Unbalanced Motion Research Articles

Inertia–gravity waves in inertially stable and unstable shear flows

An inertia–gravity wave (IGW) propagating in a vertically sheared, rotating stratified fluid interacts with the pair of inertial levels that surround the critical level. An exact expression for the form of the IGW is derived here in the case of a linear shear and used to examine this interaction in detail. This expression recovers the classical values of the transmission and reflection coefficients $|T|=\text{e}^{-{\rm\pi}{\it\mu}}$ and $|R|=0$, where ${\it\mu}^{2}=J(1+{\it\nu}^{2})-1/4$, $J$ is the Richardson number and ${\it\nu}$ the ratio between the horizontal transverse and along-shear wavenumbers. For large $J$, a WKB analysis provides an interpretation of this result in term of tunnelling: an IGW incident on the lower inertial level becomes evanescent between the inertial levels, returning to an oscillatory behaviour above the upper inertial level. The amplitude of the transmitted wave is directly related to the decay of the evanescent solution between the inertial levels. In the immediate vicinity of the critical level, the evanescent IGW is well represented by the quasi-geostrophic approximation, so that the process can be interpreted as resulting from the coupling between balanced and unbalanced motion. The exact and WKB solutions describe the so-called valve effect, a dependence of the behaviour in the region between the inertial levels on the direction of wave propagation. For $J<1$ this is shown to lead to an amplification of the wave between the inertial levels. Since the flow is inertially unstable for $J<1$, this establishes a correspondence between the inertial-level interaction and the condition for inertial instability.

A Simple Algorithm for Generating Stable Biped Walking Patterns

paper proposes a thorough algorithm that can tune the walking parameters (hip height, distance traveled by the hip, and times of single support phase SSP and double support phase DSP) to satisfy the kinematic and dynamic constraints: singularity condition at the knee joint, zero-moment point (ZMP) constraint, and unilateral contact constraints. Two walking patterns of biped locomotion have been investigated using the proposed algorithm. The distinction of these walking patterns is that the stance foot will stay fixed during the first sub-phase of the DSP for pattern 1, while it will rotate simultaneously at beginning of the DSP for pattern 2. A seven-link biped robot is simulated with the proposed algorithm. The results show that the proposed algorithm can compensate for the deviation of the ZMP trajectory due to approximate model of the pendulum model; thus balanced motion could be generated. In addition, it is shown that keeping the stance foot fixed during the first sub-phase of the DSP is necessary to evade deviation of ZMP from its desired trajectory resulting in unbalanced motion; thus, walking pattern 1 is preferred practically.

Interpreting Energy and Tracer Spectra of Upper-Ocean Turbulence in the Submesoscale Range (1–200 km)

AbstractSubmesoscale (1–200 km) wavenumber spectra of kinetic and potential energy and tracer variance are obtained from in situ observations in the Gulf Stream region and in the eastern subtropical North Pacific. In the Gulf Stream region, steep kinetic energy spectra at scales between 200 and 20 km are consistent with predictions of interior quasigeostrophic–turbulence theory, both in the mixed layer and in the thermocline. At scales below 20 km, the spectra flatten out, consistent with a growing contribution of internal-wave energy at small scales. In the subtropical North Pacific, the energy spectra are flatter and inconsistent with predictions of interior quasigeostrophic–turbulence theory. The observed spectra and their dependence on depth are also inconsistent with predictions of surface quasigeostrophic–turbulence theory for the observed ocean stratification. It appears that unbalanced motions, most likely internal tides at large scales and the internal-wave continuum at small scales, dominate the energy spectrum throughout the submesoscale range. Spectra of temperature variance along density surfaces, which are not affected by internal tides, are also inconsistent with predictions of geostrophic-turbulence theories. Reasons for this inconsistency could be the injection of energy in the submesoscale range by small-scale baroclinic instabilities or modifications of the spectra by coupling between surface and interior dynamics or by ageostrophic frontal effects.

The Motion Modeling and Simulation of the Swash-Plate Engine

In order to conduct a comprehensive analysis and optimization improvements of the swash-plate engine, it is necessary to analyze and discuss the motion of the swash-plate completely. Based on the motion model of the piston and the connecting rod of the engine, the motion tendencies were simulated. The results suggest the circumferential distance can cause vibration, noise, and rough running of the piston and the connecting rod in the operating process. Besides, the inertia forces/moments and the unbalanced motion are considered to be the main sources of the vibration and noise. On the basis of the results, we propose a “∞” guide groove, which can reduce the vibration and noise of the swash-plate engine, to improve the kinematical parameters.

Diagnosis of systematic analysis increments by using normal modes

AbstractThis paper applies the normal‐mode functions for the three‐dimensional diagnosis of systematic analysis increments in the operational systems of ECMWF and NCEP, in the NCEP/NCAR reanalyses and in the ensemble data assimilation system DART/CAM which is developed at NCAR. Non‐zero systematic increments are interpreted as the analysis system bias. The main region of tropospheric biases in all systems is the Tropics; most of the large‐scale tropical bias resides in the unbalanced (inertio‐gravity) motion with the eastward‐propagating component being dominant in some datasets. The magnitudes of tropospheric wind‐field biases in July 2007 were in the range between 1 ms−1 and 2 ms−1, illustrating the importance of diagnosing analysis systems in the Tropics where magnitudes of the large‐scale variability are of the same order. The systematic increments integrated over the whole models' atmosphere appear predominantly balanced; this balance is associated with biases in the zonally averaged balanced state and, in case of ECMWF and NCEP, with the biases at the model top levels. Spectra of analysis increments averaged over one month show that, on average, 20% to 45% of total energy in increment fields belongs to the inertio‐gravity motions. Copyright © 2010 Royal Meteorological Society

Decoupling of Balanced and Unbalanced Motions and Inertia–Gravity Wave Emission: Small versus Large Rossby Numbers

Abstract This paper provides a brief review of recent results on decoupling of fast [inertia–gravity wave (IGW)] and slow (vortex) motions at small Rossby numbers obtained in the framework of the geostrophic adjustment of localized perturbations. Special attention is paid to the IGW emission and its interpretation in the context of “spontaneous imbalance.” Several mechanisms that lead to spontaneous IGW emission and, thus, to violations of fast–slow splitting at large Rossby numbers are reviewed: Lighthill radiation, symmetric/inertial instability, and ageostrophic shear (Rossby–Kelvin) instability. New results on the saturation of symmetric instability and on the existence of Rossby–Kelvin instability in continuously stratified fluid are presented.

Stability and coupling dynamic behavior of nonlinear journal active electromagnetic bearing rotor system

The stability and coupling dynamic behavior of a journal active electromagnetic bearing rotor system are analyzed. The gyroscopic effect is considered in the rotor model. The system equations are formulated by combining equations for rotor motion and decentralized proportional integral differential (PID) controllers. A method combining the predictor-corrector mechanism and the Netwon-Raphson method is presented to calculate the critical speed at the corresponding Hopf bifurcation point of the system. For periodic motions, a continuation method combining the predictor-corrector mechanism and shooting method is presented. Nonlinear unbalanced periodic motions and their stability margins are obtained using the shooting method and established continuation method for periodic motions. With the change of control parameters, the system local stability and bifurcation behaviors are obtained using the Floquet theory. The numerical examples show that the schemes not only significantly save computing cost, but also have high precision.

Dissipation of Synoptic-Scale Flow by Small-Scale Turbulence

AbstractAlthough it is now accepted that imbalance in the atmosphere and ocean is generic, the feedback of the unbalanced motion on the balanced flow has not received much attention. In this work the parameterization problem is examined in the context of rotating stratified turbulence, that is, with a nonhydrostatic Boussinesq model. Using the normal modes as a first approximation to the balanced and unbalanced flow, the growth of ageostrophic perturbations to the quasigeostrophic flow and the associated feedback are studied. For weak stratification, there are analogies with the three-dimensionalization of decaying 2D turbulence: the growth rate of the ageostrophic perturbation follows a linear estimate, geostrophic energy is extracted from the base flow, and the associated damping on the geostrophic base flow (the “eddy viscosity”) is peaked at large horizontal scales. For strong stratification, the transfer spectra and eddy viscosities maintain this structure if there is synoptic-scale motion and the buoyancy scale is adequately resolved. This has been confirmed for global Rossby and Froude numbers of O(0.1). Implications for atmospheric and oceanic modeling are discussed.

Nonlinear adjustment of a front over escarpment

We present the results of fully nonlinear numerical simulations of the geostrophic adjustment of a pressure front over topography, represented by an escarpment with a linear slope. The results of earlier simulations in the linear regime are confirmed and new essentially nonlinear effects are found. Topography influences both fast and slow components of motion. The fast unbalanced motion corresponds to inertia-gravity waves (IGW). The IGW emitted during initial stages of adjustment break and form the localized dissipation zones. Due to topography, the IGW activity is enhanced in certain directions. The slow balanced motion corresponds to topographic Rossby waves propagating along the escarpment. As shown, at large enough nonlinearities they may trap fluid/tracer and carry it on. There are indications that nonlinear topographic waves form a soliton train during the adjustment process. If the coastal line is added to the escarpment at the shallow side (continental shelf), secondary fronts related to the propagation of the coastal Kelvin waves appear.

Mesoscale Predictability of Moist Baroclinic Waves: Convection-Permitting Experiments and Multistage Error Growth Dynamics

Abstract A recent study examined the predictability of an idealized baroclinic wave amplifying in a conditionally unstable atmosphere through numerical simulations with parameterized moist convection. It was demonstrated that with the effect of moisture included, the error starting from small random noise is characterized by upscale growth in the short-term (0–36 h) forecast of a growing synoptic-scale disturbance. The current study seeks to explore further the mesoscale error-growth dynamics in idealized moist baroclinic waves through convection-permitting experiments with model grid increments down to 3.3 km. These experiments suggest the following three-stage error-growth model: in the initial stage, the errors grow from small-scale convective instability and then quickly [O(1 h)] saturate at the convective scales. In the second stage, the character of the errors changes from that of convective-scale unbalanced motions to one more closely related to large-scale balanced motions. That is, some of the error from convective scales is retained in the balanced motions, while the rest is radiated away in the form of gravity waves. In the final stage, the large-scale (balanced) components of the errors grow with the background baroclinic instability. Through examination of the error-energy budget, it is found that buoyancy production due mostly to moist convection is comparable to shear production (nonlinear velocity advection). It is found that turning off latent heating not only dramatically decreases buoyancy production, but also reduces shear production to less than 20% of its original amplitude.

Ageostrophic instabilities in a horizontally uniform baroclinic flow along a slope

The normal modes of a horizontally uniform, vertically sheared flow over a sloping bottom are considered in two active layers underneath a deep motionless third layer. The variations of the layer thickness are assumed to be small to analyse the sixth-order eigenvalue problem for finite-Froude-number typical for oceanic currents. The dispersion curves for the Rossby waves and the Poincaré modes of inertia–gravity waves (IGW) are investigated to identify the different types of instabilities that occur if there is a pair of wave components which have almost the same Doppler-shifted frequency related to crossover of the branches when the Froude number increases. Simple criteria for ageostrophic instabilities due to a resonance between the IGW and the Rossby wave because of the thickness gradient in either the lower or middle layer, are derived. They exactly correspond to violation of sufficient Ripa's conditions for the flow stability. In both cases the growth rate and the interval of unstable wavenumbers are shown to be proportional to the square root of the corresponding gradient of the layer thickness. These types of ageostrophic instability can coexist (and with Kelvin–Helmholtz instability). However, their role in generating unbalanced motions and mixing processes in geophysical fluids appears limited due to small growth rates and narrow intervals of the unstable wavenumbers in comparison to Kelvin–Helmholtz instability.

Modeling and control of a flexible rotor system with AMB-based sustentation

In this work the modeling and basic control design process of a rotary flexible spindle hovered by Active Magnetic Bearings (AMB) whose good capabilities for machine-tool industry extensively treated in the literature is presented. The modeling takes into account the three main behavioral characteristics of such magnetically-levitated rotor: the rigid dynamics, the flexible dynamics and the rotating unbalanced motion. Besides, the gyroscopic coupling is also studied proving that in this case, its effects are not significant and can be neglected. Using this model, a stabilizing controller based on symmetry properties is successfully designed for the system and a complete experimental analysis of its performance is carried out. Also, the predictions of the model are compared with the actual measured experimental results on a laboratory set-up based on the MBC500 Rotor Dynamics. Afterwards, a brief study about some nonlinear behavior observed in the system and its effect over the system stability at the critical speed is included.

Lagrangian approach to geostrophic adjustment of frontal anomalies in a stratified fluid

Geostrophic adjustment of frontal anomalies in a rotating continuously stratified fluid is studied in the standard framework of strictly rectilinear fronts and jets. Lagrangian approach to this problem is developed allowing to analyze, in a conceptually and technically simple way, both major problems of the nonlinear adjustment: the existence of a smooth adjusted state for a given set of initial conditions and the attainability of the adjusted state during the adjustment process. Dynamical splitting into balanced (adjusted state) and unbalanced (inertia-gravity waves) motions becomes transparent in the Lagrangian approach. Conditions of existence of the balanced state in the unbounded domain are established. It is shown that nonexistence of a smooth adjusted state in the vertically bounded domains is generic and a parallel with the classical scenario of deformation frontogenesis is developed. Small perturbations around smooth adjusted states are then studied with special emphasis on the wave-trapping inside the jet/front. Trapped modes with horizontal scales comparable to the width of the jet are explicitly constructed for a barotropic jet and their evolution is studied with the help of the WKB-approximation for weakly baroclinic jets. Modifications of the standard scenario of adjustment due to subinertial (quasi-) trapped modes and implications for data analysis are discussed.

Local predictability in a simple model of atmospheric balance

Abstract. The 2 degree-of-freedom elastic pendulum equations can be considered as the lowest order analogue of interacting low-frequency (slow) Rossby-Haurwitz and high-frequency (fast) gravity waves in the atmosphere. The strength of the coupling between the low and the high frequency waves is controlled by a single coupling parameter, e, defined by the ratio of the fast and slow characteristic time scales. In this paper, efficient, high accuracy, and symplectic structure preserving numerical solutions are designed for the elastic pendulum equation in order to study the role balanced dynamics play in local predictability. To quantify changes in the local predictability, two measures are considered: the local Lyapunov number and the leading singular value of the tangent linear map. It is shown, both based on theoretical considerations and numerical experiments, that there exist regions of the phase space where the local Lyapunov number indicates exceptionally high predictability, while the dominant singular value indicates exceptionally low predictability. It is also demonstrated that the local Lyapunov number has a tendency to choose instabilities associated with balanced motions, while the dominant singular value favors instabilities related to highly unbalanced motions. The implications of these findings for atmospheric dynamics are also discussed.

T-4-2-3 Analysis of pendulum motion using a four-link mechanism

For increasing the speed of railway vehicles running through curved sections, four-link mechanisms is examined to be used on a car body tilting system. In this research, the main purpose is to reduce the over-centrifugal acceleration of the car body and also the unbalance of the wheel load. An experimental setup using the four-link mechanism is constructed, and the effects of the pendulum motion on these factors are then analyzed. This paper consists of two parts including steady-state analysis of unbalanced motion at a curved section and dynamic analysis of the pendulum motion. The experiment clarifies that (1) the steady state values of the over-centrifugal acceleration and wheel load unbalance are over-centrifugal acceleration and wheel load unbalance are greatly influenced by the link parameters, and (2) the natural frequency of the tilting system also changes corresponding to the link mechanism. A mathematical model is examined to explain these experimental results, considering geometric relation and equation of motion of this vehicle body tilting system.

Motion Balance Filtering

This paper presents a new technique called motion balance filtering, which corrects an unbalanced motion to a balanced one while preserving the original motion characteristics as much as possible. Differently from previous approaches that deal only with the balance of static posture, we solve the problem of balancing a dynamic motion. We achieve dynamic balance by analyzing and controlling the trajectory of the zero moment point (ZMP). Our algorithm consists of three steps. First, it analyzes the ZMP trajectory to find out the duration in which dynamic balance is violated. Dynamic imbalance is identified by the ZMP trajectory segments lying out of the supporting area. Next, the algorithm modifies the ZMP trajectory by projecting it into the supporting area. Finally, it generates the balanced motion that satisfies the new ZMP constraint. This process is formulated as a constrained optimization problem so that the new motion resembles the original motion as much as possible. Experiments prove that our motion balance filtering algorithm is a useful method to add physical realism to a kinematically edited motion.

Piecewise Frontogenesis from a Potential Vorticity Perspective: Methodology and a Case Study

Output from a numerical simulation by the University of Wisconsin Nonhydrostatic Modeling System is used to investigate the evolution of the surface frontal structure of an intense winter cyclone that occurred off the east coast of North America. Using the model data as input to the piecewise potential vorticity (PV) inversion method developed by Davis and Emanuel, the total surface frontogenesis is partitioned into contributions from discrete PV anomalies. These contributions are calculated using the nondivergent balanced winds associated with PV anomalies in the upper, interior, and surface layers as well as the balanced irrotational winds in what is termed piecewise frontogenesis. The cyclone of interest developed into a nearly stationary, well-occluded storm characterized by a cutoff PV anomaly at the tropopause and near-surface frontogenesis concentrated in the warm frontal zone. During the open wave stage of the cyclone life cycle, this frontogenesis was produced largely by the balanced flow, particularly that portion of it associated with the upper-level PV anomaly. Throughout the remainder of the cyclone life cycle, the upper-level contribution to the frontogenesis retained a distribution similar to that of the total frontogenesis. After the cyclone occluded, however, the frontogenesis forced by the unbalanced portion of the flow became increasingly significant despite the persistent smallness of the unbalanced winds. In fact, a substantial portion of the lower-tropospheric frontogenesis observed in the occluded quadrant, including that along a developing bent-back front, was eventually associated with the unbalanced flow. These results suggest a potentially significant role is played by unbalanced motions in the development and maintenance of lowertropospheric frontal structure in midlatitude cyclones.

The role of neural mechanisms of attention in solving the binding problem.

We thank L. Chelazzi, V. Ferrera, P. Fries, S. Kastner, and A. Rossi for helpful comments on the manuscript.

Simplification of potential vorcticity and mesoscale quasi-balanced dynamics model

The physical characteristics of mesoscale are analyzed, and results show that the unbalanced forced motion is the fundamental cause, which leads to the evolution of some important mesoscale weather systems. In this paper, an alternative asymptotic expansion method, which is quite different from the conventional Rossby-number expansion, is used to simplify the potential vorticity equation. And the quasi-balanced (QB) model based on nonlinear balance equation is derived. The QB model, which is in analogy with the quasi-geostrophic model, can describe the fundamental characteristics of the mesoscale accurately and may be used as the basis of theoretical studies on the mesoscale atmospheric dynamics.

Origins and Evolution of Imbalance in Synoptic-Scale Baroclinic Wave Life Cycles

Abstract A set of balance equations is derived that is appropriate for analysis of the three-dimensional anelastic system and is based on expansions in Rossby and Froude number similar to those employed in the study of the shallow-water equations by Spall and McWilliams. Terms that constitute the usual balance equations are formally retained here in addition to non-Boussinesq terms of the same order arising from the vertical variation of the background density field. The authors apply the derived set of equations diagnostically to the analysis of three-dimensional, anelastic numerical simulations of a synoptic-scale baroclinic wave. Of particular interest in this analysis is the degree to which and the time at which the flow becomes appreciably unbalanced, as well as the form of the imbalance itself. Unbalanced motions are here defined as departures from solutions of the balance equations. Application of this analysis procedure allows us to identify two classes of unbalanced motion, respectively: 1) unbal...