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    Fig. 1.

    Absolute vorticity (thin) and zonal velocity (thick) for the basic state with η0 = 4 × 10−5 s−1 on the β plane (solid) and the basic states with η0 = 10−5 s−1 (dotted) and (dashed) on the f plane. The thick dots mark the initial vortex positions for the model runs of Table 2.

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    Fig. 2.

    Profiles of the (top) vorticity and (bottom) tangential velocity for the SUB, BROAD, ANTI, and GAUSS vortices.

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    Fig. 3.

    The circulation induced by a cyclone (white circle) displaces the interface between an area of negative (dark gray) and positive (light gray) absolute vorticity. Two vorticity anomalies are created at the interface with associated circulations (black circles and arrows) that advect the cyclone toward the front.

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    Fig. 4.

    Vorticity distribution 2, 4, 7, 10, 15, and 20 days after the start of the calculation for model runs (a) WG, (b) WB, (c) WA, and (d) WS. The black dots mark the cyclone position after every full day.

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    Fig. 5.

    Evolution of the cyclone translation velocity (thick black line) in model run WG (u0 = 15 m s−1), showing the (top) zonal and (bottom) meridional components. In addition, contributions to the velocity at the cyclone center due to the basic state (dashed) and the frontal waves (dashed–dotted) are displayed for the time span in which these quantities can be determined reliably (see section 3c).

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    Fig. 6.

    Meridional velocity components due to the cyclone (dashed) and the frontal wave (dashed–dotted), and their sum (solid), in run WG at (x, y) = [xXc, yf(x)] for t = 24 h.

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    Fig. 7.

    Time evolution of |bk(m)| for modes (left) m = 1 and (right) m = 4 of the model runs WG (solid), WB (dashed), WS (dotted), and WA (dashed–dotted).

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    Fig. 8.

    Meridional velocity at y = 0 km induced by cyclone and anticyclones υc(x, 0) + υa(x, 0) (top) in the initial state and (bottom) after 4 days for the runs WA, WB, WS, and WG. Note the different scales along the velocity axis.

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    Fig. 9.

    Vorticity distributions after 4 days for runs WA (linear shear), BA (parabolic shear), and BA2 (parabolic shear, initial vortex position at the velocity minimum).

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    Fig. 10.

    Time evolution of the meridional cyclone position for model runs BG, WG, FG, and SG.

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    Fig. 11.

    (top) Evolution of |m(t)| [see Eq. (8)] for modes m = 1, … , 5 during the time period before the cyclone comes close to the front [t < tf; see Eq. (20)] in runs WG, FG, and BG. For each time, the mode with the wavelength closest to the resonant wavelength is marked with a black dot. (bottom) Time evolution of ϕk(m)(t) for models runs WG, FG, and BG. Parts for which |bk(m)(t)| < 10 km are omitted. The dotted lines enclose the region −π/2 < ϕk(m) < π/2 in which m is able to grow. For ϕk(m) < −π/2 or ϕk(m) > π/2 the wave is damped.

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    Fig. 12.

    Frontal deformation spectrum for runs BG, WG, and FG. The amplitudes |m| are averaged over 15 < t < 20 days.

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    Fig. 13.

    Time evolution of the meridional cyclone position for model runs BB, WB, and FB.

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    Fig. 14.

    Frontal position in model runs WB (solid) and WB-m (dashed) after 12 days.

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The Resonant Interaction of a Tropical Cyclone and a Tropopause Front in a Barotropic Model. Part I: Zonally Oriented Front

Leonhard ScheckInstitute for Meteorology and Climate Research, Karlsruhe Institute of Technology, Karlsruhe, Germany

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Sarah C. JonesInstitute for Meteorology and Climate Research, Karlsruhe Institute of Technology, Karlsruhe, Germany

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Martin JuckesBritish Atmospheric Data Centre, Rutherford Appleton Laboratory, Chilton, United Kingdom

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Abstract

The interaction of a tropical cyclone and a zonally aligned tropopause front is investigated in an idealized framework. A nondivergent barotropic model is used in which the front is represented by a vorticity step, giving a jetlike velocity profile. The excitation of frontal waves by a cyclone located south of the front and the impact of the wave flow on the cyclone motion is studied for different representations of the cyclone and the jet. The evolution from the initial wave excitation until after the cyclone has crossed the front is discussed. The interaction becomes stronger with increasing jet speed. For cyclone representations containing negative relative vorticity, anticyclones develop and can influence the excitation of frontal waves significantly. Resonant frontal waves propagating with a phase speed matching the zonal translation speed of the cyclone are decisive for the interaction. The frontal wave spectrum excited by a cyclone on the front is dominated by waves that are in resonance in the initial phase. These waves have the largest impact on the cyclone motion.

Corresponding author address: Leonhard Scheck, Karlsruher Institut für Technologie, Institut für Meteorologie und Klimaforschung, Kaiserstr. 12, 76131 Karlsruhe, Germany. Email: leonhard.scheck@kit.edu

Abstract

The interaction of a tropical cyclone and a zonally aligned tropopause front is investigated in an idealized framework. A nondivergent barotropic model is used in which the front is represented by a vorticity step, giving a jetlike velocity profile. The excitation of frontal waves by a cyclone located south of the front and the impact of the wave flow on the cyclone motion is studied for different representations of the cyclone and the jet. The evolution from the initial wave excitation until after the cyclone has crossed the front is discussed. The interaction becomes stronger with increasing jet speed. For cyclone representations containing negative relative vorticity, anticyclones develop and can influence the excitation of frontal waves significantly. Resonant frontal waves propagating with a phase speed matching the zonal translation speed of the cyclone are decisive for the interaction. The frontal wave spectrum excited by a cyclone on the front is dominated by waves that are in resonance in the initial phase. These waves have the largest impact on the cyclone motion.

Corresponding author address: Leonhard Scheck, Karlsruher Institut für Technologie, Institut für Meteorologie und Klimaforschung, Kaiserstr. 12, 76131 Karlsruhe, Germany. Email: leonhard.scheck@kit.edu

1. Introduction

The interaction of a tropical cyclone with the midlatitude flow during extratropical transition (ET) is frequently associated with low forecast skill both of the ET event and of the midlatitude flow downstream (Harr et al. 2008; Anwender et al. 2008). During the early stages of the interaction the tropical cyclone can modify the flow upstream of the tropical cyclone (Bosart and Lackmann 1995) and contribute to enhanced ridging downstream (Henderson et al. 1999; Riemer and Jones 2010). These modifications can feed back onto the motion and structural changes of the ET system. Subsequently, the excitation of a Rossby wave train on the midlatitude tropopause can lead to downstream development (Harr and Dea 2009). The relative importance of these different features can vary significantly among ET cases, as can the reduction in forecast skill. Thus it is important to understand which factors control the perturbation of the tropopause by a tropical cyclone and the feedback onto the tropical cyclone itself.

Idealized modeling studies provide a suitable framework for isolating these factors. Riemer et al. (2008) used an idealized configuration in a full-physics model. They showed that the initial excitation of a Rossby wave train by a tropical cyclone is associated with ridge building due to its divergent outflow impinging on the tropopause. Subsequently, the balanced anticyclone of the outflow plays an important role in allowing phase-locking between the tropical cyclone and the developing ridge, as well as in the amplification of the downstream trough. At later times, the cyclonic circulation contributes to the ridge building and to the wrap-up of the upstream trough. Thus, both the balanced cyclonic circulation and the upper-level anticyclone are important in the excitation of a Rossby wave train. Despite the idealized initial conditions of Riemer et al. (2008), the use of a complex model limits the extent to which specific mechanisms could be investigated, due to both the computational requirements for the large domain and the difficulties involved in disentangling the various physical and dynamical processes.

In this study we investigate the mechanisms that determine the interaction between the midlatitude flow and a tropical cyclone in an idealized, two-dimensional vortex dynamics framework. Thus we can isolate and interpret the dynamical evolution as well as study the sensitivity to the structure of vortex and front in detail. For this purpose we model the upper-level jet stream as a region of enhanced vorticity gradient or a vorticity interface, which can be considered as the tropopause front.

Similar configurations were used in two studies that are not directly related to tropical cyclones. Bell (1990) presented analytic linear solutions describing the interaction between a point vortex and an interface separating two regions of uniform potential vorticity. Using a nonlinear hemispheric primitive equation model, Schwierz et al. (2004) studied the effect of an isolated topographic feature on a vorticity interface in a linear barotropic β-plane model and the interaction between a zonally aligned jet stream and PV anomalies located poleward of the jet stream. In both studies a resonance condition is identified that allows sustained growth of frontal waves.

Here we investigate the interaction of vortices representing tropical cyclones with a zonally aligned upper-level jet stream in a nonlinear nondivergent barotropic model. We study the implications of the resonance condition for the interaction strength and the influence of the vortex structure on the excitation of frontal waves. The impact of preexisting frontal waves on the vortex motion and the predictability of cyclone tracks will be investigated in a second paper (Scheck et al. 2011, hereafter Part II).

The paper is organized as follows. After a short description of the numerical setup in section 2, a brief discussion of the jet and vortex representation is given in section 3. In section 4 we discuss the vortex–jet interaction for different vortices and jet types. Finally, section 5 contains a summary of our results.

2. Numerical methods

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. A third-order upwind scheme is used to compute the advection terms in combination with a second-order Runge–Kutta scheme for the time integration, which results in a stable numerical scheme (Baldauf 2008). The streamfunction is calculated in Fourier space (Press et al. 1986, chapter 20, section 20.4.1) and the velocity components are computed from the streamfunction using centered differences. The constant viscosity parameter ν is chosen so that it is high enough to suppress small-scale noise while influencing the larger-scale flow as little as possible.

In our calculations the numerical domain extends Lx = 27 000 km in the zonal direction and Ly = 9000 km (in some cases 18 000 km) in the meridional direction. Typically 3000 × 1001 grid points are used (corresponding to a grid length of 9 km). As tested for several cases, running the simulation with a grid length of 4.5 km yields only small differences. We use a channel geometry with periodic boundaries in the zonal direction and walls at y = ±Ly/2. The value of β is 2 × 10−11 m−1 s−1, which corresponds to a latitude of approximately 30°.

3. Representations of fronts and tropical cyclones

The barotropic model is initialized by adding the relative vorticity field of a vortex to the absolute vorticity background that represents the front. Several vortices with different tangential wind profiles and several front representations with different values of the jet speed are used.

a. 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 = 0. 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 profile
i1520-0469-68-3-405-e1
which gives rise to the zonal velocity profile
i1520-0469-68-3-405-e2
where u0 = u(0) is the jet speed.

On the f plane (β = 0) the velocity profile contains two regions with constant meridional shear of opposite sign (Fig. 1). Two cases are considered: a relatively weak front with η0 = 10−5 s−1, which corresponds to a jet speed of u0 = 15 m s−1, and a front with and a jet speed u0 = 40 m s−1. The values of η0 and u0 are chosen such that u(1500 km) = 0 m s−1 in both cases.

On the β plane Eq. (2) yields a velocity profile with varying meridional shear. The wind speed is maximum at y = 0 km and has two minima at y = ±dmin (Fig. 1). For appropriate values of η0, the wind profile between y = ±dmin resembles a realistic jet profile. For u(dmin) = 0 m s−1, the velocity at the interface is u0 = (½)η02/β and dmin = η0/β. For η0 = 4 × 10−5 s−1, the jet velocity is 40 m s−1 and dmin is 2000 km. For higher values of η0, both the maximum velocity and the distance between the minima (2dmin) increase.

An infinitely sharp vorticity step cannot be modeled with the numerical methods used for this study. Therefore we replace the vorticity step by a narrow smooth transition zone. Following Melander et al. (1987), we replace Eq. (1) by
i1520-0469-68-3-405-e3
with the function f (s) = exp{−ks−1 exp[1/(s − 1)]} and k = exp(2) ln(2)/2. As approximately 90% of the variation of ηnum occurs within [−D/2, D/2], the parameter D can be regarded as the width of the transition zone. We choose D = 100 km.

In the course of the simulation the interface is displaced away from y = 0 km (see next section). We denote the meridional interface position (where η = 0 s−1) for a given longitude and time as yf(x, t). For small displacements, the evolution of these perturbations can be computed using the linearized barotropic vorticity equation, which yields wavelike solutions (see, e.g., Vallis 2006, chapter 6, section 6.2.3).1 In the following we will summarize the well-known properties of these “edge waves,” which can be described as Rossby waves trapped on an interface.

Substituting a perturbation wave function of the form
i1520-0469-68-3-405-e4
(where ak = constant) into the linearized barotropic vorticity equation yields the dispersion relation
i1520-0469-68-3-405-e5
where ω is the angular frequency and k = 2π/λ is the wavenumber. Thus, the phase velocity
i1520-0469-68-3-405-e6
decreases linearly with increasing wavelength and approaches the jet velocity in the limit λ → 0. The group velocity ∂ω/∂k = u0 is constant. The frontal displacement yf giving rise to the perturbation streamfunction [Eq. (4)] is
i1520-0469-68-3-405-e7
with bk = 2kak/η0. Therefore the circulation caused by a frontal wave is proportional to the frontal displacement amplitude bk and the magnitude of the vorticity jump η0 and decays exponentially in meridional direction with an e-folding scale that is proportional to the wavelength [see Eq. (4)]. Thus, for a given frontal displacement amplitude, increasing the wavelength or increasing the jet speed will cause a stronger wave circulation.
Because of the finite resolution and the periodic boundary conditions in our model, arbitrary frontal waves can be decomposed into components associated with a discrete spectrum of wavenumbers k(m) = 2πm/Lx, so that the frontal displacement at zonal position xn = nLx/Nx can be written as
i1520-0469-68-3-405-e8
where m is the mode number and m the corresponding complex wave amplitude.

b. Tropical cyclone

To study the influence of the cyclone structure on the interaction with a jet we use several vortex types (Table 1; Fig. 2). The maximum tangential wind is approximately 40 m s−1 for all of the vortices, but the radius of maximum wind and the radial decay scale vary. The corresponding relative vorticity distributions differ, in particular in the amount of fluid with negative vorticity. In real tropical cyclones the diabatic heating in the eyewall leads to a continuous reduction of upper-level potential vorticity and the formation of an upper-level anticyclone. This process cannot be modeled in a barotropic model. However, by comparing results for the different vortex types we can obtain information on how the negative vorticity sheared from the vortex (a process that takes place in real tropical cyclones also) may become important in the scenarios considered.

The simplest vortex used, the GAUSS vortex, has a Gaussian radial vorticity profile
i1520-0469-68-3-405-e9
where s = r/r0. The SUD vortex from Smith et al. (1990) is defined by the tangential wind
i1520-0469-68-3-405-e10
The BROAD vortex from Jones (1995) has
i1520-0469-68-3-405-e11
Finally, a vortex that we will denote as ANTI consists of an SUD cyclone superimposed onto a broader, but weaker SUD anticyclone.

In the SUD and BROAD vortices the relative vorticity is negative at larger radii. This outer part “shields” the circulation that would be induced by the inner, positive vorticity part alone. For the SUD, BROAD, and ANTI vortices the initial integrated vorticity is very small. Consequently, the velocities induced far from the vortex center are small. For the GAUSS vortex the velocity profile falls off rather slowly—at large radius as 1/r (Fig. 2).

c. Vorticity decomposition

To distinguish the influence of various vorticity components on the wind field we decompose the absolute vorticity
i1520-0469-68-3-405-e12
into the basic-state vorticity η and the perturbation vorticity η′. The latter contains components related to the frontal waves ηf , the cyclone ηc, and the rest, consisting mainly of the anticyclones and filaments of negative vorticity ηa. The frontal wave component is defined as all of the vorticity between the interface and its undisturbed location so that ηf = ηξf with
i1520-0469-68-3-405-e13
where yf(x, t) is the front displacement. The cyclone component is ηc = (η′ − ηf)ξc, where
i1520-0469-68-3-405-e14
with r = and rc = 600 km. Here (Xc, Yc) are the coordinates of the cyclone center, which we define as the location where η′ − ηf is maximum. The coordinates are obtained by fitting a quadratic function as described in Smith et al. (1990). Finally, ηa = η′ − ηfηc. For each of these components i ϵ {f, c, a} we compute the corresponding streamfunction Ψi, fulfilling ΔΨi = ηi, and the velocity components ui = −∂Ψi/∂y and υi = ∂Ψi/∂x. This decomposition method ceases to work when the cyclone comes close to the interface and deforms it so strongly that η[x, yf(x)] = 0 no longer has a unique solution yf(x).

4. Cyclone–front interaction

We performed model runs for all combinations of the vortex types and front representations discussed above. A list of model runs with the relevant parameters is given in Table 2. Before we investigate the influence of the vortex and front parameters in detail, we will first give an overview of the basic processes that play a role in all of the model runs.

a. Basic evolution

The circulation of the cyclone displaces the interface between the two regions of constant absolute vorticity, generating a trough upstream and a ridge downstream of the cyclone (Fig. 3). The vorticity anomalies associated with the trough and the ridge induce circulations that contribute to the steering flow advecting the cyclone. If the vorticity step at the interface is positive (the setup considered in this study), the cyclone is advected northward (Fig. 3). An anticyclone would cause the inverse geometry with an upstream ridge and a downstream trough and would therefore be advected southward. The direction of the vortex motion does not depend on whether the vortex is located south or north of the front (i.e., a cyclone located south of the front is attracted by the front, whereas north of the front the cyclone is repelled).

In model run WG a Gaussian vortex is placed 1500 km south of a weak front with η0 = 10−5 s−1. At t = 2 days a weak trough is visible upstream of the cyclone and a more pronounced ridge is located downstream (Fig. 4a). The circulation caused by the growing frontal waves results in a northward acceleration of the cyclone (Fig. 5, bottom panel), which begins at the start of the calculation and is nearly constant for the first few days. When the cyclone is advected closer to the front, the zonal velocity of the basic-state flow at the cyclone center u(Xc, Yc) increases. Initially the zonal translation velocity of the cyclone is well determined by the basic-state flow at the cyclone center, Ucu(Xc, Yc). However, when the amplitude of the frontal wave increases, the circulation induced by the wave reduces the zonal velocity of the cyclone, compared to the basic-state flow (Fig. 5, upper panel). After 7 days the cyclone is close to the vorticity interface and has a zonal velocity of about 5 m s−1. This is considerably less than the jet speed of 15 m s−1.

The effect limiting the translation velocity of the cyclone is related to the shape of the front. Initially the frontal displacement caused by the vortex circulation is antisymmetric [yf(x) = −yf(−x)] and leads to a trough upstream and a ridge downstream of the cyclone, which are of equal amplitude (as sketched in Fig. 3). However, the meridional velocity component at the front caused by the ridge and trough vorticity anomalies is symmetric [υf(x) = υf(−x)]. Between the trough and the ridge axis the antisymmetric meridional flow component due to the cyclone and the symmetric component caused by the ridge and the trough are parallel downstream and antiparallel upstream of the cyclone, respectively (Fig. 6). Therefore, the ridge downstream of the cyclone grows much faster than the upstream trough (Fig. 4a). The resulting asymmetric vorticity distribution [similar to the stationary limit discussed in Bell (1990)] leads to a negative zonal velocity component uf(Xc, Yc) at the cyclone center, so that the zonal translation velocity remains well below the basic-state velocity at the cyclone position.

When the cyclone reaches the front after about 7 days, it moves into the downstream ridge and starts to “wind up” the interface (see Fig. 4a for t > 7 days). The cyclone moves toward the center of the anticyclonic circulation. Close to the center of the ridge the flow caused by the ridge anomaly is weak and the cyclone is not advected further northward. The continuous decline of the meridional cyclone translation velocity during the winding up process is visible in Fig. 5 from t ≈ 5 days to t ≈ 10 days. At t ≈ 10 days the meridional translation velocity has become negative, which is caused by the circulation associated with the downstream trough located between x ≈ 5000 and 9000 km (Fig. 4a). In the subsequent evolution, the cyclone closes in on this slow-moving trough. The latter becomes narrower and weaker under the influence of the cyclone circulation, which moves the vorticity interface northward in the western part of the trough. The cyclone circulation also strengthens the negative vorticity anomaly directly downstream of the cyclone and creates a positive anomaly upstream of the cyclone (Fig. 4a). These processes result in a northward advection of the cyclone from t ≈ 12 days. The zonal translation velocity of the cyclone declines for t > 15 days as the cyclone moves slowly further northward into regions with lower basic-state velocity.

The frontal waves excited by the cyclone spread downstream with a group speed equal to the jet speed (see section 3a). The troughs and ridges, however, move slowly. Only the waves that are generated when the cyclone is close to the vorticity interface move faster and are characterized by smaller wavelengths [in accordance with Eq. (6)].

The basic processes that determine the cyclone–jet interaction in run WG are present in all of the runs. However, their effects vary depending on the cyclone structure and the representation of the front. Additional processes can play an important role also, as described in the following.

A comparison of model run WG with model runs WB, WA, and WS, which have the same front parameters but different vortices, reveals the influence of the vortex structure on the cyclone–front interaction. The BROAD, ANTI, and SUD vortices (in runs WB, WA, and WS, respectively) all have a ring of negative vorticity around the cyclonic core (Fig. 2). Because of the shear of the basic-state flow, the outer part of the vortex containing fluid with negative relative vorticity is partially removed from the vortex and forms two anticyclones that move away from the cyclonic vortex (t ≤ 4 days in Figs. 4b–d). After 4 days the anticyclones have moved more than 1000 km away from the cyclone and the integrated vorticity of the cyclone remains nearly constant in the subsequent evolution. As will be discussed later, the anticyclones can influence significantly the vortex motion in the initial phase (section 4b) and the excitation of frontal waves (section 4d).

As the SUD vortex in model run WS is more compact than the BROAD vortex in run WB (Fig. 2), the wave excitation and the advection of the cyclone toward the front proceeds more slowly in run WS. The cyclone arrives at y = 0 km about 5 days later in run WS and the frontal waves are weaker (Figs. 4b,d). The advection of the cyclone toward the front is delayed even more in model run WA with the ANTI vortex. In this case the front intersects initially with the anticyclonic part of the velocity profile (see Fig. 2). Thus, a frontal wave with a trough downstream and a (very weak) ridge upstream of the cyclone’s longitude is generated (Fig. 4c; t ≤ 4 days) and the cyclone is repelled from the front. After about 4 days, when a significant part of the fluid with negative vorticity has been removed from the cyclone and has formed two anticyclones, the circulation at the front and near the zonal position of the cyclone has become cyclonic. Therefore run WA develops a trough upstream and a ridge downstream of the cyclone like the other models (Fig. 4; t ≥ 7 days) and the cyclone is advected toward the front.

Up to now only f-plane model runs with a relatively weak (u0 = 15 m s−1) jets have been discussed. The f- and β-plane runs with stronger jets (u0 = 40 m s−1 or u0 = 60 m s−1 in runs starting with F, S, and B; see Table 2) show basically the same behavior as the runs with weak jets. However, because of the stronger meridional shear of the basic state, the formation of the anticyclones occurs earlier. After 2 days the anticyclones are already well separated from the cyclone. Furthermore, the frontal wave spectrum and the interaction strength vary with the jet speed (see section 4c).

b. Wave excitation and vortex evolution in the initial phase

The different vortex types cause differences in the wave excitation and cyclone motion from the start of the calculations. The wavelength and amplitude of the frontal waves excited by the vortex circulation depend on the separation of vortex and front |Yc| and the vortex structure. The meridional velocity υ(x, 0) at the front associated with a cyclone with tangential velocity profile υT(r) located at (0, Yc) is given by
i1520-0469-68-3-405-e15
For a point vortex with velocity profile ∝ 1/r, this yields (employing a sine transformation), from which it can be seen that all large wavelengths are excited with similar strength and that wavelengths smaller than Yc are suppressed.
A frontal deformation
i1520-0469-68-3-405-e16
with wavenumber k and time-dependent amplitude bk, and phase ϕk has an associated flow that is described by a streamfunction
i1520-0469-68-3-405-e17
where ak = bkη0/(2k). The growth of this wave component due to a stationary point vortex with circulation Γc is given by
i1520-0469-68-3-405-e18
for the streamfunction (Schwierz et al. 2004), and thus
i1520-0469-68-3-405-e19
for the frontal displacement. Here ϕk = k(XcXk) is the phase of the wave relative to the vortex, where Xk is defined as the first location upstream of the cyclone (XkXc), for which yf,k = 0 and ∂yf,k/∂x > 0. Thus ϕk = 0 corresponds to the case of fastest northward advection shown in Fig. 3. As long as wave and cyclone stay in phase (ϕk = constant) and the cyclone–front distance varies only slowly, Eq. (18) predicts a linear growth of ak. The phases and amplitudes can be computed from the complex amplitudes of Eq. (8) (which can be determined from the model results) using the relation m = (½)bk exp(k), where k = 2πm/Lx.

For the Gaussian vortex the vorticity is concentrated in a region that is much smaller than the initial distance to the interface and the vorticity profile remains nearly constant throughout the calculations. Therefore the results for a point vortex should approximately describe the behavior of the Gaussian vortex, as long as the vortex does not come too close to the interface. For each mode the amplitude should grow linearly, as long as the phases ϕk vary only slowly [see Eq. (19)]. The highest growth rates are expected for the modes with lowest wavenumbers, due to the factor exp(−k|Yc|) in Eq. (19).

The Gaussian vortex in run WG is able to excite frontal waves with large wavelengths right from the start of the model run (see m = 1 mode in Fig. 7). The wave amplitude grows nearly linearly for all modes in the first few days and for higher mode index m, the growth rate is lower (Fig. 7). The induced meridional flow at the vortex center due to the frontal waves, υf(Xc, Yc) (see section 3c), increases linearly with time also, and the cyclone is advected toward the front with linearly increasing speed (Fig. 5).

In contrast to the Gaussian vortex, the other vortices have more complex, evolving vorticity structures that lead to additional effects. For the SUD, BROAD, and ANTI vortices, the integrated relative vorticity is zero and thus the circulation is restricted to the area in which the vorticity is nonnegligible. Owing to the limited range of the circulation these vortices are initially not able to excite the modes with the lowest wavenumbers. Thus the m = 1 mode in runs WB, WA, and WS is suppressed, compared to run WG (Fig. 7).

The circulation of the SUD vortex at the front is negligible at t = 0 days, and thus the growth of higher modes is suppressed initially also. However, when a part of the fluid with negative vorticity has been sheared from the vortex, the shielding effect of this fluid vanishes gradually and the vortex is able to influence larger scales. After 4 days the circulation of the SUD vortex at the front exceeds locally that of a GAUSS vortex and influences a zonal range of several 1000 km (Fig. 8). This range is not sufficiently large to allow for a significant growth of the m = 1 mode (with a wavelength of 27 000 km) but permits higher modes to grow with a similar growth rate as in run WG (see mode m = 4 in Fig. 7).

The BROAD and ANTI vortices cause a circulation at the front at t = 0 days because their vorticity distributions extend beyond y = 0 km. For these vortices the zonal region of influence increases as the anticyclones form, in contrast to the GAUSS vortex, whose circulation remains almost unchanged (Fig. 8). Thus higher modes are able to grow in runs WB and WA from t = 0 days (Fig. 7).

The initial wave growth rates are identical in the runs with weaker and stronger jets for the same vortex type, since for negligible changes in the phase ϕk [Eq. (19)] only the cyclone circulation is important for this process. In contrast, the initial acceleration of the cyclone depends on the strength of the frontal vorticity anomalies, and thus on the vorticity step η0. Therefore, the initial acceleration is stronger for runs with faster jets, for instance dVc/dt ≈ 1 m s−2 in run WG and ≈2.7 m s−2 in run FG.

In the β-plane runs the excitation of frontal waves in the initial phase proceeds in a very similar manner to the f-plane cases. However, the initial acceleration of the cyclone, which occurs on the f plane only because of the circulation associated with the growing frontal waves, is dominated by a different effect on the β plane. The varying meridional shear in the β-plane models leads to an asymmetric distribution of negative vorticity around the cyclone. The anticyclone that forms downstream becomes more pronounced than the one upstream of the cyclone. Furthermore, the cyclone is not located on the line connecting the two anticyclones (see run BA in Fig. 9). Therefore, the circulations of the two anticyclones do not cancel but cause a flow with a northward component at the cyclone center. A vorticity decomposition (see section 3c) for run BA shows that in the first few days this flow is stronger than the circulation caused by the frontal waves. Therefore the cyclone is advected toward the front from the initial time, in contrast to run WA. The largest effect is obtained if the vortex is initially located at the velocity minimum. Here the fluid with negative vorticity is only transported in the downstream direction, which results in a one-sided negative vorticity distribution dominated by a single anticyclone (run BA2; Fig. 9). Because of the stronger acceleration by the single anticyclone, the cyclone in run BA2 reaches the front earlier than the one in run BA (Table 2).

c. Resonant interaction

For the discussion of the initial wave excitation the factor cos(ϕk) in Eq. (19) can be neglected and the initial acceleration is proportional to the vorticity jump η0. However, there is no such simple relation between η0 (or the jet speed) and the interaction strength at later times. An indicator for the strength of the interaction is given by the time it takes the cyclone to come close to the front,2
i1520-0469-68-3-405-e20
where we choose Y0 = −200 km. Figure 10 shows that even for the simplest cases (the f-plane runs with a Gaussian vortex) the relation between interaction strength and η0 is nonlinear. In runs FG and WG, which differ by a factor in η0 (and therefore also in the jet speed u0), the cyclone reaches the front at almost the same time, after about 7 days, whereas in run SG with an even higher jet speed the front is reached after 5 days. To analyze this behavior, and the influence of the jet profile on the cyclone–jet interaction in general, it is necessary to investigate the impact of a varying phase ϕk between cyclone and jet.
Unless the zonal translation velocity Uc of the cyclone is higher than u0, a resonant frontal wave exists—a wave propagating with a phase speed c = u0η0λ/2π that matches Uc (Schwierz et al. 2004). The wavelength of this resonant wave is given as
i1520-0469-68-3-405-e21
Assuming that the cyclone is advected with the background flow, Ucu(Yc) yields
i1520-0469-68-3-405-e22
Thus, in the f-plane runs the resonant wavelength is proportional to the cyclone–front separation and on the β plane the resonant wavelength is smaller (for the same cyclone–front separation and vorticity jump). For a constant cyclone translation velocity, the time scale on which a phase shift π develops between a wave of wavenumber m (with wavelength λm and phase speed cm) and the cyclone is given by
i1520-0469-68-3-405-e23
This phase shift corresponds to a transition from maximum northward to maximum southward advection of the cyclone by the wave circulation. Here we neglect the fact that the excitation of waves due to the cyclone can change the phase.

From Eq. (6) it follows that ∂c/∂λη0; that is, for larger vorticity jumps (and thus faster jets) the phase velocity changes faster with the wavelength. Thus, for the wave modes that are not in resonance with the cyclone, τm decreases with increasing jet speed. For example, the phase speed of the m = 2 mode is about 3 times larger for the F runs than for the W runs so that for a cyclone at rest (Uc = 0) the time scale is τ2 ≈ 12 days in the W runs (slower jet) but only τ2 ≈ 4 days in the F runs (faster jet).

Returning to the question as to why the interaction is of similar strength in runs WG and FG, although η0 is much higher in run FG, it is instructive to consider the time evolution of frontal wave amplitudes and phases. In run WG the modes with the largest wavelengths are continuously excited by the cyclone and contribute to the cyclone acceleration until tf ≈ 7 days (Fig. 11), when the cyclone reaches the front.3 The time scales τm are larger than or of the same order as tf for the modes m < 10, so that the corresponding phases ϕm vary only slowly and stay within the range −π/2 < ϕk < π/2 (Fig. 11). This leads to a northward acceleration of the cyclone and a further excitation of the wave component. Thus, the resonance condition does not play an important role in this case. Both resonant and nonresonant modes are able to grow for t < tf .

In run FG, however, τm is much smaller for nonresonant modes. The modes m = 1, 2 can grow only for 2–3 days, until their phase shift with respect to the cyclone becomes so large that they are damped by the cyclone circulation (Fig. 11). For instance, the mode m = 1 is able to grow only until t ≈ 2 days when its phase becomes larger than ϕ1 = π/2. Subsequently, the m = 1 wave is damped until its amplitude is zero at t ≈ 4 days and another growth period starts. Because the lowest modes cannot be excited effectively, the displacement of the front is considerably less pronounced in run FG than in run WG. Only the amplitudes of the modes m = 3 and 4, which are close to resonance, grow continuously in run FG in the first 5 days and reach amplitudes similar to those in run WG.

In run WG the modes m = 1 and 2 provide the largest contribution to the cyclone advection by the frontal wave circulation [due to the factor exp(−|ky|) in Eq. (17)]. These modes are missing in run FG, which should lead to a weaker interaction compared to run WG. However, the vorticity jump η0 is larger in run FG, so that for the same interface displacement a stronger vorticity anomaly is generated in run FG than in run WG. For the specific setup considered here, the additional modes contributing to the cyclone advection in run WG have roughly the same effect on tf as the larger value of η0 in run FG. In run SG η0 is even higher and, consequently, the waves of modes m = 1 and 2 lose phase coherence with the cyclone even earlier. However, as these modes remain only weakly excited in both FG and SG, only the higher value of η0 in run SG makes a difference and causes a stronger meridional advection of the cyclone, so that it reaches the front earlier (Fig. 10). Similarly, for a run with a value of η0 lower than in run WG the interaction would become weaker, because in both cases the same modes (the lowest modes) would dominate the meridional cyclone advection.

In the β-plane run BG tf ≈ 10 days (Fig. 10) so that the interaction is weaker than in the f-plane runs WG, FG (which has the same u0), and SG (which has the same η0). As in runs FG and SG, only the modes that are close to resonance can be continuously excited in run BG. However, because of the different shape of the jet velocity profile the resonant wavelength is smaller in the β-plane runs [Eq. (22)] and therefore it is the m = 5 mode (and not m = 3 and 4) that attains the largest amplitude in run BG, while the other modes lose phase coherence with the cyclone (Fig. 11). According to Eqs. (17) and (18), both the excitation of waves and the frontal wave circulation are weaker for larger wavenumbers, which explains the weak interaction in run BG.

In runs FG, SG, and BG the spectrum of frontal waves excited in the course of the cyclone–jet interaction is dominated by the modes that are in resonance for the longest time—the ones with wavelengths close to the initial resonant wavelength. In runs FG and SG these are the modes m = 3 and m = 4, and in run BG the modes m = 5 and m = 6. (Fig. 12). When the cyclone is advected toward the front, it is accelerated in the zonal direction and the resonant wavelength decreases. However, in the first few days the zonal velocity of the cyclone and thus also the resonant wavelength changes only slowly. When the cyclone comes closer to the front, the resonant wavelength varies much more quickly, leaving less time for the excitation of modes with smaller wavelengths.4 When the cyclone crosses the interface, it stays close to y = 0 km for several days and moves with a zonal translation speed of about 50%–70% of the jet speed (see, e.g., Fig. 5). The frontal waves caused by the cyclone during this period are visible as a secondary peak around m = 9–10 in the deformation spectrum of runs FG and BG (Fig. 12).

Up to now only the resonant interaction of Gaussian vortices with jets was considered. The resonance condition plays an important role for the other vortex types also. However, the negative vorticity shielding of these vortices causes additional effects. For instance, Fig. 13 shows that, in contrast to the runs with Gaussian vortices (Fig. 10), the interaction for BROAD vortices is considerably weaker for the W jet than for the F jet. The reason for this is that, unlike in run WG, the mode m = 1 cannot be excited effectively in run WB because of the shielding effect and thus cannot contribute to the cyclone advection. Therefore, the higher value of η0 in FB is the decisive factor. In the β-plane run BB tf has about the same value as in the f-plane run FB. This is also in contrast to the Gaussian runs, where the interaction in BG is much weaker than in FG because of the smaller resonant wavelength. In this case the initial acceleration of the cyclone by the asymmetric distribution of negative vorticity fluid caused by the varying shear is responsible for the difference (see section 4b).

Finally, we consider how the results presented above depend on the size of the numerical domain. The meridional domain size must be sufficiently large to account for the slow meridional decay of the wave streamfunction [Eq. (17)] for large wavelengths. For Ly = 18 000 km the numerical phase velocity of the m = 1 mode agrees well with the theoretical value [Eq. (6)]. For Ly = 9000 km the numerical phase velocity for m = 1 is 30% higher, whereas the phase velocities of the modes m > 1 are still well reproduced. This indicates that Ly = 9000 km can be acceptable for model runs, in which the m = 1 mode has only a weak influence. For instance, model run BG, in which the excitation of the m = 1 mode is suppressed due to the resonance condition, shows only minor differences when Ly is reduced from 18 000 to 9000 km—after 8 days the average difference in the front positions yf is less than 30 km. In model runs with BROAD, SUD, or ANTI vortices the excitation of the m = 1 mode is hampered even further by the influence of the anticyclones. Therefore, it seems justified to use Ly = 9000 km in these cases in order to reduce the computational cost and to adopt Ly = 18 000 km only for the runs with the Gauss vortex.

When the zonal domain size is increased, wave modes with larger wavelengths can be excited. For cases in which the mode with the largest wavelength plays an important role, this could change the results significantly. However, this should be the case only in model run WG and possibly in model run FG. In all other model runs the excitation of the modes with the largest wavelengths is suppressed because of the resonance condition (τm is small for small m, if cm < Uc) and/or the presence of anticyclones. Therefore, the large wavelength modes have no significant influence in these runs. Increasing Lx in these model runs should not lead to significant changes in the frontal waves or the cyclone track.

d. Wave excitation by anticyclones

As the integrated vorticity in the initial state is zero for the SUD, BROAD, and ANTI vortices, the integrated vorticity of the two anticyclones is the same size but of opposite sign to that of the cyclone. Therefore, a significant impact of the anticyclones on the frontal wave excitation should be possible.

The basic evolution of the two anticyclones is similar for all of the runs. Here run WB (Fig. 4b) is used as an example. Initially, because of the circulation of the cyclone, the upstream anticyclone moves away from the front and is left behind in the slow or even eastward-moving basic-state flow. The downstream anticyclone is brought closer to the front by the cyclone circulation and is then advected away from the cyclone by the faster zonal flow near the front. Therefore, the cyclone-induced flow moving the anticyclone northward becomes weaker. On the other hand, the anticyclone induces frontal waves, which cause a southward-directed circulation at the anticyclone center. When the amplitude of these waves increases, the anticyclone eventually moves away from the front. Thereby the zonal translation velocity of the anticyclone is reduced and becomes similar to that of the cyclone. The cyclone–anticyclone separation is approximately constant for several days and the meridional positions of cyclone and anticyclone are similar (see Fig. 4b, t = 7 days). Finally, the anticyclone moves further away from the front and is overtaken by the cyclone.

The upstream anticyclone never comes close to the front and has no significant influence on the frontal waves. However, the downstream anticyclone is closer to the front than the cyclone for several days (in run WB for t < 10 days). During this period both cyclone and anticyclone contribute to the growth of the ridge between them. Furthermore, the anticyclone is instrumental in the formation of the trough downstream of this ridge (as has been observed also for more realistic models; see Riemer et al. 2008). Without the anticyclone (e.g., in run WG) the trough is formed only due to the propagation of the frontal wave that has been excited further upstream by the cyclone. The trough is rather weak in run WG at t = 7 days but quite pronounced in run WB at t = 10 days, at the times the cyclones reach the front in both models (Figs. 4a,b).

The effect of the anticyclone on the wave excitation is demonstrated more clearly by a comparison of runs WB and WB-m. Run WB-m is started from the same initial conditions as run WB, but after 3 days the downstream anticyclone is removed.5 At this time the anticyclone has moved sufficiently far from the cyclone to be removed without affecting the cyclone and the frontal waves are still rather weak. In the subsequent evolution the meridional extents of the first trough forming downstream of the cyclone and the next ridge are considerably larger in run WB than in WB-m (Fig. 14). Similar phases of joint wave excitation by cyclone and anticyclone can be found in other model runs containing anticyclones (see, e.g., runs WS and WA in Fig. 4, t = 7 days), and also in runs with faster jets.

5. Summary

We investigated the interaction of a tropical cyclone and a tropopause front using an idealized model. In a barotropic framework a vorticity step is used to model the front, giving rise to a jetlike velocity profile. The cyclone circulation excites Rossby waves that are trapped on the vorticity interface and can propagate only in the zonal direction. The displacement of the vorticity interface in these frontal waves causes vorticity anomalies that have an associated flow that influences the motion of the cyclone. A northward or southward advection of the cyclone can result, depending on its position relative to the wave troughs and ridges.

The circulation of a cyclone located south of the front creates a trough upstream and a ridge downstream of its location—a configuration that leads to a northward advection of the cyclone. The combined effect of the cyclone circulation and the flow associated with the vorticity anomalies at the front causes the ridge to grow much faster than the trough. The resulting asymmetric frontal deformation leads to an easterly flow component at the cyclone center that reduces the translation speed of the cyclone, compared to the background flow. Thus, even when a cyclone crosses the vorticity interface, its translation speed remains well below the jet speed.

Several different representations of a cyclone were used, including vortices with a core of positive vorticity surrounded by a ring of negative vorticity (mimicking the anticyclonic outflow in real tropical cyclones). The ring is partly sheared from the core by the background flow and forms two anticyclones. If the background shear is nonlinear, the cyclone is advected northward during this process. The negative vorticity of the anticyclones shields the circulation that would be induced by the remaining vortex, which contains mostly positive vorticity. Because of this shielding effect, the excitation of frontal waves with large wavelengths is suppressed. In the subsequent evolution one of the anticyclones is advected closer to the front. This can enhance the formation of troughs and ridges downstream of the cyclone significantly.

Resonant waves (i.e., waves with a phase speed close to the zonal translation speed of the cyclone) turn out to be of particular importance. A continuous acceleration of the cyclone in one direction and a continuous amplification or damping of the wave are possible only for these waves. In general, the spectrum of frontal waves excited by the cyclone is dominated by those wave modes that are initially in resonance with the cyclone, and therefore these modes have the strongest influence on the cyclone motion. For slow jets the resonance condition is less restricting and nonresonant large-wavelength modes contribute significantly to the interaction. For higher jet speed the excitation of these modes is suppressed. Nevertheless, the interaction strength is enhanced with increasing jet speed because the frontal vorticity anomalies become stronger.

In this idealized study we have identified some of the fundamental mechanisms that determine the interaction of a tropical cyclone with a straight front. A fact known to be important for the extratropical transition is the location of the tropical cyclone relative to midlatitude troughs and ridges. Thus, in Part II we extend this study by including frontal waves in the initial conditions.

Acknowledgments

This study was carried out as part of the German Research Foundation (DFG) Priority Program SPP 1276 “MetStröm: Multiple Scales in Fluid Mechanics and Meteorology.” We acknowledge the support of Tobias Hahn, Björn Rocker, and Vincent Heuveline of the Institute of Applied and Numerical Mathematics, Karlsruhe Institute of Technology, for assistance in implementing the model on graphics hardware. This study benefitted from discussions with Huw Davies. We are grateful to Pat Harr, Chris Rozoff, and an anonymous reviewer for their helpful comments on a previous version of the manuscript.

REFERENCES

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Fig. 1.
Fig. 1.

Absolute vorticity (thin) and zonal velocity (thick) for the basic state with η0 = 4 × 10−5 s−1 on the β plane (solid) and the basic states with η0 = 10−5 s−1 (dotted) and (dashed) on the f plane. The thick dots mark the initial vortex positions for the model runs of Table 2.

Citation: Journal of the Atmospheric Sciences 68, 3; 10.1175/2010JAS3482.1

Fig. 2.
Fig. 2.

Profiles of the (top) vorticity and (bottom) tangential velocity for the SUB, BROAD, ANTI, and GAUSS vortices.

Citation: Journal of the Atmospheric Sciences 68, 3; 10.1175/2010JAS3482.1

Fig. 3.
Fig. 3.

The circulation induced by a cyclone (white circle) displaces the interface between an area of negative (dark gray) and positive (light gray) absolute vorticity. Two vorticity anomalies are created at the interface with associated circulations (black circles and arrows) that advect the cyclone toward the front.

Citation: Journal of the Atmospheric Sciences 68, 3; 10.1175/2010JAS3482.1

Fig. 4.
Fig. 4.

Vorticity distribution 2, 4, 7, 10, 15, and 20 days after the start of the calculation for model runs (a) WG, (b) WB, (c) WA, and (d) WS. The black dots mark the cyclone position after every full day.

Citation: Journal of the Atmospheric Sciences 68, 3; 10.1175/2010JAS3482.1

Fig. 5.
Fig. 5.

Evolution of the cyclone translation velocity (thick black line) in model run WG (u0 = 15 m s−1), showing the (top) zonal and (bottom) meridional components. In addition, contributions to the velocity at the cyclone center due to the basic state (dashed) and the frontal waves (dashed–dotted) are displayed for the time span in which these quantities can be determined reliably (see section 3c).

Citation: Journal of the Atmospheric Sciences 68, 3; 10.1175/2010JAS3482.1

Fig. 6.
Fig. 6.

Meridional velocity components due to the cyclone (dashed) and the frontal wave (dashed–dotted), and their sum (solid), in run WG at (x, y) = [xXc, yf(x)] for t = 24 h.

Citation: Journal of the Atmospheric Sciences 68, 3; 10.1175/2010JAS3482.1

Fig. 7.
Fig. 7.

Time evolution of |bk(m)| for modes (left) m = 1 and (right) m = 4 of the model runs WG (solid), WB (dashed), WS (dotted), and WA (dashed–dotted).

Citation: Journal of the Atmospheric Sciences 68, 3; 10.1175/2010JAS3482.1

Fig. 8.
Fig. 8.

Meridional velocity at y = 0 km induced by cyclone and anticyclones υc(x, 0) + υa(x, 0) (top) in the initial state and (bottom) after 4 days for the runs WA, WB, WS, and WG. Note the different scales along the velocity axis.

Citation: Journal of the Atmospheric Sciences 68, 3; 10.1175/2010JAS3482.1

Fig. 9.
Fig. 9.

Vorticity distributions after 4 days for runs WA (linear shear), BA (parabolic shear), and BA2 (parabolic shear, initial vortex position at the velocity minimum).

Citation: Journal of the Atmospheric Sciences 68, 3; 10.1175/2010JAS3482.1

Fig. 10.
Fig. 10.

Time evolution of the meridional cyclone position for model runs BG, WG, FG, and SG.

Citation: Journal of the Atmospheric Sciences 68, 3; 10.1175/2010JAS3482.1

Fig. 11.
Fig. 11.

(top) Evolution of |m(t)| [see Eq. (8)] for modes m = 1, … , 5 during the time period before the cyclone comes close to the front [t < tf; see Eq. (20)] in runs WG, FG, and BG. For each time, the mode with the wavelength closest to the resonant wavelength is marked with a black dot. (bottom) Time evolution of ϕk(m)(t) for models runs WG, FG, and BG. Parts for which |bk(m)(t)| < 10 km are omitted. The dotted lines enclose the region −π/2 < ϕk(m) < π/2 in which m is able to grow. For ϕk(m) < −π/2 or ϕk(m) > π/2 the wave is damped.

Citation: Journal of the Atmospheric Sciences 68, 3; 10.1175/2010JAS3482.1

Fig. 12.
Fig. 12.

Frontal deformation spectrum for runs BG, WG, and FG. The amplitudes |m| are averaged over 15 < t < 20 days.

Citation: Journal of the Atmospheric Sciences 68, 3; 10.1175/2010JAS3482.1

Fig. 13.
Fig. 13.

Time evolution of the meridional cyclone position for model runs BB, WB, and FB.

Citation: Journal of the Atmospheric Sciences 68, 3; 10.1175/2010JAS3482.1

Fig. 14.
Fig. 14.

Frontal position in model runs WB (solid) and WB-m (dashed) after 12 days.

Citation: Journal of the Atmospheric Sciences 68, 3; 10.1175/2010JAS3482.1

Table 1.

Vortex parameters.

Table 1.
Table 2.

Parameters for the runs discussed in section 4 (see section 4d for the difference between runs WB and WB-m). The time tf required by the cyclone to reach the front is given in the last column.

Table 2.

1

The basic strategy is to find first general solutions for the perturbation streamfunction with ΔΨ = 0 in the two regions y > 0 and y < 0. Then the parameters of these two solutions are chosen such that pressure is continuous and the mass flux across the interface is zero, which leads to Eqs. (4) and (5).

2

It is not trivial to determine the exact time the cyclone crosses the vorticity interface, as the latter is strongly deformed during this process.

3

For the m = 1 mode this holds only for t < 5 days.

4

In this phase the asymmetric frontal displacement reduces the zonal cyclone speed relative to the basic-state flow (see section 4a), which prevents an even more rapid decrease of the resonant wavelength.

5

This is accomplished by setting η = −η0 in a region encompassing the anticyclone and recalculating the streamfunction.

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