Baroclinic Wave Packets Research Articles

Rogue-wave solutions for an inhomogeneous nonlinear system in a geophysical fluid or inhomogeneous optical medium

Under investigation in this paper is an inhomogeneous nonlinear system, which describes the marginally-unstable baroclinic wave packets in a geophysical fluid or ultra-short pulses in nonlinear optics with certain inhomogeneous medium existing. By virtue of a kind of the Darboux transformation, under the Painlevé integrable condition, the first- and second-order bright and dark rogue-wave solutions are derived. Properties of the first- and second-order bright and dark rogue waves with α(t), which measures the state of the basic flow, and β(t), representing the interaction of the wave packet and mean flow, are graphically presented and analyzed: α(t) and β(t) have no influence on the wave packet, but affect the correction of the basic flow. When we choose α(t) as a constant and linear function, respectively, the shapes of the first- and second-order dark rogue waves change, and the peak heights and widths of them alter with the value of β(t) changing.

Semirational solutions and baseband modulational instability of the AB system in fluid mechanics

Under investigation in this paper is the AB system describing marginally unstable baroclinic wave packets in geophysical fluids. By means of the n -fold modified Darboux transformation, the semirational solutions in terms of the determinants of the AB system are derived. These solutions, which are a combination of rational and exponential functions, can be used to model the nonlinear superposition of the Akhmediev breathers (or the Kuznetsov-Ma breathers) and the rogue waves. The k -order rogue wave of the AB system is produced by the interaction between the l-order rogue wave with $\frac{1}{2}(k-l)(k+l+1)$ neighboring elements in the $(k-l)$ -order breathers $(0<l<k)$ . The proper values of eigenvalue $ \lambda$ and shift $ S_{0}$ are also the requirement for generating the higher-order rogue waves. The link between the baseband modulational instability and the existence condition of these rogue waves is revealed.

The canonical AB system: conservation laws and soliton solutions

Under investigation in this paper is the canonical AB system, which describes the marginally unstable baroclinic wave packets in the geophysical fluids and ultra-short pulses in nonlinear optics. Through symbolic computation, conservation laws are derived. Furthermore, by virtue of the Darboux transformation, the explicit multi-soliton solutions are generated. Figures are plotted to reveal the following dynamic features of the solitons: (1) Elastic interactions of two one-peak bright solitons, of two one-peak dark solitons and of two two-peak dark solitons; (2) Parallel propagations of two one-peak bright solitons, of two one-peak dark solitons and of two two-peak dark solitons; (3) Propagations of three types of bound solitons: periodic propagation of bound solitons taking on contrary trends, mutual attractions and repulsions of two bight bound solitons and of two dark bound solitons.

Rogue wave solutions of AB system

In this paper, the generalized Darboux transformation is established to the AB system, which mainly describes marginally unstable baroclinic wave packets in geophysical fluids and ultra-short pulses in nonlinear optics. A unified formula of Nth-order rogue wave solution for the AB system is found by the direct iterative rule. In particular, rogue wave solutions possessing several free parameters from first to second order are calculated. The dynamic properties of rogue waves in the AB system are shown through some figures.

Dynamic behaviors of the breather solutions for the AB system in fluid mechanics

Under investigation in this paper is the AB system, which describes marginally unstable baroclinic wave packets in geophysical fluids. Through symbolic computation, Lax pair and conservation laws are derived and the Darboux transformation is constructed for this system. Furthermore, three types of breathers on the continuous wave (cw) background are generated via the obtained Darboux transformation. The following contents are mainly discussed by figures plotted: (1) Modulation instability processes of the Akhmediev breathers in the presence of small perturbations; (2) Propagations characteristics of Ma solitons; (3) Dynamic features of the breathers evolving periodically along the straight line with a certain angle of z-axis and t-axis.

Interaction of North Atlantic baroclinic wave packets and the Mediterranean storm track

AbstractObservational studies show that the main energy source for developing cyclones is due to downstream radiation from existing upstream disturbances. This process is involved in the propagation of coherent wave packets linking the North Atlantic and Mediterranean storm tracks in winter. To understand the link, the Rossby wave packets have been analyzed for the winter season 2004/05 by means of wave envelope, energetic, and wave‐activity diagnostics. Two specifically selected cases are presented: one with clear propagation of the wave packet to the Mediterranean region from the North Atlantic storm track, and the other with zonal propagation of the wave packets along the central latitude of the North Atlantic storm track. The dynamical effects leading to the distinct propagation characteristics of the two cases are discussed. It is shown that the two cases make clear impacts on the horizontal flux of wave activity across a domain surrounding the Mediterranean storm track.

Integrability aspects and soliton solutions for an inhomogeneous nonlinear system with symbolic computation

Under investigation in this paper is an inhomogeneous nonlinear system, which describes the marginally unstable baroclinic wave packets in geophysical fluids and ultra-short pulses in nonlinear optics under inhomogeneous media. Through symbolic computation, the Painlevé integrable condition, Lax pair and conservation laws are derived for this system. Furthermore, by virtue of the Darboux transformation, the explicit multi-soliton solutions are generated. Figures are plotted to reveal the following dynamic features of the solitons: (1) Parallel propagation of solitons: separation distance of the two parallel solitons depends on the value of |Im(λ1)|-|Im(λ2)| (where λ1 and λ2 are the spectrum parameters); (2) Periodic propagation of bound solitons: periodic bound solitons taking on contrary trends, and mutual attractions and repulsions of two bright bound solitons; (3) Elastic interactions of two one-peak bright solitons and of two one-peak dark solitons.

Observation of first and second baroclinic mode Yanai waves in the ocean

AbstractThe structure and propagation of oceanic Yanai waves (also known as mixed Rossby–gravity waves) are investigated by analysis of satellite‐derived sea‐surface‐height (SSH) data. Significant spectral peaks along the dispersion curves of first and second baroclinic mode Yanai waves are identified by the wavenumber–frequency spectral analysis of SSH for the period 10–17 days. The spatial structure and propagation of these modes are described by an analysis of SSH time series filtered in the frequency–wavenumber domain that includes strong signals along the dispersion curves but excludes those of 17‐day tropical instability waves. The difference in meridional structure between the first and second baroclinic modes, which is consistent with theory, is evident. The slow eastward propagation of first and second baroclinic mode Yanai wave packets, which is consistent with the group velocity derived from theory, is also isolated from the SSH data. Year‐to‐year variability of Yanai wave activity is compared with that of the El Niño–Southern Oscillation (ENSO), showing a moderate correlation with ENSO Modoki. © Published in 2011 by John Wiley &amp; Sons Ltd.

Characteristics of Baroclinic Wave Packets during Strong and Weak Stratospheric Polar Vortex Events

Abstract This study tested a numerical and theoretical prediction that the stratosphere and troposphere are coupled through the effect of stratospheric vertical wind shear on baroclinic waves. Wavelengths, phase speeds, and background quasigeostrophic potential vorticity gradients were analyzed over the Pacific and Atlantic during strong and weak stratospheric polar vortex events and interpreted in terms of the counterpropagating Rossby wave perspective on baroclinic instability. Effects of zonal variations in the background flow were included in the analysis of phase speeds. Observed changes in wave packet average wavelength and phase speed support the vertical shear hypothesis for stratosphere–troposphere coupling; however, changes in the intrinsic phase speed contradict the hypothesis. This inconsistency was resolved by considering the change in zonal wind speed in the lower stratosphere, which accounts for most of the change in phase speeds during strong and weak vortex events. Changes in the average wavelengths and meridional wave activity flux are also consistent with this modified hypothesis involving the stratospheric zonal wind. The results demonstrate that a simple mechanism for stratosphere–troposphere coupling can be found in the observational record.

Baroclinic wave packets in an extended quasigeostrophic two‐layer model

An extended quasigeostrophic two‐layer model including nonlinear Ekman pumping and nonlinear interface perturbation is used to numerically study the formation and maintenance of baroclinic wave packets. With an initial zonal jet flow of north–south symmetry and small random initial perturbations wave packets of finite amplitude are produced and maintained with an equilibrated wave packet‐zonal flow system exhibiting simultaneously downstream‐upstream asymmetry and north–south asymmetry, the later of which has not been observed in the past simple quasigeostrophic two‐layer modeling studies. The northern upstream part of the wave packets is stronger than its southern upstream counterpart and therefore the peak of time‐and‐zonal‐average eddy kinetic energy associated with the wave packets is shifted poleward while the peak of the time‐mean zonal jet flow is shifted equatorward, which is in agreement with the observations.

A Local Energetics Analysis of the Life Cycle Differences between Consecutive, Explosively Deepening, Continental Cyclones

Abstract Local energetics diagnostics of the life cycles of consecutive, explosively deepening, extratropical cyclones that migrated across central North America in April 2001 are presented. Both storms developed rapidly and followed nearly identical tracks through the region. Despite similar mature-stage intensities, the two storms underwent vastly different evolutions during cyclolysis; the first decayed as rapidly as it had developed, and the second decayed very slowly. Examination of the volume-integrated eddy kinetic energy (EKE) budget for each storm reveals that the sea level pressure minimum associated with the first cyclone developed well after its associated EKE center had reached its maximum intensity. In contrast, the second cyclone’s sea level pressure minimum developed much more in concert with the development of its associated EKE center. As a consequence, the first cyclone began losing EKE through downstream energy fluxes even as it was developing at the surface, whereas the second cyclone did not disperse EKE downstream until later in its life cycle. Consideration of the EKE budget results in terms of baroclinic wave packets demonstrates that the first cyclone developed and decayed on the upstream edge of a wave packet, whereas the second cyclone developed in the midst of a wave packet, only decaying once it had reached the upstream edge. Thus, it is suggested that postmature phase decay is dynamically linked to a cyclone’s position in a given wave packet.

Envelope Solitary Waves and Periodic Waves in the AB Equations

This article deals with the envelope solitary waves and periodic waves in the AB equations that serve as model equations describing marginally unstable baroclinic wave packets in geophysical fluids and also ultra‐short pulses in nonlinear optics.An envelope solitary wave has a width proportional to its velocity and inversely proportional to its amplitude. The velocity of the envelope solitary wave is partially dependent on its amplitude in the sense that the amplitude determines the upper or lower limit of the velocity. When two envelope solitary waves collide, they survive the collision and retain their identities except for a shift in the positions of both the envelopes and the carrier waves.The periodic wave solutions in sine wave form may be stable or unstable depending upon the wave parameters. When the sine wave is destabilized by small perturbations, its long‐time evolution shows a Fermi–Pasta–Ulam‐type oscillation.

The Structure of Baroclinic Wave Packets

Abstract In this paper, the three-dimensional structure of baroclinic wave packets is studied using composites from ECMWF reanalysis data. Eddy variances and covariances associated with wave packets are examined in terms of the eddy energy budget and wave activity and fluxes, and the large-scale, phase-independent flow component associated with wave packets is studied using composites of the low zonal wavenumber component of the flow. The results suggest that the sources of upper-level eddy energy–wave activity associated with the wave packets are located over the central and upstream portion of the wave packet, and the downstream part of the wave packet is entirely maintained by radiation of energy–wave activity fluxes from the upstream end. The results are entirely consistent with previous studies of the energetics of individual wave evolution cases. The composites also show that the basic flow response to eddy transports associated with the wave packets consists mainly of a poleward shift of the jet ne...

Mechanisms for Wave Packet Formation and Maintenance in a Quasigeostrophic Two-Layer Model

Abstract A quasigeostrophic, two-layer, β-plane channel model is used to investigate the dynamics of baroclinic wave packets. A series of experiments are performed in which an unstable flow is maintained by lower-level Ekman friction and radiative relaxation toward a temperature profile that corresponds to a broad parabolic upper-level jet. The final statistically steady state achieved in each experiment is found to depend on the magnitude of the hyperdiffusivity ν0 and the supercriticality, which is controlled by β. The most important qualitative difference in such states between experiments is found to be the degree to which a waveguide in the upper level is found to develop. The mechanism for this upper-level waveguide development is the mixing effect of the eddies at the flanks of the jet, which leads to a strong potential vorticity gradient at the center of the channel, with well-mixed regions to the north and south. Two distinct regimes with different qualitative behavior are observed and illustrate...

Propagation of envelope solitons in baroclinic atmosphere

The propagation of finite amplitude baroclinic wave packets in the two-layer model is investigated by using the multiple-scale method. It is shown that the propagation of the wave packets can be described by the so-called unstable nonlinear Schrodinger equation which possesses envelope soliton solutions. The speeds of the solitons are independent of their amplitudes.while the width of the solitons is directly proportional to their speeds but inversely proportional to their amplitudes.

Collision Interactions of Solitons in a Baroclinic Atmosphere

Abstract In this paper the interactions between two marginally unstable baroclinic wave packets in the two-layer Phillips model are investigated by using the multiple-scale method. It is shown that the interactions can be described by a set of two coupled nonlinear Schrodinger equations. Except for two special cases the equations have only four invariants of motion and cannot be solved by the inverse scattering method. The equations are solved numerically to study the collision interactions between two solitons. It is found that though the coefficients in the equations are fixed, the behavior of the two solitons may be quite different, which is closely related to the initial states of the two solitons (the speeds and the amplitudes of the solitons well before the interactions). For some initial conditions the collision interactions may be soliton-like in that the properties of the two solitons change very little, while for other initial conditions some “inelastic” phenomena are observed: one soliton may b...

Tropical Intraseasonal Oscillation, Super Cloud Clusters, and Cumulus Convection Schemes

A new framework for interpreting the origin of the tropical intraseasonal oscillation (TISO), which avoids the speed and scale selection problems in the previous theories, is proposed in this study. In this interpretation TISO is viewed as an oscillation driven by an eastward moving convective region. This convective region consists of one or more super cloud clusters originating in the Indian Ocean and terminating in mid-Pacific, and is then followed by another convective region arising in the Indian Ocean in a period of 40-50 days. Additionally, a formal analogy is pointed out between super cloud clusters and the middle-latitude baroclinic wave packets. This study includes a simulation of TISO in a 2D model to support our interpretation. Experiments were conducted with four different convection schemes. The authors advocate that the successful simulation of TISO depends on the successful simulation of super clouds cluster, which in turn depends on the successful simulation of the life cycle of cloud clusters, which further in turn depends on the choice of cumulus convection scheme. What makes a cumulus convection scheme successful in simulating TISO is discussed.

Low-Frequency Synoptic-Eddy Activity in the Pacific Storm Track

The nature of low-frequency variations in synoptic-eddy activity over the North Pacific is examined in a general circulation model (GCM). A comparison with observations reveals that the GCM produces realistic time mean and low-frequency synoptic-eddy forcing of the 200-mb zonal wind. In the time mean, this forcing, which is computed as the divergence of the extended Eliassen–Palm (E–P) flux, shows an east-west dipole structure that tends to reduce the zonal wind over the western North Pacific and tends to enhance it to the east. This structure is consistent with the general picture of the life cycle of baroclinic waves, which show strong upward and eastward propagation in the western and central Pacific and meridional propagation to the east. The western and central Pacific synoptic-eddy forcing is dominated by the convergence of the baroclinic component of the E–P flux divergence, while over the eastern Pacific the divergence of the barotropic component is important. The dominant component of the low-frequency “envelope” (periods > 10 days) of synoptic-eddy forcing, computed as the first empirical orthogonal function (EOF) modulates the time mean synoptic-eddy forcing. This modulation is associated with low-frequency changes in the intensity of the synoptic-eddy activity and is only weakly tied to fluctuations in the low-frequency flow. Composites of the hemispheric distribution of synoptic-eddy forcing in the GCM, based on the extremes of the dominant Pacific EOF, show a seesaw behavior with enhanced eddy forcing in the North Pacific basin associated with suppressed forcing in the North Atlantic basin, and vice versa. The link between the Pacific and the Atlantic basins appears to be due to the presence of eastward-traveling baroclinic wave packets that travel around the globe with a period of about 10 days. Some evidence is found for a similar behavior in the observations.

Baroclinic Wave Packets in Models and Observations

Coherent baroclinic wave packets are present in the Southern Hemisphere, most clearly in the summer season. These coherent packets are also found in a hierarchy of models of nonlinear baroclinic instability-a two-layer quasigeostrophic (QG) model on a β-plane, a two-level primitive equation (PE) model, and a general circulation model. The flows are chaotic, but the packet itself can remain remarkably coherent, despite the complex evolution of the flow within the packet. In both QG and PE models, the packets become more robust as the supercriticality of the flow is reduced. In both models and the observations, the packets move with a group velocity that is greater than the phase speed of the individual disturbances, so that these disturbances exhibit downstream development. The structure of the baroclinic waves in the packet as a function of longitude resembles the life cycles of sinusoidal baroclinic waves as a function of time. More than one packet can exist in the domain at the same time. In the QG model, the number of packets increases in a systematic way as the length of the channel increases.

Is the Midlatitude Zonal Flow Absolutely Unstable?

Abstract An analysis is performed of the growth and propagation of unstable baroclinic wave packets in relatively realistic midlatitude zonal currents. The absolute growth rates are calculated, incorporating the effects of both Ekman friction and barotropic shear. It is found that these effects do not alter earlier conclusions (based on simpler models)that a finite easterly low-level wind is required for absolute instability in models with continuous shear. Ekman friction exacerbates the situation, shifting the absolute instability threshold from essentially zero surface wind to negative surface wind. While absolute instability of the midlatitude terrestrial jets thus seems unlikely, the conclusion stands that the atmosphere is at least near the boundary of absolute instability, especially under conditions prevailing in the oceanic storm tracks. In consequence, slow-moving short waves with growth rates below the maximum normal-mode growth rate typically contribute more to the amplification of wave packets...