The local group velocity is defined as 5cg,x≡∂Ω∂k=c+2β∗k2K4cg,y≡∂Ω∂l=2β∗klK4, where c=ω/k denotes the zonal phase speed. Correspondingly, a ray is defined as the trajectory traced by a fluid parcel advected with the group velocity, satisfying 6DgxDT≡∂Ω∂k=cg,x,DgyDT≡∂Ω∂l=cg,y, where DgDT=∂∂T+cg⋅∇ represents material derivative following the group velocity vector cg. Additionally, the following kinematic relations hold 7DgkDT≡-∂Ω∂X=0,DglDT≡-∂Ω∂Y,DgωDT≡∂Ω∂T=0. Note that ray tracing theory essentially describes the trajectory of the wave packet for an almost-plane wave with given initial zonal and meridional wavenumbers and specified initial location. However, as Harnik (2002) suggested, it cannot capture nonlocal effects such as reflection, tunneling, and damping because they occur when the characteristic wavelength is on the order of the meridional scales of the wave guide and the evanescent regions. This limitation of ray tracing may be addressed by defining a wave activity packet as a component of the wave that propagates at the wave activity velocity, where this velocity is defined as the Eliassen–Palm flux divided by the wave activity density.

As Lighthill (2001) suggested, a wave packet evolving in position and wavenumber following Eqs. () and () may be interpreted as a “quasi–particle” whose energy–momentum relationship at every spatial point precisely mirrors the local frequency–wavenumber relationship (i.e., the local dispersion relation). This equivalence implies that wave packets exhibit particle–like behavior, propagating along ray trajectories defined by the group velocity. Consequently, one can trace a wave packet's evolution from one location to another using Eqs. () and () without solving the full wave equations (Vallis, 2017). Ray tracing proves invaluable for two complementary reasons: First, it provides critical information on the spatial distribution of the wavenumber vector; second, it demonstrates that wave energy transport occurs along these ray trajectories (Lighthill, 2001).

Equation () demonstrates that zonal wavenumber (k) and frequency (ω) are conserved along a ray trajectory, as the dispersion relation does not explicitly depend on X and T. In contrast, meridional wavenumber (l) evolves dynamically along the ray path, leading to two critical meridional locations for ray propagation: a turning point (y=yt) where l=0 and meridional group velocity vanishes (cg,y=0) , halting southward/northward propagation; a critical point (y=yc) , characterized by divergent meridional wavenumber (l2→∞) where group velocity aligns with zonal phase speed (cg=ci) . Notably, the WKB solutions (Eqs. and ) break down near turning points, where amplitude exhibit Airy function behavior in the meridional direction (Hoskins and Karoly, 1981). To reconcile this, asymptotic matching techniques employ a “patching” wave function in the turning point neighborhood, connecting the WKB solution in the “classical” region (l2>ε where ε≪1) with the Airy function solution (e.g., Griffiths and Schroeter, 2018). Post–reflection from the turning point, the WKB approximation regains validity, enabling continued application of ray tracing beyond the critical locus.

Previous studies (e.g., Li et al., 2021a; Yang and Hoskins, 1996) have categorized three distinct propagation regimes for Rossby wave packets. The evolution–dispersion (ED) regime, bounded between a single turning point and a critical point, typically exhibits initial propagation toward the turning point, followed by reflection and asymptotic approach to the critical point. Within this regime, Li et al. (2021a) observed potential transient amplification of both wave energy and amplitude during the turning point approach. The wave guide (WG) regime, confined between two turning points, supports quasi–periodic ray reflections that enable long–distance propagation with minimal meridional wavenumber variation. Note that WG regime predicted by ray tracing theory may be a poor predictor for fairly narrow and strong jets where WKB approximation becomes invalid (Wirth, 2020). In contrast, the bidirectional dissipation (BD) regime, spanning dual critical points, drives unidirectional ray propagation toward any a critical point without reflection.

When the meridional gradient of background absolute vorticity vanishes (β∗=0, a condition corresponding to the Rayleigh–Kuo necessary instability criterion) at an inflection point (y=yi) , the dispersion relation Eq. () simplifies to ω=u‾k, rendering meridional wavenumber mathematically indeterminate. However, by applying the L'Hospital's rule to resolve this indeterminate form, we derive a finite critical total wavenumber Ki (meridional shear of the basic flow is generally not set to zero at the inflection point, that is, du‾/dyy=yi≠0) satisfying 8Ki2≡lim⁡y→yiβ∗u‾-c=lim⁡y→yidβ∗/dydu‾/dy=-d3u‾/dy3y=yidu‾/dyy=yi<∞, where Ki>k ensures real meridional wavenumber solutions (li=±Ki2-k2) and physically admissible wave–like structures. At y=yi, zonal group velocity converges to zonal phase speed (cg,x=c) while meridional group velocity vanishes identically (cg,y=0) , confining ray trajectories to purely zonal propagation along the inflection point.

Note that a fundamental discrepancy exists between the propagation regimes of ray tracing theory and the wave geometry framework of overreflection theory, despite their shared dependence on critical and inflection points. In overreflection dynamics, incident waves can propagate through the evanescent region (where the index of refraction is negative) to get overreflection (e.g., Lindzen and Tung, 1978; Lindzen, 1988; Harnik and Heifetz, 2007). However, ray tracing theory predicts that rays can never enter such an evanescent region no matter whether the inflection point exists or not. Furthermore, rays can arrive at the critical point in finite time only if the critical point coincides with the inflection point. Pedlosky (1987) elucidates a key constraint: marginally stable waves, adjacent to unstable waves, must exhibit either vanishing Reynolds stress jumps or their self–canceling summation. This condition holds unless the meridional gradient of absolute vorticity vanishes at yc, that is, β∗yc=0.

Taking the first–order asymptotic approximation, the wave action conservation equation is expressed as: 9∂A∂T+∇⋅cgA=0, where A≡E/ω′ denotes wave action density, ω′=ω-u‾k is intrinsic frequency, and the wave energy density E is defined as 10E=14K2Ψ02. Along ray trajectories, Eq. () becomes 111ADgADT=-∇⋅cg. This implies that the divergence of the group velocity (∇⋅cg) serves as the sole source term governing variations in wave action density. However, direct evaluation of ∇⋅cg using group velocity expression (Eq. ) remains impracticable as determining the partial derivatives in ∇⋅cg requires knowledge of neighboring ray solutions (Lighthill, 2001).

Precise diagnostics of wave packet energy require solving the wave action conservation equation (Eq. or Eq. ). In this framework, a wave packet propagating along a ray is conceptualized as a “quasi–particle” (e.g., Lighthill, 2001), where the spatial extent of this “quasi–particle” exceeds the wavelength scale while remaining negligible compared to the slowly varying background flow. Consequently, when viewed at the basic flow scale, the wave packet manifests as a dimensionless point (e.g., Bretherton and Garrett, 1968). Therefore, while ray tracing theory accurately describes the wave packet's path, it cannot resolve the intra–packet wave structure (e.g., Vallis, 2017). This scale separation implies that the divergence of the group velocity vector depends exclusively on variations in its magnitude – owing to no lateral energy fluxes (Li et al., 2021a) – and can be expressed as 12∇⋅cg=lim⁡δT→0cgT+δT-cgTcgδT=1cgDgcgDT=Dgln⁡cgDT, where T denotes time, δT is a small time interval, and cg=cg,x2+cg,y2 represents the magnitude of the group velocity. It is critical to emphasize that this approximation (Eq. ) is derived at the basic flow scale, explicitly neglecting the wave packet's finite spatial extent.

Substituting Eq. () into Eq. () yields 13AtA0=cg0cgt, where A0 and cg0 denote the initial wave action density and initial group velocity magnitude, respectively. This relation reveals that the rate of change in wave action density is inversely proportional to the rate of group velocity magnitude variation along the ray trajectory. By invoking the definition of wave action density, we immediately derive 14EtE0=ω′tω′0cg0cgt, where E0 and ω′0 represent the initial wave energy density and initial intrinsic frequency, respectively. This equation further demonstrates that the rate of change in wave energy density scales directly with the rate of intrinsic frequency variation and inversely with the rate of group velocity magnitude variation along the ray path.

Notably, these relations can be alternatively derived using the plane wave approximation: for a slowly varying wave train locally approximating a plane wave (constant amplitude), wave action density (and hence wave energy density) should exhibit no local variation, reducing Eq. () to ∇⋅Acg=0. This implies that E/ω′cg is a solenoidal vector (Lighthill, 2001), meaning the flux of wave action through a thin ray tube remains conserved. Consequently, E/ω′cg is constant along the ray, yielding E∼ω′/cg – a result fully consistent with the earlier derivation.

According to the observed distribution of propagative areas for rays analyzed by Li et al. (2021a), the ED regime in the northern hemisphere, which is enclosed by a southern critical point and a northern turning point, is mainly located south of the jet center. For some zonal wavenumber, the turning point may be located north of the jet center. Therefore, let's first consider a simplest case where a ray with a fixed zonal wavenumber propagates from an initial location y0 toward a northern turning point at yt (cg,y>0 and hence l>0) that is situated south of a westerly jet with β∗>0 (cf. Fig. 1a, b). Besides, β∗ varies slowly to ensure the meridional wavenumber monotonically decreases from its initial value to zero when the ray moves from the initial location to the northern turning point.

With above preliminaries, kl>0 and du‾/dY>0 so that the sign of Eq. () is only determined by the sign of λ. According to Eq. (), λ=0 requires its numerator equals zero, that is, 21c2-2k2K2u‾-c2=0. The solution is 22cλ0y=2kKy+2ku‾y. Note that cλ0 varies along ray trajectories since u‾ and K varies along ray trajectories. Furthermore, when the ray moves toward the northern turning point from its initial location, u‾ is monotonically increasing (the ray is moving toward the jet center) while K is monotonically decreasing (caused by monotonically decreasing meridional wavenumber). Correspondingly, cλ0 will monotonically increases from its minimum at the initial location to its maximum at the turning point, that is, 23ca≡2kKy0+2ku‾y0=cλ0y0≤cλ0y≤cλ0yt=2kKyt+2ku‾yt≡cb, where ca and cb denote the minimum and maximum, respectively (Fig. 1c).

If the zonal phase speed of the wave (labeled as c) is slower than ca (illustrated by the dashed vertical line in Fig. 1c), we know c<cλ0y holds on when the ray moves from its initial location to the turning point (note that c does not vary along the ray), which indicates λ<0 and hence DgE/DT<0, that is, wave energy will monotonically decrease to a minimum when the ray arrives at the turning point where λ=0 due to zero meridional wavenumber (Fig. 1d). While the WKB approximation breaks down within a neighborhood to the turning point (e.g., yt-δ,yt where δ≪1) , it remains asymptotically valid at the boundary of the neighborhood (at y=yt-δ) . This dichotomy implies that stationary wave energy cannot physically attain the theoretical minimum value predicted at the turning point; however, this minimum serves as a critical indicator of energy thresholds prior to reflection.

If the zonal phase speed of the wave is faster than the minimum but slower than the maximum (ca<c<cb, illustrated by the dash–dotted vertical line in Fig. 1c), there exists an intermediate location ym within the range y0,yt to ensure c=cλ0ym . When the ray moves from y0 to ym, λ>0; when the ray arrives at ym, λ=0; and when the ray continues to move from ym to yt, λ<0. It means wave energy will first intensify to a maximum at ym before decaying to a minimum at yt (Fig. 1e). If the zonal phase speed of the wave is faster than cb (illustrated by the dotted vertical line in Fig. 1c), we have c>cλ0y , which means λ>0 and hence DgE/DT>0 when the ray moves from from y0 to yt. Correspondingly, wave energy will monotonically increase to a maximum when the ray arrives at yt where λ=0 due to zero meridional wavenumber (Fig. 1f).

When the ray arrives at the turning point, the local dispersion relation Eq. () reduces to 24cyt=u‾yt-β∗ytk2. Note that cyt prescribes the possible upper limit of the zonal phase speed of a wave whose ray can propagate in an ED regime. If c>cyt, the ray will not be reflected by any turning point along its trajectory. Since β∗>0, we have cyt<u‾yt, indicating that the upper limit zonal phase speed is slower than the basic wind speed at the turning point and, naturally, slower than the maximum wind speed at the jet center.

To analyze amplitude variation, let us consider the scenario where the zonal phase speed falls within the interval ca,cb without loss of generality. Since the wave energy reaches a maximum at ym, we have 25DgEDTy=ym=14DgK2DTΨ02y=ym+14DgΨ02DTK2y=ym=0, by applying Eq. () – the definition of the wave energy density. The first term on the RHS is negative since the meridional wavenumber decreases monotonically. The second term must therefore be positive to enable the wave energy to reach its maximum at ym. Consequently, the amplitude increases at ym. When the ray arrives at yt, the first term vanishes, and thus the second term must also vanish simultaneously, meaning that amplitude attains its maximum at the turning point. This demonstrates a monotonic increase in amplitude throughout the ray's northward trajectory (Fig. 1d, e, f).

Let's now consider a slightly complex case where the turning point lies north of the jet center, with other conditions remaining unchanged. Without loss of generality, we analyze the scenario where the zonal phase speed falls within the interval ca,cb. As the ray propagates from y0 to ym, wave energy increases to a maximum at ym. When the ray travels from ym to the jet center (say, yz) , wave energy declines from the maximum to a minimum at yz where zero wind shear can also contribute to this minimum. As the ray proceeds from yz to yt, wave energy rebounds from the minimum to another maximum due to both negative wind shear and λ. Compared with the previous case, the most notable distinction is that wave energy can attain two distinct maxima as the ray propagates from the initial location to the turning point. Since analysis methodology remains consistent, we do not explicitly analyze this case.

To summarize, the evolution of wave energy and amplitude along a ray propagating in an ED regime (note that we focus here on the case where the northern turning point lies south of the jet center) depends on the wave's initial zonal phase speed. If the zonal phase speed is slower (c<ca), wave energy undergoes monotonic decay to a minimum value at the turning point (yt), while amplitude conversely amplifies monotonically to its maximum at this turning point. If the zonal phase speed is intermediate (ca<c<cb), energy exhibits nonmonotonic evolution – first intensifying to a peak at a specific point (ym) within the region y0,ym before decaying to a minimum at yt within the region ym,yt while amplitude monotonically amplifies to its maximum at yt. If the zonal phase speed is faster (c>cb) , both energy and amplitude undergo concurrent monotonic amplification, reaching their respective maxima at yt, where maximum intensification occurs as the zonal phase speed approaches its upper bound cyt (assuming cyt>cb). Note that we focus on a general theoretical analysis and thus do not prescribe an explicit profile of the westerly jet in this section. Readers who are not familiar with the above mathematical details can find specific examples for a classical westerly jet prototype in Sect. 4.

Notably, a wave with zonal phase speed in a range can develop significantly. This is strongly consistent with the bounds on zonal phase speed in the normal mode instability theory (e.g., Howard, 1961; Pedlosky, 1987). From a mathematical standpoint, rays initiating propagation from critical points – where the meridional wavenumber diverges (l2→∞) – exhibit unbounded amplification of both energy and amplitude. This divergence, however, violates the scale separation prerequisite of the WKB approximation. Physical realizability therefore imposes constraints on meridional wavenumber as well as zonal wavenumber to ensure large–scale wave dynamics.

Let's consider a case where a ray, with an initial meridional wavenumber l0, is just moving from an initial point y0 to an inflection point yi where the critical total wavenumber is Ki and corresponding meridional wavenumber is ±li according to Eq. (). When the ray arrives at yi, meridional group velocity equals zero and zonal group velocity equals zonal phase speed due to β∗=0 there. It means the ray will horizontally move along yi with a constant speed c. According to Eq. (), wave energy at yi can be expressed as 26E∼exp⁡σiT, where σi=2kliKi2du‾dYy=yi is a constant growth rate, the sign of which depends on the critical meridional wavenumber and meridional shear of the basic flow. Notably, a positive (negative) growth rate demonstrates exponential increase (decrease) of wave energy, and hence amplitude. The unlimited amplification of both wave energy and amplitude undoubtedly means modal instability, consistent with the critical layer problem in the classic instability theory (e.g., Pedlosky, 1987).

When β1∗>0, the dispersion relation reduces to 31c1=u‾1yt-β1∗ytk12≡cytyt, at turning point yt. Below we will consider cyt as a function of yt. Substituting expressions of u‾1 and β1∗, it is easy to show that if k12<8 (or k1<8≡kc≈2.83. Corresponding wavelength is around 2200 km), cyt has three extrema, two of which, labeled as cM, are the same maxima and the left of which, labeled as cm, is the local minimum (see Appendix B), that is, 32cm≡cyt0=uM-β1+2uMk12cM≡cytyM=k12+4uM24-1k12β1k12+4-uM6, where yM=ln⁡12k12+4±12k12+4-1. If k12>8, only cm survives and it becomes a maximum. Besides, to make sure a ray can arrive at a turning point, cM should be larger than zero if there are three extrema and cm should be larger than zero if there are only one extremum. At critical point yc, the dispersion relation reduces to 33c1=u‾1yc≡cpyc. Note that we have considered cp as a function of yc. Due to the same form with the basic flow, the critical point curve cp has a maximum at y1=0, which equals the magnitude of the westerly jet.

The meridional areas bounded by the turning points (the solid curve in Fig. 3) and the critical points (denoted by the dash-dotted curve in Fig. 3) consist of the propagative regions (Fig. 3), featured by the ED, WG, and BD regimes. When β1∗>0, if k<kc (Fig. 3a), rays with zonal phase speed in the range 0,cM propagate within the ED region. Note that in this range, there exists a subrange cm,cM where rays can propagate in both ED and WG regions. This depends on the initial location of a ray. Since we mainly focus on the propagation feature in the ED regime in this investigation, we do not explicitly analyze the cases where rays propagate in the WG regime. On the other hand, rays with zonal phase speed in range cM,cp propagate in the BD region. By contrast, if k>kc (Fig. 3b), rays with zonal phase speed in 0,cm propagate within the ED region, while those in cm,cp fall into the BD region.

When β1∗<0, intersection points between turning and critical points emerge at β1∗=0. We focus on our analysis to regions (y1<0) with positive wind shear. Let the two intersection points be located at -yi and -yj (-yi<-yj<0), with corresponding zonal phase speeds labeled ci and cj (ci<cj) , as illustrated in Fig. 3c and d. When -yj<-yM, the propagation scenario (Fig. 3c) mirrors the β1∗>0 case in Fig. 3a, that is, rays with zonal phase speed in 0,cM propagates within the ED region (note that we have neglected the WG region). Conversely, when -yj>-yM, the zonal phase speed range defining the ED regions shrinks to 0,cj while the remaining cj,cM range transfers to a small WG region (Fig. 3d).

Building on the preceding analysis, the zonal phase speed range defining the ED region is determined by critical thresholds cm, cM, or cj, which we collectively denote as cmax⁡ for simplicity (where cmax⁡ corresponds to cm, cM, or cj depending on the basic flow configuration). Consistent with prior theoretical analysis in Sect. 3, this interval 0,cmax⁡ can be partitioned into three contiguous subranges – 0,ca , ca,cb, and cb,cmax⁡ – as illustrated in Fig. 4. Within these subranges, wave energy evolution follows distinct patterns: in 0,ca, energy decays monotonically to a minimum at the turning point; in ca,cb, it first amplifies to a maximum before decaying to a local minimum at the turning point; and in cb,cmax⁡, it amplifies monotonically, peaking at the turning point. Amplitude, by contrast, demonstrates uniform monotonic growth across the entire 0,cmax⁡ range, reaching its maximum precisely at the turning point. The amplification of wave energy toward the turning points aligns with transient growth mechanisms in barotropic Rossby waves within uniformly sheared basic flows (e.g., Tung, 1983; Yamagata, 1976). The present study extends these findings to more generalized basic flows, where analytical solutions for initial–value problems remain intractable, thereby enhancing the applicability of transient growth theory to non–uniform shear environments.

Figure 4 further reveals that both wave energy and amplitude at the turning point are monotonically increasing functions of zonal phase speed, which is consistent with the theoretical prediction in Sect. 3. For a wave with fixed zonal wavenumber, this implies that maximum transient growth of both energy and amplitude occurs when the zonal phase speed reaches cmax⁡. In other words, cmax⁡ corresponds to the zonal phase speed at which waves with fixed wavenumbers exhibit the most significant development (Fig. 4a) or even transition to potential instability (Fig. 4b). To focus our analysis, we examine the evolution of wave energy and amplitude for rays with zonal phase speed equal to cmax⁡, which is readily identifiable (Fig. 5) owing to the analytical westerly jet model. As illustrated in Fig. 5, the cmax⁡k1,uM=0 curve exhibits monotonic decay with increasing zonal wavenumber. Below this curve, the dashed contour indicates that rays do not encounter any turning point along their trajectories. Above the curve, the farther a point lies from the curve, the more rapidly the energy amplification rate becomes. Additionally, a second critical curve (cM=cj) emerges (red solid curve in Fig. 5), above which the amplification rate become independent of zonal wavenumber and instead increases monotonically with the intensity of the westerly jet. This configuration corresponds to scenarios where cmax⁡ assumes the value of cj rather than cm or cM as depicted in Fig. 3d.

Figure 6 further illustrated the maxima of wave energy (Fig. 6a) and amplitude (Fig. 6b) at turning points for ray with zonal phase speed cmax⁡. Given the amplitude's evolutionary pattern closely mirrors wave energy but with reduced magnitude, we focus our analysis on amplitude dynamics. For weak jet intensities – particularly when uM<2.428 (precluding normal mode instability) – transient growth upon reaching the turning point remains moderate. Consequently, Rossby waves exhibit moderate growth without transitioning to instability. By contrast, for jet intensities exceeding a critical threshold (e.g., uM>3.0) , an optimal zonal wavenumber (or corresponding wavelength) emerges, coinciding with the strongest transient amplitude growth. For instance, when uM=4.0, a ray with k1=1 (wavelength is around 6000 km) amplifies to around 7.4 times its initial amplitude upon reaching the turning point (wave energy has a larger amplification). Although amplitude decays post–reflection from the turning point, the transient amplification is sufficient large to potentially trigger nonmodal instability. As jet intensity increases, the optimal zonal wavenumber shifts upward. When uM=5.0, this optimal wavenumber corresponds to k1≃1.1 (wavelength is around 5500 km). Notably, the optimal zonal wavenumber lies above the cj=cM defined boundary from Fig. 5 (reproduced in Fig. 6 for comparison). This indicates that the optimal zonal wavenumber resides within the ray propagation regime characterized in Fig. 3d.

When a ray traverses a critical layer – a phenomenon extensively studied in geophysical fluid dynamics (e.g., Pedlosky, 1987) – we also focus on the region south of the jet center where meridional shear is positive. Consider a ray (solid curve in Fig. 7a) with zonal phase seed cmax⁡ (equivalent to cj) propagating southward (cg,y<0 hence initial meridional wavenumber l0<0) from an initial location -y0 toward the inflection point -yi (also the critical point). Along this trajectory, the negative meridional wavenumber (solid curve in Fig. 7b) increases monotonically to the negative critical value -li, then stabilizes as dictated by Eq. (). This negative wavenumber corresponds to a negative growth rate, driving decay of both wave energy and amplitude (solid curve in Fig. 7c). Similarly, a northward–propagating ray (dashed curve in Fig. 7a) originating from -y0 must also exhibit a negative meridional wavenumber (dashed curve in Fig. 7b) to satisfy the positive meridional group velocity constraint. Upon reaching the inflection point -yi, this negative critical meridional wavenumber -li similarly induces a negative growth rate, resulting in declining energy and amplitude (dashed curve in Fig. 7c). In contrast, rays propagating southward from an initial location -y0 (-y0<-yi) encounter a southern turning point, where they undergo significant transient amplification. This pre–turning–point growth – sufficient to trigger nonmodal instability – renders reflection dynamics at the turning point less critical to the overall instability process.

Consider a ray originating exactly at the inflection point (-y0=-yj). Owing to cg,y=0, the ray propagates zonally with the zonal group velocity that equals the zonal phase speed cmax⁡ (Fig. 8a). Although the initial meridional wavenumber l0 at the inflection point is unconstrained, its evolution remains governed by the second equation in Eq. (), that is, DglDT≡-∂Ω∂Y. Within the prototype westerly jet, l0<li drives DglDT>0, while l0>li induces DglDT<0. Consequently, as Fig. 8b illustrated, if l0<li (dashed curve) or l0>li (solid curve), the meridional wavenumber approaches li; if l0<-li (dash–dotted curve), it monotonically decrease toward negative infinity. Correspondingly, wave energy exhibits exponential growth when meridional wavenumber approaches li (solid and dashed curves in Fig. 8c) but decays exponentially when the meridional wavenumber tends toward negative infinity (dash–dotted curve in Fig. 8c).