Nonlinear stratified spin-up

2002 ◽  
Vol 473 ◽  
pp. 211-244 ◽  
Author(s):  
LEIF N. THOMAS ◽  
PETER B. RHINES
Keyword(s):  
Boundary Layer ◽  
Heat Flux ◽  
Wind Stress ◽  
Thermal Boundary Layer ◽  
Ekman Layer ◽  
Ekman Transport ◽  
Ekman Pumping ◽  
Vertical Diffusion ◽  
Vertical Vorticity ◽  
Horizontal Advection

Both a weakly nonlinear analytic theory and direct numerical simulation are used to document processes involved during the spin-up of a rotating stratified fluid driven by wind-stress forcing for time periods less than a homogeneous spin-up time. The strength of the wind forcing, characterized by the Rossby number ε, is small enough (i.e. ε[Lt ]1) that a regular perturbation expansion in ε can be performed yet large enough (more specifically, ε∝E1/2, where E is the Ekman number) that higher-order effects of vertical diffusion and horizontal advection of momentum/density are comparable in magnitude. Cases of strong stratification, where the Burger number S is equal to one, with zero heat flux at the upper boundary are considered. The Ekman transport calculated to O(ε) decreases with increasing absolute vorticity. In contrast to nonlinear barotropic spin-up, vortex stretching in the interior is predominantly linear, as vertical advection negates stretching of interior relative vorticity, yet is driven by Ekman pumping modified by nonlinearity. As vertical vorticity is generated during the spin-up of the fluid, the vertical vorticity feeds back on the Ekman pumping/suction, enhancing pumping and vortex squashing while reducing suction and vortex stretching. This feedback mechanism causes anticyclonic vorticity to grow more rapidly than cyclonic vorticity. Strict application of the zero-heat-flux boundary condition leads to the growth of a diffusive thermal boundary layer E−1/4 times thicker than the Ekman layer embedded within it. In the Ekman layer, vertical diffusion of heat balances horizontal advection of temperature by extracting heat from the thermal boundary layer beneath. The flux of heat extracted from the top of the thermal boundary layer by this mechanism is proportional to the product of the Ekman transport and the horizontal gradient of the temperature at the surface. The cooling caused by this heat flux generates density inversions and intensifies lateral density gradients where the wind-stress curl is negative. These thermal gradients make the potential vorticity strongly negative, conditioning the fluid for ensuing symmetric instability which greatly modifies the spin-up process.

scholarly journals Effects of a Shallow Pycnocline and Surface Meltwater on Sea Ice–Ocean Drag and Turbulent Heat Flux

2014 ◽  
Vol 44 (8) ◽  
pp. 2176-2190 ◽  
Author(s):  
Achim Randelhoff ◽  
Arild Sundfjord ◽  
Angelika H. H. Renner
Keyword(s):  
Boundary Layer ◽  
Heat Flux ◽  
Wind Stress ◽  
Wave Drag ◽  
Ekman Transport ◽  
Surface Velocity ◽  
Turbulent Shear ◽  
Turbulent Heat ◽  
Neutral Stability ◽  
Flux Parameterization

Abstract Comprehensive boundary layer measurements from a drift station on first-year ice in the late summer of 2012 in the Nansen basin, when stable stratification in the upper ocean extended all the way to the surface, are analyzed. Observed quadratic ice–ocean drag coefficients, based on measurements of wind stress, are roughly 3.6 × 10−3, consistent with neutral-stability Rossby similarity scaling. The turning angles of 32°–39° between surface velocity and stress are larger than Rossby similarity predicts and obey a different scaling. This can be explained by the shallow pycnocline forcing the Ekman transport into a thin layer and modeled roughly employing a simple first-order correction to Rossby similarity. Turbulent shear stress in the ice–ocean boundary layer is on average 3 times smaller than the estimate based on wind stress, possibly because internal wave drag was significant. This lowers vertical scalar fluxes by 38% compared to a scenario where turbulent stress accounts for the total drag. The authors measure an average upward ocean–ice heat flux of 10 W m−2, which is 50% smaller than predicted by a bulk heat flux parameterization. This reduction is attributed to additional sources of heat and freshwater that alter the ice–ocean interface salt balance. This study shows that a commonly used bulk heat flux parameterization is a special case of a simple downgradient parameterization allowing for a modified interface salt budget. For similar wind forcing, observed ice–ocean fluxes of heat and salt were 40%–100% larger when the ice-relative current approached from a nearby pressure ridge keel than otherwise.

scholarly journals Heat transfer in rotating Rayleigh–Bénard convection with rough plates

2017 ◽  
Vol 830 ◽  
Author(s):  
Pranav Joshi ◽  
Hadi Rajaei ◽  
Rudie P. J. Kunnen ◽  
Herman J. H. Clercx
Keyword(s):  
Heat Transfer ◽  
Boundary Layer ◽  
Ekman Layer ◽  
Ekman Pumping ◽  
Ekman Boundary Layer ◽  
Bénard Convection ◽  
Roughness Elements ◽  
Benard Convection ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard

This experimental study focuses on the effect of horizontal boundaries with pyramid-shaped roughness elements on the heat transfer in rotating Rayleigh–Bénard convection. It is shown that the Ekman pumping mechanism, which is responsible for the heat transfer enhancement under rotation in the case of smooth top and bottom surfaces, is unaffected by the roughness as long as the Ekman layer thickness $\unicode[STIX]{x1D6FF}_{E}$ is significantly larger than the roughness height $k$. As the rotation rate increases, and thus $\unicode[STIX]{x1D6FF}_{E}$ decreases, the roughness elements penetrate the radially inward flow in the interior of the Ekman boundary layer that feeds the columnar Ekman vortices. This perturbation generates additional thermal disturbances which are found to increase the heat transfer efficiency even further. However, when $\unicode[STIX]{x1D6FF}_{E}\approx k$, the Ekman boundary layer is strongly perturbed by the roughness elements and the Ekman pumping mechanism is suppressed. The results suggest that the Ekman pumping is re-established for $\unicode[STIX]{x1D6FF}_{E}\ll k$ as the faces of the pyramidal roughness elements then act locally as a sloping boundary on which an Ekman layer can be formed.

scholarly journals Destruction of Potential Vorticity by Winds

2005 ◽  
Vol 35 (12) ◽  
pp. 2457-2466 ◽  
Author(s):  
Leif N. Thomas
Keyword(s):  
Boundary Layer ◽  
Wind Stress ◽  
Potential Vorticity ◽  
Ekman Layer ◽  
Buoyancy Flux ◽  
Coriolis Parameter ◽  
Mode Water ◽  
Ocean Fronts ◽  
Secondary Circulations ◽  
Frictional Forces

Abstract The destruction of potential vorticity (PV) at ocean fronts by wind stress–driven frictional forces is examined using PV flux formalism and numerical simulations. When a front is forced by “downfront” winds, that is, winds blowing in the direction of the frontal jet, a nonadvective frictional PV flux that is upward at the sea surface is induced. The flux extracts PV out of the ocean, leading to the formation of a boundary layer thicker than the Ekman layer, with nearly zero PV and nonzero stratification. The PV reduction is not only active in the Ekman layer but is transmitted through the boundary layer via secondary circulations that exchange low PV from the Ekman layer with high PV from the pycnocline. Extraction of PV from the pycnocline by the secondary circulations results in an upward advective PV flux at the base of the boundary layer that scales with the surface, nonadvective, frictional PV flux and that leads to the deepening of the layer. At fronts forced by both downfront winds and a destabilizing atmospheric buoyancy flux FBatm, the critical parameter that determines whether the wind or the buoyancy flux is the dominant cause for PV destruction is (H/δe)(FBwind/FBatm), where H and δe are the mixed layer and Ekman layer depths, FBwind = S2τo/(ρof ), S2 is the magnitude of the lateral buoyancy gradient of the front, τo is the downfront component of the wind stress, ρo is a reference density, and f is the Coriolis parameter. When this parameter is greater than 1, PV destruction by winds dominates and may play an important role in the formation of mode water.

scholarly journals Seasonal variability of the Ekman transport and pumping in the upwelling system off central-northern Chile (∼  30° S) based on a high-resolution atmospheric regional model (WRF)

Ocean Science ◽  
2016 ◽  
Vol 12 (5) ◽  
pp. 1049-1065 ◽  
Author(s):  
Luis Bravo ◽  
Marcel Ramos ◽  
Orlando Astudillo ◽  
Boris Dewitte ◽  
Katerina Goubanova
Keyword(s):  
High Resolution ◽  
Wind Stress ◽  
Seasonal Variability ◽  
Coastal Upwelling ◽  
Ekman Transport ◽  
Northern Chile ◽  
Ekman Pumping ◽  
Relative Contribution ◽  
Alongshore Wind Stress ◽  
Alongshore Wind

Abstract. Two physical mechanisms can contribute to coastal upwelling in eastern boundary current systems: offshore Ekman transport due to the predominant alongshore wind stress and Ekman pumping due to the cyclonic wind stress curl, mainly caused by the abrupt decrease in wind stress (drop-off) in a cross-shore band of 100 km. This wind drop-off is thought to be an ubiquitous feature in coastal upwelling systems and to regulate the relative contribution of both mechanisms. It has been poorly studied along the central-northern Chile region because of the lack in wind measurements along the shoreline and of the relatively low resolution of the available atmospheric reanalysis. Here, the seasonal variability in Ekman transport, Ekman pumping and their relative contribution to total upwelling along the central-northern Chile region (∼  30° S) is evaluated from a high-resolution atmospheric model simulation. As a first step, the simulation is validated from satellite observations, which indicates a realistic representation of the spatial and temporal variability of the wind along the coast by the model. The model outputs are then used to document the fine-scale structures in the wind stress and wind curl in relation to the topographic features along the coast (headlands and embayments). Both wind stress and wind curl had a clear seasonal variability with annual and semiannual components. Alongshore wind stress maximum peak occurred in spring, second increase was in fall and minimum in winter. When a threshold of −3  ×  10−5 s−1 for the across-shore gradient of alongshore wind was considered to define the region from which the winds decrease toward the coast, the wind drop-off length scale varied between 8 and 45 km. The relative contribution of the coastal divergence and Ekman pumping to the vertical transport along the coast, considering the estimated wind drop-off length, indicated meridional alternation between both mechanisms, modulated by orography and the intricate coastline. Roughly, coastal divergence predominated in areas with low orography and headlands. Ekman pumping was higher in regions with high orography and the presence of embayments along the coast. In the study region, the vertical transport induced by coastal divergence and Ekman pumping represented 60 and 40 % of the total upwelling transport, respectively. The potential role of Ekman pumping on the spatial structure of sea surface temperature is also discussed.

Free convection beneath a heated horizontal plate in a rapidly rotating system

2007 ◽  
Vol 586 ◽  
pp. 491-506
Author(s):  
ROBERT J. WHITTAKER ◽  
JOHN R. LISTER
Keyword(s):  
Boundary Layer ◽  
Thermal Boundary Layer ◽  
Ekman Layer ◽  
Outer Edge ◽  
Horizontal Plate ◽  
Rotating System ◽  
Plate Temperature ◽  
System Equation ◽  
Simple Boundary ◽  
Simple Boundary Condition

Laminar flow beneath a finite heated horizontal plate in a rapidly rotating system is considered in both axisymmetric and planar geometries. In particular, we examine the case where the Ekman layer is confined well within a much deeper (yet still thin) thermal boundary layer. This situation corresponds to the regime E−3/2 ≪ Ra ≪ E−5/2, where E and Ra are the natural Ekman and Rayleigh numbers for the system (equation (2.6)). The outward flux of buoyant fluid from beneath the plate occurs primarily in the Ekman layer, while outward flow in the thicker thermal boundary layer is inhibited by a dominant thermal-wind balance. The O(Ra−1/2E−3/4 thickness of the thermal boundary layer is determined by a balance between Ekman suction and diffusion. There are several possible asymptotic regimes near the outer edge of the plate, differing only by logarithmic factors, but in all cases the edge corresponds to a simple boundary condition on the interior flow. With a uniform plate temperature, the dimensionless heat transfer (equation (7.6)) is given by a Nusselt number $\Nu\,{\sim} \tfrac{1}{2}\Ra^{1/2}\Ek^{3/4}[\ln (\Ra^{-1} \Ek ^{-5/2})]^{1/2}$. The solution for a uniform plate heat flux is also presented.

scholarly journals The wind-driven overturning circulation of the World Ocean

2005 ◽  
Vol 2 (5) ◽  
pp. 473-505 ◽  
Author(s):  
K. Döös
Keyword(s):  
Boundary Layer ◽  
Wind Stress ◽  
World Ocean ◽  
Conveyor Belt ◽  
Meridional Overturning Circulation ◽  
Ekman Transport ◽  
Western Boundary ◽  
Overturning Circulation ◽  
The World ◽  
Meridional Overturning

Abstract. The wind driven aspects of the meridional overturning circulation of the world ocean and the Conveyor Belt is studied making use of a simple analytical model. The model consists of three reduced gravity layers with an inviscid Sverdrupian interior and a western boundary layer. The net north-south exchange is made possible by setting appropriate western boundary conditions, so that most of the transport is confined to the western boundary layer, while the interior is the Sverdrupian solution to the wind stress. The flow across the equator is made possible by the change of potential vorticity by the Rayleigh friction in the western boundary layer, which is sufficient to permit water and the Conveyor Belt to cross the equator. The cross-equatorial flow is driven by a weak meridional pressure gradient in opposite direction in the two layers on the equator at the western boundary. The model is applied to the World Ocean with a realistic wind stress. The amplitude of the Conveyor Belt is set by the northward Ekman transport in the Southern Ocean and the outcropping latitude of the NADW. It is in this way possible to set the amount of NADW that is pumped up from the deep ocean and driven northward by the wind and converted in the surface layer into less dense water by choosing the outcropping latitude and the depth of the layers at the western boundary. The model has proved to be able to simulate many of the key features of the Conveyor Belt and the meridional overturning cells of the World Ocean. This despite that there is no deep ocan mixing and that the water mass conversions in the this model are made at the surface.

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