Flow structures at an idealized bifurcation : a numerical experiment

River bifurcations are key nodes within braided river systems controlling the flow and sediment partitioning and therefore the dynamics of the river braiding process. Recent research has shown that certain geometrical configurations induce instabilities that lead to downstream mid-channel bar formation and the formation of bifurcations. However, we currently have a poor understanding of the flow division process within bifurcations and the flow dynamics in the downstream bifurcates, both of which are needed to understand bifurcation stability. This paper presents results of a numerical sensitivity experiment undertaken using computational fluid dynamics (CFD) with the purpose of understanding the flow dynamics of a series of idealized bifurcations. A geometric sensitivity analysis is undertaken for a range of channel slopes (0.005 to 0.03), bifurcation angles (22 degrees to 42 degrees) and a restricted set of inflow conditions based upon simulating flow through meander bends with different curvature on the flow field dynamics through the bifurcation. The results demonstrate that the overall slope of the bifurcation affects the velocity of flow through the bifurcation and when slope asymmetry is introduced, the flow structures in the bifurcation are modified. In terms of bifurcation evolution the most important observation appears to be that once slope asymmetry is greater than 0.2 the flow within the steep bifurcate shows potential instability and the potential for alternate channel bar formation. Bifurcation angle also defines the flow structures within the bifurcation with an increase in bifurcation angle increasing the flow velocity down both bifurcates. However, redistributive effects of secondary circulation caused by upstream curvature can very easily counter the effects of local bifurcation characteristics. Copyright (C) 2011 John Wiley & Sons, Ltd.

GIS-based debris flow source and runout susceptibility assessment from DEM data - a case study in NW Nicaragua

In October 1998, Hurricane Mitch triggered numerous landslides (mainly debris flows) in Honduras and Nicaragua, resulting in a high death toll and in considerable damage to property. The potential application of relatively simple and affordable spatial prediction models for landslide hazard mapping in developing countries was studied. Our attention was focused on a region in NW Nicaragua, one of the most severely hit places during the Mitch event. A landslide map was obtained at 1:10 000 scale in a Geographic Information System (GIS) environment from the interpretation of aerial photographs and detailed field work. In this map the terrain failure zones were distinguished from the areas within the reach of the mobilized materials. A Digital Elevation Model (DEM) with 20 m×20 m of pixel size was also employed in the study area. A comparative analysis of the terrain failures caused by Hurricane Mitch and a selection of 4 terrain factors extracted from the DEM which, contributed to the terrain instability, was carried out. Land propensity to failure was determined with the aid of a bivariate analysis and GIS tools in a terrain failure susceptibility map. In order to estimate the areas that could be affected by the path or deposition of the mobilized materials, we considered the fact that under intense rainfall events debris flows tend to travel long distances following the maximum slope and merging with the drainage network. Using the TauDEM extension for ArcGIS software we generated automatically flow lines following the maximum slope in the DEM starting from the areas prone to failure in the terrain failure susceptibility map. The areas crossed by the flow lines from each terrain failure susceptibility class correspond to the runout susceptibility classes represented in a runout susceptibility map. The study of terrain failure and runout susceptibility enabled us to obtain a spatial prediction for landslides, which could contribute to landslide risk mitigation.

A subcritical instability of wave-driven alongshore currents

The development of shear instabilities of a wave-driven alongshore current is investigated. In particular, we use weakly nonlinear theory to investigate the possibility that such instabilities, which have been observed at various sites on the U.S. coast and in the laboratory, can grow in linearly stable flows as a subcritical bifurcation by resonant triad interaction, as first suggested by Shrira eta/. [1997]. We examine a realistic longshore current profile and include the effects of eddy viscosity and bottom friction. We show that according to the weakly nonlinear theory, resonance is possible and that these linearly stable flows may exhibit explosive instabilities. We show that this phenomenon may occur also when there is only approximate resonance, which is more likely in nature. Furthermore, the size of the perturbation that is required to trigger the instability is shown in some circumstances to be consistent with the size of naturally occurring perturbations. Finally, we consider the differences between the present case examined and the more idealized case of Shrira et a/. [ 1997]. It is shown that there is a possibility of coupling between triads, due to the richer modal structure in more realistic flows, which may act to stabilize the flow and act against the development of subcritical bifurcations. Extensive numerical tests are called for.

Aerodynamical instability of web

Diplomityön tavoitteena oli tutkia miten ilman turbulenttisuus vaikuttaa tasaisesti liikkuvan rainan tilaan. Yhtenä sovelluskohteena teollisuudessa voidaan mainita esimerkiksi leiju-kuivain. Tiedetään, että konenopeuksien kasvu ja siitä johtuva ilmavirran nopeuden kasvu aiheuttaa voimavaikutuksia rainaan ja voi aiheuttaa lepatusta. Lepatus johtaa dynaamiseen epästabiilisuuteen, joka voidaan havaita, kun lineaarinen systeemi tulee epävakaaksi ja joh-taa epälineaariseen, rajoitettuun värähtelyyn. Lepatus huonontaa tuotteiden laatua ja voi johtaa ratakatkoihin. Työssä on esitetty tietoa ilman ja rainan vuorovaikutuksesta, jota hyödyntämällä voidaan kehittää yksinkertaistettu malli, jonka avulla liikkuvaa rainaa voidaan simuloida kuivaimes-sa. Kaasufaasin virtausyhtälöt on ratkaistu eri turbulenttimalleja käyttäen. Myös viskoelas-tisen rainan muodonmuutosta on tarkasteltu. Koska rainalle ei ole kirjallisuudesta saatavilla tarkkoja fysikaalisia ja mekaanisia arvoja, näitä ominaisuuksia testattiin eri arvoilla, jotta rainan käyttäytymistä jännityksen alaisena voidaan tarkastella. Näiden ominaisuuksien tun-teminen on ensiarvoisen tärkeää määritettäessä rainan aeroviskoelastista käyttäytymistä. Virtaussimulointi on kallista ja aikaa vievää. Tämä tarkoittaa uusien tutkimusmenetelmien omaksumista. Tässä työssä vaihtoehtoisena lähestymistapana on esitetty yksinkertaistettu malli, joka sisältää ilman ja rainan vuorovaikutusta kuvaavat ominaisuudet. Mallin avulla saadaan tietoa epälineaarisuuden ja turbulenssin vaikutuksesta sekä monimutkaisesta yh-teydestä stabiilisuuden ja ulkoisesti aikaansaadun värähtelyn sekä itse aiheutetun värähtelyn välillä. Työn lopussa on esitetty havainnollinen esimerkki, jolla voidaan kuvata olosuhteita, jossa rainan tasainen liike muuttuu epävakaaksi. Kun turbulenttisuudesta johtuva painevaih-telu ylittää tietyn rajan, rainan värähtely kasvaa muuttuen satunnaisesta järjestäytyneeksi. Saaduttulokset osoittavat, että turbulenttisuudella on suuri vaikutus eikä sitä voi jättää huomioimatta. Myös rainan viskoelastiset ominaisuudet tulee huomioida, jotta rainan käyt-täytymistä voidaan kuvata tarkasti.

The counter-propagating Rossby-wave perspective on baroclinic instability. I: Mathematical basis

It is shown that Bretherton's view of baroclinic instability as the interaction of two counter-propagating Rossby waves (CRWs) can be extended to a general zonal flow and to a general dynamical system based on material conservation of potential vorticity (PV). The two CRWs have zero tilt with both altitude and latitude and are constructed from a pair of growing and decaying normal modes. One CRW has generally large amplitude in regions of positive meridional PV gradient and propagates westwards relative to the flow in such regions. Conversely, the other CRW has large amplitude in regions of negative PV gradient and propagates eastward relative to the zonal flow there. Two methods of construction are described. In the first, more heuristic, method a ‘home-base’ is chosen for each CRW and the other CRW is defined to have zero PV there. Consideration of the PV equation at the two home-bases gives ‘CRW equations’ quantifying the evolution of the amplitudes and phases of both CRWs. They involve only three coefficients describing the mutual interaction of the waves and their self-propagation speeds. These coefficients relate to PV anomalies formed by meridional fluid displacements and the wind induced by these anomalies at the home-bases. In the second method, the CRWs are defined by orthogonality constraints with respect to wave activity and energy growth, avoiding the subjective choice of home-bases. Using these constraints, the same form of CRW equations are obtained from global integrals of the PV equation, but the three coefficients are global integrals that are not so readily described by ‘PV-thinking’ arguments. Each CRW could not continue to exist alone, but together they can describe the time development of any flow whose initial conditions can be described by the pair of growing and decaying normal modes, including the possibility of a super-modal growth rate for a short period. A phase-locking configuration (and normal-mode growth) is possible only if the PV gradient takes opposite signs and the mean zonal wind and the PV gradient are positively correlated in the two distinct regions where the wave activity of each CRW is concentrated. These are easily interpreted local versions of the integral conditions for instability given by Charney and Stern and by Fjørtoft.

The counter-propagating Rossby-wave perspective on baroclinic instability. Part III: Primitive-equation disturbances on the sphere

Baroclinic instability of perturbations described by the linearized primitive quations, growing on steady zonal jets on the sphere, can be understood in terms of the interaction of pairs of counter-propagating Rossby waves (CRWs). The CRWs can be viewed as the basic components of the dynamical system where the Hamiltonian is the pseudoenergy and each CRW has a zonal coordinate and pseudomomentum. The theory holds for adiabatic frictionless flow to the extent that truncated forms of pseudomomentum and pseudoenergy are globally conserved. These forms focus attention on Rossby wave activity. Normal mode (NM) dispersion relations for realistic jets are explained in terms of the two CRWs associated with each unstable NM pair. Although derived from the NMs, CRWs have the conceptual advantage that their structure is zonally untilted, and can be anticipated given only the basic state. Moreover, their zonal propagation, phase-locking and mutual interaction can all be understood by ‘PV-thinking’ applied at only two ‘home-bases’—potential vorticity (PV) anomalies at one home-base induce circulation anomalies, both locally and at the other home-base, which in turn can advect the PV gradient and modify PV anomalies there. At short wavelengths the upper CRW is focused in the mid-troposphere just above the steering level of the NM, but at longer wavelengths the upper CRW has a second wave-activity maximum at the tropopause. In the absence of meridional shear, CRW behaviour is very similar to that of Charney modes, while shear results in a meridional slant with height of the air-parcel displacement-structures of CRWs in sympathy with basic-state zonal angular-velocity surfaces. A consequence of this slant is that baroclinically growing eddies (on jets broader than the Rossby radius) must tilt downshear in the horizontal, giving rise to up-gradient momentum fluxes that tend to accelerate the barotropic component of the jet.

Impact of Tibetan Orography and Heating on the Summer Flow over Asia

The influence on the summer flow over Asia of both the orographic and thermal forcing of the Tibetan Plateau is investigated using a sequence of idealised experiments with a global primitive equation model. The zonally averaged flow is prescribed and both realistic and idealised orography and heating are used. There is some similarity between the responses to the two forcings when applied separately. The upper tropospheric Tibetan anticyclone is predominantly forced by the heating but also weakly by the orography. Below this, both forcings tend to give air descending in an equatorward anticyclonic circulation down the isentropes to the west and rising in a similar poleward circulation to the east. However the heating-only response has a strong ascending southwesterly flow that is guided around the south and south-east of the orography when it is included. On the northern side, the westerly flow over the orography gives ascent on the upslope and descent on the downslope. It is found that heating over the Plateau leads to a potential vorticity (PV) minimum and that if it is sufficiently strong the flow is unstable, producing a quasi-biweekly oscillation. During this oscillation the Tibetan anticyclone changes between a single centre over the southwestern side of the Plateau and a split/double structure with centres over China and the Middle East. These characteristics are similar to observed variability in the region. Associated with this quasi-biweekly oscillation are significant variations in the strength of the ascent over the Plateau and the Rossby wave pattern over the North Pacific. The origin of the variability is instability associated with the zonally extended potential vorticity PV minimum on a θ-surface, as proposed by Hsu and Plumb (2000). This minimum is due to the tendency to reduce the PV above the heating over the Plateau and to advection by the consequent anticyclone of high PV around from the east and low PV to the west. The deep convection to the south and southeast of the Plateau tends to suppress the quasi-biweekly oscillation because the low PV produced above it acts to reduce the meridional PV gradient reversal. The occurrence of the oscillation depends on the relative magnitude of the heating in the two regions.

Stochastic resonance in a nonlinear model of a rotating, stratified shear flow, with a simple stochastic inertia-gravity wave parameterization

We report on a numerical study of the impact of short, fast inertia-gravity waves on the large-scale, slowly-evolving flow with which they co-exist. A nonlinear quasi-geostrophic numerical model of a stratified shear flow is used to simulate, at reasonably high resolution, the evolution of a large-scale mode which grows due to baroclinic instability and equilibrates at finite amplitude. Ageostrophic inertia-gravity modes are filtered out of the model by construction, but their effects on the balanced flow are incorporated using a simple stochastic parameterization of the potential vorticity anomalies which they induce. The model simulates a rotating, two-layer annulus laboratory experiment, in which we recently observed systematic inertia-gravity wave generation by an evolving, large-scale flow. We find that the impact of the small-amplitude stochastic contribution to the potential vorticity tendency, on the model balanced flow, is generally small, as expected. In certain circumstances, however, the parameterized fast waves can exert a dominant influence. In a flow which is baroclinically-unstable to a range of zonal wavenumbers, and in which there is a close match between the growth rates of the multiple modes, the stochastic waves can strongly affect wavenumber selection. This is illustrated by a flow in which the parameterized fast modes dramatically re-partition the probability-density function for equilibrated large-scale zonal wavenumber. In a second case study, the stochastic perturbations are shown to force spontaneous wavenumber transitions in the large-scale flow, which do not occur in their absence. These phenomena are due to a stochastic resonance effect. They add to the evidence that deterministic parameterizations in general circulation models, of subgrid-scale processes such as gravity wave drag, cannot always adequately capture the full details of the nonlinear interaction.

On the initiation of surface waves by turbulent shear flow

An analytical model is developed for the initial stage of surface wave generation at an air-water interface by a turbulent shear flow in either the air or in the water. The model treats the problem of wave growth departing from a flat interface and is relevant for small waves whose forcing is dominated by turbulent pressure fluctuations. The wave growth is predicted using the linearised and inviscid equations of motion, essentially following Phillips [Phillips, O.M., 1957. On the generation of waves by turbulent wind. J. Fluid Mech. 2, 417-445], but the pressure fluctuations that generate the waves are treated as unsteady and related to the turbulent velocity field using the rapid-distortion treatment of Durbin [Durbin, P.A., 1978. Rapid distortion theory of turbulent flows. PhD thesis, University of Cambridge]. This model, which assumes a constant mean shear rate F, can be viewed as the simplest representation of an oceanic or atmospheric boundary layer. For turbulent flows in the air and in the water producing pressure fluctuations of similar magnitude, the waves generated by turbulence in the water are found to be considerably steeper than those generated by turbulence in the air. For resonant waves, this is shown to be due to the shorter decorrelation time of turbulent pressure in the air (estimated as proportional to 1/Gamma), because of the higher shear rate existing in the air flow, and due to the smaller length scale of the turbulence in the water. Non-resonant waves generated by turbulence in the water, although being somewhat gentler, are still steeper than resonant waves generated by turbulence in the air. Hence, it is suggested that turbulence in the water may have a more important role than previously thought in the initiation of the surface waves that are subsequently amplified by feedback instability mechanisms.

Generation of inertia-gravity waves in the rotating thermal annulus by a localised boundary layer instability

Waves with periods shorter than the inertial period exist in the atmosphere (as inertia-gravity waves) and in the oceans (as Poincaré and internal gravity waves). Such waves owe their origin to various mechanisms, but of particular interest are those arising either from local secondary instabilities or spontaneous emission due to loss of balance. These phenomena have been studied in the laboratory, both in the mechanically-forced and the thermally-forced rotating annulus. Their generation mechanisms, especially in the latter system, have not yet been fully understood, however. Here we examine short period waves in a numerical model of the rotating thermal annulus, and show how the results are consistent with those from earlier laboratory experiments. We then show how these waves are consistent with being inertia-gravity waves generated by a localised instability within the thermal boundary layer, the location of which is determined by regions of strong shear and downwelling at certain points within a large-scale baroclinic wave flow. The resulting instability launches small-scale inertia-gravity waves into the geostrophic interior of the flow. Their behaviour is captured in fully nonlinear numerical simulations in a finite-difference, 3D Boussinesq Navier-Stokes model. Such a mechanism has many similarities with those responsible for launching small- and meso-scale inertia-gravity waves in the atmosphere from fronts and local convection.

On nonlinear symmetric stability and the nonlinear saturation of symmetric instability

A nonlinear symmetric stability theorem is derived in the context of the f-plane Boussinesq equations, recovering an earlier result of Xu within a more general framework. The theorem applies to symmetric disturbances to a baroclinic basic flow, the disturbances having arbitrary structure and magnitude. The criteria for nonlinear stability are virtually identical to those for linear stability. As in Xu, the nonlinear stability theorem can be used to obtain rigorous upper bounds on the saturation amplitude of symmetric instabilities. In a simple example, the bounds are found to compare favorably with heuristic parcel-based estimates in both the hydrostatic and non-hydrostatic limits.

Nonlinear stability of multilayer quasi-geostrophic flow

New nonlinear stability theorems are derived for disturbances to steady basic flows in the context of the multilayer quasi-geostrophic equations. These theorems are analogues of Arnol’d's second stability theorem, the latter applying to the two-dimensional Euler equations. Explicit upper bounds are obtained on both the disturbance energy and disturbance potential enstrophy in terms of the initial disturbance fields. An important feature of the present analysis is that the disturbances are allowed to have non-zero circulation. While Arnol’d's stability method relies on the energy–Casimir invariant being sign-definite, the new criteria can be applied to cases where it is sign-indefinite because of the disturbance circulations. A version of Andrews’ theorem is established for this problem, and uniform potential vorticity flow is shown to be nonlinearly stable. The special case of two-layer flow is treated in detail, with particular attention paid to the Phillips model of baroclinic instability. It is found that the short-wave portion of the marginal stability curve found in linear theory is precisely captured by the new nonlinear stability criteria.

Nonlinear saturation of baroclinic instability: part III: bounds on the energy

Rigorous upper bounds are derived on the saturation amplitude of baroclinic instability in the two-layer model. The bounds apply to the eddy energy and are obtained by appealing to a finite amplitude conservation law for the disturbance pseudoenergy. These bounds are to be distinguished from those derived in Part I of this study, which employed a pseudomomentum conservation law and provided bounds on the eddy potential enstrophy. The bounds apply to conservative (inviscid, unforced) flow, as well as to forced-dissipative flow when the dissipation is proportional to the potential vorticity. Bounds on the eddy energy are worked out for a general class of unstable westerly jets. In the special case of the Phillips model of baroclinic instability, and in the limit of infinitesimal initial eddy amplitude, the bound states that the eddy energy cannot exceed ϵβ2/6F where ϵ = (U − Ucrit)/Ucrit is the relative supercriticality. This bound captures the essential dynamical scalings (i.e., the dependence on ϵ, β, and F) of the saturation amplitudes predicted by weakly nonlinear theory, as well as exhibiting remarkable quantitative agreement with those predictions, and is also consistent with heuristic baroclinic adjustment estimates.

Nonlinear saturation of baroclinic instability: part II: continuously stratified fluid

Rigorous upper bounds are derived that limit the finite-amplitude growth of arbitrary nonzonal disturbances to an unstable baroclinic zonal flow in a continuously stratified, quasi-geostrophic, semi-infinite fluid. Bounds are obtained bath on the depth-integrated eddy potential enstrophy and on the eddy available potential energy (APE) at the ground. The method used to derive the bounds is essentially analogous to that used in Part I of this study for the two-layer model: it relies on the existence of a nonlinear Liapunov (normed) stability theorem, which is a finite-amplitude generalization of the Charney-Stern theorem. As in Part I, the bounds are valid both for conservative (unforced, inviscid) flow, as well as for forced-dissipative flow when the dissipation is proportional to the potential vorticity in the interior, and to the potential temperature at the ground. The character of the results depends on the dimensionless external parameter γ = f02ξ/β0N2H, where ξ is the maximum vertical shear of the zonal wind, H is the density scale height, and the other symbols have their usual meaning. When γ ≫ 1, corresponding to “deep” unstable modes (vertical scale ≈H), the bound on the eddy potential enstrophy is just the total potential enstrophy in the system; but when γ≪1, corresponding to ‘shallow’ unstable modes (vertical scale ≈γH), the eddy potential enstrophy can be bounded well below the total amount available in the system. In neither case can the bound on the eddy APE prevent a complete neutralization of the surface temperature gradient which is in accord with numerical experience. For the special case of the Charney model of baroclinic instability, and in the limit of infinitesimal initial eddy disturbance amplitude, the bound states that the dimensionless eddy potential enstrophy cannot exceed (γ + 1)2/24&gamma2h when γ ≥ 1, or 1/6;&gammah when γ ≤ 1; here h = HN/f0L is the dimensionless scale height and L is the width of the channel. These bounds are very similar to (though of course generally larger than) ad hoc estimates based on baroclinic-adjustment arguments. The possibility of using these kinds of bounds for eddy-amplitude closure in a transient-eddy parameterization scheme is also discussed.

Nonlinear saturation of baroclinic instability: part I: the two-layer model

A rigorous bound is derived which limits the finite-amplitude growth of arbitrary nonzonal disturbances to an unstable baroclinic zonal flow within the context of the two-layer model. The bound is valid for conservative (unforced) flow, as well as for forced-dissipative flow that when the dissipation is proportional to the potential vorticity. The method used to derive the bound relies on the existence of a nonlinear Liapunov (normed) stability theorem for subcritical flows, which is a finite-amplitude generalization of the Charney-Stern theorem. For the special case of the Philips model of baroclinic instability, and in the limit of infinitesimal initial nonzonal disturbance amplitude, an improved form of the bound is possible which states that the potential enstrophy of the nonzonal flow cannot exceed ϵβ2, where ϵ = (U − Ucrit)/Ucrit is the (relative) supereriticality. This upper bound turns out to be extremely similar to the maximum predicted by the weakly nonlinear theory. For unforced flow with ϵ < 1, the bound demonstrates that the nonzonal flow cannot contain all of the potential enstrophy in the system; hence in this range of initial supercriticality the total flow must remain, in a certain sense, “close” to a zonal state.