On the normal mode instability of modons and Wu-Verkley waves

Abstract

The normal mode instability of steady Wu-Verkley (1993) wave and modons by Verkley (1984, 1987, 1990) and Neven (1992) is considered. All these flows are solutions to the vorticity equation governing the motion of an ideal incompressible fluid on a rotating sphere. A conservation law for infinitesimal perturbations to each solution is derived and used to obtain a necessary condition for its exponential instability. By these conditions, Fjörtoft's (1953) average spectral number of the amplitude of an unstable mode must be equal to a specific number that depends on the degree of the solution in its inner and outer regions as well as on spectral distribution of the mode energy in these regions. Some properties of the conditions for different types of modons are discussed. The maximum growth (and decay) rate of the modes is estimated, and the orthogonality of the amplitude of each unstable, decaying, or non-stationary mode to the basic solution is shown in the energy inner product.

The new instability conditions confine the unstable disturbances of the WV wave and modon to a hypersurface in the perturbation space and allow interpretation of their energy structure. They are also useful both in estimating the maximum growth rate of unstable modes and in testing the numerical algorithms designed for the linear stability study.     

Effects of plant roots on the soil erosion rate under simulated rainfall with high kinetic energy

ABSTRACT

This study examined the effects of herbaceous plant roots on interrill erosion using two herbaceous species: clover (Trifolium repens) and oats (Avena sativa). We developed a simple rainfall simulator with relatively high normalized kinetic energy (KE; 23.2 J m^?2 mm^?1). Under simulated rainfall, we measured eroded soil for 42 boxes with various amounts of aboveground and belowground biomass. Aboveground vegetation had a significant effect on the soil erosion rate (SER). We found a clear negative relationship between the percent vegetation cover (c) and the SER. In contrast, plant roots showed no effects on the SER. The SER was not significantly different between the boxes with and without plant roots under similar c conditions. Thus, plant roots could have less of an effect on the SER under higher KE conditions.

Editor M.C. Acreman Associate editor N. Verhoest     

Application of piezoelectric transducers in simulated rainfall erosivity assessment

Abstract

Rainfall simulators have often been used to mimic natural rainfall for studies of various land-surface and water interaction processes. The characteristics of the simulated rainfall are the main indicators used to judge the performance of the rainfall simulators. The aim of this study is to investigate the potential of piezoelectric transducers for measuring and evaluating a dripper-type simulated rainfall drop-size distribution (DSD) and kinetic energy (KE). The directly measured KE was significantly correlated with the estimated KE using the drop-size distribution and empirical rain drop fall velocity relationships. This result emphasizes the potential use of the piezoelectric sensor to directly measure and evaluate rainfall kinetic energy. Also, the relationship between rainfall intensity and KE showed good patterns of agreement between simulated rainfall and natural rainfall.

Citation Abd Elbasit, M. A. M., Yasuda, H. & Salmi, A. (2011) Application of piezoelectric transducers in simulated rainfall erosivity assessment. Hydrol. Sci. J. 56(1), 187–194.     

Maintenance of the mean kinetic energy in the global ocean by the barotropic and baroclinic energy routes: the roles of JEBAR and Ekman dynamics

In order to determine the maintenance mechanisms of the currents of the global ocean, this study investigates the budget of the annual mean kinetic energy (KE) in a high-resolution (0.1° × 0.1°) semi-global ocean simulation. The analysis is based on a separation of the mean KE using the barotropic (i.e., depth-averaged) and baroclinic (the residual) components of velocity. The barotropic and baroclinic KEs dominate in higher and lower latitudes, respectively, with their global average being comparable to each other. The working rates of wind forcing on the barotropic and baroclinic circulations in the global ocean are 243 and 747 gigawatts, respectively. This study presents at least three new results for the budget of the barotropic KE. Firstly, an energy diagram is rederived to show that the work of the barotropic component of the horizontal pressure gradient (HPG) is connected to the work related to the joint effect of baroclinicity and bottom relief (JEBAR), and then to the budget of potential energy (PE). Secondly, the model analysis shows that the globally averaged work of the barotropic HPG (which is connected to the work related to JEBAR and then to the budget of the PE) is nearly zero. This indicates that the wind- and buoyancy-induced barotropic circulations in the global ocean are of the same strength with opposite sign. Thirdly, it is found that the work of the wind forcing on the barotropic component of the simulated Antarctic Circumpolar Current (ACC) is canceled by the combined effect, in equal measure, of the work of the barotropic HPG and the work of dissipative processes for mean KE. This result makes a significant contribution to the discussion on the depth-integrated momentum balance of the ACC. The barotropic KE is dissipated by the effects of bottom frictional stress, lateral frictional stress, and the Reynolds stress, of which more than half is attributed to an unexpectedly large contribution from biharmonic horizontal friction. Future studies should pay more attention to the role of biharmonic friction used in high-resolution numerical models.     

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.     

Comparison of three types of laser optical disdrometers under natural rainfall conditions

ABSTRACT

Optical disdrometers can be used to estimate rainfall erosivity; however, the relative accuracy of different disdrometers is unclear. This study compared three types of optical laser-based disdrometers to quantify differences in measured rainfall characteristics and to develop correction factors for kinetic energy (KE). Two identical PWS100 (Campbell Scientific), one Laser Precipitation Monitor (Thies Clima) and a first-generation Parsivel (OTT) were collocated with a weighing rain gauge (OTT Pluvio²) at a site in Austria. All disdrometers underestimated total rainfall compared to the rain gauge with relative biases from 2% to 29%. Differences in drop size distribution and velocity resulted in different KE estimates. By applying a linear regression to the KE–intensity relationship of each disdrometer, a correction factor for KE between the disdrometers was developed. This factor ranged from 1.15 to 1.36 and allowed comparison of KE between different disdrometer types despite differences in measured drop size and velocity.     

On the normal mode instability of harmonic waves on a sphere

Abstract

The normal mode instability of harmonic waves in an ideal incompressible fluid on a rotating sphere is analytically studied. By the harmonic wave is meant a Legendrepolynomial flow αP_n(μ) (n ≥ 1) and steady Rossby-Haurwitz wave of set F ₁ ⊕ H_n where H_n is the subspace of homogeneous spherical polynomials of the degree n(n ≥ 2), and F ₁ is the one-dimensional subspace generated by the Legendre-polynomial P₁(μ). A necessary condition for the normal mode instability of the harmonic wave is obtained. By this condition, Fjörtoft's (1953) average spectral number of the amplitude of each unstable mode must be equal to . It is noted that flow αP_n (μ) is Liapunov (and hence, exponentially and algebraically) stable to all the disturbances whose zonal wavenumber m satisfies condition |m| ≥ n. The bounds of the growth rate of unstable normal modes are estimated as well. It is also shown that the amplitude of each unstable, decaying or non-stationary mode is orthogonal to the harmonic wave.

The new instability condition can be useful in the search of unstable perturbations to a harmonic wave and on trials of numerical stability study algorithms. For a Legendre-polynomial flow, it complements Kuo's (1949) condition in the sense that while the latter is related to the basic flow structure; the former characterizes the structure of a growing perturbation.     

Time–frequency analysis to profile hydrological regimes: application to Haiti

Abstract

Comprehensive characterization of its flow rates is prerequisite to a proper understanding and water management of a given hydrological region. Several studies question the soundness of stationarity in time series and suggest the need for a quantification of the events and non-stationary features in flow rate time series. In this study, we combine statistical and time–frequency (TF) analyses to characterize and classify the flow rates of an understudied region, namely Haiti. Wavelet transforms and cyclostationarity analyses were combined with principal component analysis and hierarchical clustering to identify three groups of hydrological regimes in the country, suggesting similar management: (1) relatively stable flow rates with TF behaviour; (2) periodic and strongly seasonal flow rates; and (3) unstable flow rates. We argue that the TF methodology can yield additional information in regard to flow events and multiscale behaviour, even for short records. Flow rate characterization would benefit from the exhaustive approach described here.

EDITOR Z.W. Kundzewicz ASSOCIATE EDITOR E. Toth     

层结海洋中小振幅内行进波的演变和破碎

采用高精度的拟谱方法,数值模拟了层结海洋中小振幅内行进波的演变和破碎过程.在演变过程中,导致内波破碎的PSI不稳定机制在共振相互作用中逐渐占据主导地位,能量从初级波向低频、高波数运动缓慢传递并形成一次级波包,随即破碎发生.破碎后产生的层化湍流引起的强烈混合以及湍流间歇性可从总能量和涡度峰度随时间的变化趋势看出.我们分析了层化湍流的一些统计特性,包括动能和有效位能沿垂向波数ky的功率谱.结果表明,动能和有效位能谱都存在一个谱段满足k-3y律,且分别可表示为01N4k-3y和02N4k-3y(N为Brunt Visl频率),通常称其为浮力子区.另外,我们分析了Cox数(湍流扩散系数与分子扩散系数之比),在层化湍流维持在一定强度时,计算结果和由海洋内区观测(远离内波强生成源和复杂地形)所推测的结论较为吻合.     

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.     

A local stability analysis of astronomical disks

Abstract

A fifth-order dispersion relation describing the local stability of a differentially rotating flow against small perturbations is derived. Finite viscosity and conductivity and both vertical (parallel to the rotation axis) and radial gradients in density, temperature and pressure are included. A general form is assumed for the equation of state, although this is not exploited in the paper. A number of special cases are studied: with negligible viscosity and conductivity, it is shown that modes can often be separated into two high frequency (modified acoustic), two intermediate frequency (combined inertial and internal waves) and a low frequency mode. In convectively unstable situations the intermediate frequency modes may be replaced by a damped/growing pair of instablities. Various criteria for mode excitation are given. It is shown that viscosity always inhibits instability at very short wavelengths, while non-zero conductivity may destabilize the flow. At intermediate wavelengths viscosity could also play a destabilizing role. A parameter study of the effects of fluctuations in the conductivity shows that it could cause mode excitation under certain circumstances.     

基态位涡径向分布对浅水涡旋系统动力稳定性的影响分析

本文基于正压浅水模型,分析基态位涡（Potential Vorticity：PV）结构对热带气旋（Tropical Cyclone：TC）类涡旋系统稳定性及其波动特征的影响.通过引入基态PV结构参数：宽度δ（眼墙内外边界涡度发生陡变的半径长度之比）和中空度γ（眼心相对涡度与内核区域平均相对涡度之比）,设计具有相同基流最大切向风速和最大风速半径的170组不同基态PV环结构的敏感性试验,并讨论了不同基态PV结构下涡旋系统最不稳定波数（the most unstable wavenumber：MUWN）和系统最不稳定模态（the most unstable mode of System：MUMS）的特征频率及其不稳定增长率的大小.结果指出：当PV环较宽,系统表现为低波数最不稳定,相应的MUMS为低频波且增长率小;当PV环较窄,系统表现为高波数不稳定,且PV环越实最不稳定波数越高;当PV环窄且空时,MUMS均为中高频波动,且不稳定增长率随PV环的宽度变窄和中空度变空而明显增大.分析典型PV结构下系统演变特征可知,当PV环较宽,MUMS表现为具有平衡约束的低频波动的线性不稳定特征;当PV环趋向窄且空时,MUMS的平衡性约束趋向弱化,同时不稳定增长表现为明显的指数型增长.进一步讨论系统内部非对称结构的形成和传播机制发现,对于弱不稳定的PV环来说,低波数波最不稳定的特征波动具有典型涡旋Rossby波特征;而对于强不稳定的PV环来说,高波数不稳定的特征波动混合波性质明显.     

Spatial growth of gulf stream meanders

Abstract

An attractive explanation for the observed spatial growth of the Gulf Stream meanders is that the meanders are spatially growing unstable waves. The results of a calculation based on a simple two-layer model of baroclinically unstable flow presented here support this idea. The model is a familiar one with the energy for the growth of the meander perturbations coming from the potential energy available in the geostrophic tilt of the interface between the two layers due to their velocity shear. In order to distinguish between spatial and temporal growth, it IS necessary to assume that the meanders are generated in a localized region, or equivalently, that the meanders are upstream disturbances which are amplified as they enter a region of unstable flow. This assumption is implemented mathematically through the use of a Green's function which governs the propagation of the meanders. Analysis of the spatial and temporal characteristics of the Green's function leads to a criterion which must he satisfied if the meanders arc to grow spatially. This criterion is that the mean flow velocity must be sufficiently greater than the velocity shear, U_m > √2 U_s, in order to have spatial growth. This simply means that the growing meanders must be washed downstream faster than they spread upstream, or equivalently the spatial growth is due to downstream advection of growing disturbances. The actual Gulf Stream flow is in fair agreement with this criterion.     

Probabilistic evaluation of seismic performance of 3‐story 3D one‐ and two‐way steel moment‐frame structures

This paper presents the results of a probabilistic evaluation of the seismic performance of 3D steel moment‐frame structures. Two types of framing system are considered: one‐way frames typical of construction in the United States and two‐way frames typical of construction in Japan. For each framing system, four types of beam–column connections are considered: pre‐Northridge welded‐flange bolted‐web, post‐Northridge welded‐flange welded‐web, reduced‐beam‐section, and bolted‐flange‐plate connections. A suite of earthquake ground motions is used to compute the annual probability of exceedence (APE) for a series of drift demand levels and for member plastic‐rotation capacity. Results are compared for the different framing systems and connection details. It is found that the two‐way frames, which have a larger initial stiffness and strength than the one‐way frames for the same beam and column volumes, have a smaller APE for small drift demands for which members exhibit no or minimal yielding, but have a larger APE for large drift demands for which members exhibit large plastic rotations. However, the one‐way frames, which typically comprise a few seismic frames with large‐sized members that have relatively small rotation capacities, may have a larger APE for member failure. The probabilistic approach presented in this study may be used to determine the most appropriate frame configuration to meet an owner's performance objectives. Copyright © 2008 John Wiley & Sons, Ltd.     

Seasonal variations in internal wave energy in a Scottish sea loch

Near-continuous observations of an internal wave field were made over a period of 13 months at a location in Inchmarnock Water at the northern end of the Clyde Sea. This paper sets out to determine the seasonal form of the energy density of the internal wave field at this location based on the hypothesis that it varies smoothly throughout the year, being greater in summer than in winter. The mooring was maintained between June 1999 and July 2000 in 150-m water with seven deployments. Estimates of kinetic and potential energy density were derived from Acoustic Doppler Current Profiler (ADCP) and vertical temperature profiles respectively. Both were shown to vary on time scales less than 1 month with median values of mean kinetic energy (KE) density0.5 J m^–3 and for mean potential energy (PE) density0.01 J m^–3. The energy of the internal wave field was found to be continuous and without a clear seasonal form. Further, it was also always non-zero with intermittent peaks of much higher energy. In the late autumn the system experienced complete vertical overturning driven by local convective processes destroying the thermocline and causing a reduction in the overall KE density.Responsible Editor: Jens KappeubergOrginally presented as a poster at PECS 2002, Hamburg Germany     

Zonal propagation of kinetic energy and convection in the South China Sea and Indian monsoon regions in boreal summer

As early as in the 1980s, Chinese scientists hadfirst proposed that there exits two summer monsoonsystems in Asia, namely the East Asian summer mon-soon (EASM) and the Indian summer monsoon(ISM)[1-4]. The two monsoon systems are quite dif-ferent in characteristics. Since then, such issue andconclusion had been documented and approved by alot of studies in the past two decades, and was appliedin the guideline of the South China Sea summer mon-soon experiment (SCSMEX), which was undertak…     

Convection and gravity waves in two layer models. I. Overstable modes driven in conducting boundary layers

Abstract

Stability analysis is formulated for a two-layer fluid model in which the upper and lower layers are convectively stable and unstable, respectively. With discontinuities in viscosity and conductivity at the interface, the exchange of stability does not generally hold and overstability is possible. A detailed analytical treatment is presented for the case of small viscosity and conductivity in which viscous and conducting boundary layers are formed at the interface.

The usual damping effect due to the energy dissipation by viscosity and thermal conductivity exists irrespective of whether the mode is the convection or the gravity wave, but, for larger horizontal wave lengths, the effect of the boundary layer can become more important. The jump in the thermal conductivity in the boundary layer can give rise to overstability of the gravity wave in agreement with Souffrin and Spiegel (1967). The jump in the viscosity provides a self-catalytic action for the unstable flow if the viscosity is assumed to be the nonlinear turbulent viscosity due to the motion itself. The effect, however, is not strong enough to overcome the usual viscous damping.     

Hydro-elastic wave proliferation over an impermeable seabed with bottom deformation

ABSTRACT

A hydro-elastic frame has been considered to investigate the proliferation of waves over small base deformation on an infinitely extended flexible seabed. The flexible base surface is assumed as a thin elastic plate of very small thickness and it depends on the Euler–Bernoulli beam equation. For any particular frequency, there are two different modes of time-harmonic propagating wave exists rather than one mode of propagating wave along the positive horizontal direction. The waves with smaller wavenumber spread along the free-surface of the sea (say, free-surface mode) and the waves with higher wavenumber spread along the flexible base surface (say, flexural mode). A simplified perturbation approach is utilised to bring down the entire equations which govern the original boundary value problem (bvp) to a less complex bvp for the first-order velocity potential function. The first-order potential function along with the first-order reflection and transmission coefficients for both modes are calculated by a procedure based upon Fourier transform approach. A shape of sinusoidal swells flexible base surface is taken as an example to approve the scientific results. It is observed that when the train of normal incident propagating wave spreads over base distortion because of either the free-surface unsettling influence or the flexural wave movement in the sea, the reflected and transmitted energy are always feasible to be exchanged from one particular wave mode to another wave mode. Furthermore, we notice that the realistic changes in the flexural rigidity behaviour on the flexible base surface of the sea have a significant effect on the problem of water wave proliferation over small base deformation. Moreover, the energy conservation equation is derived with the help of the Green's integral theorem. The results for the values of reflection and transmission coefficients obtained for both the free-surface unsettling influence as well as flexural wave movement in the fluid are found to satisfy the energy conservation equation almost accurately.     

Some effects of a mean zonal thermocline gradient on planetary equatorial waves

Planetary equatorial waves are studied with the shallow water equations in the presence of a mean zonal thermocline gradient. The interactions between this gradient and waves are represented by three non-linear terms in the equations: one in the wind-forcing formulation in the x-momentum equation, and two for the advection of mass and divergence of the velocity field in the continuity equation. When the mean gradient is imposed but small, these three (linearized) terms will perturb the behavior of the equatorial waves. This paper gives a simple analytic treatment of this problem.The equatorial Kelvin mode is first solved with all three contributions, using a Wentzel-Kramers-Brillouin method. The Kelvin mode shows a spatial or/and temporal growth when the thermocline gradient is negative which is the usual situation in the equatorial Pacific ocean (deep thermocline in the west and shallow in the east). The more robust and efficient contribution comes from the advection term.The single effect of the advection of the mean zonal thermocline gradient is then studied for the Kelvin and planetary Rossby modes. The Kelvin mode remains unstable (damped), while the Rossby modes appear damped (unstable) for a negative (positive) thermocline gradient.