Topographic influence on Gulf Stream meandering

1 Introduction

The Gulf Stream and its eddies play a significant role in the physical and biological properties on the continental shelf as well as in heat transport in the Atlantic and thereby the climate system structure. It also has local impacts on both temperature and ecosystems near its path, including providing warming to countries on the eastern side of the North Atlantic (Palter, 2015).

Beyond its large-scale role in climate and ecosystems, the Gulf Stream exhibits complex mesoscale dynamics marked by the presence of meanders, vortices, and ring structures. These features are not mere byproducts of the current but play an active role in enhancing lateral transport processes, particularly the distribution of nutrients and pollutants on regional scales (Wiebe et al., 1976; Flierl and Davis, 1993; Gaube and McGillicuddy Jr, 2017). Understanding the formation and evolution of such structures is therefore essential to gain insight into the fine-scale mechanisms that govern the Gulf Stream’s behavior and its interaction with its surrounding environment.

On its travel up the coast from Florida to Cape Hatteras, the Gulf Stream is predominantly on the shelf with the topographic rise to the coast on its left. As it exits into deep water, it is still not far from the continental slope. We can expect that the meandering, eddy formation, and resulting large-scale turbulence will be affected by the topographically induced vortex stretching.

Since the 1960s, studies focusing on the Gulf Stream have increased significantly (Fofonoff, 1981). As the understanding of the geophysical dynamics governing the Gulf Stream improves and as numerical models become increasingly sophisticated, it becomes possible to incorporate a wider range of physical parameters into its study. Although the role of ocean bottom topography in triggering instabilities along the current has been investigated using high-resolution regional models (Gula et al., 2015a,b), in this work we focus on the fundamental mechanism of instability using a combination of analytical analysis and idealized modeling.

Several studies have examined the influence of topography on the stability of coastal and boundary-current systems. Mysak et al. (1981) used a two-layer QG model to investigate the combined effect of barotropic and baroclinic instability on a coastal current under different lateral boundary conditions. Their analysis involved a similar geometry to that considered here but did not include an off-shelf, isolated jet structure. Dewar and Leonov (2004) and Leonov and Dewar (2008) examined shelf waves in a similar system consisting of a two-layer offshore region adjacent to a single-layer shelf, but without an unstable jet. Studies that explore the influence of topography on the stability of coastal jets with barotropic or 1.5-layer shallow water models include, for example, Li and McClimans (2000) and Poulin and Flierl (2005), respectively. When the jet is running in the same direction as the topographic waves, it can result in or enhance the instability, whereas it damps the growth when it is opposite. Although we employ a different model, aspects of this behavior arise in our work. Lozier et al. (2002) looked at the problem with a very shallow shelf-break jet, while Ezer (2017) considered both idealized and realistic topography but did not explore the parameter regime in terms of topographic slope and closeness to the current. That is the goal of this paper—to provide insight into how these two factors influence the instability and finite amplitude growth.

We will use the two-layer quasi-geostrophic (QG) model to investigate jet–topography interaction. The warning in Poulin and Flierl (2005) that asymmetries in the shallow water problem are not present in QG suggests that details would differ between the two analyses. However, we expect that our results are indicative of the nature of the instability and meandering of a baroclinic jet as it interacts with topography. We begin in Section 2 by discussing the general instability problem both in terms of an integral–differential matrix equation and as the displacement of multiple potential vorticity (PV) contours.

Then, we treat a series of examples consisting of a zonal jet U_j(y) and a topography with a northward transition from deep offshore to a shallow “shelf”. In Section 3, we examine the instability for cases when the topography is confined to the bottom layer (examples 1–3). Example 1 includes a baroclinic jet with a topographic step in this bottom layer. This allows us to see how separation between the jet and the topography affects the growth as well as interactions between the deep westward flow and topographic Rossby waves. In example 2, the jet is blocked from the shallow region (although the topography is still in the bottom layer), allowing exploration of the effects of asymmetry and the absence of jet flow along the topographic step. Example 3 looks at a smooth sech²(y/w) top layer flow with no deep jet and smooth topography; this configuration is used in the nonlinear numerical studies. In Section 4 (example 4), we use the same piece-wise constant PV ansatz as the first two cases but consider what happens when the topography blocks the bottom layer entirely and extends into the top layer [as in Mysak et al. (1981); Thompson (1993); Dewar and Leonov (2004)]. Because the shelf flow is now purely barotropic and governed by different dynamics than the offshore flow, the instability problem cannot be approached in quite the same way. This case gives insight in the impact of topography rising rapidly from the deep ocean to the continental shelf. We summarize our findings in Section 5.

Finally, in Section 6, we discuss a series of numerical experiments to see how the eddy energy levels off and what the structures look like for different jet-to-topography distances. Section 6.2 considers topography, which is a ridge or seamount chain that crosses the jet rather than being parallel to it; these give some idea of how the New England Seamount chain alters the development of the instability. Figure 1 gives a somewhat abstract picture of the flow and topography that will be addressed; it has both the top “Gulf Stream” jet, a deeper undercurrent (not in every calculation), and the nearby topographic slope.

2 Linear stability

Quasi-geostrophic dynamics simplify the stability problem and the nonlinear evolution to a single variable, the QG potential vorticity, q(x,y,z,t):

qt+J(ψ,q)=0

with an inversion relation

ℒψ+ℬ=q

where ℬ is the contribution from topography. In the two-layer problem, with the topography in the bottom layer, the non-dimensional form is

q=[q1(x,y,t)q2(x,y,t)], ℒ=[∇2−F1F1F2∇2−F2], ψ=[ψ1ψ2], ℬ=[0B(y)]

The length and time scales are Rd=g′H1H2/(H1+H2)/f and R_d/U_max, respectively, where U_max is the maximum speed of the basic state jet flow. H₁ and H₂ are the mean depths of the top and bottom layers. The parameters, written in terms of the depth ratio d = H₁/H₂, are F₁ = 1/(1 + d) and F₂ = d/(1 + d), and the dimensional topographic height, B_dim, contributes

B(y)=fBdimRdH2U

to the non-dimensional PV. We will take d = 0.2 throughout.

Defining q in terms of the rest state (no velocities) plus the part associated with the jet plus the fluctuations yields

q=q¯(y)+q′(x,y,t)=ℬ(y)+Q(y)+q′(x,y,t) and ψ=Ψ+ψ′

with

ℒΨ=Q or ℒU=−Qy and ℒψ′=q′

The evolution of the fluctuations is

∂∂tq′+J(Ψ,q′)+J(ψ′,Q+ℬ)+J(ψ′,q′)=0

which can be expressed in terms of the inversion operator G=ℒ−1

∂∂tq′+J(Ψ,q′)+J(Gq′,Q+ℬ+q′)=0

For zonal flows and topography, the linearized perturbation can be written as q′(x,y,t)=q′(x−ct,y) with c and q′ potentially complex; the resulting equation is simply

(1) Uq′+QyG(k)q′=cq′

with the eigenvalue c determining the stability. The superscript (k) marks the value of the wavenumber being used in the inversion. The translational invariance in x means that we can examine each member of the Fourier series separately, q′(x,y,t)=q′exp(ık[x−ct]), and find the growth rate σ=kℑ(c(k))

If we express this for the two-layer, deep topography model

Uij(y)qj′(y)+(Qy+ℬy)im∫dy′ Gmj(k)(y−y′)qj′(y′)=cqj′(y)

(with the U and Qy+ℬy being diagonal matrices). The Green’s function is found from

ℒim(k)Gmj(k)(y−y′)=δijδ(y−y′)

(see Appendix A).

Our problem is then a standard matrix eigenvalue problem

(2) Mq′=cq′M=U+(Qy+By) G(k)diaɡ(U)=−G(0)diaɡ(Qy)

If, for example, we have 50 y points in two layers, then the matrices are 200 × 200, with U, Q_y, B_y diagonal. G⁽^k⁾ just differs from G⁽⁰⁾ in the form of the operator; for the latter ∇2→∂2∂y2.

However, there is another way to pose the problem: suppose we represent q′ as the anomalies produced by the displacements of the Q contours. We use the linear form (Nakamura and Zhu, 2010) discusses the appropriate nonlinear form)

(3) q(x,y,t)=ℒψ+ℬ(y)≃ℬ(y−η)+Q(y−η)

so that the small amplitude form is

ℒψ≃Q(y)−(Qy+ℬy)η

and the inversion formulas are

(4) ℒU=−Qy , ℒψ′=−(Qy+ℬy)η

The dynamical equation recognizes that η is a material line

ηt+Uηx=ψx

and we recover Equation 1 but as

[U+G(k)(Qy+By)]η=cη

which is the same result we would get by substituting q′=−(Qy+By)η in (1). The approaches are equivalent but posing it as Equations 2, 4 can help keep the rest state part’s role obvious.

In the stability sections, we will mostly deal with two non-dimensional parameters: the height of the topography Δ_Tand the distance from the center of the jet to the topography y_T. We discuss the dimensional forms in Section 6; for reference, when y_T = 1, the topographic center is 35 km from the jet center and Δ₁ = 1 corresponds to topographic height of about 30% of the layer thickness—1 km when in the bottom layer, but 3.5 km when it extends into the top layer. This can be compared to the Gulf Stream, which meanders within 50 km of the shelf in the South Atlantic Bight (Gula et al., 2015b).

Figure 2 shows the different configurations used in the stability calculation; examples 1 and 2 have piece-wise constant potential vorticities, example 3 has a smooth jet, and example 4, where the topography extends into the top layer, provides new insights.

3 Topography in the bottom layer

Using piece-wise constant PV patches can simplify the problem to bring out the nature of the topographic influence on the stability; the gradients of PV, ∂∂yQi, are now a set of delta functions Δ_jδ(y − y_j). Subscripts on Q_ior ψ_iindicate the layer, while those on Δ_jgive the PV jump at the interface number j, not which layer it is in. Δ_Tis the PV jump arising from the topography. Then, Equation 1 implies q′ will also be a set of delta functions. The equations are still Equation (1) but may deal with only three or four points, resulting in a low-order matrix eigenvalue problem (Equation 2). The system consists of three regions: an offshore region where y< 0, a nearshore region between the jet and the topographic step 0< y< y_T, and a shelf region where y > y_T. In this configuration, the topography exists only in the bottom layer. The stability of the system will depend on the boundary conditions imposed on the basic-state jet at the topography, which we can fix to some value at y = y_T or allow to decay freely onto the “shelf”.

3.1 Example 1: jet extends into the shallow region

The jet is taken to be purely baroclinic with U1=e−|y|, U2=−de−|y|, which implies Q₁_y= 2δ(y), Q₂_y= −2dδ(y) (see Figure 2, top). Now, PV anomalies are generated as the interface in the fluid separating the higher PV to the north of the step to the lower value to the south wiggles onto and off of the topography. Thus, we have

diaɡ(Qy+By)=[2−2dΔT], diaɡ(U)=[U1(0)U2(0)U2(yT)]=[1−d−de−yT]

and

G=[G11(0)G12(0)G12(yT)G21(0)G22(0)G22(yT)G21(yT)Gyt22(yT)G22(0)]

We have two parameters to explore: the size of the topographic step Δ_T and the separation y_T. Typical growth rate σ=ℑ(kc) vs. k curves show two different instabilities. We can get a feel for these by looking at the three frontal waves treated as independent c_j = M_jj. For large k, one mode limits to c = U₁(0) = 1 and decreases for smaller k, the second to c = U₂(0) = −d (and increases), and the third to −de_−|*y*| plus a topographic Rossby wave term which becomes more and more negative as k decreases. Thus, the last two modes’ ℜ(c) values intersect and interact for short waves—the modes seen in Figure 3 near k ≃ 1.7, 2.2, 8. For smaller k, these can no longer synchronize, and the flow is not unstable. Then, when k< 1, the top and bottom layer jet modes can phase-lock and reinforce each other: the baroclinic instability appears. Detailed growth rate calculations (Figure 3) show the two modes as well as inhibition of the baroclinic instability when the topography’s positive PV jump is close to the deep jet’s negative one. This is most noticeable when Δ_T > −Δ₂.

In summary:

• The top-layer jet front has a strong eastward flow (U₁(0) = 1) and, since Δ₁ > 0, relative westward wave propagation.
• The bottom-layer jet front has westward flow U₂(0) = −d and relative eastward propagation Δ_2< 0. The baroclinic instability is associated with the interaction of the two fronts at y = 0.

The topographic wave sees weaker westward flow U2(yT)=−de−|yT| and a positive PV jump Δ_T so that the waves are westward. For k → ∞, c₂ approaches −d from above while c₃ approaches −de−|yT| from below. The short-wave instability is a horizontal shear mode in the bottom layer, occurring near where these individual c_j’s cross.

Zeroing out the first row and column of G gives growth rates like those in the short-wave filament in Figure 3, verifying that it is indeed interactions between the deep westward flow and the topographic wave. Unlike the usual shear instability where we would expect ky_T to be order one, the scale here remains small because the U₂(y_t) − U₂(0) changes very little as the separation increases. The Doppler-shifted wave in the deep jet interface with c ≃ −d − 2dG₂₂(0) matches Δ_TG₂₂(0) (k = 1.456, 2.217 for Δ_T = 0.2, 0.5).

3.2 Example 2: no jet in the shallow region

We next examine a setup in which top- and bottom-layer PV jumps at the topography allow the basic state velocities to vanish at y = y_T (Figure 2, second row). This structure is like example 4 when the topography penetrates into the top layer. The PV jumps are

∂∂yQ1=Δ1δ(y)+Δ3δ(y−yT)∂∂yQ2=Δ2δ(y)+(Δ4+ΔT)δ(y−yT)

For the zonal flow to be purely baroclinic in y < 0 and U₁, U₂ both vanish at y = y_T, it must have U₂ = −dU₁ with no barotropic component. Therefore, the top-layer velocity satisfies

∂2∂y2U1−U1=−Δ1δ(y)−Δ3δ(y−yT)

giving

(5) U1={eyy≤0sinh (yT−y)sinh (yT)0≤y≤yT0y≥yT

and

(6) Δ1=1+cosh(yT)sinh(yT), Δ3=−1sinh (yT)

The bottom-layer velocity is simply U₂ = −dU₁ with Δ₂ = −dΔ₁, Δ₄ = −dΔ₃.

The linear stability problem from Equation (2) has

diaɡ(Qy+By)=[Δ1−dΔ2Δ3−dΔ3+ΔT], diaɡ(U)=[1−d00]

and

G=[G11(0)G12(0)G11(yT)G12(yT)G21(0)G22(0)G21(yT)G22(yT)G11(yt)G12(yT)G11(0)G12(0)G21(yt)G22(yT)G21(0)G22(0)]

Because the flow has a very strong baroclinic shear layer for small y_T, the growth rate reaches 3.2 with y_T = 1/16, whereas the baroclinic instability dominates at large separations peaks (as before) at 0.074 as y_T ≥ order 3 (Figure 4). Unlike, the previous case, there is a long-wave cutoff on the baroclinic instability. This is most noticeable for smaller y_T where Δ₃ ≃ −Δ₁ so that the long waves nearly cancel the PV jumps in each layer. For larger y_T, this case resembles the first one, with the baroclinic instability and the deep jet/topographic wave instability at small scales. .

3.3 Example 3: smooth jet

For smooth jets (Figure 2, third row), we can still use the integral formulation:

Ui(y)=−∫dy′Gij(0)(y−y′)∂∂y′Qj(y′)

Ui(y)q′i(y)−∂∂y[Qi(y)+ℬi(y)]∫dy′Gij(k)(y−y′)q′j(y′)=cq′i(y)

Discretizing (using a simple sum over the values at the grid points) turns these into matrix equations; given U_i(y), we can invert the matrix to find ∂∂yQi(y), resulting in a standard, albeit large matrix eigenvalue problem for c.

However, we can get some ideas just from the Raleigh sufficient conditions for stability. Our model velocity and topography are

U1=sech2(y/w), U2=0, ℬ(y)=Btanh([y−yT]/wT)

where w is the width of the jet and w_T is the width of the topography. These give expressions for the Q_y functions,

∂∂yQ1=−∂2∂y2U1+F1U1=1w2[6 sech4(y/w)−4 sech2(y/w)]+F1sech2(y/w)∂∂y[Q2+ℬ]=−F2U1+ℬy=−F2 sech2(y/w)+BwTsech2([y−yT]/wT)

Having the topography north or south of the jet will not matter in this case—the symmetry in U(y) and ∂∂yQ show that replacing y by −y will give the same results as replacing y_T by −y_T (Supplement 1). The Green’s function matrices in the earlier examples have the same character: they depend on |y_T|, not its sign. If the topographic gradient or the jet were not symmetric, of course, this would no longer be true.

The sufficient conditions for stability are simply that the gradients of both must be nonnegative (since the top-layer jet has positive values at y = 0). Thus, we need

a. ∂∂yQ1≥0:w≥2/F1=2.19 for F₁ = 1/(1 + d)

b. baroclinic F2 sech2(y/w)≤BwTsech2([y−yT]/wT) for all y. This implies (Figure 5)

(7) B˜>sech2(y/w)sech2([y−yT]/wT), B˜=BF2wT

c. The generalized Rayleigh condition for baroclinic stability must hold for all y so that B˜ is constrained by the three parameters. However, if we look at large y, this limits to

B˜>exp(−2yT/wT)exp(2y[1/wT−1/w])

so we require w_T ≥ w to ensure stability: narrow topography cannot stop the sign reversal in the top- and bottom-layer PV gradients in the jet wings.

Note that we will not have the topographic Rossby-wave instability in the bottom layer; that requires a westward deep jet flow to synchronize the eastward propagating (but advected westward) waves on the negative PV gradient and depth with the westward waves on the slope. Figure 6 shows that the topographic slope, which, in the deep layer is opposite in sign to the F₂U₁ baroclinic term, reduces the growth rates when the jet is over the slope. For narrow jets (width less than 2.2R_d), we see growth associated with the horizontal shear (condition (a) not satisfied); for wider jets with high-enough topographic step, condition (b) is also satisfied, and the linear growth rate drops to zero.

We can determine the topographic amplitude required to stabilize the jet for various distances y_T and widths (Figure 6) using arguments based on the stability conditions:

• At y_T → 0, we need B˜≥sech2(y/w)/sech2(y/ w T), which implies B˜>1 and w_T ≥ w.
• For w = w_T, we can take y∗=y/wT→−∞; then, the r.h.s. of (7) limits to

B˜≥exp(2yT/wT−2y∗)exp(−2y∗)=exp(2yT/wT)

• giving a straight line in log(B˜) vs. y_T and is pretty close when y_T > order 2.

• as w_T/w becomes greater than one, B˜ exp (−2yT∗ also limits to a function for large yT∗ which depends on w_T/w; i.e., B˜≃A( w T/w)exp(2 y T ∗). We can estimate A by considering the large yT∗ limit; then, the minimum of the right side of Equation (7) occurs at approximately

exp(ry∗)=r−1r+1, r=wT/w

• and we can use this to get a fairly good estimate when y_T > 1.5w_T or so.

In any case, these basically show that reasonable-sized topography will have very little influence on the baroclinic mode when the jet is more than the width of the topography away in the deep water.

4 Topography extending into the top layer

Next, we look at the instability when the topography blocks the bottom layer completely and extends into the top layer as in Thompson (1993) (see Figure 2, fourth row).

The offshore region, y < 0, and the nearshore region, 0 < y < y_T, remain unchanged from example 2, but the shelf y > y_Thas no bottom layer; there is now only a single layer (the flow is barotropic) with a bottom topography, Δ_T. We can no longer extend the zonal flow over the shelf and have it decay to the north since (if it is constant PV)

−∂2∂y2U=0.

We choose to have U = 0 for y ≥ y_T; this provides matching conditions related to the PV jump in the fluid at y = y_T. We still have a free parameter since the bottom-layer jet velocity along the wall is not specified. We take U₂ = 0 at y = y_T; in this case, the U₁, U₂ velocities match those in example 2: a purely baroclinic flow with no velocity at y_T and a Δ₃ jump to make ∂∂yU1=0 for y > y_T. Supplement 2 deals with the choice ∂∂yU2(yT)=0; the growth rate results are similar. Equations 8, 9 imply that the zonal flows (when we take U₂(y_T) = 0) and the jumps in PV associated with the discontinuities in (∂∂yU Δ₁, Δ₂, Δ₃) are the same as those in Equations 5 and 6. We can figure out the forms by writing the top- and bottom-layer PV equations as in Equation (3):

(8) ∇2ψ1+F1(ψ2−ψ1)ℋ(yt−y)+ΔTℋ(y−yT)=Δ1ℋ(y−η1(x,t))+(Δ3+ΔT)ℋ(y−yT−η3(x,t))∇2ψ2+F2(ψ1−ψ2)=Δ2ℋ(y−η2(x,t))

The first Heaviside function is just to mark the change in character of the top-layer equation on the shelf. The basic state equations and perturbation equations become

∂2∂y2Ψ1+F1(Ψ2−Ψ1)ℋ(yt−y)=Δ1ℋ(y)+Δ3ℋ(y−yT)∂2∂y2Ψ2+F2(Ψ1−Ψ2)=Δ2ℋ(y)∇2ψ1′+F1(ψ2′−ψ1′)ℋ(yt−y)=−Δ1δ(y)η1−(Δ3+ΔT)δ(y−yt)η3∇2ψ2′+F2(ψ1′−ψ2′)=−Δ2δ(y)η2

The zonal flows (when we take U₂(y_T) = 0) and the jumps in PV associated with the discontinuities in ∂∂yU(Δ1,Δ2,Δ3) are the same as those in (Equations 5, 6).

We can write the perturbation solution in the top layer in terms of seven constants. In the perturbation equations, we also have the wavenumber k and the baroclinic version K=1+k2.

(9) ψ1′={a1eky+a2eKyy≤0a3eky+a4eKy+a5e−ky+a6e−Ky0<y<yTa7e−kyy≥yTψ2′={a1eky−da2eKyy≤0a3eky−da4eKy+a5e−ky−da6e−Ky0<y<yTnoney≥yT

We also have seven conditions to apply; using [ϕ(y0)]−+=limϵ→0[ϕ(y0+ϵ)−ϕ(y0−ϵ)] to represent the step, they are

[ψ1′(0)]−+=0, [ψ2′(0)]−+=0, [ψ1′(yT)]−+=0, ψ2′(yT)=0

and

[∂∂yψ1′(0)]−+=−Δ1η1, [∂∂yψ2′(0)]−+=−Δ2η2, [∂∂yψ1′(yT)]−+=−(Δ3+ΔT)η3

These determine the constants a₁…a₇ so that we can express ψ1′(0), ψ2′(0), ψ1′(yT) in terms of η₁, η₂, η₃. This defines the equivalent of −G(k)Qy. It can be done with tedious computations (or a less tedious version by using Maxima) or, more transparently by splitting the problem into a 3 × 7 matrix times a 7 × 7 matrix times a 7 × 3 matrix (Supplement 2).

For small jet–topography separations, the horizontal shear between the top-layer jet and the PV jump in the fluid at y = y_Tdominates and also inhibits the baroclinic mode. However, this mode disappears when y_T > sinh−¹(1/Δ_T) ≡ y_Tc, resulting in an effective PV jump seen by the perturbations Δ₃ + Δ_T becoming positive—the same sign as Δ₁. The baroclinic growth rate then returns to the pure baroclinic jet value as y_T >> 1 (Figure 7).

We can gain a clearer understanding of the instabilities in each region in Figure 7 by systematically zeroing out the contribution due to each PV front in the interaction matrix (Figure 8, top row) and by examining the relative η amplitudes in the unstable mode’s eigenvector (Figure 8, bottom row). Region 2, with k ∼ 1 and y_{T <} y_Tc (=1.44 for Δ_T = 0.5), is a shear instability between the top layer jet front and the one over the topography; the modal amplitudes of η₁ and η₃ are much larger than η₂. Region 3, with y_T > y_Tc and k ∼ 0.6, is the baroclinic instability¹ of the jet, somewhat inhibited by the topography; the amplitude of η₃ becomes negligible as y_T increases. Region 4 (k > 1 and y_T > y_Tc) has a baroclinic instability between the front over the topography and the bottom layer jet front. The modal amplitudes in the k = 1.005 case at y_T ∼ 1.5 do not make this as obvious—the instability involves the mutual interaction between η₂ and η₃, but η₁ is pushed along as well. The growth rates and therefore the interaction between the shelf and the bottom layer jet are weak; it is hard for the waves on the two fronts to synchronize. Region 1 is the area where no pair of PV fronts can, by themselves, amplify: it is truly an interaction between all three fronts. The three amplitudes are comparable for k = 0.6 and y_{T <} y_Tc. The interaction with η₃ gives rise to the long-wave cutoff of the baroclinic instability shown in Figure 7.

In terms of the separation y_T and Δ_T, the growth rate for Δ_T = 0.5 changes from the shear mode mostly when y_T< y_Tc (dotted line in Figure 7) to the baroclinic instability for y_T > y_Tc. The height of the topography required to stabilize the shear mode increases dramatically and unrealistically, as y_T gets small, since Δ₃ = −1/sinh(y_T).

5 Summary of instability calculations

For topography in the bottom layer, when the jet extends smoothly across the topography, we find that:

•With a westward counter-current in the deep layer, there can be a shear instability with waves on the jet coupling with topographic Rossby waves. The top-bottom layer baroclinic instability is weakened.

•For smooth, top-layer-only jets, there can be a weak shear instability if it is narrow enough compared to R_d so that the PV gradient changes sign. When the topography is under the jet and is strong enough and wide enough (w_T > w), it can suppress the baroclinic instability.

•When the jet flow vanishes at y_T, there can be a shear layer instability between the PV jumps at y = 0 and y_T.

When the topography extends into the top layer (providing a wall in the deep layer and a step in the top layer—Eqn. 8), the jet velocity will be independent of y on the barotropic shelf if the PV there is constant. Choosing that value to be zero and U₂(y_T) = 0 also implies that there must be a negative PV jump, Δ₃, in the top layer; this competes with the positive topographic Δ_T.

•When the topography is close to the jet, the shear in the basic state flow is very large, Δ₃ overwhelms Δ_T, and the PV jump seen by the perturbations changes sign. The necessary condition for horizontal shear instability is met and can be at least as strong as the baroclinic instability (and stronger when the separation is comparable to R_d).

•For larger separations, Δ₃ → 0, the inhibition from the shear instability decreases and the system approaches a purely baroclinic instability. The case with topography in the bottom layer, but U = 0 north of the topography, is similar. In some cases, there are weak short-wave instabilities in narrow k ranges, but higher growth rates exist in the long-wave band, so these are unlikely to play much of a role.

6 Finite amplitude development

To investigate the behavior after the perturbations become finite amplitude, we use a two-layer QG channel model, with the topography in the bottom layer (the situation in Section 4 presents a more difficult numerical problem). The model will be discussed in dimensional form² with the basic-state jet U₁ = U³ sech²(y/w), U₂ = 0, and the topography fB tanh ([y − y_T]/w_T) with B = h_T/H₂. The default parameters are f = 8 day⁻¹ (latitude 39.5° N), R_d = 35 km, w = 50 km, U₀ = 80 km/d, w_T = 80 km, and d = H₁/H₂ = 0.2, and the channel width is 1,500 km. The model is run for 365 days.

Numerically, the grid is 256 × 128, the time-stepping is second-order Adams–Bashforth, and advection uses centered differences with a Van Leer (1974) correction to prevent oscillations. q and ψ are split into the zonal mean and the fluctuations; the latter are inverted from q′ to ψ′ with a sine transform in y and a regular FFT in x. The zonal average q (not including the topography) is inverted with a cosine transform so that the mean zonal velocity along the walls is zero. The latter ensures conservation of mass:

∫q1=−F2∫(ψ1−ψ2)+∮u·dl

Since the term on the left is conserved and the last on the right is 0, this ensures that the integral of layer thickness anomalies h=(f/g′)(ψ1−ψ2) is conserved.

6.1 Jet parallel to topography

For reference, we show two snapshots without topography; the initial perturbation displaces the jet centerline by

0.025 km[sin(10πx/L)+sin(7πx/L)+sin(3πx/L)+sin(2πx/L)]

so that multiple waves are excited. The perturbations become visible by approximately day 50, reaching a peak-to-trough amplitude of around 20 km for the largest meander. This grows to 100 km by day 60, and meanders begin to break and cut off eddies by day 90 (Figure 9). By 1 year, the mean flow has spread so that the jet is not visible and the potential energy fueling the instability has been depleted enough that the domain-averaged eddy energy −12H1〈ψ1q1〉+H2〈ψ2q2〉H1+H2, although still variable, is neither growing nor decaying (Figure 10).

When the topography with B = 0.1 is directly under the jet, the instability is suppressed and effectively cuts off for B = 0.1 as shown in Figure 10.

When the topography is far from the jet (480 km = 6w_T), the snapshots show that the initial development barely feels the topography, and the growth matches the case without topography. However, later on, the eddies in either layer do not penetrate significantly into the shallow region. The energy is fed back into the zonal mean, generating a vertically sheared flow over the topography. This flow is pretty stable (Figure 11).

For closer topography (160 km = 2w_T), the bottom layer clearly feels the topography at 90 days, and the top-layer eddies become blocked early on. Zonal mean flows form again over the topography, and the eddy energy decreases (Figure 12).

The growth rates of perturbation energy from linear stability calculations (Figure 10, log plot), although having very similar trends with y_T, tend to be larger than the numerical model rates; this presumably arises from dissipation associated with the rather low resolution and the timestepping. For the case y_T s= 0, the model domain had to be reduced by 1/5 in each direction to see the instability from the horizontal change in sign of the PV gradient in the sech² jet. For the finite separations, the growth seems to overshoot what might be the final equilibrium, though that is not well defined in these spindown experiments. This indicates that some transient growth (Farrell (1987)) may be occurring with the damped mode initially canceling the growing one; as it decays, the energy increases faster, and the alteration in the mean flow may catch up slowly.

6.2 Jet crossing a ridge

For this example, relevant, in a rough way, to the influence of the New England Seamount Chain, the current is crossing a ridge:

ℬdim=fB exp(−0.5[x/wT]2) with B=htopH2

In these experiments, we include the β-effect with the value for our latitude, β = 1.5 × 10⁻³ km⁻¹ d⁻¹. Figure 13 shows the full potential vorticities (including topography) when the ridge height is 60% of the bottom layer thickness and w_T= 200 km. The periodicity mostly masks the upstream/downstream asymmetry, and the mean zonal flow which develops in the bottom layer is westward (though not very regular), so we do not expect to see a Rossby wave wake. However, it is clear that the top layer flow is affected by the ridge with a strong deflection of the jet to the south and then back north as it goes over the ridge.

6.3 Jet crossing seamount chain

The topography for this experiment is a line of seven seamounts across the channel

ℬdim=fB12[1+sin(14πy/W)]exp(−0.5[(x−W)/45]2)

with B = 0.6. Figure 14 suggests that the seamount chain acts much like a ridge. It shows some initial asymmetry between the upstream and downstream region as the perturbation develops and then a clear flow southward upstream and back northward downstream at larger times when the energy has stopped growing.

6.4 Summary of numerical experiments

The numerical experiments with the jet flowing east and the bottom shallowing to the north imply the following:

• The nonlinear evolution is affected by the topography even when it is far enough from the jet that the linear growth rate is essentially the same as without the slope, but the eddies impinge on the topography and begin to develop a weaker, but wider, top-layer zonal flow over the topography; in effect, the jet moves to center on topography where it is nearly stabilized, and the eddy energy weakens. The potential energy extracted from the original instability goes partially into the new flow over the topography rather than all ending up in the eddies.
• Topography can stabilize the jet when the sufficient condition for the baroclinic mode is satisfied. The instability from the horizontal shear seems to be damped by numerical dissipation in the lower resolution model, but manifests at higher resolution. It is doubtful that it plays a significant role in Gulf Stream meandering and ring formation. The small wavelength of these disturbances are likely to result instead in filaments and submesoscale vortices. The topography, on the other hand, clearly slows the growth when the jet is near it and limits the eddy energy in the shallower region even though the slope is in the bottom layer (Figure 15).
• The ridge and seamount chain experiments show that, even with bottom layer topography, the top-layer jet can be significantly deflected.
• These runs, especially the ridge/seamount ones, suggest that studying these processes with inflow/outflow conditions is warranted.

7 Discussion

While the models in this paper are highly idealized, they point to the importance of nearby topography in the instabilities and eddy formation of baroclinic jets such as the Gulf Stream and the Kuroshio. When the jet is in deep water, with the slope up to shallower water on its left, the baroclinic instability is weakened; however, if the jet velocity goes to zero rapidly at the shelf edge, it can become vulnerable to a horizontal shear instability. This is especially noticeable when the shelf is very shallow and does not have jet flow. With stratification, however, the decrease in jet velocity may not be as strong, but we can still expect to see filaments and smaller eddies in the onshore side.

Once the topography is far enough away, the linear baroclinic instability growth rate recovers; however, the numerical experiments suggest that the turbulence is stronger to the south of the jet than between it and the topography. Eddy formation from the jet is more common to the south than the north when the topography is close enough (roughly 300 km). Warm-core rings generally have shorter lifetimes than their cold-core cousins (Richardson, 1983), and there is some suggestion (Auer, 1987) that their formation rate may be slightly slower.

Of course, the Gulf Stream and Kuroshio spend time in shallow water, or very close to the slope, before going out to sea (Webster, 1961; Todd, 2021). For the QG bottom layer topography calculations, both stability and numeric will give the same results for the jet in deep water with the shelf on its left or in shallow water with the drop-off to its right (Supplement 1). Proximity to the slope suggests that the meanders will generally be small and will not generate cutoff eddies. The wall on the left will alter the meandering on the shoreward side as well. However, in many situations, the symmetry will not exist: e.g., the topography extending into the top layer, a wall, or non-QG effects, but it suggests that the turbulence from the instability when the Gulf Stream is in shallow water will be pretty much confined to the vicinity of the jet.

It is also the case that the jet distance from the topography and the latter’s height and width vary significantly along its track. The simplification we have made needs to be revisited, taking into account spatial variation [c.f., Spall and Pedlosky (2008)].

Generative AI statement

The author(s) declared that generative AI was not used in the creation of this manuscript.

Any alternative text (alt text) provided alongside figures in this article has been generated by Frontiers with the support of artificial intelligence and reasonable efforts have been made to ensure accuracy, including review by the authors wherever possible. If you identify any issues, please contact us.

Publisher’s note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

Supplementary material

The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/fmars.2026.1929138/full#supplementary-material

Supplementary material Supplementary material at the source

Footnotes

Edited by: Andrew Poje, The City University of New York, United States

Reviewed by: Jean Reinaud, University of St Andrews, United Kingdom

Erick Raul Olvera-Prado, Autonomous University of Baja California, Mexico

Footnotes

The cutoff for the jet’s baroclinic instability is k = 0.84.

Yes, the basic units are not usual but are natural for the phenomena being studied; for reference, 80 km/day = .926 m/s, 8 day⁻¹ = 9.26 × 10⁵ s⁻¹, and β (used later) is 1.76 × 10⁻¹¹ m⁻¹s⁻¹.