The life cycle of a tropical cyclone (TC) is typically divided into several stages including early genesis, tropical disturbance, tropical depression, tropical storm, hurricane, and finally the dissipation. Among these five stages, the tropical cyclogenesis (TCG), defined as a period during which a weak atmospheric disturbance grows into a mesoscale tropical depression with a close isobar and a maximum surface wind of >17 m s-1 , is perhaps the least understood due to its unorganized structure as well as ill-defined characteristics of TCs. During this genesis period (typically 2–5 d), synergetic interactions among various dynamical and thermodynamic processes at different scales can result in an eventually self-sustained, warm-core vortex before the subsequent intensification can take place. These early formation processes are so intricate that no single or distinct mechanism could operate for all TCs, rendering the genesis forecasting very challenging in practice. Such a multi-faceted nature of TCG is the main factor preventing us from obtaining a complete understanding of TC formation and development at present.

Early studies by provided a list of necessary climatological conditions for TCG to occur, which include (i) an underlying warm sea surface temperature (SST) of at least 26 ∘C, (ii) a finite-amplitude low-level cyclonic disturbance, (iii) weak vertical wind shear, (iv) a tropical upper tropospheric trough, and (v) a moist lower to middle troposphere. While the above conditions for genesis have been well documented in numerous observational and modeling studies since then, it is intriguing that the actual number of TCs composes a small fraction of the cases that meet all these conditions in the tropical region every year. Moreover, TCG varies wildly among different ocean basins due to relative importance of large-scale disturbances, local forcings, and surface conditions, thus inheriting strong regional characteristics that common criteria may not be applied everywhere. For example, TC genesis in the North Atlantic basin often shows a strong connection to active tropical waves originating from the South African jet . In the northwestern Pacific basin, studies by , , , , , and showed that the genesis is mostly related to the intertropical convergence zone (ITCZ) and monsoon activities. In the northeastern Pacific, vortex interaction associated with the topographic and tropical waves seems to generate abundant disturbances that act as the seeds of TC genesis .

Other large-scale conditions that can interfere with TCG have been also reported in previous studies such as the Saharan air layer (SAL; ), upper-level potential vorticity anomalies , mixed Rossby–gravity waves , the ITCZ breakdown , or multiple vortex merges . Along with this diverse nature of genesis in different basins, observational and modeling studies of TC development have shown that the evolution of tropical disturbances during the early genesis stage often encompasses a wide range of scales from convective-scale hot towers and mesoscale convective systems to large-scale quasi-balanced lifting and cloud–radiation feedbacks e.g.,. In this regard, TCG is a truly multi-scale process and the relative importance of different mechanisms must be carefully examined when studying the TC genesis in real atmospheric conditions.

Recent efforts in the TC genesis research have been shifted from examining local mechanisms to a broader perspective of how environmental conditions can produce and maintain TC disturbances during TC early development . The most current attempt in quantifying the large-scale factors governing the genesis in the North Atlantic basin focuses on the so-called “pouch” conceptual model, which treats an early TC embryo as a protected region within large-scale easterly waves . To some extent, this pouch idea can be considered as an advance of the requirement of an incipient disturbance for genesis to occur that was originally put forth by . Much of the development along this “pouch” idea has been on tracking wave packets in the co-moving frame required to protect the mid-level disturbances (the so-called Kelvin cat-eye in ).

Despite much progress over recent decades, several outstanding issues in the TC genesis study still remain. From the global perspective, a particular question of what is the maximum number of TCs that the Earth's tropical atmosphere can form and support in any given day has not been adequately addressed. Answering this question will help explain a long-standing question of why the Earth has only a specific number of ∼100 TCs globally every year. A recent modeling study of the global TC formation by demonstrated that the daily number of genesis events is indeed intriguingly bounded (<10), even in a perfect environment. This number is quite consistent with a simple scale analysis based on the typical scale of TCs with a diameter ∼3000 km, which shows that there should have been less than 14 TCs on the Earth's atmosphere at any given time, assuming that the radius of the Earth is ∼6400 km. Using idealized simulations for a tropical channel, showed in fact that genesis occurs in episodes of 7–10 storms each time with a frequency between the episodes of 12–16 d. This episodic development at the global scale as well as the upper bound of ∼10 storms for each episode as obtained from these idealized experiments suggests that there must have some large-scale environmental conditions or intrinsic properties of the tropical dynamics, which control the genesis processes beyond the basin-specific mechanisms as discussed in .

While recent advances in global numerical models can reasonably capture the very early stage of the TCG and serve as guidance for operational genesis forecasts, analytical models of TC development have been confined mostly to the later stage of TC development such that the axisymmetric characteristics of disturbances could be employed. The axisymmetry is critical for the theoretical purposes, because it reduces the Navier–Stokes equations to a set of approximated equations for which some balance constraints and simplifications can be employed.

Given various basin-specific mechanisms that could produce TCs beyond axisymmetric models for an individual TC, the main objective of this study is to focus specifically on a large-scale mechanism behind the formation of tropical disturbances associated with ITCZ breakdown. This special pathway is very typical at the global scales whereby converging winds from the two hemispheres could set up the right environment for large-scale stability to develop . Indeed, satellite observations often show that the ITCZ tends to undulate and break into a series of mesoscale vortices, some of which may eventually grow into TCs . This is especially apparent in the WPAC basin where early studies by showed that TC genesis primarily occurs along the ITCZ, which accounts for nearly 80 % of genesis occurrences in this area.

Although the ITCZ breakdown appears to be a slow process as compared to other pathways such as vortex mergers e.g., or tropical easterly waves e.g.,, it is an inherent property of the tropical atmosphere at the global scale that could provide a source of large-scale disturbances responsible for TCG. To minimize the complication due to the basin-specific features, we thus limit our study of the global TC formation to an idealized aqua-planet configuration to facilitate the analytical analyses in this study.

The rest of the paper is organized as follows. In the next section, an analytical model for the large-scale TC genesis based on the ITCZ breakdown model is presented. Section 3 presents detailed analyses of the principle of exchange of stabilities for the ITCZ model as well as the stability analyses of the dynamical transition. Numerical examination will be discussed in Sect. 4, and concluding remarks are given in the final section.

A unique characteristic of the ITCZ that provides a favorable environment for TC genesis to occur is the highly unstable zone along the ITCZ where trade winds from the two hemispheres converge. Such a zone with strong horizontal shear is well documented along the tropical belt where the potential vorticity gradient changes sign, providing a necessary condition for disturbances to grow according to Rayleigh's theorem . Thus, a disturbance embedded within the ITCZ can trigger a nonlinear growth and extract the energy from the background, resulting in a potential amplification of the disturbance with time.

Because of such a dominant role of the ITCZ in the global TC formation, a natural model for global TCG should take into account the large-scale ITCZ breakdown processes. This ITCZ breakdown model is particularly suitable for an aqua-planet that does not have other triggering mechanisms such as land–sea interaction or terrain effects. For this reason, we will consider the ITCZ breakdown as a starting model for the TC genesis at the global scale in this study. Inspired by the modeling studies of the ITCZ breakdown based on the shallow water equation e.g.,, we examine a similar model for the ITCZ dynamics on the horizontal plane for which the governing equation for the ITCZ can be reduced to an equation for the potential vorticity as follows. 1dΔψdt=νeΔ2ψ+F-αΔψ-β∂ψ∂x, where the horizontal streamfunction ψ has been introduced as a result of the continuity equation, νe is horizontal eddy viscosity, α is a relaxation time, Δ is the Laplacian operator, and F is an external force that represents either a source/sink of mass within the ITCZ

In , the relaxation time α is proportional to the ratio of the Ekman depth DE over the depth of the fluid H, while the external forcing term F can be treated as a large-scale vorticity source.

While the stability analyses in the previous section could show an upper bound on possible unstable modes, the structure of these unstable modes as well as the subsequent effects of the nonlinear terms have not been discussed. The existence of an eigenvalue with a zero real part at R=R* immediately imposes the condition that the stability analyses must require all nonlinear terms so that the behaviors of dynamical transitions can be captured. Depending on the multiplicity of the eigenvalues when the PES conditions are met, one can, however, reduce the full nonlinear system (Eq. ) to a set of ordinary differential equations on a central manifold at R=R* and construct a central manifold function to examine the bifurcation and the structure of new states with all nonlinear terms included. Standard procedures of stability analyses on the central manifold for the ITCZ model show that there exists indeed a supercritical Hopf bifurcation for R>R*, with a new stable state approximated as follows see, 39ψ=ψS+ℜ(ρn,1)|ℜ(A)|12fn(x,y,t)+h.o.t., where h.o.t. denotes higher order terms, provided that the nondimensional parameter R is sufficiently close to R*, i.e., 0<R-R*R*≪1.

Using a higher-order approximation around the critical point on the extended central manifold, one can obtain a better manifold function that better approximates ψ for R>R*. Nonetheless, the smooth behaviors of the eigenvector at R=R* for the supercritical Hopf bifurcation suffices to indicate that the structure of the solution at R=R* can represent well the behavior of the new stable solution near R=R* as dictated by the dynamical transition theorem in . Technically, either Hopf bifurcation or double Hopf bifurcation may appear, depending on the transition multiplicity at the critical value. This subtlety will introduce much more complex analysis of the bifurcation and transversal intersections of the parameter planes that we will not present herein.

While these higher-order derivations of the central manifold function require some technical details that are beyond the scope of this study, it is possible to approach the transition dynamics by numerically solving the eigenvalue problem (Eq. ). Specifically, we notice that the x dependence can be obtained by simply searching for the first unstable mode m as R approaches the critical value R*. Using this numerical approach to find the critical value of R*, the entire spectrum of eigenvectors associated with the potential new stable states after the dynamical transition can be found for each set of large-scale environmental parameters. We note again at this point that the exact mode m at which the eigenvector becomes first unstable is dependent on R as shown in Figs.  and . The only constraint we can be certain of is that |m|<k. Thus, a new stable mode for R>R* emerged under the case of the supercritical Hopf bifurcation could inherit all structure of a zonal mode |m|<k at R=R*. This numerical approach is powerful, as it allows one to search for not only the critical parameter R* at which the PES conditions are ensured, but also the structure of new stable states for any value of R>R* after the bifurcation point.

To illustrate the results from this numerical approach, we consider the following set of the large-scale environmental conditions in the typical tropical region: Ly∼1000km,U0∼10ms-1,α∼10-6s-1,β∼10-11m-1s-1,ν=1000m2s-1, which result in a Rossby number ϵ≈0.5 and an Ekman number E≈0.05. With the further use of Eq. () for the steady state and noting that U0=∂ψS/∂y, one then obtains an estimation for the forcing amplitude γ≈7×10-10s-2. From the definition of the nondimensional number R, we then get R≈4.8, which is above the critical value R*≈4 for m=9 as shown in Fig. . Thus, the PES conditions are satisfied, and a new stable structure must emerge after the dynamical transition as a consequence of the supercritical Hopf bifurcation. As such, the eigenvalue problem (Eq. ) can be solved for the first eigenvector and its dual eigenvector, given the value R=4.8. Note that this estimation of R is most sensitive to the strength of the shear flow U0, the beta effect β, and the scale Ly but not on the diffusion coefficient ν. To some extent, this insensitivity of the dynamical transition on the diffusion is expected, because the large-scale eddy diffusion process is often negligible.

For this numerical method, we use a Legendre–Galerkin method where the unknown fields are expanded using a basis of N polynomials, which are compact-combinations of the Legendre polynomials satisfying the four boundary conditions (Eq. ; see , for the details of this numerical method). For the convergence of the numerical scheme, N=100 is sufficient. Once the eigenvector problem is solved, a further approximation on the central manifold can be applied so that the stability of new stable states can be examined around the critical point on the central manifold.

Figure  shows a new state as a result of the dynamical transition for R>R* under the supercritical Hopf bifurcation case that is obtained from the numerical procedures described above. Among several significant features of this numerical solution, the most noteworthy one is that the new state possesses a large-scale structure consistent with the ITCZ breakdown as observed in the idealized simulations by . Specifically, the tropical channel contains 10 large-scale disturbances; each has the horizontal scale of about 5000 km and could serve as embryos for the subsequent TC formation. Furthermore, the disturbances in this new state move to the left with a period of T∼3.2116 (i.e., T≈4 d in the physical dimensional unit) as a result of the nonzero imaginary part of the eigenvalues. For different domain configurations such as different planets or future climates with different tropical width, the unstable structure and/or wave speed may be different, and so the maximum number of the TC-favorable disturbances will change as well.

That these large-scale structure of disturbances move to the left with a timescale of ∼4 d as a consequence of the dynamical transition shown above is interesting, because these westward-moving disturbances are to some extent similar to easterly waves in the real atmosphere. While it is entirely possible that these easterly waves are a mode of the equatorial mixed-Rossby waves, it should be noted that the numerical procedure of finding new stable modes on the central manifold presented in this study does not allow us to separate different modes of easterly waves. As such, these easterly waves could be a combination of different modes of westward-moving Rossby waves and mixed-gravity waves that we may not be able to separate. In any case, the easterly waves that are often associated with TC genesis can be now seen as a natural consequence of the dynamical transition, even for barotropic flows. Such a consistency between the observed and theoretical estimation of the large-scale modes in the tropical region suggests that the barotropic instability and its inherent nonlinear dynamics can account for the preconditioning environment for TC genesis.

As a final remark, the large-scale structure shown in Fig.  does not itself impose that the disturbances have to grow and turn into TCs. Instead, these structures are simply new stable periodic solutions associated with the supercritical Hopf bifurcation after the dynamical transition occurs. That is, for R<R*, the stable structure is the steady state as given in Fig. 2, whereas the new stable structure shown in Fig.  will emerge after R>R*. As soon as these stable structures emerge, subsequent dynamic-thermodynamic feedback may be triggered and result in further growth of the disturbances within each wave. For a larger value of R, the stability of the periodic state may no longer be ensured, because the central manifold function must be reevaluated and a new structure may arise. The subsequent intensification of any tropical disturbances as a result of the new unstable structure would require additional detailed physics that are, however, not the focus of this work and so will not be discussed hereinafter. In this regard, the new stable periodic state shown in Fig.  can serve only as a preconditioning environment for incipient disturbances to grow.

In this study, we examined the dynamical mechanisms underlying the large-scale formation of tropical cyclones (TCs) using a barotropic model for the intertropical convergence zone (ITCZ). Assuming a forcing function that mimics the mass sink/source in the ITCZ, it was shown that the large-scale steady flow (i.e., the critical point or equilibrium) in the ITCZ model loses its stability for a bounded range of the wavenumber |m|<k if large-scale environmental conditions including the magnitude of the mean flow, the Ekman number, and the Rossby number satisfy certain constraints. That the number of the unstable modes in the tropical region is upper bounded is a very significant result, because it could offer an explanation for a fundamental question of why the Earth's tropical atmosphere can support a limited number of TCs globally each year.

Using the principle of exchange of stabilities condition for the ITCZ model, we found that the ITCZ model undergoes a bifurcation with associated dynamical transition at the critical point, which helps determine the maximum number of TC disturbances that the Earth's atmosphere can generate. Specifically, a theoretical estimation of the largest zonal wavenumber k that can allow an unstable structure as a result of the ITCZ breakdown is k=12, assuming the typical scales of the Earth's tropical channel in which the zonal scale of the tropical channel is about an order of magnitude larger than the width of the channel. Such a dynamical constraint on the maximum number of TC disturbances is remarkable, as it suggest an intrinsic large-scale mechanism that controls the climatology of the TC numbers beyond the basin-specific features as recently noted in . Of interest is that this constraint on the largest wavenumber of the unstable eigenmodes imposes not only an upper limit on the number of TC disturbances in the tropical region, but also a lower bound on the size of TC disturbances. This lower bound on the size of the tropical disturbances may help explain why TCs cannot be arbitrarily small but must be larger than a certain limit in the tropical region.

To verify our theoretical analyses, a numerical method is used to search for the new structure on the central manifold of the ITCZ model as the model parameter R is larger than a critical value R*. Here, the key parameter R controlling the bifurcation in our ITCZ model is given by R=γϵπEπ4+Aπ2, where γ is a parameter measuring the strength of the ITCZ mass sink/source, A is the parameter measuring the effect of surface drag, ϵ is a parameter measuring the mean zonal flow, and E is the Ekman number representing the eddy viscosity. Our numerical results confirmed that for R>R*, a new large-scale state emerges as a result of the supercritical Hopf bifurcation whose structure depends on the value R. For R sufficiently close to the critical value R*, the new periodic state possesses a type of wave motion with two groups of symmetric disturbances across the Equator. These new stable periodic solutions describe a type of westward-moving disturbances within the ITCZ, very similar to the classical easterly waves in the tropical region. These findings obtained from the ITCZ breakdown model in this study thus provide new insights into the formation of TC disturbances in the Earth's tropical atmosphere, as well as a rigorous mathematical proof for the bounded number of TCs at the global scale.