Thermal convection in differentially rotating systems

Abstract

The onset of convection in a cylindrical fluid annulus is analyzed in the case when the cylindrical walls are rotating differentially, a temperature gradient in the radial direction is applied, and the centrifugal force dominates over gravity. The small gap approximation is used and no-slip conditions on the cylindrical walls are assumed. It is found that over a considerable range of the parameter space either convection rolls aligned with the axis of rotation or rolls in the perpendicular (azimuthal) direction are preferred. It is shown that by a suitable redefinition of parameters, results for finite amplitude Taylor vortices and for convection rolls in the presence of shear can be applied to the present problem. Weakly nonlinear results for transverse rolls in a Couette flow indicate the possibility of subcritical bifurcation for Prandtl numbers P less than 0.82. Heat and momentum transports are derived as functions of P and the problem of interaction between transverse and longitudinal rolls is considered. The relevance of the analysis for problems of convection in planetary and stellar atmospheres is briefly discussed.     

A case study of storm commencement and recovery plasmaspheric electric fields near L=2.5 at equinox

Data from the VLF Doppler experiment at Faraday, Antarctica (65○ S, 64○ W) are used to study the penetration of the high-latitude convection electric field to lower latitudes during severely disturbed conditions. Alterations of the electric field at L-values within the range 2.0 - 2.7 are studied for two cases at equinox (10 - 12 September 1986 and 1 - 3 May 1986). The recovery of the electric field is found to be approximately an exponential function of time. Values for the equatorial meridional E×B drift velocity, inferred from the data, are used as inputs to a model of the plasmasphere and ionosphere. The model and experimental results are used to investigate the post-storm alteration of ionospheric coupling processes. The magnitude of the effect of ionosphere-plasmasphere coupling fluxes on N_mF2 values and the O⁺-H⁺ transition height is dependent on the local time of storm commencement, and on the orientation of the electric field. The coupling fluxes appear to have a maximum influence on ionospheric content during the main phase of geomagnetic activity that produces outward motion of plasmaspheric whistler ducts.     

The equatorial dynamics of a shallow,homogeneous ocean

Abstract

It is shown that the linear equatorial dynamics of a shallow ocean is characterized by two boundary layers of width γ^? L and γL (γ is the Ekman number of the flow, assumed small, and L is a horizontal dimension of the basin). In the γ^? layer stress in the bottom Ekman layer is comparable to that in the surface Ekman layer. In the γ layer vertical friction is important throughout the depth of the ocean. Should the Rossby number ? be so large as to invalidate a linear theory (? > γ^5/3), then inertial effects become important at a distance ?^2/5 L from the equator. The role played in the circulation of the basin by the non-linear equatorial current first studied by Charney (1960) is shown to be similar to that of the γ layer of the linear theory. Though lateral friction is unimportant in a linear model of the flow, shear layers at the equator are found to be a necessary feature of non-linear flow.     

Steady flows at the top of the core from geomagnetic field models: The steady motions theorem

Abstract

It is demonstrated that the steady tangential velocity v_s at the closed surface δK of a perfect fluid conductor bounded by a rigid, impenetrable exterior can be uniquely determined from knowledge of the normal component of the time varying magnetic flux density B _n, on δK. In the context of a simple earth model consisting of an electrically insulating mantle surrounding a perfectly conducting core, the assumption of steady flow provides enough extra information to eliminate the toroidal ambiguity in B _nv and to allow derivation of a unique, global flow at the top of the core from a model of the geomagnetic field.     

Mechanisms of the meridional heat transport in the Southern Ocean

The Southern Ocean (SO) transports heat towards Antarctica and plays an important role in determining the heat budget of the Antarctic climate system. A global ocean data synthesis product at eddy-permitting resolution from the Estimating the Circulation and Climate of the Ocean, Phase II (ECCO2) project is used to estimate the meridional heat transport (MHT) in the SO and to analyze its mechanisms. Despite the intense eddy activity, we demonstrate that most of the poleward MHT in the SO is due to the time-mean fields of the meridional velocity, V, and potential temperature, θ. This is because the mean circulation in the SO is not strictly zonal. The Antarctic Circumpolar Current carries warm waters from the region south of the Agulhas Retroflection to the lower latitudes of the Drake Passage and the Malvinas Current carries cold waters northward along the Argentinian shelf. Correlations between the time-varying fields of V and θ (defined as transient processes) significantly contribute to the horizontal-gyre heat transport, but not the overturning heat transport. In the highly energetic regions of the Agulhas Retroflection and the Brazil-Malvinas Confluence the contribution of the horizontal transient processes to the total MHT exceeds the contribution of the mean horizontal flow. We show that the southward total MHT is mainly maintained by the meridional excursion of the mean geostrophic horizontal shear flow (i.e., deviation from the zonal average) associated with the Antarctic Circumpolar Current that balances the equatorward MHT due to the Ekman transport and provides a net poleward MHT in the SO. The Indian sector of the SO serves as the main pathway for the poleward MHT.     

On the inadequacies of Long's model for steady two-dimensional stratified flows

Nonlinear analysis of two-dimensional steady flows with density stratification in the presence of gravity is considered. Inadequacies of Long's model for steady stratified flow over topography are explored. These include occurrence of closed streamline regions and waves propagating upstream. The usual requirements in Long's model of constant dynamic pressure and constant vertical density gradient in the upstream condition are believed to be the cause of these inadequacies. In this article, we consider a relaxation of these requirements, and also provide a systematic framework to accomplish this. As illustrations of this generalized formulation, exact solutions are given for the following two special flow configurations: the stratified flow over a barrier in an infinite channel; the stratified flow due to a line sink in an infinite channel. These solutions exhibit again closed-streamline regions as well as waves propagating upstream. The persistence of these inadequacies in the generalized Long's model appears to indicate that they are not quite consequences of the assumptions of constant dynamic pressure and constant vertical density gradient in Long's model, contrary to previous belief.

On the other hand, solutions admitted by the generalized Long's model show that departures from Long's model become small as the flow becomes more and more supercritical. They provide a nonlinear mechanism for the generation of columnar disturbances upstream of the obstacle and lead in subcritical flows to qualitatively different streamline topological patterns involving saddle points, which may describe the lee-wave-breaking process in subcritical flows and could serve as seats of turbulence in real flows. The occurrences of upstream disturbances in the presence of lee-wave-breaking activity described by the present solution are in accord with the experiments of Long (Long, R.R., “Some aspects of the flow of stratified fluids, Part 3. Continuous density gradients”, Tellus 7, 341--357 (1955)) and Davis (Davis, R.E., “The two-dimensional flow of a stratified fluid over an obstacle”, J. Fluid Mech. 36, 127–143 ()).     

On the probability of exceeding a given river flow threshold

ABSTRACT

To obtain estimates of the probability that a river flow will exceed a given threshold at time t + 1, given the flow value at time t, two stochastic models are considered: a filtered Poisson process and a diffusion process with jumps. Estimates derived from linear regression are also considered. The model parameters are assumed to depend on the flow value. An application to the Delaware River is presented.

Editor D. Koutsoyiannis; Associate editor S. Grimaldi     

On topographic core-mantle coupling

Abstract

The purpose of this note is two-fold: to draw attention to a perplexing difficulty connected with topographic core-mantle coupling, and to suggest tentatively an explanation. The difficulty is an apparent conflict between the most comprehensive theory of the coupling so far attempted (Anufriev and Braginsky, 1975a, b, 1977a, b) and recent explicit calculations based on magnetic and seismic information (Speith et al., 1986). It is argued that asymmetric deviations from Anufriev and Braginsky's basically axisymmetric model of the underlying core flow are capable of resolving the difficulty.     

Wave-induced topographic formstress in baroclinic channel flow

Large-scale zonal flow driven across submarine topography establishes standing Rossby waves. In the presence of stratification, the wave pattern can be represented by barotropic and baroclinic Rossby waves of mixed planetary topographic nature, which are locked to the topography. In the balance of momentum, the wave pattern manifests itself as topographic formstress. This wave-induced formstress has the net effect of braking the flow and reducing the zonal transport. Locally, it may lead to acceleration, and the parts induced by the barotropic and baroclinic waves may have opposing effects. This flow regime occurs in the circumpolar flow around Antarctica. The different roles that the wave-induced formstress plays in homogeneous and stratified flows through a zonal channel are analyzed with the BARBI (BARotropic-Baroclinic-Interaction ocean model, Olbers and Eden, J Phys Oceanogr 33:2719–2737, 2003) model. It is used in complete form and in a low-order version to clarify the different regimes. It is shown that the barotropic formstress arises by topographic locking due to viscous friction and the baroclinic one due to eddy-induced density advection. For the sinusoidal topography used in this study, the transport obeys a law in which friction and wave-induced formstress act as additive resistances, and windstress, the effect of Ekman pumping on the density stratification, and the buoyancy forcing (diapycnal mixing of the stratified water column) of the potential energy stored in the stratification act as additive forcing functions. The dependence of the resistance on the system parameters (lateral viscosity ε, lateral diffusivity κ of eddy density advection, Rossby radius λ, and topography height δ) as well as the dependence of transport on the forcing functions are determined. While the current intensity in a channel with homogeneous density decreases from the viscous flat bottom case in an inverse quadratic law ~δ ^–2 with increasing topography height and always depends on ε, a stratified system runs into a saturated state in which the transport becomes independent of δ and ε and is determined by the density diffusivity κ rather than the viscosity: κ/λ ² acts as a vertical eddy viscosity, and the transport is λ ²/κ times the applied forcing. Critical values for the topographic heights in these regimes are identified.     

The Navier–Stokes-α equations revisited

The Navier–Stokes-α equation is a regularised form of the Euler equation that has been employed in representing the sub-grid scales in large-eddy simulations. Determined efforts have been made to place it on a secure deductive foundation. This requires two steps to be completed. The first is fundamental and consists of establishing from the equations governing the fluid flow, a relationship between two velocities called by Holm (Chaos, 2002a, 12, 518) the “filtered” and “unfiltered” velocities. The second consists of the relation between these two velocities. Until now, the preferred route to the first objective has been variational, by varying the action using Hamilton's principle. Soward and Roberts (J. Fluid Mech., 2008, 604, 297) followed that variational route and established the existence of an important but unwelcome term omitted by Holm in his derivation. It is shown here that the Soward and Roberts result may be derived from Euler's equation by a direct approach with considerably greater efficiency. Holm achieved the second objective by making a “Taylor hypothesis”, which we use here to evaluate the unwelcome term missing from his analysis of the first step. The resulting model equations differ from those of Holm's α model, and the attractive mean Kelvin's circulation theorem that follows from his α equations is no longer valid. For that reason, we call the term omitted by Holm unwelcome.     

Hydrologic indicators of hydraulic conditions that drive flow–biota relationships

Abstract

This article tests the association between streamflow alteration and the alteration of ecologically significant hydraulic environments. There has been a recent shift in environmental flow assessments to develop rapid desktop-based approaches that are applicable in a regional context. Streamflow statistics (e.g. minimum monthly flow) are often chosen to predict the impact of streamflow alteration on aquatic ecosystems. The assumption that the flow–biota relationship will be obscured by the effect of how streamflow interacts with channel morphology is often acknowledged, but not quantified. In this study, streamflow statistics are derived for 19 reaches in four river systems in Victoria, Australia. Hydraulic metrics were used to quantify ecologically significant surface flow conditions (Froude number) and the area of bench inundation, shallow and deep water. Multivariate analysis was used to investigate the correlation between streamflow statistics altered with regulation and the hydraulic metrics. It was found that streamflow statistics have a weak correlation to surface flow condition and the area of shallow water under natural streamflow conditions. The results show that hydrologic statistics have limited utility in quantifying changes in hydraulic environments. A similar magnitude of flow alteration can produce diverse hydraulic results. The confounding influence of channel morphology prevents streamflow statistics being an adequate surrogate for the assessment of hydraulic alteration. Modelling flow–biota relationships in a regional context is limited by the inadequacy of streamflow statistics to model ecologically significant hydraulic function. Improving knowledge of ecohydraulically significant hydrologic statistics will improve the effectiveness of environmental flow planning to sustain instream habitat conditions. A probabilistic approach is required to enable a risk-based approach to desktop generalization of flow–biota relations.

Editor Z.W. Kundzewicz; Guest editor M. Acreman

Citation Turner, M. and Stewardson, M., 2014. Hydrologic indicators of hydraulic conditions that drive flow–biota relationships. Hydrological Sciences Journal, 59 (3–4), 659–672.     

Scattering of rossby waves by partial barriers

Abstract

The scattering of Rossby waves by partial barriers is studied. It is found that only a small fraction of the wave energy will penetrate the gap between South America and Antarctica if the wave period is less than a month.     

On the evolution and stability of interfacial ripples on and off the critical frequency

Abstract

Under consideration are interfaces between two media of different densities and which arise from the interaction between the Mth and Nth harmonics of the motion where 1 ≤ N < M. By means of the method of multiple scales in both space and time a pair of nonlinear coupled partial differential equations is derived which model the progression of the interface. The equations contain a detuning parameter [sgrave] which allow imperfections in the resonance to be taken into account. Stokes-type sinusoidal solutions to the equations were sought. It was found that solutions exist for all values of the interaction ratio M/N. In some situations interfaces exist at both exact and near resonance; while in others they are destroyed by amplifications in the detuning. In yet others, a quantity of detuning is actually necessary for the profiles to exist. In all cases, even when the parameters are fixed, a very large class of interface profiles is possible. Finally, the stability of the profiles is studied. It is found that some are quite stable, even to perturbations with wavenumbers close to the main flow.     

Converted-wave seismology in anisotropic media Revisited,Part II: Application to parameter estimation

In transversely isotropic media with a vertical symmetry axis (VTI), the converted-wave (C-wave) moveout over intermediate-to-far offsets is determined by four parameters. These are the C-wave stacking velocity V _C2, the vertical and effective velocity ratios γ ₀and γ _eff, and the anisotropic parameter X _eff. We refer to the four parameters as the C-wave stacking velocity model. The purpose of C-wave velocity analysis is to determine this stacking velocity model. The C-wave stacking velocity model V _C2, γ ₀, γ _geff, and X _eff can be determined from P- and C-wave reflection moveout data. However, error propagation is a severe problem in C-wave reflection-moveout inversion. The current short-spread stacking velocity as deduced from hyperbolic moveout does not provide sufficient accuracy to yield meaningful inverted values for the anisotropic parameters. The non-hyperbolic moveout over intermediate-offsets (x/z from 1.0 to 1.5) is no longer negligible and can be quantified using a background γ. Non-hyperbolic analysis with a γ correction over the intermediate offsets can yield V _C2 with errors less than 1% for noise free data. The procedure is very robust, allowing initial guesses of γ with up to 20% errors. It is also applicable for vertically inhomogeneous anisotropic media. This improved accuracy makes it possible to estimate anisotropic parameters using 4C seismic data. Two practical work flows are presented for this purpose: the double-scanning flow and the single-scanning flow. Applications to synthetic and real data show that the two flows yield results with similar accuracy but the single-scanning flow is more efficient than the double-scanning flow. This work is funded by the Edinburgh Anisotropy Project of the British Geological Survey. First Author Li Xiangyang, he is currently a professorial research seismologist (Grade 6) and technical director of the Edinburgh Anisotropy Project in the British Geological Survey. He also holds a honorary professorship multicomponent seismology at the School of Geosciences, University of Edinburgh. He received his BSc(1982) in Geophysics from Changchun Geological Institute, China, an MSc (1984) in applied geophysics from East China Petroleum Institute (now known as the China University of Petroleum), and a PhD (1992) in seismology from the University of Edinburgh. During 1984–1987, he worked as a lecturer with the East China Petroleum Institute. Since 1991, he has been employed by the British Geological Survey. His research interests include seismic anisotropy and multicomponent seismology.     

再论各向异性介质转换波地震学,第二部分:参数计算研究

在具有垂直对称轴横向各向同性介质中，利用四种参数来确定中间至远偏移距转换波（C-波）动校正。它们是C-波叠加速度Vc2，垂直速度比和有效速度比γ0和γeff以及各向异性参数χeff。我们将这四种参数作为C波叠加速度模型。C-波速度分析的目的就是确定这种叠加速度模型。C-波叠加速度模型Vc2，γ0，γeff，和χeff可以由P-波和C-波反射动校正资料获得。然而错误的传播是C-波反射动校正反演中的严重问题。当前短排列叠加速度由于是从双曲线动校正推算而得，因而其精度不足以为各向异性参数提供有意义的反演值。中间偏移非双曲线动校正不再被人们所勿略，而是可以用一个背景γ加以量化。非双曲线分析通过中间偏移距的γ校正量可以产生Vc2，若数据不含燥音，其误差小于1％。方法稳健，允许γ启始假定值的误差达20％。该方法也适用垂直非均匀各向异性介质。精度的提高使能够用4分量地震资料计算各向异性参数。为此提出了两种工作流程：双扫描和单扫描流程。理论数据和实际数据的应用表明这两种流程得出的结果其精度相似，但是单扫描流程比双扫描更有效。     

Uncertainty and sensitivity analysis of runoff and sediment yield in a small agricultural watershed with KINEROS2 / Analyse d'incertitude et de sensibilité des simulations d'écoulement et de transport solide de KINEROS2 dans un petit bassin versant agricole

Abstract

Using the Monte Carlo (MC) method, this paper derives arithmetic and geometric means and associated variances of the net capillary drive parameter, G, that appears in the Parlange infiltration model, as a function of soil texture and antecedent soil moisture content. Approximate expressions for the arithmetic and geometric statistics of G are also obtained, which compare favourably with MC generated ones. This paper also applies the MC method to evaluate parameter sensitivity and predictive uncertainty of the distributed runoff and erosion model KINEROS2 in a small experimental watershed. The MC simulations of flow and sediment related variables show that those parameters which impart the greatest uncertainty to KINEROS2 model outputs are not necessarily the most sensitive ones. Soil hydraulic conductivity and wetting front net capillary drive, followed by initial effective relative saturation, dominated uncertainties of flow and sediment discharge model outputs at the watershed outlet. Model predictive uncertainty measured by the coefficient of variation decreased with rainfall intensity, thus implying improved model reliability for larger rainfall events. The antecedent relative saturation was the most sensitive parameter in all but the peak arrival times, followed by the overland plane roughness coefficient. Among the sediment related parameters, the median particle size and hydraulic erosion parameters dominated sediment model output uncertainty and sensitivity. Effect of rain splash erosion coefficient was negligible. Comparison of medians from MC simulations and simulations by direct substitution of average parameters with observed flow rates and sediment discharges indicates that KINEROS2 can be applied to ungauged watersheds and still produce runoff and sediment yield predictions within order of magnitude of accuracy.     

Estimating the internal pressure gradient errors in a σ-coordinate ocean model for the Nordic Seas

Internal pressure gradient estimation is problematic in σ-coordinate ocean models and models based on more generalised topography following coordinate systems. Artificial pressure gradients in these models may create artificial flow. In recent literature, several methods for reducing the errors in the estimated internal pressure gradients are suggested. A basin with a bell-shaped seamount in the middle has often been applied as a test case. To supplement the findings from these more idealised experiments, the internal pressure gradient errors in a σ-coordinate ocean model for the Nordic Seas are discussed in the present paper. Three methods for estimating internal pressure gradients are applied in these experiments. The sensitivity of the results to the subtraction of background stratification and to the horizontal viscosity are also investigated. For the extended Nordic Seas case, basin scale modes dominate after a few days of simulation. The errors in the transports across some sections may be larger than 1 Sv (1 Sv = 10⁶ m³ s^− 1) in these studies with 16-km grid resolution. The order of magnitude of the errors in the transports of Atlantic water into the Nordic Seas is approximately 0.5 Sv or between 5 and 10 % of recent transport estimates based on measurements. The results do not indicate that the errors are generally reduced if the background stratification is subtracted when estimating internal pressure gradients in terrain following models. However, the results from the experiments initialised with the background stratification show that the erroneous flows may be reduced considerably by using more recent techniques for estimating internal pressure gradients, especially for higher values of horizontal viscosity.     

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.