Over Steep Topography

JOURNAL OF GEOPHYSCAL RESEARCH, VOL. 88, NO. C1, PAGES 743-750, JANUARY 20, 1983 On the Formaton of Surface to Bottom Fronts Over Steep Topography Y. HSUEH AND BENOT CUSHMAN-ROSN! Department of Oceanography, Florda State Unversty, Tallahassee, Florda 32306 mposed horzontal densty dfferences n a laterally unbounded rotatng flud across a bottom of varable depth lead to the formaton of surface to bottom fronts wth enhanced asymmetry. n a twolayer flud, the down-slope spllage of heavy flud leads to greater lower layer penetraton than the extent of upper layer advance n the opposte drecton. These fnte ampltude movements generate barotropc flows that have the same drecton as the velocty n the layer that thns as a result of the movement. Lower-layer movements up the depth gradent enhance the generaton of lower-layer flow due to vortex foreshortenng and favor a barotropc flow n the same drecton. n contrast, lower-layer movements down the depth gradent enhance the upper-layer flow for the same reason and favor a barotropc flow n the drecton of that flow. Extended down-slope movement of heavy fluds also leads to the formaton of solated heavy flud lens over the bottom, wthn whch the flow s remnscent of observed bottom currents such as that assocated wth the cold, dense Norwegan Sea water south of the Denmark Strat (Smth, 1976). n addton to bodly dsplacng the front n the Ekman sense, Unform surface stress drected wth deep (shallow) flud to the rght flattens (sharpens) the front. The dsplacemert of the front as a whole also gves rse to addtonal vortex stretchng that leads to further devaton from symmetry n the cross-front dstrbuton of the barotropc transport. 1. NTRODUCTON t has been noted that on the contnental margns of the ocean, there s often a complete separaton of relatvely fresh coastal water from the deep water further offshore across a densty front that extends from the sea surface to the sea bottom n an S-shaped curve [Csanady, 1978]. The recently observed front over the nner shelf off the Yangtze Rver estuary (Fgure 1) and the shelf break front off the eastern North Amercan contnental shelf [see Csanady, 1978, Fgure 2] are good examples of ths knd of separaton. The Yangtze Rver front separates relatvely fresh nearshore water from waters on the contnental shelf wth a salnty value of 32%o. Contrary to rver plume fronts n ths area whch are fast movng, t s steady, persstent, and n near geostrophc balance wth the flow. (Velocty measurements, not shown, confrm th e latter.) Comparng observed currents wth geostrophc calculatons for the shelfslope front south of New England, Flagg [ 1977] also found that the cross-shelf momentum balance s approxmately geostrophc wthn the front. t appears that fronts that are persstent n tme and form the boundares between dstnct water types are fea- tures of an equlbrum that nvolves the mutual adjustment to geostrophy of the two neghborng bodes of water. ndeed, Csanady [1971] was successful n descrbng the sprng thermoclne behavor n Lake Ontaro wth a smple geostrophc adjustment model of fnte ampltude that conserves potental vortcty. More recently, Csanady [1978] extended ths work to nclude the effects of wnd on the frontal adjustment processes. Csanady's model s, however, restrcted to cases n whch the sea floor underlyng the front s flat. n contrast, the contnental shelf s sloped. n partcula r, the slope of the sea floor ncreases rapdly near the shelf break where some of the major fronts occur. Mesoscale Ar-Sea nteracton Group, Florda State Unversty. CoPyrght 1983 by the Amercan Geophyscal Unon. Paper number 2C1541. 0148-022783002C- 1541 $05.00 743 Flagg and Beardsley [1978] have shown that for an overlyng Margules-tYpe planar front, an ncrease bottom slope tends to stablze the flow completely. Although t remans to be seen whether the cross-front shear n the along'front flow that s not ncluded n the planar front model does not change ths stablzng tendency, the presence of a Steep topography s clearly a feature of consderable nfluence n the frontal flow dynamcs. n the context of the formaton of the front as the product of a geostrophc adjustment process, the sgnfcance of a steep bottom topography les n the potental vortcty gradent t mposes upon the movement of the overlyng water. Because of ths, potental vortcty bookkeepng over regons of sharp topographc change, such as along the contnental shelf break requres a knowledge of the dsplacement that each ndvdual water column undergoes before the equlbrum s reached. Taggng water columns by ther dsplacements has been su ggested n lterature [Csanady, 1978]. The purpose of the present work s to provde a formalsm to the dea and to utlze t to consder the formaton of surface to bottom fronts over steep topography n a geostrophc adjustment process n two dmensons wth no along-front varatons. The work establshes a framework n whch external, spatal- ly varable, momentum sourcesuch as wnds and mesoscale eddes may be consdered. n addton, t also sets the stage for an examnaton of the stablty O f surface to bottom fronts over the contnental shelf break that are not necessarly of the Margules-type. n the followng, the mathematcal development begns wth takng the well-known approach of assumng a hypothetcal vertcal barrer that ntally keeps apart the two fluds found on ether sdes of a densty front n the adjusted state [see, for example, Stommel and Verons, 1980] (The hypothetcal barrer dea was frst utlzed explctly by Csanady [1978]). The barrer s then removed to allow geostrophc adjustment to proceed. The end products are the locaton and shape of the front as functons of a set of gven physcal parametersuch as the baroclnc deformaton radus and the bottom slope. The use of a hypothetcal

744 HSUEH AND CUSHMAN-RoSN' FORMATON OF FRONTS OVER TOPOGRAPHY 1111981 1111881 HOUR 0140 0033 2315 2154 2047 STATON NUMBERS 8106 8107 8109 8117 8114,,, 0 2 0-6'0 4o 6 KLOMETERS Fg. 1. A salnty front off the Yangtze Rver mouth along 120ø28'E. The observed temperature shows a smlar pattern (not shown) and ranges from 14øC near the surface to around 16øC below the front. The tme s local. Durng ths perod, the wnds were about 8 ms to the west. barrer to mpose an ntal horzontal densty dfference s, of course, a vald practce only f the mechansm by whch densty contrast s formed n nature operates on tme scales short compared to the geostrophc adjustment tme whch s approxmately a day. The coolng of coastal waters due to passage of severe cold fronts s one such mechansm. 2. MODEL Consder two mmscble, ncompressble, homogeneous fluds of denstes p and p2(p2 > p ) that are kept apart ntally by a vertcal barrer over a steep topography. On ether sde of ths regon of topographc change, the bottom s flat (Fgure 2). There s no along-shelf varaton. Upon removal of the barrer, the two fluds flow aganst each other toward a state of equlbrum. t wll be assumed that through the adjustment process, frcton and mxng are neglgble, and potental vortcty s conserved. n the adjusted state, the two fluds are therefore separated by an nterface (a densty front) that ntersects both the free surface and the bottom. The resultng flow s along-front and n geostrophc balance. The crux of the problem s to calculate the fnal adjusted state from the ntal confguraton of the two-flud system. For mathematcal convenence, let the cross-front dstance (x) be measured from the pont of termnaton of the lower layer. The poston of the barrer, x = -a, and the pont of termnaton of the upper layer, x = -a - b, are thus both unknown. Let the bottom topography be gven by z = zb(x) wth z measured upward from the flat bottom to the left n Fgure 2. The ntal water depth s thus gven by H(x) = Ho -zb(x). Let the equlbrum thckness of the lghter (upper) flud be h(x) and that of the denser (lower) flud h2(x). n order to keep track of the cross-topography movement of the water columns, dsplacement functons *h(x) and *h(x) are ntroduced, respectvely, for the upper and lower layers. The dsplacement functon represents the crosstopography dstance a water column travels before t arrves at x n the equlbrum state. The conservaton of volume through the adjustment process thus requres the followng:.e., hj(x)dx = H[x-,lj(x)]d(x- tj) j = 1, 2 hj(x) = H[x -,j(x)](1- *b' ) j = 1, 2 (1) Upon ntegratng the along-front momentum equaton n the Lagrangan sense of followng a water column, the generaton of the along-front flow n the adjusted state can be seen clearly: v = -fn,' + j = l, 2 (2) H[x - bex) ] where f s the Corols parameter, v the along-front velocty of the jth flud, and j = pj- 1 f [t, x - (x)]dt, j = 1, 2 The functon represents the acton of an along-front wnd stress, as an mpulse exerted before the water column undergoes any depth change. The duraton T of the mpulse should not exceed the total adjustment tme whch s on the order of a day. The adjustment s acheved by the propagaton of gravty waves radatng away from the zone of adjustment. Typcally the fastest gravty wave travels one radus of deformaton n about a perod f-1, about one half of a day at mddle lattudes. One half of a day s thus a lower bound for the adjustment tme. The wnd-stress mpulse that could be consdered s restrcted to that whch arses from storms and rapd passages of atmospherc fronts. Other forcng functons that could be consdered n a smlar manner nclude cross-front gradent of the Reynolds stress u'v' that can arse from mesoscale eddes offshore. However, the long tme scale of these eddes, whch s on the order of tens of days, complcates the matter and precludes ther nco oraton except n extreme cases n whch a sudden change n cross-front flux of along-front momentum occurs n a tme shorter than day. n vew of the strngent requrement for short actng duratons, the mpulse representaton must reman an over-dealzaton. Equatons (1) and (2) ensure the conservaton of potental vortcty. Upon elmnatng E from (1) and (2), t follows mmedately that vx+f f +- h H[x- (x)] h H(x- ) x where the layer ndex j s suppressed. H o ',--;' 'F' ' -X O-a ~a-b a ' hl Pl Ho-H1...., H1 0 2L -Xo-a (x) Fg. 2. A two-layer flud model of a surface to bottom front over a bottom of varable depth. The flud s unbounded laterally and rotatng wth the earth.

HSUEH AND CUSHMAN-RoSN: FORMATON OF FRONTS OVER TOPOGRAPHY 745 The cross-front geostrophy n the adjusted state requres n addton the followng: 1. n the regon () of flat bottom to the fght, x -> 2L - a -- X0, fvl = ghlx (3) 2. n the sngle layer regon () over the fght flank of the topography, 0 -< x -< 2L - a - x0, fv = g(hl - H)x (4) 3. n the two-layer regon (), -a - b -< x -< 0, fvl = g(h + h2- H), (5) fv2 = g[(1 - e)h + h2- H]. (6) 4. n the sngle layer regon (V) over the left flank of the topography, - a - x0 -< x -< - a - b, fv2- g(h2 - H), (7) 5. n the regon (V) offlat bottom to the left, x -< -a -x0, fv2 = gh (8) Here and everywhere else n the artcle the subscrpt x ndcates ordnary dfferentaton wth respect to the crossfront coordnate, and e = (p2 - pl)p2. 3. METHOD OF SOLUTON Upon assumng that the adjusted flow vanshes far away from the frontal zone, the system of equatons (1)-(8) may be solved for the *j, h, v, and constants a and b. The method of soluton employs a smple shootng technque. n what follows, the computatonal scheme s outlned for the case where the wnd stress s unform; the therefore are constants and equal, represented by. Generalzaton to spatally varyng wnds s straghtforward. t s convenent at ths pont to scale the cross-front dstance wth the barotropc deformaton radus (gho)!2f based on the constant depth to the left of the front and to scale the along-front velocty and depth wth (gho) m and H0, respectvely. A convenent scale for the wnd stress s thus pjho(gho)l2t. The dmensonless equatons are precsely (1)-(8) wth the parameters T, p, f, and g replaced wth unty. Hereupon (1)-(8) wll be referred to n ther dmensonless form. Note that no dmensonless number appears n the equatons. However, the rato of the slope wdth (2L) to the barotropc deformaton radus does enter n the dmensonless depth profle and forms the only nondmensonal number of the problem. The ntegraton begns wth a guess for the dstance a between the barrer and the lower layer nose and proceeds from regon where an ampltude C! forms another guess. The tral value for a s guded by the baroclnc deformaton radus based upon half the ntal depth at the barrer (x0 away from the left end of the slope) and C1 s generally small on the order of 10-2 or less. Once a value for C s gven, the soluton for regon s completely determned as h = 1 -H + C e -qx (9) *1 = qcl e-qx + (1 - H1) (10) where q = (1 - H1)- 2. n regon, the par of frst order equatons to be solved s the followng: *t: 1 - hlh(x - *1) (11) h l = Hx- *1 d- lh(x- *) (12) The second equaton comes from elmnatng U 1 from (2) and (4), and stands for the mpulse functon arsng from a gven wnd-stress forcng. These equatons can be ntegrated numercally from values of h l and *1 at x = 2L - a - x0 whch are now known. The ntegraton proceeds untl x = 0 s reached where the calculaton on the bass of the next set of equatons begns. n the next, two-layer regon, the governng equatons can be wrtten as follows: *! = 1 - hlh(x- *!) (13) *2 = 1 - h2h(x - *2) (14) h t = (*2- *1 d- )E (15) h L = Hx + [(1 - e)*! - *2 - A+ elh(x- *1)]e, (16) where A = l[1h(x - * ) - 1H(x - *2)]. Note the dependence upon elh(x - *1) n addton to A, the latter of whch only s found n Csanady [ 1978]. Ths extra term s due to the presence of a free surface. The e terms n the numerator dsappear when the rgd ld assumpton s made, as s done by Csanady [1978]. The ntegraton n ths regon commences at x - 0 wth calculated values of h and *1 and the followng boundary condtons that preval at the lower layer nose: h2 = 0 *2 = a (17) Ths ntegraton s carred forward untl h l = 0 s reached. The upper layer boundary condton at ths pont then requres that x + a and * (x) be equal, both beng -b, negatve of the dstance between the barrer and the poston of the upper layer nose. Snce the ntal choce for a and C may not be correct, an error el results at ths pont that s gven by el =*l(wherehl =0)-x(wherehl =0)-a (18) The computaton for h2 and *2 then proceeds nto regon V where the equatons are as follows: *2 = 1 - h2h(x - *2) (19) h2 = H,- *2 + lh(x- *2) (20) When x = -a - Xo s reached, h2 and *2 must become solutons n regon V whch are smply h2 = 1 - C2 ex (21) *2 = C2e x + (22) A second error e2 s thus obtaned wth the calculated values of h2 and *2: e2 = h2 + *2 - - 1 (23) By mnmzng el and e2 wth a systematc tral for values of a and C l, a fnal soluton can be found that satsfes all the constrants. (n the case of a lower-layer lens formaton, the ntegraton begns wth a guess for the leadng edge poston, x - a l, of the lens and s frst carred out untl h2-0. The calculaton

746 HSUEH AND CUSHMAN-RoSN: FORMATON OF FRONTS OVER TOPOGRAPHY then proceeds through an ntervenng sngle-layer regon untl the shelf break s reached. The ntegraton wthn the two-layer regon that follows begns at the shelf break where t s assumed that h 2 = 0, on account of the fact that lowerlayer separaton tends to occur here. n accordance wth ths physcal assumpton, the dsplacement experenced by the flud found at the shelf break n the adjusted state must be such that the orgnal volume of flud between tself and the barrer s precsely that of the solated lens. Ths determnes *72 at the shelf break and allows the ntegraton wthn the second two-layer regon to proceed untl h l = 0. At ths pont, el s found from (18) wth a = 0, snce n ths case, the orgn s chosen to concde wth the shelf break. Beyond ths pont, the ntegraton proceeds entrely wthn th6 heavy flud untl the begnnng of the flat bottom regon where (23) agan yelds e2. See later Fgure 12 for an example.) (A), a- 0.088, b = 0.080 t \ t \ ',, TO.O 4. REsu TS Wth the procedure developed above, frontal zone shapes and transports can be computed for an ntally vertcal separaton of two fluds over any bottom topography, provded that the bottom s flat far to the left and to the rght. For the present paper, examples nclude cases of flat bottom and constant-slope bottom to varous degrees. The densty dfference s chosen to be 10ot unts, a purposeful exaggeraton by a factor of about 10 to facltate the demonstraton of effects of surface modes and the dscusson of barotropc transports. A densty dfference of lot unt s more representatve of coastal water stuatons. t leads to narrower front zones and weaker barotropc transports, but, qualtatvely, all results are dentcal to those presented here. The ntal barrer s placed at the top of the slope as major sempermanent fronts are often found along contnental shelf edges. Coolng can take place qute rapdly so that waters of dfferent densty can be found sde by sde. The barrer s thus chosen as a vertcal plane partton separatng the two waters. Although varous shapes can be chosen for barrers, the sngle vertcal partton elmnates the need to study the effects of the ntal barrer confguraton on the fnal stage. Two classes of adjustment processes are treated, wth and wthout wnd-stress forcng. For the cases wth wnd-stress forcng, a constant along-front wnd stress of 10 dynecm 2 s assumed to have blown for one day before the adjustment process s allowed to begn. The magntude of the mpulse s thus 8.64 x 105 cm2s. The convergence crtera are el -< 10-3 and e2 --< 10-3 for all computatons. For convenence of presentaton, as a rule, frontal zone wdths and penetraton dstances a and b are gven n unts of XgHof wth H0 = 40 m. Transports are, however, gven n unts of V'gHo where H0 = 40 m when the shallow regon s to the left and H0 = 200 m when the shallow water s to the rght. Frontal Zone Shape and Transports n Absence of External Forcng The specal case wth a flat bottom and no wnd-stress forcng s treated frst. Snce the depth H s a constant n ths case, the ntal potental vortcty s everywhere the same; the problem can thus be solved wthout nvolvng the ntervenng rj [Stommel and Verons, 1980], f t s further assumed that the surface s a rgd ld, t then becomes qute easy to show that the upper layer penetrates to the left just as -0.01! Fg. 3. The shape of the nterface for fluds of denstes p and p2 (p2 > p ) ntally separated by a vertcal barrer over a flat bottom. The lower layer penetrates to the rght to the pont where ts thckness, h2, vanshes. Wth that pont as the orgn, the upper layer penetrates to the pont x = -a - b. ntally, the layers of flud were separated by a vertcal barrer at x = -a. (b) Upper layer, lower layer, and total (barotropc) transports marked by T, T2, and TB, respectvely, for the free surface case. The transports are gven n unts of (gho3)l2 wth Ho = 40 m. far as the lower layer does to the fght and the layers are of equal thckness at equal dstance to ether sde of the barrer and that at the poston of the orgnal barrer, the veloctes are equal n magntude and opposte n drecton. Consequently, the total (barotropc) transport (T = h Vl + h2v2) s, n ths case, antsymmetrc about the pont of separaton. To ths relatvely smple pcture of a fgd-ld case, the present calculaton provdes a contrast. Fgure 3 shows the calculated frontal zone shape and transports over a flat bottom wthout the fgd-ld approxmaton. The shft away from the barrer of the pont of vanshng T, the nequalty of a and b, and the lopsded T dstrbuton are all evdence of a free surface undulaton that contrbutes to a barotropc transport n the negatve y drecton. The source of ths partcular flow les n the cross-front pressure gradent force, maxmal at the bottom and zero at the surface, pushes the fluds to the rght, makng a > b, and generates a postve dsplacement that nduces a negatve along-front velocty. n contrast, the surface coolng problem treated by Stommel and Verons [1980], s characterzed by a pressure gradent force that s from rght to left at the surface and decreases to zero at the bottom. On the average ths force pushes the fluds to the left. Consequently, n ther case, an opposte (postve) barotropc along-front flow s contrbutng to the transport asymmetry and the upper layer penetrates farther to the left than does the lower layer to the fght (b > a n the notaton of the present paper.) Note that the wdth of the frontal zone n Fgure 3 s 0.168 compared to the baroclnc

_ ß HSUEH AND CUSHMAN-RoSN.' FORMATON OF FRONTS OVER TOPOGRAPHY 747 (A) [ -\ a- o. 151!k b: 0.09,! \ \\\ \ \ \! \, J \ ' 0.01 ß, (A) a= 0.089,' b-o.o,! " "0.001 -a T2 Fg. 4. Same as Fgure 3 except ntally the flud of densty p s bounded below by a varable depth bottom wth a constant slope of 1.6 x 10-3 that extends horzontally from the barrer poston to a pont 100 km to the rght where a deep flat bottom s joned. j-o.001 Fg. 5. Same as Fgure 3 except ntally the flud of densty p2 s bounded below by a varable depth bottom wth a constant slope of 1.6 x 10-3 that extends horzontally from the barrer poston to a pont 100 km to the left where a deep flat bottom s joned. The transports are gven n unt of (gho3)l2 wth Ho = 200 m. deformaton radus of 0.071, based upon the half depth H2. Note that, although barotropc effects are undenable, the actual surface can hardly be dstngushed from a smooth horzontal plane. The ncluson of topography under the front renders the ndspensable. Due to gravty, penetraton of the deep layer, when t s down-slope, s much more extensve than n the flat bottom case. Smlarly, the penetraton of deep flud nto the shallow regon s curtaled n comparson. Fgure 4 shows the fnal adjusted state wth the heavy flud ntally lmted to the shallow regon by a barrer at the top of a slopng bottom that jons a deep regon of constant depth to the rght. The lower layer penetraton (a = 0.151) s nearly twce that n the flat bottom case. Both the lower layer transport (T2 = h2v2) and the upper layer transport (T = hlv) ncrease by an apprecable amount; the former due to an ncreased v2, the latter due prmarly to an ncrease n depth. (The appearance of a cusp n both the T2 and TB curves s due to the dscontnuty n bottom slope at the top of the topography.) Note that n contrast to the flat bottom case, the zero of TB occurs to the rght of the barrer. Ths zero shft s a drect result of a varable bottom depth. The upper layer flud undergoes a much more drastc decrease n depth than t does n the flat-bottom case. Conservaton of potental vortcty thus calls for an ncreased postve along-front velocty, and ths leads to a postve T that domnates the total transport over the shelf break. When the heavy flud occupes the deep regon, the movement of the deep layer onto the shelf s aganst gravty and therefore severely lmted. As a consequence, compared wth the flat bottom case, the wdth of the frontal zone s reduced (a + b = 0.168) and all transports are smaller by a factor of about 2. Fgure 5 presents these results graphcally. The foreshortenng of water columns gong up-slope dom- nates the total transport wth the generaton of a strong lower-layer along-front flow n the negatve y drecton. ncrease n the steepness of the bottom slope accentuates these gravtatonal effects. Fgure 6 summarzes the results for a case smlar to that whch s represented n Fgure 4 except that the slope s now twce as steep. There s over 50% ncrease n the horzontal dstance of penetraton (a = (A) a- 0.234 \\ b-o.,o2 \ \ \ 0.05-0. O5 Fg. 6. Same as Fgure 4 except the bottom slope s 3.2 x 10-3 and the foot of the slope s at a pont 50 km to the rght of the barrer poston. the transports are gven n unts of (gho3)l2 wth Ho = 40 m.

748 HSUEH AND CUSHMAN-RoSN: FORMATON OF FRONTS OVER TOPOGRAPHY wnd-stress mpulse = T y(y) dt -- H ø ', densty P2 Densty densty Front 1 2 Fg. 7. A schematc dagram of stress-nduced movements of a two-layer flud over a flat bottom. The two layers were ntally separated by a vertcal barrer (dashed). The mposton of a surface (wnd) stress mpulse,, gves rse to a unform flud dsplacement, Ho, n the Ekman sense n addton to movements (h and2) due to gravty. The net dsplacements (rt and *2) n the two layers combne to create an nterface that s centered about an magnary vertcal lne (dashed-dotted) a dstance Ho from the barrer poston. 0.234) of the heavy flud down the steeper slope. The frontal zone wdth s also ncreased by nearly 40%. The lower-layer depth s nearly zero at the shelf break. As a consequence, the total transport s domnated by the upper layer flow to a greater extent than s the case wth a more gentle slope. Frontal Zone Shape and Transports Wth Wnd-Stress Forcng The role of an along-front wnd-stress mpulse s prmarly that of nducng an extra amount of water column dsplacement n addton to that due to gravty. n an unbounded twoflud system over a flat bottom, a unform wnd-stress mpulse generates no flow snce a water column that undergoes lateral dsplacements experences no change n depth and therefore no change n relatve vortcty. Such a unform mpulse s, however, capable of movng the front as a rgd body. Fgure 7 s an llustraton of ths partcular effect. Upon the removal of the vertcal barrer (dashed lne), the fluds move to the left by a unform amount gven by -Ho ( < 0). The gravtatonal flow then proceeds n both the upper and lower layers untl a front s formed, centered about a vertcal lne (dot-dashed) -Ho to the left of the orgnal barrer. n the upper layer, the mpulse-nduced movement Ho s n the same drecton as the movement ( < 0) due to gravty; hence a greater net penetraton, + Ho. n the lower layer, the two movements are opposte to each other; hence a lesser net penetraton, 2 + Ho ( 2 > 0). The end result s smply a dsplaced front. Smlar arguments can also be used to show that a dfference n wnd-stress mpulses appled to the two fluds results n a narrowng of the frontal zone when - 12 > 0 and a broadenng of the frontal zone when - 12 < 0. Csanady [1978] arrved at smlar conclusons mathematcally. Ths change of frontal zone wdth represents the prmary effect of wnd-stress mpulses on fronts. n fact, for a rgd-ld case, As the only way mpulse functons enter ( 13)-(16). When the bottom s slopng, the wnd-stress mpulse affects the generaton of along-front flow n a subtle way. the dsplacement t entals adds to the dstance by whch a water column must be traced back n order to determne the relevant ntal depth. (See (11), for example.) n ths man- P net, t forces a certan amount of vortex stretchng n addton to that whch accompanes the gravtatonal flow. Fgure 8 shows the calculated frontal shape and transports for the same geometry as that represented n Fgure 5. A constant negatve, y-drected wnd-stress mpulse of the magntude mentoned before s now appled. The front as a whole s moved to the left. As the forcng s more effectve over the shallow regon than t s over the deep the mpulse dfference A = H(x - q ) - H(x - *2) s n ths case negatve. The frontal zone wdth (0.181) s greater than that n Fgure 5 (0.168) as s expected. Because of wnd-forced dsplacements, much deeper flud over the slope s nvolved than otherwse, resultng n a two-fold ncrease n T2 compared wth the same quantty n Fgure 5. There s no apprecable change n the magntude of T. The upper-layer vortex stretchng due to movement of the front as a whole can, however, be seen to contrbute to a slghtly postve barotropc transport to the rght of the frontal regon, whereas n the frontal regon the total transport s negatve, wth the wnd. As the drecton of the wnd-stress s reversed, the front s moved onto the shallow flat, drawng greater amount of deep flud and causng even greater amount of vortex foreshortenng. Ths, as shown n Fgure 9 leads to a negatve transport whch s now opposed to the wnd. To the left, a barotropc transport s formed over the slope due to wnd-forced vortex foreshortenng. As expected for A> 0, the frontal zone wdth (0.152) s now less than that n Fgure 5. Smlar features are notceable when the heavy flud occupes the shallow regon (as n Fgure 4). When a negatve wnd stress s mposed, the front s dsplaced entrely onto the shallow regon and, snce As postve and there s no down-slope spllng on the flat, the front zone wdth s reduced sharply from 0.242 to 0.155 (Fgure 10). The profles of the transports are smlar to those n Fgure 9 except that 0.003 _; -0.003 Fg. 8. Same as Fgure 5 except a surface (wnd) stress mpulse n the negatve y drecton s ncluded. The magntude of the appled mpulse s 8.64 x 105 cm2s. A= H(x - q )- H(x - r2) < 0.

HSUEH AND CUSHMAN-RoSN' FORMATON OF FRONTS OVER TOPOGRAPHY 749 (A) o - 0.198 b--q046 (B}!!,, 0.002 (A) ' 0=0.267 ' k b- 0.023, \ \ ' ' 0.05 TB -OO02 Fg. 9. Same as Fgure 8 except the stress s y drected. A> 0. t-0.05 Fg. 11. Same as Fgure 10 except the stress s y drected. A< 0. the negatve y-drected transport to the left of the frontal zone s here merely a part of the gradual decay of a smlar lower-layer transport due to vortex foreshortenng beneath the front. The postve along-front velocty shows a slght ncrease to the rght. Ths s prmarly a combned result of wnd and gravtatonally nduced vortex foreshortenng that are both generatng postve along-front flows. When the wnd-stress mpulse s appled n reverse, A< 0 the front zone wdth ncreases to 0.290 (Fgure 11). The transport profles are smlar to those n Fgure 4 except that the extended lower-layer movement down-slope now gves rse to an ncreased amount of vortex foreshortenng n the upper layer. Ths leads to a postve upper-layer flow that domnates the total frontal transport. The sharp decay of the postve along-front flow just to the rght of the front zone s notceable. Ths curosty occurs because whle the gravtatonally nduced dsplacement generates vortex foreshortenng and therefore postve along-front flow, the wnd-nduced dsplacement generates just the opposte. 5. DSCUSSON (A),,,,,, o =-0.017 " b: 0.172 ", }0.02 TB - Fg. 10. Same as Fgure 4 except a surface (wnd) stress mpulse n the negatve y drecton s ncluded. The magntude of the appled mpulse s 8.64 x 105 cm2s. L = H(x - rh) - H(x - *2) > 0. 2 The study s largely motvated by the observed rchness n mesoscale features along an often dstnct boundary between the contnental-shelf and the deep-sea waters. Off the west Florda coast, nfrared mage showed at least on one occason (February 22, 1981), mmedately after a cold front passage, an erupton n patches of fresh coastal water seaward over the open ocean [Hsueh et al., 1982]. t s not hard to magne that at hgh lattudes, the cold coastal water would also spll over a deepenng bottom to form solated lens under a relatvely warm body of open ocean water. The separaton of a part of the flud formng an extended front appears to be a possble mechansm for the creaton of a populaton of these mesoscale features. A case n pont s the calculated lower-layer depth n the case whose results are presented n Fgure 6. The movement of the heavy flud s remnscent of severely cooled contnental shelf water spllng down the upper contnental slope. t s sgnfcant that the lower-layer depth over the break s a mere 2 m n the graph shown. Wth a greater down-slope push by gravty, due ether to a steeper slope or to a greater densty dfference, a lower-layer separaton n the neghborhood of the shelf break can be antcpated. The separaton creates a new surface-to-bottom front on the shelf and an solated heavy flud lens on the lower slope. Computaton for just such a case s made n whch the geometry s the same as

750 HSUEH AND CUSHMAN-ROSN' FORMATON OF FRONTS OVER TOPOGRAPHY (A), 02 ' al = 0.;.'502 b = 0.063 az= 0 bz = -0.147 Fg. 12. Same as Fgure 6 except the densty dfference s s doubled. The orgn of the x axs now concdes wth the vertcal barrer poston at the shelf break. The forward edge of the lens s found at the pont x = a, and the tralng edge at x = b. The rest of the lower layer flud s separated from the upper layer by a shelf front that ntersects the surface at the pont x -- b2. The ntersecton of the shelf front wth the bottom s fxed at the shelf break (x = a2). The transports are gven n unts of (gno3)l2 wth H0 = 40 m. 6. CONCLUSON The formaton of surface-to-bottom fronts over a varable bottom s treated as a geostrophc adjustment problem nvolvng two fluds ntally separated by a vertcal barrer. The fluds are otherwse unbounded laterally. Upon the removal of the barrer, gravtatonal flow that conserves potental vortcty develops n both fluds. t s ths flow that helps acheve the fnal state of equlbrum. The sloped bottom thus enhances the penetraton of the heavy flud when movement s down-slope and nhbts such penetraton when the movement s up-slope. The addton of a wnd-stress mpulse affects the adjustment n several ways. t ndrectly nduces a barotropc flow by forcng flud columns across topography. The dfference (mpulse over the lghter flud to the rght mnus that over the heaver flud to the left) n mpulses appled to the two fluds drectly affects the wdth of the frontal zone. Wnd-stress mpulses that force flud column movemento dverge (dfference < 0) result n a broadenng of the frontal zone. Wndstress mpulses that force flud column movement to converge (dfference > 0) result n a narrowng of the frontal zone. n the hypothetcal, laterally unbounded system, an addtonal effect of the wnd s to bodly dsplace the front, a possblty excluded n a rgd-ld treatment [see Csanady, 1978]. Acknowledgments. Ths research was supported by the Offce of Naval Research under contracts N00014-82-K0082 and N00014-81- K-0251 and by the Natonal Scence Foundaton under grant OCE80-15656. The manuscrpt was typed and fgures drafted by Beverly Morrs. Donna Arnold drafted Fgure 12 and helped wth the fnal draft. The computaton s made on computer facltes supported by the Offce of Naval Research under contract N00014-80-C-0076. n Fgure 6 excepthat the densty dfference s doubled. The ntersecton of the new front wth the bottom s assumed to REFERENCES occur at the shelf break. The value of the dsplacement Csanady, G. T., On the equlbrum shape of the thermoclne the functon r2 at the shelf break s thus determned by the shore zone, J. Phys. Oceanogr., 1,263-270, 1971. volume of the solated lens. The computaton proceeds as Csanady, G. T., Wnd effects on surface to bottom fronts, J. outlned before. The results are presented n Fgure 12. The Geophys. Res., 83, 4633-4640, 1978. orgn of the x axs s now placed at the shelf break. An Flagg, C. N., The knematcs and dynamcs of the New England solated lens of the lower layer flud s found over the lower contnental shelf and shelfslope front, Ph.D. thess, Mass. nst. of Technol., Cambrdge, Mass., 1977. slope. The wdth of the lens s gven by a - b - 0.239 Flagg, C. N., and R. C. Beardsley, On the stablty of the shelf compared to the baroclnc deformaton radus of 0.085, waterslope water front south of New England, J. Geophys. Res., based upon the maxmum lens thckness of 0.731. 83, 4623-4631, 1978. The separaton of a part of the upper flud can also be Hsueh, Y., G. O. Marmorno, and L. L. Vansant, Numercal model studes of the wnter-storm response of the West Florda Shelf, J. calculated n a smlar manner. But here, the erecton of a Phys. Oceanogr., 12, 1037-1050, 1982. coastal boundary wthn a barotropc deformaton radus of Smth, P. C., Baroclnc nstablty n the Denmark Strat overflow, the frontal zone may be essental [see Csanady, 1978]. The J. Phys. Oceanogr., 6, 355-371, 1976. presence of a coastal wall wthn a barotropc deformaton Stommel, H., and G. Verons, Barotropc response to coolng, J. radus of the front can also restrct the movement of the front Geophys. Res., 85, 6661-6666, 1980. as a rgd body and alter the shapng of the frontal zone n response to wnd. t s concevable that n the stuaton of Fgure 8, the blockng effect of a coastal wall n the shallow (Receved July 19, 1982' regon could lead to a near-bottom return flow that tend to revsed September 20, 1982; shft the front foot back toward the shallow (nshore) sde. accepted October 1, 1982.)