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924 VOLUME 32J O U R N A L O F P H Y S I C A L O C E A N O G R A P H Y

q 2002 American Meteorological Society

Instability of a Shelfbreak Front

M. SUSAN LOZIER

Earth and Ocean Sciences, Duke University, Durham, North Carolina

MARK S. C. REED

North Carolina Supercomputing Center, Research Triangle Park, North Carolina

GLEN G. GAWARKIEWICZ

Woods Hole Oceanographic Institution, Woods Hole, Massachusetts

(Manuscript received 11 July 2000, in final form 26 July 2001)

ABSTRACT

In an attempt to understand whether local instabilities can account for the observed frontal variability in theMiddle Atlantic Bight, a linear stability analysis was conducted for a wide range of background density andvelocity fields. Three-dimensional perturbations superposed on a continuously stratified shelfbreak front wereinvestigated using the hydrostatic primitive equations. Model results indicate that the shelfbreak frontal jet isunstable over the wide parameter range dictated by the observed velocity and density structure. Model growthrates, on the order of one day, and wavelengths of ;10–50 km compare favorably to observations, suggestingthat local baroclinic/barotropic instabilities are a likely source for the strong temporal and spatial variability ofthe shelfbreak front in the Middle Atlantic Bight.

1. Introduction

Offshore of the eastern United States a sharp tem-perature and salinity front at the continental shelf breakseparates the relatively cool and fresh shelf waters fromthe warm and salty waters of the North Atlantic. In theMiddle Atlantic Bight this shelfbreak front is a highlydynamical feature that is often severely contorted bylarge amplitude meandering (Beardsley et al. 1985).Aligned with this front is an alongshore current that isgeostrophic, to first order (Flagg 1977). The strong tem-poral and spatial variability of this frontal jet is evidentfrom the meandering paths of 10-m drifters that werereleased near Georges Bank in 1996 (Fig. 1), as part ofthe Global Oceans Ecosystems Dynamics (GLOBEC)Northwest Atlantic program (R. Beardsley and R. Lime-burner 2000, personal communication). These drifterswere entrained into the shelfbreak jet from the onshoreside of the front and then generally advected down-stream to the southwest, collectively demonstrating thecontinuity of the frontal jet from Georges Bank to CapeHatteras. Along their path through the Middle AtlanticBight, drifters were entrained from, and detrained to,

Corresponding author address: Dr. M. Susan Lozier, Division ofEarth and Ocean Sciences, Box 90227, Duke University, Durham,NC 27708-0227.E-mail: [email protected]

both sides of the current, although offshore exchangewas predominant (Lozier and Gawarkiewicz 2001).From Lozier and Gawarkiewicz’s analysis of 10-m and40-m drifter data over a three-year period (1995–97), itis apparent that the meandering is restricted neither byseason nor by locale along the shelf break. Evidence ofsuch ubiquitous meandering and frontal exchange re-news speculation that a possible source for the observedvariability in the Middle Atlantic Bight is local frontalinstability. Such a line of study was first pursued overtwenty years ago by Flagg and Beardsley (1978), whoused a two-layer front in geostrophic balance to studythe stability of the front south of New England. Forrealistic topography beneath the front, Flagg andBeardsley found that the most unstable modes had char-acteristic e-folding times greater than 50 days. Thus,they concluded that some other mechanism must be re-sponsible for the observed wavelike features on the NewEngland shelf/slope. In a later study, Gawarkiewicz(1991) also employed a layered model to simulate thewintertime and summertime fronts south of New Eng-land. This study, which differed from the Flagg andBeardsley (1978) study in that it used unbounded bottomtopography offshore, produced much shorter character-istic growth scales (on the order of 4–10 days), partic-ularly when a subsurface front (which is characteristicof the summer frontal configuration in the Middle At-

MARCH 2002 925L O Z I E R E T A L .

FIG. 1. Spaghetti diagram of 10-m drifters released in the vicinity of Georges Bank during 1996 as part of the GLOBECprogram. Deployment locations are marked with a red star. All drifter positions were low-pass filtered following the methodused by Brink et al. (2000) for drifters in the California Current. [This figure is reproduced from Lozier and Gawarkiewicz(2001).]

lantic Bight) was used as the background density field.While these times are short enough so that an initialperturbation could grow rapidly enough to be of con-sequence, observations suggest that growth is more rap-id, with timescales on the order of 1–2 days (Garvineet al. 1988). Gawarkiewicz’s study (1991) illustrated thedependence of shelfbreak frontal instabilities on thebackground stratification. Such dependence suggeststhat layered models, though analytically tractable, pos-sibly limit the range of baroclinic and barotropic insta-bilities in a frontal region.

In an effort to further explore the full range of frontal

instabilities at a shelfbreak front, we employ a spectralmethod that was developed by Moore and Peltier (1987)to solve for the linear instabilities of a background geo-strophic jet that is characterized by continuous verticaland cross-stream stratification. Moore and Peltier usedthe method to study the stability of two-dimensionalatmospheric fronts perturbed by disturbances that aregoverned by the linearized primitive equations. Theyfound that primitive equation dynamics captured modesthat were either absent or distorted by approximatedequations (i.e., semigeostrophic and quasigeostrophicmomentum equations). The method and the model dy-


namics used by Moore and Peltier have been employedto study oceanic fronts. Barth (1994) and Samelson(1993) successfully used the method to study the sta-bility of coastal upwelling jets and subtropical conver-gence zones, respectively. More pertinent to the presentstudy are the studies of Xue and Mellor (1993) andMorgan (1997), which investigated the instability ofcontinuously stratified oceanic jets overlying a slopingbottom. Xue and Mellor (1993), modifying the Mooreand Peltier model to include topography, studied theeffect of downstream topographic changes on the sta-bility of the Gulf Stream in the South Atlantic Bight.Morgan (1997), using primitive equations approximatedby a finite-difference solution method developed byProehl (1996), studied the stability characteristics of ashelfbreak frontal jet in the vicinity of Georges Bank.Using climatological data (Linder and Gawarkiewicz1998) to define seasonal structure, Morgan found a typ-ical winter jet in the vicinity of Georges Bank to bemore unstable than a typical summer jet, with doublingtimes of ;4 and ;6 days, respectively.

The primary goal of this work is to use a linearizedprimitive equation stability model to ascertain whetherfrontal instabilities can account for the observed tem-poral and spatial variability within the Middle AtlanticBight. Primitive equation dynamics are particularly im-portant in the region of the Middle Atlantic Bight wherelarge Rossby number flows and outcropping isopycnalsrender traditional quasigeostrophic stability theory in-appropriate. Recent field work in the Middle Atlantichas shown that significant changes in frontal jet strengthand structure can occur within a matter of days (Fra-tantoni et al. 2001). The magnitude of these changes inthe horizontal and vertical velocity shear over a fewdays can be greater than the seasonal change that hasbeen determined from climatological data (Linder andGawarkiewicz 1998). Thus, a stability study contrastinga climatological winter and climatological summer jetwould not illuminate the full range of instabilities pre-sent in the Middle Atlantic Bight since it would notcover the full range of background states. Our intent isto assess the stability characteristics for an envelope ofprobable jet structures. Thus, we have chosen to conducta thorough parameter study of a shelfbreak jet, with thebounds of the parameter study set by the range of en-vironmental conditions within the Middle AtlanticBight, as assessed by a recent field program (Gawar-kiewicz et al. 2001), as well as by Linder and Gawar-kiewicz’s climatology. The purpose of our parameterstudy is to understand the dependence of stability char-acteristics on the vertical and horizontal shear of thebackground density and velocity fields. For investiga-tions into the dependence of stability characteristics ona bathymetric slope, the reader is referred to past studies(e.g., Orlanski 1969; Flagg and Beardsley 1978; Xueand Mellor 1993; and references therein).

In sum, the objective of this study is to answer threemain questions: 1) What background density and ve-

locity structures will yield an unstable jet at a shelfbreakfront? 2) What are the temporal and spatial scales ofthe dominant unstable modes? 3) Do the predicted tem-poral and spatial scales approximate those found in theMiddle Atlantic Bight? Overall, this study aims to im-prove our understanding of the mechanisms that controlthe exchange of heat, freshwater, sediment, and otherproperties between the coastal ocean and the openocean. In the next section, the model equations andbackground fields are presented and the model solution,convergence, and verification are discussed. Section 3contains the results of the parameter study using anidealized shelfbreak jet. Model results using backgroundjets derived from synoptic and climatological data inthe Middle Atlantic Bight are presented in section 4,followed by the summary and conclusions in section 5.

2. Methods

Our computation of the linear instabilities of a back-ground geostrophic jet with continuous stratification isbased on the method developed by Moore and Peltier(1987) and modified by Xue and Mellor (1993) to in-clude a topographic slope. In this section we will brieflydescribe the model equations and solution methods. Thereader is referred to Moore and Peltier (1987) and Xueand Mellor (1993) for more details.

a. Model equations and solution method

The model used in this study consists of a steadybackground current flowing along the slope of idealizedbathymetry (Fig. 2a). The model uses a Cartesian co-ordinate system, where x is the offshore axis (positiveoffshore), y the alongshore axis (positive upstream), andz the vertical axis (positive upward). Bathymetry is giv-en by h(x) and the alongshore background flow is givenby V(x, z). [Functional forms for h(x) and V(x, z) aregiven in section 2b.] The background flow is in thermalwind balance with a mean density field, r (x, z), ac-cording to f oVz 5 Bx, where f o is a constant Coriolisparameter and B is the mean buoyancy defined by B 52gr (x, z)/ro, with ro the reference density and g thegravitational acceleration. In order to assess the stabilityof this basic state, three-dimensional velocity and den-sity perturbations are superposed onto a two-dimen-sional background velocity and density field. The evo-lution of the perturbations is governed by the hydrostaticprimitive equations, linearized about a geostrophicbackground state:

u 1 Vu 2 f y 5 2p (2.1)t y o x

y 1 uV 1 Vy 1 wV 1 f u 5 2p (2.2)t x y z o y

0 5 2b 1 p (2.3)z

u 1 y 1 w 5 0 (2.4)x y z

b 1 uB 1 Vb 1 wB 5 0. (2.5)t x y z


FIG. 2. (a) Schematic of model jet and bathymetry used in this study. V(x, z) and h(x) are given in section 2b. All model jets are centeredat x 5 50 km and z 5 0. (b) Envelope of model jets used in the parameter study. This schematic illustrates the range of widths and depthsused for the background jet by showing the 10 cm s21 isotach for the jet. Stratification (rshallow) and the maximum velocity (Vo) were alsovaried for the model runs.

The Cartesian components of the perturbation velocityare (u, y, w), p is perturbation pressure divided by ro,and b is the perturbation buoyancy. Equations (2.1)–(2.3) are the equations of motion with no frictional orexternal forces, while Eqs. (2.4) and (2.5), derived fromthe conservation of mass, represent the continuity equa-tion for an incompressible fluid and the conservation ofdensity, respectively. Boundary conditions stipulate thatnormal flow across solid boundaries is zero [u 5 0 atx 5 0 and w 5 2uhx at z 5 2h(x)] and that disturbancesvanish at distances far from the coast. Additionally, therigid-lid approximation (w 5 0 at z 5 0) is imposed.The system is assumed to be inviscid with respect tovertical mixing. Recent observations at the shelfbreakfront have suggested that vertical diffusion coefficientsare quite small (Rehmann and Duda 2000). With suchlimited diffusion, an estimate for the spindown time ison the order of 50 days, much larger than either thewave periods or growth rates that, as will be shown,were obtained in this study.

A mapping is applied to Eqs. (2.1)–(2.5) in order totransform the irregular grid created by the bathymetryinto a regular rectangular grid (in the x and z directions).Such a mapping facilitates the use of basis functions,which are described below. The mapping, applied ac-cording to z 5 1 1 z/h(x), yields a new vertical velocity,v, given by v 5 w 2 (z 2 1)uhx (Xue and Mellor1993). With this mapping it is convenient to define trans-port variables: (u*, y*, b*) 5 (uh, yh, bh). Additionally,all variables are cast in dimensionless form using thefollowing scaling:

x9 5 x/L y9 5 y/L z9 5 z t9 5 t fo o o

u9 5 u*/u y9 5 y*/u v9 5 (L /u )vo o o o

h9 5 h /H p9 5 (H /u f L )po o o o o

2b9 5 (H /u f L )b* B9 5 B /H N V9 5 V/V ,o o o o o o o

where Lo and Ho are the horizontal and vertical lengthscales, respectively, uo is the typical perturbation ve-locity times Ho, is the Brunt–Vaisala frequency, and2N o

Vo is the maximum of the background velocity, V(x, z).For all model runs Lo 5 100 km, Ho 5 200 m, and f o

5 9.37 3 1025 s21 (corresponding to a latitude of 408N);Vo and are varied for the model runs, as explained2N o

in section 2b. Solutions for the three-dimensional per-turbations are sought in the form

i(st1ly)(u9, v9) 5 Re[(u*(x, z), v*(x, z))e ] (2.6)i(st1ly)(y9, p9, b9) 5 Re[i(y*(x, z), p*(x, z), b*(x, z))e ],

(2.7)

where s is the complex frequency (s 5 sr 1 isi), l isthe alongfront wavenumber, the starred variables are thestructure functions in x and z, and Re[ ] denotes the realpart of the expression inside the brackets. Substitutionof Eqs. (2.6) and (2.7) into (2.1)–(2.5) yields

(s 1 R lV )u* 2 y*o

5 2hp* 1 (z 2 1)h b* (2.8)x x

(s 1 R lV )y* 2 u*(1 2 R V ) 2 R V v*o o x o z

5 2lhp* (2.9)


2b* 1 p* 5 0 (2.10)z

u* 2 ly* 1 v* 5 0 (2.11)x z

(s 1 R lV )b* 2 SB u* 2 SB v* 5 0, (2.12)o x z

where Ro 5 Vo/ f oLo is the Rossby number for the back-ground flow, and S, the Burger number, is defined as[NoHo/( f oLo)]2. For the numerical calculation of Eqs.(2.8)–(2.12) the background flow and the perturbationsare confined to a domain bounded by x 5 0, x 5 100km, the sea surface (z 5 1) and sea floor (z 5 0) (Fig.2a). The size of the domain was chosen to balance theneed for computational affordability with the need forall perturbations to be vanishingly small at the channelwalls. Additionally, to provide for no normal flow, u*is set to 0 at x 5 100 km. With these boundary con-ditions, Eqs. (2.8)–(2.12) define the problem with acomplex frequency given by s and functional formsgiven by Eqs. (2.6) and (2.7). This set of equations istransformed into spectral space via a combination ofGalerkin and Fourier colocation schemes. To obtain asolution, structure functions, for example, u*(x, z), arespectrally decomposed into orthogonal sets of trigo-nometric basis functions in the vertical (z) and cross-shelf (x) directions, as detailed in Xue and Mellor(1993). For numerical solution these expansions aretruncated at a finite number of spectral modes, M. Theseexpansions are substituted into (2.8)–(2.12) and manip-ulated to produce a standard matrix eigenvalue equation,which is solved by standard linear algebra methods fora specified background field, truncation level, and wave-number. Model solutions yield growth rate, 2si, phasespeed, 2sr/l, and modal structure of the instabilities ateach wavenumber.

b. Background fields

The background velocity field, modeled as a cross-shelf Gaussian wave form that exponentially decayswith depth, is expressed as

2V(x, z) 5 V exp[2s (x 2 x ) 2 s (z 2 z )],o x o z o

where sx 5 4ln(Vo/0.1 m s21)/ and sz 5 ln(Vo/0.1 m2xd

s21)/zd are formulated such that the 10 cm s21 isotachwill fall at a distance xd/2 from xo and zd from zo. Forall model runs xo 5 50 km, that is, the center of the jetis placed at x 5 50 km, as explained earlier and shownin Fig. 2a, and zo 5 0 m, that is, the maximum jetvelocity is at the surface. The width and depth of the10 cm s21 isotach are specified for each model run bythe variables xd and zd, respectively. These variables,along with changes to Vo, are adjusted to produce arange of jet structures that are generally consistent withthe frontal jet in the Middle Atlantic Bight. Because thebasic state is in thermal wind balance, the backgrounddensity field, B(x, z), is set by the velocity field and thespecification of density at the offshore boundary. At theoffshore boundary the stratification is assumed to be

linear (Bz is constant) for all model runs, with the deep-est isopycnal set at r 5 1.0275 g cm23 (Linder 1996).To create changes in the background stratification, theuppermost isopycnal, rshallow, is varied at the easternboundary.

The governing equations for the perturbation [Eqs.(2.8)–(2.12)] depend on the horizontal and vertical shearof the background density and velocity fields, as wellas on the strength of the velocity field. Since the hor-izontal shear of the buoyancy field (Bx) is linearly re-lated to the vertical shear of the velocity field (Vz) viathe thermal wind relationship, there are four backgroundconditions to vary for the parameter study (Vx, Vz, Vo,and Bz). In order to produce a manageable number ofjet variations, we have chosen to approach this param-eter study by varying separately the jet width (whichfor a constant Vo will principally vary Vx, but also Bz),the jet depth (which for a constant Vo will principallyvary Vz, but also Bz), and the background stratification(Bz), each for two basic states, one with Vo 5 30 cms21 and the other with Vo 5 60 cm s21. Each basic statehas a jet width of 20 km, a jet depth of 70 m (bothmeasured by the location of the 10 cm s21 isotach), andrshallow 5 1.022 g cm23. For each basic state, the jetwidth is varied from 15 to 40 km, while holding the jetdepth and offshore boundary stratification constant, thejet depth is varied from 30 to 120 m, while holding thejet width and the offshore boundary stratification con-stant, and rshallow is varied from 1.020 to 1.026 g cm23,while holding the jet width and depth constant. Thisparameter range, dictated by synoptic and climatologicaldata in the Middle Atlantic Bight, results in 34 test cases,which will be presented in section 3. The envelope ofjet widths and depths used in this study is shown in Fig.2b. It is important to note that for our present study wehave chosen to keep the frontal jet’s centerline positionfixed, to maintain constant stratification at the offshoreboundary and to maintain symmetrical horizontal ve-locity shear. Each of these constraints is made for thepurpose of simplification and is expected to be variedin our ongoing study of this frontal system.

Finally, the bathymetry used in our model study wasdetermined from the fit of a hyperbolic tangent func-tional form (Xue and Mellor 1993) to the bathymetricmeasurements in the Nantucket Shoals region (Linder1996). This fit is given by

h(x) 5 H 1 0.5(H 2 H )(1 1 tanh[(x 2 x )/a]),s d s m

where Hs (shelf depth) 5 60 m, Hd (maximum domaindepth) 5 200 m, xm (location of maximum slope) 5 50km, and a (lateral extent of slope) 5 15 km. While theoffshore asymptotic depth in the Middle Atlantic Bightis on the order of 2000 m, we have used 200 m in ourmodel study in order to reduce the computational grid.A sensitivity study to offshore depth found negligibledifferences between model runs using 200 m as the off-shore depth and those using depths in excess of 200 m.


FIG. 3. Convergence plot showing growth rate as a function of the mode cutoff, M, for selectedwavenumbers, as denoted in the insert.

c. Model convergence and computation

Past studies using this spectral model have found thatM 5 28 was necessary for model convergence (Xue andMellor 1993; Barth 1994; Samelson 1993). However,these studies were restricted to relatively small wave-numbers. Our choice of the wavenumber domain to ex-plore was dictated by the observed spatial variability inthe Middle Atlantic Bight, which is on the order of 10–50 km. In order to ascertain the number of spectralmodes required to adequately resolve the physicalmodes at these wavelengths, we conducted a conver-gence study for a representative baroclinic jet. We ranthis convergence test from l 5 1 to l 5 100, corre-sponding to wavelengths (l) from ;600 to 6 km (l 52pLo/l), using M 5 20 to M 5 60, in increments of 4.Results from this convergence test are shown for se-lected wavenumbers in Fig. 3. As expected, convergencefor the low wavenumber modes was achieved for a rel-atively small M, while convergence at the high wave-number end required considerably more modes. In orderto reach convergence for wavenumbers up to 60 (l ;10 km), it was necessary to use 44 spectral modes. Be-yond this wavenumber, convergence was approachedwith M 5 60, but not achieved. Due to the high com-putational costs associated with high spectral modecomputations, we decided to use M 5 44 for all com-putations and limit our analysis to the wavenumberrange from 1 to 60. With this cutoff we exclude per-turbations with wavelengths less than 10 km. Since ouremphasis is principally on spatial scales on the order of10–50 km this restriction does not significantly hamperour study.

The computational cost of running this model is sig-nificant, mainly because the size of the eigenvalue ma-trix becomes quite large for the high truncation levelnecessary to investigate the stability of large wavenum-bers. The choice of M 5 44 results in significantly lon-ger run times than those of previous studies since therun-time scales as approximately M 6 (Moore and Peltier1987). To efficiently investigate the instabilities at largewavenumbers the performance of the model was im-proved by implementing a parallel version of the modelcode, which uses the Message Passing Interface (MPI)library to perform communication between processors.The resultant code scales well with processor numberand is highly portable, running on either shared or dis-tributed memory architectures. Finally, verification testsreported in Xue and Mellor (1993) were repeated usingthe topography and stratification specific to our study.In all cases, modes that arise due to computational errorshave growth rates several orders of magnitude smallerthan those associated with the physical modes of thesystem. Therefore, errors in the computational schemedo not significantly affect our model results.

3. Parameter study of an idealized jet

In this section the growth rate and phase speed of themost unstable mode will be presented as a function ofwavenumber for each of the 34 background configu-rations. Representative modal structures will also beshown. As mentioned earlier, each background config-uration is determined by a jet width, jet depth, Vo, andrshallow. We have chosen to characterize the resultant two-


dimensional V, Vx, Vz, and Bz fields by two nondimen-sional numbers, namely, a characteristic Rossby number,Ro 5 | Vx/ f o | max and Burger number, S 5 ^ /2 2N Ho o

( )&max, as well as by | Vz | max and Vo. The brackets2 2f Lo o

are used to indicate depth averaging. (Note that we nowuse a definition for Ro that distinguishes between jetswith differing horizontal shears.) These parameters pro-vide a meaningful, if not complete, description of thebackground field.

In addition to a classification of the basic state, eachresultant perturbation can be characterized by the degreeto which it obtains energy for its growth. Basically,modes can derive energy from the available potentialenergy created by the sloping isopycnals, the horizontalshear of the mean flow (a Rayleigh instability), and/orthe vertical shear across the isopycnals (a Kelvin–Helm-holtz instability). The former is a baroclinic instability,while the latter two are barotropic instabilities. In orderto classify the perturbations, it is customary to analyzethe kinetic and potential energy budgets for both themean and perturbation flows. Previous studies (Barth1994; Xue and Mellor 1993) using this model have con-ducted such energy analyses to quantify the energysource for the unstable perturbations. Though our focusis not specifically on the energy sources for these per-turbations, we do note that, based on these past studiesand on the parameter range of our study, we expect thatthe unstable perturbations in this study will derive theirenergy from both the baroclinic and barotropic fields.A measure of the importance of available potential en-ergy as the energy source relative to kinetic energy isgiven by the ratio of the internal Rossby radius of de-formation, ri (where ri 5 NoD/ f o, with D the verticalscale of the background flow defined as Vo/ | Vz | max) tothe horizontal length scale of the basic flow field, Lh

(where Lh 5 Vo/ | Vx | max). If Lh k ri (Lh K ri), then itis expected that baroclinic (barotropic) instability willpredominate. In prior studies the reciprocal of this ratio,Lh/ri, has been referred to as the internal Kelvin number(Krauss 1973). For the 34 cases presented here, therange for ri/Lh is from 0.43 (Lh 5 22.4 km, ri 5 9.6km) to 1.62 (Lh 5 11.3 km, ri 5 18.3 km). Thus, it isexpected that neither baroclinic nor barotropic instabil-ities will exclusively govern the perturbations’ growth.Furthermore, it is expected that the barotropic instabil-ities will result primarily from a Rayleigh-type insta-bility since Kelvin–Helmholtz instabilities are likely tooccur when the Richardson number of the flow ( Bz/2f o

) is ,1. In all but one of the 34 cases in this study2Bx

the Richardson number exceeds 1, reflecting the rela-tively weak vertical shear of these background states.

Finally, as mentioned earlier, the use of primitiveequation dynamics allows for the expression of a non-geostrophic (or ageostrophic) mode, which Moore andPeltier (1987) referred to as a cyclogenesis mode. In astudy of coastal jets and fronts, Barth (1994) found theoceanic equivalent of this mode, which he termed afrontal mode, to derive its energy primarily from the

potential energy of the basic state. Borrowing Barth’snomenclature, we will refer to this nongeostrophic modeas a frontal instability in our subsequent discussions.

a. Horizontal shear variations

1) Vo 5 60 cm s21

In order to test the sensitivity of the stability charac-teristics to the horizontal shear of the background jet,model runs were made with varying jet widths. Thewidth of a Vo 5 60 cm s21 jet was varied from 15 to20 to 30 to 40 km, yielding Rossby numbers rangingfrom 0.96 to 0.366. The growth rates and phase speedsfor each of the four jets are shown in Fig. 4a as afunction of wavenumber. Overall, it is evident that eachof these jets is unstable over the entire range of thechosen wavenumber space. For each of the four curves,the growth rates generally increase with increasingwavenumber. The fastest growth rates over this wave-number range are on the order of 1 day for perturbationswith wavelengths of approximately 10–20 km. Sincethe horizontal scale of the background jet (Lh 5 7, 9,13, and 18 km for the 15-, 20-, 30-, and 40-km jets,respectively) is smaller than the Rossby deformationradii (ri 5 9, 9.7, 11, and 12 km), it is expected thatbarotropic instabilities will contribute to the growth ofthe perturbation energy. In fact, as the Rossby numberincreases, the growth rate increases over the entire rangeof wavenumber space (Fig. 4a), confirming the impor-tance of horizontal shear as an energy source for thisinstability.

The phase speeds in Fig. 4a are bounded from 0 toVo, in accordance with the semicircle theorem (Pedlosky1979). Also evident from Fig. 4a is the overall increasein the magnitude of the phase speed as wavenumberincreases. In other words, the shorter waves tend totravel faster downstream. Such a pattern is consistentwith the behavior of baroclinic (Eady) modes where thephase speed of the instability is approximately the ve-locity of the background jet at the depth of the Rossbyheight, HR, which is given by f o/Nol (Gill 1982). Es-sentially, the steering level of the perturbation [whereV(z) 5 cr] is located at the Rossby height. As the Rossbyheight decreases with increasing l, the phase speed in-creases in magnitude since the background speed in-creases toward z 5 0. A quantification of this relation-ship will be given in the following paragraph. Finally,it is noted that the relationship between phase speed andwavenumber is characterized by stepwise changes,which correspond to changes in the modal structure ofthe perturbation, as discussed next.

The amplitude of the structure functions for u, y, andb were analyzed for each wavenumber to determine themodal structures for the dominant perturbations. For thejet with a width of 15 km, there were five identifiablemodal structures. Each mode was characterized by near-ly uniform structure for u, y, and b and by a nearly


FIG. 4. (a) Growth rate and phase speed as a function of wavenumber for jets with varying widths. Each jethas Vo 5 60 cm s21. As shown in the gray inset, the four cases principally differ in their Rossby number. (b)As above but for jets with Vo 5 30 cm s21.


FIG. 5. The modal structure for a jet with a width of 15 km, depth of 70 m, Vo 5 60 cm s21, and rshallow 5 1.022 gm cm23. The amplitudeof the perturbation downstream velocity, y, is shown for (a) l 5 11 (;57 km), (b) l 5 23 (;27 km), and (c) l 5 59 (;11 km). The growthrate and phase speed for each of these wavenumbers is shown in Fig. 4a. The lowest contour and the contour interval for this and allsubsequent modal structures is 0.05 (in nondimensional units). (d) The cross-shelf flux of buoyancy, ^ub&, associated with the l 5 11 case.Shown in (a)–(d) and all subsequent plots of the background jet are the background isopycnals, denoted in sigma units.

constant phase speed over a range of wavenumbers. Aswitch in modes as the wavenumber changed was notedby abrupt changes in the structure of u, y, and b, by astepwise change in the phase speed for the perturbationand, at times, by changes in the slope of the growth ratecurve. Because the five modes for the 15-km jet havefairly similar structures, only three of them are shownin Fig. 5. Although the u, y, and b structures were an-alyzed, for the sake of brevity only the y perturbationis reproduced here. The u and b structure functions are

generally similar to the y structure function. As evidentfrom an examination of these modal structures in Fig.5, all modes are approximately centered at 50 km, thecenter of the background jet. Additionally, all modesare surface-intensified, with the degree of surface trap-ping increasing as the wavenumber increases.

From all 34 model runs a consistent pattern for themodal structures emerged. At low wavenumbers themodal structure has horizontal scales commensuratewith the jet width, and the vertical decay scale of the


perturbation is approximately the same as the depth ofthe basic flow field. Following this mode, at higher wav-enumbers, is a series of modes that are all quite similarin that they are center and surface trapped (i.e., quitenarrow and shallow). These characteristics agree withBarth’s (1994) description of the modal structures thatresulted from his analysis. Basically, the low wave-number mode exhibits the behavior of a classical geo-strophic instability, while the high wavenumber modesexhibit the characteristics of the ageostrophic or frontalinstability. The dominant wavelength for the geostroph-ic mode is approximated by 2pri, while the shorterwavelengths of the frontal modes are much closer inhorizontal scale to Lh (Barth 1994).

Although the high wavenumber modes are nongeo-strophic modes, it is interesting that the decreasing ver-tical scale with increasing wavenumber is consistentwith the behavior of a classical baroclinic instability(the Eady mode), where the e-folding depth of the per-turbation is given by the Rossby height, HR (Gill 1982),as defined above. Calculation of the Rossby heights forthe modes shown in Fig. 5 yields HR 5 40, 19, and 8m for the l 5 11, 23, and 59 cases, respectively. Froman inspection of the modes in Fig. 5 it is apparent thatthe decay scale for the perturbation is in close agreementwith these estimated Rossby heights, with a sharp de-crease in height as the perturbation wavenumber in-creases. Furthermore, the phase speeds correspondingto these l 5 11, 23, and 59 cases approximately matchthe magnitude of the background jet velocity at theRossby heights, consistent with baroclinic modes, asdiscussed above. As an example, the phase speed forthe l 5 11 perturbation (with HR 5 40 m) is expectedto be ;222 cm s21 [calculated using the expression forV(x, z) in section 2b with x 5 0 and z 5 240 m]. Asseen in Fig. 4a, this estimation is quite close to thecomputed phase speed of ;225 cm s21.

As there is no mean cross-shelf flow, it is the per-turbations that are responsible for cross-frontal ex-change of both buoyancy and momentum. The cross-shelf buoyancy flux and the cross-shelf momentum fluxare calculated as ^ub& and ^uy&, respectively, where theangle brackets denote an average over one wavelength.As this analysis is restricted by the linearization of thegoverning dynamics, the magnitude of the cross-shelffluxes cannot be assessed. However, the utility of thiscomputation lies in the ability to assess the structureand sign of the fluxes. Not surprisingly, given the sur-face-trapped modal structures in Figs. 5a–c, the calcu-lated buoyancy flux for the l 5 11 case is also surface-trapped (as shown in Fig. 5d). As with the modal struc-tures, this flux becomes increasingly surface-intensifiedas the wavelength increases. The sign of the flux ispositive, indicating a net offshore downgradient flux ofbuoyancy. (While the density gradient, ]r/]x, is positive,the buoyancy gradient, ]b/]x, is negative.) This down-gradient flux is consistent with a baroclinic instabilitywhere the perturbations extract their energy from the

mean buoyancy field. The cross-shelf momentum flux(not shown) has a structure quite similar to that for thebuoyancy flux (Fig. 5d). However, this flux does notshow exclusively downgradient structure relative to thebackground velocity field, V, suggesting a lesser rolefor barotropic instability for this mode. A counter case,where it appears that the barotropic instability domi-nates, will be discussed in section 3a(2). As mentionedearlier, an energetics analysis could ascertain the relativeimportance of these instabilities. The concentration ofthese model fluxes over a limited depth range has im-plications for the measurement of shelfbreak fluxes.Garvine et al. (1989) found that heat and salt fluxesfrom summer measurements were concentrated withina depth range of 20 to 50 m, immediately beneath theseasonal pycnocline. The model similarly suggests thatbuoyancy fluxes may be limited in vertical extent, al-though the model basic state creates surface-trappedfluxes. Further work with a model configuration con-taining realistic summer stratification might reveal con-centration of cross-shelf fluxes below the surface.

For the background jets with greater widths (and cor-respondingly smaller Rossby numbers), the same trendas with the narrow jets was observed for the modalstructures. As the wavenumber of the perturbation in-creases, its structure becomes increasingly surfacetrapped. Only the structure of the first mode (i.e., thegeostrophic mode found at the low wavenumber end ofthe range) is noticeably sensitive to changes in the widthof the jet, although the growth rates at all wavenumbersare quite sensitive to jet width changes. Figure 6 showsthe change in the low wavenumber structure as the jetwidth increases from 20 to 30 to 40 km. These structuresare to be compared with the low wavenumber modalstructure for the 15-km jet, shown in Fig. 5a. It is evidentfrom a comparison of these modal structures that as thejet widens, the perturbation widens proportionally (con-sistent with the increase in Lh), but, more surprisingly,the perturbation maximum shifts strongly offshore.While the perturbation structure itself remains centeredover the background jet axis (at 50 km offshore) theperturbation y becomes increasingly asymmetric as thefrontal width increases. When the jet is 40 km wide, itsperturbation maximum has shifted approximately 10 kmoffshore from its position at a jet width of 15 km. Froman inspection of Fig. 6, it is apparent that the center ofthe perturbation tracks the offshore outcropping of is-opycnals, or the density front (Barth 1994). Such a per-turbation would tend to alter any symmetry in the back-ground frontal jet and perhaps lead to asymmetricalcross-frontal exchange. This would particularly applyto relatively wide frontal jets, where the density frontis significantly displaced from the velocity center.

2) Vo 5 30 cm s21

Using Vo 5 30 cm s21, the set of runs discussed in thepreceding section was repeated. Jet widths from 15 to


FIG. 6. Modal structures for the amplitude of the per-turbation downstream velocity, y, for a jet with Vo 5 60cm s21, a depth of 70 m, rshallow 5 1.022 gm cm23, andwidths of (a) 20 km, (b) 30 km, and (c) 40 km. For eachof these widths the dominant low wavenumber mode,which occurs at l 5 7, 7, and 11 for (a), (b), and (c),respectively, is shown.

40 km now yield Rossby numbers ranging from 0.379to 0.144. (This contrasts to the Rossby number rangefrom 0.96 to 0.366 for the Vo 5 60 cm s21 runs.) Overall,as seen by comparing Fig. 4b to Fig. 4a, there is asignificant drop in growth rates as Vo decreases. Themaximum growth rates for the slow jet, which occur atthe high end of the wavenumber range, yield e-foldingtimes of 3 days. Even though these rates are approxi-mately five times slower than the growth rates for theVo 5 60 cm s21 jet, they are still of sufficient magnitudeto lead to observable instabilities. While these resultsgenerally show an increase in growth rate with increas-ing Rossby number, it is apparent that the correlationbetween growth rate and Rossby number is not as strongfor the low Rossby number cases as it is with the high

Rossby number cases. A comparison of the 30-km jetto the 40-km jet in Fig. 4b illustrates this point. At highwavenumbers (l . 45) the higher Rossby number jet(30 km) has smaller growth rates. Though the offshorestratification is held constant in each of these runs, theBurger number, S, does vary with jet width since, ac-cording to the thermal wind balance, variations in thevelocity field create a shifting of the isopycnals fromtheir nearly horizontal plane at the offshore boundary.Associated with increasing Rossby number is a decreas-ing Burger number since the isopycnal slope increasesto accommodate the increased shear. As will be shownin section 3b, as S decreases, with all else constant, theperturbation growth rate generally increases. Thus, thetwo effects (an increasing Ro and a decreasing S) should


act in tandem to produce increasing growth rates as thejet narrows. While this holds true at the low wave-number end, it does not hold for high wavenumbers.This discrepancy may result from our use of scalars tocharacterize the two-dimensional background fieldswhen clearly their structure is more complex. Addi-tionally, it may be that the frontal instability is sensitiveonly to the baroclinic structure in the near vicinity ofthe density front and our choice of parameters fails toaccount for these changes. However, as will be dem-onstrated in subsequent sections, the use of scalars issufficient to explain the trends for other variations inthis study, so their utility is not discounted because ofthis unexplained behavior at high wavenumbers.

It is interesting to note that the differences in thegrowth rate curves between the slower background jetand its faster counterpart cannot solely be explained interms of the generally higher Rossby number changesassociated with the faster jet. A comparison of the Vo

5 60 cm s21 jet with a width of 40 km (Fig. 4a) to theVo 5 30 cm s21 jet with a width of 15 km (Fig. 4b)highlights this point. For the former the Rossby numberis 0.366, while for the latter the Rossby number is 0.379.This ;4% change in Rossby number cannot accountfor the large difference in the growth rates of these twojets. Overall, the Vo 5 60 cm s21 jets have much greatergrowth rates than the slower jets, a testimony to theimportance of the background velocity’s advection ofthe perturbation density and velocity, as expressed inEqs. (2.2) and (2.5).

The phase speeds for the perturbations (Fig. 4b) arereduced by half for the slow jet, compared to the fastjet, in accordance with the phase speeds being boundedby 0 and Vo. Again, the phase speeds in Fig. 4b showthe stepwise change from one physical mode to the next.The trend for the faster growing modes to have smallerphase speeds generally holds for these Vo 5 30 cm s21

cases, except in the vicinity of wavenumber 10 wherea mode with a particularly slow phase speed appears tobe dominant for the 15- and 20-km jets and near wave-number 18, where there is a local decrease in the phasespeed for the 30- and 40-km jets. As will be shownbelow, this local decrease is associated with a mode thatwas not found in the faster jet runs.

The modal structures for the Vo 5 30 cm s21 jetsshow a similar pattern to the Vo 5 60 cm s21 pertur-bation structures in that changes in the background con-figuration bring significant changes to only the lowwavenumber, or geostrophic, mode. While changes inthe background horizontal shear bring discernible andsignificant changes to the growth rates and phase speedsfor all wavenumbers, the surface-trapped modal struc-tures at high wavenumbers are apparently insensitive tothe parameter changes made in this study. This insen-sitivity may result from the fact that the changes to thebackground field are on scales that exceed the pertur-bation length scales, particularly at high wavenumbers.Shown in Fig. 7 are the changes in the low wavenumber

perturbation structure as the width of the Vo 5 30 cms21 jet changes. It is apparent that the perturbation struc-tures associated with the slower jet are considerablywider and deeper than those associated with the fasterjet (Figs. 5 and 6). While the width of the perturbationdoes increase in tandem with the increase in Lh, therelationship is only approximate. The deeper penetrationof the modes that result from the slower jets is consistentwith the increase in Rossby height associated with thedecrease in No (and thus in Burger number). The Rossbyheights for the 15-, 20-, 30-, and 40-km jets are cal-culated to be 48, 46, 44, and 42 m, respectively. Whilethese depths roughly approximate the decay scales ob-served in Fig. 7, it is not at all clear that the decay scaleactually decreases with increasing jet width, as sug-gested by the calculated Rossby heights. Thus, whilethe dependence of the Rossby height on wavenumber,l, appears to be robust, the dependence on No does nothold for most, if not all, of our model runs. However,this discrepancy could result from the fact that the Ross-by height is based on a depth-averaged No. Finally, itis noted that, in addition to increases in the width anddepth of the perturbation, it is evident that the modesfor the slower jet have structure along the sloping ba-thymetry. The magnitude of this perturbation is smallrelative to the surface perturbation, but it appears to bea robust signal over each of the jet widths.

The trend for the wider jets to have their perturbationmaximum shifted offshore with the density front holdsalso for these Vo 5 30 cm s21 jets. In fact, this trendis even more pronounced for the slow jet cases, as seenby the shift in the 40-km jet. In Fig. 7d the perturbationmaximum is located at approximately 70 km offshore,or 20 km seaward of the jet’s center. At this distancethe perturbation would be at the offshore edge of thebackground jet.

As mentioned above, a new modal structure appears forthis slow jet and is associated with very slow phase speeds.The modal structure shown in Fig. 8a is representative ofthe mode present in each of the 15-, 20-, 30-, and 40-kmjets in the wavenumber range of 10–25. This double lobemode is centered at the midpoint of the background jet,yet its magnitude is stronger on the offshore side. Runsmade with a flat bottom show that this asymmetry is solelythe result of the sloping bottom, implying a possible con-tribution from topographic wave motion. This mode is alsointeresting in that its cross-shelf momentum flux (Fig. 8b)shows a nice signature of downgradient behavior. The twolobes, split at the Vmax location, are of opposite sign, sincethe mean velocity gradient changes sign at Vmax. Such astructure implies the increasing importance of the baro-tropic contribution to the instability. Morgan (1997, hisFig. 3.22) shows that, as the relative vorticity increaseson the offshore side of the front, the horizontal Reynoldsstress contribution to the buoyancy flux becomes increas-ingly important. The asymmetry of the buoyancy flux im-plies a preference for offshore exchange, which was noted


FIG. 7. Modal structures for the amplitude of the perturbation downstream velocity, y, for a jet with Vo 5 30 cm s21, a depthof 70 m, rshallow 5 1.022 gm cm23, and widths of (a) 15 km, (b) 20 km, (c) 30 km, and (d) 40 km. For each of these widthsthe dominant low wavenumber mode, which occurs at l 5 5, 3, 5, and 3 for (a), (b), (c), and (d), respectively, is shown.

by Lozier and Gawarkiewicz (2001) from Lagrangianmeasurements of frontal exchange.

b. Stratification variations

From Rossby height considerations, increasing (de-creasing) No should reduce (increase) the vertical scaleof the modal structures and increase (decrease) the phasespeeds [since the Rossby height will be located higher(lower) in the water column]. To explore these effects,a change in the background stratification was achievedwithout attendant changes in the horizontal or verticalgradient of velocity. Thus, the influence of stratificationchanges on the stability of the background jet is morestraightforward than the other parameter changes. For

each Vo case, changes were made to rshallow for both ahigh Rossby number (jet width 5 15 km) and a lowRossby number (jet width 5 40 km) case. Since therewas such a dependence of the stability characteristicson Rossby number (section 3a), we felt it necessary toexplore stratification changes for the high and low endof this parameter. While our study included both the Vo

5 60 cm s21 and Vo 5 30 cm s21 cases, only the formerwill be reported here since the latter cases differ sub-stantially from the Vo 5 60 cm s21 cases only in themagnitude of the growth rates, as explained in the priorsection.

The growth rate and phase speed curves for the 15-km and 40-km jet are shown in Figs. 9a and 9b, re-spectively. As the stratification is decreased, there is an


FIG. 8. (a) Modal structure (at l 5 9) for the perturbation downstream velocity, y, for a jet with Vo 5 30 cm s21, a depth of70 m, rshallow 5 1.022 gm cm23, and a width of 20 km. (b) The associated cross-shelf flux of momentum, ^uy&.

increase in the growth rate of the dominant mode ateach wavenumber. However, this destabilizing effect isonly substantial for the higher wavenumber modes forthe high Rossby number jet (Fig. 9a). At wavenumbersgreater than ;14 for the fast jet (where there is a modalshift for all four cases), the growth rate curves for thevarious stratifications diverge, with the least stratifiedjet showing a remarkable increase in growth rate. In-terestingly, this divergence occurs approximately wherethe perturbation wavelength matches 2pri, the scale thatseparates the geostrophic mode from the frontal modes.

At the high wavenumber end in Fig. 9a, all growthrates are considerably less than 1 day, even those as-sociated with a sharply stratified water column (whereS 5 0.249). It is apparent that these rapid growth ratesare attributable to the high Rossby number associatedwith each of these cases. This influence is evident froman inspection of Fig. 9b, where the growth rates are lessthan a day over the entire Rossby number range, yetthe offshore specification of rshallow is the same as it isin Fig. 9a. It is noted that since the density structureaway from the offshore boundary is dictated by thethermal wind relation, our measure of S does vary ineach case. However, the S variation alone cannot ac-count for the differences in the growth rate curves. Forinstance, a comparison of the S 5 0.249 case in Fig. 9ato the S 5 0.260 case in Fig. 9b shows that the highRossby number jet (in Fig. 9a) always has the largergrowth rate, particularly at low wavenumbers. That thegrowth rates for these two cases (S 5 0.249 and S 50.260) become more similar at higher wavenumbers re-flects the increasing baroclinicity of the higher wave-number modes and, as a result, their strong dependenceon the background stratification. Likewise, the weak de-

pendence on stratification for the low wavenumbermode (Fig. 9a) suggests that a primary source of energyfor this mode is the horizontal shear.

The weak dependence of the growth rates on the back-ground stratification in Fig. 9b is interesting to note.While this weak dependence is evident only at the lowwavenumber end in Fig. 9a, it is in place for the entirewavenumber range in Fig. 9b. Such a feature may in-dicate that the lateral velocity shear fields are more im-portant in establishing the stability characteristics atthese relatively high Burger numbers.

It is apparent from an inspection of Figs. 9a and 9bthat stratification has very little effect on the phasespeeds. As with the previous model runs, the magnitudeof the phase speed increases as wavenumber increases.Thus, consistent with our earlier assessment, the phasespeed shows the expected dependence on l, accordingto Rossby height considerations, but does not displaythe expected dependence on stratification. Finally, aninspection of the modes for the cases shown in Figs. 9aand 9b reveals essentially the same pattern for modalstructures as explained in section 3a. An inspection ofthe perturbation u, y, and b fields shows that the struc-ture functions for the geostrophic mode deepen as No

increases, but the frontal modes are essentially insen-sitive to stratification changes.

c. Vertical shear variations

To complete our parameter study, the runs made byvarying the depth of the background jet are presentedin this section. With Vo at either 60 or 30 cm s21 avariation in depth creates variation in Vz, which has notbeen explicitly tested in the previous runs.


←

FIG. 9. Growth rate and phase speed as a function of wavenumber for Vo 5 60 cm s21 jets with depths of 70 m. (a) Stratification changesfor high Rossby number jets (width 5 15 km). (b) Stratification changes for low Rossby number jets (width 5 40 km). As shown in thegray inset, the four cases in (a) and (b) differ only in their Burger number.

1) Vo 5 60 cm s21

Changing the depth from 30 to 120 m creates significantdifferences in the growth rates of the dominant insta-bilities, as seen in Fig. 10a. For these runs, the Rossbynumber of the background flow remains a constant, yetS varies significantly as Vz is changed. In fact, S variesover a range comparable to the range tested in section3b, where the stratification effect was explicitly tested.In this set of runs (Fig. 10a) a decrease in growth rateas the jet deepens is apparent despite the fact that thestratification decreases as the jet deepens (which wouldtend to increase the growth rate). Apparently, the de-crease in Vz, which is in tandem with rx changes, over-rides the destabilizing effect of a weakly stratified watercolumn. Since this parameter changes the baroclinicityof the background jet, it is no surprise that the surface-trapped modes, at high wavenumbers, are the most af-fected. It is important to note that for all jets (with depthsfrom 30 to 120 m) the model growth rates are less than;2 days for wavelengths smaller than approximately60 km (l 5 10).

Phase speed does show some sensitivity to changesin the jet depth (Fig. 10a), yet the relationship is notconsistent over the entire wavenumber range. At somewavenumbers a large Vz corresponds to small phasespeeds, while at other wavenumbers a large Vz corre-sponds to large phase speeds. Modal structures followthe same trend as discussed with the previous parameterruns, with a tendency for deeper jets to have deeperperturbations for the low wavenumber mode and for theshallow jets to have shallower perturbations. Thesechanges are expected for the geostrophic mode. Addi-tionally, the increase in the depth of the perturbation isconsistent with a decreasing S, which results in a largerHR. Shown in Fig. 11a is a low wavenumber mode forthe deepest jet studied (120 m). Its center is situated atx 5 50 km, as before, yet now its structure extends to;50 m and its width is greater than the mode shownin Fig. 7, to which it should be compared.

2) Vo 5 30 cm s21

Shown in Fig. 10b is the sensitivity of a slow jet tochanges in Vz. The pattern is the same as for the fastjet, yet the overall growth rates and phase speeds arereduced in magnitude. For this slow jet the modal struc-ture for a low wavenumber mode with a deep jet isshown in Fig. 11b. Here the feature of interest is thestructure near the shelf break. While most of the modesfound in this parameter study have their structure con-centrated near the surface, it is interesting to note thatcertain configurations can give significant structure else-

where in the water column, consistent with observa-tions.

d. Summary of parameter study

To better understand the parameter space spanned bythe Middle Atlantic Bight shelfbreak flow field, a lowerresolution parameter study was performed. For the pur-poses of this study, the simulation was performed withM 5 32 instead of the M 5 44 that was used in therest of this work. The shortened simulation run timeallows a more complete exploration of parameter spaceand gives results that are qualitatively similar. In orderto illustrate the simultaneous dependence of the growthrates on Ro, S, and Vz (with Vo fixed at 60 cm s21), athree-dimensional plot is employed (Fig. 12). Our aimwith this analysis is to give a more complete and ac-curate picture of the relationship between the variables.Some general trends are noticeable in Fig. 12. For ex-ample, higher Rossby numbers uniformly show highergrowth rates. A general trend of decreasing growth ratesfor increased stratification can be seen, as evidenced bythe several sets of data for fixed horizontal (Rossby)and vertical (Vz) shear. By looking at planes of fixedRossby number, it can be seen that for a given Burgernumber, increasing vertical shear is associated withmore unstable disturbances. Note that for low Rossbynumber cases there does not appear to be enough energyin the system to drive high growth rates, even when thevertical shear is large. It should be pointed out that,although the physical characteristics of the backgroundstate, namely width and depth and initial backgroundstratification, were varied uniformly, this did not resultin a uniform distribution of points in parameter space.However, the sampled areas of parameter space repre-sent those regions associated with observed Middle At-lantic Bight fronts.

4. Instability of the shelfbreak frontal jet in theMiddle Atlantic Bight

The intent of the parameter study in section 3 was toestablish the stability for an envelope of possible shelf-break frontal jets in the Middle Atlantic Bight. In thissection we seek to establish the stability characteristicsfor a model jet based on the climatological velocity anddensity fields (Linder 1996) and for a model jet basedon a composite of synoptic sections that have beenstream-coordinate averaged (Fratantoni et al. 2001). Tofacilitate a comparison with our parameter runs, we havechosen to use the same analytical formulation for thevelocity field (section 2b) for the two jets in this section.


FIG. 11. Modal structures for the perturbation downstream velocity, y, for a jet with a width of 20 km, rshallow 5 1.022 gmcm23, a depth of 120 m, and (a) Vo 5 60 cm s21 and (b) Vo 5 30 cm s21. For both (a) and (b) the dominant low wavenumbermode is shown (l 5 5).

←

FIG. 10. (a) Growth rate and phase speed as a function of wavenumber for jets with varying depths. Each jet has Vo 5 60 cm s21. As shownin the gray inset, the four cases shown differ in their values of S and | Vz | max. (b) As for (a) but for jets with Vo 5 30 cm s21.

We have selected a jet width (20 km), jet depth (70 m),and Vo (25 cm s21) from a composite of the winter andsummer climatological fields given by Linder (1996).Based on the climatological density fields the offshoredensity profile varies linearly from 1.024 to 1.0275 gmcm23. The resultant stability profile for a jet with thesecharacteristics (labeled CLIM) is shown in Fig. 13. Alsoshown are the results for a model jet whose width (40km), depth (90 m), and Vo (40 cm s21) are based on thestream-coordinate averaged jet that Fratantoni et al.(2001) constructed from eight synoptic sections acrossthe shelfbreak frontal jet south of Nantucket Shoals from1997 to 1998. Based on the synoptic density fields theoffshore density profile was set to vary linearly from1.0244 to 1.027 gm cm23. As seen in Fig. 13, both jetsare unstable over the range of wavenumbers tested, withe-folding scales of between 2 and 3 days for the mostunstable perturbations of the stream-coordinate aver-aged jet and between 3 and 4 days for the climatologicaljet. The faster growth rates for the stream-coordinate-averaged jet are attributed almost entirely to the fasterspeed of this jet. While one would expect the stream-coordinate-averaged jet to have a higher Rossby number,the width of this jet (40 km) precludes such a charac-teristic. In fact, the Rossby numbers for the two jets arenearly identical, thus precluding the possibility thatRossby number differences could account for the dif-ferences in the stability results. While it is arguable

which of these jets, if either, is representative of anaveraged basic state in the Middle Atlantic Bight, thesemodel results show that, even for an averaged profile,the frontal jet in the Middle Atlantic Bight is unstableto perturbations that have substantial growth rates.

5. Summary and conclusions

A stability analysis of a two-dimensional geostrophicjet overlying shelfbreak topography has found the jet tobe unstable to perturbations for a wide range of back-ground conditions. While earlier studies of shelfbreakfrontal instabilities found growth rates to be prohibi-tively small to be of consequence in the energetic shelf-break region, the model growth rates from this studyare on the order of 1 day. Such rapid growth wouldclearly lead to substantial temporal and spatial vari-ability for the shelfbreak front. The inclusion of con-tinuous horizontal and vertical shear for the backgrounddensity and velocity field, as well as the use of primitiveequation dynamics, are believed to be responsible forthe capture of physical modes with rapid growth rates.While the perturbations with the shortest wavelengthsgenerally had higher growth rates, often the growth ratecurves were relatively flat at high wavenumbers, sug-gesting that a range of wavelengths might be present inthe shelfbreak vicinity rather than one dominant wave-length. Indeed, past observations of spatial variability


FIG. 12. Summary of the parameter runs for a Vo 5 60 cm s21 jet. The dependence of growth rate on Rossbynumber, Burger number, and | Vz | max are shown. The position of the sphere represents the point in three-dimensionalparameter space, while the size and color of the sphere (from blue, smallest, to red, largest) represent the relativemagnitude of the maximum growth rate for the single fastest growing wavenumber (e.g., small spheres denote smallgrowth rates). These simulations were performed with a modal cutoff of 32 instead of the 44 that was used in therest of this work. Additionally, wavenumber space was explored from 1 to 59, for every odd wavenumber.

FIG. 13. Growth rate and phase speed as a function of wavenumber for an idealized climatological frontal jet (Linderand Gawarkiewicz 1996) and an idealized stream-coordinate-averaged frontal jet (Fratantoni et al. 2001).


in the Middle Atlantic Bight have not narrowly definedthe dominant spatial scale of the eddy motions. Collec-tively, these past studies have reported a range of from10 to 75 km for the the spatial scale, a range that isconsistent with our model results.

Because the shelfbreak frontal jet in the Middle At-lantic Bight exhibits a large degree of variability aboutits climatological mean winter and summer states, anenvelope of possible synoptic jet structures was tested ina parameter study. Jet width, depth, and velocity maxi-mum were varied, along with stratification, in an attemptto isolate the relative importance of the Rossby number,the vertical velocity shear, the maximum velocity, andthe Burger number to the stability of the background jet.Changes to the background fields were based on the rangeof conditions present in the Middle Atlantic Bight. Over-all, growth rates increased as the jet’s maximum velocityincreased, as the Rossby number increased, and as thejet’s velocity shear in the vertical increased. A decreasein growth rate resulted from an increase in the jet’s strat-ification. Thus, a fast, narrow, shallow jet with relativelyweak stratification maximizes the growth rate of the un-stable perturbations. Of all the parameters tested, themodel growth rates were the least sensitive to changesin the background stratification. This result runs some-what counter to the notion that a summer jet is morestable than a winter jet because of the strong stratificationprovided by seasonal heating. All things being constant,these model results do show this to be true but, if thesummer jet also happens to be faster or narrower or shal-lower than the winter jet, it is likely to be more unstable.Thus, seasonal differences in stability may not be con-trolled solely by seasonal changes in stratification if theseasons also bring significant changes in the width anddepth and strength of the jet. Considering the strong sen-sitivity of the stability characteristics to the structure ofthe velocity field, it is likely that changes in the stabilityof the frontal jet on the order of days may occur if thebackground jet is altered due to ring interactions, slopewater variability, riverine input, and/or local wind events.It seems likely that the stability may change more rapidlyon synoptic timescales than on seasonal timescales. Thissuggestion is made with the caveat that we have con-ducted our model study with the constraint of uniformstratification. Nonconstant stratification might alter thisgeneralization.

While the results presented here provide a consistentframework for anticipating stability characteristics ofshelfbreak fronts, it is important to recognize some ofthe limitations of the model in light of recent obser-vations in the Middle Atlantic Bight. The basic stateused in our study assumes a purely geostrophic balance,without frictional boundary layers, and hence does notcontain significant convergences or divergences. Recentobservational work (e.g., Houghton and Visbeck 1998)suggests that there are significant convergences withinthe bottom boundary layer that result in upwelling alongfrontal isopycnals. These effects are not included in the

present basic state, nor are they easily accommodatedwithin linear eigenvalue problems. In his study on thestability of coastal upwelling jets, Barth (1989) includedthe effects of bottom friction on the evolution of insta-bilities, but the friction was introduced into the pertur-bation quantities and not the basic state. Furthermore,with this formulation, Barth found that the perturbationswere not damped much by friction since the e-foldingtimes for dissipation exceeded the perturbation growthrates. Our present model also lacks a mean cross-shelfflow, which is commonly observed in the Middle At-lantic Bight. The structure of the cross-shelf flow wouldbe dependent on the details of the vertical mixing withinthe boundary layer, as well as the lateral shear of theoverlying (geostrophic) alongshelf flow.

Another interesting issue relative to observations isthe possible existence of bottom-trapped modes thatmight lead to variability of the foot of the front, wherethe frontal isopycnals intersect the bottom. Gawarkie-wicz (1991) found bottom-trapped modes in a two-layermodel, but these were rarely the most unstable modes.Given the surface-trapped jet considered in this study,it is not surprising that the most unstable modes werealso surface-trapped. Basic states with stronger shelfflows in the vicinity of the foot of the front or withnonconstant stratification might lead to more energeticbottom-trapped modes. However, these modes would,in the actual front, likely be affected by the convergencepatterns and dissipative processes near the foot of thefront, neither of which are included within the presentmodel.

The extension of this work to include realistic frontalfeatures such as shear asymmetry, variations in the on-shore/offshore location of the front, nonuniform strat-ification, and a subsurface jet maximum are essential tofurther the applicability of these results to the observedfields. Finally, it is noted that, while the range of con-ditions tested in this study was based on observationsin the Middle Atlantic Bight, we believe this specificitydoes not limit the general application of this work toother geographical regions.

Acknowledgments. The authors are grateful to H. Xuefor providing the model code and patiently answeringquestions about its use. Thanks are also extended to R.Pickart, P. Fratantoni, and R. Limeburner for providingdata input to this modeling study. The authors are ap-preciative of the input provided by R. Garvine and ananonymous reviewer. Computational resources for thisstudy were provided by the North Carolina Supercom-puting Center. Finally, MSL and GGG gratefully ac-knowledge support from the Office of Naval Research(N00014-96-1-0150 and N00015-98-1-0059, respec-tively) for this study.

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