The nonlinear effects on the characteristics of gravity wave packets: dispersion and polarization relations

By analyzing the results of the numerical simulations of nonlinear propagation of three Gaussian gravity-wave packets in isothermal atmosphere individually, the nonlinear effects on the characteristics of gravity waves are studied quantitatively. The analyses show that during the nonlinear propagation of gravity wave packets the mean flows are accelerated and the vertical wavelengths show clear reduction due to nonlinearity. On the other hand, though nonlinear effects exist, the time variations of the frequencies of gravity wave packets are close to those derived from the dispersion relation and the amplitude and phase relations of wave-associated disturbance components are consistent with the predictions of the polarization relation of gravity waves. This indicates that the dispersion and polarization relations based on the linear gravity wave theory can be applied extensively in the nonlinear region.     

Seismic inversion with generalized Radon transform based on local second-order approximation of scattered field in acoustic media

Sound velocity inversion problem based on scattering theory is formulated in terms of a nonlinear integral equation associated with scattered field. Because of its nonlinearity, in practice, linearization algorisms (Born/single scattering approximation) are widely used to obtain an approximate inversion solution. However, the linearized strategy is not congruent with seismic wave propagation mechanics in strong perturbation (heterogeneous) medium. In order to partially dispense with the weak perturbation assumption of the Born approximation, we present a new approach from the following two steps: firstly, to handle the forward scattering by taking into account the second-order Born approximation, which is related to generalized Radon transform (GRT) about quadratic scattering potential; then to derive a nonlinear quadratic inversion formula by resorting to inverse GRT. In our formulation, there is a significant quadratic term regarding scattering potential, and it can provide an amplitude correction for inversion results beyond standard linear inversion. The numerical experiments demonstrate that the linear single scattering inversion is only good in amplitude for relative velocity perturbation ( \( \delta_{c}/c_{0} \) ) of background media up to 10 %, and its inversion errors are unacceptable for the perturbation beyond 10 %. In contrast, the quadratic inversion can give more accurate amplitude-preserved recovery for the perturbation up to 40 %. Our inversion scheme is able to manage double scattering effects by estimating a transmission factor from an integral over a small area, and therefore, only a small portion of computational time is added to the original linear migration/inversion process.     

中层耗散大气中Rossby波包的非线性相互作用理论

采用弱非线性近似得出中层耗散大气连续谱Ｒｏｓｓｂｙ波包的非线性时空演化方程，讨论了Ｒｏｓｓｂｙ波包的三波相互作用问题．数值计算表明，耗散和非线性的共同效应决定了Ｒｏｓｓｂｙ波包的演变．当一个Ｒｏｓｓｂｙ波包通过大气传播时，它的振幅若超过某个阈值，空间尺度分别比它大和比它小的两个次级Ｒｏｓｓｂｙ波包的振幅会随时间增长．特别当这两个次级波包同时随时空变化时，仅当主波的振幅超过一个更大的阈值，且其群速度介于两次级波包的群速度之间时，两次级波包的振幅才会随时空同时增长，即出现绝对不稳定现象，耗散和３个波包的频率失配都会增大不稳定的阈值．     

损耗大气中重力波的非线性相互作用理论

基于中高层大气重力波动力学是由非线性过程和损耗过程共同决定的物理思想,本文采用弱非线性相互作用近似,推导出损耗大气中重力波的非线性相互作用方程.这组方程是研究固定相位和随机相位重力波相互作用问题的出发点.通过引入平均振幅,我们得到了损耗情况下离散重力波的三波相互作用方程,该方程描述了重力波波包非线性时空演变的规律.作为该方程的一个具体应用,我们考虑了由于波-波相互作用产生的不稳定性.当一大尺度大振幅的主重力波通过大气传播时,非线性相互作用可能导致两个次级波振幅随时间指数增长.由于分子损耗和频率失配,主波的振幅必须大于一个阈值,这种指数增长才可能出现.共振条件满足时,阈值变为最小.频率失配还会导致次级波本征频率发生改变,改变的大小是频率失配值的一半.     

The nonlinear stabilization of a zonal shear flow instability

Abstract

The development of initially small perturbations in a weakly supercritical zonal shear flow on a β-plane is studied. Two different scenarios of evolution are possible. If the supercriticality is sufficiently small, the growth of a perturbation is stopped in the viscous critical layer regime; for this case the evolution equation (corrected by the inclusion of a quintic nonlinearity) is derived. At greater supercriticality the nonlinearity cannot stop the growth of the perturbation in a linear (viscous or unsteady) critical layer regime, and the evolution is more complicated. Transition to a nonlinear critical layer regime leads to a reduction in the growth rate and to a slowing (but not a stopping) of the increase in amplitude, A. These are connected to the formation of a plateau (S=constant) of width L=O(A ^½) in the profile of absolute vorticity, S. Careful analysis reveals that the growth in amplitude ceases only when the whole instability domain (where the slope of unperturbed S-profile is positive) becomes covered again by the plateau.     

Characteristics of the response of isothermal atmospheres with nonlinear waves

Abstract

We have studied the behavior of nonlinear waves in isothermal atmospheres using an improved beam scheme method. It is shown that although the magnitude of the mass leakage due to side beams is significant, nevertheless in a motion close to hydrostatic equilibrium it can be suppressed, if special density distributions and/or suitable mesh intervals are selected. We adopt as the parameters for our model of the atmospheres M=1 M[odot], R=50 R[odot] and T=4864 K. The time variation of the physical quantities in each cell is found for several oscillation periods. The critical frequency for the model atmosphere in the case of waves with a weak nonlinearity seems to be consistent with that in the linear theory.     

A Formal Theory for Vortex Rossby Waves and Vortex Evolution

A formal theory is presented for the balanced evolution of a small-amplitude, small-scale wave field in the presence of an axisymmetric vortex initially in gradient-wind balance and the accompanying changes induced in the vortex by the azimuthally averaged wave fluxes. The theory is a multi-parameter, asymptotic perturbation expansion for the conservative, rotating, f-plane, shallow-water equations. It extends previous work on Rossby-wave dynamics in vortices and more generally provides a new perspective on wave/mean-flow interaction in finite Rossby-number regimes. Some illustrative solutions are presented for a perturbed vortex undergoing axisymmetrization.     

Barotropic instability of weakly non-parallel zonal flows

Abstract

Barotropic instability of weakly non-parallel zonal flows with localized intense shear regions is investigated numerically. The numerical integrations of the linear stability problem reveal the existence of unstable localized wave packets whose spatial structure and eigenfrequencies depend on two parameters which measure the degree of supercriticality and the zonal length-scale of the shear region. The results indicate that the structure of the instability is determined by conditions that ensure the decay of the wave packet at infinity and the transition from long to short waves across a turning point (critical layer) region which is controlled by non-parallel effects. The controlling influence exerted by the weak non-parallel effects on the evolution of the instability underlines the weakness of the parallel flow assumption which can be used locally, away from critical layers, as a diagnostic tool only.     

The nonlinear spin-up of a stratified ocean

Abstract

The adjustment of a nonlinear, quasigeostrophic, stratified ocean to an impulsively applied wind stress is investigated under the assumption that barotropic advection of vortex tube length is the most important nonlinearity. The present study complements the steady state theories which have recently appeared, and extends earlier, dissipationless, linear models.

In terms of Sverdrup transport, the equation for baroclinic evolution is a forced advection-diffusion equation. Solutions of this equation subject to a “tilted disk” Ekman divergence are obtained analytically for the case of no diffusion and numerically otherwise. The similarity between the present equation and that of a forced barotropic fluid with bottom topography is shown.

Barotropic flow, which is assumed to mature instantly, can reverse the tendency for westward propagation, and thus produce regions of closed geostrophic contours. Inside these regions, dissipation, or equivalently the eddy field, plays a central role. We assume that eddy mixing effects a lateral, down-gradient diffusion of potential vorticity; hence, within the closed geostrophic contours, our model approaches a state of uniform potential vorticity. The solutions also extend the steady-state theories, which require weak diffusion, by demonstrating that homogenization occurs for moderately strong diffusion.

The evoiution of potential vorticity and the thermocline are examined, and it is shown that the adjustment time of the model is governed by dissipation, rather than baroclinic wave propagation as in linear theories. If dissipation is weak, spin-up of a nonlinear ocean may take several times that predicted by linear models, which agrees with analyses of eddy-resolving general circulation models. The inclusion of a western boundary current may accelerate this process, although dissipation will still play a central role.     

Nonlinear dynamics of a stellar dynamo in a spherical shell

Abstract

A simple mean-field model of a nonlinear stellar dynamo is considered, in which dynamo action is supposed to occur in a spherical shell, and where the only nonlinearity retained is the influence of the Lorentz forces on the zonal flow field. The equations are simplified by truncating in the radial direction, while full latitudinal dependence is retained. The resulting nonlinear p.d.e.'s in latitude and time are solved numerically, and it is found that while regular dynamo wave type solutions are stable when the dynamo number D is sufficiently close to its critical value, there is a wide variety of stable solutions at larger values of D. Furthermore, two different types of dynamo can coexist at the same parameter values. Implications for fields in late-type stars are discussed.     

Solitary Rossby Waves in Zonally Varying Jet Flows

The dynamics of solitary Rossby waves (SRWs) embedded in a meridionally sheared, zonally varying background flow are examined using a non-divergent barotropic model centered on a midlatitude g -plane. The zonally varying background flow, which is produced by an external potential vorticity (PV) forcing, yields a modified Korteweg-de Vries (K-dV) equation that governs the spatial-temporal evolution of a disturbance field that contains both Rossby wave packets and SRWs. The modified K-dV equation differs from the classical equation in that the zonally varying background flow, which varies on the same scale as the disturbance field, directly affects the disturbance linear translation speed and linear growth characteristics. In the limit of a locally parallel background flow, equations governing the amplitude and propagation characteristics of SRWs are derived analytically. These equations show, for example, that a sufficiently large (small) translation speed and/or a sufficiently weak (strong) background zonal shear favor transmission (reflection) of the SRW through (from) the jet. Conservation equations are derived showing that time changes in the domain averaged amplitude ("mass") or squared amplitude ("momentum") are due to zonal variation in both the linear, long-wave phase speed and linear growth; dispersion and nonlinearity do not affect the "mass" or "momentum". Provided (1) the background PV forcing is sufficiently small, or (2) the background PV forcing is meridionally symmetric and the disturbance is a SRW, the dynamics of the disturbance field is Hamiltonian and mass and energy are thus conserved. Numerical solutions of the K-dV equation show that the zonally varying background flow yields three general classes of behavior: reflection, transmission, or trapping. Within each class there exists SRWs and Rossby wave packets. SRWs that become trapped within the zonally localized jet region may exhibit the following behaviors: (1) an oscillatory decay to a steady state at the jet center, (2) the creation of additional SRWs within the jet region, or (3) a steady-state wherein the solution has a smoothed step-like structure located downstream along the jet axis.     

Nonlinear aspects of long shelf waves

Abstract

Continental shelf waves are examined in the long wavelength limit, and the effects of weak topographic dispersion calculated. These dispersive effects are then balanced against nonlinear terms and a Korteweg-de Vries equation is derived to describe the evolution of the wave amplitude. Two particular cases are worked in detail.     

A steady nonlinear and singular vortex Rossby wave within a rapidly rotating vortex. Part II: Application to geophysical vortices

In a previous paper, Caillol [Geophys. Astrophys. Fluid Dyn., 2014, 108] investigated the steady nonlinear vortical structure of a singular vortex Rossby mode that has survived to a strong critical-layer-like interaction with a linearly stable, columnar, axisymmetric and dry vortex. We presented a general theory for this wave/mean flow interaction through the nonlinear critical layer theory and calculated the mean azimuthal and axial winds induced at the critical radius at the end of this interaction in the final stage. We here apply that theory to rapidly rotating geophysical vortices: tropical cyclones, cold-air mesocyclones and tornadoes. We find that the numerous assumptions invoked in that paper agree well with the reality of those intense vortices. We also find that in spite of a lack of moist-convection modelling, this dry vortex is fairly well accelerated at the critical radius by such a shear wave with a magnitude of order the square root of the damped-wave amplitude. The intensification level strongly depends on the aspect ratio, height of the system: rapid vortex and parent vortex, over core radius. The thinner the vortex is, the sharper the intensification is. This result is in sharp contrast to the numerous numerical simulations on VR wave/vortex interactions that yield a much smaller intensification of order the square of the wave amplitude. This weakly nonlinear approach nevertheless fails to model small vertical wavelength VR wave/vortex interactions for their related asymptotic expansions are divergent and for they yield strongly nonlinear VR waves coupled with evolving critical layers whose extent can no longer be considered as thin.     

Linear and weakly nonlinear internal wave theories applied to “morning glory” waves

Abstract

Detailed comparisons are made between the predictions of Benjamin's weakly nonlinear theory for internal solitary waves in fluids of great depth, with observational data on solitary wave-type disturbances in the lower atmosphere associated with the “morning glory” phenomenon.

It is shown that, while the theory is not wholly unreasonable, neither is it completely satisfactory. In particular, although the calculated wave speeds are generally close to those observed, they are no improvement on those based on linear long wave theory; at the same time the predicted wave half-widths are too large by a factor of two to three. The limitations of the theory appear to be associated with the requirement that wave half-widths are much less than the total fluid depth, a condition not satisfied in the atmospheric case. However, the alternative theory for shallow fluids, based on the Korteweg-de Vries equation is found to be even more unsuitable.

Our analyses highlight some of the problems in comparing theory with observations and bring to the fore some of the present limitations of the data for such purposes.     

中层大气Rossby波与惯性重力波的非线性相互作用 （Ⅰ）相互作用方程

从包含Ｒｏｓｓｂｙ波和惯性重力波的大气运动方程组出发，采用弱非线性相互作用近似，推导出耗散大气中这两种尺度相差很大的波动之间的非线性相互作用方程．以此为基础，得到了描述窄角谱Ｒｏｓｓｂｙ波包和惯性重力波包的非线性时空演变规律的三波相互作用方程．数值分析表明，当一个Ｒｏｓｓｂｙ波包与两个惯性重力波包发生相互作用时，两个惯性重力波包之间进行快速的能量交换，同时与Ｒｏｓｓｂｙ波包之间进行缓慢的能量传输．从时间尺度上讲，惯性重力波可以看作Ｒｏｓｓｂｙ波包运动的背景噪声，因此上述非线性相互作用过程可以理解为大尺度Ｒｏｓｓｂｙ波包与背景噪声之间的能量交换过程．     

Nonlinear waves on a coupled density front

Abstract

It is shown that the inclusion of the nonlinear terms in the equations of motion of a coupled density front of zero potential vorticity results in wave solutions which merely propagate with time. The linear theory, on the other hand, predicts an exponential temporal growth. The nonlinear equation admits steady solutions representing standing waves whereas if the nonlinear terms are omitted no steady solutions exist. The general initial value problem is difficult to solve numerically since the linear problem is ill posed.

In addition we prove that the general similarity solution of the nonlinear equation tends to zero for large times, at any point in space, regardless of the initial condition.     

Nonlinear galactic dynamos

Abstract

We present trial calculations for a simple nonlinear ‘‘thick disk'’ galaxy dynamo. The nonlinearity is a simple α-quenching. Our strictly axisymmetric solution demonstrates the possibility of a nonlinear interaction between modes of opposite parity. We suggest that a three dimensional model might exhibit a similar persistent interaction between axisymmetric and non-axisymmetric modes.     

Inversion of Travel Times in Weakly Anisotropic Rock Samples

Based on the perturbation theory, inversion formulae for travel time of qP and qS waves in arbitrary weak anisotropy media are presented. The inversion formulae are linear expressions of elastic parameters expressed in terms of weak anisotropy (WA) parameters. The formulae of qS₁ and qS₂ waves have the same form and they can be used without identifying which wave is considered. A synthetic experiment similar to the measurement of rock sample in the laboratory is carried out to illustrate the efficiency of the presented inversion formulae. Two data sets for qP wave travel time from rock samples in the laboratory are inverted and 15 WA parameters are obtained.     

Dielectrophoretic,thermal instability in a spherical shell of fluid

Abstract

This paper is concerned with the dielectrophoretic instability of a spherical shell of fluid. A dielectric fluid, contained in a spherical shell, with rigid boundaries is subjected to a simultaneous radial temperature gradient and radial a.c. electric field. Through the dependence of the dielectric constant on temperature, the fluid experiences a body force somewhat analogous to that of gravity acting on a fluid with density variations. Linear perturbation theory and the assumption of exchange of stabilities lead to an eighth order differential equation in radial dependence of the perturbation temperature. The solution to this equation, satisfying appropriate boundary conditions, yields a critical value of the electrical Rayleigh number and corresponding critical wave number at which convective motion begins. The dependence of each critical number is presented as a function of the gap size and temperature gradient. In the limit of zero shell thickness both the critical Rayleigh number and critical wave number agree with results for the case in the infinite plane problem.     

Vlasov simulations of electron heating by Langmuir turbulence near the critical altitude in the radiation-modified ionosphere

One-dimensional Vlasov equations are solved numerically for conditions appropriate to the ionospheric F-region during the initial stages of HF-radiation modification experiments at two altitudes: one at the critical altitude, the other approximately 1.5 km lower. Numerical simulations of wave growth and saturation with self-consistent evolution of particle distributions are run past the point at which a statistically steady state is reached. At the critical altitude the wave turbulence is dominated by coherent collapsing wave packets or ‘cavitons’ and at the lower altitude by a combination of coherent (strong) and incoherent (weak) turbulence. Our results are consistent with the predictions of Hanssen et al. [Journal of Geophysical Research, 97, 12,073 (1992)]. Semi-open boundary conditions, in which a small fraction of the hot electrons generated by interactions with the strong localized caviton fields are replaced by electrons from the cool background distribution, are employed to model a heated region of finite length that is large compared to the simulation domain. The resultant steady-state electron distributions are characterized by power-law tails of hot electrons superposed on an approximately Maxwellian bulk distribution. The Langmuir-wave dissipation spectra are found to be in good agreement with predictions based on linear Landau damping on the nonthermal electron tails.