a. Numerical methods and grid

In this study we use a nondivergent barotropic model on the β plane with second-order diffusion. We employ an explicit finite difference code to solve these equations numerically (see Part I for details).

In our calculations the numerical domain extends L_x = 27 000 km in the zonal direction and L_y = 9000 km in the meridional direction. For the model runs of Part I typically 3000 × 1001 grid points were used (corresponding to a grid length of 9 km). To allow for the large number of simulations that are required to explore the parameter space, most of the model runs performed for this study are computed with a reduced resolution of 1500 × 501 zones. This results in some damping of the cyclones but does not alter the results discussed below qualitatively (as tested for several cases).

We use a channel geometry with periodic boundaries in the zonal direction and walls at y = ±L_y/2. The value of β is 2 × 10⁻¹¹ m⁻¹ s⁻¹, which corresponds to a latitude of approximately 30°.

b. Front and cyclone representations

The barotropic model is initialized by adding the relative vorticity field of a vortex to the absolute vorticity background that represents the front. A simple model for a front in a barotropic framework (e.g., Bell 1990; Schwierz et al. 2004) is given by two regions of constant absolute vorticity η = f ± η₀ separated by an interface at y = y_f(x). The Coriolis parameter is approximated as f = f₀ + βy. For convenience, the constant f₀, which has no influence on the calculations, is set to zero. Thus we adopt an absolute vorticity distribution

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An infinitely sharp vorticity step cannot be modeled with the numerical methods used for this study. As described in more detail in Part I, we therefore replace the vorticity step by a narrow smooth transition zone with a width of about 100 km.

For a zonally aligned jet (y_f ≡ 0 km), Eq. (1) gives rise to the zonal velocity profile

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where u₀ = u(0km) is the jet speed (see Fig. 1 in Part I).

For the model runs in this study we set the jet parameters to u₀ = 40 m s⁻¹ and η₀ = 4 × 10⁻⁵ s⁻¹ and use the SUD vortex described in Part I, which reaches the maximum wind speed of 40 m s⁻¹ at a radius of 100 km. The start position of the vortex is 1500 km south of the front. Thus the initial state is set up as in model run BS from Part I, except that frontal waves are already present at t = 0 days.

Frontal waves are modeled as a sinusoidal frontal deformation:

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where k_m = 2πm/L_x is the wavenumber, ϕ_mⁱ the initial wave phase, and b_mⁱ the initial wave amplitude of mode m. In some cases several wave modes are excited.

A frontal wave of mode m propagates with a phase speed

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where λ_m = L_x/m is the wavelength (see Part I).

The circulation associated with the propagating frontal wave is given by the zonal and meridional components

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and

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respectively, with a velocity amplitude ν_m = b_mⁱη₀/2. Unless otherwise noted, we use a frontal displacement amplitude b_mⁱ = 250 km. For wave mode m = 5 this corresponds to an initial steering flow of 1.7 m s⁻¹ at the center of the cyclone.

As the initial zonal cyclone position is X_c = 0, a value of ϕ_mⁱ = 0 corresponds to the configuration with a trough upstream and a ridge downstream of the cyclone, which causes a northward flow at the cyclone center. The values ϕ_mⁱ = π/2, π, and 3π/2 correspond to eastward, southward, and westward flows, respectively. Note that in the meridional direction the circulation is damped exponentially with an e-folding scale of λ_m/2π.

The flow steering the cyclone contains contributions associated with the vorticity step [Eq. (2)], the preexisting frontal wave, the frontal waves excited by the cyclone in the course of the model run, and the anticyclones sheared from the vortex in the initial phase (see Part I).

a. Sensitivity to the wave phase

As a first step we investigate calculation set BS5-X, in which the wavelength of the preexisting frontal wave, λ₅ = L_x/5 = 5400 km, is close to the initial resonant wavelength. In Fig. 1 the evolution of the cyclone position is shown for the cases ϕ₅ⁱ = 0, π, 1.43π, and 1.44π and for the run with an unperturbed front (run BS). Not unexpectedly, the cyclone in run BS5-0.0 is advected toward the front faster than in run BS and crosses the interface 4 days earlier (Fig. 1a). The cyclone in run BS5-1.0, which is initially advected southward by the wave circulation, reaches the front 7 days later than in run BS. However, BS5-1.0 is not the most extreme case in terms of the time needed to reach the front. In model run BS5-1.43 the cyclone does not cross the interface at all in the 20 days included in the calculation.

An unexpected sensitivity to the initial conditions is found when comparing runs BS5-1.43 and BS5-1.44, in which the preexisting frontal wave is shifted only λ₅ × 0.01 = 54 km farther upstream. The start positions of the cyclones in these runs are close to the trough axis (ϕ₅ⁱ = 1.5π). After about 5 days the meridional locations of the cyclones in the two runs diverge dramatically. While in run BS5-1.43 the cyclone moves southward, it is advected rapidly northward in run BS5-1.44 and crosses the front at about the same time as in the unperturbed case BS. The huge differences in the temporal evolution of the meridional cyclone coordinate lead also to vastly different zonal cyclone positions (Fig. 1b). The cyclones that approach the front are advected downstream with translation speeds of up to 30 m s⁻¹, whereas the cyclones that stay farther away from the front attain only rather low zonal translation speeds. The difference in the cyclone position of the runs BS5-1.43 and BS5-1.44 is initially very small but increases rapidly from t ≈ 7 days. For t > 9 days the distance grows by about 2000 km day⁻¹. Thus, a preexisting frontal wave can have a strong influence on the cyclone track and in some cases this influence can be very sensitive to the position of the cyclone with respect to the wave.

The sensitivity of the cyclone track to the initial conditions varies strongly with the initial wave phase ϕ₅ⁱ. For phases that are associated with a northward advection of the cyclone, the sensitivity is rather low. In all these cases the cyclone crosses y = 0 km after 6–7 days (Fig. 2). In contrast, in cases where the cyclone is initially advected southward the sensitivity is much higher and cases can be found in which the cyclone stays south of the front for more than 20 days (Fig. 2). The most interesting feature, however, is that there are several critical phases at which the solution changes abruptly—most notably at ϕ₅ⁱ = 1.44π, between the runs BS5-1.44 and BS5-1.43, but also around ϕ₅ⁱ ≈ π Further model runs have shown that at these critical phases shifting the frontal wave by a distance smaller than the grid length (in this case Δx = 18 km) is sufficient to switch between vastly different solutions, (e.g., BS5-1.43-like solutions in which the cyclone remains south of the front and solutions in which the cyclone reaches the front in less than 10 days, as in run BS5-1.44). This behavior can be interpreted as a bifurcation in the solution space.

To investigate the cause of the bifurcations it is sufficient to consider only the effect of the freely propagating preexisting wave on the cyclone motion and neglect the changes in wave phase and amplitude that are caused by the cyclonic and anticyclonic circulations, as these become important only after some days. According to this simplification, the motion of the cyclone relative to the basic-state flow is determined by the phase of the preexisting wave at the zonal position of the cyclone:

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For −π/2 < ϕ_c,m < π/2 (around a trough–ridge transition) cyclones are advected northward by the wave circulation; otherwise they are advected southward. South of troughs (π < ϕ_c,m < 2π) cyclones are advected eastward relative to the basic-state flow, and south of ridges (0 < ϕ_c,m < π) they are advected westward. The phase ϕ_c,m is indicated by the shading in Fig. 3, which displays the evolution of the zonal cyclone position for all runs of set BS5-X.

The cyclones starting at phases ϕ₅ⁱ < 0.5π or ϕ₅ⁱ > 1.5π are initially advected northward, thereby increasing their zonal translation velocity sufficiently to stay in the northward advecting range (white background in Fig. 3) for several days and reach the front within 9 days. Cyclones starting at phases slightly smaller than ϕ₅ⁱ = 1.5π [i.e., upstream of the trough center, where the preexisting frontal wave causes a southward directed flow (gray background) initially] are still able to overtake the trough and reach the northward advection zone. This is possible for two reasons. At t = 0 days the cyclones are already moving with a translation speed close to the phase speed of the wave (5.6 m s⁻¹) because the wave circulation is eastward near the trough center and thus U_c(t = 0 days) ≈ u(Y_c)+ ν₅ = 4.4 m s⁻¹. The increase in zonal velocity required to overtake the ridge is provided by the anticyclones, which cause initially a northward advection (see Part I) and thus also a zonal acceleration of the cyclone.

However, at a critical phase ϕ_crit ≈ 1.44π the cyclone attains a zonal velocity matching the phase speed of the wave at a time when it is south of the center of the trough (i.e., on the trough axis). In this position the meridional velocity component at the cyclone center caused by the wave is zero. Thus, the cyclone remains at the same meridional position and its zonal translation speed remains constant and equal to the phase speed of the wave. In the frame of reference of the wave this is a stationary situation—cyclone and wave move with the same zonal velocity and their separation remains constant. In runs with ϕ₅ⁱ = ϕ_crit the cyclone can stay close to this position for a rather long time. Very small changes in the initial wave phase determine whether the cyclone eventually moves toward the next trough–ridge transition downstream of the trough, where it is attracted by the front, or toward the ridge–trough transition upstream of the trough, where it is repelled from the front. The fact that very small changes in the initial state determine which of two vastly different solutions will occur gives rise to the bifurcation.

Neglecting the influence of the anticyclones and the frontal waves excited by the cyclone, the point at which the bifurcation occurs fulfills the conditions V_c = υ_f(X_c, Y_c) = 0 and

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There are actually two points south of the front that satisfy these conditions, as sketched in Fig. 4: a point T south of the trough center (ϕ_c = 3π/2) where the background flow and the circulation of the preexisting frontal wave are parallel, and a point R south of the ridge center (ϕ_c = π/2), where they are antiparallel. For the set of calculations considered here Eq. (8) yields meridional coordinates Y_cT ≈ −1400 km and Y_cR ≈ −1100 km for points T and R, respectively. The bifurcation at ϕ ≈ 1.44π occurs close to the trough center and is thus clearly related to point T. The bifurcation at ϕ ≈ π (see Fig. 2) is also located close to the trough center.¹ No unusual spreading of the tracks is visible for cyclones that attain positions south of the ridge center (dotted lines in Fig. 3), so there are no bifurcations that occur close to point R.

The fact that bifurcations occur at point T but not at point R is related to the different flows in the vicinity of the two points. Close to point T the solutions diverge. As sketched in Fig. 4, cyclones that are located slightly downstream of point T are attracted toward the front and accelerated in the zonal direction, whereas cyclones slightly upstream of point T are advected farther away from the front and slow down. Point T can be approached from the southeast because a cyclone located downstream and farther away from the front than point T moves more slowly than the point itself and experiences a northward directed flow component from the wave. Both effects will bring the cyclone closer to point T.

In contrast to point T, point R cannot be reached by cyclones from the south. To get closer to the front, the cyclone must be located upstream of point R. However, in this position the cyclone is moving more slowly than point R, so that by the time the cyclone has reached the meridional position of point R, the latter has already propagated downstream. If a cyclone is located downstream and south of point R, it will move southward and slow down until it is overtaken by point R. Subsequently this cyclone will move toward the front and also pass point R on the upstream side (see Fig. 4). Thus point R cannot be reached by a cyclone moving northward, but the flow around point R causes cyclones to revolve around it (in the frame of reference of the frontal wave).

In accordance with these considerations, cyclone tracks diverging near point T and cyclones tracks orbiting around point R are visible in Fig. 5 (top panel), which displays the cyclone tracks of simulation set BS5-X in the frame moving with the wave. These cyclone tracks are influenced not only by the preexisting frontal wave, but also by frontal waves excited by the cyclone and by the anticyclones. The locations of points R and T indicated in Fig. 5, however, are computed under the simplifying assumption that the influence of the anticyclones and of the additional frontal waves excited by the cyclone can be neglected. Thus it is assumed that the cyclone is advected like a tracer particle in a velocity field (u_p, υ_p) that is a superposition of the stationary jet and the flow caused by the propagating preexisting frontal wave. The limits of this approximation are evident from Fig. 5 (bottom panel), which displays cyclone tracks together with contour lines of the streamfunction Ψ_p that gives rise to (u_p, υ_p) in the frame of the wave. The points R and T are located at maxima and saddle points of Ψ_p, respectively. Overall the streamlines are similar to the cyclone tracks—they diverge near point T and orbit around point R, and the angle between the direction of the cyclone motion and the closest streamline is relatively small for most of the time and most of the tracks.

However, clear differences between the tracks and the streamlines are visible. These differences are related to the anticyclones and the frontal waves excited by the cyclone. During the formation of the anticyclones in the first few days the cyclone is advected northward (see Part I). Because of this effect the tracks diverge at a point slightly upstream of the position marked with T, where the southward flow component caused by the wave compensates for the northward flow due to the anticyclones. At later times, and in particular when the cyclone comes close to the interface, the waves excited by the cyclone can no longer be neglected compared to the preexisting wave, and the tracks no longer follow the streamlines. Therefore, considering the cyclone as a tracer particle can explain some basic features of the tracks, but this is not useful to determine the cyclone tracks quantitatively. When the amplitude of the preexisting frontal wave is reduced, the cyclone-induced waves become important earlier and eventually the bifurcation should vanish. However, the bifurcation of tracks near point T is still clearly visible even for a preexisting wave amplitude of only 100 km.

Figure 5 illustrates also why a bifurcation occurs at ϕ₅ⁱ ≈ π (see Fig. 2). The cyclone starting at ϕ₅ⁱ = π circles around point R, approaches point T from the northwest, and is deflected toward the front. The cyclone starting at ϕ₅ⁱ = 1.02π takes the same route initially but is then deflected to the south. For slightly higher values of ϕ₅ⁱ the cyclone is advected northward before it can come closer to point T. Thus, a rather large area downstream of point R cannot be reached by any cyclone.

b. Influence of perturbation wavelength

The wavelength of the m = 5 frontal wave mode used above is slightly smaller than the initial resonant wavelength and the phase speed is slightly faster than the initial zonal translation velocity of the cyclone. In the following, the influence of waves with larger or smaller wavelengths is discussed and cases are investigated in which several wave modes are excited. For this purpose, several additional sets of calculations are considered (see Table 1 for a list of all sets discussed in this section). In particular, we consider several calculation sets similar to BS5-X, but with frontal waves of the modes m = 4, 7, 10, and 15.

The wave amplitude b_mⁱ = 250 km is the same in all runs, which means that at a fixed distance from the front the modes with the largest wavelengths are associated with the highest wind speeds [see Eqs. (5) and (6)]. Therefore the m = 4 wave has a strong impact on the cyclone track—the time required by the cyclone to approach the front varies between 8 and 14 days (Fig. 6). However, for all phases the solution changes smoothly with ϕ_mⁱ and there are no bifurcations. This fundamental difference is related to the fact that m = 5 waves propagate in the same direction as the fluid itself, whereas m = 4 waves propagate in the upstream direction. The bifurcations for m = 5 can arise because two kinds of solutions can develop from very similar initial conditions: one in which the cyclone overtakes the wave, and one in which the wave overtakes the cyclone. This situation can be encountered only if the cyclone can actually obtain a zonal translation speed matching the phase speed of the wave. As the zonal translation speed of the cyclone is close to the zonal speed of the background flow (unless the cyclone is close to the front), this requires that the phase speed is contained in the zonal speed profile of the jet. This is not the case for m = 4(c₄ < 0,but u > 0), so the cyclone will always overtake the wave. These results indicate that preexisting frontal waves with wavelengths larger than the initial resonant wavelength can have a strong influence on the cyclone track, but they are not important for the sensitivity of the track with respect to small changes in the initial conditions.

For smaller wavelengths [i.e., higher values of m and faster (positive) phase speeds], U_c = c_m is again possible. With decreasing wavelength the frontal wave circulation decays faster in the meridional direction [Eqs. (5) and (6)] and the meridional positions of points R and T move closer to the front. Thus the strongest effects of preexisting frontal waves with smaller wavelengths, including bifurcations, are expected to occur only after the cyclone has moved sufficiently close to the front. Indeed, for the m ≥ 7 cases the cyclone position in the first 7 days hardly varies for different phases ϕ_mⁱ. During this period the cyclone motion is not yet influenced significantly by the preexisting frontal wave, but the cyclone circulation excites waves with the current resonant wavelength. This wave component causes a northward acceleration of the cyclone, making it considerably more difficult for the preexisting wave to advect the cyclone southward. Consequently, the range of phases for which the cyclone does not cross the front during the simulated 20 days becomes narrower. While for m = 5 a considerable fraction ϕ₅ⁱ ∈ [0.85π, 1.43π] causes the cyclone to reach the front only after 10 days or later (Fig. 2), this is the case for m = 7 only when ϕ₇ⁱ ∈ [0.6π, 0.7π]. For higher mode numbers there is more time for the cyclone to excite frontal waves before the bifurcation process takes place. Thus the influence of the preexisting wave becomes weaker compared to the other waves, and the impact of the bifurcation process on the cyclone track becomes less severe. At m = 10 the bifurcation process is able to change the time required by the cyclone to reach the front by about 3 days (Fig. 7) but does not lead to such strongly diverging evolutions as in the case of runs BS5-1.43 and BS5-1.44 (Fig. 1).

For preexisting frontal waves with very small wavelengths and high phase speeds, bifurcations become impossible again because the wave is always faster than the cyclone. In our calculations with a jet speed u₀ = 40 m s⁻¹ the maximum zonal translation velocity obtained by the cyclone is about 30 m s⁻¹, which corresponds to m ≈ 15 for the β-plane cases discussed here (see Fig. 13 in Part I). For m = 10 bifurcations are still present whereas for m = 15 no bifurcations are encountered (Fig. 6). Thus, a rather narrow band of wavelengths smaller than the initial resonant wavelength has a strong influence on the sensitivity of the solution on the initial conditions. Assuming that the zonal translation velocity of the cyclone is in the range [0, u₀/2], bifurcations can only be caused by waves with a phase speed in the same range. The corresponding range of wavelengths can be computed using the relation u₀ = η₀²/2β (see Part I, section 3a) and the dispersion relation [Part I, Eq. (6)]. The maximum wavelength (i.e., the wavelength of the wave with phase speed zero) is thus λ_b = π, which amounts to about 6000 and 8000 km for jet speeds of 40 and 70 m s⁻¹, respectively. The minimum wavelength is about λ_b/2.

To examine whether bifurcations can still occur when preexisting waves containing several wave modes are present, several further sets of calculations are considered. In a first experiment, set BS5a-X, the preexisting frontal wave consists of an m = 4 and an m = 5 component with equal amplitudes (250 km). The phase of the m = 5 component is varied continuously, whereas for the m = 4 mode only some phases are investigated. We show only the case ϕ₄ⁱ = 1.5π, which should be the least favorable for the occurrence of bifurcations, as the m = 4 wave causes the maximum northward advection for this phase. Yet bifurcations are present even in this case (Fig. 8). The bifurcation is suppressed only when the amplitude of the m = 5 mode is reduced to 100 km and ϕ₄ⁱ = 1.5π (set BS5b-X, Fig. 8). If either the amplitude is increased or a more favorable phase is chosen, bifurcations are present.

Finally, in set BS5c-X four wave modes (m = 4, 5, 6, 8) are excited in the preexisting frontal wave, each with an amplitude of 250 km. In this case several bifurcations are present (Fig. 8). These results demonstrate that bifurcations due to waves that are in resonance with the cyclone at some time are not easily suppressed by the presence of other resonant or nonresonant waves.

c. Influence of frontal wave–induced shear

In the previous sections we showed that the main features of the interaction between the cyclone and the frontal wave can be explained by treating the cyclone as a point vortex. A secondary effect was the modification of the cyclone structure by the background shear associated with the jet, leading to the formation of anticyclones. The tropopause evolution, however, is sensitive to the interaction with the upper-level anticyclone of a tropical cyclone that has a much larger horizontal scale than the cyclonic core. Thus it is important to consider the structural evolution of an anticyclone as a function of its position relative to the frontal wave. Because of its large horizontal scale, the anticyclone is more sensitive to the horizontal shear of the jet and the frontal wave than the cyclonic vortex considered previously.

We investigate this interaction with a set of calculations, BN5-X, for which the anticyclonic part of the ANTI vortex (see section 3b in Part I) is used. In this anticyclonic vortex (υ_max = −8 m s⁻¹, r_max = 500 km) the velocity gradient is of a size comparable to the background shear. Furthermore, for the m = 5 frontal wave with amplitude 250 km in BN5-X the shear of the frontal wave circulation is also of the same order as the background shear. Consequently, the vortex structure can be changed considerably because of the shear and the changes depend on the initial position of the vortex.

Two extreme cases are shown in Fig. 9. In run BN5-0.6 the anticyclone is initially located south of a ridge, in the vicinity of point R. In this region the shear of the frontal wave circulation partially compensates for the background shear. The shape of the anticyclone is basically preserved as it circles around point R (Fig. 9, top panel). In run BN5-1.5 the anticyclone starts south of a trough, close to point T. Because of the strong shear deformation caused by the jet profile and the wave circulation in this region, the vortex is deformed to an elongated shape and then disrupted into two filaments within several days. The two filaments are advected to the northeast and the southwest, respectively, and end up south of the two neighboring ridges (Fig. 9, bottom panel). In this model run the anticyclone cannot influence the frontal wave substantially, as its vorticity distribution soon extends over a zonal range comparable to the wavelength of the frontal wave. In contrast, the anticyclone in run BN5-0.6, which remains compact and stays in phase with the wave, can influence the wave more strongly and causes a phase shift of about 500 km in 10 days (Fig. 9, top panel).