Finite-amplitude mountain waves in a compressible atmosphere including the effects of dissipation

Abstract

Adiabatic, two-dimensional, steady-state finite-amplitude, hydrostatic gravity waves produced by flow over a ridge are considered. Nonlinear self advection steepens the wave until the streamlines attain a vertical slope at a critical height z_c. The height z_c , where this occurs, depends on the ridge crest height and adiabatic expansion of the atmosphere. Dissipation is introduced in order to balance nonlinear self advection, and to maintain a marginal state above z_c. The approach is to assume that the wave is inviscid except in a thin layer, small compared to a vertical wavelength, where dissipation cannot be neglected. The solutions in each region are matched to obtain a continuous solution for the streamline displacement δ. Solutions are presented for different values of the nondimensional dissipation parameter β. Eddy viscosity coefficients and the thickness of the dissipative layer are expressed as functions of β, and their magnitudes are compared to other theoretical evaluations and to values inferred from radar measurements of the stratosphere.

The Fourier spectrum of the solution for z ≫ z_c is shown to decay exponentially at large vertical wave numbers n. In comparison, a spectral decay law n ^?-8/3 characterizes the marginal state of the wave at z = z_c .     

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.     

Energy spectra associated with long's model for steady finite-amplitude flow over orography

Abstract

The process of wave steepening in Long's model of steady, two-dimensional stably stratified flow over orography is examined. Under conditions of the long-wave approximation, and constant values of the background static stability and basic flow, Long's equation is cast into the form of a nonlinear advection equation. Spectral properties of this latter equation, which could be useful for the interpretation of data analyses under mountain wave conditions, are presented. The principal features, that apply at the onset of convective instability (density constant with height), are:

i) a power spectrum for available potential energy that exhibits a minus eight-thirds decay, in terms of the vertical wavenumber k _z ^-;

ii) a rate of energy transfer across the spectrum that is inversely proportional to the wavenumber for large k _z ^-;

iii) an equipartition between the kinetic energy of the horizontal motion and the available potential energy, under the longwave approximation, although all the disturbance energy is kinetic at the point where convective instability is initiated. It is also shown that features i) and ii) apply to more general conditions that are appropriate to Long's model, not just the long-wave approximation. Application to fully turbulent flow or to conditions at the onset of shearing instability are not considered to be warranted, since the development only applies to conditions at the onset of convective instability.     

Characterisation of Fulvic Acids and Glycyrrhizic Acid by Time‐of‐Flight Secondary Ion Mass Spectrometry

Fulvic acids of different origin, spray deposited on polished silicon after dissolution in high‐purity water without any additives, were analysed by time‐of‐flight secondary ion mass spectrometry (TOF‐SIMS) in combination with oblique 24...36 keV SF₅⁺ ion bombardment. The observed, highly reproducible mass spectra cover more than five orders of magnitude in dynamic range, without background subtraction. Apart from lines due to atomic ions and low‐mass ion fragments, the mass spectra exhibit broad maxima between m/z 200...350, mainly due to a beat‐like superposition of lines at every single mass number, up to at least m/z 400. In the negative ion spectra the beats have a spacing of m/z 14, corresponding to a CH₂‐unit. The high‐mass tails of the spectra extend well beyond m/z 5000, with similar slopes in the positive and the negative ion mass spectra. The negative spectra appear to be less affected by fragmentation products than the positive spectra. Fulvic acids (FAs) of different origin show distinctly different spectra, with mean masses ranging between m/z ≈ 450 and 580 (for a low‐mass cut‐off m/z 150). To further verify the ability of TOF‐SIMS to detect molecules and clusters with masses significantly above the maxima of the FA spectra, samples of glycyrrhizic acid (GA, as GA ammonium salt with molecular weight 840) were also analysed. Parent ions as well as multimers (GA)_n were observed as positive and negative ions, up to n = 4 (m/z 3320). The results are compared with spectra recently obtained by other mass spectrometric techniques.     

Magnetic equilibria resulting from a helically symmetric kink instability

Abstract

This paper explores magnetic equilibria which could result from the kink instability in a cylindrical magnetic flux tube. We examine a variety of cylindrical magnetic equilibria which are susceptible to the kink, and simulate its evolution in a frictional fluid. We assume that the evolution takes place under conditions of helical symmetry, so the problem becomes effectively two-dimensional. The initial cylindrical equilibrium field is specified in terms of its twist function k(r) = B _θ/(rB_z ) and for a variety of k(r) functions we calculate linear growth rates for the kink instability, assuming that it develops under helical symmetry with pitch τ. We find that the growth rate is sensitive to the value of τ.

We simulate nonlinear evolution of the kink using a Lagrangian frictional code which constrains the field to have helical symmetry of a given pitch τ. Ideal MHD is assumed and the plasma pressure is taken to be small in order to mimic conditions in the solar corona. In some cases the flux tube evolves to a new smooth helically symmetric equilibrium which involves a relatively small change in the maximum electric current. In other cases there is evidence of current-sheet formation.     

Nonlinear development of phase-mixed alfvén waves

Abstract

We derive an equation governing the nonlinear propagation of a linearly polarized Alfvén wave in a two-dimensional, anisotropic, slightly compressible, highly magnetized, viscous plasma, where nonlinearities arise from the interaction of the Alfvén wave with fast and slow magnetoacoustic waves. The phase mixing of such a wave has been suggested as a mechanism for heating the outer solar atmosphere (Heyvaerts and Priest, 1983).

We find that cubic wave damping dominates shear linear dissipation whenever the Alfvén wave velocity amplitude δv_y exceeds a few times ten metres per second. In the nonlinear regime, phase-mixed waves are marginally stable, while non-phase-mixed waves of wavenumber k_a are damped over a timescale k_uRe ₀|δ v_y/v_A |^?2, Re ₀ being the Reynolds number corresponding to the Braginskij viscosity coefficient η₀ and v_A the Alfvén speed. Dissipation is most effective where β = (v_s /v_A) ² ≈ 1, v_s being the speed of sound.     

Penetrative convection in the planetary mantle

Abstract

The hydrodynamic equations for thermal convection in a plane layer of viscous, heat conducting fluid are scaled using the normalization of Ostrach (1965) in which the magnitude of the non-dimensional group τ = gαd/c_p determines the importance of compression work and viscous dissipation in the energy balance of the flow. A linear asymptotic theory valid in the limit τ → ∞ is constructed for the Bénard problem and this is shown to be analogous to Couette flow between contra-rotating cylinders. For sufficiently large τ the flow becomes penetrative. This fact is illustrated for homogeneous fluids by the numerical integration of a set of coupled 1st order differential equations, both for the Bénard and internally heated configurations. The effect of viscosity and thermal conductivity in-homogeneity on the depth of penetration of the main cell in the circulation pattern are assessed and it is concluded that such interactions may be sufficient to effectively limit the depth extent of mantle convection. Finally a discussion of the effect of phase transitions is given following the technique of Busse and Schubert (1971).     

The equatorial dynamics of a deep homogeneous ocean

Abstract

The flow properties of an homogeneous fluid which is bounded by two concentric spheres and two meridional planes which intersect along a diameter of the spheres are investigated. The spheres rotate about this diameter with slightly different angular velocities. As in the axisymmetric case studied by Proudman (1956) and Stewartson (1966) the viscous terms in the equations of motion are important only in boundary layers on the spheres and on the cylinder C which circumscribes the inner sphere and which has generators parallel to the axis of rotation, provided the Ekman number E is small. In the inviscid region the velocities are independent of the coordinate measuring distance along the axis of rotation and are much weaker, by a factor 0(E ^½), than the velocities in the Ekman layer on the driving surface (outer sphere). (It is assumed that the reference frame is fixed in the slower rotating inner sphere.) If the separation of the spheres is small compared to their radii then the asymmetric circulation inside C is characterized by an intense jet along the western wall. Loss of fluid from this jet sustains the eastward and northward flow in the inviscid interior where motion is driven by the suction of the Ekman layer on the outer sphere. (Geophysical conventions have been adopted.) Outside C an intense current is present on the eastern, not western, wall while motion in the inviscid region is westward, and away from the axis of rotation. Though there is no transport across C in the inviscid region, the meridional transport of the Ekman layer on the outer sphere is continuous across C and increases, through suction, as the equator is approached until it drains into an eastward flowing equatorial current of width 0(E ^1/7). The eastern boundary current outside C and shear layers on C carry this fluid to the intersection of C and the western wall where it feeds the western boundary current inside C.

The relation between this study and the experiments of Baker and Robinson (1970) is discussed.     

Fundamental considerations for the design of non-linear viscous dampers

An Erratum has been published for this article in Earthquake Engineering and Structural Dynamics 2000; 29(7):1076. Two interrelated issues related to the design of non-linear viscous dampers are considered in this paper: structural velocities and equivalent viscous damping. As the effectiveness of non-linear viscous dampers is highly dependent on operating velocities, it is important to have reliable estimates of the true velocity in the device. This should be based on the actual relative structural velocity and not the commonly misused spectral pseudo-velocity. This is because if spectral pseudo-velocities (PSV) are used, they are based on design displacements (S_v=ω₀S_d) and are thus fundamentally different from the actual relative structural velocity. This paper examines the difference between these two velocities, and based on an extensive study of historical earthquake motions proposes empirical relations that permit the designer to transform the well-known spectral pseudo-velocity to an actual relative structural velocity for use in design. Non-linear static analysis procedures recommended in current guidelines for the design of structural systems with supplement damping devices are based on converting rate-dependent device properties into equivalent viscous damping properties based on an equivalent energy consumption approach. Owing to the non-linear velocity dependence of supplemental devices, an alternative approach for converting energy dissipation into equivalent viscous damping is advanced in this paper that is based upon power consumption considerations. The concept of a normalized damper capacity (ϵ) is then introduced and a simple design procedure which incorporates power equivalent linear damping based on actual structural velocities is presented. Copyright © 1999 John Wiley & Sons, Ltd.     

Sulla turbolenza delle correnti marine nello stretto di messina

Riassunto Gli Autori espongono i risultati di un tentativo di valutazione del grado di turbolenza verticale delle correnti marine davanti alle imboccature Nord e Sud dello Stretto di Messina. I valori massimi ottenuti per il coefficiente di diffusione verticale della salinitàK _z sono dell'ordine di 10³–10⁴ cm²/sec.

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Summary The Authors present the results of some calculations carried out to obtain values of the vertical turbulence (by means of the coefficient of eddy-diffusion of salinityK _z ) of the currents in the Strait of Messina. The maximum values ofK _z are of the order of 10³–10⁴ cm²/sec.

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Comunicazione presentata il 24 Aprile 1957 alla Quinta Assemblea Generale della «Società Italiana di Geofisica e Meteorologia» (Genova: 23–25 Aprile 1957).     

Space discontinuous Galerkin method for shallow water flows—kinetic and HLLC flux,and potential vorticity generation

In this paper, a second order space discontinuous Galerkin (DG) method is presented for the numerical solution of inviscid shallow water flows over varying bottom topography. Novel in the implementation is the use of HLLC and kinetic numerical fluxes¹ in combination with a dissipation operator, applied only locally around discontinuities to limit spurious numerical oscillations. Numerical solutions over (non-)uniform meshes are verified against exact solutions; the numerical error in the L₂-norm and the convergence of the solution are computed. Bore–vortex interactions are studied analytically and numerically to validate the model; these include bores as “breaking waves” in a channel and a bore traveling over a conical and Gaussian hump. In these complex numerical test cases, we correctly predict the generation of potential vorticity by non-uniform bores. Finally, we successfully validate the numerical model against measurements of steady oblique hydraulic jumps in a channel with a contraction. In the latter case, the kinetic flux is shown to be more robust.     

Internal waves in a randomly stratified fluid

Abstract

We discuss the propagation of internal waves in a rotating stratified unbounded fluid with randomly varying stability frequency, N. The first order smoothing approximation is used to derive the dispersion relation for the mean wave field when N is of the form N ² = N _o ²(1 + ?μ), where μ is a centered stationary random function of either depth (z) or time (t), N _o = constant and O < ?² ≦ 1. Expressions are then derived for the change in phase speed and growth rate due to the random fluctuations μ; in particular, attention is focused on the behaviour of these expressions for short and long correlation lengths (case μ = μ(z)) and times (case μ = μ(t)). For the case μ = μ(z), which represents a model for the temperature and salinity fine-structure in the ocean, the appropriate statistics of the fluctuations observed at station P (50°N, 145°W) have been incorporated into the theory to estimate the actual importance of the effects due to these random fluctuations. It is found that the phase speed of the mean wave decreases significantly if (i) the wavelength is short compared to g/N_o ² or (ii) the wave number vector is essentially horizontal and the wave frequency is very close to N _o. Also, the random fluctuations cause a significant growth (decay) in the amplitude of a wave propagating upwards (downwards) through a depth of a few kilometers. However, in the direction of energy propagation, the kinetic energy is conserved. Finally, it is shown that the average effect of the depth dependent fluctuations at station P is to slightly decrease the stability frequency and the magnitude of the group velocity.     

Asymptotic regimes in unstable miscible displacements in random porous media

We study two asymptotic regimes of unstable miscible displacements in porous media, in the two limits, where a permeability-modified aspect ratio, R_L=L/H(k_v/k_h)^1/2, becomes large or small, respectively. The first limit is known as transverse (or vertical) equilibrium, the second leads to the problem of non-communicating layers (the Dykstra–Parsons problem). In either case, the problem reduces to the solution of a single integro-differential equation. Although at opposite limits of the parameter R_L, the two regimes coincide in the case of equal viscosities, M=1. By comparison with high-resolution simulation we investigate the validity of these two approximations. The evolution of transverse averages, particularly under viscous fingering conditions, depends on R_L. We investigate the development of a model to describe viscous fingering in weakly heterogeneous porous media under transverse equilibrium conditions, and compare with the various existing empirical models (such as the Koval, Todd–Longstaff and Fayers models).     

Buoyant boundary layer flows- an analogy to the Ekman spiral

Abstract

Flow details inside the buoyant boundary layer in the heat-up process of a contained, stably stratified, fluid are presented. Numerical solutions were obtained for the heatup problem in a cylinder considered by Sakurai and Matsuda (1972). By plotting the scaled vertical velocity W versus the scaled temperature θ as functions of the normal distance from the sidewall, the precise shape of the buoyant layer spiral is constructed. The analogy between this spiral and the Ekman spiral in rotating fluids is apparent. As the Rayleigh number Ra increases, the magnitude of the scaled vertical velocity increases substantially, but the scaled temperature does not vary appreciably. The buoyant layer thickness is determined by measuring the zero-crossing normal distance for the vertical velocity. The buoyant layer suction increases significantly as Ra increases. The effects of vertical level and of time on the qualitative behavior of buoyant layer flows are found to be small. The buoyant layer flows decay over the heat-up time scale t _n ; t _h characterizes the time span over which the overall adjustment process in the inviscid interior region is accomplished. This work clarifies that the analogy between heat-up and spin-up, which has been known to exist in the main body of inviscid fluid, applies equally well to the boundary layer regions.     

Self‐centering structural systems with combination of hysteretic and viscous energy dissipations

This paper presents an innovative set of high‐seismic‐resistant structural systems termed Advanced Flag‐Shaped (AFS) systems, where self‐centering elements are used with combinations of various alternative energy dissipation elements (hysteretic, viscous or visco‐elasto‐plastic) in series and/or in parallel. AFS systems is developed using the rationale of combining velocity‐dependent with displacement‐dependent energy dissipation for self‐centering systems, particularly to counteract near‐fault earthquakes. Non‐linear time‐history analyses (NLTHA) on a set of four single‐degree‐of‐freedom (SDOF) systems under a suite of 20 far‐field and 20 near‐fault ground motions are used to compare the seismic performance of AFS systems with the conventional systems. It is shown that AFS systems with a combination in parallel of hysteretic and viscous energy dissipations achieved greater performance in terms of the three performance indices. Furthermore, the use of friction slip in series of viscous energy dissipation is shown to limit the peak response acceleration and induced base‐shear. An extensive parametric analysis is carried out to investigate the influence of two design parameters, λ₁ and λ₂ on the response of SDOF AFS systems with initial periods ranging from 0.2 to 3.0 s and with various strength levels when subjected to far‐field and near‐fault earthquakes. For the design of self‐centering systems with combined hysteretic and viscous energy dissipation (AFS) systems, λ₁ is recommended to be in the range of 0.8–1.6 while λ₂ to be between 0.25 and 0.75 to ensure sufficient self‐centering and energy dissipation capacities, respectively. Copyright © 2010 John Wiley & Sons, Ltd.     

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.     

Wavefield downward extrapolation for migration velocity analysis on Marmousi data-set

The authors present a method for estimation of interval velocities using the downward continuation of the wavefield to perform layer-stripping migration velocity analysis. The generalized, phase-shift migration MG(F-K) in wavenumber-frequency domain was used for fulltime downward extrapolation of the wavefield. Such downward depth extrapolation accounts for strong changes of velocity in lateral and vertical directions and helps in correct positioning of the wavefield image in complex structures. Determination of velocity is the recursive process which means that the wavefield on depth level z _n−1 (n = 0, 1, ...) is an input data-set for determination of velocity on level z _n . The velocity ν [x, z _n − z _n−1] can be thus treated as interval velocity in Δz _n = z _n − z _n−1 step. This method was tested on synthetic Marmousi data-set and showed satisfactory results for complex, inhomogeneous media.     

Theoretical eduction and numerical simulation researches on the relationship between resistivity and water saturation of waterflood oil zone

In the process of water displacing oil, the relationship between resistivity and water saturation is the fundament of the quantitative research on the waterflooded grade and the remaining oil saturation with well logging data. A large number of core analysis data and production data are cumulated in the process of oil field exploitation, which offers the basis for the above research. This paper educed two methods from the Archie equation and material balance theory to calculate the quantitative relationships between R _z and S _w, and between R _t and S _w. The relationships set up by the two methods are similar to those set up by the real core measurements. The results can be used to analyze influencing factors and determine saturation quantitatively.

     

Shallow water tidal currents in close proximity to the seafloor and boundary-induced turbulence

Velocity measurements with vertical resolution 0.02 m were conducted in the lowest 0.5 m of the water column using acoustic Doppler current profiler (ADCP) at a test site in the western part of the East China Sea. The friction velocity u _* and the turbulent kinetic energy dissipation rate ε _wl(ζ) profiles were calculated using log-layer fits; ζ is the height above the bottom. During a semidiurnal tidal cycle, u _* was found to vary in the range (1–7) × 10⁻³ m/s. The law-of-the-wall dissipation profiles ε _wl(ζ) were consistent with the dissipation profiles ε _mc(ζ) evaluated using independent microstructure measurements of small-scale shear, except in the presence of westward currents. It was hypothesized that an isolated bathymetric rise (25 m height at a 50-m seafloor) located to the east of the measurement site is responsible for the latter. Calculation of the depth integrated internal tide generating body force in the region showed that the flanks of the rise are hotspots of internal wave energy that may locally produce a significant turbulent zone while emitting tidal and shorter nonlinear internal waves. This distant topographic source of turbulence may enhance the microstructure-based dissipation levels ε _mc(ζ) in the bottom boundary layer (BBL) beyond the dissipation ε _wl(ζ) associated with purely locally generated turbulence by skin currents. Semi-empirical estimates for dissipation at a distance from the bathymetric rise agree well with the BBL values of ε _mc measured 15 km upslope.