The Influence of Topography on the Stability of Shelfbreak Fronts

M. SUSAN LOZIER

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

MARK S. C. REED

North Carolina Supercomputing Center, Research Triangle Park, North Carolina

(Manuscript received 26 August 2003, in final form 31 August 2004)

ABSTRACT

In an attempt to understand the degree to which the stability of a shelfbreak front, characterized bycontinuous horizontal and vertical shear, is affected by topography, a linear stability analysis was conductedfor a range of frontal jets and bottom-slope configurations. Three-dimensional perturbations superposed ona continuously stratified shelfbreak front were investigated using linearized, hydrostatic primitive equations.For all model runs in the study, the frontal instability mode, which is the fastest-growing mode for abaroclinic flow, was not influenced by the bottom: Retrograde, prograde, and flat-bottom jets all share thesame stability characteristics. In contrast, weakly baroclinic jets are strongly influenced by bottom topog-raphy. The presence of a bottom slope stabilizes prograde jets and destabilizes retrograde jets, a differenceattributed to the orientation of the isopycnals relative to the bottom slope. Temporal and/or downstreamchanges in the bottom slope and/or background stratification are shown to produce sizeable changes in theinstability of a weakly baroclinic jet.

1. Introduction

Shelfbreak currents are generally characterized bysharp property fronts that delineate coastal from openocean waters. Significant temporal variability and spa-tial variability of these fronts lead to the mixing andexchange of coastal and open ocean waters. As such, anunderstanding of the source of this variability wouldimprove our estimates of onshore/offshore exchangeand our ability to predict frontal evolution, a predictiondesirable for a host of applications, including those as-sociated with navigation, fisheries, weather forecasting,and national defense. A much-studied shelfbreak cur-rent that exemplifies this highly variable nature is situ-ated in the Middle Atlantic Bight (Beardsley et al.1985). Over 20 years ago it was suggested that frontalinstabilities were responsible for the observed variabil-ity of this front (Flagg and Beardsley 1978); however,until recently, instability models were unable to cap-ture modes with appreciable growth rates (Flagg andBeardsley 1978; Gawarkiewicz 1991) that matched ob-servational growth rates (Garvine et al. 1988). Studying

the stability of a continuously stratified shelfbreak fron-tal current governed by linearized primitive equations,Lozier et al. (2002; hereinafter referred to as LRG)found unstable modes with growth rates on the order of1 day over a wide range of idealized background con-ditions applicable to the Middle Atlantic Bight. Lend-ing credence to the supposition that instabilities are asource of the variability associated with the Middle At-lantic Bight shelfbreak current are two recent observa-tional studies. From a study of ADCP data collectedover an 18-month period in the shelfbreak region southof New England, Fratantoni and Pickart (2003) con-cluded that the most probable cause of the observedmesoscale variability was baroclinic instability of theshelfbreak jet. Local wind forcing and tidal rectifica-tion, as well as the offshore effects of Gulf Stream rings,topographic Rossby waves, and Gulf Stream meander-ing, were all discounted as possible sources for the ob-served variability. Furthermore, from an analysis of sur-face drifters that were entrained upstream of theMiddle Atlantic Bight, Lozier and Gawarkiewicz(2001) found ubiquitous meandering and cross-frontalexchange that were restricted by neither locale nor sea-son and which existed even in the absence of GulfStream rings—all characteristics consistent with frontalinstability.

Though the LRG study focused specifically on theMiddle Atlantic Bight, with model topography basedon the shelfbreak bathymetry just south of Nantucket

Corresponding author address: M. Susan Lozier, NicholasSchool of the Environment and Earth Sciences, Division of Earthand Ocean Sciences, Box 90230, Duke University, Durham, NC20078-0227.E-mail: mslozier@duke.edu

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© 2005 American Meteorological Society

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Shoals, a wide range of background conditions for thevelocity and density fields were examined. A parameterstudy was conducted by varying the strength, width, anddepth of the background velocity field and the extent ofdensity stratification, a range of conditions believed tobe sufficient to extend the applicability of the results toother shelfbreak frontal currents. However, shelfbreakcurrents are characterized not only by differences inbackground conditions of the velocity and density field,but also by their underlying topography. Shelfbreakcurrents vary according to the steepness of the conti-nental slope, as well as by their flow direction relativeto the slope. Prograde currents flow such that their cy-clonic flank is in shallow (onshore) water, while retro-grade jets flow with their cyclonic flank in the deeper(offshore) water [Mooers et al. 1978; see Li and McCli-mans (2000) for an alternate definition]. Past studieshave indicated that both of these factors, slope steep-ness and flow direction relative to the bathymetricslope, can influence flow instability (e.g., Orlanski 1969;Blumsack and Gierasch 1972; Li and McClimans 2000).Thus, the extent to which the LRG results are generallyapplicable to other shelfbreak frontal currents is dic-tated by the effect of topography on the stability ofthose currents. Past studies on the influence of topog-raphy on oceanic flows have generally been restrictedto either barotropic or baroclinic flows and conductedwith approximated dynamics. The overall goal of thisstudy is to assess the influence of topography on thestability of a shelfbreak frontal current characterized bycontinuous horizontal and vertical shear and governedby primitive equation dynamics.

To characterize the frontal instabilities, a linearizedprimitive equation stability model is employed to de-termine the three-dimensional propagation of pertur-bations superposed on a unidirectional mean flow field,which varies continuously across stream and withdepth. The role of topography in stabilizing or destabi-lizing the flow is investigated for both retrograde andprograde jets. In addition, the effect of stratification onsuch stability is investigated. The relevant backgroundfor this study is given in the next section. Methods areoutlined in section 3, followed by a discussion of resultsin section 4 and a summary in section 5.

2. Background

In a seminal study almost 40 years ago, Pedlosky(1964) derived a necessary condition for the instabilityof continuously stratified, quasigeostrophic flows char-acterized by both horizontal and vertical shear. Follow-ing the formulation used in Pedlosky (1979), we con-sider a background zonal flow, defined by a geostrophicstreamfunction �(y, z) superposed with a perturbationstreamfunction given by �(x, y, z, t), so that the evolu-tion of the flow field is given by �(x, y, z, t) � �(y, z)� �(x, y, z, t). In the absence of friction and for small-amplitude perturbations,

�E

�t� �

�E���

�t� �

0

z0��1

1

�0�u��U

�y� ��

��

�y � dy dz,

�2.1�

where E( ) and E represent the total perturbation en-ergy and the total mean energy, respectively. The meanand perturbation potential temperatures are given by and �, respectively, and u (�) represents the zonal (me-ridional) perturbation velocity. The mean zonal veloc-ity is denoted by U, 0 is the background density, andthe top and bottom boundaries are at z � z0 and z � 0,respectively. All quantities are expressed in nondimen-sional form. Conversion of mean to perturbation en-ergy (and vice versa) depends upon perturbation mo-mentum and heat fluxes acting upon the backgroundvelocity and potential temperature fields. For pertur-bation energy to increase with time, the perturbation oreddy fluxes must be downgradient such that there is adecrease in the mean gradient fields. The perturbationfluxes can be expressed in terms of the backgroundfield and the meridional displacement of fluid elements,�(x, y, z, t). In particular, the perturbation heat flux atthe bottom boundary can be expressed as

�� � ��U

�z� S

��B

�y � ��2�2�t

at z � 0, �2.2�

where �B is the nondimensional height of the bathym-etry as measured from z � 0 (�B � 0) and S is theBurger number. For an unstable field, where the me-ridional displacements increase with time, this formu-lation shows that the sign of the perturbation heat fluxdepends upon the difference between the backgroundvelocity shear and the bottom slope. Expressing all theperturbation fluxes as a function of � and imposing theconstraint that in the absence of friction the x-averagedzonal momentum is preserved, a necessary conditionfor instability can be derived (Pedlosky 1979):

0 � �0

z0��1

1

�0

�o

�y � �

�t�2� dy dz

� ��1

1 ��0

S

�U

�z

�

�t�2�

z�z0

dy

� ��1

1 ��0�S�1�U

�z�

��B

�y � �

�t�2�

z�0dy, �2.3�

where �0 is the background potential vorticity. Asnoted by Pedlosky (1964, 1979), the vanishing of thisintegral is a necessary condition for instability. Such acondition can be met in a number of ways, dependingupon the background flow and the flow configuration(viz., the vertical shear) at the top and bottom bound-aries. Our interest is focused on the bottom boundarywhere it is apparent that the bottom slope plays a sig-nificant role in this integral constraint. This role can beunderstood in a geometric context when the thermal

1024 J 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 VOLUME 35

wind relationship is used to rewrite the innermost term(in parentheses) of the final term in Eq. (2.3) as

�S�1�U

�z�

��B

�y � �L

D ���z

�y ��

��hB

�y � at z � 0,

�2.4�

** * *

where L(D) is the horizontal (vertical) length scale, � isthe Rossby number, hB is the dimensional bathymetry(as measured from z � 0), (�z*/�y*)*

is the slope of theisotherms in the basic state, and the terms on the right-hand side are all dimensional, as denoted by the aster-isk. The stability of the flow can now be seen to dependon the bottom slope relative to the slope of the iso-therms. The link between this geometry and the stabil-ity process is made by combining Eqs. (2.2) and (2.4) sothat the perturbation heat flux at the bottom boundarycan be expressed as

�� � C���z

�y ��

��hB

�y � ��2

�tat z � 0, �2.5�*

* * *

where C is a positive constant. Thus, whether the per-turbation flux is downgradient (contributing to thegrowth of the perturbation energy) or upgradient (con-tributing to the growth of the energy of the mean flow)depends directly upon the geometry of the isothermsrelative to the bottom boundary. This relationship willbe made more explicit in the following sections.

The incorporation of topography in early studies offlow instability necessarily used approximated dynam-ics and approximated shear; that is, the flow fields werepurely barotropic, purely baroclinic, or layered in thevertical direction. Studies on the topographic influenceon shelfbreak frontal instability also fell into these cat-egories. Using a two-layer model and the geostrophicmomentum approximation (Hoskins 1975), Flagg andBeardsley (1978) found a rapid decrease in growth rateswith increasing bottom slopes for the baroclinic insta-bility of a retrograde jet, a decrease so large as to makethe growth rates unrealistic. Using the same approxi-mated dynamics but with a more sophisticated layeredmodel, Gawarkiewicz (1991) also found that topogra-phy stabilized a retrograde jet. In their study of a purelybarotropic slope current, Li and McClimans (2000)found that even mild slopes stabilized both retrogradeand prograde jets. Recent advances in numerical meth-ods (Moore and Peltier 1987) have allowed for the so-lution of instability equations using primitive equationdynamics and continuous shear in both the horizontaland vertical flow fields (Barth 1994; Xue and Mellor1993; Samelson 1993). With such a stability model, Xueand Mellor (1993) studied the Gulf Stream flow in theSouth Atlantic Bight and found that topography stabi-lized the mixed baroclinic/barotropic instability of thisprograde jet. Barth (1994) used a similar model to studythe stability of coastal upwelling jets and found that

topography had little influence on a frontal instabilitymode captured by the primitive equation dynamics(Moore and Peltier 1989), whereas topography gener-ally stabilized the traditional baroclinic mode (at lowwavenumbers) of a prograde jet. Our work followsfrom these studies in that we employ a jet characterizedby continuous horizontal and vertical shear and gov-erned by primitive equation dynamics, yet our focus isexclusively on the influence of topography on the sta-bility of a shelfbreak frontal current for which jet ori-entation, slope strength and stratification are varied.

3. 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.The reader is referred to Moore and Peltier (1987), Xueand Mellor (1993), and LRG for details on this methodand computation.

a. Model equations and solution method

The model used in this study consists of a steadybackground current flowing along the slope of idealizedbathymetry (Fig. 1). The model uses a Cartesian coor-dinate system, where x is the offshore axis (positiveoffshore), y is the alongshore axis (positive upstream),and z is the vertical axis (positive upward). Note thatthis coordinate system differs from that presented inthe last section, in which the system used by Pedlosky(1964, 1979) was preserved. For this formulation ofthe shelfbreak jet, our preference is to orient the y axisin the alongshore direction. Here, water depth is givenby h(x) and the alongshore background flow is givenby V(x, z). The background flow is in thermal windbalance with a mean density field (x, z) according tof0Vz � Bx, where f0 is a constant Coriolis parameter, Bis the mean buoyancy defined by B � �g (x, z)/ 0, with 0 being the reference density and g the gravitationalacceleration. To assess the stability of this basic state,three-dimensional velocity and density perturbationsare superposed onto a two-dimensional background ve-locity and density field. The evolution of the perturba-tions is governed by the hydrostatic primitive equa-tions, linearized about a geostrophic background state:

ut � Vuy � f0� � ��x, �3.1�

�t � uVx � V�y � wVz � f0u � ��y, �3.2�

0 � �b � �z, �3.3�

ux � �y � wz � 0, and �3.4�

bt � uBx � Vby � wBz � 0. �3.5�

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The Cartesian components of the perturbation velocityare (u, �, w); � is perturbation pressure divided by 0,and b is the perturbation buoyancy. Equations (3.1)–(3.3) are the equations of motion with no frictional orexternal forces, and Eqs. (3.4) and (3.5), derived fromthe conservation of mass, represent the continuityequation for an incompressible fluid and the conserva-tion of density, respectively. Boundary conditions stip-ulate that normal flow across solid boundaries is zero[u � 0 at x � 0 and w � �uhx at z � �h(x)] and thatdisturbances vanish at distances far from the coast. Inaddition, the rigid-lid approximation (w � 0 at z � 0) isimposed. [Note: For our model, we set z � 0 at the seasurface, which differs from the formulation used in sec-tion 2 in which z � 0 defines the bottom boundary. Wechose to follow the Pedlosky (1979) formulation in sec-tion 2.]

To transform the irregular grid created by thebathymetry into a regular rectangular grid (in the x andz directions), a mapping is applied to Eqs. (3.1)–(3.5).Such a mapping facilitates the use of basis functions,which are described below. The mapping, applied ac-cording to � � 1 � z/h(x), yields a new vertical velocity� given by � � w � (� � 1)uhx (Xue and Mellor 1993).With this mapping it is convenient to define transportvariables: (u*, �*, b*) � (uh, �h, bh). In addition, allvariables are cast in dimensionless form using the fol-lowing scaling: x� � x/L0, y� � y/L0, ����, t� � tf0, u� �u*/u0, �� � �*/u0, �� � (L0/u0)�, h� � h/H0, �� � �H0/(u0 f0L0), b� � b*H0/(u0 f0L0), B� � B/(H0N2

0), and V� �V/V0, where L0 and H0 are the horizontal and verticallength scales, respectively; u0 is the typical perturbation

velocity times H0; N0 is the Brunt–Väisälä frequency;and V0 is the maximum of the background velocityV(x, z). For all model runs L0 � 100 km, H0 � 200 m,and f0 � 9.37 � 10�5 s�1 (corresponding to a latitude of40°N). Solutions for the three-dimensional perturba-tions are sought in the form:

�u�, �� � Re��u*�x, ��, *�x, ��� ei��t�ly�� and �3.6�

���, ��, b�� � Re�i��*�x, ��, �*�x, ��, b*�x, ��� ei��t�ly��, �3.7�

where � is the complex frequency (� � �r � i�i), l is thealongfront wavenumber, the starred variables are thestructure functions in x and z, and Re{} denotes the realpart of the expression inside the curly braces. Substitu-tion of Eqs. (3.6) and (3.7) into Eqs. (3.1)–(3.5) yields

�� � R0lV�u* � �* � �h�*x � �� � 1�hxb*,

�3.8�

�� � R0lV��* � u*�1 � R0Vx�

� R0V� * � �lh�*, �3.9�

�b* � �*� � 0, �3.10�

u*x � l�* � *� � 0, and �3.11�

�� � R0lV�b* � SBxu* � SB� * � 0, �3.12�

where R0 � V0/( f0L0) is the Rossby number for thebackground flow and S is the Burger number, definedas [N0H0/( f0L0)]2. For the numerical calculation of Eqs.(3.8)–(3.12), the background flow and the perturbations

FIG. 1. Schematic of retrograde model jet and bathymetry used in this study. Definitions ofV(x, z) and h(x) are given in section 3b. All model jets are centered at x � 50 km.

1026 J 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 VOLUME 35

are confined to a domain bounded by x � 0, x � 100km, the sea surface (� � 1), and the seafloor (� � 0)(Fig. 1). The size of the domain was chosen to balancethe need for computational affordability with the needfor all perturbations to be vanishingly small at the chan-nel walls. In addition, to provide for no normal flow, u*is set to 0 at x � 100 km. With these boundary condi-tions, Eqs. (3.8)–(3.12) form an eigenvalue problemwith a complex frequency given by � and eigenmodesgiven by Eqs. (3.6) and (3.7). The solution of this eigen-value problem is achieved with a combination of Galer-kin and Fourier collocation schemes. To obtain a solu-tion, structure functions [e.g., u*(x, z)] are spectrallydecomposed into orthogonal sets of trigonometric basisfunctions in the vertical (�) and cross-shelf (x) direc-tions, as detailed in Xue and Mellor (1993). For numeri-cal solution these expansions are truncated at a finitenumber of spectral modes M. These expansions aresubstituted into Eqs. (3.8)–(3.12) and manipulated toproduce a standard matrix eigenvalue equation, whichis solved by standard linear algebra methods for a speci-fied background field, truncation level, and wavenum-ber. All model runs for this study used M � 44. Readersare referred to LRG for a discussion of model conver-gence. Model solutions yield growth rate (��i), phasespeed (��r/l), and modal structure of the instabilitiesat each wavenumber.

b. Jet velocity profiles and bathymetry

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

V�x, z� � V0 exp���x�x � x0�2 � �z�z � z0��, �3.13�

where �x [�4 ln(V0/0.1 m s�1)/x2d] and �z [�ln(V0/0.1

m s�1)/zd] are formulated such that the 10 cm s�1 iso-tach will fall at distances xd/2 from x0 and zd from z0.For all model runs, the center of the jet is placed at x �50 km (x0 � 50 km) and the maximum jet velocity is atthe surface (z0 � 0 m). The width and depth of the 10cm s�1 isotach are specified for each model run by thevariables xd and zd, respectively. The background den-sity field B(x, z) is set by the velocity field and thespecification of density at the offshore boundary andthe condition of thermal wind balance. At the offshoreboundary, the stratification is assumed to be linear (Bz

is constant) for all model runs, with the deepest isopy-cnal set at � 1.0275 g cm�3 (Linder 1996). To createchanges in the background stratification, the uppermostisopycnal shallow is varied at the eastern boundary. Forbrevity, values of shallow will be expressed using sigma-tunits.

Using these parameters, two qualitatively differenttypes of jets were produced: an idealized Middle At-lantic Bight frontal jet with strong vertical shear andone with very weak vertical shear, such that it could becharacterized as a weakly baroclinic jet. The former

was studied extensively in LRG, where it was shownthat the modal structures associated with this jet areoften surface-trapped, except for the very lowest ofwave numbers. To probe the interaction of unstablemodes and the bottom slope, the latter jet was alsochosen for this study. The same functional form, asgiven above, was used for both jets. They differ only intheir value of �z [Eq. (3.13)], which controls thestrength of the vertical shear.

The bathymetry used in our model study was deter-mined from the fit of a hyperbolic tangent functionalform (Xue and Mellor 1993) to the bathymetric mea-surements in the Nantucket Shoals region (Linder1996). This fit is given by

h�x� � Hs � 0.5�Hd � Hs��1 � tanh��x � xm�����,

�3.14�

where Hs (shelf depth) � 60 m, Hd (maximum domaindepth) � 200 m, xm (location of maximum slope) � 50km, and � (lateral extent of slope) � 15 km. The off-shore asymptotic depth in the Middle Atlantic Bight ison the order of 2000 m, but we have used 200 m in ourmodel study to reduce the computational grid. A sen-sitivity study to offshore depth found negligible differ-ences between model runs using 200 m as the offshoredepth and those using depths in excess of 200 m. Tostudy model sensitivity to bottom slope and to isolatequantities that are significant along the seafloor, thebathymetry of the model was modified by varying � inEq. (3.14) above. The range of bathymetric slopes, pro-duced by varying � from 5 km (steep) to a value of 25km (less steep), is shown in Fig. 2. It is important tonote that for our study, as for LRG, we have chosen tokeep the frontal jet’s centerline position fixed, to main-tain constant stratification at the offshore boundary andto maintain symmetrical horizontal velocity shear. Eachof these constraints is made for the purpose of simpli-fication and could be relaxed in future studies of thisfrontal system.

c. Prograde and retrograde jets

To understand how flow direction relative to the bot-tom slope affects flow stability, the velocity and densityfields of the retrograde jets were constructed to bemirror images to the prograde fields, with the only ex-ception being the orientation of the isopycnals to thebottom slope. Given the structure of the density fieldimposed by the thermal wind balance, isopycnals asso-ciated with a retrograde surface-intensified jet slope inthe opposite direction from the bathymetric slope (Fig.2) while the isopycnals associated with a surface-inten-sified prograde jet slope in the same direction as thebathymetric slope. To create equivalent interior fields,the velocity field was simply reversed while the follow-ing procedure was used for the density fields. The den-sity field for the retrograde jet was computed by speci-

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fying the boundary conditions along the eastern borderand then integrating westward using the thermal windrelation. The density values obtained along the westernboundary for the retrograde jet were then specified asthe eastern boundary conditions for the prograde jet. Awestward integration from these boundary conditionsyielded a “mirror image” of the retrograde densityfield.

4. Results

In this section we first present the model results forthe weakly baroclinic jet, illustrating the effect of to-pography on both retrograde and prograde jets markedby either weak or strong stratification. Second, the re-sults of the strongly baroclinic jet are presented. Asmentioned above, in all cases the retrograde and pro-grade jets were constructed with equivalent potentialvorticity fields in the interior. Thus, the necessary con-dition for instability [Eq. (2.3)] will be met equally bythe retrograde jet and its prograde counterpart in termsof interior fields and the upper boundary condition.Thus, differences between the two jets will be attrib-uted to differences in the orientation of their isopycnalsrelative to the bathymetric slope. We will specificallystudy how this difference affects the growth of the per-

turbation field. Our focus will be on the relationshipexpressed in Eq. (2.5), which for our study (using iso-pycnal surfaces rather than isentropic surfaces and analongshore jet rather than a purely zonal jet) is rewrit-ten as

u� � �C���z

�x��

��hB

�x � ��2

�t, �4.1�

where C is a positive constant, �hB/�x � 0 for our modelconfiguration (hB � Hd � h; see Fig. 4, described later)and (�z/�x) denotes the slope of the background iso-pycnals. For the following discussion, it is useful to notethat the background density field for a retrograde (pro-grade) jet has � /�x � 0 (� /�x � 0), as illustrated in Fig.2 for a retrograde jet, yielding positive (negative) iso-pycnal slopes (�z/�x) for the retrograde (prograde) jet.

As has been demonstrated from energy analyses inpast studies (Xue and Mellor 1993; Barth 1994), it isexpected that the unstable perturbations in this studywill derive their energy from both the vertical and hori-zontal shear. We can use the ratio of the internalRossby radius of deformation ri to the horizontal lengthscale of the flow field Lh as a measure of the impor-tance of available potential energy relative to kineticenergy as the energy source for the perturbation growth(LRG). For all runs in this study with the weakly baro-

FIG. 2. Velocity and isopycnals typical of (left) a weakly baroclinic jet and (right) a strongly baroclinic jet for the standard bathymetry(shaded gray). Also shown (with black lines) are the bathymetry profiles that result by varying the slope parameter � [see Eq. (3.14)]from 5 km (steepest) to 25 km (least steep) in increments of 5 km. Note how the slope of the isopycnals is opposed to the slope of theunderlying bathymetry for this retrograde jet. Annotated velocity contours are at 0.2 m s�1. Isopycnals are denoted by their sigma-tunits.

1028 J 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 VOLUME 35

clinic jet, Lh � ri, indicating the predominance of thebarotropic instability. For those runs with the stronglybaroclinic jet, it is expected that, because Lh � ri, nei-ther baroclinic or barotropic instabilities will exclu-sively govern the perturbations’ growth. In accord withthis expectation, an assessment of the backgroundfields shows that the mean kinetic energy is an order ofmagnitude larger (smaller) than the mean available po-tential energy for the weakly (strongly) baroclinic jets.

a. Weakly baroclinic jet

A stability analysis on a weakly stratified flow ( shallow

� 27.25) was conducted using three different topogra-phies. In all three cases, the structure of the jet re-mained unchanged; however, either the underlyingbathymetry was changed or the jet’s flow direction rela-tive to the bathymetry was changed. In one case the jetwas retrograde, in another case it was prograde, and inanother case the bottom was flat. For all three cases,the jet is unstable over a range of wavenumbers (Fig. 3),with the maximum growth rate occurring betweenwavenumbers 11 and 13, corresponding to approxi-mately 50 km. (For all of the weakly baroclinic jets,significant growth rates were restricted to the wave-number range of 0–30.) Of note is the distinct differ-ence in the magnitude of the growth rates for each ofthe three cases. For this weakly stratified jet, the retro-grade jet is more unstable than that for the flat-bottomcase, which is more unstable than the prograde jet. It isapparent that, for weakly stratified jets, topography de-stabilizes retrograde jets and stabilizes prograde jets.

As seen from Eq. (4.1), downgradient perturbationfluxes are consistent with the growth of the perturba-tions (��2/�t � 0) for both the flat-bottom retrogradeand prograde jets. With the addition of topography(�hB/�x � 0), the perturbation flux for the retrogradejet becomes more negative; thus the downgradient fluxis increased, resulting in a more unstable jet as evi-denced in Fig. 3. For the prograde jet, however, theaddition of a topographic slope, where |(�z/�x) | �|�hB/�x | , creates upgradient fluxes, which act to stabi-lize the flow, as seen in Fig. 3. Calculation of the kineticand potential energy exchange between the mean andperturbations fields shows that the addition of topog-raphy increases the conversion of mean potential en-ergy to eddy potential energy for the retrograde jet, yetfor the prograde jet the topographic addition creates atransfer from the eddy potential energy to the meanpotential energy, consistent with the upgradient fluxesdiscussed above. The difference in the conversion be-tween the mean and eddy potential energies is reflectedin the growth-rate differences for these three cases.

The difference in the topographic effect on the ret-rograde and prograde jet is illustrated qualitatively inFig. 4. Because there is no flow normal to the boundary,the perturbation flow is constrained to lie parallel withthe topographic slope. Thus, the degree to which veloc-ity perturbations can create perturbation density fluxesis strongly dependent upon the slope of the isopycnals.For instance, when the isopycnal slope matches the to-pographic slope, the perturbation density flux will tendto zero because the perturbation velocity is essentially

FIG. 3. Plot of growth rate as a function of wavenumber for a weakly baroclinic jet ( shallow� 27.25; S � 0.006) under three different bathymetric conditions: a flat bottom, the jet in theretrograde direction, and the jet in the prograde direction. Note that for a flat bottom thegrowth-rate characteristics for prograde and retrograde jets are identical; hence for flat bot-toms only the retrograde version is shown in this and all subsequent figures.

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along a density surface. However, as the “wedge” be-tween the bottom and the isopycnal “opens,” the per-turbation density fluxes become nonnegligible becausethe perturbation velocity at the boundary now crosses adensity surface. In Fig. 4, the upper (lower) line repre-sents a sloping isopycnal for a retrograde (prograde)jet. When topography is added, it is apparent thatdowngradient fluxes for the retrograde jet will increaseand those for the prograde jet will decrease. The effectof this differing geometry is apparent in Fig. 5 wherethe perturbation density fluxes (averaged over onewavelength) u associated with the fastest-growingwave for the retrograde and prograde jets shown in Fig.3 are presented. The retrograde jet is characterized bybottom-intensified downgradient fluxes (u � 0), andthe prograde jet is characterized by relatively weak up-gradient fluxes (also u � 0 and also bottom-inten-sified). These fields are consistent with the relationshipexpressed in Eq. (4.1).

Based on the arguments above, a steepening of thebottom slope is expected to further destabilize a retro-grade jet while further stabilizing a prograde jet. To testthis expectation, flow characteristics for both retro-grade and prograde jets were held constant while thebottom slope was changed. Shown in Fig. 6 are theresults of a study in which � was varied to create moreand less steep slopes. As evident from Fig. 6a, a steep-ening of the slope destabilized the retrograde jet; aninspection of Fig. 6b shows the reverse: increasing bot-tom slopes are associated with decreasing growth ratesfor prograde jets. It is important to keep in mind thatthe perturbations are of mixed nature; that is, energy isgained through baroclinic/barotropic instabilities. Inboth the retrograde and prograde cases, an increasingbottom slope should lead to the stabilization of the

barotropic instability. Given that the retrograde jet isdestabilized, we surmise that the strong downgradientdensity fluxes that would increase with increasing slopeare sufficient to offset the barotropic tendency for sta-bilization. Overall, it is important to note that a rela-tively small change in the bottom slope has a largerimpact on the prograde jet than on the retrograde jet,perhaps because the barotropic and baroclinic instabili-ties both act to stabilize the flow.

Using the geometric argument discussed above, it isreadily apparent how a change in the background strati-fication will affect the stability of these jets. Given thatthe slope of the isopycnal is determined by

��z

�x��

� ����

�x�����

�z�, �4.2�

FIG. 4. Schematic showing the effect of a bottom slope andstratification on the orientation of the isopycnals to the bottom.

FIG. 5. (top) Cross section of the perturbation density flux u for a weakly baroclinic retrograde jet, calculated for the fastest-growing wave; (bottom) the prograde jet. For both of these jets, shallow � 27.25 and S � 0.006. Depth in meters is given on thevertical axis. The right color bar denotes the scale for the densitycontours, in sigma-t units; the horizontal bar gives the scale for u ,in dimensionless units. For clarity, flux contours near 0 have beenomitted.

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Fig 5 live 4/C

an increase in the stratification will act to reduce themagnitude of the isopycnal slope for both the retro-grade and prograde jet, as shown in Fig. 4. An increasein stratification essentially flattens the isopycnals forboth types of jets, such that a convergence of the sta-bility characteristics at high stratification is expected.To check the validity of this statement, the backgroundstratification was increased for the retrograde and pro-grade jets used to produce Fig. 3. As seen in Fig. 7a, theretrograde (prograde) jet is indeed stabilized (destabi-lized) as the stratification is increased.1 The growthrates of the two jets converge for strong stratification,when both jets have essentially the same geometric con-figuration for the isopycnals and bottom slope. Thisresult implies that the barotropic instability for theseretrograde and prograde jets is equally affected by thetopography, consistent with the Li and McClimans(2000) study of a purely barotropic shelfbreak jet.

Though not shown here, when the highly stratified ret-rograde and prograde jets are run with a flat bottom,the growth rate increases, reinforcing the result thattopography stabilizes a barotropic instability. Of inter-est is that even though these jets have strong horizontalshear (R0 � 0.72), their growth rates are affected byrelatively small changes in the orientation of the isopy-cnals relative to the bottom slope. It is clear that therelease of potential energy by the perturbation fluxes atthe bottom has a significant effect, though in no caseswas the flow completely stabilized by the addition oftopography. However, growth rates can vary by morethan a factor of 2 on the basis of bottom isopycnalorientation alone (Fig. 7a). Thus, even for weakly baro-clinic jets, a small change in the background isopycnalslope can produce growth-rate changes that are effec-tively distinct (�1 day as compared with 3 days; Fig.7a). In fact, such changes in the growth rate are com-parable in magnitude to those when the horizontalshear of the background flow is doubled (LRG). Thus,relatively small temporal and/or downstream changesin the background structure of the shelfbreak jet couldhave large consequences for the resultant variability.Of importance is that these changes would dependupon whether the jet was prograde or retrograde.

A calculation of the energy conversions for each ofthe model runs confirms the expectation that changes inthe orientation of the isopycnals relative to the bathym-etry affect the energy conversions, in turn affecting theperturbation growth rate. Though the available poten-tial energy is small relative to the mean kinetic energy

1 The prograde 26.0 jet appears to be an exception to the statedhypothesis that increasing stratification destabilizes weakly baro-clinic prograde jets. However, such a hypothesis holds only forthose modes that are influenced by the bottom, as explained ear-lier. As shown in LRG, stratification always acts to stabilize asurface-trapped mode. The abrupt change in the prograde 27.25growth rate curve at l � 9 (Fig. 7a), along with changes in thephase speed and modal structure (not shown), indicate that thefastest-growing mode has switched from a predominantly baro-tropic mode to a surface-trapped mode, where the bottom nolonger matters. Here then the stratification acts to stabilize the jetand we are left with a 27.25 result that is more stable than the 26.0jet.

FIG. 6. Growth rates as a function of wavenumber for (top) retrograde and (bottom)prograde jets ( shallow � 27.25; S � 0.006) with varying bottom slopes. The lateral extent of thebathymetry, � in Eq. (3.14), was varied from 5 km (more steep) to 25 km (less steep) inincrements of 5.

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for the weakly baroclinic jets, it is not zero, and, asevidenced by the changes in the growth rates when themean kinetic energy is held constant, its release is sig-nificant to the resultant instability. A change in theisopycnal slope relative to the bathymetric slope pro-duces significant changes in the horizontal heat fluxterm (indicating the transfer of mean potential energyto eddy potential energy). Changes in this energy con-version are reflected in the growth-rate changes.

To test the intuitive understanding of the effect ofstratification changes on flow stability, illustrated sche-matically in Fig. 4 and explained above, the right-handside of Eq. (4.1) is evaluated as a function of cross-stream distance for each of the model jet configurationsshown in Fig. 7a. As seen in Fig. 7b, the magnitude ofthe perturbation density fluxes is maximized near thejet center, primarily because the bottom slope is maxi-mized there, but also because the isopycnal slopes arealso maximized there. For the retrograde jet, the down-gradient perturbation flux (u � 0) decreases in mag-nitude as stratification is increased, thus stabilizing theflow. An increase in stratification for the prograde jet,however, increases the magnitude of the perturbationflux, destabilizing the flow. The convergence, as notedin the growth-rate curves, is also apparent for the per-turbation fluxes at strong stratification (Fig. 7b). How-ever, we note that while the increase in stratification forthe prograde jet essentially opens the wedge betweenthe bottom slope and the isopycnals, thus creating alarger magnitude for the perturbation density flux, thedirection of the fluxes are upgradient.

Because baroclinic instability is characterized by en-ergy exchanges in which the mean available potentialenergy is lost to the eddy potential energy, which is thentransferred to the eddy kinetic energy, it is instructiveto examine the structure of the cross-shelf momentumfluxes, u�, for these unstable jets. Shown in Fig. 8 arethe cross-shelf momentum fluxes associated with thefastest-growing wave for some of the weakly andstrongly stratified retrograde and prograde jets studiedin Fig. 7. The flux fields exhibit a signature of down-gradient behavior: the two lobes, split at the Vmax loca-tion, are of opposite sign, because the mean velocitygradient changes sign at Vmax. Such a structure impliesthe importance of the barotropic instability for thishigh-Rossby-number flow (R0 � 0.72). For the weaklystratified jet, the fluxes are nearly uniform in the ver-tical direction, with a slight surface intensification. Theretrograde jet, which has been destabilized by topogra-phy, has a relatively stronger flux field than the pro-grade jet, which was stabilized by topography. Of im-portance is that the fluxes for these weakly stratifiedjets are present at the bottom, in contrast to the fluxesfor the strongly stratified retrograde and prograde jet.For these jets, the same doubled-lobed structure, cen-tered on Vmax, is present; however, the flux field isstrongly surface intensified such that there is no signa-ture at the bottom. In effect, stratification has insulatedthe perturbation from the influence of the underlyingbathymetry. Such behavior is reminiscent of baroclinic(Eady) modes in which the depth scale of the pertur-bation is essentially the Rossby height HR, given by

FIG. 7. (a) Growth rates as a function of wavenumber for weakly baroclinic jets with varyingstratifications ( shallow � 27.25, 26.0, and 20.0, with S � 0.006, 0.033, and 0.164, respectively)for both retrograde (R) and prograde (P) jets. (b) The slope term, �[(�z/�x) � �h/�x],corresponding to the jets in (a), calculated at the bottom boundary. The solid black line depictsthe cross-shelf bathymetric slope �hB /�x.

1032 J 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 VOLUME 35

f0/(N0l) (Gill 1982). As the stratification increases, theRossby height, where the steering level of the pertur-bation is located, decreases and the perturbation“moves off” the bottom (Barth 1994). (For the depen-dence of HR on l, the alongfront wavenumber, refer toLRG.) It is noted also that the perturbation momentumflux is greater for the retrograde jet than for the pro-grade jet, perhaps an indication of the stronger pertur-bation density flux that results for that jet. Also, theoffshore intensification of the perturbation momentumflux for the highly stratified jets suggests the possibilityof an asymmetry in cross-frontal exchange.

In summary, topography stabilizes a weakly baro-clinic prograde jet and destabilizes a weakly baroclinicretrograde jet. However, as stratification is increased,

the prograde jet is destabilized and the retrograde jet isstabilized, resulting in no difference in the growth ratesbetween these jet types. These results essentially showthat the geometry of the isopycnal relative to the bot-tom slope either promotes or stifles perturbation den-sity fluxes that feed the growth of the instability. Thisresult, predicted for quasigeostrophic dynamics, isshown to hold for a range of stratifications and bottomslopes for retrograde and prograde jets governed byprimitive equation dynamics. It is interesting that thebottom density fluxes have an impact on the flow sta-bility despite the fact that the flow is strongly barotro-pic. Different runs in which the background Rossbynumber varied gave qualitatively the same results; thatis, the retrograde/prograde jet dependence on topogra-

FIG. 8. (a) Cross section of the perturbation momentum flux u� for a weakly baroclinic retrograde jet with low backgroundstratification ( shallow � 27.25; S � 0.006) calculated for the fastest-growing wave. (b) As for (a) but for a prograde jet. (c) Similar to(a) but now for high background stratification ( shallow � 20.0; S � 0.164). (d) As in (c) but for a prograde jet. Depth in meters is givenon the vertical axis.

JUNE 2005 L O Z I E R A N D R E E D 1033

Fig 8 live 4/C

phy and stratification was preserved. Also, the effect ofthe Rossby number on the stability characteristics wasunchanged from that reported in LRG; that is, topog-raphy was not a modifier of this behavior.

b. Strongly baroclinic jet

The effect of topography on flow instability wastested for a model jet with considerable baroclinicstructure. Unlike the jets studied in the previous sec-tion, these baroclinic jets are characterized by stronghorizontal and vertical shear. We chose to study thestandard jet configuration used in LRG’s parameterstudy: a jet with a depth of 70 m, Vmax of 60 cm s�1, andR0 of 0.72, for both weak and strong stratification. Asshown in Fig. 9, the growth-rate curves for the flat-bot-tom case, the retrograde case, and the prograde caseexhibit only small differences—particularly so for theweakly stratified jet. As was demonstrated in LRG, theperturbation modes for these baroclinic jets are surfacetrapped, concentrated where there is appreciable verti-cal shear (according to their Rossby height, as discussedearlier). Because the perturbation velocities near thesloping boundary would be negligible, it follows thatany perturbation density flux would be too weak toaffect the overall stability of the jet. It is interesting tonote that the effect of increasing stratification for eachof these baroclinic jets is to stabilize the flow (see theparameter study conducted in LRG), unlike the casesdiscussed above in which increasing stratification pro-duced a more (less) stable retrograde (prograde) jet.Because a flattening of the isopycnals generally reduces

the potential energy available to feed an instability, re-duced growth rates are expected. This expectation,however, is altered when considering the perturbationfluxes near the bottom and their dependence on theorientation of the isopycnal slope relative to the bottomslope, as explained in the previous section.

In addition to being insensitive to the bottom slope,the growth-rate curves for these strongly baroclinic jetsdiffer from those shown in Figs. 3 and 6 in that thegrowth rate increases with increasing wavenumber and,especially for the low-stratification runs, the maximumgrowth rate lies beyond the l � 30 cutoff. As demon-strated by LRG and Barth (1994), the instability of jetswith appreciable vertical shear is dominated by a fron-tal instability mode characterized by smaller wave-lengths and smaller vertical scale than the traditionalbaroclinic (Eady) mode (Moore and Peltier 1989). Inall model runs for both the retrograde and the progradejets, this frontal instability mode was insensitive to thepresence of the bottom slope and to changes in theorientation of the background isopycnals to the bottomslope. As noted in section 2, Barth (1994) also notedthis insensitivity in his study of coastal upwelling jets.Furthermore, in all parameter studies run in LRG, thefrontal instability mode dominated—that is, it had alarger growth rate than the baroclinic mode. Thus, weare left to conclude that for those jets with relativelystrong vertical shear, such that the Rossby height �water depth, the stability of the jet will be insensitive tothe bottom. Our results have shown that the baroclinicmode is sensitive to the bottom configuration, but only

FIG. 9. (a) Plot of growth rate as a function of wavenumber for a strongly baroclinic jet withlow stratification ( shallow � 27.25) under three different bathymetric conditions: a flat bottom,a jet in the retrograde direction, and a jet in the prograde direction. (b) As above, but for casesof high stratification ( shallow � 20.00).

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for some jet/slope configurations. Topography has beenshown to influence the stability of weakly baroclinicjets, characterized by weak stratification. An increase inthe stratification reduces, and eventually eliminates,this dependence. In addition, an increase in the verticalvelocity shear such that the jet is more surface intensi-fied reduces and then eliminates the effect of topogra-phy on stratification. In both cases, it is the contributionof perturbation density fluxes near the boundary to theinstability that has been altered. The degree to whichthe stratification needs to be increased to eliminate anytopographic influence depends strongly upon thestrength of the flow field near the bottom boundary.This dependence is shown in Fig. 10 in which the dif-ference in growth rate between a prograde jet and aretrograde jet is plotted as a function of zd (the depth ofthe 10 cm s�1 isotach) for varying background stratifi-cations at the eastern boundary. [Note that for thosecases in which zd exceeds the water depth, the implica-tion is that 10 cm s�1 would have fallen at zd given thestructure of V(z).] For weakly stratified flows ( shallow

� 27.25 and 26.0), the effect of topography is seen toincrease as zd is increased. In other words, the geometryof the sloping isopycnals at the bottom boundary influ-ences the growth rate for those jets with appreciablevelocity shear near the bottom. On the other hand,strongly stratified jets ( shallow � 24.0, 22.0, and 20.0)are not affected by the underlying topography over arange of depths. For these cases, the Rossby height issufficiently removed from the bottom such that the per-turbation is insulated from any topographic effects.

5. Summary

To ascertain the effect of topography on a shelfbreakfrontal current, a linearized primitive equation stabilitymodel was employed to determine the three-dimen-sional propagation of perturbations superposed on atwo-dimensional mean flow field, with continuousshear across the current and with depth. The effect ofstratification and bottom slope changes was investi-gated for both retrograde currents and prograde cur-rents. In no instance was a frontal instability mode af-fected by the presence of the bottom slope, a resultattributable to the strong surface trapping of thesemodes. Their Rossby height is sufficiently small to iso-late the perturbation from any bottom effect. Onlywhen the water depth approaches the Rossby height(on the order of 10–20 km) would one expect the bot-tom topography to influence the stability characteristicsof these frontal modes. Because a prior study (LRG)found that these modes generally have faster growthrates than the traditional baroclinic modes, it is ex-pected that for strongly baroclinic shelfbreak flows thetopographic influence is negligible. On the other hand,this study has demonstrated that for jets characterizedby weak vertical shear (jets referred to in this study asweakly baroclinic) the bottom slope has a strong influ-ence. Although the stabilizing effect of topography onpurely barotropic jets is well known, our work hasshown that sloping isopycnals associated with weak ver-tical shear of the jets can influence the growth rate ofthe instability. We specifically show that the orientation

FIG. 10. A semilog plot of the average rms difference in growth rate of the retrograde andprograde jets per wavenumber over the range of wavenumbers 0–20, illustrating the combinedimpact of jet depth and background stratification on instability. The horizontal axis is zd, asgiven in Eq. (3.13), and reflects the vertical shear of the jet.

JUNE 2005 L O Z I E R A N D R E E D 1035

of the isopycnal to the bottom slope sets the magnitudeof the perturbation density flux, which can exert a sta-bilizing or destabilizing influence on the flow field. Forrelatively weak stratification, the presence of a bottomslope destabilizes a retrograde jet and stabilizes a pro-grade jet. In addition, as the stratification of the retro-grade jet is increased, its stability is increased while thestability of the prograde jet is decreased. The change inthe growth rate from these stratification changes is aslarge as the change in growth rate when the Rossbynumber of the background flow is substantiallychanged. Thus, one could expect the instability charac-teristics of a weakly baroclinic jet to change either spa-tially or temporally as the frontal contrast between theoffshore and onshore waters increases or decreases thecross-stream density gradient. Such influence of thebottom on the stability of the flow can be diminishedand even negated by an increase in the backgroundstratification. Such an increase effectively decreases thepenetration of the perturbation into the water column,essentially isolating the perturbation from any effect ofthe bottom. Thus, we conclude that only for weaklybaroclinic jets with relatively weak stratification willthere be a significant influence of the topography onthe stability of the flow field.

We note that our study is limited by two simplifyingassumptions: symmetric cross-stream shear and uni-form stratification at the eastern boundary. The latterassumption is considered to have the most influence onthese results. The presence of a strong mixed layer,common in the winter months, could potentially createunstable modes that were not surface trapped. In suchcases, it is likely that the topographic influence wouldbe larger than that found here where the fastest-growing modes were all surface trapped. The incorpo-ration of a mixed layer into this instability model is theplanned focus of a future study.

Acknowledgments. The authors thank G. Gawark-iewicz and L. Pratt for their valuable input to this work.MSL gratefully acknowledges support from the Officeof Naval Research (N00014-01-1-0260). Computationalresources for the study were provided by the NorthCarolina Supercomputing Center.

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