Zonal jets and cyclone–anticyclone asymmetry in decaying rotating turbulence: laboratory experiments and numerical simulations

The problem of zonal jet formation and cyclone–anticyclone asymmetry in decaying rotating turbulence is addressed using both laboratory experiments and numerical simulations with a high-resolution shallow water model in a spherical geometry. Experiments are performed at different Rossby and Froude numbers and applying a rigid wall as meridional boundary in the numerical scheme to mimic the experimental apparatus. The formation of a zonally banded flow pattern, i.e. meridionally confined easterly/westerly jets, has observed; both experimental and numerical results confirmed that this tendency is favoured by high-planetary vorticity gradients. Also, in the experiments characterized by large rotation speeds and small Rossby deformation radius, an initial symmetric distribution of relative vorticity is found to evolve towards a dominance of anticyclonic structures, indicating a breaking of the cyclone–anticyclone symmetry. This aspect has deepened by numerically analysing the sensitivity of the temporal variations of the asymmetry index with respect to the position of the meridional confinement as well as the effect of relaxing the divergence of the fluid (i.e. non-divergent case) to zero. Results suggested that experiments characterized by the higher rotation speed and the lower fluid thickness are better reproduced by a divergent model with a high-latitude meridional boundary.     

Locally-induced nonlinear modes and multiple equilibria in planetary fluids

It is demonstrated that nonlinear Rossby modes, such as modons and IG eddies, can be excited in planetary fluids by a sufficiently strong forcing of potential vorticity. When a weak forcing is balanced with a weak dissipation, two (linear and nonlinear) equilibrium states can be produced, depending on the initial condition. When the fluid is inviscid, a sufficiently strong steady forcing may generate a sequence of propagating nonlinear eddies. A weak forcing, by contrast, only generates linear Rossby waves. The criterion which divides the high amplitude nonlinear state and the low amplitude linear state may be interpreted in terms of a ratio of a time necessary to force the eddy to a time for a fluid particle to circulate about the nonlinear eddy once.     

正压大气模式下具有β效应与地形效应的Rossby孤立波

正压大气模式下,采用摄动方法和时空伸长变换推导了具有β效应、地形效应和耗散的mKdV-Burgers方程,得到Rossby孤立波振幅的演变满足带有β效应,地形与耗散的mKdV-Burgersm方程的结论.说明β效应、地形效应是诱导Rossby孤立波的重要因素.     

The topographically forced component of the atmospheric motion

Summary The zonally asymmetric stationary component of the general circulation is studied for small Rossby number without the beta-plane approximation. The equations for this component are linearized about a mean flow. An analytic solution for the meridional wind is found when the zonal wind and static stability of the mean flow are independent of the vertical coordinate. The solution is used to compute the transports of angular momentum and heat. The angular momentum transports give rise to a net convergence of the order of Rossby number and are balanced by the zonal mean Coriolis torque. However, the heat transports vanish at this order of magnitude.     

Instability of planetary waves in a high vertical resolution model

Abstract

A high vertical resolution model is used to examine the instability of a baroclinic zonal flow and a finite amplitude topographically forced wave. Two families of unstable modes are found, consisting of zonally propagating most unstable modes, and stationary unstable modes. The former have time scale and spatial structure similar to baroclinic synoptic disturbances, but are localized in space due to interaction with the zonally asymmetric forcing. These modes transport heat efficiently in both the zonal and meridional directions. The second family of stationary unstable modes has characteristics of modes of low frequency variability of the atmosphere. They have time scales of 10 days and longer, and are of planetary scale with an equivalent barotropic vertical structure. The horizontal structure resembles blocking flows. They are maintained by available potential energy of the basic wave, and have large zonal heat fluxes. The results for both families of modes are interpreted in terms of an interaction between forcing and baroclinic instability to create favoured regions for eddy development. Applications to baroclinic planetary waves are also considered.     

Generalized dynamic analysis of structural single rocking walls (SRWs)

The investigation of structural single rocking walls (SRWs) continues to gain interest as they produce self-centering lateral load responses with reduced structural damage. The simple rocking model with modifications has been shown to capture these responses accurately if the SRW and its underlying base are infinitely rigid. This paper advances previous rocking models by accounting for (1) the inelastic actions at or near the base of the SRW and (2) the flexural responses within the wall. Included in the proposed advancements are hysteretic and inherent viscous damping associated with these two deformation components so that the total dynamic responses of SRWs can be captured with good accuracy. A system of nonlinear equations of motion is developed, in which the rocking base is discretized into fibers using a zero-length element to locate the associated compressive deformations and damage. The flexural deformations of the rocking body are captured using an elastic term, while the impact events are modeled using impulse-momentum equations. Comparisons with experiments of structural precast concrete and masonry SRWs show that the proposed approach accurately estimates the dynamic responses of different SRWs with and without unbonded posttensioning, for various dynamic excitations and degrees of hysteretic action. Using the proposed approach, a numerical investigation employs different configurations of structural SRWs to quantify the various sources of energy loss, including hysteretic action and impact damping, during various horizontal ground motions.     

纬向非均匀基流对大气长波调整的作用

大气长波的发展和演变影响着大气的可预报性,并对提高天气预报和气候预测水平有重要的意义.在影响大气长波演变的因子中,除波与波非线性相互作用外,基流的作用也非常重要.本文利用非均匀基本场下Rossby波运动方程,通过数值求解,分析了基本场结构和初始场对Rossby波演变的影响,揭示了纬向非均匀基本场对长波调整的作用.研究结果表明:基流纬向非均匀时,线性Rossby波也会出现长波调整现象,基流随纬向变化是长波发生调整的又一个重要机制;大气长波调整对波动的初始振幅不敏感,但基本场振幅影响着长波调整能否出现和出现的时间;基本场纬向平均西风基流的大小除影响波动传播的速度和方向外,还影响长波调整出现的时间和规律;长波调整的出现还与基本场和初始场的结构有关,不同基本场时,波动是否发生调整、向高波数还是向低波数调整都决定于基本场结构,相同基本场时,不同初始结构的波动也有着不同的演变过程.     

Amplification and energy transformation of the magnetized Rossby waves in the ionosphere with an inhomogeneous zonal wind. I. A theory

The generation and further dynamics of the planetary magnetized Rossby waves and inertial waves in a dissipative ionosphere in the presence of a smooth inhomogeneous zonal wind (shear flow) have been studied. The magnetized Rossby waves are caused by the interaction with the spatially inhomogeneous geomagnetic field and represent the ionospheric manifestations of usual tropospheric Rossby waves. The effective linear mechanism of amplification and mutual transformation of the Rossby and inertial waves has been revealed. For shear flows, the operators of linear problems are not self-adjoint, and the corresponding eigenfunctions are non-orthogonal; therefore, a canonical modal approach is of little use in studying such motions. It becomes necessary to apply the so-called nonmodal mathematical analysis, which has actively been developed for the last years. The nonmodal approach makes it possible to reveal that the transformation of wave-like disturbances in shear flows is caused by the nonorthogonality of eigenfunctions in the problem of linear dynamics. Thus, there appear a new degree of the system freedom and a new way of disturbance evolution in the medium.     

Shear flow driven magnetized planetary wave structures in the ionosphere

The generation and further linear and nonlinear dynamics of planetary ultra-low-frequency (ULF) waves are investigated in the rotating dissipative ionosphere in the presence of inhomogeneous zonal wind (shear flow). Planetary ULF magnetized Rossby type waves appear as a result of interaction of the medium with the spatially inhomogeneous geomagnetic field. An effective linear mechanism responsible for the intensification and mutual transformation of large scale magnetized Rossby type and small scale inertial waves is found. For shear flows, the operators of the linear problem are not self-conjugate, and therefore the eigenfunctions of the problem may not be orthogonal and can hardly be studied by the canonical modal approach. Hence, it becomes necessary to use the so-called nonmodal mathematical analysis. The nonmodal analysis shows that the transformation of wave disturbances in shear flows is due to the non-orthogonality of eigenfunctions of the problem in the conditions of linear dynamics. Using numerical modeling, the peculiar features of the interaction of waves with the background flow as well as the mutual transformation of wave disturbances are illustrated in the ionosphere. It has been shown that the shear flow driven wave perturbations effectively extract an energy of the shear flow increasing the own energy and amplitude. These perturbations undergo self-organization in the form of the nonlinear solitary vortex structures due to nonlinear twisting of the perturbation’s front. Depending on the features of the velocity profiles of the shear flows the nonlinear vortex structures can be either monopole vortices or vortex streets and vortex chains.     

The effects of vertical shear and stratification on stationary rossby waves

Abstract

The generation of stationary Rossby waves by sources of potential vorticity in a westerly flow is examined here in the context of a two-layer, quasi-geostrophic, β-plane model. The response in each layer consists of a combination of a barotropic Rossby wave disturbance that extends far downstream of the source, and a baroclinic disturbance which is evanescent or wave-like in character, depending on the shear and degree of stratification. Contributions from each of these modes in each layer are strongly dependent on the basic flows in each layer; the degree of stratification; and the depths of the two layers. The lower layer response is dominated by an evanescent baroclinic mode when the upper layer westerlies are much larger than those in the lower layer. In this case, weak stationary Rossby waves of large wavelengths are confined to the upper layer and the disturbance in the lower layer is confined to the source region.

Increasing the upper layer flow (with the lower layer flow fixed) increases the Rossby wavelength and decreases the amplitude. Decreasing the lower layer flow (with the upper layer flow fixed) decreases the wavelength and increases the amplitude. Stratification increases the contribution from the barotropic wave-like mode and causes the response to be confined to the lower layer.

The finite amplitude response to westerly flow over two sources of potential vorticity is also considered. In this case stationary Rossby waves induced by both sources interact to reinforce or diminish the downstream wave pattern depending on the separation distance of the sources relative to the Rossby wavelength. For fixed separation distance, enhancement of the downstreatm Rossby waves will only occur for a narrow range of flow variables and stratification.     

On steady and travelling waves over bottom topography in the model of homogeneous flow in a beta-plane channel

Abstract

A spectral low-order model is proposed in order to investigate some effects of bottom corrugation on the dynamics of forced and free Rossby waves. The analysis of the interaction between the waves and the topographic modes in the linear version of the model shows that the natural frequencies lie between the corresponding Rossby wave frequencies for a flat bottom and those applying in the “topographic limit” when the beta-effect is zero. There is a possibility of standing or eastward-travelling free waves when the integrated topograhic effect exceeds the planetary beta-effect.

The nonlinear interactions between forced waves in the presence of topography and the beta-effect give rise to a steady dynamical mode correlated to the topographic mode. The periodic solution that includes this steady wave is stable when the forcing field moves to the West with relatively large phase speed. The energy of this solution may be transferred to the steady zonal shear flow if the spatial scale of this zonal mode exceeds the scale of the directly forced large-scale dynamical mode.     

On Rossby wave propagation in a meridionally stratified channel

Rossby wave propagation in the presence of a nonseparable Brunt-Väisälä frequency,N(y,z), and the associated geostrophic zonal flow,U(y,z), is examined in this paper. The usual quasi-geostrophic potential vorticity equation only includes vertical variations in Brunt-Väisälä frequency (i.e.N(z)). We derive a linearised quasi-geostrophic potential vorticity equation which explicitly includesN(y, z), where variations inN may occur on the internal Rossby radius length scale. A mixed layer distribution that monotonically deepens in the poleward direction leads to a nonseparableN(y,z). The resulting meridional pressure gradient is balanced by an eastward zonal geostrophic flow.By assuming mixed layer depth changes occur slowly, relative to a typical horizontal wavelength of a Rossby wave, a local analysis is presented. The Rossby wave is found to have a strongly modulated meridional wavenumber,l, with amplitude proportional to |l|^–1/2. To elucidate whether the modulations of the Rossby wave are caused by the horizontal variations inN orU we also consider the cases where eitherN orU vary horizontally. Mixed layer depth changes lead to largestl where the mixed layer is deepest, whereasl is reduced in magnitude whereU is nonzero. When bothU(y,z) andN(y,z) are present, the two effects compete with one another, the outcome determined by the size of |c|/U _max, wherec is the Rossby wave phase speed. Finally, the slowly varying assumption required for the analytical approach is removed by employing a numerical model. The numerical model is suitable for studying Rossby wave propagation in a rectangular zonal channel with generalN(y, z) andU(y, z).     

Barotropic unstable modes in zonal and meridional channel on the beta-plane

Abstract

The linear stability of a non-divergent barotropic parallel shear flow in a zonal and a non-zonal channel on the β plane was examined numerically. When the channel is non-zonal, the governing equation is slightly modified from the Orr-Sommerfeld equation. Numerical solutions were obtained by solving the discretized linear perturbation equation as an eigenvalue problem of a matrix. When the channel is zonal and lateral viscosity is neglected the problem is reduced to the ordinary barotropic instability problem described by Kuo's (1949) equation. The discrepancy between the stability properties of westward and eastward flows, which have been indicated by earlier studies, was reconfirmed. It has also been suggested that the unstable modes are closely related to the continuous modes discretized by a finite differential approximation. When the channel is non-zonal, the properties of unstable modes were quite different from those of the zonal problem in that: (1) The phase speed of the unstable modes can exceed the maximum value of the basic flow speed; (2) The unstable modes are not accompanied by their conjugate mode; and (3) The basic flow without an inflection point can be unstable. The dispersion relation and the spatial structure of the unstable modes suggested that, irrespective of the orientation of the channel, they have close relation to the neutral modes (Rossby channel modes) which are the solutions in the absence of a basic shear flow. The features mentioned above are not dependent on whether or not the flow velocity at the boundary is zero.     

Truncated model of a barotropic atmosphere with a periodic forcing

The effect of variations in time of the zonal flow is investigated by the study of a simplified truncated model of a barotropic atmosphere in the presence of an oscillating zonal forcing. Long-time numerical simulations of a triadic model in spherical geometry are carried out for various values of both the frequency and the amplitude of the oscillating part of the zonal forcing. It is found that the reaction of the system to simple sinusoidal forcing is characterized, as happens for strongly nonlinear systems, by complicated trajectories in the phase-space and that the spectrum of the zonal component is much more complicated than that of the forcing function, with interesting relative maxima in the range of very low climatological frequencies. Moreover it is shown that, for proper values of both the frequency and the amplitude of the sinusoidally oscillating part of the forcing function, our simplified model of the large-scale planetary circulation oscillates between an essentially zonal regime (a flow pattern dominated by the zonal flow component) and a wave regime (a flow pattern characterized by significant values of the meridional component of the velocity field associated with the wave components). The transitions between the two regimes are strongly asymmetric: in fact, the time needed for a wave-like flow to evolve into an essentially zonal one is, in the limit of our model, typically 4 to 5 times greater than that needed for the inverse transition. The results are intuitively interpreted in the limit of very long periods of the oscillations of the forcing function. Other interesting features of the results are considered.     

The generation of baroclinic rossby waves by meridional oscillations of a zonal wind stress

Abstract

This paper investigates the generation of linear baroclinic Rossby waves by meridional oscillations of a climatological zonal wind stress in a reduced gravity ocean bounded by an eastern coastline. Using a power series technique an analytical solution is derived for the interfacial displacement. It is found that for a given period of oscillation of the zonal winds, a finite number of propagating Rossby waves will be generated with frequencies equal to a harmonic of the forcing frequency. The number of propagating modes increases with increasing period of the wind stress. In addition to the propagating waves the complete solution for the interfacial displacement consists of a rapidly convergent infinite sum of evanescent terms. The displacement field is calculated for atmospheric forcing parameters typical of those found at mid-latitudes. Further, it is shown that a near resonant response can be generated using atmospheric parameter values typical of those found over the North Pacific.     

Effects of wave-wave and wave-mean flow interactions on the evolution of a baroclinic wave

Abstract

The weakly nonlinear evolution of a free baroclinic wave in the presence of slightly supercritical, vertically sheared zonal flow and a forced stationary wave field that consists of a single zonal scale and an arbitrary number of meridional harmonics is examined within the context of the conventional two-layer model. The presence of the (planetary-scale) stationary wave introduces zonal variations in the supercriticality and is shown to alter the growth rate and asymptotic equilibrium of the (synoptic-scale) baroclinic wave via two distinct mechanisms: The first is due to the direct interaction of the stationary wave with the shorter synoptic wave (wave-wave mechanism), and the second is due to the interaction of the synoptic wave with that portion of the mean field that is corrected by the zonally rectified stationary wave fluxes (wave-mean mechanism). These mechanisms can oppose or augment each other depending on the amplitude and spatial structure of the stationary wave field. If the stationary wave field is confined primarily to the upper (lower) layer and consists of only the gravest cross-stream mode, conditions are favorable (unfavorable) for nonzero equilibrium of the free wave.

In addition to the time dependent heat flux generated by baroclinic growth of the free wave, its interaction with a stationary wave field consisting of two or more meridional harmonics generates time dependent heat fluxes that vary with period of the free wave. However, if the stationary wave field contains several meridional harmonics of sufficiently large amplitude, the free baroclinic wave is destroyed.     

Diffusive destabilisation of the baroclinic circular vortex

Abstract

It is shown that, even for vanishingly small diffusivities of momentum and heat, a rotating stratified zonal shear flow is more unstable to zonally symmetric disturbances than would be indicated by the classical inviscid adiabatic criterion, unless σ, the Prandtl number, = 1. Both monotonic instability, and growing oscillations ("overstability") are involved, the former determining the stability criterion and having the higher growth rates. The more σ differs from 1, the larger the region in parameter space for which the flow is stable by the classical criterion, but actually unstable.

If the baroclinity is sufficiently great for the classical criterion also to indicate instability, the corresponding inviscid adiabatic modes usually have the numerically highest growth rates. An exception is the case of small isotherm slope and small σ.

A single normal mode of the linearized theory is also, formally, a finite amplitude solution; however, no theoretical attempt is made to assess the effect of finite amplitude in general. But, in a following paper, viscous overturning (the mechanism giving rise to the sub‐classical monotonic instability when σ > 1) is shown to play an important role at finite amplitude in certain examples of nonlinear steady thermally‐driven axisymmetric flow of water in a rotating annulus. Irrespective of whether analogous mechanisms turn out to be identifiable and important in large‐scale nature, it appears then that a Prandtl‐type parameter should enter the discussion of any attempt to make laboratory or numerical models of zonally‐symmetric baroclinic geophysical or astrophysical flows.     

切变基本纬向流中非线性赤道Rossby长波

为了解决观测和理论研究中的一些问题以及更好地了解热带大气动力学 ,有必要进一步研究基本气流的变化对大气中赤道Rossby波动的影响 .本文研究分析基本气流对赤道Rossby长波的影响 ,利用一个简单赤道 β平面浅水模式和摄动法 ,研究纬向基本气流切变中非线性赤道Rossby波 ,推导出在切变基本纬向流中赤道Rossby长波振幅演变所满足的非线性KdV方程并得到其孤立波解 .分析表明 ,孤立波存在的必要条件是基本气流有切变 ,而且基流切变不能太强 ,否则将产生正压不稳定 .     

热带大洋对纬向和经向风应力的联合响应

考虑了经向风应力和纬向风应力联合作用下热带大洋的响应问题.结果表明,只有一阶的经向风应力或具有辐合辐散的经向风应力才对最后的速度场和位势场造成影响.零阶的扰动温跃层和纬圈流受风应力的直接驱动和Kelvin波、Rossby短波的影响,而Rossby短波由经向风应力直接造成;二阶模则受风应力的直接驱动和Rossby短波的作用,同时经向风应力也产生了附加的Rossby短波.另外,在西边界处存在很强的暖水补充到赤道的现象,经向风应力有使暖水向赤道输送的作用,而西风应力使西边界处的暖水向东输送.