Chaotic mixing of tracer and vorticity by modulated travelling Rossby waves

Abstract

We consider the mixing of passive tracers and vorticity by temporally fluctuating large scale flows in two dimensions. In analyzing this problem, we employ modern developments stemming from properties of Hamiltonian chaos in the particle trajectories; these developments generally come under the heading “chaotic advection” or “Lagrangian turbulence.” A review of the salient properties of this kind of mixing, and the mathematics used to analyze it, is presented in the context of passive tracer mixing by a vacillating barotropic Rossby wave. We then take up the characterization of subtler aspects of the mixing. It is shown the chaotic advection produces very nonlocal mixing which cannot be represented by eddy diffusivity. Also, the power spectrum of the tracer field is found to be k ^{? l} at shortwaves—precisely as for mixing by homogeneous, isotropic two dimensional turbulence,—even though the physics of the present case is very different. We have produced two independent arguments accounting for this behavior.

We then examine integrations of the unforced barotropic vorticity equation with initial conditions chosen to give a large scale streamline geometry similar to that analyzed in the passive case. It is found that vorticity mixing proceeds along lines similar to passive tracer mixing. Broad regions of homogenized vorticity ultimately surround the separatrices of the large scale streamline pattern, with vorticity gradients limited to nonchaotic regions (regions of tori) in the corresponding passive problem.

Vorticity in the chaotic zone takes the form of an arrangement of strands which become progressively finer in scale and progressively more densely packed; this process transfers enstrophy to small scales. Although the enstrophy cascade is entirely controlled by the large scale wave, the shortwave enstrophy spectrum ultimately takes on the classical k ^{? l} form. If one accepts that the enstrophy cascade is indeed mediated by chaotic advection, this is the expected behavior. The extreme form of nonlocality (in wavenumber space) manifest in this example casts some doubt on the traditional picture of enstrophy cascade in the Atmosphere, which is based on homogeneous two dimensional turbulence theory. We advance the conjecture that these transfers are in large measure attributable to large scale, low frequency, planetary waves.

Upscale energy transfers amplifying the large scale wave do indeed occur in the course of the above-described process. However, the energy transfer is complete long before vorticity mixing has gotten very far, and therefore has little to do with chaotic advection. In this sense, the vorticity involved in the enstrophy cascade is “fossil vorticity,” which has already given up its energy to the large scale.

We conclude with some speculations concerning statistical mechanics of two dimensional flow, prompted by our finding that flows with identical initial energy and enstrophy can culminate in very different final states. We also outline prospects for further applications of chaotic mixing in atmospheric problems.     

时空变化风场中声重波波包的演变

本文从可压缩情况下的流体力学方程出发,采用多重尺度方法和W.K.B.方法,研究时空变化风场中声重波波包的演变过程。文中导出了声重波波长、相速、传播方向以及振幅的演变方程。结果表明,沿群路径上波长、相速和传播方向的变化取决于风场的时空变化以及风场与波的相对方位。此外,波振幅的演变方程还表明:在时空慢变风场中,声重波的波作用量具有守恒性,声重波波包能量增加伴随波频率增高,声重波波包能量减小伴随频率降低。能量变化的实质是,风场的时变性引起了波流相互作用。     

Instability of a geostrophic front and its energetics

Abstract

The instability of a current with a geostrophic surface density front is investigated by means of a reduced gravity model having a velocity profile with nearly uniform potential vorticity. It is shown that currents are unstable when the mean potential vorticity decreases toward the surface front at the critical point of the frontal trapped waves investigated by Paldor (1983). This instability is identical with that demonstrated by Killworth (1983) in the longwave limit.

The cross-stream component of mass flux and the rates of energy conversions among the five energy forms defined by Orlanski (1968) are also calculated. The main results are as follows, (a) The mass flux toward the surface front is positive near the front and negative around the critical point. The positive mass flux near the front does not vanish at the position of the undisturbed surface front, so that the mean position of the front moves outward and the region of the strong current spreads. (b) The potential energy of the mean flow integrated over the fluid is released through the work done by the force of the pressure gradient of the mean flow on the fluid, and is converted into the kinetic energy of the mean flow. (c) In the critical layer, the mean flow is rapidly accelerated with the growth of the unstable wave. This acceleration is caused by the rapid phase shift of the unstable wave in the critical layer.     

A one-dimensional turbulence model; Enstrophy cascades and small-scale intermittency in decaying turbulence

Abstract

Severe unidirectional Fourier truncation of the equations for 2-D incompressible flow leads to a system of three coupled PDEs in one space dimension with the same quadratic invariants as the original set (i.e. energy and enstrophy). Numerically generated equilibria for inviscid, truncated versions of the reduced system are well approximated by Kraichnan's energy-enstrophy equipartition spectra. Viscous calculations for decaying turbulence at moderate resolution (1024 degrees of freedom) also appear to be consistent with a direct, k ^?3, enstrophy cascading inertial range when the dissipation is small. Dissipation range intermittency in the form of spatially intermittent enstrophy dissipation with occasional strong bursts producing linear phase locking is also observed. In contrast to full 2-D simulations, no tendency towards the emergence of isolated, coherent vorticity structures is observed. The model consequently mimics some, but not all, of the properties of the full 2-D set.     

Several effects of baroclinic currents on the three‐dimensional propagation of inertial‐internal waves†

A ray theory is applied to the problem of three‐dimensional propagation of inertial‐internal waves in the presence of a mean baroclinic current which does not vary in the downstream coordinate. As time increases, the Doppler‐shifted wave frequency, or intrinsic frequency, tends to a limiting value determined by the horizontal and vertical variations of the mean current and density fields. The limiting value of the intrinsic frequency determines critical surfaces where energy is transferred to the mean motion. Also, the group velocity tends to the mean current velocity, and the phase velocity tends to be oriented towards or away from the core of the mean current, depending upon whether the wave is either initially propagating with a wave number component antiparallel or parallel to the mean current.     

Role of wave–mean flow interaction in sun–climate connections: Historical overview and some new interpretations and results

Quasi-decadal variations in solar irradiance – termed the 11-year solar cycle (SC) – have been linked to variations in a variety of atmospheric circulation features, including the polar vortex, the Brewer–Dobson circulation, and the quasi-biennial oscillation. These features share an underlying commonality: they are all rooted in wave–mean flow interaction. The purpose of this paper is to provide a historical overview of the connection between the SC and wave–mean flow interaction and to propose a more complete theoretical framework for solar modulated wave–mean flow interaction that includes both zonal-mean and zonally asymmetric ozone as intermediaries for communicating variations in solar spectral irradiance to the climate system. We solve a quasi-geostrophic model using the WKB formalism to highlight the physics connecting the SC to planetary wave-drag. Numerical results show the importance of the zonally asymmetric ozone field in mediating the effects of solar variability to the wave-driven circulation in the middle atmosphere.     

Wave to wave and wave to zonal mean flow kinetic energy exchanges during contrasting monsoon years

Fourier analysis of the monthly mean northern hemispheric geopotential heights for the levels 700 mb and 300 mb are undertaken for the months of April through to August. The wave to wave and wave to zonal mean flow kinetic energy interactions are computed for specified latitude bands of the northern hemisphere during the pre-monsoon period (April to May) and monsoon period (June through to August) for bad monsoon years (1972, 1974, 1979) and for years of good monsoon rainfall over India (1967, 1973, 1977). Planetary scale waves (waves 1 to 4) are the major kinetic energy source in the upper atmosphere during the monsoon months. Waves 1 and 2 in particular are a greater source of kinetic energy to other waves via both wave to wave interactions as well as wave to zonal mean flow interactions in good monsoon years than in bad monsoon years. The zonal mean flow shows significantly larger gains in the kinetic energy with a strengthening of zonal westerlies in good monsoon years than in bad monsoon years.     

Dynamics and spectra of cumulant update closures for two-dimensional turbulence

Abstract

Non-Markovian closure theories are compared with ensemble averaged direct numerical simulations (DNS) for decaying two-dimensional turbulence at large scale Reynolds numbers ranging from ≈ 50 to ≈ 4000. The closures, as well as DNS, are formulated for discrete wave numbers relevant to flows on the doubly periodic domain and are compared with the results of continuous wave number closures. The direct interaction approximation (DIA), self-consistent field theory (SCFT) and local energy-transfer theory (LET) closures are also compared with cumulant update versions of these closures (CUDIA, CUSCFT, CULET). The cumulant update closures are shown to have comparable performance to the standard closures but are much more efficient allowing long time integrations.

The discrete wave number closures perform considerably better than continuous wave number closures as far as evolved energy and transfer spectra and skewness are concerned. The discrete wave number closures are in reasonable agreement with DNS in the energy containing range of the large scales for Reynolds numbers ranging from ≈ 50 to ≈ 4000. The closures tend to underestimate the enstrophy flux to high wave numbers, increasingly so with increasing Reynolds number, resulting in underestimation of small-scale kinetic energy.     

Ion--neutral friction as a model for magnetic relaxation

It is generally admitted that a plasma in the absence of forcing will relax to a minimum energy state compatible with appropriate constraints. Usually this is a force-free state, which, in two dimensions, implies a potential magnetic field except by the possible presence of current sheets. The precise mechanism of this relaxation, and in particular the plasma velocity, are generally ignored. There exists, however, a physically well-defined process that should produce magnetic relaxation: ion--neutral (or ambipolar) friction. While there is no guarantee of the existence of a limit of this process when t?→?∞, there exists a family of sequential limits for whom the Lorentz force tends to zero. To analyze the configuration of these limit states, we study the evolution of several moments of the magnetic energy. We prove that for as long as the enstrophy remains bounded, the current density energy also remains bounded in two dimensions: this excludes all classical configurations of current sheets across which the magnetic field reverses direction. Hence, these sheets cannot be the limit of ion--neutral diffusion unless the flow becomes increasingly irregular over time.     

Impact of partial steps and momentum advection schemes in a global ocean circulation model at eddy-permitting resolution

Series of sensitivity tests were performed with a z-coordinate, global eddy-permitting (1/4°) ocean/sea-ice model (the ORCA-R025 model configuration developed for the DRAKKAR project) to carefully evaluate the impact of recent state-of-the-art numerical schemes on model solutions. The combination of an energy–enstrophy conserving (EEN) scheme for momentum advection with a partial step (PS) representation of the bottom topography yields significant improvements in the mean circulation. Well known biases in the representation of western boundary currents, such as in the Atlantic the detachment of the Gulf Stream, the path of the North Atlantic Current, the location of the Confluence, and the strength of the Zapiola Eddy in the south Atlantic, are partly corrected. Similar improvements are found in the Pacific, Indian, and Southern Oceans, and characteristics of the mean flow are generally much closer to observations. Comparisons with other state-of-the-art models show that the ORCA-R025 configuration generally performs better at similar resolution. In addition, the model solution is often comparable to solutions obtained at 1/6 or 1/10° resolution in some aspects concerning mean flow patterns and distribution of eddy kinetic energy. Although the reasons for these improvements are not analyzed in detail in this paper, evidence is shown that the combination of EEN with PS reduces numerical noise near the bottom, which is likely to affect current–topography interactions in a systematic way. We conclude that significant corrections of the mean biases presently seen in general circulation model solutions at eddy-permitting resolution can still be expected from the development of numerical methods, which represent an alternative to increasing resolution.     

Wave-mean flow interaction and its relationship with the atmospheric energy cycle with diabatic heating

In a linear system, the wave characteristics depend strongly on the distributions with height of wind ve-locity and static stability. The simplest case is for con-stant wind and static stability (e.g., isothermal atmos-phere with pressure and density exponentially de-creasing with height). In such circumstances there is no convergence or divergence in wave energy flux, therefore, no energy exchange between the wave and mean flow. In the atmosphere wind speed varies with increasing height, inte…     

Statistical properties of an enstrophy conserving finite element discretisation for the stochastic quasi-geostrophic equation

ABSTRACT

A framework of variational principles for stochastic fluid dynamics was presented by Holm, and these stochastic equations were also derived by Cotter, Gottwald and Holm. We present a conforming finite element discretisation for the stochastic quasi-geostrophic equation that was derived from this framework. The discretisation preserves the first two moments of potential vorticity, i.e. the mean potential vorticity and the enstrophy. Following the work of Dubinkina and Frank, who investigated the statistical mechanics of discretisations of the deterministic quasi-geostrophic equation, we investigate the statistical mechanics of our discretisation of the stochastic quasi-geostrophic equation. We compare the statistical properties of our discretisation with the Gibbs distribution under assumption of these conserved quantities, finding that there is an agreement between the statistics under a wide range of set-ups.     

Baroclinic nonlinear exchanges of energy and potential enstrophy in the quasi-geostrophic two-layer model

Abstract

Merilees and Warn's (1975) nonlinear interaction analysis of two-dimensional nondivergent flow is extended to examine the quasi-geostrophic two-layer model. Two sets of triads exist in this model (Salmon, 1978). The purely barotropic triads are the same as the triads examined by Merilees and Warn. Baroclinic-barotropic triads are found to exchange more energy or potential enstrophy with smaller or larger scales depending on the scale of motion as compared with the internal Rossby deformation radius and the relative wavenumber position of baroclinic and barotropic components.     

夏季东北亚阻塞高压年际变化的一个物理机制

根据实际观测资料反演获得描述大气环流演变的空间谱函数后,从改进的高截断谱模式途径出发研究了夏季东北亚阻塞高压年际变化的物理机制.结果表明,前期外部热源强迫的空间分布大致为El Nio型分布时,外部热力强迫导致大气环流演变中波波相互作用主要表现为纬向2波的相互作用;波流相互作用主要表现为经向2波和3波与反映基本流中的经向1波的相互作用.这样使得500 hPa高度场上东北亚地区为一相对正异常区,为夏季东北亚阻塞的频繁发生提供了有利的大气环流背景.而前期外部热源强迫大致为La Nia型分布时,外部热力强迫则导致大气环流演变中波波相互作用主要表现为纬向1波的相互作用;波流相互作用主要表现为经向2波和4波与反映基本流中的经向2波的相互作用.从而使得500 hPa高度场上帆北亚地区出现相对负异常,抑制了夏季东北亚阻塞的发生.     

Development of bedforms due to waves blocked by a counter current

Experiments are conducted in a laboratory flume on the propagation of a surface wave against unidirectional flow with a sediment bed. This article presents the spatial variation of bedforms induced by the wave-blocking phenomenon by a suitably tuned uniform fluid flow and a counter-propagating wave. The occurrence of wave-blocking is confirmed by finding a critical wave frequency in a particular flow discharge in which the waves are effectively blocked and is established using the linear dispersion relation. The purpose of this work is to identify wave-blocking and its influence on the development of bedforms over the sediment bed. Interestingly bedform signatures are observed at a transition of bedforms in three zones, with asymmetric ripples having a steeper slope downstream face induced by the incoming current, followed by flat sand bars beneath the wave-blocking zone and more symmetric ripples below the wave-dominated region at the downstream. This phenomenon suggests that the sediment bed is segmented into three different regions of bed geometry along the flow. The deviations of mean flows, Reynolds stresses, turbulent kinetic energy, and power spectral density due to the wave-blocking phenomenon are presented along the non-uniform flow over sediment bed. The bottom shear stress, bed roughness and stochastic nature of the bedform features are also discussed. The results are of relevance to engineers and geoscientists concerned with contemporary process as well as those interested in the interpretation of palaeoenvironmental conditions from fossil bedforms. © 2019 John Wiley & Sons, Ltd.     

Wave-induced setup of the mean surface over a sloping beach

A new theoretical approach for the wave-induced setup over a sloping beach is presented that takes into consideration the explicit variations of the surface waves due to bottom slope and viscosity. In this way, the wave forcing of the mean Lagrangian volume fluxes is calculated without assuming that the local depth is constant. The analysis is valid in the region outside the surf zone and is based on the shallow-water assumption. A novel approach for separating the viscous damping of the waves from the frictional damping of the mean flow is introduced, where the mean Eulerian velocity is applied in the bottom stress for the mean fluxes. In the case where the onshore Lagrangian mean transport is zero, a new formula is derived for the Eulerian mean free surface slope, in which the effects of bottom slope, viscous wave damping and frictional bottom drag on the mean flow are clearly identified. The analysis suggests that viscous damping of the waves and frictional dissipation of the Eulerian near-bed return flow could lead to setup outside the surf zone.     

On the derivation of the energy flux of linear magnetohydrodynamic waves

Abstract

New light is shed on the derivation of the energy flux of the linear MHD waves. It is shown that, according to a suggestion of Lighthill, the usual perturbation procedure, which starts from the general expression for the energy flux, need not be supplemented by an averaging procedure. As a result, it is shown that to second order in the wave amplitude, a quantity identifiable as the wave energy flux is conserved. Some of the subtleties inherent in the derivation of the pertubation energy equation are discussed.     

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.     

Rossby wave resonance in the presence of a nonlinear critical layer

Abstract

The behavior of Rossby waves on a shear flow in the presence of a nonlinear critical layer is studied, with particular emphasis on the role played by the critical layer in a Rossby wave resonance mechanism. Previous steady analyses are extended to the resonant case and it is found that the forced wave dominates the solution, provided the flow configuration is not resonant for the higher harmonics induced by the critical layer. Numerical simulations for the forced initial value problem show that the solution evolves towards the analysed steady state when conditions are resonant for the forced wave, and demonstrate some of the complications that arise when they are resonant for higher harmonics. In relating the initial value and steady problems, it is argued that the time dependent solution does not require the large mean flow distortion that Haberman (1972) found to be necessary outside the critical layer in the steady case.