Quasigeostrophic planetary waves in a two-layer ocean with one-dimensional periodic bottom topography

Although the study of topographic effects on the Rossby waves in a stratified ocean has a long history, the wave property over a periodic bottom topography whose lateral scale is comparable to the wavelength is still not clear. The present paper treats this problem in a two-layer ocean with one-dimensional periodic bottom topography by a simple numerical method, in which no restriction on the wavelength and/or the horizontal scale of the topography is required. The dispersion diagram is obtained for a wavenumber range of [?π/L _b , π/L _b ], where L _b is the periodic length of the topography. When the topographic?β?is not negligible compared to the planetary β, the Rossby wave solutions around the wavenumbers which satisfy the resonant condition among the waves and topography disappear and separate into an infinite number of discrete modes. For convenience, each mode is numbered in order of frequency. As topographic height is increased, the high frequency barotropic Rossby wave (mode 1) becomes a topographic mode which can exist even on the f plane, and the highfrequency baroclinic mode (mode 2) becomes a surface intensified mode. Behaviors of low frequency modes are somewhat complicated. When the topographic amplitude is small, the low frequency baroclinic modes tend to be bottom trapped and the low frequency barotropic modes tend to be surface intensified. As topographic amplitude further increases, the relation between the mode number and vertical structure changes. This change can be attributed to the increase of the frequency of the topographic mode with the topographic amplitude.     

On the influence of stratification and tidal forcing upon mixing in sill regions

A cross-sectional non-hydrostatic model with idealized topography was used to examine the processes influencing tidal mixing in the region of sills. Initial calculations with appropriate parameters for the sill at the entrance to Loch Etive showed that the model could reproduce the main features of the observed mixing in the region. In particular, the hydraulic jump in the sill region was reproduced, as was an intense mid-water jet that was observed to separate from the lee side of the sill. Shear instabilities associated with the jet appeared to be a source of mixing within the thermocline. In addition, internal lee waves were generated on the lee side of the sill, with the observed amplification because of trapping during the flood stage. Their magnitude and hence the mixing increased with increasing Froude number (F _r). In the case of vertically varying buoyancy frequency, its value near the sill top determined the F _r number, with its value below influencing internal waves magnitude at depth. At high F _r values particularly with strong currents, short waves and overturning occurred.     

Rotating flow over long shallow ridges

Abstract

The flow of a rotating homogeneous, incompressible fluid past a long ridge is investigated. An analysis is presented for flows in which E ? 1, Ro ～ E^½, H/D ～ E⁰, h/D ～ E^½ and cosα ～ E⁰ where E is the Ekman number, Ro the Rossby number, H/D the fluid depth to ridge width ratio, h/D the ridge height to ridge width ratio and α the angle between the free stream flow and a line perpendicular to the ridge axis. The analysis includes effects of the nonlinear inertial terms. Particular examples of a ridge of triangular cross section and a sinusoidal topography are investigated in some detail. Experiments are presented for a triangular ridge which are in good agreement with the theory.     

Spectra of seismic force reduction factors of MDOF systems normalized by two characteristic periods

Seismic force reduction factor(SFRF) spectra of shear-type multi-degree-of-freedom(MDOF) structures are investigated. The modified Clough model, capable of considering the strength-degradation/hardening and stiffnessdegradation, is adopted. The SFRF mean spectra using 102 earthquake records on a typical site soil type(type C) are constructed with the period abscissa being divided into three period ranges to maintain the peak features at the two sitespecific characteristic periods. Based on a large number of results, it is found that the peak value of SFRF spectra may also exist for MDOF, induced by large high-mode contributions to elastic base shear, besides the mentioned two peak values. The variations of the stiffness ratio λk and the strength ratio λF of the top to bottom story are both considered. It is found that the SFRFs for λF≤λk are smaller than those for λF λk. A SFRF modification factor for MDOF systems is proposed with respect to SDOF. It is found that this factor is significantly affected by the story number and ductility. With a specific λF(= λk0.75), SFRF mean spectra are constructed and simple solutions are presented for MDOF systems. For frames satisfying the strong column/weak beam requirement, an approximate treatment in the MDOF shear-beam model is to assign a post-limit stiffness 15%-35% of the initial stiffness to the hysteretic curve. SFRF spectra for MDOF systems with 0.2 and 0.3 times the post-limit stiffness are remarkably larger than those without post-limit stiffness. Thus, the findings that frames with beam hinges have smaller ductility demand are explained through the large post-limit stiffness.     

A laboratory study of baroclinic instability

Abstract

Experimental and theoretical results are presented for a simple system which exhibits baroclinic instability. We consider the motion of two immiscible fluids with densities ρ ₁ and ρ ₂ contained in a cylinder rotating with angular frequency ω. The motion is driven by a contact lid rotating with frequency ω + ω. In this paper ω, ω, 2(ρ ₂ – ρ ₁)/(ρ ₂ + ρ ₁), and the geometry are such that the interface does not intersect the “ground” (e.g. an almost horizontal boundary). The motions are described by two-layer quasi-geostrophic equations which are identical, except perhaps for the presence of interfacial friction and tension, with those used in meteorology and oceanography. For small enough internal Froude number F = 4ω² L ²/(g(Δρ/ρ)H) or small enough Rossby number ? = ω/2ω the flow is steady and axisymmetric, the velocity field in each layer being determined primarily by frictional effects in top, bottom, and interfacial Ekman layers. For certain (F, ?) the flow becomes non-axisymmetric. The transition points for the case where the basic potential vorticity gradient is due to interface slope alone have been carefully measured and are in very good agreement with a linear instability theory which neglects sidewall effects. Some preliminary observations of supercritical motion, which include repeatable amplitude and wavenumber vacillation, are reported.     

Experiment on morphological and hydraulic adjustments of step-pool unit to flow increase

Exceptional flood events with a return period of about 50 years can be destructive to step-pool channel segments. However, field investigations and flume experiments have not examined the hydraulic and morphological feedbacks of step-pool morphology during unsteady hydrographs of exceptional flood events. We performed a series of flume experiments with a manually constructed step model, perturbed with three hydrographs that varied in the rate of water supply change. The bed texture, topography, flow regimes, surface flow field and water depth were characterized and measured as the flow rate was increased during the experiments. A distinct pool feature emerged downstream of the manually constructed step when the flow rate exceeded the threshold scaled to the peaks of ordinary flood events in well-graded mountain streams. The pool feature was modified in several different ways with flow rate increase. The bed surface steadily coarsened, micro-bedforms developed and became more pronounced, the bed topography became more spatially complex based on analysis using the Hurst exponent, and last, pool depth steadily increased. Pool modification was also linked to the flow regime: the impinging jet regime led to grain size segmentation in the pool while the jump regime contributed to decelerating flow velocity. The steeper rising limb of hydrograph led to a less developed pool feature, with smaller sized micro-bedforms in the pool bottom to outlet, and higher discharge threshold for distinct coarsening and scouring in the pool. The estimated energy dissipation within the step-pool unit decreased as a power function from low to high flow, quantified as the ratio h_c/H_S, where h_c is the critical water depth and H_S is scour depth. Our results highlight the interaction between morphology, hydraulics, and energy dissipation of step-pool unit and the crucial role of hydrograph shape on the interaction during flow increase © 2019 John Wiley & Sons, Ltd.     

The effects of alongshore variation in bottom topography on a boundary current—(topographically induced upwelling)

A theory which describes the constant f-plane flow of a steady inviscid baroclinic boundary current over a continental margin with a bathymetry that varies slowly in the alongshore but rapidly in the offshore directions is developed in the parameter regime (L_D/L)² ≤ Ro 1, where L_D is the internal deformation radius, L the horizontal length scale, and Ro the Rossby number. To lowest order in the Rossby number the flow is along isobaths with speed q_o = V_u(h,z)|Vh|/α, where V_u(h,z) is the upstream speed, α the upstream bottom slope at depth h, and Vh the bottom slope downstream at depth h. The lowest order flow produces a variation in the vertical component of relative vorticity along the isobath as the magnitude and direction of Vh vary in the downstream direction. The variation of vorticity requires a vertical as well as a cross-isobath flow at first order in the Rossby number. The first order vertical velocity is computed from the vorticity equation in terms of upstream conditions and downstream variations of the bathymetry. The density, pressure, and cross-isobath flow at first order in the Rossby number are then calculated. It is shown that in the cyclonic region of current (d/dh(V_u/α) > 0), if the isobaths diverge in the downstream direction ((∂/∂s)|Vh| < 0), then upwelling and onshore flow occur. The theory is applied to the northeastern Florida shelf to explain bottom temperature observations.     

A laboratory study of formative conditions for characteristic ripple patterns associated with a change in wave conditions

A wave‐?ume experiment was conducted to examine the formative condition for three types of distinctive bedforms that emerged through deformation of existing ripples due to waning wave power. They were ripple marks with: (1) a single secondary crest, (2) double secondary crests, and (3) a rounded crest. Data were analysed using two parameters, kh and d₀/λ_*, where k is the wave number, h is the water depth, d₀ is the near‐bottom orbital diameter, and λ_* is the spacing of existing ripples. The former quantity, kh, was employed as a surrogate for the degree of ?ow asymmetry. The result showed that ripples with secondary crests developed under a rather symmetrical ?ow ?eld with kh ? 0·7, if d₀/λ_* ? 1·2, whereas rounded‐crest ripples emerged under asymmetrical ?ow ?eld with kh ? 0·7, if d₀/λ_* ? ?2·9 kh + 3·2. The number of secondary crests, which initially occurred in each trough, was single if d₀/λ_* ? 0·8, or double if d₀/λ_* ? 0·8. Copyright © 2004 John Wiley & Sons, Ltd.     

Structural features of the subauroral ionosphere during the origination of the polarization jet

Narrow jets of rapid westward ion drifts were registered near the plasmapause projection at the F-region altitudes on the Cosmoc-184 satellite and were called “a polarization jet.” In this work, the effect of this polarization jet on the ionospheric structure has been studied, using a three-dimensional model of the high-latitude ionosphere, when strong local magnetospheric electric fields were originated. The calculations indicated that a narrow trough in the latitudinal variations in the electron density at the F-region maximum was formed in the zone where the electric field was switched on. This trough was more pronounced in the early evening hours, when the electron background density was still high, and was less distinct at low back-ground levels during premidnight hours. A comparison of the calculations and experimental data indicated that they were in good agreement with one another, which made it possible to state that the polarization jet was the main mechanism by which narrow electron density troughs were formed in the subauroral ionosphere.     

The influence of mesoscale topography on the stability and growth rates of a two-layer model of the open ocean

Abstract

The influence of mesoscale topography on the baroclinic instability of a two-layer model of the open ocean is considered. For westward velocities in the top layer (U), and for a sinusoidal topography independent of x or longitude (a cross-stream topography), the critical value of U (U_c ) leading to instability is the same as when there is no topography. The wavelength of the unstable perturbation corresponding to U _c is shortened. For a given wavevector (k) of the perturbation the system becomes stable (as also in the absence of topography) for large values of |U|. The minimum value of the shear leading to stability is, however, significantly reduced by the topography.

For sufficiently large values of the height of the topographic features, instabilities appear which are localized within a narrow range of the shear. These instabilities are studied for a topography that depends both on x and y.

For a cross-stream topography the growth rates are somewhat smaller than those without topography and they depend only weakly on k_y . For the topographies considered here which depend both on x and y, perturbations with different values of k_y can again have roughly the same growth rate.

In the case of stable oscillations, variations in the eddy energy with very long periods are made possible by the coexistence of topographic modes with closely lying periods.     

Quasi-geostrophic shallow-water vortex–patch equilibria and their stability

We examine the equilibrium form, properties, stability and nonlinear evolution of steadily-rotating simply-connected vortex patches in the single-layer quasi-geostrophic model of geophysical fluid dynamics. This model, valid for rotating shallow-water flow in the limit of small Rossby and Froude numbers, has an intrinsic length scale L _D called the “Rossby deformation length” relating the strength of the stratification to that of the background rotation. Here, we generate steadily-rotating vortex equilibria for a wide range of γ?=?L/L _D , where L is the typical horizontal length scale of the vortex. We vary both γ (over the range 0.02?≤?γ?≤?10) and the vortex aspect ratio λ (over the range 0?<?λ?<?1). We find two modes of instability arising at sufficiently small aspect ratio λ?<?λ _c (γ): an asymmetric (dominantly wave 3) mode at small γ (or large L _D ) and a symmetric (dominantly wave 4) mode at large γ (or small L _D ). At marginal stability, the asymmetric mode dominates for γ???3, while the symmetric mode dominates for γ???3. The nonlinear evolution of weakly-perturbed unstable equilibria results in major structural changes, in most cases producing two dominant vortex patches and thin, quasi-passive filaments. Overall, the nonlinear evolution can be classified into three principal types: (1) vacillations for a limited range of aspect ratios λ when 5?≤?γ?≤?6, (2) filamentation and a single-dominant vortex for γ???1, and (3) vortex splitting – asymmetric for 1???γ???4 and symmetric for γ???4.     

Froude number effects on flow over topography

Abstract

The response or a depth independent two layer flow to an underlying topographic irregularity is studied for flows in which the square of the internal Froude number exceeds the Rossby number. Irrespective of the magnitude of the Rossby number, rotation is important for such flows. The flow generally adjusts so that the thickness of the lower layer is nearly constant. However small anomalies from the constant thickness are found to extend to very large distances from the topography when the Rossby number exceeds unity.     

A numerical study of detached shear layers in a rotating sliced cylinder

Abstract

The low Rossby number flow in a rotating cylinder with an inclined bottom, of small slope, is examined when part of the lid of the container is rotating at a slightly different rate. The resulting flow is calculated numerically by solving the governing equations for the two-dimensional geostrophic motion which approximates the flow in most of the fluid including the inertially-modified E ^¼ -layers. The presence of ageostrophic regions, on the container walls and beneath the velocity discontinuity on the lid, is accounted for in the governing equations and their boundary conditions. This study supplements previous work on this configuration, in which the zero Rossby number flow was calculated and experimental results were presented, by enabling a direct comparison to be made between the results of the low Rossby number theory and the experiments. The numerical results for a range of Rossby and Ekman numbers compare well with those from the experiments despite a severe limitation on the size of the Rossby number arising from the analysis in the ageostrophic part of the detached shear layer.     

Experiments in ocean circulation modelling

Abstract

In a laboratory model ocean, fluid in a rotating tank of varying depth is subjected to “wind-stress”, For a certain range of the parameters, Ekman number E and Rossby number R, a homogeneous fluid displays steady, westward intensified flow. For the same range of E and R, a two-layer fluid can have baroclinic instabilities. The parameter range for the various kinds of instabilities is mapped in a regime diagram. The northward transport in the western boundary current is measured as it varies with Rossby number for both homogeneous and two-layer fluid.     

Spin-up of a two-layer fluid in a rotating cylinder

Abstract

we report the results of experiments on the spin-up of two layers of immiscible fluid with a free upper surface in a rotating cylinder over a wide range of internal Froude numbers. Observations of the evolution of the velocity field by particle tracking indicates that spin-up of the azimuthal velocity in the upper layer take much longer than in a homogeneous fluid. Initially, spin-up occurs at a rate comparable to that of homogeneous fluid but, at high internal Froude number, a second phase follows in which the remaining lative motion decays much more slowly. Quantitative comparison of these measurements to the theory of Pedlosky (1967) shows good agreement.

Visualization of the interface displacement during spin-up detected the presence of transient azimuthal variations in the interface elevation over a wide range of Froude (F), Ekman (E), and Rossby (ε) number. nalysis of the occurrence of the asymmetric variations using the parameter space (Q, F), where Q = E ^1/2/ε, suggested by the baroclinic instability theory and experiments of Hart (1972), showed that the flow was stable for Q > 0.06 with no discernable dependence on F. This result, together with the prediction of Pedlosky's theory that radial gradient of potential vorticity in the two layers have opposite signs, suggests at the baroclinic instability mechanism was responsible for the asymmetries. The location and timing of these instabilities may account for the discrepancies between the observations and the Pedlosky (1967) theory.     

A comparison of numerical simulations of eddy generation performed with a two- and a three-layer quasi-geostrophic model of oceanic mesoscale eddies

Abstract

The generation of eddies by a large-scale flow over mesoscale topography is studied with the help of two- and three-layer nonlinear quasi-geostrophic models of the open ocean. The equations are integrated forward in time with no eddies present initially. For a given time, the displacement of the interface between layers two and three (ζ) tends to a well-defined limit (function of the horizontal spatial coordinates) as ρ ₃ - ρ ₂ → 0 (ρ_r is the density of layer r). Even for values of α[= (ρ ₃ - ρ ₂)/(ρ ₂ - ρ ₁)] ^as small as 0.01 the potential energy due to ζ is not negligible and it can reach, in some cases, a considerable fraction of the total eddy energy.     

Sediment transferring function of the lower reaches of the Yellow River influenced by drainage basin factors and human activities

Since 1986, with a sharp decrease in water dis-charges, the Yellow River has entered a period charac-terized by low discharges and seasonally occurring dry-ups[1,2]. Since 1999, more strict management of water diversion has been imposed, and therefore the dry-ups have been well under control. However, the lower reaches of the Yellow River is still predominated by low-discharges, and has become a man-induced shrinking river. In the past 40 years, significant effect of soil and water conservat…     

Evolution of droplets in subsea oil and gas blowouts: Development and validation of the numerical model VDROP-J

The droplet size distribution of dispersed phase (oil and/or gas) in submerged buoyant jets was addressed in this work using a numerical model, VDROP-J. A brief literature review on jets and plumes allows the development of average equations for the change of jet velocity, dilution, and mixing energy as function of distance from the orifice. The model VDROP-J was then calibrated to jets emanating from orifices ranging in diameter, D, from 0.5 mm to 0.12 m, and in cross-section average jet velocity at the orifice ranging from 1.5 m/s to 27 m/s. The d₅₀/D obtained from the model (where d₅₀ is the volume median diameter of droplets) correlated very well with data, with an R² = 0.99. Finally, the VDROP-J model was used to predict the droplet size distribution from Deepwater Horizon blowouts. The droplet size distribution from the blowout is of great importance to the fate and transport of the spilled oil in marine environment.     

Seasonal and annual variation of CO2 flux above a broad-leaved Korean pine mixed forest

Long-term measurement of carbon metabolism of old-growth forests is critical to predict their behaviors and to reduce the uncertainties of carbon accounting under changing climate. Eddy covariance technology was applied to investigate the long-term carbon exchange over a 200 year-old Chinese broad-leaved Korean pine mixed forest in the Changbai Mountains (128°28′E and 42°24′N, Jilin Province, P. R. China) since August 2002. On the data obtained with open-path eddy covariance system and CO₂ profile measurement system from Jan. 2003 to Dec. 2004, this paper reports (i) annual and seasonal variation of F _NEE, F _GPP and R _E; (ii) regulation of environmental factors on phase and amplitude of ecosystem CO₂ uptake and release Corrections due to storage and friction velocity were applied to the eddy carbon flux.

L_AI and soil temperature determined the seasonal and annual dynamics of F_GPP and R_E separately. V_PD and air temperature regulated ecosystem photosynthesis at finer scales in growing seasons. Water condition at the root zone exerted a significant influence on ecosystem maintenance carbon metabolism of this forest in winter.

The forest was a net sink of atmospheric CO₂ and sequestered −449 g C·m⁻² during the study period; −278 and −171 gC·m⁻² for 2003 and 2004 respectively. F _GPP and F _RE over 2003 and 2004 were −1332, −1294 g C·m⁻². and 1054, 1124 g C·m⁻² respectively. This study shows that old-growth forest can be a strong net carbon sink of atmospheric CO₂.

There was significant seasonal and annual variation in carbon metabolism. In winter, there was weak photosynthesis while the ecosystem emitted CO₂. Carbon exchanges were active in spring and fall but contributed little to carbon sequestration on an annual scale. The summer is the most significant season as far as ecosystem carbon balance is concerned. The 90 days of summer contributed 66.9, 68.9% of F _GPP, and 60.4, 62.1% of R _E of the entire year.