OCTOBER 1995 K ARSTEN ET AL. 2391Stability Characteristics of Deep-Water Replacement in the Strait of Georgia RICHARD H. KARSTEN AND GORDON E. SWATERSApplied Mathematics Institute, Department of Mathematical Sciences, University of Alberta, Edmonton, Alberta, Canada RICHARD E. THOMSONInstitute of Ocean Sciences, Sidney, British Columbia, Canada(Manuscript received 6 July 1994, in final form 21 March 1995)ABSTRACT It has been suggested that low-frequency current fluctuations in the southern Strait of Georgia are the resultof baroclinic instability. However, data extracted from cyclesonde and fixed current meter moorings suggest thatthe conditions for baroclinic instability are highly variable in space and time. It has been recently discoveredthat there are summertime bottom-intensified gravity currents with fortnightly and monthly periods associatedwith the introduction of salty waters from the Juan de Fuca Strait during periods of neap tides. These currentsare the dominant mechanism for deep-water renewal in the Strait of Georgia, It is argued that these currents arebaroclinically unstable and that the stability characteristics are reasonably consistent with the observed structureof the low-frequency current fluctuations. The episodic nature of these unstable bottom flows may help to explainthe spatial and temporal variability of the low-frequency current fluctuations observed in the Strait of Georgia.1. Introduction The origin and evolution of the low-frequency variability in the Strait of Georgia (SOG) are not completely understood (e.g., see the discussion in LeBlond1983). In particular, many explanations have been proposed for the low-frequency current fluctuations thatoccur in the southern region of the SOG. The energyassociated with these currents is known to be as important as that associated with the diurnal and semidiurnal tides (Chang et al. 1976). Helbig and Mysak(1976) initially suggested that the low-frequency variability could be described by bottom-trapped topographic planetary waves. However, a later analysis ofavailable SOG data (Yao et al. 1982) showed that thevertical structure of the observed current fluctuationswas inconsistent with the Helbig and Mysak model.Yao et al. (1982) also examined and dismissed the possibility that the low-frequency variability could be described by internal Kelvin waves. Shortly afterward, Yao et al. (1985) proposed thatthe low-frequency current fluctuations were the resultof a quasigeostrophic instability associated with the observed mean flow. However, the agreement betweenthe theory and the observations was problematic sincethere were clearly times in the year when the obser Corresponding author address: Dr. Gordon E. Swaters, Dept. ofMath., Applied Math. Inst., University of Alberta, 632 Central Academic Building, Edmonton, Alberta T6G 2G 1, Canada.E-mail: gordon @ hamal.math.ualberta.cavations did not satisfy various necessary conditions forquasigeostrophic instability (e.g., there were statistically insignificant correlations between the density andvelocity fluctuations and there was no evidence of upward phase propagation of the disturbances), but thetheory predicted instability nevertheless (Yao et al.1985; Stacey et al. 1991). In a series of subsequent papers, Stacey and colleagues (Stacey et al. 1987; Stacey et al. 1988; Stacey et al. 1991) presented and analyzed data takenfrom an array of cyclesonde and fixed current meterswith sufficient spatial coverage to resolve the horizontal and vertical structure of the low-frequencyvariability in the southern part of the SOG. Their results were surprising in several respects. The fluctuations appear to have a relatively small horizontallength scale on the order of about 10 km (Stacey etal. 1987; see also Yao et al. 1985). Based on objectively produced streamfunction maps, Stacey et al.(1988) concluded that there was "unmistakable evidence'' for the formation of subsurface eddies associated with the current fluctuations. Stacey et al. (1991) addressed the issue of identifying an energy source for these low-frequency motions. Based on the Stacey et al. (1987) dataset, it wasdetermined that the correlation between the density andcurrent fluctuations was statistically significant, and ofthe correct sign for baroclinic instability, below approximately 160 m of depth. However, the appropriatecorrelations for instability were not observed at allmoorings in the region examined. Stacey et al. ( 1991 )c 1995 American Meteorological Society2392 JOURNAL OF PHYSICAL OCEANOGRAPHY VO~.UME25 concluded that if baroclinic instability is occurring, it is horizontally localized in space. It would seem clear that if the observed subsurface eddies are the result of the baroclinic instability of a background current, it is an instability that occurs at depth but which is not uniformly distributed over the entire horizontal region or over the entire year. The problem then is to identify an unstable flow feature in the SOG that has these qualitative characteristics. LeBlond et al. ( 1991 ) have established the existencein the SOG of summertime bottom-intensified currentsthat have fortnightly and monthly periods. These currents are the result of the introduction of salty watersfrom the Juan de Fuca Strait during periods of neaptides and are not formed during the winter months.They provide the dominant mechanism for deep-waterrenewal in the SOG. These currents flow northward onthe eastern side of the SOG in the form of an elongated,bottom-trapped gravity current that is transversely confined in the across-strait direction. These currents therefore possess the property that they are not uniformlydistributed in space and time. The principal purpose ofthis paper is to show that these gravity currents areunstable and that the stability characteristics of the mostunstable modes are consistent with the observations. Mesoscale gravity currents are formed when densewater is formed or otherwise released in a shallow sea,such as a shelf region, and settles to the bottom. If thebottom is sloping, then the combined influences of theCoriolis and buoyancy stresses may force the currentto be transversely constrained and flow, in the NorthernHemisphere, with the direction of locally increasingbottom height to its right. Swaters (1991) developed anonquasigeostrophic baroclinic instability theory formesoscale gravity currents on a sloping bottom. Theinstability mechanism modeled by Swaters is the release of the available gravitatiohal potential energy associated with a pool of relatively dense water sittingdirectly on a sloping bottom surrounded by relativelylighter water. As such, this instability mechanism is,phenomenologically, completely different than theshear-based instability associated with a buoyancydriven current containing lighter water sitting on top ofa finite lower layer (e.g., Paldor and Killworth 1987).The Swaters theory describes a purely baroclinic instability in that it filters out the shear-based instability andexclusively models the convective destabilization of amesoscale gravity current on a sloping bottom. Moreover, this instability model does not make theassumption, implicit in quasigeostrophic theory, thatthe dynamic deflections in the thickness of the gravitycurrent are small in comparison to its scale height. Byallowing for finite-amplitude deformations in the current height, the Swaters theory can describe the instability of gravity currents with isopycnals that intersectthe bottom. In addition, the Swaters model does notrequire a zero in the transverse potential vorticity gradient for instability. The intrinsically baroclinic instability of the Swaters model differs from the nonbaroclinic instability identified by Griffiths et al. (1982)associated with a coupling of the two fronts in a mesoscale gravity current. [For a discussion comparingthese two models, see Swaters (1991).] Numerical simulations based on the primitive equations (M. Kawase1994, personal communication) suggest that the convective instability mechanism is two orders of magnitude more important than any other instability mechanism for mesoscale gravity currents. Our results show that a model gravity current thatreproduces the principal qualitative features of thedeep-water replacement current in the SOG is baroclinically unstable. It turns out that the most importantparameters in the stability calculation are the nondimensional width of the front relative to the internaldeformation radius and a parameter, denoted p, thatmeasures the ratio of the maximum current height tothe slope of the ambient topography. Physically, theparameter/~ has a straightforward interpretation in thatit measures, roughly speaking, the ratio of the destabilizing influence of baroclinic vortex-tube stretchingin the nonfrontal layer to the stabilizing influence ofthe sloping bottom, which acts as a topographic /5plane. The plan of this paper is as follows. In section 2, wepresent the model and derive the linear stability equations and boundary conditions. In section 3, we brieflypresent some general stability properties of the linearstability and normal-mode equations. In section 4, wepresent our instability calculation for a model mesoscale gravity current on a wedge-shaped bottom. In section 5, we discuss how the solution depends on thevarious parameters and apply the model to the SOG.The paper is summarized in section 6.2. Problem formulationa. The governing equations Since the derivation of the equations is very similarto that described by Swaters (1991), we will be briefin our presentation here. The basic model we assumeis an f-plane, two-layer shallow-water system (bothlayers are assumed hydrostatic and homogeneous) withvarying cross-channel topography (see Fig. 1 ). If we denote the geostrophic pressure in the upperlayer by ~(x, y, t), the height of the gravity current byh(x, y, t), and the variable bottom topography byh~(y), the nondimensional governing equations can bewritten in the form A~/, + h~(~7.~ + hx) +/~J(~/, At/) = 0, (2.1) h, - h~,.h~, + #J(rl, h) ~ O, (2.2)where J(A, B) = AxBy - AyBx, with x and y thealongchannel and cross-channel coordinates, respectively. The leading-order velocity fields in the upperand lower layers, denoted u~(x, y, t) and u2(x, y, t),OCT(mER 1995 K ARSTEN ET AL. 2393Pl,U_i Z ~, ~o/2~..~x ~:-B HFiG. 1, The geometry of the two-layer system, in a channel withcross-channel topography given by hs(y) and walls at y = -B, D.respectively, and the geostrophic pressure in the lowerlayer, denoted p(x, y, t), are related to r/(x, y, t) andh(x, y, t) via the relations u] = & x Vr/, (2.3) u2 = -h&g~ +/.t~3 x ~7(r/+ h), (2.4) p = ha +/a(r/+ h), (2.5)where the parameter t~ m h,/hR, with h,, a scale heightfor the gravity current and h~,, a scale height for thevariable bottom topography. The parameter/.t can be interpreted as measuring theratio of the destabilizing influence of baroclinic vortextube stretching to the stabilizing influence of the variable bottom topography that acts as a background vorticity gradient (i.e., a topographic/~ plane). As it turnsout, for a given along-channel mode, a minimum/.t isrequired for instability. If we denote the projection on the plane z: 0 of aparticular intersection of the front height with the bottom by 4>(x, y, t) = 0, then the kinematic condition isgiven by 4>, - h~v 4>~, +/~J(r/+ h, 4>) = 0, (2.6)on 4> = 0, and the frontal height must satisfy h = 0 (2.7)on 4> = 0. Note that the determination of the evolutionof 4>(x, y, t) is part of the problem. The no-normal flowcondition at the channel walls (see Fig. 1 ) is given by r/x=0 on y=-B,D. (2.8) Equation (2.1) [actually (2.1) + (2.2)] expressesthe conservation of potential vorticity in the nonfrontallayer. Equation (2.2) expresses mass conservation forthe gravity current with a geostrophically determinedvelocity field, which includes the effects of an upperlayer and the gravitational acceleration associated withthe sloping bottom. The derivation of these governingequations is identical to the formal asymptotic reduction of the two-layer, shallow-water equations presented in Swaters ( 1991 ), except that the bottom slopeparameter s. in $waters ( 1991 ) would need to be replaced by h~./L (where L is the horizontal length scale)to reflect the general topographic height retained here.This derivation is not included here and the reader isreferred to Swaters ( 1991 ) for further details.Linear stability equations and boundaryconditionsIt is straightforward to verify that (2.1)- (2.8) possess the exact steady alongchannel solution ~ho(y), a, < y < a2 h=[0, -B < y < a, or a2 < y < D, r/: r/0(y), 4>, = y - a, (2.9) 4> = 4>2 Y a2.In order to focus attention on the baroclinic instabilityof the gravity current and filter out any barotropic instability in the nonfrontal layer, we set % ~ 0. The linear stability equations for this flow configuration are obtained by substituting h = ho(y) + h'(x, y, t), ~7 = rf (x, y, t), 4>, = y - a, - 4>~(x, t), 4> = 4>2 = Y - a2 - 4>~(x, t), (2.10)into the model equations and neglecting all quadraticand higher-order perturbation (primed) terms. In theregion a~ < y < a2, the linearized equations take thefo_rm (henceforth, we shall drop the prime notation forthe perturbation fields) A~t '~- hBy(llx + hx) = O, (2.11) h, - havh~ +/~h0~ r/x --- 0. (2.12)In the nonfrontal regions, -B < y < a~ and a2 < y< D, the stability problem for the channel water issimply At/, + hm~/~ = O. (2.13)The linearized and Taylor expanded boundary conditions are given by h + ho.~4>,.2 = 0 on y = a,.2, (2.14) on y = at.2, (2.15)2394 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME25 ~x=0 on y=-B,D. (2.16)We also impose the condition that the channel-waterperturbation pressure and normal mass flux be continuous at y = a~ and y = a:. We have made the assumption that the current associated with the deep-water replacement in the SOGcan be modeled with a function to(y) that extends infinitely in the alongchannel direction. This is obviouslyan extremely crude model-for a flow, which, in reality,is more like a pulse than a steady current. However,even though the water mass associated with the deepwater renewal extends finitely in both the alongchanneland cross-channel directions, the data clearly indicatethat the cross-channel length scale is much smaller thanthe alongchannel length scale, so that this approximation is reasonable for the instability calculation presented here. The data presented in Stacey et al. (1987) and furtherdiscussed by LeBlond et al. ( 1991 ) clearly indicate thatthe strong currents associated with the deep-water renewal are not uniformly distributed across the strait. Ifone uses the distance between the cyclesonde stationswhere the current was strongly observed and thosewhere it was not so strongly observed, a crude estimateis that the cross-channel width of the current is no morethan about 20 km. On the other hand, the LeBlond et al. ( 1991 ) analysisof the data suggests that each pulse of deep-water renewal lasts continuously for about 10 days and propagates northward at a speed of about 18 cm s-t. Geometrical considerations aside, this suggests that thealongchannel length scale of the pulse is about 150 kin,which is roughly on the order of 10 times the currentwidth.c. The normal-mode equations The normal-mode equations are obtained by assuming alongchannel propagating solutions of the form[~/, h, qb,, qb] = [~/(y), fi(y), fb,, $21 x exp[ik(x - ct)] + c.c., (2.17)where c.c. means complex conjugate, k is the alongchannel wavenumber, and c is the alongfront complexphase speed. Substitution of (2.17) into (2.11) and(~2.12) yields [ after dropping the tildes and eliminatingh(y) in (2.18) using (2.19)] [ _~h,~hOV]c + C(~yy -- k2~7) -- hBy -~- hl~ ]~ = O, (2.18) phoy h ~- -- 7, (2.19) c + hByin the frontal region at < y < a2. We solve (2.18) for~/(y) and then compute h (y) using (2.19) in this region.In the nonfrontal regions where -B < y < at and a2< y < D, the problem for ~(y) is given by c(~/yy - k2~/) - h~y~? = 0. (2.20) The boundary conditions at y = ate2 for the normalmodes can be written in the form h + h0.vqb~,2 --- 0, (2.21) (C + hBy)~bt,2 = -/~, (2.22)where we have eliminated the h(at.2) term in (2.22)using (2.14). The boundary condition (2.16) becomessimply ~=0 on y=-B,D. (2.23)In addition, we also require that the pressure and normal mass flux in the channel water be continuous acrossthe frontal boundaries. Since there is no mean flow inthe upper layer this is equivalent to requiring that r/yand ~ be continuous at y = at.2.3. General stability characteristics In this section, we derive some general stability results for the model equations. These results are usefulin analyzing the detailed stability calculations presented in section 4. The reader who is not interested inthese general derivations may go directly to section 4.a. Perturbation energetics If (2.11 ) and (2.13 ) are multiplied by ~(x, y, t) andsubsequently integrated over -B < y < D and 0 < x< h, where 3, is the alongchannel wavelength of theperturbation, it follows (after integration by parts) thatthe averaged perturbation kinetic energy of the nonfrontal layer satisfies of/ -- (~'~-V~/)dy = -2 ht~vvth)dy, (3.1)Ot ~ , 'where ((,)) = x-'(*)&. It follows ~at if b~oclinic instability occurs, thenon average the co,elation between ~e pen~bationcross-ch~nel velocity in the channel water, ~e frontalheight ~omaly, and ~e ch~nel-bottom slope must benegative. If we assume thin on average the channel bottom slopes downw~d towed ~e center of the channel,then, as found in Swaters ( 1991 ), a necess~ conditionfor instability is a net tr~spo~ of heat towed ~e ch~nel w~ls. Since the bottom slope is equiv~ent m atopographic B plane, we c~ inte~mt a net flux of heattowed the wMls, that is, up the sloped bottom, asequivalent to the no~hw~d flux of heat required in~dlatimde baroclinic instability (~Blond and Mys~,section 44).OCTOBE~ 1995 K ARSTEN ET AL. 2395 If (2.12) is multiplied by he~ h, then we can form thebalance __O ~'--h2&' (h2)dy=-2 h&(u,h)dy. (3.2) Ot ~tha.,. ho,. 'This expression and (3.1) can be combined to form (3.3)It immediately follows ~at instability c~ only occurif ha~(y)ho~(y) > 0 for some vMue ofy - (a~, a2).Conversely, it follows that if ha,. (y)ho~ (y) ~ 0 for MIy ~ (a~, a2), then the front is line~ly stable.b. General stability results for the normal modes By multiplying (2.18) and (2.20) by the complexconjugate of r/(y), integrating the result over -B < y< D, and adding the two equations together, it is possible to form the balance (after integration by parts): ~ clrbl2 + ck2 + hay + O(y)l. th~vho~ B I c + h~,.I2 1~[2 dy = 0, (3.4)where O(y) = 1 for a~ < y < a2 and O(y) = 0 for -B< y ~< a~ and a2 ~< y < D, and c* is the complexconjugate of c = cR + ict. The imaginary and real partsof (3.4) are given by, respectively,c, f_~[iv~l~ + (k2. O(y)gh&ho~~_~ )(3.5) ~ [ O(y)gh&hov], , = - 1 + i7; h-~2 Jn*'lr/12dy' (3.6) B We see ag~n that from (3.5) a necess~y conditionfor instability is ~at ha~ (y)ho~ (y) > 0 for some valueof a~ < y < a2. Consequently, assuming ~at instabilityoccurs, we may set max [h&(y)ho.~(y)] = T~ > 0, (3.7) yG(alwhere 7 > 0, which will be used momentarily. Assuming that instability occurs, the expression inside the curly brackets in (3.5) must be identically zeroand can be rearranged into the formwheref j,'- ha~ h0~ = ~Q(3.8) Q = (l%l2 + k2lrll~)dy, (3.9) Bwhich is just the kinetic energy of the perturbation fieldin the channel water. However, using (3.7) it followsfrom (3.8) that Ic + ~ ~T2 'which can be rearranged to imply min I c + h~l~ ~ t. t3`aQ-~ Inl~dy,Y~(al,a2) ' I ~</.tT 2Q-' Irl[2dy, B (t~2~~-' ak2rll2dy, k2 -(3.10) If we now assume that max ha~ (y) = a~ and miny~(al,a2) y~(al,a2) (3.11)where -oo < cr2 ~ oq ~ 0% it follows from (3.10) thatif instability occurs, then the complex phase speed mustlie in the region defined by /.t3`2 (c,+a~)2+c~2~<-~-, if c,<-a~, /.tT2 ct2 ~< -~-, if -at ~< C, ~ --a2, (3.12) /.tT2 (cn + or2)2 + c~2 <~ -~, if ca >where c~ ~> 0 and 3,2 is determined from (3.7). Theregion represents a rectangle of length a~ - a2 with aquarter circle on each end, with the height of the rectangle and the radius of the circles given by t. tl/23,/k.Note that the area of the region will increase with increasing/~, but will decrease with increasing alongfrontwavenumber k. Also, it follows from (3.12) that thegrowth rate satisfies cr = kc~ ~< 3,/.d/2. (3.13) We can get alternative bounds on the real part of thephase speed as follows. Assuming instability occurs,2396 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME25we may use (3.8) to simplify (3.6). The result can bewritten in the form O( y ) ~hB,, hocR=-(2Q)-'f_-R[l+ i~--~&,2" ]ha,.,rll2dy. (3.14)From (3.14) a second set of bounds on the real phasespeed can be determined. These bounds together with(3.12) can be used to infer the existence of a highwavenumber cutoff. The details are similar to thosepresented in Swaters (1991) and are omitted. In thenext section, we will explicitly compute the high wavenumber cutoff for a wedge-shaped bottom profile.4. Stability calculation for a parabolic gravity current on a wedge-shaped bottom The available data is unfortunately not able to resolve the detailed cross-channel shape of the gravitycurrent associated with the deep-water replacement inSOG. Nevertheless, much can be learned from determining the stability characteristics associated with amodel gravity current given by(4.1)where I m a~ - a~ is the width and a ~ (a~ + a2)/2 isthe midpoint of the unperturbed current. Note thatho(a) = 1, which means that the scale current heightis chosen as the maximum dimensional height of theunperturbed current. The model (4.1) allows for thegravity current height to intersect the bottom in a coupled-front configuration. In section 5, we will choosevalues of a and I so that our model will closely approximate the current described in the volume flux estimates of LeBlond et al. ( 1991 ). The bottom bathymetry in the region of the SOGwhere the Stacey et al. (1987) dataset was collectedcan be reasonably well modeled with a wedge-shapedbottom of the form ~ -s~y, -B < y < Oha(y) = (4.2) I. s2y, 0 < y < D.We consider only the case where the gravity currentlies entirely on one side of the channel, that is, when itlies entirely in either the y < 0 or y > 0 region. Thecase where the current spans y = 0 will not be discussedbecause the linearization of the problem fails across thediscontinuity in hay at y = 0. Since the problem is symmetric in y we will study only the case where a2 ~ 0.This implies that we may choose ha, = Sl*L, so that s~= (L/ha.)s~* = 1. Values for the parameters s2, B, andD will be discussed in section 5.a. Derivation of the dispersion relation Substitution of hB(y) given by (4.2) and ho(y) givenby (4.1) into the normal-mode equations (2.18) and(2.20) yields the equations tlyy -- k2 -- ~ T~ -~- O, for -- B < y < al (4.3) ~ - - - + c--~c ~_ i)12 ~/=0, '' C fora~ <y<a2a2<y<0(4.4)(4.5) c'- - - ~/= 0, for Tlyy-k2q-~Tl=O, for 0<y<O, (4.6)with the boundary conditions ~/=0 on y=-B,D, (4.7)and the pressure and mass flux matching conditions [~] ~- [9~y] ~ 0 on y = a~,2. (4.8) The general solution to (4.4) may be written in the formn(y) = C,A,[~(y)] + C2B~[~(y)], for a, < y < a2, (4.9)where &.(() and Bi(() are Airy functions (Abramowitz and Stegun 1972, section 10.4) with argument~(y) given by + c c(c- 1)/2 ' (4.~0)and where C~ and C2 are, as yet, undetermined coefficients. Using (4.7), the solutions to (4.3), (4.5), and(4,6) respectively, may be written in the form ] (y + B)], for -B<y<a~, (4.11)V(y)=C4exp - k2-7] y]+Cs [( xexp k2-7,] yl , for a2 < y < O, (4.12) + for 0<y<D, (4.13)OC*OSER 1995 K ARSTEN ET AL. 2397where C~, C4, C5, and Co are additional, as yet, undetermined coefficients. We take our branch cut in thecomplex plane along the negative real axis. The application of the matching conditions(4.8) is straightforward and leads to a system ofsix homogeneous equations in the unknown coefficients C~, C2, C3, C4, Cs, and C6. This systemcan be most conveniently written in the matrixform M.C = O, (4.14)where C = (C~, C2, C~, C4, C5, C6)T (a column vector)and M is a 6 x 6 matrix determined by (4.9), (4. ll),(4.12), and (4.13). For a nontrivial solution of (4.14), we require det(M) = 0, (4.15)which forms the complex dispersion relationship forthe normal-mode solutions. We may consider that(4.15) implicitly defines a seven-parameter dispersionrelationship of the form c = c(k, t~, a, l, s2, B, D). (4.16)(In the above formulas, we can use a~ = a - l/2and a2 = a + I/2 since the front is symmetricabout a .) Assuming that (4.15) is solved, we may determinethe coefficients C2, C3, C4, C5, and Co as functions ofthe single free coefficient Ci. With r/(y) determined asdescribed here, the perturbation frontal thickness h(y)will be given by [ see (2.19) ] 8/~(y - a)~7(y) h(y) = - 12(c- 1) ' (4.17)in the region a~ < y < a2, and the amplitude of theperturbation frontal boundaries will be given by [ see(2.22)1 qb~.2 - --, (4.18) c-1at y = a~ and a2, respectively. Finally, with the complexnormal-mode amplitudes determined, the real valuedsolutions are obtained by substituting these amplitudefunctions into (2.17).b. High wavenumber cutoff and minimum interaction parameter estimates For this model, it is possible to explicitly determinea semicircle of instability, a high wavenumber cutoff,and a minimum tt needed for instability based on thetheory developed in section 3b. For the parabolic front(4.1), it follows using (3.7) that the region of instability described by (3.12) reduces to the semicircle region given by(cR - 1 )2 + c~2 ~< 4__~_~ lk2 '(4.19)which gives the growth rate bound 2/.ZI/2a ~< l~/~'~. (4.20) It follows from (3.14) that the real part of the phasespeed will satisfy 1ca = ~ + (2Q)-' x ~ Irll~dy- s2 Inl2dy , (4.21)which implies that 1 s2 1 1 ~ - 2k--~ ~< cn ~< ~ + 2k---~. (4.22)However, it also follows from (4.19) that the real partof the complex phase speed of an unstable mode mustlie in the interval 2b~I/2 2kt1/21 - kl~-~-~ ~< ce ~< 1 + kp/-~-~. (4.23) Clearly, for sufficiently large alongfront wavenumbers(for given 3t, l) the intervals (4.22) and (4.23) will bedisjoint since the interval in (4.22) collapses to a smallneighborhood centered at ca = 1/2 and (4.23) collapsesto a small neighborhood centered at ca = 1. Consequently,it follows that instabihty can only occur when 12 1 + >~ 1 kl~/2 , (4.24)which is the necessary and sufficient condition for theintersection of the two intervals (4.22) and (4.23) tobe nonempty. The inequality (4.24) can be rearrangedto imply that the wavenumber of an unstable modemust satisfy 2/~~/2 + (l + 4/.~)~/2 k ~< kmax = ll/2 (4.25)The value of kmax given by (4.25) is an overestimate ofthe actual high wavenumber cutoff. Given a particular wavenumber, the estimated minimum tz needed for instability as determined by (4.24)is given by l(k2 - 1)2 /-~m in -- 16k2 , (4.26)if k > 1 and tZm~, = 0. for 0 ~< k ~< 1. The value ofin (4.26) is an underestimate.2398 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME255. Description of the solutions and application to the Strait of Georgiaa. General discussionparaboliccurrent In this section, we choose "typical" values for the'parameters in (4.16). Using these values, we presentthe solution to the dispersion relation and discuss thestability characteristics. We also examine the effect ofchanging the parameters. As it turns out, the singlemost important parameter that influences the stabilitycharacteristics is the baroclinic stretching pfirameter/~. We will choose the typical parameter values to reflect the physical characteristics of the strait and thecurrent at the cyclesonde station where the current wasmost strongly observed [see station number 3 in LeBlond et al. (1991)]. The nondimensionalization schemeused to derive (see Swaters 1991 for details) the governing model equations (2.10 and (2.2) uses the internal deformation radius based on the total mean depthof the nonfrontal layer and the Nof speed (Nof 1983)as the horizontal length and velocity scales, respectively. Taking the averaged observed velocity as theNof speed gives the velocity scale of U ~ 18 cm s-l.Using a representative depth of the strait, H m 300 m,and an average bottom slope s~* ~-- 9 m km-I at theobservation point, we obtain an estimate of the horizontal length scale, L = (HU/fos~*)I/2 ~ 7 km. Timeis scaled advectively giving a timescale of approximately T m 11 hours. We will denote dimensionalquantities with an asterisk; for example, the nondimensional phase velocity is denoted c and the dimensionalanalogue is c*, and so on. We will model the strait as a channel of width 28 kmwith its center 21 km from the eastern wall. The bottom254380505 gravity current 7 0 -1.75 -8.75 ,-15.75 -21 Cross-Channel Direction FIG. 2. Geometry of the gravity current model used for the Straitof Georgia in dimensional units. The horizontal lengths are in kilometers and the depths in meters.255volume fluxcurrent280305 ~330 ~355 -1.75 -8.75 -15.75 Cross-Channel Direction FIG. 3. A comparison of the parabolic gravity current model usedin this paper with the volume-flux current used in LeBlond et al.(1991). The horizontal lengths are in kilometers and the depths inmeters.slope on the eastern and western side of the strait aretaken to be 9 and 18 m km-t, respectively. Based onestimates in LeBlond et al. ( 1991 ), we choose the maximum current height to be 65 m and the current widthto be 14 km. We will assume that the current lies completely on the eastern side of the channel with its centerlocated at the station 3 cyclesonde (see Stacey et al.1987 or LeBlond et al. 1991 ), which is about 8.75 kmfrom the center of the channel. In Fig. 2, we show thegeometry used to model the gravity current in the SOG. LeBlond et al. ( 1991 ), in estimating the volume fluxof the gravity current, used a model current with a trapezoidal cross section. In Fig. 3, we compare our parabolic current profile to their trapezoidal current profile.Using this model, we estimate that typical nondimensional parameter values would be given by/~= 1.0, a=-1.25, s2=2.0,./=2.0, B=3.0, D=I.0. In Fig. 4, we present the nondimensional growth rate,frequency, and phase speed obtained from the dispersion relationship versus the wavenumber for the typicalparameter values as given above. The graph is plottedover the interval 0 ~< k ~< kmax, where kmax is given by(4.25). Note that the actual high wavenumber cutoff isless than that given by (4.25). The most unstab!.e modeoccurs at k m 1.1, with the COrresponding growth ratea ~, 0.46, phase velocity cR ~, 0.65, and frequency w~ 0.72. In dimensional terms, this corresponds to awavelength of approximately 40 km, an e-folding timescale of approximately 1 day, a phase speed of approximately 12 cm s-I, and a period of approximately 4days. The stability boundary (i.e., the actual high wavenumber cutoff) corresponds to a wavelength of aboutOCTOBER 1995 K ARSTEN ET AL. 23990.5 i i i i i .0.40.20.10.0 0.5 1.0 1.5 2.0 2.5 Wavenumber, k3.0 3.50.80.60.40.20,0 , I i I I f ~ I ~ I ~ I ~ 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 Wavenumber, k3.02.52.01.50.50.0 0.0 I ' I ' I ' I I I ', I , f I I I I ~ I ~ I . I 0.5 1.0 1.5 2.0 2.5 3.0 WaYenumber, k3.5 FiG. 4. Graphs of the nondimensional (a) growth rate, (b)frequency, and (c) along-channel phase speed versus thealong-channel wavenumber k for ~ = 1.0. The range of thegraph is given by 0 ~< k ~ L,ax, where km~x is given by (4.25).24 km. Thus, all modes with wavelength longer thanabout 24 km are unstable. This is a relatively rapidly growing instability. Asthis mode grows, nonlinear effects eventually come todominate the evolution. These nonlinear effects couldlead to further destabilization and even smaller lengthscale eddylike features could develop or they could stabilize the instability. Assuming that the stabilized eddylike anomalies would have length scales of about ahalf-wavelength, our calculations suggest the finite-amplitude anomalies would have a length scale of about20 km. This value is somewhat larger than the objectiveanalysis of Stacey et al. (1988) would suggest, but isreasonably consistent nonetheless given our uncertainty in the parameter estimates and the sensitivity ofthe calculation to the parameter values. Next we examine how the stability characteristicschange as the parameters are varied. We will concentrate on how parameter variations lead to changes inthe high wavenumber cutoff and the wavenumber andgrowth rate of the most unstable mode. The parameter /~ has the most direct impact onthe stability characteristics. This parameter reflectschanges in the scale frontal height and bottom topography and therefore, reflects changes in two of the morevariable aspects of the flow geometry that we had toestimate. In Fig. 5a, we plot the calculated maximumgrowth rate wavenumber, the high wavenumber cutoff2400 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME255.04.54.03.53.02.50.50.0~ 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 Interaction Parameter,1.21.11.00.90.80.70.60.50'.40.30.20.10.0 0.0~ I I I i I ~ I ~ I ~ I t -~ ;i I i I I I , I ~ I ~ I i 0.5 1.0 1.5 2.0 2.5 3.0 Interaction Parameter, gFIG. 5. (a) (from bottom to top) The wavenumber of the most unstable mode, the actual stability boundary, and the predictedhigh wavenumber cutoff as given by (4.25) versus the parameter it. (b) The maximum growth rate versus it.3.5(i.e., the actual stability boundary) and the predictedhigh wavenumber cutoff as given by (4.25) versus theparameter/~. For a given value of/z, all wavenumbers less thanthe wavenumber of the stability boundary correspondto unstable modes. Hence, as /z increases there is anincreasing band of wavenumbers that are unstable. Inaddition, as/~ increases we see that the wavenumber ofthe most unstable mode also increases, which impliesthat the length scale of the instabilities would decreasewith increasing/~. The parameter )z can be directly related to the slope of the bottom since h. h, I~ - hB, s ~* L '[see the discussion after (4.2)]. Thus, reducing the slopes ~* increases/~, and we conclude that decreasing the localbottom slope decreases the wavelength of the most unstable mode. This is of interest because the current arraydeployed by Stacey et al. (1987) is located in a regionwhere the local bottom slope is decreasing in the northward direction. We suggest that the small-scale subsurface eddy features observed in the objective analysis ofStacey et al. (1988) may correspond to instabilities of thesort described here, which rapidly develop into smallerscale structures due to the locally decreasing bottom slopeas the destabilized pulses of deep water move northwardon the eastem side of the strait. - Note also that Fig. 5a clearly shows that the actualhigh wavenumber cutoff is less than the predicted oneas expected, but it also shows that (4.25) gives a relatively accurate prediction of kraal. The high wavenumber cutoff and most unstable mode wavenumber growlike )z~/: as (4.25) suggests. In Fig. 5b, we plot the maximum growth ra~e versusthe parameter/~. Clearly, increasing the value of/~ increases the growth rate of the most unstable mode. Themaximum growth rate appears to grow linearly in ~L forthese small values of/~ rather than like/~/2, as suggestedby (4.20). This implies that a higher front or a flatterbottom would result in a more rapidly developing instability at a shorter wavelength. For example at/~ = 2.0 or3.0, values that could reasonably occur in the flat centralsection of the SOG (see LeBlond et al. 1991 ), the wavelength of the most unstable mode becomes approximately29 and 24.km, respectively, and the e-folding time isabout 14 and 10 hours, respectively. Thus, it is plausiblethat once the deep-water replacement current goes unstable and the anomalies propagate northward, the alongchoannel variations in the bottom slope may act to increasegrowth rates and shorten the length scales of the perturbations leading to smaller length scale current fluctuations. This is a scenario that is completely consistent withthe suggestion of Stacey et al.. (1991) that the low-frequency current fluctuations look more like geostrophicturbulence than wavelike features. It should be noted thatas )z becomes very large,/~ >~ 8, a second mode of insta~bility develops with an even more complicated horizontalstructure. This second mode is not likely to occur in theSOG, and the reader is referred to Swaters (1991) forfurther details. Changing the width of the front also has a noticeableeffect on the stability characteristics. In Fig. 6, we examine the effect of the frontal width on the growth rateof the most unstable mode. We see that as I increasesOCTOBER 1995 K ARSTEN ET AL. 24010.500.45 0.40rr 0.35'~ 0.30~ 0.25E'-', 0.20E~ 0.15 0.100.050.00 , I [ I i I [ I [ I [ I i 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 Current Width, Q FiG. 6. Graph of the maximum growth rate versus the current width parameter l.the growth rate of the most unstable mode also increases. We conclude that broader gravity currents experience a more rapid destabilization than narrowercurrents for the same maximum current height. Note that at first appearance, (4.20) and Fig. 6 contradict each other. From Fig. 6, we see that the growthrate decreases as l decreases, but (4.20) suggests theupper bound increases. The figure shows the correctbehavior since if no front exists; that is, l = 0, thenthere is no source of instability in the model. The problem with (4.20) is that it has been derived assuming c/#: 0. In fact, it follows from (3.5) that in the limit l -0; that is, a~ -, a2, that c/-- 0 since a2lim f, hBv hov.2~., , I;7~-~1~ I~l~dy f, --8y + 4(al + a2) = lim 1 ,2 a2~., I c + 112 , (a2 - a,)2= lim Inl2~y a2fa [-8y + 4(a, + a2)llwl2dy.2~., I c + I I 2 (a2 -- al)21Ic + 112 a2 --4(a2 -- a,)l~(a2)l2 + 4 f. Ir~12dy Ix lim 2(a2 -- al) 1Ic + 112 (-21r/(a')12 + 21r/(a')12) = O.This fact together with (3.5) implies that c~Q = 0,where Q is given by (3.9), which in turn implies c~-= 0 for a nontrivial mode. Changing the placement of the channel walls at y-- -B and y = D, the placement of the center of thefront at y = a, and the western bottom slope, s2, haslittle effect on the solutions. Only when D - 0, or s2 -0% and the down slope edge of the front nears the centerof the channel, is there a noticeable effect. In thesesituations the western wall, or steeply sloped bottom,prevents the slumping action of the current, and thusinhibits instability. These are extreme situations and notof relevance to the SOG. Thus, we see that it is theparameter/~ and, to a lesser extent, the frontal width 1that are most important in determining the characteristics of the solution.b. Spatial structure of the unstable modes In this section we describe the spatial structure of themost unstable mode for the typical parameter values.To effectively illustrate what the shape of an unstablegravity current is according to our theory, we will introduce a sufficiently large perturbation amplitude. Inwhat follows we have chosen the free coefficient Cl in(4.9) so that the maximum perturbation height is approximately one. In Fig. 7a, we depict the perturbation pressure fieldin the upper layer. The anomalies take the form ofalternating cyclones and anticyclones. The wave fieldcan be thought of as essentially an amplifying topographic planetary wave. The local extrema in thepressure field are centered near the downslope edgeof the unperturbed front located at y -- a2 = -0.25.This reflects the fact that the amplitude of the instability is maximized at precisely the location wherethere is the maximum release of local available potential energy. The wave field propagates in the positive x direction. In Fig. 7b, we depict the total height of the perturbed gravity current. The most important feature tonote here is that the current boundary on the offshoreor downslope side is substantially deformed in comparison to the onshore or upslope side. This reflects,again, the fact that the energetics of the instability isthe release of the available potential energy associated with the downslope "slumping" of the gravitycurrent. Note also, that if we compare Fig. 7a withFig. 7b, the local extrema in the upper layer are displaced slightly in the positive x direction. This, ofcourse, reflects nothing more than a "westward"phase shift with height associated with an unstablebaroclinic wave (recall the sloping bottom acts likea topographic fi plane).6. Summary and conclusions We have presented a theoretical calculation for thebaroclinic instability of the mesoscale gravity current2402 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME25 , -0. 5. , 0.05 , u.~l-, ,- ,,, / 0 0 ,,,/,-0.~5,,,,,,, .... .o.~5 .... ,,,,,, \\~ ,,~ IIIII" ,, k IIIII ~ IIIIIo IIIll~ ~ ........ II~XX /1111 -1.0 ..., .... ,,,',-..,//,~ F/,,,, '-- /' ' -3.0 I , I I I I I I -6.0 -4.0 -2.0 0.0 i iI i i/','2','"", V, "I L ~',h',',,',.' [ I I II I 2.0 4.0 6.0Along-Channel Coordinate, x 1.0 0.5~0.0c~ -0.50'~ -1.0 (~ -1 ..5 m -2.0~o0 -2.5 -3.0 -6.0I I I I I I I I I I I -4.0 -2.0 0.0 2.0 4.0 6.0 Along-Channel Coordinate, x FIG. 7. Horizontal contour plots for the most unstable mode for/~ = 1. (a) The perturbation pressure. (The contour interval is 0.05.) (b)The total frontal height. (The contour interval is 0.1.) The instability appears as a series of growing, propagating anticyclones on the downslopeside of the gravity current.associated with deep-water replacement (LeBlond etal. 1991 ) in the Strait of Georgia. Our model is basedon the nonquasigeostrophic theory developed by Swaters ( 1991 ) for the convective instability of a densitydriven current on a sloping bottom. This is a model thatexplicitly filters out horizontal shear-based instabilitiesand the destabilization associated with a coupling oftwo lateral free streamlines. Our calculations indicate that the deep-water replacement current is unstable. For parameter values reflective of the data as seen at cyclesonde station 3 inStacey et al. (1987) or LeBlond et al. ( 1991 ), our calculations suggest a most unstable mode with a wavelength of about 40 km and an e-folding growth timescale of about one day. The wavelength is somewhatlarger than the length scales suggested by the objectiveanalysis of Stacey et al. (1988). However, the wavelength of the most unstable mode is rather sensitive tothe choice of parameter values, and it can be arguedthat for environmental parameter values located nearbut not exactly at cyclesonde station 3 in Stacey et al.(1987) the wavelength of the most unstable mode isreduced by a factor of almost 2. The rapid timescale ofthe instability is encouraging since it suggests there issufficient time for the finite-amplitude development ofthese unstable modes into small-scale eddylike anomalies with a characteristic radius of approximately 10km. This is an estimate that is close to the objectiveanalysis results of Stacey et al. (1988) for the subsurface anomalies. The origin and structure of the low-frequency currentfluctuations in the SOG are not completely understood.In particular, the spatial and temporal variability of thefluctuations has not been adequately resolved. Manyexplanations have been proposed for these fluctuationsranging from bottom-trapped vorticity waves or Kelvinwaves to classical baroclinic/barotropic instability. Allof these explanations have been problematic to somedegree. It may be that there is no single dominant source forthese low-frequency fluctuations. However, we suggestthat the rapid destabilization of the current associatedwith the deep-water replacement does address severalaspects of the observations particularly in regards tothe observed horizontally localized and seasonally variable nature of the conditions for baroclinic instabilityand the fact that the instability seems to occur at depth. Whether or not, in fact, the convective instability ofthe water mass associated with the deep-water replacement in the SOG is the primary source of these fluctuations cannot be definitely resolved with the availabledatasets. Clearly, an observational program designed tomonitor the evolution in time and space of the this water mass would go a long way to answer this question.Acknowledgments. Preparation of this manuscript. was supported in part by a research grant awarded bythe Natural Sciences and Engineering Research Council of Canada (NSERC) and a science subventionawarded by the Department of Fisheries and Oceans ofCanada to G. E. S. and a NSERC Postgraduate Scholarship awarded to R. H. K. We also thank the refereesfor their comments, which led to a substantially improved paper.OCTOBER 1995 K ARSTEN ET AL. 2403REFERENCESAbramowitz, M., and J. A. Stegun, 1972: Handbook of MathematicalFunctions. Dover Press, 1046 pp.Chang, P., S. Pond, and S. Tabata, 1976: Subsurface currents in the Strait of Georgia, west of Sturgeon Bank. J. Fish. Res. Board Can., 33, 2218-2241.Griffiths, R. W., P. D. Killworth, and M. E. Stem, 1982: Ageostrophic instability of ocean currents. J. Fluid Mech., 117, 343-377.Helbig, J. A., and L. A. Mysak, 1976: Strait of Georgia oscillations: Low frequency currents and topographic planetary waves. J. Fish. Res. Board Can., 33, 2329-2339.LeBlond, P. H., 1983: The Strait of Georgia: Functional anatomy of a coastal sea. Can. J. Fish Aquat. Sc-, 40, 1033-1063. - and L. A. Mysak, 1978: Waves in the Ocean. Elsevier, 602 pp. ., H. Ma, F. Doherty, and S. Pond, 1991: Deep and intermediate water replacement in the Strait of Georgia. Atmos.-Ocean, 29, 288-312.Nof, D., 1983: The translation of isolated cold eddies on a sloping bottom. Deep-Sea Res., 30(2A), 171 - 182.Paldor, N., and P. D. Killworth, 1987: Instabilities of a two-layercoupled front. Deep-Sea Res., 34, 1525-! 539.Stacey, M. W., S. Pond, P. H. LeBlond, H. J. Freeland, and D. M. Farmer, 1987: An analysis of the low-frequency current fluctu ations in the Strait of Georgia, from June 1984 until January 1985. J. Phys. Oceanogr., 17, 326-342. , --, and , 1988: An objective analysis of the low-fre quency currents in the Strait of Georgia. Atmos.-Ocean, 26, 1 15. , --, and ,1991: Flow dynamics in the Strait of Georgia, British Columbia. Atmos.-Ocean, 29, 1 - 13.Swaters- G. E., 1991: On the baroclinic instability of cold-core cou pled density fronts on a sloping continental shelf. J. Fluid Mech., 224, 361-382.Yao, T., S. Pond, and L. A. Mysak, 1982: Low frequency subsurface current and density fluctuations in the Strait of Georgia. Atmos. Ocean, 20, 340-356. , --, and . , 1985: Profiles of low frequency subsurface current fluctuations in the Strait of Georgia during 1981 and 1982. J. Geophys. Res., 90, 7189-7198.

## Abstract

It has been suggested that low-frequency current fluctuations in the southern Strait of Georgia are the result of baroclinic instability. However, data extracted from cyclesonde and fixed current meter moorings suggest that the conditions for baroclinic instability are highly variable in space and time. It has been recently discovered that there are summertime bottom-intensified gravity currents with fortnightly and monthly periods associated with the introduction of salty waters from the Juan de Fuca Strait during periods of neap tides. These currents are the dominant mechanism for deep-water renewal in the Strait of Georgia. It is argued that these currents are baroclinically unstable and that the stability characteristics are reasonably consistent with the observed structure of the low-frequency current fluctuations. The episodic nature of these unstable bottom flows may help to explain the spatial and temporal variability of the low-frequency current fluctuations observed in the Strait of Georgia.