Convection in rotating annular channels heated from below. Part 2. Transitions from steady flow to turbulence

We report the results of fully three-dimensional numerical simulations of nonlinear convection in a Boussinesq fluid in an annular channel rotating about a vertical axis with lateral no-slip or stress-free sidewalls, stress-free top and bottom, uniformly heated from below, a problem first studied by Davies-Jones and Gilman (1971 Davies-Jones, RP and Gilman, PA. 1971. Convection in a rotating annulus uniformly heated from below.. J. Fluid Mech., 46: 65–81.  [Google Scholar]) and Gilman (1973 Gilman, PA. 1973. Convection in a rotating annulus uniformly heated from below. Part 2. Nonlinear results. J. Fluid Mech., 57: 381–400.  [Google Scholar]). A substantial range of the Rayleigh number R (R_c≤R≤O(100 R_c)), where R_c denotes the critical value at the onset of convection) is considered. It is found that the wall-localized convection mode, unaffected by the velocity boundary condition imposed on the sidewalls, is nonlinearly robust. Both directions of travelling waves, one propagating against the sense of rotation near the outer sidewall and the other propagating in the same sense as the rotation in the vicinity of the inner sidewall, are always present in the nonlinear solutions. In contrast to nonlinear convection in a rotating Bénard layer, neither convection rolls nor the Küpper–Lortz instability can exist in a rotating annular channel because of the effect of the sidewalls. It is the nonlinear interaction between the wall-localized modes and the internal mode that plays an essential role in determining the nonlinear properties of convection in a rotating annular channel. Our studies reveal systematically the various nonlinear phenomena, from steady travelling waves trapped in the vicinities of the sidewalls to convective turbulence exhibiting columnar structure.     

Convection in rotating annular channels heated from below. Part 3: Experimental boundary conditions

Results are presented from both linear stability analysis and numerical simulations of three-dimensional nonlinear convection in a Boussinesq fluid in an annular channel, under experimental boundary conditions, rotating about a vertical axis uniformly heated from below. The focus is placed on the Prandtl number Pr = 7.0, representing liquid water at room temperature. The linear analysis shows that, when the aspect ratio is sufficiently small, there exists only one stationary mode that occupies the whole fluid container. When the aspect ratio is moderate or large, however, there exist three different linear solutions: (i) the outer sidewall-localized traveling wave propagating against the sense of rotation; (ii) the inner sidewall-localized traveling wave propagating in the same sense as rotation; and (iii) both the counter-traveling waves occurring simultaneously. Guided by the result of the linear stability analysis, fully three-dimensional simulations are then performed for a channel with a moderate aspect ratio. It is found that neither the prograde nor the retrograde mode is physically realizable near threshold and beyond. The dynamics of nonlinear convection in a rotating channel are chiefly characterized by the interaction between the sidewall-localized waves and the interior convection cells/rolls, producing an interesting and unusual nonlinear phenomenon. In order to compare with the classical Rayleigh–Bénard problem without vertical sidewalls, we also study linear and nonlinear convection at exactly the same parameters but in an infinitely extended layer with periodic horizontal conditions. This reveals that both the linear instability and nonlinear convection in a rotating channel are characteristically different from those in a rotating layer with periodic horizontal conditions.     

Convection in a rotating,laterally heated annulus transition to lower symmetry

Abstract

The radial temperature differences at which the transition from lower symmetry to the wave regime and the transition from the wave regime to lower symmetry occur have been measured for rotation rates ≦2rad/sec. It was found that the temperature differences at which the transitions occur differ for a fixed rotation rate, depending on whether the radial temperature difference is either increased or decreased with time. There is hysteresis in the transition at lower symmetry.     

Vortical and wave modes in 3D rotating stratified flows: random large-scale forcing

Utilizing an eigenfunction decomposition, we study the growth and spectra of energy in the vortical (geostrophic) and wave (ageostrophic) modes of a three-dimensional (3D) rotating stratified fluid as a function of ε = f/N, where f is the Coriolis parameter and N is the Brunt–Vaisala frequency. Throughout, we employ a random large-scale forcing in a unit aspect ratio domain and set these parameters such that the Froude and Rossby numbers are roughly comparable and much less than unity. Working in regimes characterized by moderate Burger numbers, i.e. Bu = 1/ε² < 1 or Bu ≥ 1, our results indicate profound change in the character of vortical and wave mode interactions with respect to Bu = 1. Indeed, previous analytical work concerning the qualitatively different nature of these interactions has been in limiting conditions of rotation or stratification domination (i.e. when Bu ? 1 or Bu ? 1, respectively). As with the reference state of ε = 1, for ε < 1 the wave mode energy saturates quite quickly and the ensuing forward cascade continues to act as an efficient means of dissipating ageostrophic energy. Further, these saturated spectra steepen as ε decreases: we see a shift from k ^?1 to k ^?5/3 scaling for k _f < k < k _d (where k _f and k _d are the forcing and dissipation scales, respectively). On the other hand, when ε > 1 the wave mode energy never saturates and comes to dominate the total energy in the system. In fact, in a sense the wave modes behave in an asymmetric manner about ε = 1. With regard to the vortical modes, for ε ≤ 1, the signatures of 3D quasigeostrophy are clearly evident. Specifically, we see a k ^?3 scaling for k _f < k < k _d and, in accord with an inverse transfer of energy, the vortical mode energy never saturates but rather increases for all k < k _f . In contrast, for ε > 1 and increasing, the vortical modes contain a progressively smaller fraction of the total energy indicating that the 3D quasigeostrophic subsystem, though always present, plays an energetically smaller role in the overall dynamics. Combining the vortical and wave modes, the total energy for k > k _f and ε ≤ 1 shows a transition as k increases wherein the vortical modes contain a large portion of the energy at large scales, while the wave modes dominate at smaller scales. There is no such transition when ε > 1 and the wave modes dominate the total energy for all k > k _f .     

Instability of flows near the onset of convection in a rotating layer with stress-free horizontal boundaries

We investigate instability of convective flows of simple structure (rolls, standing and travelling waves) in a rotating layer with stress-free horizontal boundaries near the onset of convection. We show that the flows are always unstable to perturbations, which are linear combinations of large-scale modes and short-scale modes, whose wave numbers are close to those of the perturbed flows. Depending on asymptotic relations of small parameters α (the difference between the wave number of perturbed flows and the critical wave number for the onset of convection) and ε (ε² being the overcriticality and the perturbed flow amplitude being O(ε)), either small-angle or Eckhaus instability is prevailing. In the case of small-angle instability for rolls the largest growth rate scales as ε^8/5, in agreement with results of Cox and Matthews (Cox, S.M. and Matthews, P.C., Instability of rotating convection. J. Fluid. Mech., 2000, 403, 153–172) obtained for rolls with k = k _c . For waves, the largest growth rate is of the order ε^4/3. In the case of Eckhaus instability the growth rate is of the order of α².     

Mean-field equations for weakly nonlinear two-scale perturbations of forced hydromagnetic convection in a rotating layer

We consider stability of regimes of hydromagnetic thermal convection in a rotating horizontal layer with free electrically-conducting boundaries, to perturbations involving large spatial and temporal scales. Equations governing the evolution of weakly nonlinear mean perturbations are derived under the assumption that the α-effect is insignificant in the leading-order (e.g. due to a symmetry of the system). The mean-field equations generalise the standard equations of hydromagnetic convection: New terms emerge – a second-order linear operator representing the combined eddy diffusivity and quadratic terms associated with the eddy advection. If the perturbed CHM regime is nonsteady and insignificance of the α-effect in the system does not rely on the presence of a spatial symmetry, the combined eddy diffusivity operator also involves a nonlocal pseudodifferential operator. If the perturbed CHM state is almost symmetric, α-effect terms appear in the mean-field equations as well. Near a point of a symmetry-breaking bifurcation, cubic nonlinearity emerges in the equations. All the new terms are in general anisotropic. A method for evaluation of their coefficients is presented; it requires solution of a significantly smaller number of auxiliary problems than in a straightforward approach.     

Climate-driven trends in mean and high flows from a network of reference stations in Ireland

Abstract

This paper introduces a reference hydrometric network for Ireland and examines the derived flow archive for evidence of climate-driven trends in mean and high river flows. The Mann-Kendall and Theil-Sen tests are applied to eight hydroclimatic indicators for fixed and variable (start and end date) records. Spatial coherence and similarities of trends with rainfall suggest they are climate driven; however, large temporal variability makes it difficult to discern widely-expected anthropogenic climate change signals at this point in time. Trends in summer mean flows and recent winter means are at odds with those expected for anthropogenic climate change. High-flow indicators show strong and persistent positive trends, are less affected by variability and may provide earlier climate change signals than mean flows. The results highlight the caution required in using fixed periods of record for trend analysis, recognizing the trade-off between record length, network density and geographic coverage.

Editor Z.W. Kundzewicz; Associate editor H. Lins

Citation Murphy, C., Harrigan, S., Hall, J., and Wilby, R.L., 2013. Climate-driven trends in mean and high flows from a network of reference stations in Ireland. Hydrological Sciences Journal, 58 (4), 755–772.     

A steady nonlinear and singular vortex Rossby wave within a rapidly rotating vortex. Part II: Application to geophysical vortices

In a previous paper, Caillol [Geophys. Astrophys. Fluid Dyn., 2014, 108] investigated the steady nonlinear vortical structure of a singular vortex Rossby mode that has survived to a strong critical-layer-like interaction with a linearly stable, columnar, axisymmetric and dry vortex. We presented a general theory for this wave/mean flow interaction through the nonlinear critical layer theory and calculated the mean azimuthal and axial winds induced at the critical radius at the end of this interaction in the final stage. We here apply that theory to rapidly rotating geophysical vortices: tropical cyclones, cold-air mesocyclones and tornadoes. We find that the numerous assumptions invoked in that paper agree well with the reality of those intense vortices. We also find that in spite of a lack of moist-convection modelling, this dry vortex is fairly well accelerated at the critical radius by such a shear wave with a magnitude of order the square root of the damped-wave amplitude. The intensification level strongly depends on the aspect ratio, height of the system: rapid vortex and parent vortex, over core radius. The thinner the vortex is, the sharper the intensification is. This result is in sharp contrast to the numerous numerical simulations on VR wave/vortex interactions that yield a much smaller intensification of order the square of the wave amplitude. This weakly nonlinear approach nevertheless fails to model small vertical wavelength VR wave/vortex interactions for their related asymptotic expansions are divergent and for they yield strongly nonlinear VR waves coupled with evolving critical layers whose extent can no longer be considered as thin.     

The influence of debris flows on channels and valley floors in the Oregon Coast Range,U.S.A.

Debris flows are one of the most important processes which influence the morphology of channels and valley floors in the Oregon Coast Range. Debris flows that initiate in bedrock hollows at heads of first-order basins erode the long-accumulated sediment and organic debris from the floors of headwater, first- and second-order channels. This material is deposited on valley floors in the form of fans, levees, and terraces. In channels, deposits of debris flows control the distribution of boulders. The stochastic nature of sediment supply to alluvial channels by debris flows promotes cycling between channel aggradation which results in a gravel-bed morphology, and channel degradation which results in a mixed bedrock- and boulder-bed morphology. Temporal and spatial variability of channel-bed morphology is expected in other landscapes where debris flows are an important process.     

Coupling mechanisms in double sandbar systems. Part 1: patterns and physical explanation

Crescentic sandbars and rip channels along wave‐dominated sandy beaches are relevant to understand localized beach and dune erosion during storms. In recent years, a paradigm shift from hydrodynamic template models to self‐organization mechanisms occurred to explain the formation of these rhythmic features. In double sandbar systems, both the inner‐ and outer‐bar rip channels and crescentic planshapes are now believed to be free instabilities of the nearshore system arising through self‐organization mechanisms alone. However, the occasional occurrence of one or two inner‐bar rip channels within one outer‐bar crescent suggests a forced, morphologically coupled origin. Here we use a nonlinear morphodynamic model to show that alongshore variability in outer‐bar depth, and the relative importance of wave breaking versus wave focussing by refraction across the outer bar, is crucial to the inner‐bar rip channel development. The coupling patterns simulated by our model are similar to those observed in the field. Morphological coupling requires a template in the morphology (outer‐bar geometry) which, through the positive feedback between flow, sediment transport and the evolving morphology (that is, self‐organization) enforces the development of coupling patterns. We therefore introduce a novel mechanism that blurs the distinction between self‐organization and template mechanisms. This mechanism may also be extended to explain the dynamics of other nearshore patterns, such as beach cusps. The impact of this novel mechanism on the alongshore variability of inner‐bar rip channels is investigated in the companion paper. Copyright © 2010 John Wiley and Sons, Ltd.     

The 26 December 2004 Sumatra Tsunami: Analysis of Tide Gauge Data from the World Ocean Part 1. Indian Ocean and South Africa

The M_w = 9.3 megathrust earthquake of December 26, 2004 off the northwest coast of Sumatra in the Indian Ocean generated a catastrophic tsunami that was recorded by a large number of tide gauges throughout the World Ocean. Part 1 of our study of this event examines tide gauge measurements from the Indian Ocean region, at sites located from a few hundred to several thousand kilometers from the source area. Statistical characteristics of the tsunami waves, including wave height, duration, and arrival time, are determined, along with spectral properties of the tsunami records.     

Seismic design and testing of the bottom vertical boundary elements in steel plate shear walls. Part 1: design methodology

This research investigates the seismic design method and the cyclic inelastic behavior of the bottom column, also called the vertical boundary element (VBE), in steel plate shear walls (SPSWs). This study consists of two parts. This Part 1 paper discusses the anticipated pushover responses for properly designed SPSWs and the possible inelastic responses of the bottom VBE at various levels of inter‐story drift. Considering both the tension field action of the infill panel and the sway action of the boundary frame, this study develops a simplified method to compute the flexural and shear demands in the bottom VBE. Based on the superposition method, this approach considers various plastic hinge forming locations at different levels of inter‐story drift. One of the key performance‐based design objectives is to ensure that the top ends of the bottom VBEs remain elastic when the SPSWs are subjected to the maximum considered earthquake. This paper presents the comprehensive design procedures for the bottom VBE. Furthermore, this study conducted cyclic performance evaluation tests of three full‐scale two‐story SPSWs at the Taiwan National Center for Research on Earthquake Engineering in 2011 to validate the effectiveness of the proposed design methods. The experimental program, cyclic inelastic responses of the SPSWs and bottom VBEs, and numerical simulations are presented in Part 2. Copyright © 2014 John Wiley & Sons, Ltd.     

Groundwater ‐ the disregarded component in lake water and nutrient budgets. Part 1: effects of groundwater on hydrology

Lake eutrophication is a large and growing problem in many parts of the world, commonly due to anthropogenic sources of nutrients. Improved quantification of nutrient inputs is required to address this problem, including better determination of exchanges between groundwater and lakes. This first of a two‐part review provides a brief history of the evolution of the study of groundwater exchange with lakes, followed by a listing of the most commonly used methods for quantifying this exchange. Rates of exchange between lakes and groundwater compiled from the literature are statistically summarized for both exfiltration (flow from groundwater to a lake) and infiltration (flow from a lake to groundwater), including per cent contribution of groundwater to lake‐water budgets. Reported rates of exchange between groundwater and lakes span more than five orders of magnitude. Median exfiltration is 0.74 cm/day, and median infiltration is 0.60 cm/day. Exfiltration ranges from near 0% to 94% of input terms in lake‐water budgets, and infiltration ranges from near 0% to 91% of loss terms. Median values for exfiltration and infiltration as percentages of input and loss terms of lake‐water budgets are 25% and 35%, respectively. Quantification of the groundwater term is somewhat method dependent, indicating that calculating the groundwater component with multiple methods can provide a better understanding of the accuracy of estimates. The importance of exfiltration to a lake budget ranges widely for lakes less than about 100 ha in area but generally decreases with increasing lake area, particularly for lakes that exceed 100 ha in area. No such relation is evident for lakes where infiltration occurs, perhaps because of the smaller sample size. Copyright © 2014 John Wiley & Sons, Ltd.     

A marching in space and time (MAST) solver of the shallow water equations. Part I: The 1D model

A new approach is presented for the numerical solution of the complete 1D Saint-Venant equations. At each time step, the governing system of partial differential equations (PDEs) is split, using a fractional time step methodology, into a convective prediction system and a diffusive correction system. Convective prediction system is further split into a convective prediction and a convective correction system, according to a specified approximated potential. If a scalar exact potential of the flow field exists, correction vanishes and the solution of the convective correction system is the same solution of the prediction system. Both convective prediction and correction systems are shown to have at each x − t point a single characteristic line, and a corresponding eigenvalue equal to the local velocity. A marching in space and time (MAST) technique is used for the solution of the two systems. MAST solves a system of two ordinary differential equations (ODEs) in each computational cell, using for the time discretization a self-adjusting fraction of the original time step. The computational cells are ordered and solved according to the decreasing value of the potential in the convective prediction step and to the increasing value of the same potential in the convective correction step. The diffusive correction system is solved using an implicit scheme, that leads to the solution of a large linear system, with the same order of the cell number, but sparse, symmetric and well conditioned. The numerical model shows unconditional stability with regard of the Courant–Friedrichs–Levi (CFL) number, requires no special treatment of the source terms and a computational effort almost proportional to the cell number. Several tests have been carried out and results of the proposed scheme are in good agreement with analytical solutions, as well as with experimental data.     

Sensitivity of runoff and soil erosion to climate change in two Mediterranean watersheds. Part II: assessing impacts from changes in storm rainfall,soil moisture and vegetation cover

The impacts of climate change on storm runoff and erosion in Mediterranean watersheds are difficult to assess due to the expected increase in storm frequency coupled with a decrease in total rainfall and soil moisture, added to positive or negative changes to different types of vegetation cover. This report, the second part of a two‐part article, addresses this issue by analysing the sensitivity of runoff and erosion to incremental degrees of change (from ? 20 to + 20%) to storm rainfall, pre‐storm soil moisture, and vegetation cover, in two Mediterranean watersheds, using the MEFIDIS model. The main results point to the high sensitivity of storm runoff and peak runoff rates to changes in storm rainfall (2·2% per 1% change) and, to a lesser degree, to soil water content (?1·2% per 1% change). Catchment sediment yield shows a greater sensitivity than within‐watershed erosion rates to both parameters: 7·8 versus 4·0% per 1% change for storm rainfall, and ? 4·9 versus ? 2·3% per 1% change for soil water content, indicating an increase in sensitivity with spatial scale due to changes to sediment connectivity within the catchment. Runoff and erosion showed a relatively low sensitivity to changes in vegetation cover. Finally, the shallow soils in one of the catchments led to a greater sensitivity to changes in storm rainfall and soil moisture. Overall, the results indicate that decreasing soil moisture levels caused by climate change could be sufficient to offset the impact of greater storm intensity in Mediterranean watersheds. Copyright © 2009 John Wiley & Sons, Ltd.