistent with Bennett's [1973] results for a nonlinear Kelvin wave. The assumption of potential vorticity conservation rests on assuming weak dissipation and that the changes were not due to the advection of a new water mass but rather a deepening of the already existing one. For a pure baroclinic signal the surface displacement related to (1) is g'h ( - -,• g(• + •) •xp (-•/n), (•.) with the lower-layer, alongshore velocity, g'H u- Ri(H + n) 'rl•' exp (-y/R), (3) and the upper-layer, alongshore velocity, U - -hu/H, (4) to order 9'/9, assumed small [LeBlond and Mysak, 1979, sections 16 and 24]. For now, ignore the region near the escarpment, shown shaded in Figure 2a, and consider the flow further north. If a distance of 5 Rossby radii is far enough from the escarpment hat no lower-layer fluid from south of the escarpment has been advected that far, then the potential vorticity of the fluid will not have changed (where small dissipation has been as- sumed away from the escarpment). Therefore we can assume a cross-shore shape similar to the disturbance south of the escarpment but with a smaller baroclinic Rossby radius R2 due to the shallower depth. Inter- action with the escarpment also excites a barotropic Kelvin mode and a long double Kelvin mode out along the escarpment. So northward of the escarpment, where r/c and r/t are the baroclinic and barotropic com- ponents, respectively, and a is the barotropic Rossby radius, with a 2 - g(H + h2)/?. The other variables are given by -g' h2 g(n + n•.) TM •P (-•/n•) H+h2 + n• ,•, •p (-•/•), (•) g'H • • (- y/ S• ) +.("+ •) ah•.f r•, exp (-y/a,), (7) U __ (8) to order 9'/g [LeBlond and Mysak, 1979, sections 16 and 24]. Now consider the flow in the interaction region near the escarpment. In this region the initial bore generates transient waves (evanescent Poincar• and topographic waves) and a long double Kelvin wave along the escarp- ment [Johnson, 1990] which settles down to a steady current flowing offshore. Along the coast the initial bore generates transient Kelvin waves and a long Kelvin wave which settles down to a steady coastal current. Instead of considering the details of the complicated transient and steady flow, we will restrict attention to the mo- mentum balance along the wall. The first momentum equation for the upper layer is ou ou ou •( ot + v y•-• +v• - Iv - -g• + •,,, (•) 24,980 ALLEN AND HSIEH: ESCARPMENT EFFECT ON EL NI•IO COASTAL CURRENT and for the lower layer is Ou Ou Ou .f v -- -g - + Ft (10) ot + + ' where F represents the frictional effects. Consider a few inertial periods after the head of the bore has passed, with the flow having reached an almost steady state. Along the wall v, V _-- 0, as there can be no flow through the wall. Thus along the wall, ou oi_ ou U•-• + g Oa• - at + F=, (11) Ou O( g, O,• _ Ou u •-• + g •-• + Oa• - Ot +Ft. (12) These equations can be integrated across the escarp- ment from a position-X0 several (say 5) Rossby radii upstream to a position Xz several (say 5) Rossby radii downstream. As the left-hand sides are exact differ- entials, if we assume the right-hand sides to be 0, the integrals can be evaluated without knowing the details of the flow in the region of the escarpment. The integral of the left-hand side of (11) is xo 2 Xl -Xo (•3) which can be expanded by substituting the expressions (2), (4), (6) and (8). The integral of the left-hand side of (12) is •?c - •7,,, 1- 2hh2(H- •?,,,) + 0 • (16a) saa ] r/c rho 1 h 2(1-rho/H ) ' (16b) wi•h Ah = h- h• and assuming H/h • I1 - n/HI that is, •ocannot approach H. Note that for small upper layer depth, even for relatively large pycnocline displacements, the change over the escarpment is small. The last term on the right-hand side of (16b) with the small factor H/h2 removed (i.e., multiplied by h2/H) is contoured as a function of the other two parameters Ah/h and •oH in Figure 4. Due to nonlineartry, the solution for a depressed py- cnocline (negative •o is very different from that of an elevated pycnocline (positive •0). In the negative •o case, here is a small-amplitude gain as can be seen from (16a), whereas in the positive •ocase there is a small-amplitude loss. The difference is due to the dy- namics at the depth change. As the depth decreases (h becomes maller), the magnitude of the upper layer flow U decreases (from (3) and (4)). With a negative pycn- ocline displacement •o (3) and (4) imply U is positive, that is, northward flow. Thus the depth decrease leads to a smaller positive U, that is, flow deceleration. The pressure force required for the flow deceleration implies higher pressure and sea level (but lower pycnocline) to the north. Hence the original negative pycnocline dis- f_x• ( Ou O(g,O•?) u 2 x• , Xo 2 -Xo (14) which can be expanded by substituting the expressions (1), (2), (3), (5), (6)and (7). The integration gives two equations for the magnitudes r/t (which is small) and r/•. Solving for r/• gives •7• - H- h:• -1+ 1+ h•H (H-h2)(H-h) 2) 1/2] +' hh2 H2 rk. , (15) to order (g,/g)Z/2 and is accurate to the order of i f_•c• 0u Y + As Ft must remove energy, (15) is an overestimate of the magnitude of •. The values of Ft should be of the same order in the region of the escarpment as away from it, so the loss due to this term is similar to the losses seen all along the coast. A simple solution is obtained in the limit of small upper layer depth compared to the lower layer depth: 0.0 -0.9 •400 ,. • .200_ ! i ....... •.100_ ' , -.. \ -... • ...... 100• \ "'.050. ' ,, _ -----_.. '" - ..... .050 -, ,. .005 ..................... 005.......................... 005 ............... ...................... 005 ............................ 005 .......... ,, ,,' - .005 / -.o5o -.0o5-.o•o • , o.o Ah 0.4 h Figure 4. Contour plot of the amplitude loss ((•, - n•)/n•.), scaled up by h2/H for case 1, flow over a steep depth decrease, assuming H/h2 small and H/A << I 1 - rl,,,/HI • (equation (16b)). Solid contours are in in- crements of 0.1; dotted contours are as labeled. Nega- tive values of amplitude loss imply an amplitude gain. ALLEN AND HSIEH: ESCARPMENT EFFECT ON EL NIfO COASTAL CURRENT 24,981 0.3 0.25 0.2 0.15 ! , ] , , i , , , , , ..... Approx. solution i i • i i i i i • i i 80100 300 1000 Upper layer thickness H (m) 30 , ,, I ........ ', ß Exact solution ' - .... Approx. solution 10 _.o 3 .-- 1. E ' • •..,•.•,.,..•_ _ _., _ _ _• _ •_ •_ 0.:3 • b) I I ] I I I I I i I I 0.1 ' 80100 300 1000 Upper layer thickness H (m) F;gure 5. Comparison between the approximate solu- tion (dashed curve) and the full solution (solid curve) for case 1, flow over a steep depth decrease where the in- cident amplitude r/,0 is (a) -75 rn and (b) +75 m. Kelvin wave amplitude gain/loss (in percent) is plotted versus upper-layer thickness H in case a/case b. Other param- eters are given in the text. placement will be slightly intensified by the depth de- crease to the north. In the case of a positive pycnocline displacement U is negative. The depth decrease, which reduces the magnitude of U, leads to a less negative U to the north. Hence the pressure force must push southward to give an accelerated, more negative U to the south, implying, again, higher pressure and sea level (but lower pycno- cline) to the north. Hence the original positive pycno- cline displacement will be reduced by the depth decrease to the north. These results can also be seen from the conservation equation (13), where the approximate balance of the two terms on the right-hand side implies that the sea level must increase to the north as U 2 is reduced by the depth decrease. As the flow is principally baroclinic, a negative surface elevation change corresponds to a posi- five interface elevation change. Thus negative V• values are reinforced and positive V• values are decreased. In conclusion, the warm E1 Nifio-generated coastal cur- rent should have a slight amplitude gain after passing the Mendocino escarpment. A routine, but somewhat tedious energy calculation, confirms that the small-amplitude gain does not imply an energy gain. Owing to the smaller Rossby radius north of the escarpment, the energy decreases regard- less of the sign of V•. At steady state, in a frictionless model, the energy balance is such that the inflow energy from the coastal current equals the outflow energy from the downstream coastal current and from the offshore current along the escarpment. The exact solution (15) and the approximate solution (16b) are compared by plotting the amplitude gain (in percent) as a function of the upper layer thickness H, while holding constant V• at -75 m, h at 4000 m, and h2 at 3000 m (Figure 5a). The amplitude gain is very small (< 0.3%) as the upper layer thickness is varied from 76 to 1000 m. For the case of V• - +75 m, the amplitude loss increases as H decreases (Figure 5b). Generally, the approximate solution (dashed curve) agrees well with the exact solution (solid curve), but as H -+ V•, the approximate solution approaches infinity, though the exact solution stays finite. Flow over a Steep Depth Increase The previous analysis cannot hold for flow down an escarpment as it implies an increase in energy. Thus the friction term F must not be small. In particular, besides the waves generated in case 1, there will be a swift boundary current governed by nonlinear, fric- tional and inertial effects close to the coast [Johnson, !985; Allen, 1988; Wajsowicz, 1991]. However, in this case flow is not diverted out along the escarpment, and volume flux must be conserved in the region of the es- carpment [Allen, 1996]. The short double Kelvin waves that propagate out along the escarpment do not carry any volume flux [Johnson, 1993]. The incoming volume flux in the upper layer is •o øø g' hH g' h •. (H- •7)Udy - -f(H + h)•7• + 2f(H+h) •7,,,, (17) where . and U are given by (1) and (4), respectively. The incoming lower-layer flux is j•o • g' hH g' H • (h + v)udy - f(H + h)• + 2 f(H + h) •' (18) •o order (g'/g)X/• and where . and u are given by (1) and (3), respectively. Assuming •ha• potential vorfici•y advection is con- fined within a region of say 5 Rossby radii of •he es- carpment, outside •ha• region •he outgoing volume flux in •he upper layer is 24,982 ALLEN AND HSIEH' ESCARPMENT EFFECT ON EL NI•TO COASTAL CURRENT g(H + h9.)Hrlt g' h9. Hrl• h•.f f(H + h•.) + 2f(H +h=)' (19) to order g'/g with r/and V giYen by (5) and (8), respec- tively. The outõoinõ lower-layer flux is oo + (20) •o order (g'/g)X/•', with.and u given by (5) and (?), respectively. The lower-layer incoming volume flux (18) must equal the outgoing volume flux (20) and similarly in the up- per layer (equations (17) and (19)). Using the same notation as in the previous section, equating the fluxes and solving for r/• between the two resulting equations gives (21) In the linear limit, rho• 0, (21) gives + H) )  ) , in agreement with Wajsowicz [1991], showing that there is finite attenuation even as a linear disturbance crosses a steep escarpment to deeper water. In the linear limit, (15) shows that there is no attenuation as a linear dis- turbance crosses a steep escarpment to shallower water, again in agreement with Wajsowicz [1991]. Assuming small upper-layer depth, that is, H/h •( 1 and assuming H/h <( 11 - •7,,,/HI •', then (23a) HAh (1-•./2H) ] (23b) r• • n• I h2 h ( -rho/H) ' with Ah = h•.-h. In (23b), for small upper-layer depth, the decrease in amplitude is proportional to the ratio of the upper-layer depth to the lower-layer depth (in the same way as for case 1) and is therefore itself small. Unlike case 1, amplitude loss occurs for both positive and negative rho The relative amplitude loss, with the small factor H/h2 removed(i.e., multiplied by h:•/H), is plotted as a function of Ah/h and rhoH in Figure 6. The amplitude loss, as a function of the upper layer thickness H, is illustrated for h - 3000 m, h•. - 4000 0.7 , , , , ß 0000 . 0 .2 -0.9 -' - i .20•00 o.o Ah h Figure 6. Contour plot of the amplitude loss ((rh. - r/•)/•/,•), scaled up by h,./H for case 2, flow over a steep depth increase, assuming H/h2 small and H/h << [1- rl•/H[ •' (equation (23b)). Contours are in increments of 0.1. m and rho = -75 m (Figure 7a) and ,•,o = +75 m (Fig- ure 7b). The amplitude loss is generally much higher than for case 1. Also in contrast to case 1, the ampli- tude loss eventually rises as H increases: shows that the amplitude loss increases linearly with H when rl•/H << 0, in contrast to case 1 (16b), where the ampli- tude loss approaches a constant when rh0/H << 0. The approximate solution (dashed curve) generally overes- timates the amplitude loss. For positive r/,•, as H -• r/w, again the approximate solution approaches infinity, while the exact solution stays finite (Figure 7b). Applications Let us estimate the effects of the Mendocino escarp- ment on anomalous coastal currents during an E1 Nifio. With h = 4000 m, h•. = 3000 m, H = 150 m, and rh0 = -75 m, the gain in the amplitude (15) is 0.21%. If the pycnocline were raised by 75 m instead of being depressed uring an E1 Nifio, the amplitude loss would have been 0.59%. In contrast, if the disturbance were propagating into deeper water (case 2), that is, from 3000 to 4000 m, the amplitude loss (21) would have been 1.0% and 1.7% for rh0 = -75 m and r/w = +75 m, respectively, much larger than the corresponding values in case i. The very small values for all the amplitude changes are due to the shallow upper layer compared to the lower layer, leading to a relatively large distance between the interface and the uneven bottom. If we examine the effects of a smaller amplitude dis- turbance, say r/• = -10 m, the amplitude gain is 0.039% in case 1 versus an amplitude loss of 1.16% in ALLEN AND HSIEH: ESCARPMENT EFFECT ON EL NI•IO COASTAL CURRENT 24,983 lO o.1 ' ' i : Exact solution --x--approx. solution 80 100 3•0 1000 Upper layer thickness H (m) 3 . =. 3 b) ß 1 ß ß I : Exact solution ..... approx. solution ß , I 80100 i i i i i 3•0 1000 Upper layer thickness H (m) Figure ?. Comparison between the approximate solu- tion (dashed curve) and the full solution (solid curve) for case 2, flow over a steep depth increase where the incident amplitude r/• is (a)-75 m and (b) +75 m. Am- plitude loss (in percent) is plotted versus upper-layer thickness H. Other parameters are given in the text. case 2; for r/w - +10 m, the reduction in amplitude is 0.045% in case 1 and 1.23% in case 2, that is, the effect is now much larõer in case 2 than in case 1 by a factor of almost 30. This chanõe is a reflection of the fact that unlike case 1 the percentaõe amplitude loss for case 2 is nonzero even when r/w --> 0 (22). The exact solutions found for cases 1 and 2 are also valid for bottom water flowinõ over escarpments. Bottom waters formed at hiõh latitudes flow equa- torward as western boundary currents [Storereel and Arons, 1960]. The presence of escarpments and canyons should greatly enhance the spreading of the bottom water away from the western boundaries, and in the Southern Ocean, away from Antarctica. In Figure 8 the amplitude loss for case 1 is plotted as a function of the incident interface amplitude r/w, with H - 3500 m, h = 500 m, and h2 = 1000 m. For case 2 (dashed curve), with H - 3500 m, h - 1000 m, and h2 -- 500 m, the amplitude loss (about 40%) is much higher than for case 1. As •/w decreases, the percentage amplitude loss for case 2 increases lightly, in contrast o case 1, where the percentage loss drops steeply. These results compare well with the numerical model results of Waj- sotoicz [1991]. Unlike for the E1 Nifio-generated coastal current, an escarpment results in a substantial amplitude loss for a bottom-water current, as the interface is located much closer to the uneven bottom. Figure 9 illustrates how the amplitude loss is affected by varying the distance between the interface and the ocean bottom, while keep- ing the mean ocean depth constant for (a) r/w - -100 m and (b) r/w - 9-100 m. In general, as the interface is placed farther from the ocean bottom, the ampli- tude loss (or gain) decreases. A minor exception occurs when the interface is located very far from the bottom, so that the upper layer depth H is small enough to start approaching the interface displacement r/w, thereby in- creasing the amplitude loss (case 1, Figures 9b and 5). Conclusions A nonlinear, two-layer, f plane shallow-water model was used in this study. In the real world the baroclinic Rossby radius changes ubstantially with latitude, and together with the presence of continental shelf topogra- phy leads to substantial changes in the coastal trapped wave solution with latitude [Brink, 1982]. These and the beta effect were not included in our model. From our theory, the amplitude changes in the region of the Mendocino escarpment suffered by the steady coastal current after the passage of the Kelvin wave bore during an E1 Nifio was found to be surprisingly small. lOO ! i i i i i i i i i i : Case 1 - -. -- Case 2 0.1 ' ' " ' • ' ''! ' ' I 0 30 100 300 Incident amplitudes! w(m) Figure 8. Amplitude loss (in percent) in a bottom- water coastal current flowing over a steep depth de- crease (case 1, solid curve) and flowing over a steep depth increase (case 2, dashed curve) as a function of its incident amplitude. The upper-layer depth H is 3500 m, and the average lower depth, (h 9- h2)/2, is only 750 m. The escarpment is 500 m high. 24,984 ALLEN AND HSIEH: ESCARPMENT EFFECT ON EL NI•IO COASTAL CURRENT • : Case 1 6 0 ',, --.--Case 2 40 20 -20 . 'a) -40 ''''! .... • .... • .... • .... ! .... I .... • .... 0 1000 2000 3000 4000 Distance between interface and ocean bottom (m) '-. -' Case 1 •' ' -. - -. - - Case 2 • 10 '-...... '0 " E I • < b) , , , , ! t • • , ! .... I , ,, , I , , , , ! .... i .... I , , , , 0.1 0 1000 2000 3000 4000 Distance between interface and ocean bottom (m) F]õure 9. The amplitude loss (in percent) plotted as a function of the distance between the interface and the average ocean bottom, for case 1 (steep depth decrease) m and (b) •7,• - +100 m. The depth change at the escarpment is 500 m, the mean ocean depth is 4000 m, and the interface displacement is 100 m at the coastal wall. Negative values of amplitude loss imply an ampli- tude gain. Even more surprisingly, there was a very slight ampli- tude gain, instead of a loss! The small change resulted from the large spatial separation between the interface and the uneven ocean bottom. In contrast, when ap- plying our theory to bottom water traveling over an es- carpment, the amplitude change was substantial, as the interface was located much closer to the ocean bottom. For Kelvin wave bores descending an escarpment (case 2), the amplitude losses for the steady flow were generally much higher than for bores ascending an es- carpment (case 1). In case 2, no long offshore double Kelvin mode was generated as in case 1, but frictional effects became very important, hence the much larger amplitude loss. In case 2 the percentage amplitude loss did not approach 0 even when the incident wave am- plitude approached O, again in contrast to case 1. An- other interesting contrast is that in case 2, both positive and negative pycnocline displacements were reduced in magnitude after passing the escarpment to the north, whereas in case 1 the negative pycnocline displacements (warm currents) were slightly intensified, while the pos- itive pycnocline displacements (cold currents) were re- duced. Acknowledgments. We wish to thank Joe Tam for as- sistance with the contour plots. This research was supported by grants from the Natural Sciences and Engineering Re- search Council of Canada. References Allen, S. E., Rossby adjustment over a slope, Ph.D. thesis, 206 pp., Cambridge Univ., Cambridge, England, 1988. Allen, S. E., Rossby adjustment over a slope in a homoge- neous fluid, J. Phys. Oceanogr., •6, 1646-1654, 1996. Bennett, J. R., A theory of large-amplitude Kelvin waves, J. Phys. Oceanogr., 3, 57-60, 1973. Brink, K. H., A comparison of long coastal trapped wave theory with observations off Peru, J. Phys. Oceanogr., 897-913, 1982. Enfield, D. B., and J. S. Allen, On the structure and dynam- ics of monthly mean sea level anomalies along the Pacific coast of North and South America, J. Phys. Oceanogr., /0, 557-578, 1980. Gill, A. E., M. K. Davey, E. R. Johnson, and P. F. Linden, Rossby adjustment over a step, J. Mar. Res., 44, 713-738, 1986. Huyer, A., and R. L. Smith, The signature of E1 Nifio off Oregon 1982-1983, J. Geoph•ls. Res., 90, 7133-7142, 1985. Johnson, E. R., Topographic waves and the evolution of coastal currents, J. Fluid Mech., 160, 499-509, 1985. Johnson, E. R., The low-frequency scattering of Kelvin waves by stepped topography, J. Fluid Mech., •15, 23- 44, 1990. Johnson, E. R., The low-frequency scattering of Kelvin waves by continuous topography, J. Fluid Mech., 173-201, 1993. Johnson, E. R., and M. K. Davey, Free-surface adjustment and topographic waves in coastal currents, J. Fluid Mech., •19, 273-289, 1990. Killworth, P. D., How much of a baroclinic coastal Kelvin wave gets over a ridge?, J. Phys. Oceanogr., 19, 321-341, 1989a. Killworth, P. D., Transmission of a two-layer coastal Kelvin wave over a ridge., J. Phys. Oceanogr., 19, 1131-1148, 1989b. LeBlond, P. H., and L. A. Mysak, Waves in the Ocean, 602 pp., Elsevier Sci., New York, 1979. Longuet-Higgins, M. S., On the trapping of waves along a discontinuity of depth in a rotating ocean, J. Fluid Mech., 31,417-434, 1968a. Longuet-Higgins, M. S., Double Kelvin waves with continu- ous depth profiles, J. Fluid Mech., 34, 49-80, 1968b. Pares-Sierra, A., and J. J. O'Brien, The seasonal and the interannual variability of the California current system: A numerical model, J. Geophys. Res., 9•, 3159-3180, 1989. Philander, S., El Nifio, La Ni•a, and the Southern Oscilla- tion, 293 pp., Academic, San Diego, Calif., 1990. Rhines, P. B., Slow oscillations in an ocean of varying depth, I, Abrupt topography, J. Fluid Mech., 37, 161-189, 1969. Stommel, H., and A. B. Arons, On the abyssal circulation of the world ocean, II, An idealized model of the circulation ALLEN AND HSIEH: ESCARPMENT EFFECT ON EL NI•TO COASTAL CURRENT 24,985 pattern and amplitude in oceanic basins, Deep Sea Res., 6, 217-233, 1960. Wajsowicz, R. C., On stratified flow over a ridge intersecting coastlines, J. Phys. Oceanogr., 21, 1407-1437, 1991. Wallace, J., and D. Gutzler, Telecormections in the geopo- tential height fields during the northern hemisphere win- ter, Mon. Weather Rev., 109, 784-812, 1981. Willmort, A. J., Forced double Kelvin waves in a stratified ocean, J. Mar. Res., •42, 319-358, 1984. Willmort, A. J., and R. H. J. Grimshaw, The evolution of coastal currents over a wedge-shaped escarpment, Geo- phys. Astrophys. Fluid Dyn., 57, 19-48, 1991. Willmort, A. J., and E. R. Johnson, On geostrophic ad- justment of a two-layer, uniformly rotating fluid in the presence of a step escarpment, J. Mar. Res., 53, 49-77, 1995. Wood, I. R., and J. E. Simpson, Jumps in layered miscible fluids, J. Fluid Mech., 140, 329-342, 1984. S. E. Allen and W. W. Hsieh, Oceanography, De- partment of Earth and Ocean Sciences, University of British Columbia, 6270 University Boulevard, Vancou- ver, B.C., Canada, V6T 1Z4. (e-maih allen@eos.ubc.ca; hsieh@eos.ubc.ca) (Received February 16, 1996; revised April 18, 1997; accepted May 29, 1997.)


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