1. Introduction
Many currents flow along coastal boundaries. These currents have both negative and positive vorticity, being usually unstable and affected by waves or eddylike disturbances. Large amplitude unstable waves have been observed in infrared images from satellites and ice pack ladder observations. Some waves exhibit a regular wave train with crests breaking on the upstream side (e.g., Ohshima and Wakatsuchi 1990; Wakatsuchi and Ohshima 1990), while others display mushroomlike shapes consisting of two vortices having opposite signs that advect themselves offshore far from the coastal boundary (e.g., Ikeda et al. 1984a,b; Ikeda and Emery 1984; Strub et al. 1991; McCreary et al. 1991).
Ohshima and Wakatsuchi (1990) used a numerical shallow-water model to conclude that regular wavelike disturbances in the Soya Warm Current were generated through barotropic instability. In the California Current System, the shapes of disturbances in the boundary current exhibit seasonal variability. Strub et al. (1991) presented three conceptual models to explain the observed filament behavior. The three models consisted of the squirts model, which demonstrated the offshore transportation of cold coastal deep water by vortex pairs; the mesoscale eddy field model consisting of a number of mesoscale eddies embedded in a southward boundary current; and the meandering jet model, in which the southward boundary current in the upper layer meandered both offshore and onshore. It was noted that differences in the shapes of the disturbances also play an important role in the coastal water mixing and offshore transport of the coastal water, which in turn affect biological activity.
The primary purpose of the present study is to clarify the physical reasons that cause disturbances in boundary currents to exhibit several distinct characteristics. To extract the essence of the nonlinear behavior of an unstable wave, the contour dynamics model was adopted for analysis, which focuses on the dynamics of a finite number of piecewise uniform vorticity or potential vorticity regions (e.g., Zabusky et al. 1979). In the first step of the study, pure two-dimensional barotropic flow without any external forcing or dissipation was examined. The baroclinic effect, which may play an important role in this problem, will be considered in a forthcoming paper. Similar studies on the behavior of the vorticity front near the boundary were conducted by Pullin (1981), Stern and Pratt (1985), Stern (1986), and Kubokawa (1991). Pullin (1981) and Stern and Pratt (1985) treated pure two-dimensional flows, while Stern (1986) and Kubokawa (1991) used a 1½-layer quasigeostrophic model, motivated by the squirts in the California Current System and the behavior of the outflow from Tsugaru Strait. Since they considered only one (potential) vorticity front, however, the currents were linearly stable. Ohshima and Wakatsuchi (1990) suggested that the existence of two vorticity regions having opposite signs was essential to simulate the observed current behavior. Therefore, in the present paper, consideration is only given to currents consisting of three piecewise uniform vorticity regions, separated by two vorticity fronts, being the minimal condition to include linear instability. In general, actual oceanic boundary currents are driven and maintained by local wind forcing (e.g., McCreary et al. 1991), or buoyancy forcing (e.g., Ohshima and Wakatsuchi 1990; Seung and Yoon 1995). The current profile can be established by lateral and bottom friction. In the present study, such external forcing and dissipation are not considered. These effects are important when the timescale of the evolution of the disturbances is comparable to that of the external forcing. When the timescale is short, however, the evolution can be primarily governed by free motion. Therefore, the method of the present study should be one of the best ways as a first step in this kind of study.
An outline of both the contour dynamics model and linear stability are presented in section 2. An unstable wave solution is used as the initial value and the nonlinear behavior of the waves is classified into three distinct regimes in section 3. In section 4, the dependence of the evolution on the vorticity distribution and wavelength of the disturbance is examined. It was founded that changes in the evolutional manner occur as the wavelength of the disturbance was varied, even if other parameters were fixed. In section 5, a simple theoretical interpretation of the classification is then presented using a point vortex model. Summarization of the results of the present study is given in section 6.
2. The model and linear stability
a. The model
b. Linear stability
3. Typical pattern of nonlinear evolution
In this section, an examination is made of the nonlinear evolution of linearly unstable barotropic boundary currents in a periodic domain. Numerical experiments are performed following the initial condition, in which the linearly unstable wave described in the preceding section is superimposed upon the vorticity fronts of the basic current, L̄1 and L̄2. Since the flow is purely two-dimensional, a typical scale does not exist. Only the ratios |q1/q2| (=q∗) and L̄1/(L̄2 − L̄1) (=L̄∗) characterize the basic current. The periodic domain is set equal to nλ (n being an integer). The initial state is determined by L̄∗, q∗, λ, and the initial amplitude of the disturbance. Here, values of q2 and L̄2 are set to unity, and the timescale is normalized by |L̄2/q2(L̄2 − L̄1)|. The velocity for y > L̄2 is assumed to be zero in the basic state without loss of generality; that is, U2 = 0 in the linear stability analysis in the preceding section. In the present case, the current flows in the positive x direction.
A total of 150 numerical experiments were carried out with different parameters, resulting in the discovery of different kinds of evolution. The evolutions were classified into three regimes referred to as the breaking wave, the vortex pair, and the boundary trapped vortex. An outline of the typical behavior of the unstable waves in these three regimes and the procedure of regime classification are now discussed. In many cases, the classification of the evolution is determined only by the temporal behavior of the phase difference between the vorticity centers [ξ = k(x
In the following cases, the basic currents are chosen such that the velocity vanishes at both y = 0 and y = L̄2, resulting in a zero total basic current circulation. The numerical parameters of the contour surgery and the node redistribution are described in the figure captions.
a. Breaking wave regime
Figure 2 shows the temporal nonlinear evolution of the fastest growing unstable wave (wavenumber 3.92), superimposed on the basic current with values of L̄∗ = 3.0 and q∗ = 0.33 (experiment 1). The shear on the offshore side of the current axis is stronger than that on the onshore side. The dispersion relations for the real and imaginary parts of the phase speed are shown in Fig. 3.
Initially, the unstable wave grows according to linear theory as described in the preceding section, maintaining a phase difference when the vorticity center on the offshore side x
After examining the mixing of fluid due to nonlinear evolution, it was found that vigorous mixing occurred around the pool of positive vorticity via filamentation of the negative vorticity fluid. The extent of the mixing was limited to within the width of the basic current. Offshore transport of fluid was therefore not effective. In contrast, the fluid on the offshore side (strong shear) exhibited very little mixing since most of the fluid with strong shear formed a large eddy.
For the case shown in Fig. 2, the regime can be easily identified according to the temporal change of the phase relationship between the two vorticity centers. As will be described in the next section, the evolution changes from the breaking wave to the vortex pair regime as the wave length or the periodic length increases. The procedure used to distinguish between the two regimes at the border will be mentioned later.
The identification and characteristics of the breaking wave regime are summarized as follows:
The amplitude of the disturbance reaches a maximum when the phase difference between x
c1 and xc2 The evolution is divided into the growing and damping stages.
Both vorticity regions remain near the boundary, with no effective offshore transportation.
The onshore fluid evolves into a filament structure.
The evolution of the breaking wave regime occurs only when the magnitude of the shear on the offshore side is greater than that on the coastal side.
b. Vortex pair regime
Figure 5 shows the nonlinear evolution of the unstable current for values of L̄∗ = 1.0 and q∗ = 1.0, which results in a symmetric basic current profile around the current axis. The wavenumber of the initial unstable wave is k = 2.59 (experiment 2), which yields the fastest growing mode. The dispersion relations for the real and imaginary parts of the phase speed are shown in Fig. 6. The unstable wave initially grows in a manner similar to that in the breaking wave regime. When the frontal displacement becomes sufficiently large, however, the vortex pair is cut off from the boundary current at t ≈ 16.0 and advected offshore, far from the boundary. The direction of movement of the vortex pair gradually changes from an offshore-downstream direction to a pure offshore direction, and the phase lag of x
In the above situation, the vorticity rows are established in the final stage. There is, however, another situation in which the vortex pair continues to propagate but does not form rows of vortices (an example is shown in Fig. 13). In the latter case, the phase lag of x
In this manner, it is not always possible to distinguish the vortex pair regime from the breaking wave regime from only the temporal behavior of the phase relationship between the two vorticity centers. In this event, attention is focused on the behavior of the onshore fluid. The evolution is classified as the vortex pair if the onshore fluid is cut off and evolves into a pool structure. On the other hand, the evolution is classified as the breaking wave if the onshore fluid evolves into a filament structure. In several cases, when q∗ ≤ 1 and L∗ ≤ 1, most of the onshore fluid evolves into a filament structure, but the remaining fluid evolves into a pool and forms a dipole structure (Fig. 8). In such cases, it is difficult to classify the evolution as a specific single regime. An intermediate regime between the breaking wave and vortex pair (e.g., Fig. 17c) is then introduced. The identification and characteristics of the vortex pair regime are summarized as follows:
Large amplitude disturbances are cut off from the boundary currents and evolve into dipole eddies, which move offshore due to self-induced flow and carry the coastal water found in the boundary currents.
The phase relationship of the growing stage is maintained for a long period of time.
The onshore fluid evolves into a pool structure.
The evolution of the vortex pair regime can occur when the magnitude of the shear on the offshore side is greater than or equal to that on the coastal side.
c. Boundary trapped vortex regime
Figure 9 shows the evolution of the fastest growing unstable waves (λ = 2.48) superimposed on the basic current for values of L̄∗ = 0.33 and q∗ = 3.0 (experiment 3) in which a strong shear exists on the coastal side of the current axis. The dispersion relations for the real and imaginary parts of the phase speed are given in Fig. 10.
In this case, the evolution is also divided into growing and damping stages similar to that in the breaking wave regime. The temporal behavior of the phase relationship between x
Another difference between the boundary trapped vortex regime and the breaking wave regime is that the region of negative vorticity near the boundary maintains contact with the coastal boundary. Therefore, the mixing via filamentation as seen in the breaking wave regime cannot be found in the onshore region having strong shear, but is found in the offshore region with weak shear.
The identification and characteristics of the boundary trapped vortex regime are summarized as follows:
The evolution is divided into growing and damping stages.
The amplitude of the disturbance reaches a maximum when the phase difference between x
c1 c2 The onshore fluid does not become detached from the side boundary.
Both vorticity regions remain near the boundary.
The boundary trapped vortex type evolution occurs only when the magnitude of the shear on the coastal side is greater than that on the offshore side.
4. Dependence on the basic current profile and the wavelength of the initial disturbance
Discussion is now centered on the dependence of the evolution on the vorticity ratio q∗, the frontal ratio of the basic currents L̄∗, and the wavenumber of the unstable wave. In the preceding section, the typical patterns of the evolution were described for boundary currents having zero total circulation. Figure 12 shows the evolutional map in the q∗–k domain for zero total circulation currents in which q∗ = 1/L̄∗. The shaded region and broken line denote the linearly unstable parameter region and the parameters yielding the fastest growing wave, respectively. When q∗ is fixed, the evolution tends to be classified as a breaking wave as L̄∗ increases, while tending to be classified as a boundary trapped vortex as L̄∗ decreases. On the other hand, when L̄∗ is fixed the evolution tends to be classified as a breaking wave as q∗ decreases, and as a boundary trapped vortex as q∗ increases. The vortex pair type evolution is found, not only when q∗ ≈ 1.0 (L̄∗ ≈ 1.0), but also when the wavelength of the unstable wave exceeds that of the fastest growing unstable wave, even if q∗ is less than unity. This implies that the qualitative change in the evolution from the breaking wave to the vortex pair occurs at a certain value of λ as λ increases, even if the basic current has the same velocity profile.
An example demonstrating this dependency on wavelength is shown in Fig. 13 (experiment 4). The basic current is the same as in experiment 1 (Fig. 3), that is, L̄∗ = 3.0 and q∗ = 0.33, but the wavelength is doubled, that is, k = 1.96, which is the first subharmonic of the fastest growing unstable wave. As in experiment 1, a vorticity pool is generated on the offshore side of the positive vorticity region, and a large amplitude wave breaks on the upstream side at t = 9.0. The phase relationship between the two vorticity regions, however, does not change after the breaking occurs (Fig. 14). The vorticity pool surrounds itself with fluid of negative vorticity. After this process occurs repeatedly, a dipole structure is organized in the final stage at t = 24.0. The only difference between experiments 1 and 4 is the wavelength. In experiment 1, the wavelength is sufficiently short to cause interactions between the positive vorticity pool generated in the offshore shear region and the wave crest on L1 in the upstream domain, making it possible to have the phase relationship of the damping wave. In experiment 4, however, the wavelength is too long to result in such interactions. Therefore, the phase relationship in the growing wave is maintained while the fluid constituting the boundary current is carried away from the boundary by the vortex pair.
Although only the subharmonic wave has so far been considered as the initial disturbance, a similar evolution can occur even if the initial disturbance consists of this wave and the fastest growing wave (experiment 5, Fig. 15). Initially, the fastest growing wave governs the evolution, and two pools of positive vorticity are generated in one periodic domain. The asymmetry between the crests of the fastest growing wave, which is due to the subharmonic wave, causes a difference in the propagation speeds of the wave crests. The current induced by the large amplitude crest attracts the crest of the smaller amplitude wave, and the smaller wave crest finally merges into the larger. As a result, a dipole structure is created and the evolution falls in the vortex pair regime. This result suggests that if a large amplitude longwave disturbance exists, caused by coastal topography or other elements, the evolution falls in qualitatively different features than expected from the fastest growing unstable wave. A similar result was obtained by the numerical study of the California Current system (Ikeda et al. 1984a,b; Ikeda and Emery 1984). In these studies, it was concluded that small-scale disturbances induced by the coastal topography cascaded upscale due to nonlinear interactions between the primary wave and subharmonics. This phenomenon, however, does not always occur. For example, the initial amplitude of the unstable wave is sufficiently small (experiment 6, Fig. 16), the evolution occurs as when only the most unstable disturbance is observed.
To this point, discussion has been focused on the dependence of the evolution of unstable currents on q∗, L̄∗, and k for zero total circulation. Next, the case of nonzero total circulation is examined. Figures 17a–c show the distribution of the regimes in q∗–k parameter space for the values of L̄∗ = 3.0, 1.0, and 0.33 respectively. From these figures, it is found that the dependency has the same tendency regardless of the value of total circulation. An exception to this is found in Fig. 17a at q = 0.5 and k = 2.0. In this case, short waves generated through nonlinear interactions govern the evolution. As a result, the evolution falls into the breaking wave regime.
Another interesting phenomenon is found for values of q∗ in the range when the 1/L̄∗ < q∗ < 1, that is, when the current near the boundary flows in the opposite direction to that on the current axis and the shear on the offshore side is equal to or greater than that on the coastal side. Since the shear on the coastal side must be stronger than that in the offshore region for the disturbance to demonstrate the boundary trapped vortex behavior mentioned in section 4, only the breaking wave and the vortex pair regimes occur. Under these conditions, the vorticity region constituting the countercurrent is barely affected by the nonlinear disturbance. The region of positive vorticity on the offshore side can only move the negative vorticity fluid, which flows in the same direction as the current axis; the fluid of negative vorticity near the coast remains almost undisturbed. An example of breaking wave behavior is shown in Fig. 18, for the values of L̄∗ = 7.0, q∗ = 0.33, and k = 7.84 (experiment 7). The basic current has the same vorticity distribution and wavenumber as in experiment 1 (Fig. 3) if the vorticity region consisting of the countercurrent is eliminated and the wavenumber renormalized by the resulting current width (=0.5). The evolution is very similar to that in Fig. 3 over the region y > 0.5. Another example of vortex pair behavior is shown in Fig. 19 in which L̄∗ = 3.0, q∗ = 1.0, and k = 5.18 (experiment 8). The basic current has the same vorticity distribution and wavenumber as in experiment 2 (Fig. 4) if the vorticity region consisting of the countercurrent is eliminated and the wavenumber renormalized by the resulting current width (=0.5). The evolution is very similar to that in Fig. 6 over the region y > 0.5.
The existence of a side boundary is important for the nonlinear evolution classified as the boundary trapped vortex regime, but is of less importance for the two other regimes. Even without the side boundary, the two other regimes are possible. This is the reason the phenomenon described above is not seen when the shear in the onshore region is greater than that in the offshore region.
5. Theoretical interpretation of the regime classification using a point vortices model
The temporal change of the phase relationship between the two vorticity centers in the constant vorticity regions was important during nonlinear evolution, being a basis of the regime classification. In this section, in order to extract the dynamics of the nonlinear evolution, use is made of a model in which finite area vorticity regions are approximated by point vortices. This model is too ideal to predict the exact evolution of an unstable wave, since the point vortices have no area of vorticity. On the other hand, the model can be used to clarify why the three regimes appear and for a better understanding of the qualitative dependence of the evolution on the parameters, such as wavelength and vorticity ratio.
The model used here considers two point vortices having circulation Γ1 and Γ2 that are located at (x1 + nλ, y1) and (x2 + nλ, y2), as illustrated in Fig. 20. The global vorticity centroid in y is an invariant quantity in this system. The invariant quantities deduce the freedom of motion in y. The location of the two point vortices in y can then be specified by only one variable. In x, on the other hand, since a periodic domain is considered, only the relative location in x is important. As a result, the number of variables can be reduced from four to only two phase variables by specifying the locations of the two point vortices. In addition, the total Hamiltonian of the system is also an invariant quantity. Therefore, the motion of the point vortices must lie upon a Hamiltonian curve in phase space. The motion of point vortices can be clearly classified by the geometrical pattern of the Hamiltonian curve, similar to the analysis in Aref (1979). The outline of the model used here is described in appendix B.
a. Typical geometrical pattern of the Hamiltonian curve
Figure 21c shows the motion of the point vortices for the values of Γ1/λ = −0.75, Γ2/λ = 1.0, λ = 2.094, and I = π(Γ1y1 + Γ2y2)/λ = πyG(Γ1 + Γ2)/λ = 1.865 in phase space. Here, ξ denotes the phase difference between x
1) Breaking wave trajectory
In Fig. 21c, closed trajectories can be found around the elliptic point eπ. The motion on these trajectories can be characterized as follows:
The maximum offshore distance of the trajectory occurs at ξ = ±π.
The maximum offshore distance of the trajectory is always less than yG when y1 = y2 = yG. This implies that the offshore movement is slight and the vortices remain near the boundary.
Therefore, a point vortex initially located on the offshore side always remains located on the offshore side of the other vortex.
These characteristics are similar to those of the breaking wave type evolution. Therefore, these trajectories can be classified as the breaking wave trajectories. The breaking wave trajectory is not only characterized by the closed trajectory around eπ but also by the phase difference at the maximum offshore distance. For example, in Fig. 21a (λ = 1.257) open breaking wave trajectories can be found in the region between S0 and Sπ. The difference between the closed and open trajectories depends on the effect of a side boundary. The effect of the side boundary is important for the closed trajectory but not for the open trajectory. Therefore, motion with an open trajectory can appear without a side boundary. These differences were also found in the evolution of unstable waves (not shown in the figure).
2) Vortex pair trajectory
The maximum offshore distance of the breaking wave trajectory cannot be located beyond yG. Motion of the trajectories inside S0 and between S0 and Sπ extends from near the ξ axis (y1 = 0) over yG (Fig. 21c). This implies that large offshore movement of the point vortices can occur in these regions. This motion can then be classified as vortex pair trajectories. The trajectories in these two regions are characterized as follows:
The maximum offshore distance of the trajectory occurs at ξ = 0.
The maximum offshore distance of the trajectory is greater than yG. On the other hand, the minimum offshore distance is less than yG.
The vortex pair trajectories also display closed and open patterns, with the difference between them depending on the effect of a side boundary, as was found in the breaking wave case. Open trajectories require a side boundary, while closed trajectories can appear with no side boundary.
In Fig. 21c, the separatrix Sπ defines the border between the breaking wave and vortex pair regions. Motion on Sπ can reach the hyperbolic point hπ, which results in a solution consisting of rows of vortices. When the vortices are defined by point vortices, the solution of rows of vortices can be unstable, even if infinitesimal disturbances are added. In the case of finite-area vortices, rows of vortices can maintain their geometrical distribution by changing their shape, as was mentioned in section 3. The point vortex model used here cannot demonstrate this effect. This is a limitation of the point vortex model when comparing it with the results of nonlinear evolution. It also could be a possible reason for the difficulty in distinguishing between the breaking wave and vortex pair regimes near their border, only by the temporal behavior of the phase relationship between the vorticity centers. In addition to this limitation, the point vortex model cannot demonstrate merger and fission of the vortices. In such cases, it is also difficult to compare the motion of the point vortices with nonlinear evolution. Although it is possible to increase the number of point vortices, the motion would then become chaotic, and the phase diagram method would not be applicable.
3) Boundary trapped vortex trajectory
The trajectories in the remaining regions in Fig. 21c, that is, outside of the separatrix Sπ and between η = −|I| and S0, are characterized as follows:
The point vortex initially located on the offshore side is always located on the offshore side of the other vortex.
The maximum offshore distance occurs at ξ = 0.
These features are the same as those of the boundary trapped vortex evolution. These trajectories are then classified as boundary trapped vortex trajectories. The second item given above is similar to the breaking wave trajectory and implies that the maximum offshore distance is short. The maximum offshore distance, however, occurs at ξ = 0 (second item). The second item can also be found in the characteristics of the vortex pair trajectory. The relative location in y between the two point vortices, however, does not change. This is the major difference between the boundary trapped vortex and vortex pair trajectories. In the discussion given below, it will be demonstrated that only boundary trapped vortex trajectories can appear when yG is negative.
There are two regions where boundary trapped vortex trajectories exist. The difference between the trajectories in the first region and those in the second is the relative location of the two point vortices in y. In the first region, the negative point vortex is always located on the offshore side of the positive point vortex. This situation corresponds to that of the numerical experiments associated with the boundary trapped vortex evolution. While in the second region, the positive point vortex is located on the coastal side. In both cases, the vortex near the boundary couples stronger to its mirror image than the other vortex having the opposite sign of circulation. Thus, the vortex near the boundary surpasses the other and the maximum offshore distance of the two vortices occurs at ξ = 0.
b. Dependence on periodic length
In the contour dynamics experiments, it was found that the nonlinear evolution of unstable waves strongly depends on the wavelength. That is, short waves tend to exhibit a backward breaking pattern, while long waves tend to form a vortex pair, even with identical basic current profiles. When the basic current profiles are the same, the Hamiltonian per unit length H/λ, the invariant quantity I, or yG must have the same magnitude if the amplitude of the disturbance is small enough.
An examination was conducted on the dependence of the Hamiltonian curves on the periodic length by varying λ under conditions that values of Γ1/λ, Γ2/λ, and I are fixed. Figures 21a–e display the Hamiltonian contours having the same parameters except for the periodic length. When λ = 1.257 (Fig. 21a), the separatrix S0 lies around eπ and Sπ around e0. As λ increases, the two separatrices S0 and Sπ combine at λ = 1.829 (Fig. 21b) or, in other words, separatrix reconnection occurs. Thus, the region between S0 and Sπ found in Fig. 21a, where the trajectories show their maximum displacement at ξ = π, disappears. When λ exceeds 1.829 (Fig. 21c, λ = 2.094), the region having characteristics of the vortex pair appears between S0 and Sπ. As λ increases to λ = 3.142 (Fig. 21d), this region increases, but the closed region around eπ, having the characteristics of the breaking wave, becomes smaller. As λ further increases to a value of λ = 6.283 (Fig. 21e), the closed region on ξ = ±π disappears due to the absence of the elliptic point eπ and hyperbolic point hπ. The entire space is then occupied by vortex pair trajectories except for the region between the ξ axis and S0 and the region outside of Sπ. This is the physical reason why long waves tend to fall into the vortex pair evolution.
c. Dependence on the sign of yG
The breaking wave region can appear when all the elliptic and hyperbolic points e0, eπ, h0, and hπ are located in the ξ–η domain. The vortex pair region requires the existence of e0 and h0. These points can appear in the ξ–η domain only when the value of yG is positive. These points, however, are not important for the boundary trapped vortex trajectory. Therefore, when yG < 0, only the boundary trapped vortex trajectory can appear. For example, in Fig. 22, boundary trapped vortex trajectories occupy the entire space because the value of yG is negative.
6. Summary and discussion
In the present paper, an examination was conducted of the free nonlinear evolution of unstable barotropic boundary currents consisting of three piecewise uniform vorticity regions by varying the vorticity distribution and the wavelength of the disturbance.
The following is a summary of the present study. The evolutional manner was classified into three qualitatively distinct regimes, referred to as the breaking wave, vortex pair, and boundary trapped vortex regimes. The first breaking wave regime is characterized by the breaking of the wave crest on the upstream side of the vortex. Strong mixing can occur around the vorticity pool within the region determined by the width of the basic current, but the mass of coastal water cannot be carried far away from the boundary. The second vortex pair regime is characterized by the formation of a dipole structure consisting of two vortices having opposite signs. The coastal water mass trapped by the vortex pair is advected far away from the boundary. The third boundary trapped vortex regime is characterized by the meandering of the current in which the vortex generated by the unstable currents remains trapped by the coastal boundary. Therefore, offshore transport of coastal water rarely occurs in this regime.
Although only simple linearly unstable boundary currents were employed, the nonlinear evolution well demonstrated the possible patterns observed in the real ocean. For example, the patterns of the disturbances in the California Current System exhibit a seasonal variability that strongly depends on the basic current profile forced by the local wind stress (McCreary et al. 1991). Strub et al. (1991) examined three observed patterns and referred to them as mesoscale eddy field, squirts, and meandering jet. These three patterns are qualitatively similar to the three evolution regimes classified in the present study. When examining other boundary currents, possible patterns of nonlinear evolution can also fall into one of the three patterns mentioned here. For example, observation of the Tsushima warm currents flowing along the Japanese coastline reveal dipole eddy detachments that occur in late summer to fall, while meander is observed during the winter (Seung and Yoon 1995).
In the present study, not only has the classification of the evolution been presented but also a discussion of dynamical differences among the three regimes. In both the breaking wave and vortex pair regimes, the pool of positive vorticity generated in the offshore region moves the coastal region fluid in the offshore-downstream direction. As this occurs, the phase lag of the vorticity center on the offshore side of the current axis x
In the boundary trapped vortex regime the change in the phase relationship occurs in a manner similar to that in the breaking wave regime, but with some differences. In the boundary trapped vortex regime the phase difference between the vorticity centers decreases and changes from that of the growing to damping stage when the phase difference is zero. In the breaking wave regime, on the other hand, the change occurs as the lag increases. Subsequently, the pool of positive vorticity in the offshore region precedes the pool of the negative vorticity on the coastal side and the amplitude decreases.
The dependence of the manner of evolution on the parameters governing linearly unstable currents is as follows. The evolution is determined by the vorticity ratio q∗ rather than the ratio of the width of the vorticity regions L̄∗ in the initial state. As q∗ increases, the manner of evolution tends to fall in the boundary trapped vortex regime. On the other hand, as q∗ decreases, the evolution tends to be classified as the breaking wave regime. The vortex pair regime occurs when the parameter range falls between the boundary trapped vortex and the breaking wave regime. The most noteworthy result is the dependence of the evolution on the wavelength of the unstable wave. Even if the basic currents have the same velocity profiles, that is, the same vorticity distributions, the evolution changes from the breaking wave to the vortex pair regime as the wavelength increases. The same phenomenon occurs when the initial disturbance is composed of multiple unstable waves having different wavelengths with initial amplitudes sufficiently large to cause asymmetry in the growth among the wave crests of the faster growing modes.
In the present paper, attention was mainly focused on the temporal behavior of the phase relationship between the vorticity centers to classify the evolution. From this point of view, to examine the dynamics of the differences among the three regimes of evolution, use was made of a model in which finite area vorticity regions were approximated by point vortices. In spite of the simplicity of the model, the regimes of motion associated with the point vortices were classified into the same three regimes as found in the nonlinear evolution. The dependency of the motion of the point vortices on the parameters also displayed the same tendency. Thus, the point vortices model can help in the regime classification of the nonlinear evolution and will offer a new point of view in understanding the nonlinear behavior of unstable waves.
The present study emphasized the importance of the temporal evolution of the phase relationship between vorticity centers in a piecewise uniform vorticity region having opposite signs of vorticity. The concept of this paper can be applied not only to barotropic boundary currents but also to various geophysical fluid instabilities that include baroclinicity. Examples of these can be found in baroclinic boundary currents, extensions of the western boundary currents (free jet), and the instability of a baroclinic vortex.
Acknowledgments
The authors sincerely thank Professor Masaki Takematsu for his encouragement. The present paper is a part of a Ph.D. thesis (K. Shimada) submitted to Kyushu University.
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APPENDIX A
Numerical Method
In the numerical calculations, the vorticity fronts y = L1(x, t) and y = L2(x, t) are resolved by a finite number of Lagrangian points. Their temporal evolutions are calculated by advecting these points with numerically evaluated velocities through line integrals along the vorticity fronts. The line integration used here is the same method as that in Wu et al. (1984). The elongation of vorticity fronts occurs as unstable waves grow. Since spatial accuracy in the contour dynamics calculation depends on the internodal distance between adjacent nodes (μ), a node redistribution is required here, a linear interpolation is used to interpolate between adjacent nodes when the internodal distance is greater than a predefined maximum value (μmax). To perform long time integrations of the nonlinear behavior of the unstable currents, a contour surgery algorithm is required to avoid an immense increase the number of calculated nodes. The contour surgery algorithm used here, which is slightly different from that in Fig. 4 of Dritschel (1988), is as follows. The distance between contours dij is defined as the distance between the middle point of a line segment (Si,i+1) and the intersection of a line perpendicular from the middle point and another line segment (Sj,j+1). When the intersection does not fall on the other line segment, contour surgery is not performed. When both dij and dji are less than a predefined distance δ the two line segments are combined as depicted in Fig. A1. By introducing this algorithm, sufficiently thin filaments are removed and the merger of the vortices can then be visualized. The sensitivity of the present calculation is estimated by examining the evolution of the circulation or the conservation of the area of the piecewise uniform vorticity regions. Selection of the other parameters characterizing the flow, as in experiment 1 is made, since the filamentation and merger frequently occurs. Fig. A2 shows the dependency of μ, δ, and time step (Δt) respectively on the evolution. As pointed out in Dritschel (1988), smaller values of μ and δ lead to greater accuracy. Accuracy is fairly well maintained when Δt is less than 0.125. Taking these concepts into consideration, use is made of adequate numerical parameter values in this study. In fact, the evolution of experiment 1 is visibly the same as when making use of parameters yielding greater accuracy.
APPENDIX B
Equation of the Point Vortices
The motion of point vortices in a fluid and/or geophysical fluid has been investigated theoretically and/or numerically by many researchers (e.g., Batchelor 1970; Novikov 1975; Acton 1976; Aref 1979; Gryanik 1983; Hogg and Stommel 1985a,b; Legg and Marshall 1993; Funakoshi 1995). Here, only a brief introduction of the equations of point vortices motion is given.
Schematic illustration of the model configuration. The boundary currents consist of three piesewise constant vorticity regions, separated by the vorticity fronts y = L1 and y = L2.
Citation: Journal of Physical Oceanography 27, 7; 10.1175/1520-0485(1997)027<1326:NEOLUB>2.0.CO;2
Temporal evolution of the vorticity fronts (breaking wave regime, experiment 1). The solid and broken lines denote L1 and L2, respectively. Here, L̄∗ = 3.0, q∗ = 0.33, k = 3.92 (most unstable wave), and the initial amplitude is 0.025. The numerical parameters are μmin = 0.025, μmax = 0.05, δ = 0.005, and Δt = 0.0125.
Citation: Journal of Physical Oceanography 27, 7; 10.1175/1520-0485(1997)027<1326:NEOLUB>2.0.CO;2
(a) The real (solid line) and imaginary (bold line) parts of the phase speed as a function of wavenumber for the cases of L̄∗ = 3.0 and q∗ = 0.33. (b) The growth rate.
Citation: Journal of Physical Oceanography 27, 7; 10.1175/1520-0485(1997)027<1326:NEOLUB>2.0.CO;2
Temporal evolution of the phase difference ξ = k(x
Citation: Journal of Physical Oceanography 27, 7; 10.1175/1520-0485(1997)027<1326:NEOLUB>2.0.CO;2
Temporal evolution of the vorticity fronts (vortex pair regime, experiment 2). L̄∗ = 1.0, q∗ = 1.0, k = 2.59 (most unstable wave) and the initial amplitude 0.050. μmin = 0.02, μmax = 0.06, δ = 0.01, Δt = 0.01.
Citation: Journal of Physical Oceanography 27, 7; 10.1175/1520-0485(1997)027<1326:NEOLUB>2.0.CO;2
(a) The real (solid line) and imaginary (bold line) parts of the phase speed as a function of wavenumber for the cases of L̄∗ = 1.0 and q∗ = 1.0. (b) The growth rate.
Citation: Journal of Physical Oceanography 27, 7; 10.1175/1520-0485(1997)027<1326:NEOLUB>2.0.CO;2
Temporal evolution of the phase difference ξ and the displacement of the two vorticity centers in the y component Δyi (i = 1, 2) for the values of L̄∗ = 1.0, q∗ = 1.0, k = 2.59 (most unstable wave), and the initial amplitude of 0.025. Detailed explanation of the figure is given in Fig. 4.
Citation: Journal of Physical Oceanography 27, 7; 10.1175/1520-0485(1997)027<1326:NEOLUB>2.0.CO;2
Temporal evolution of the vorticity fronts for the values of L̄∗ = 0.33, q∗ = 1.0, k = 1.0 (breaking wave/vortex pair intermediate regime), and the initial amplitude of 0.05. The numerical parameters are μmin = 0.05, μmax = 0.15, δ = 0.01, and Δt = 0.375.
Citation: Journal of Physical Oceanography 27, 7; 10.1175/1520-0485(1997)027<1326:NEOLUB>2.0.CO;2
Temporal evolution of the vorticity fronts for the values of L̄∗ = 0.33, q∗ = 3.0, k = 2.48 (boundary trapped vortex regime, experiment 3), and the initial amplitude of 0.025. The numerical parameters are μmin = 0.025, μmax = 0.05, δ = 0.005, and Δt = 0.0125.
Citation: Journal of Physical Oceanography 27, 7; 10.1175/1520-0485(1997)027<1326:NEOLUB>2.0.CO;2
(a) The real (solid line) and imaginary (bold line) parts of the phase speed as a function of wavenumber for the cases of L̄∗ = 0.33 and q∗ = 3.0. (b) The growth rate.
Citation: Journal of Physical Oceanography 27, 7; 10.1175/1520-0485(1997)027<1326:NEOLUB>2.0.CO;2
Temporal evolution of the phase difference ξ and the displacement of the two vorticity centers in the y component Δyi (i = 1, 2) for the values of L̄∗ = 0.33, q∗ = 3.0, k = 2.48 (most unstable wave), and the initial amplitude of 0.025. Detailed explanation of the figure is given in Fig. 4.
Citation: Journal of Physical Oceanography 27, 7; 10.1175/1520-0485(1997)027<1326:NEOLUB>2.0.CO;2
Classification of the manner of evolution in q∗–k space for L̄∗ = 3.0. Here, B denotes the breaking wave, V the vortex pair, and T the boundary trapped vortex. Instability occurs in the hatched region. The broken line denotes the fastest growing wave.
Citation: Journal of Physical Oceanography 27, 7; 10.1175/1520-0485(1997)027<1326:NEOLUB>2.0.CO;2
Temporal evolution of the vorticity fronts (experiment 4). The basic flow is as in experiment 1 (Fig. 2). The first subharmonic of the most unstable wave is superimposed on the basic flow. Here, values of L̄∗ = 3.0, q∗ = 0.33, k = 1.96, and the initial amplitude of 0.025 are employed. The numerical parameters are μmin = 0.025, μmax = 0.1, δ = 0.005, and Δt = 0.025.
Citation: Journal of Physical Oceanography 27, 7; 10.1175/1520-0485(1997)027<1326:NEOLUB>2.0.CO;2
Temporal evolution of the phase difference ξ and the displacement of the two vorticity centers in the y component Δyi (i = 1, 2) for the values of L̄∗ = 3.0, q∗ = 0.33, k = 1.96 (first subharmonics of the fastest growing mode), and the initial amplitude of 0.025. Detailed explanation of the figure is given in Fig. 4.
Citation: Journal of Physical Oceanography 27, 7; 10.1175/1520-0485(1997)027<1326:NEOLUB>2.0.CO;2
Temporal evolution of the vorticity fronts (experiment 5). The basic flow is as in experiments 1 and 4. The most unstable wave and its first subharmonic are superimposed on the basic flow. Here, values of L̄∗ = 3.0, q∗ = 0.33, k = 3.92, 1.96, and the initial amplitude 0.100 are employed. The numerical parameters are μmin = 0.025, μmax = 0.1, δ = 0.01, and Δt = 0.025.
Citation: Journal of Physical Oceanography 27, 7; 10.1175/1520-0485(1997)027<1326:NEOLUB>2.0.CO;2
As in Fig. 15 except for the initial amplitude 0.025 (experiment 6).
Citation: Journal of Physical Oceanography 27, 7; 10.1175/1520-0485(1997)027<1326:NEOLUB>2.0.CO;2
Classification of the manner of evolution in q∗–k space for the values of (a: top) L̄∗ = 3.0, (b: middle) L̄∗ = 1.0, and (c: bottom) L̄∗ = 0.33. In the figures, B denotes the breaking wave, V the vortex pair, T the boundary trapped vortex, and BV the breaking wave/vortex pair intermediate regime. Instability occurs in the hatched region. The broken line denotes the fastest growing wave.
Citation: Journal of Physical Oceanography 27, 7; 10.1175/1520-0485(1997)027<1326:NEOLUB>2.0.CO;2
Temporal evolution of the vorticity fronts (experiment 7) for the values of L̄∗ = 7.0, q∗ = 0.33, k = 7.84, and the initial amplitude of 0.025. The numerical parameters are μmin = 0.0125, μmax = 0.025, δ = 0.0025, and Δt = 0.00625.
Citation: Journal of Physical Oceanography 27, 7; 10.1175/1520-0485(1997)027<1326:NEOLUB>2.0.CO;2
Temporal evolution of the vorticity fronts (experiment 8) for the values of L̄∗ = 3.0, q∗ = 0.33, k = 5.18, and the initial amplitude 0.025. The numerical parameters are μmin = 0.01, μmax = 0.025, δ = 0.005, and Δt = 0.0125.
Citation: Journal of Physical Oceanography 27, 7; 10.1175/1520-0485(1997)027<1326:NEOLUB>2.0.CO;2
Schematic illustration of the configuation of the point vortices. Two point vortices exist with opposite sign of circulation per unit domain.
Citation: Journal of Physical Oceanography 27, 7; 10.1175/1520-0485(1997)027<1326:NEOLUB>2.0.CO;2
Contours of the Hamiltonian exp(H/λ) in phase space ξ–η. Here, values of Γ1/λ = −0.75, Γ2/λ = 1.0, and I = 1.865 (yG = 2.375) are employed for the periodic lengths (a) λ = 1.257, (b) λ = 1.829, (c) λ = 2.094, (d) λ = 3.142, and (e) λ = 6.283. The symbols ★ and ☆ denote hyperboric points and elliptic points, respectively; Sπ denotes separatries, which are accompanied by a hyperboric point hπ located at ξ = ±π, and S0 denotes separatries accompanied by a hyperboric point h0 located at ξ = 0.
Citation: Journal of Physical Oceanography 27, 7; 10.1175/1520-0485(1997)027<1326:NEOLUB>2.0.CO;2
(Continued)
Citation: Journal of Physical Oceanography 27, 7; 10.1175/1520-0485(1997)027<1326:NEOLUB>2.0.CO;2
Contours of the Hamiltonian exp(H/λ) in phase space ξ–η for the values of Γ1/λ = −1.5, Γ2/λ = 1.0, λ = 2.094, and I = 1.178 (yG = −0.750).
Citation: Journal of Physical Oceanography 27, 7; 10.1175/1520-0485(1997)027<1326:NEOLUB>2.0.CO;2
Fig. A1. Schematic illustration of typical contour surgery. For details, see text (appendix A).
Citation: Journal of Physical Oceanography 27, 7; 10.1175/1520-0485(1997)027<1326:NEOLUB>2.0.CO;2
Fig. A2. The temporal change of constant vorticity area, which is a conserved quantity and an indicator of numerical accuracy. The figures are drawn for the dependence on μmin and μmax, dependence on δ, and dependence on Δt.
Citation: Journal of Physical Oceanography 27, 7; 10.1175/1520-0485(1997)027<1326:NEOLUB>2.0.CO;2
Fig. A2. (Continued)
Citation: Journal of Physical Oceanography 27, 7; 10.1175/1520-0485(1997)027<1326:NEOLUB>2.0.CO;2
