Downstream Wave Research Articles

Nonlinear free surface flows past a semi-infinite flat plate in water of finite depth

We consider the steady free surface two-dimensional flow past a semi-infinite flat plate in water of a constant finite depth. The fluid is assumed to be inviscid, incompressible and the flow is irrotational; surface tension at the free surface is neglected. Our concern is with the periodic waves generated downstream of the plate edge. These can be characterized by a depth-based Froude number F and the depth d (draft) of the depressed plate. Previous analytical studies have been restricted to small values of d and subcritical flows F<1, when linearized theory can be used. Here, our main concern here is with the critical regime, F≈1, where we use a weakly nonlinear long-wave analysis, For finite values of d, we solve the fully nonlinear problem numerically using a boundary integral equation method. Our results confirm previous studies and extend these to the critical regime. As d increases, our numerical results demonstrate that the downstream waves approach the highest possible wave. We also find some wave-free solutions, which when compared to the weakly nonlinear results are essentially just one-half of a solitary wave.

Prediction of intake noise of an automotive engine in run-up condition

It is very important to predict the radiated noise from the engine intake system for the effective noise control and virtual prototyping of in-cavity and outdoor noise of a vehicle. To this end, one should precisely measure the in-duct acoustic source parameters of the intake system, viz., source strength and source impedance. Usually, the noise radiation characteristics need to be expressed as a function of engine speed. In this study, acoustic source parameters of an engine intake system under engine run-up condition were measured by using the direct method. Direct method employed two external loudspeakers, turned on simultaneously, and three microphones for the separation of upstream and downstream wave components. It was noted that the frequency spectra of source impedance hardly changes with the increase of engine speed. Utilizing this fact, source strength under the engine run-up condition was calculated by assuming invariant source impedance. Predicted insertion loss and radiated sound pressure level using the measured source parameters were compared with those of measured data and predicted data using several idealized source models, which have been adopted for the calculations. A reasonably good agreement was observed between measured sound spectra at the intake orifice and predicted one using the measured source data. It was shown that the source data obtained by the present method yielded a far better prediction accuracy than those by the idealized source models.

Changes in the large‐scale features associated with the Indian summer monsoon in the recent decades

AbstractThe inter‐annual variability of the Indian summer monsoon rainfall (ISMR) has been examined in association with the variability of surface temperature, the mean sea level pressure (MSLP), the tropospheric geopotential height, and wind patterns over the globe to study the changes in the large‐scale features associated with Indian summer monsoon (ISM) in the recent decades. The data for the study have been considered from 1949–2005 (for a period of 57 years). It has been found that, during the recent decades, the El Nino‐Southern Oscillation (ENSO) relationship over ISMR has weakened while the northwest (NW) of North Atlantic sea surface temperature (SST) relationship has increased. A physical mechanism for the relationship of North Atlantic SST with ISMR is proposed, by which during the excess years of ISMR, the SST over NW of North Atlantic is above normal. The rise in SST is the consequence of intensification and a cell‐type shaped Azores high with isobars parallel to the east coast of North America, which drives a subtropical oceanic gyre (Gulf stream) transporting equatorial warm water NW of North Atlantic. The Azores high act as a source of downstream Rossby wave train over Eurasian region having barotropic structure consisting of a successive pressure trough and ridge. The Rossby wave may be associated with intensifying the low surface pressure anomaly over the Iranian landmass, which intensify the monsoon current over the Indian region. Copyright © 2008 Royal Meteorological Society

Influence of Coolant Flow Direction on Criticality Waves

Two models for criticality wave systems are presented, distinguished by the coolant flow direction with respect to the wave propagation direction, namely, an upstream wave (UW) model and a downstream wave (DW) model. It is shown that in the DW case the reactor power and wave velocity are considerably lower than those in the UW case, but the burnup is much higher.

Lie Symmetry Preservation and Shock-Capturing Methods

It is known that some traditional shock-capturing schemes, including Godunov- or Roe-type schemes, generate downstream oscillatory wave patterns that cannot be effectively dampened by the dissipation methods they provide. The theme of this paper is to understand the formation and behavior at these upstream/downstream patterns from the point view of the symmetry inherent in the PDE. This paper investigates symmetry breaking and spatial oscillations in finite difference solutions for the inviscid Burgers and one-dimensional Navier-Stokes shock equations using the Lax-Wendroff, TVD, and nonoscillatory, nonfree parameter dissipative schemes. Symmetry breaking is considered as a candidate for the inherent cause of the nonphysical oscillations in the vicinity of the shock.

Catastrophic flooding from Glacial Lake Wisconsin

Glacial Lake Wisconsin was a large proglacial lake that formed along the southern margin of the Laurentide Ice Sheet during the Wisconsin glaciation. It was formed when ice of the Green Bay Lobe came into contact with the Baraboo Hills in southwestern Wisconsin and blocked the south-flowing Wisconsin River. During early glacial recession, the ice dam failed catastrophically and the lake drained in about a week. Despite early recognition of the former lake and the likelihood that it failed catastrophically, outflow rates during the failure have not been previously evaluated. Estimates based on step-backwater modeling indicate that peak discharge was between 3.6 and 5.3 × 10 4 m 3/s in the lower Wisconsin River. As an alternate method, we used a previously derived empirical relationship between lake volume and peak discharge for dam-break events. From a digital elevation model altered to incorporate isostatic depression, we estimated the lake volume to be 87 km 3 just prior to dam breach, suggesting that the flooding magnitude was as high as 1.5 × 10 5 m 3/s at the outlet. Adjusting these results for downstream flood wave attenuation gives a discharge of around 4.4 × 10 4 m 3/s in the lower reach, which closely matches the results of the step-backwater modeling. These estimates of discharge from the catastrophic failure of ice-marginal lakes improve our understanding of the processes that have produced the morphology and behavior of present-day upper Midwest river systems.

The Evolution of Organizational Forms in Russian Retailing

Conceptual schemes of the organizational ecology are applied to reveal major trends in the evolution of trading formats in Russian retailing at present. They include the increasing share of modern stores, downstream waves of exploring the market niches from premium to economic market segments, and the spread of the multiformat strategy. The role of new trading formats in Russia is compared with that in the countries of Central and Eastern Europe. Scenarios of future development in the retailing sector are presented.

Turbulence transmission in parallel relativistic shocks using ray tracing

We apply a semi-classical approach of handling waves as quasiparticle gas in a slowly varying flow – analogous to ray tracing – to calculate the Alfvén wave transmission parameters, the resulting cross-helicity of the waves and the scattering-centre compression ratio, for cases where the shock thickness is large enough for the turbulent waves in the plasma to see the transition of the background flow parameters as smooth and slowly varying. For nonrelativistic shocks the wave transmission produces similar effects on the downstream turbulence and the scattering-centre compression ratio as does the transmission through a step shock: the downstream Alfvén waves propagate predominantly towards the shock in the local plasma frame and, thus, the scattering-centre compression ratio is larger than the gas compression ratio. For thick relativistic shocks, however, we find qualitative differences with respect to the step-shock case: for low-Alfvénic-Mach-number shocks the downstream waves propagate predominantly away from the shock, and the scattering-centre compression ratio is lower than that of the gas. Thus, when taken into account, the Alfvén wave transmission can decrease the efficiency of the first-order Fermi acceleration in a thick relativistic shock.

An experimental investigation of the propagation mechanism of critical deflagration waves that lead to the onset of detonation

The reflection of a CJ detonation from a perforated plate is used to generate high speed deflagrations downstream in order to investigate the critical conditions that lead to the onset of detonation. Different perforated plates were used to control the turbulence in the downstream deflagration waves. Streak Schlieren photography, ionization probes and pressure transducers are used to monitor the flow field and the transition to detonation. Stoichiometric mixtures of acetylene–oxygen and propane–oxygen were tested at low initial pressures. In some cases, acetylene–oxygen was diluted with 80% argon in order to render the mixture more “stable” (i.e., more regular detonation cell structure). The results show that prior to successful detonation initiation, a deflagration is formed that propagates at about half the CJ detonation velocity of the mixture. This “critical” deflagration (which propagates at a relatively constant velocity for a certain duration prior to the onset of detonation) is comprised of a leading shock wave followed by an extended turbulent reaction zone. The critical deflagration speed is not dependent on the turbulence characteristics of the perforated plate but rather on the energetics of the mixture like a CJ detonation (i.e., the deflagration front is driven by the expansion of the combustion products). Hence, the critical deflagration is identified as a CJ deflagration. The high intensity turbulence that is required to sustain its propagation is maintained via chemical instabilities in the reaction zone due to the coupling of pressure fluctuations with the energy release. Therefore, in “unstable” mixtures, critical deflagrations can be supported for long durations, whereas in “stable” mixtures, deflagrations decay as the initial plate generated turbulence decays. The eventual onset of detonation is postulated to be a result of the amplification of pressure waves (i.e., turbulence) that leads to the formation of local explosion centers via the SWACER mechanism during the pre-detonation period.

Green Function of Steady Motion in Finite Water Depth

The Green function associated with a steady translating source on a straight horizontal course in water with finite constant depth and infinite horizontal extent satisfying the classical free surface condition is studied by decomposing it into three parts: an array of Rankine singularities A, local disturbance D, and downstream wave part S. Each of the three parts is studied by several methods. This is used to verify the numerical scheme and find the most time-efficient procedure. The method of repeated averaging of partial sums for oscillating series is efficiently used to evaluate the infinite sum of Rankine singularities and the downstream wave part. The local disturbance needed in vertical force and pitch moment calculations is most demanding. The Green function is used in combination with thin ship theory to calculate wave resistance, vertical force, pitch moment, and far-field wash for a Wigley hull. The results are compared with Tuck's (1966) slender body theory for shallow water and experimental and theoretical results of wave resistance by Everest and Hogben (1970). The agreement is satisfactory. A shallow water wave resistance ratio r expressing the ratio between wave resistance in finite depth and infinite depth is introduced as an indirect way to minimize wash. It is demonstrated that a large influence of critical depth Froude number requires the ratio between fluid depth and ship length to be small.

Synoptic and dynamic aspects of an extreme springtime Saharan dust outbreak

AbstractIn March 2004, a large‐scale, strong and persistent outbreak of Saharan dust on to the adjacent tropical and subtropical Atlantic Ocean was observed in satellite imagery. Surface observations show that the event was accompanied by unusual weather conditions across large parts of North Africa including low temperatures and strong winds over the Sahara, extreme precipitation in Libya and a heat wave on the Guinea coast. The dust outbreak was associated with an unseasonally southerly position of the Intertropical Convergence Zone and a delayed northward progression of the African monsoon.The dust front was initially related to a density current caused by strong evaporational cooling along a precipitating cloud‐band that penetrated into the northern Sahara ahead of an upper‐level trough. At later stages, massive upper‐level convergence, sinking, low‐level divergence and an explosive anticyclogenesis over northwest Africa caused strong northerly flow and a quick spreading of the dust front to the south and west. The strong pressure gradients over North Africa were further enhanced by the formation of a cyclone ahead of the upper‐trough. A dynamical analysis based on potential‐vorticity (PV) theory provides new insights into the generation of North African harmattan surges. Using a PV inversion, it is shown that the explosive anticyclogenesis was caused largely by unbalanced dynamics related to the density current, divergent outflow from the precipitation over Libya and inertial instability in the entrance region of the strong subtropical jet and tropical plume along the eastern side of the upper trough. These processes enhanced the effects of the balanced advection of negative vorticity and cold air by the large‐scale wave. Moreover, we show the importance of an explosive cyclogenesis event over the central North Atlantic for the amplification of the downstream wave associated with the penetration of the cold front into North Africa. Copyright © 2006 Royal Meteorological Society.

Low‐frequency wave characteristics in the upstream and downstream regime of the terrestrial bow shock

Collisionless shocks in space are accompanied by the low‐frequency waves of the magnetic field both upstream and downstream of the shock. However, characteristics of these waves in the plasma rest frame are poorly understood, since spatial scale such as wavelength is hardly determined using single or at best two spacecraft observations. Taking advantage of multipoint measurements of the Cluster spacecraft, we present a statistical study to reveal the rest frame properties of the waves in the terrestrial bow shock environment such as frequencies, wave numbers, phase velocities, propagation directions, polarization and transport ratios for the quasi‐parallel and the quasi‐perpendicular shock regimes and in the upstream region, the outer, the middle, and the inner magnetosheath. We find that the quasi‐parallel shock upstream waves (or the foreshock waves) exhibit properties of the fast mode wave excited by the ion beam instability. Upstream of the quasi‐perpendicular shock the waves also exhibit similar properties to the foreshock waves, but the wave mode identification needs further investigations. In the magnetosheath the waves exhibit properties which are persistent from the outer to the inner magnetosheath. They can be interpreted as the mirror mode coupled to the plasma inhomogeneity or as the slow mode. We also find an organization in the wave propagation pattern that the upstream waves propagate toward upstream, while the magnetosheath waves propagate toward the flank region and toward the magnetopause. It is concluded that wave properties are different between the upstream and the downstream regions, but they are similar between the quasi‐parallel and the quasi‐perpendicular shock regimes, suggesting that the upstream waves are not transmitted to the downstream region across the shock and that the downstream waves do not depend on the shock angle (between the upstream magnetic field and the shock normal direction).

Circumglobal Teleconnection in the Northern Hemisphere Summer*

AbstractAnalysis of the 56-yr NCEP–NCAR reanalysis data reveals a recurrent circumglobal teleconnection (CGT) pattern in the summertime midlatitude circulation of the Northern Hemisphere. This pattern represents the second leading empirical orthogonal function of interannual variability of the upper-tropospheric circulation. The CGT, having a zonal wavenumber-5 structure, is primarily positioned within a waveguide that is associated with the westerly jet stream. The spatial phases of CGT tend to lock to preferred longitudes. The geographically phase-locked patterns bear close similarity during June, August, and September, but the pattern in July shows shorter wavelengths in the North Pacific–North America sector. The CGT is accompanied by significant rainfall and surface air temperature anomalies in the continental regions of western Europe, European Russia, India, east Asia, and North America. This implies that the CGT may be a source of climate variability and predictability in the above-mentioned midlatitude regions.The CGT has significant correlations with the Indian summer monsoon (ISM) and El Niño–Southern Oscillation (ENSO). However, in normal ISM years the CGT–ENSO correlation disappears; on the other hand, in the absence of El Niño or La Niña, the CGT–ISM correlation remains significant. It is suggested that the ISM acts as a “conductor” connecting the CGT and ENSO. When the interaction between the ISM and ENSO is active, ENSO may influence northern China via the ISM and the CGT. Additionally, the variability of the CGT has no significant association with the Arctic Oscillation and the variability of the western North Pacific summer monsoon. The circulation of the wave train shows a barotropic structure everywhere except the cell located to the northwest of India, where a baroclinic circulation structure dominates. Two possible scenarios are proposed. The abnormal ISM may excite an anomalous west-central Asian high and downstream Rossby wave train extending to the North Pacific and North America. On the other hand, a wave train that is excited in the jet exit region of the North Atlantic may affect the west-central Asian high and, thus, the intensity of the ISM. It is hypothesized that the interaction between the global wave train and the ISM heat source may be instrumental in maintaining the boreal summer CGT.

Alfvén-wave transmission and test-particle acceleration in parallel relativistic shocks

Alfven-wave transmission through super-Alfvenic parallel relativistic shock waves is studied. We calculate the wave transmission coecients for given shock properties. We show (i) that the Alfven waves downstream the shock wave are propa- gating predominantly anti-parallel to the flow direction for low-Mach-number shocks, as in the case of non-relativistic shocks; and (ii) that for high-Mach-number ultra-relativistic shocks the forward and backward downstream waves are in equipartition. For low Alfvenic Mach numbers, the scattering center compression ratio of the shock, thus, becomes large and the spectral index of accelerated test particles approaches the limit ! 1 at shock waves approaching the critical value of the quasi-Newtonian Alfvenic Mach number (i.e., the ratio of upstream fluid and Alfven proper speeds), which depends on the shock properties, and equals the square root of the compression ratio at the test-wave limit. Although the inclusion of the wave electromagnetic and velocity fields to the shock jump conditions is likely to decrease the scattering-center compression ratio for shocks with critical Mach numbers, values significantly above the gas compression ratio can be expected for such shocks. Particle acceleration in weak relativistic shocks propagating in magnetized astrophysical jets may, therefore, be substantially more ecient than predicted by models neglecting turbulent electric fields.

Instability of high-frequency modes in viscoelastic plane Couette flow past a deformable wall at low and finite Reynolds number

The linear stability of plane Couette flow of an upper convected Maxwell (UCM) fluid of viscosity η , density ρ , relaxation time τ R and thickness R flowing past a linear viscoelastic solid of shear modulus G and thickness HR is analyzed using a combination of asymptotic analysis (at low Reynolds number) and numerical solution (at finite Reynolds number). The asymptotic analysis is used to analyze the effect of wall deformability on a class of high frequency modes in the UCM fluid with c ∼ O ( R e − 1 / 2 ) at R e ≪ 1 first studied by Gorodtsov and Leonov [J. Appl. Math. Mech. 31 (1967) 310–319; abbreviated GL here], who showed that these modes are stable in a rigid channel. Here, c is the wavespeed of perturbations (nondimensionalised by V), R e = ρ V R / η is the Reynolds number, V is the velocity of the moving plate. Our asymptotic results show that all the high frequency–low Re modes of GL are rendered unstable by solid wall deformability. The variation of the growth rate with the nondimensional solid elasticity parameter Γ = V η / ( G R ) shows an oscillatory behavior alternating between stable and unstable regions, and the variation for upstream traveling waves is completely antiphase with that for downstream waves. The parameter Γ is shown to be proportional to R e 1 / 4 at R e ≪ 1 for neutrally stable downstream waves. Numerical continuation shows that the instability at low Re continues to finite Re, and typically the instability ceases to exist above a critical Re. This critical Re increases with an increase in the Weissenberg number W = τ R V / R . In some cases, however, the instability continues to very high Re, and Γ ∝ R e − 1 in that limit for neutrally stable modes. Neutral stability curves in the Γ – W plane show that the predicted instability of the high frequency GL modes exists only at finite and large W, and is absent in the Newtonian fluid ( W → 0 ) limit. The asymptotic analysis for the Oldroyd-B model shows that the ratio of solvent viscosity to total viscosity of the solution should be O ( R e 1 / 2 ) in order for the instability to exist at low Re, meaning the instability is absent for realistic values of solvent viscosity at low Re. However, numerical results show that the instability of the high-frequency GL modes is present at finite Re in an Oldroyd-B fluid, and increasing solvent viscosity ratio has a stabilizing effect on the instability. Similarly, the solid to fluid viscosity ratio η r also has a stabilizing effect at finite Re, but in the limit of low Re, η r ∼ O ( R e 1 / 2 ) in order for the instability to exist. Our study thus demonstrates the presence of an instability in viscoelastic plane Couette flow past a deformable wall which is primarily due to the viscoelastic nature of the fluid, and is absent in Newtonian fluids.

Ageostrophic, anticyclonic instability of a geostrophic, barotropic boundary current

After having previously demonstrated the occurrence of ageostrophic, anticyclonic instability (AAI) for several other steady flows, in this paper we further test the hypothesis that rotating, stably stratified, shear flows are generally subject to AAI at finite Rossby (Ro) and Froude (Fr) numbers by calculating the inviscid, incompressible normal modes of a geostrophic, barotropic boundary current that vanishes toward the interior of the domain. We focus on a background current profile with no inflection point and nonvanishing absolute vorticity to exclude other known instability types. The hypothesis is again confirmed. The instability occurs for finite Ro only in an anticyclonic background flow. Its modal growth rate steeply decreases (approximately inverse exponentially, ∼e−c/Ro) as Ro→0. The downstream phase velocity of the unstable mode lies within the speed range for the background current, so the eigenmode has a near-critical layer within the zone of strong background shear. Its unstable eigenmode has an aspect ratio of downstream and vertical wave numbers, l/m, on the order of the ratio of the Coriolis and Brünt-Väisällä frequencies (as is typical for rotating, stratified flows). The maximum growth rate is independent of Fr for small Fr (the usual geophysical regime), and it decreases with large Fr, disappearing when there is no stratification. The eigenmode has a weakly decaying, oscillatory, interior far-field structure similar to an inertia-gravity wave. This supports the interpretation that the modal instability arises through the coalescence (i.e., resonance) of two stable normal-mode branches, one associated with the background shear flow and the other an inertia-gravity or Kelvin wave mode.

Waveless Gravity Flow over an Inclined Step

Waveless gravity flows over an inclined step of heightH are studied. The casesH > 0 andH < 0 are considered. It is always assumed that the flow is supercritical at infinity on the right, which ensures the existence of a solution with no downstream wave. For the case of subcritical flow, the relation between the Froude number and the step height is determined that ensures a waveless regime. An approximate analytical relation between the step height and the Froude number is obtained from an analysis of numerical data. This dependence is shown to be nearly identical for steps of any slopes. For the case of supercritical flow, it is established that the problem has a two‐parameter set of solutions. For the caseH <0, approximate analytical formulas for the free‐surface shape are obtained.

Hysteresis and precession of a swirling jet normal to a wall.

Interaction of a swirling jet with a no-slip surface has striking features of fundamental and practical interest. Different flow states and transitions among them occur at the same conditions in combustors, vortex tubes, and tornadoes. The jet axis can undergo precession and bending in combustors; this precession enhances large-scale mixing and reduces emissions of NOx. To explore the mechanisms of these phenomena, we address conically similar swirling jets normal to a wall. In addition to the Serrin model of tornadolike flows, a new model is developed where the flow is singularity free on the axis. New analytical and numerical solutions of the Navier-Stokes equations explain occurrence of multiple states and show that hysteresis is a common feature of wall-normal vortices or swirling jets no matter where sources of motion are located. Then we study the jet stability with the aid of a new approach accounting for deceleration and nonparallelism of the base flow. An appropriate transformation of variables reduces the stability problem for this strongly nonparallel flow to a set of ordinary differential equations. A particular flow whose stability is studied in detail is a half-line vortex normal to a rigid plane-a model of a tornado and of a swirling jet issuing from a nozzle in a combustor. Helical counter-rotating disturbances appear to be first growing as Reynolds number increases. Disturbance frequency changes its sign along the neutral curve while the wave number remains positive. Short disturbance waves propagate downstream and long waves propagate upstream. This helical instability causes bending of the vortex axis and its precession-the effects observed in technological flows and in tornadoes.

The role of boundary conditions in a simple model of incipient vortex breakdown

We consider incipient vortex breakdown and describe how infinitesimal perturbations may destabilize a columnar swirling jet. The framework is axisymmetric and inviscid following Wang and Rusak’s [J. Fluid Mech. 340, 177 (1997)] analysis. The goal of the present study is to relate the local properties of swirling flows in infinite pipes to their global stability properties in pipes of finite length. A spatial linear stability analysis is pursued which gives a complementary point of view to the subcritical/supercritical concept introduced by Benjamin [J. Fluid Mech. 14, 593 (1962)]. In contrast to supercritical flows which exhibit two neutral spatial branches traveling downstream and two counterpropagating evanescent spatial branches, subcritical flows exhibit a frequency range where all spatial branches are neutral, three of which travel downstream and one upstream. By using global energy budget arguments and monitoring how the upstream wave is reflected into the downstream waves and conversely, the inlet and outlet conditions are shown to drive the instability in the limit of long but finite pipes. Various inlet and outlet conditions are proposed that stabilize or destabilize the flow, depending on their ability to supply energy. The analysis demonstrates therefore that the global instability accounting for incipient vortex breakdown in Wang and Rusak’s model may arise from the combination of a locally neutral flow and suitable inlet and outlet conditions.

Alfvén-wave transmission and test-particle acceleration in parallel relativistic shocks

Alfvén-wave transmission through super-Alfvénic parallel relativistic shock waves is studied. We calculate the wave transmission coefficients for given shock properties. We show (i) that the Alfvén waves downstream the shock wave are propagating predominantly anti-parallel to the flow direction for low-Mach-number shocks, as in the case of non-relativistic shocks; and (ii) that for high-Mach-number ultra-relativistic shocks the forward and backward downstream waves are in equipartition. For low Alfvénic Mach numbers, the scattering center compression ratio of the shock, thus, becomes large and the spectral index of accelerated test particles approaches the limit at shock waves approaching the critical value of the quasi-Newtonian Alfvénic Mach number (i.e., the ratio of upstream fluid and Alfvén proper speeds), which depends on the shock properties, and equals the square root of the compression ratio at the test-wave limit. Although the inclusion of the wave electromagnetic and velocity fields to the shock jump conditions is likely to decrease the scattering-center compression ratio for shocks with critical Mach numbers, values significantly above the gas compression ratio can be expected for such shocks. Particle acceleration in weak relativistic shocks propagating in magnetized astrophysical jets may, therefore, be substantially more efficient than predicted by models neglecting turbulent electric fields.