Unstable Baroclinic Wave Research Articles

Dark-soliton asymptotics for a repulsive nonlinear system in a baroclinic flow

In geophysical hydrodynamics, baroclinic instability denotes the process in which the perturbations draw the energy from the mean flow potential power. Researchers focus their attention on the baroclinic instability in the Earth's atmosphere and oceans for the meteorological diagnosis and prediction. Under investigation in this paper is a repulsive nonlinear system modeling the marginally unstable baroclinic wave packets in a baroclinic flow. With respect to the amplitude of the baroclinic wave packet and correction to the mean flow resulting from the self-rectification of the baroclinic wave, we present a Lax pair with the changeable parameters and then derive the N-dark-dark soliton solutions, where N is a positive integer. Asymptotic analysis on the N-dark-dark solitons is processed to obtain the algebraic expressions of the N-dark-dark soliton components. We find that the obtained phase shift of each dark-dark soliton component is relevant with the N − 1 spectral parameters. Furthermore, we take N = 3 as an example and graphically illustrate the 3-dark-dark solitons, which are consistent with our asymptotic-analysis results. Our analysis may provide the explanations of the complex and variable natural mechanisms of the baroclinic instability.

On a variable-coefficient AB system in a baroclinic flow: Generalized Darboux transformation and non-autonomous localized waves

In this paper, we focus our attention on a variable-coefficient AB system, which models the marginally unstable baroclinic wave packets in a baroclinic flow. With respect to the amplitude of the baroclinic wave packet and correction to the mean flow resulting from the self-rectification of the baroclinic wave, we construct an N-fold generalized Darboux transformation (GDT) via symbolic computation, where N is a positive integer. Via the obtained GDT, several kinds of the non-autonomous localized waves, e.g., the multi-pole solitons, multi-pole breathers, rogue waves and their interactions, are investigated. We have selected four types of the variable coefficients, i.e., constant, linear about t, quadratic about t and trigonometric about t, where t is the normalized retarded time coordinate. We explore the influence of those selected variable coefficients on the non-autonomous localized waves. Moreover, we discuss the bound states among the non-autonomous multiple-pole solitons and one soliton, which exhibit the periodic attractions and repulsions between the adjacent solitons.

Effect of the Quadric Shear Zonal Flows and Beta on the Downstream Development of Certain Unstable Baroclinic Waves

Many factors are affecting the downstream development of baroclinic waves, among which zonal shear flow is one of the factors that need to be considered. In this paper, the influence of zonal shear flow and <em>β</em> on the downstream development of unstable chaotic baroclinic waves is studied from the two-layer model in a wide channel controlled by quasi-geostrophic potential vorticity equation. Through the obtained Lorentz equation, We concentrated on the influence of zonal shear flow (the second derivative of baseline zonal flow is not zero) on the downstream development of baroclinic waves. In the absence of zonal shear flow, chaotic behavior along feature points would occur, and the amplitude would change rapidly from one feature to another, that is, it would change very quickly in space. When zonal shear flow is introduced, the influence of zonal shear flow on the downstream development of unstable baroclinic waves is examined categorically. And from Lorentz’s final equation, we’re investigating a change in his solution. It is found that the zonal shear flow smoothes the solution of the equation and reduces the instability, and with the increase of zonal shear flow, the stability in space will increase gradually. The second derivative of the zonal shear flow (the quadrical shear flow) therefore has a major influence on the stability of space.

Multi-bright-dark soliton solutions to the AB system in nonlinear optics

Abstract The AB system is the basic integrable model to describe unstable baroclinic wave packets in geophysical fluids and the propagation of mesoscale gravity flows in nonlinear optics. On the basis of the spectral analysis of a Lax pair and the inverse scattering method, we establish the Riemann–Hilbert problem of the AB system. Then, the inverse problems are formulated and solved with the aid of the Riemann–Hilbert problem, from which the potentials can be reconstructed according to the asymptotic expansion of the sectional analytic function and the related symmetry relations. As an application, we obtain the multi-bright-dark soliton solutions to the AB system in the reflectionless case and discuss the dynamic behavior of elastic soliton collisions by choosing appropriate free parameters.

Mixed solutions for an AB system in geophysical fluids or nonlinear optics

To reveal and analyze the dynamics of degenerate solitons, breathers and mixed breather-rogue waves, we consider an AB system in this manuscript, which models the marginally unstable baroclinic wave packets in geophysical fluids or ultra-short pulses in nonlinear optics. Based on the generalized Darboux transformation, we construct the degenerate soliton, breather and mixed solutions, then some figures are exhibited to illustrate the collision features clearly and some interesting phenomena are shown: (a) Elastic collisions between two solitons with equal amplitude are observed; (b) Collisions between two breathers are plotted, and maximum values of the two breathers keep invariant; (c) Mixture of breather and rogue waves is illustrated. The above wave phenomena are expected to be experimentally observed and have potential applications.

Soliton solutions of the AB system via the Jacobi elliptic function expansion method

Under investigation in this paper is the AB system, which describes propagations of unstable baroclinic wave packets in geophysical fluids and ultra-short pulses in nonlinear optics. Two finite series of N degree and M degree will be proposed in terms of the extended Jacobi elliptic functions as a set of possible solutions for the AB system, the periodic solutions and new type of soliton solutions will be obtained successfully. Finally some physical properties of soliton solutions will be illustrated via graphics simulation.

Estimated Impacts of Climate Change on Eddy Meridional Moisture Transport in the Atmosphere

Research findings suggest that water (hydrological) cycle of the earth intensifies in response to climate change, since the amount of water that evaporates from the ocean and land to the atmosphere and the total water content in the air will increase with temperature. In addition, climate change affects the large-scale atmospheric circulation by, for example, altering the characteristics of extratropical transient eddies (cyclones), which play a dominant role in the meridional transport of heat, moisture, and momentum from tropical to polar latitudes. Thus, climate change also affects the planetary hydrological cycle by redistributing atmospheric moisture around the globe. Baroclinic instability, a specific type of dynamical instability of the zonal atmospheric flow, is the principal mechanism by which extratropical cyclones form and evolve. It is expected that, due to global warming, the two most fundamental dynamical quantities that control the development of baroclinic instability and the overall global atmospheric dynamics—the parameter of static stability and the meridional temperature gradient (MTG)—will undergo certain changes. As a result, climate change can affect the formation and evolution of transient extratropical eddies and, therefore, macro-exchange of heat and moisture between low and high latitudes and the global water cycle as a whole. In this paper, we explore the effect of changes in the static stability parameter and MTG caused by climate change on the annual-mean eddy meridional moisture flux (AMEMF), using the two classical atmospheric models: the mid-latitude f-plane model and the two-layer β-plane model. These models are represented in two versions: “dry,” which considers the static stability of dry air alone, and “moist,” in which effective static stability is considered as a combination of stability of dry and moist air together. Sensitivity functions were derived for these models that enable estimating the influence of infinitesimal perturbations in the parameter of static stability and MTG on the AMEMF and on large-scale eddy dynamics characterized by the growth rate of unstable baroclinic waves of various wavelengths. For the base climate change scenario, in which the surface temperature increases by 1 °C and warming of the upper troposphere outpaces warming of the lower troposphere by 2 °C (this scenario corresponds to the observed warming trend), the response of the mass-weighted vertically averaged annual mean MTG is -0.2 ℃ per 1000 km. The dry static stability increases insignificantly relative to the reference climate state, while on the other hand, the effective static stability decreases by more than 5.4%. Assuming that static stability of the atmosphere and the MTG are independent of each other (using One-factor-at-a-time approach), we estimate that the increase in AMEMF caused by change in MTG is about 4%. Change in dry static stability has little effect on AMEMF, while change in effective static stability leads to an increase in AMEMF of about 5%. Thus, neglecting atmospheric moisture in calculations of the atmospheric static stability leads to tangible differences between the results obtained using the dry and moist models. Moist models predict ~9% increase in AMEMF due to global warming. Dry models predict ~4% increase in AMEMF solely because of the change in MTG. For the base climate change scenario, the average temperature of the lower troposphere (up to ~4 km), in which the atmospheric moisture is concentrated, increases by ~1.5 ℃. This leads to an increase in specific humidity of about 10.5%. Thus, since both AMEMF and atmospheric water vapor content increase due to the influence of climate change, a rather noticeable restructuring of the global water cycle is expected.

Lie symmetries, conservation laws and solitons for the AB system with time-dependent coefficients in nonlinear optics or fluid mechanics

In this paper, the AB system with time-dependent coefficients for the ultrashort pulses in an inhomogeneous optical fibre or the marginally unstable baroclinic wave packets in an atmospheric or oceanic system is investigated via the Lie symmetry analysis. We obtain the Lie symmetries, reduced equations and group-invariant solutions. The nonlinear self-adjointness of the AB system is proved, and the conservation laws associated with the Lie symmetries are constructed. For the amplitude of the electric field in the inhomogeneous optical fibre or the amplitude of the wave packet in the atmospheric or oceanic system, and for the quantity associated with the occupation number which gives a measure of the atomic inversion in the inhomogeneous optical fibre or the quantity measuring the correction of the basic flow in the atmospheric or oceanic system, we get some solitons through the Lie symmetry transformations, whose amplitudes, widths, velocities and backgrounds are different from those of the given ones and can be adjusted via the Lie group parameters. We find a family of the ultrashort pulses propagating in the inhomogeneous optical fibre or a family of the marginally unstable baroclinic wave packets propagating in the atmospheric or oceanic system.

Dark-soliton collisions for a coupled AB system in the geophysical fluids or nonlinear optics

Under investigation in this paper is a coupled AB system, which describes the marginally unstable baroclinic wave packets in the geophysical fluids or ultra-short pulses in nonlinear optics. As the dark solitons are more resistant against various perturbations than the bright ones, we aim to investigate the dark solitons in the geophysical fluids or nonlinear optics. Dark one- and two-soliton solutions for such a system are derived based on the bilinear forms and propagations of the one solitons and collisions between the two solitons are graphically illustrated and analyzed. Further, influences of the coefficients [Formula: see text] and [Formula: see text] on the solitons are discussed, where [Formula: see text] is a parameter measuring the state of the basic flow and [Formula: see text] is the group velocity. The dark-one solitons with invariant shapes and amplitudes are viewed, and elastic collisions between the dark-two solitons are observed. Also, elastic collision between the bright and dark solitons is viewed, and the dark soliton is found to possess two peaks. [Formula: see text] is found to affect the widths of the dark-one solitons and the travelling directions of the dark-two solitons. It is shown that [Formula: see text] cannot influence shapes of [Formula: see text] and [Formula: see text], but affect the plane of the one soliton for [Formula: see text], and the two-peak dark soliton for [Formula: see text] changes to the single-peak one with the value of [Formula: see text] decreasing, where [Formula: see text] and [Formula: see text] are the packets of short waves and [Formula: see text] is the mean flow.

Collisions between the dark solitons for a nonlinear system in the geophysical fluid

Under investigation in this paper is a nonlinear system, which can be used to describe the marginally unstable baroclinic wave packets in the geophysical fluid. With the help of this nonlinear system, we study the properties of the dark solitons in the geophysical fluid. With the symbolic computation, dark one- and two-soliton solutions for such a system are obtained. Propagations of the one solitons and collisions between the two solitons are graphically shown and discussed with the parameters α and γ, where α measures the state of the basic flow and γ is the group velocity. γ is observed to affect the amplitudes of the dark one and two solitons, i.e., amplitudes of the solitons become higher with the value of γ increasing, and travelling directions of the two solitons can be influenced by γ. α is observed to affect the plane of B, but have no effect on A, where A represents the amplitude of the wave packet, and B is a quantity measuring the correction of the basic flow.

Rogue waves in baroclinic flows

We investigate an AB system, which can be used to describe marginally unstable baroclinic wave packets in a geophysical fluid. Using the generalized Darboux transformation, we obtain higher-order rogue wave solutions and analyze rogue wave propagation and interaction. We obtain bright rogue waves with one and two peaks. For the wave packet amplitude and the mean-flow correction resulting from the self-rectification of the nonlinear wave, the positions and values of the wave crests and troughs are expressed in terms of a parameter describing the state of the basic flow, in terms of a parameter responsible for the interaction of the wave packet and the mean flow, and in terms of the group velocity. We show that the interaction of the wave packet and mean flow and also the group velocity affect the propagation and interaction of the amplitude of the wave packet and the self-rectification of the nonlinear wave.

Solitons and rogue waves for a nonlinear system in the geophysical fluid

In this paper, we investigate a nonlinear system, which describes the marginally unstable baroclinic wave packets in the geophysical fluid. Based on the symbolic computation and Hirota method, bright one- and two-soliton solutions for such a system are derived. Propagation and collisions of the solitons are graphically shown and discussed with [Formula: see text], which reflects the collision between the wave packet and mean flow, [Formula: see text], which measures the state of the basic flow, and group velocity [Formula: see text]. [Formula: see text] is observed to affect the amplitudes of the solitons, and [Formula: see text] can influence the solitons’ traveling directions. By virtue of the generalized Darboux transformation, the first- and second-order rogue-wave solutions are derived. Properties of the first- and second-order rogue waves are graphically presented and analyzed: The first-order rogue waves are shown in the figures. [Formula: see text] has no effects on A, which is the amplitude of the wave packet, but with the increase of [Formula: see text], amplitude of B, which is a quantity measuring the correction of the basic flow, decreases. When [Formula: see text] is chosen differently, A and B do not keep their shapes invariant. With the value of [Formula: see text] increasing, amplitudes of A and B become larger. The second-order rogue wave is presented, from which we observe that with [Formula: see text] increasing, amplitude of B decreases, but [Formula: see text] has no effects on A. Collision features of A and B alter with the value of [Formula: see text] changing. When we make the value of [Formula: see text] larger, amplitudes of A and B increase.

Stationary nonlinear waves, superposition modes and modulational instability characteristics in the AB system

We study the AB system describing marginally unstable baroclinic wave packets in geophysical fluids and also ultra-short pulses in nonlinear optics. We show that the breather can be converted into different types of stationary nonlinear waves on constant backgrounds, including the multi-peak soliton, M-shaped soliton, W-shaped soliton and periodic wave. We also investigate the nonlinear interactions between these waves, which display some novel patterns due to the non-propagating characteristics of the solitons: (1) Two antidark solitons can produce a W-shaped soliton instead of a higher-order antidark one, (2) The interaction between an antidark soliton and a W-shaped soliton can not only generate a higher-order antidark soliton, but also form a W-shaped solion pair, (3) The interactions between an oscillation W-shaped soliton and an oscillation M-shaped soliton show the multipeak structures. We find that the transition occurs at a modulational stability region in a low perturbation frequency region.

Modulation instability and two types of non-autonomous rogue waves for the variable-coefficient AB system in fluid mechanics and nonlinear optics

Under investigation in this paper is a variable-coefficient AB (vcAB) system, which describes marginally unstable baroclinic wave packets in geophysical fluids and ultra-short pulses in nonlinear optics. The modulation instability analysis of solutions with variable coefficients in the presence of a small perturbation is studied. The modified Darboux transformation (mDT) of the vcAB system is constructed via a gauge transformation. The first-order non-autonomous rogue wave solutions of the vcAB system are presented based on the mDT. It is found that the wave amplitude of [Formula: see text] exhibits two types of structures, i.e. the double-peak structure appears if the plane-wave solution parameter [Formula: see text] is equal to zero, while selecting [Formula: see text] yields a single-peak one. Effects of the variable coefficients on the rogue waves are graphically discussed in detail. The periodic rogue wave and composite rogue wave are obtained with different inhomogeneous parameters. Additionally, the nonlinear tunneling of the rogue waves through a conventional hyperbolic nonlinear well and barrier are investigated.

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.

Nonlinear evolution of a baroclinic wave and imbalanced dissipation

AbstractWe consider the nonlinear evolution of an unstable baroclinic wave in a regime of rotating stratified flow that is of relevance to interior circulation in the oceans and in the atmosphere: a regime characterized by small large-scale Rossby and Froude numbers, a small vertical to horizontal aspect ratio and no bounding horizontal surfaces. Using high-resolution simulations of the non-hydrostatic Boussinesq equations and companion integrations of the balanced quasi-geostrophic (QG) equations, we present evidence for a local route to dissipation of balanced energy directly through interior turbulent cascades. That is, analysis of simulations presented in this study suggest that a developing baroclinic instability can lead to secondary instabilities that can cascade a small fraction of the energy forward to unbalanced scales whereas the bulk of the energy is confined to large balanced scales. Mesoscale shear and strain resulting from the hydrostatic geostrophic baroclinic instability drive frontogenesis. The fronts in turn support ageostrophic secondary circulation and instabilities. These two processes acting together lead to a quick rise in dissipation rate which then reaches a peak and begins to fall slowly when frontogenesis slows down; eventually balanced and imbalanced modes decouple. A measurement of the dissipation of balanced energy by imbalanced processes reveals that it scales exponentially with Rossby number of the base flow. We expect that this scaling will hold more generally than for the specific set-up we consider given the fundamental nature of the dynamics involved. In other results, (a) a break is seen in the total energy (TE) spectrum at small scales: while a steep $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}k^{-3}$ geostrophic scaling (where $k$ is the three-dimensional wavenumber) is seen at intermediate scales, the smaller scales display a shallower $k^{-5/3}$ scaling, reminiscent of the atmospheric spectra of Nastrom &amp; Gage and (b) at the higher of the Rossby numbers considered a minimum is seen in the vertical shear spectrum, reminiscent of similar spectra obtained using in situ measurements.

Structure, Energy, and Parameterization of Inertia–Gravity Waves in Dry and Moist Simulations of a Baroclinic Wave Life Cycle

Abstract The impact of moisture on inertia–gravity wave generation is assessed for an idealized unstable baroclinic wave using the Weather Research and Forecasting Model (WRF) in a channel on the f plane. The evolution of these waves in a moist simulation is compared with a dry simulation. The centers of action for inertia–gravity wave activity are identified as the equatorward-moving upper-level front and the poleward-progressing upper-level jet–surface front system. Four stratospheric wave packets are found, which are significantly more intense in the moist simulation and have slightly higher frequency. They are characterized by their structure and position during the baroclinic wave life cycle and are related to forcing terms in jet, front, and convection systems. By exploring the time series of mass and energy, it is shown that the release of latent heat leads to a change in enthalpy, an increase in the eddy kinetic energy, and an intensification of the inertia–gravity wave energy. The ratio of the inertia–gravity wave energy to the eddy kinetic energy is estimated to be about 1/200 for the moist simulation, which is 3 times larger than that for the dry simulation. An empirical parameterization scheme for the inertia–gravity wave energy is proposed, based on the fast large-scale ageostrophic flow associated with the jet, front, and convection. The diagnosed stratospheric inertia–gravity wave energy is well captured by this parameterization in six WRF simulations with different moisture and resolutions. The approach used to construct the parameterization may serve as a starting point for state-dependent nonorographic gravity wave drag schemes in general circulation models.

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.

A global map of meso-scale eddy diffusivities based on linear stability analysis

Using a hydrographic climatology, global maps of meso-scale eddy kinetic energy (EKE), diffusivities for mixing along isopycnals (isopycnal diffusivity) and for the advective effect of meso-scale eddies (skew diffusivity) are created using properties of the fastest growing unstable baroclinic waves and a simple ad hoc scaling of the amplitudes from linear stability theory. Amplitudes of EKE compare well with near-surface observational estimates based on satellite data and results of an eddy-permitting model, but show a low bias in regions where eddies are not generated locally but propagate into, which will likely transfer both to the diffusivities. In agreement with previous studies we find largest diffusivities in the deep Antarctic Circumpolar Current, and in the shallow western boundary and low latitude westward currents. In agreement with analytical consideration, we find that isopycnal diffusivities are increased at the depth of the steering level where unstable waves and mean flow propagate at the same speed, while skew diffusivities exhibit less vertical dependency, and that isopycnal diffusivities are roughly three times larger than skew diffusivities at the steering level. It is shown that the vertical structure of the diffusivities can be explained to a large extent by the effect of the planetary vorticity gradient which leads to a decrease of skew diffusivities at the surface (bottom) and to a downward (upward) shift of the steering level, and thus the maximum of isopycnal diffusivities, for eastward (westward) flow.