Internal Rossby Radius Research Articles

New Insights on the Formation and Breaking Mechanism of Convective Cyclonic Cones in the South Adriatic Pit during Winter 2018

Abstract The deepwater formation in the northern part of the South Adriatic Pit (Mediterranean Sea) is investigated using a unique oceanographic dataset. In situ data collected by a glider along the Bari–Dubrovnik transect captured the mixing and the spreading/restratification phase of the water column in winter 2018. After a period of about 2 weeks from the beginning of the mixing phase, a homogeneous convective area of ∼300-m depth breaks up due to the baroclinic instability process in cyclonic cones made of geostrophically adjusted fluid. The base of these cones is located at the bottom of the mixed layer, and they extend up to the theoretical critical depth Zc. These cones, with a diameter on the order of internal Rossby radius of deformation (∼6 km), populate the ∼110-km-wide convective site, develop beneath it, and have a short lifetime of weeks. Later on, the cones extend deeper and intrusion from deep layers makes their inner core denser and colder. These observed features differ from the long-lived cyclonic eddies sampled in other ocean sites and formed at the periphery of the convective area in a postconvection period. So far, to the best of our knowledge, only theoretical studies, laboratory experiments, and model simulations have been able to predict and describe our observations, and no other in situ information has yet been provided.

Role of Ocean and Atmosphere Variability in Scale‐Dependent Thermodynamic Air‐Sea Interactions

AbstractThis study investigates the influence of oceanic and atmospheric processes in extratropical thermodynamic air‐sea interactions resolved by satellite observations (OBS) and by two climate model simulations run with eddy‐resolving high‐resolution (HR) and eddy‐parameterized low‐resolution (LR) ocean components. Here, spectral methods are used to characterize the sea surface temperature (SST) and turbulent heat flux (THF) variability and co‐variability over scales between 50 and 10,000 km and 60 days to 80 years in the Pacific Ocean. The relative roles of the ocean and atmosphere are interpreted using a stochastic upper‐ocean temperature evolution model forced by noise terms representing intrinsic variability in each medium, defined using climate model data to produce realistic rather than white spectral power density distributions. The analysis of all datasets shows that the atmosphere dominates the SST and THF variability over zonal wavelengths larger than ∼2,000–2,500 km. In HR and OBS, ocean processes dominate the variability of both quantities at scales smaller than the atmospheric first internal Rossby radius of deformation (R1, ∼600–2,000 km) due to a substantial ocean forcing coinciding with a weaker atmospheric modulation of THF (and consequently of SST) than at larger scales. The ocean forcing also induces oscillations in SST and THF with periods ranging from intraseasonal to multidecadal, reflecting a red spectrum response to ocean forcing similar to that driven by atmospheric forcing. Such features are virtually absent in LR due to a weaker ocean forcing relative to HR.

Effect of Floating Offshore Wind Turbines on Atmospheric Circulation in California

In California offshore waters, sustained northwesterly winds have been identified as a key energy resource that could contribute substantially to California’s renewable energy mandate. It is these winds that drive upwelling, which is responsible for much of the primary productivity that sustains one of the richest ecosystems on the planet. The goal of this study is to quantify changes in wind fields at the sea surface as the result of offshore wind turbine deployments by use of an atmospheric model. Modeled wind fields from this study will drive an ocean circulation model. The Weather Research and Forecasting model was implemented on a regional scale along the U.S. west coast, with a higher resolution nest along the California continental shelf. Simulated arrays of offshore wind turbines were placed within call areas for wind farm development offshore of Central and Northern California. At full build-out, it was found that wind speeds at 10 m height are reduced by approximately 5%, with wakes extending approximately 200 km downwind of the nominated lease block areas. The length scale of wind speed reductions was found to be several times the internal Rossby radius of deformation, the spatial scale at which rotationally-influenced ocean circulation processes such as upwelling occur.

Near-Resonances of Superinertial and Tidal Fluctuations at Islands

AbstractA linear numerical model of an island or a tall seamount is used to explore superinertial leaky resonances forced by ambient vertically and horizontally uniform current fluctuations. The model assumes a circularly symmetric topography (including a shallow reef) and allows realistic stratification and bottom friction. As long as there is substantial stratification, a number of leaky resonances are found, and when the island’s flanks are narrow relative to the internal Rossby radius, some of the near-resonant modes resemble leaky internal Kelvin waves. Other “resonances” resemble higher radial mode long gravity waves as explored by Chambers. The near-resonances amplify the cross-reef velocities that help fuel biological activity. Results for cases with the central island replaced by a lagoon do not differ greatly from the island case which has land at the center. As an aside, insight is provided on the question of offshore boundary conditions for superinertial nearly trapped waves along a straight coast.

A comprehensive oceanographic dataset of a subpolar, mid-latitude broad fjord: Fortune Bay, Newfoundland, Canada

Abstract. While the dynamics of narrow fjords, i.e. narrow with respect to their internal Rossby radius, have been widely studied, it is only recently that interest in studying the physics of broad fjords was sparked due to their importance in glacial ice melting (in Greenland, especially). Here, we present a comprehensive set of data collected in Fortune Bay, a broad, mid-latitude fjord located on the northwest Atlantic shores. Aside from being wide (15–25 km width) and deep (600 m at its deepest), Fortune Bay also has the characteristics of having steep slopes, having weak tides and being strongly stratified from spring to fall. Thus, and since strong along-shore winds also characterize the region, this system is prone to interesting dynamics, generally taking the form of transient upwelling and downwelling travelling along its shores, similar to processes encountered in broad fjords of higher latitudes. The dataset collected to study those dynamics consists of water column physical parameters (temperature, salinity, currents and water level) and atmospheric forcing (wind speed and direction, atmospheric pressure, air temperature, and solar radiation) taken at several points around the fjord using oceanographic moorings and land-based stations. The program lasted 2 full years and achieved a good data return of 90 %, providing a comprehensive dataset not only for Fortune Bay studies but also for the field of broad fjord studies. The data are available publically from the SEANOE repository (https://doi.org/10.17882/62314; Donnet and Lazure, 2020).

Horizontal Pressure Gradient Parameterization for One‐Dimensional Lake Models

AbstractThis work presents a new method for closure of horizontally averaged 1‐D thermohydrodynamic equations in an enclosed reservoir by parameterizing the horizontal pressure gradient usually omitted in 1‐D lake models. Horizontal pressure gradient is computed using an auxiliary multilayer model where horizontal structure of speed and pressure is given by 1‐st Fourier mode. A major effect of new parameterization in 1‐D lake model is the emergence of explicitly reproduced H1 seiche modes. The parameterization is implemented in the LAKE model, with minor (2–4%) extra computational cost imposed. The model is applied to Lake Iseo (Italy), and calculated temperature series are compared to measured ones in upper, middle, and deep portions of metalimnion. The amplitude of seiche‐induced temperature oscillations well matched the observed amplitude by tuning the bottom friction coefficient only. The synoptic variability of thermocline vertical displacement caused by wind events is well reproduced by the model. The dominant peak of quasi‐diurnal period in temperature power spectrum was captured in simulations as well. Using the new parameterization of horizontal pressure gradient extends the applicability of a 1‐D lake model formulation to small lakes, which size is less than internal Rossby radius, and where pressure gradient dominates over Coriolis force.

Tidal Conversion and Dissipation at Steep Topography in a Channel Poleward of the Critical Latitude

AbstractIn high-latitude fjords and channels in the Canadian Arctic Archipelago, walls support radiating internal tides as Kelvin waves. Such waves allow for significant barotropic to baroclinic tidal energy conversion, which is otherwise small or negligible when poleward of the critical latitude. This fundamentally three-dimensional system of a subinertial channel is investigated with a suite of numerical simulations in rectangular channels of varying width featuring idealized, isolated ridges. Even in channels as wide as 5 times the internal Rossby radius, tidal conversion can remain as high as predicted by an equivalent two-dimensional, nonrotating system. Curves of tidal conversion as a function of channel width, however, do not vary monotonically. Instead, they display peaks and nulls owing to interference between the Kelvin waves along the wall and similar waves that propagate along the ridge flanks, the wavelengths of which can be estimated from linear theory to guide prediction. Because the wavelengths are comparable to width scales of Arctic channels and fjords, the interference will play a first-order role in tidal energy budgets and may consequently influence the stability of glaciers, the ventilation of deep layers, the locations of sediment deposition, and the fate of freshwater exiting the Arctic Ocean.

Comparing Eulerian and Lagrangian eddy census for a tide-less, semi-enclosed basin, the Baltic Sea

Eddies in the global and coastal ocean play a key role in mixing and transport processes. Here, we present an eddy census for the Baltic Sea covering the years 2008–2010. The eddy tracking was carried out with three different eddy detection methods (an Eulerian streamline based and two Lagrangian coherent structure-based methods). The near surface velocity fields were generated with GETM (General Estuarine Transport Model), an eddy-resolving ocean model (internal Rossby radius ≈ 5 km, grid resolution ≈ 0.6 km). The general shape of the statistics for the three detection algorithms agree well for lifetime and propagation distance. The most striking differences between the methods were visible for the eddy size, with much smaller eddy-radii detected by the Lagrangian methods. Although we only find minor variations in the monthly numbers of tracked eddies in total, we found large variations in the eddy core vorticity, with higher rotation and smaller radii during winter. This was caused by the changing stratification over the course of the year. Additionally, we found spatial changes in the ratio of cyclonic to anticyclonic rotating eddies and seasonal differences in the spatial distribution of eddies.

Wintertime Fjord‐Shelf Interaction and Ice Sheet Melting in Southeast Greenland

AbstractA realistic numerical model was constructed to simulate the oceanic conditions and circulation in a large southeast Greenland fjord (Kangerdlugssuaq) and the adjacent shelf sea region during winter 2007–2008. The major outlet glaciers in this region recently destabilized, contributing to sea level rise and ocean freshening, with increased oceanic heating a probable trigger. It is not apparent a priori whether the fjord dynamics will be influenced by rotational effects, as the fjord width is comparable to the internal Rossby radius. The modeled currents, however, describe a highly three‐dimensional system, where rotational effects are of order‐one importance. Along‐shelf wind events drive a rapid baroclinic exchange, mediated by coastally trapped waves, which propagate from the shelf to the glacier terminus along the right‐hand boundary of the fjord. The terminus was regularly exposed to around 0.5 TW of heating over the winter season. Wave energy dissipation provoked vertical mixing, generating a buoyancy flux which strengthened overturning. The coastally trapped waves also acted to strengthen the cyclonic mean flow via Stokes' drift. Although the outgoing wave was less energetic and located at the opposite sidewall, the fjord did exhibit a resonant response, suggesting that fjords of this scale can also exhibit two‐dimensional dynamics. Long periods of moderate wind stress greatly enhanced the cross‐shelf delivery of heat toward the fjord, in comparison to stronger events over short intervals. This suggests that the timescale over which the shelf wind field varies is a key parameter in dictating wintertime heat delivery from the ocean to the ice sheet.

Comparative terrestrial atmospheric circulation regimes in simplified global circulation models. Part II: Energy budgets and spectral transfers

The energetics of possible global atmospheric circulation patterns in an Earth‐like atmosphere are explored using a simplified global General Circulation Model (GCM) based on the University of Hamburg's Portable University Model for the Atmosphere (designated here as PUMA‐S), forced by linear relaxation towards a prescribed temperature field and subject to Rayleigh surface drag and hyperdiffusive dissipation. Results from a series of simulations, obtained by varying planetary rotation rate Ω with an imposed equator‐to‐pole temperature difference, were analysed to determine the structure and magnitude of the heat transport and other contributions to the energy budget for the time‐averaged, equilibrated flow. These show clear trends with rotation rate, with the most intense Lorenz energy cycle for an Earth‐sized planet occurring with a rotation rate around half that of the present‐day Earth (i.e., Ω∗ = Ω/Ω E = 1/2, where Ω E is the rotation rate of the Earth). Kinetic energy (KE) and available potential energy (APE) spectra, E K (n) and E A (n) (where n is total spherical wavenumber), also show clear trends with rotation rate, with n −3 enstrophy‐dominated spectra around Ω∗ = 1 and steeper (∼n −5) slopes in the zonal mean flow with little evidence for the n −5/3 spectrum anticipated for an inverse KE cascade. Instead, both KE and APE spectra become almost flat at scales larger than the internal Rossby radius, L d , and exhibit near‐equipartition at high wavenumbers. At Ω∗ < <1, the spectrum becomes dominated by KE with E K (n)∼(2–3)E A (n) at most wavenumbers and a slope that tends towards n −5/3 across most of the spectrum. Spectral flux calculations show that enstrophy and APE are almost always cascaded downscale, regardless of rotation rate. KE cascades are more complicated, however, with downscale transfers across almost all wavenumbers, dominated by horizontally divergent modes, for . At higher rotation rates, transfers of KE become increasingly dominated by rotational (horizontally nondivergent) components with strong upscale transfers (dominated by eddy–zonal flow interactions) for scales larger than L d and weaker downscale transfers for scales smaller than L d .

Lagrangian Dispersion Characteristics in The Western Mediterranean

Dispersion characteristics in the Western Mediterranean are analyzed using data from Coastal Ocean Dynamics Experiment (CODE) and Surface Velocity Program (SVP) surface drifters deployed in the period 1986–2017. Results are presented in terms of absolute dispersion A2 (mean-squared displacement of drifter individuals) and of relative dispersion (D2; mean square separation distance of drifter pairs). Moreover, the dispersion characteristics are estimated for different initial separation distances (D0) between particles: smaller, larger, or comparable with the internal Rossby radius of deformation. Results show the presence of a quasiballistic regime for absolute dispersion at small time scales and the nonlocal relative dispersion regime related to the submesoscale activities for scales smaller than the internal Rossby radius. At intermediate times, two anomalous absolute dispersion regimes (elliptic and hyperbolic regimes) related with the flow topology are observed, although the relative dispersion involves the Richardson and shear/ballistic regimes only for D0 smaller than the Rossby radius. During the subsequent 20–30 days, absolute dispersion shows quasirandom walk regime and relative dispersion follows the diffusive regime for scales larger than 100 km for which pair velocities are uncorrelated.

Tidally Modulated Internal Hydraulic Flow and Energetics in the Central Canadian Arctic Archipelago

AbstractThe Canadian Arctic Archipelago is a key conduit for comparatively fresh Arctic waters flowing to the Atlantic. Model estimates of the freshwater outflow, which is strongly correlated with the volume flux, contain major uncertainties because most existing models exclude tides, marginally resolve the internal Rossby radius, or both. At the same time, barotropic tidal models preclude stratified flow effects. Here we assess the relative importance of barotropic and baroclinic processes to water mass transformation, friction, and energy losses motivated by processes observed in a fine‐scale survey in the central Archipelago. A sharp separation of warmed Canada Basin water and locally formed water is observed over a long sill in a narrow channel and coincides with an internal hydraulic jump caused by the mean flow. Tidal currents, however, modulate the jump, as demonstrated by both scale analysis and a two‐dimensional simulation. The jump, together with internal tides propagating as Kelvin waves, leads to isopycnal displacements up to 50 m. The generation of these internal Kelvin waves has a leading‐order role in a regional energy budget. It is small, however, relative to bottom boundary layer dissipation, which accounts for an estimated 50% of the total tidal energy losses. Consequently, adding tides needs to be a priority for regional models.

Continental Shelf Baroclinic Instability. Part II: Oscillating Wind Forcing

AbstractContinental shelf baroclinic instability energized by fluctuating alongshore winds is treated using idealized primitive equation numerical model experiments. A spatially uniform alongshore wind, sinusoidal in time, alternately drives upwelling and downwelling and so creates highly variable, but slowly increasing, available potential energy. For all of the 30 model runs, conducted with a wide range of parameters (varying Coriolis parameter, initial stratification, bottom friction, forcing period, wind strength, and bottom slope), a baroclinic instability and subsequent eddy field develop. Model results and scalings show that the eddy kinetic energy increases with wind amplitude, forcing period, stratification, and bottom slope. The dominant alongshore length scale of the eddy field is essentially an internal Rossby radius of deformation. The resulting depth-averaged alongshore flow field is dominated by the large-scale, periodic wind forcing, while the cross-shelf flow field is dominated by the eddy variability. The result is that correlation length scales for alongshore flow are far greater than those for cross-shelf velocity. This scale discrepancy is qualitatively consistent with midshelf observations by Kundu and Allen, among others.

Degeneration of internal Kelvin waves in a continuous two-layer stratification

We explore the evolution of the gravest internal Kelvin wave in a two-layer rotating cylindrical basin, using direct numerical simulations (DNS) with a hyper-viscosity/diffusion approach to illustrate different dynamic and energetic regimes. The initial condition is derived from Csanady’s (J. Geophys. Res., vol. 72, 1967, pp. 4151–4162) conceptual model, which is adapted by allowing molecular diffusion to smooth the discontinuous idealized solution over a transition scale, ${\it\delta}_{i}$, taken to be small compared to both layer thicknesses $h_{\ell },\ell =1,2$. The different regimes are obtained by varying the initial wave amplitude, ${\it\eta}_{0}$, for the same stratification and rotation. Increasing ${\it\eta}_{0}$ increases both the tendency for wave steepening and the shear in the vicinity of the density interface. We present results across several regimes: from the damped, linear–laminar regime (DLR), for which ${\it\eta}_{0}\sim {\it\delta}_{i}$ and the Kelvin wave retains its linear character, to the nonlinear–turbulent transition regime (TR), for which the amplitude ${\it\eta}_{0}$ approaches the thickness of the (thinner) upper layer $h_{1}$, and nonlinearity and dispersion become significant, leading to hydrodynamic instabilities at the interface. In the TR, localized turbulent patches are produced by Kelvin wave breaking, i.e. shear and convective instabilities that occur at the front and tail of energetic waves within an internal Rossby radius of deformation from the boundary. The mixing and dissipation associated with the patches are characterized in terms of dimensionless turbulence intensity parameters that quantify the locally elevated dissipation rates of kinetic energy and buoyancy variance.

Eulerian Volume Transport Induced by Spatially Damped Internal Equatorial Kelvin Waves

AbstractThe Eulerian volume transport in internal equatorial Kelvin waves subject to viscous attenuation is investigated theoretically by integrating the horizontal momentum equations in the vertical. In terms of small perturbations, the time-averaged horizontal transports are determined to second order in wave steepness. The total Lagrangian volume transport in this problem consists of a Stokes transport plus an Eulerian transport. It is known that the Stokes transport, that is, the vertically integrated Stokes drift, in inviscid internal equatorial Kelvin waves vanishes identically in the rigid-lid approximation for arbitrary vertical variation of the Brunt–Väisälä frequency. The present study considers spatial wave damping due to viscosity. The Stokes transport still becomes zero, but now the radiation stresses due to decaying waves become source terms for the Eulerian mean transport. Calculations of the wave-induced Eulerian transport yield a jetlike symmetric mean flow along the equator from west to east for each baroclinic component, with compensating westward flows on both sides. The flow system scales as the internal equatorial Rossby radius in the north–south direction. The total eastward part of the Eulerian volume flux centered about the equator is estimated to about 0.2 Sv (1 Sv ≡ 106 m3 s−1) for the first baroclinic mode.

Mass transport in internal coastal Kelvin waves

We investigate theoretically the mass transport in internal coastal Kelvin waves by integrating the horizontal momentum equations in the vertical. Applying a perturbation method, the time-averaged Lagrangian horizontal fluxes are determined to second order in wave steepness. The linear wave field is expanded in the vertical using orthogonal functions. Due to the orthogonality property of these functions, formulae for the non-linear Stokes drift and the mean vertically-averaged Eulerian transport driven by the radiation stress can be derived for arbitrary vertical variation of the Brunt–Väisälä frequency N. For values of N typical of the thermocline in the Caspian Sea, the calculation of the non-linear transports yields a jet-like mean flow along the coast, limited in the off-shore direction by the internal Rossby radius. It is suggested that this wave-induced mean drift may contribute to the mean circulation in the Caspian Sea.

Bathymetric effects on estuarine plume dynamics

AbstractThe influence of bathymetry on an estuary plume at an estuary‐shelf transition is studied with a three‐dimensional ocean circulation model. To understand the response of the plume to bathymetry, several types of estuarine shapes and shelf geometries were adopted in this numerical study. The channel's shape and its width‐to‐depth aspect ratio affected the fate of the plume by determining flow characteristics inside the estuary. Moreover, the bathymetry of the shelf such as the shelf slope and the direction of a submarine channel defined the plume characteristics on the shelf. An estuarine channel with a triangular cross section generated relatively stronger exchange flows at midestuary than a rectangular cross section, which resulted in a larger surface plume over the shelf. The extension of the submarine channel onto the shelf favored increased plume water transport out to the shelf, a result of reduced frictional effects on the shelf. The orientation of the submarine channel changed the direction of the plume over the shelf, with no additional external forces. Two fronts developed at the edges of the submarine channel because of enhanced lateral shears in the flow. When the estuary was relatively wide compared to the internal Rossby radius (Kelvin number Ke ≥ 5), or when the relative strength of the freshwater discharge compared to the estuary width was weak (Rossby number Ro ≤ 0.05), the coastal plume did not expand up‐shelf. In fact, results indicated that freshwater up‐shelf transport in a coastal current, moving against Coriolis’ accelerations, was proportional to Ro.

A cyclonic gyre in an ice‐covered lake

Observations of a cyclonic gyre in an ice‐covered, midsize (< 5 km2), temperate lake are presented. Horizontal and vertical measurements of temperature and electrical conductivity measurements were collected using a conductivity‐temperature‐depth logger mounted on an autonomous underwater vehicle and additional instrumentation. These measurements revealed a cylindrical density anomaly with a radius of ∼110 m extending from the surface to ∼14 m depth. The observed radius is smaller than the internal Rossby radius of deformation (∼ 200 m), which suggests a cyclogeostrophic balance between centripetal, Coriolis, and pressure forces. The maximum azimuthal velocity, calculated assuming this balance, was ∼ 2.1 cm s−1 at 6–8 m depth. The Rossby number associated with this velocity was 1.7; this is consistent with the cyclogeostrophic assumption (i.e., Rossby number > 1) and nearly twice that of similar under‐ice eddies in the Arctic Ocean. The estimated Ekman spin‐down timescale is 1.5–15 d, but despite this, the gyre appeared to be relatively unchanged over 6 d of field observations. This persistence implies the gyre was forced over the course of the field study; however, the source of the forcing is unknown. Horizontal temperature transects at and below the bottom of the gyre revealed coherent temperature fluctuations suggestive of vertical transport associated with the gyre.

Spatial scales of mesoscale eddies from GOCI Chlorophyll-a concentration images in the East/Japan Sea

The spatial scales of mesoscale eddies are of importance to understand physio-biogeochemical processes in the East/Japan Sea. Chlorophyll-a concentration images from the Geostationary Ocean Color Imager (GOCI) revealed numerous eddies during the phytoplankton bloom in spring. These eddies were manually digitized to obtain geolocation information at the peripheries from GOCI images and then least-square fitted to each ellipse. The elliptic elements were the geolocation position of the eddy center, the rotation angle from due east, the eccentricity, the lengths of the semi-major and semi-minor axes, and the mean radius of the ellipse. The spatial scales of eddies had a mean radii ranging from 10 km to 75 km and tended to be smaller in the northern region. The scales revealed a linear trend of about −7.26 km/°N as a function of the latitude. This tendency depended on the latitudinal variation of the internal Rossby radius of deformation, which originates from the substantial difference in the density structure of the water column. The scales from the sea surface temperature image were larger by 1.30 times compared to those from ocean color image. This implies that physical processes along the periphery of the eddy affect the nutrient dynamics.

Nonlinear baroclinic equilibration at finite supercriticality

The nonlinear equilibration of finite amplitude baroclinic waves in Phillips two-layer model is investigated at finite supercriticality. The aims are to quantify the robustness and relevance of the nonlinear theory of Warn, Gauthier and Pedlosky (WGP) for the evolution of the developing baroclinic wave, and to assess the tightness of pseudomomentum and improved pseudoenergy bounds for disturbance amplitude and energy. A high-resolution numerical model is used to perform a parameter sweep in (β, W)-space, where β is the inverse criticality of the initial flow, and W is the ratio of the channel width to the (internal) Rossby radius. At low supercriticalities, the main predictions of WGP are found to be accurate at short times, but at long times the fully nonlinear results are found to diverge from WGP's solution. The mechanism for equilibration involves the elimination of the lower layer potential vorticity (PV) gradient, but as the supercriticality increases this is achieved by the roll-up of a train of opposite-signed vortices, rather than by coarse-grain PV homogenization as in WGP. Peak wave amplitudes are typically ≈90% of the maximum attainable under the pseudomomentum bound. New formulae are given for the pseudoenergy bound on disturbance energy which, unlike the WGP solution and the pseudomomentum bound, have non-trivial dependence on W. A detailed assessment is made of the extent to which these bounds are attained.