Selected Topics in Fluid Dynamics & Advanced Hydrodynamics Flows In

Total Page:16

File Type:pdf, Size:1020Kb

Selected Topics in Fluid Dynamics & Advanced Hydrodynamics Flows In Selected Topics in Fluid Dynamics & Advanced Hydrodynamics Flows in rotating systems (Dated: May 26, 2020) Recommended reading: Lautrup (Chapter 20) & Acheson (Chapter 8.5) The motivation for studying rotating flows is threefold. Firstly, it is important to assess the effect of rotation on flows in an industrial context. Secondly, and more importantly, we often witness flows in rotating frames of reference in our everyday experience. The effects of rotation are particularly pronounced in geophysical flows. As observers of these, we are necessary in a non-inertial frame of reference Even if the corrections to acceleration for such an observer are small compared to gravity, for large scale, slow flows, such as in the ocean or atmosphere, they can shape the flows we see in a considerable way. Thirdly, astrophysical flows are known to be deeply impacted by the effects of (often fast) rotation of their components. The main new players are two ficticious forces (they are ficticious from the point of view of an intertial observer, but embark on a carousel to convince yourself that they can be painfully real!): the centrifugal force and the Coriolis force. Thus, when in a non-inertial, rotating frame, we need to correct the Newton’s second law by incorporating these two forces. Here, we will only consider steady rotation, so Ω = const. For unsteady rotation, revisiting any Classical Mechanics lecture will be helpful to include dΩ=dt terms. VELOCITY AND ACCELERATION IN STEADY ROTATION In Exercises, we will discuss the derivation of the equations of motion in a steadily rotating frame of reference. Considering two frames, one inertial (x0; y0; z0) and one rotating (x; y; z). By examining the acceleration in the rotating frame, we find that a = a0 − 2Ω × u − Ω × (Ω × x) (1) Question: Estimate the magnitude of these terms (as compared to gravity as the natural scale for acceleration) for a carousel with R = 5 m and rotating at one turn per 6 seconds. Are they significant? We will now asses their significance for Earth. At typical values the centrifugal force can be estimated as Ω2R, with R being now the Earth’s radius, which compares to gravity as Ω2R=g ≈ 1=291, so it is negligibly small. It leads to a slight flattening of the rotating planet 2 perpendicularly to its rotation axis, but in the geophysical context it does not play a major role and, as we will show, it can be incorporated into effective pressure. The Coriolis force is different. Let us consider a point on the surface of the Earth at a latitude θ and introduce a local flat-Earth coordinate system with the x-axis pointing towards the East, y pointing North, and z being vertical. The rotational velocity of the Earth then is Ω = Ω(0; sin θ; cos θ). The Coriolis acceleration gC in terms of the local velocity u = dx=dt reads 0 1 uy cos θ − uz sin θ B C C B C g = 2Ω B −ux cos θ C : (2) @ A ux sin θ At the poles the Coriolis force is always horizontal, and at the equator it is always vertical for purely horizontal motion. The vertical component of the Coriolis force is very small compared to gravity. For example, for a jet aircraft flying westwards at middle latitudes at a speed close to the velocity of sound, it amounts to about 0.3% of gravity, similarly to the centrifugal force. Thus it can be neglected for most Earthly flows of interest. For typically small vertical flow velocities, we will also neglest the Coriolis force due to uz. So we will adopt an assumption that 0 1 uy B C C B C g = 2Ω? B−uxC ; (3) @ A 0 with Ω? = Ω cos θ. Concluding, in a flat-Earth coordinate system the Coriolis force indeed looks as if the Earth was rotating about the local vertical with the angular velocity Ω?. At middle ◦ − −1 latitudes θ ≈ 45 , we can take Ω? ≈ 10 4 s , rendering the typical period of rotation of a Foucault’s pedulum [1] being 2π=Ω? ≈ 34 h. FLOW IN ROTATING FRAMES Applying the modified acceleration to the Navier-Stokes equation (exercise), we can show that the resulting governing equations for the flow become @u 1 + u · ru + 2Ω × u − Ω × (Ω × x) = − rp + νr2u; (4) @t ρ r · u = 0: (5) Moreover, upon defining the effective pressure P (another exercise), the flow equations may be recast as @u 1 + u · ru = −2Ω × u − rP + νr2u; (6) @t ρ r · u = 0; (7) 3 where for the calculation of the Coriolis term we can only use the local, vertical angular velocity Ω?. Dimensionless numbers: Reynolds, Rossby, and Ekman It is again interesting to study the limits of this equation. Taking the flow velocity scale U and the length scale L, we can define, as usually, the Reynolds number UL Re = : (8) ν In the limit of large Reynolds number, Re 1, the advective term dominates over the viscous term, and we can generally neglect the effects of viscosity and the last term in Eq. (6). Remember though, that this approximation will generally only work far away from boundaries, where boundary layers will be formed! From the remaining terms, it is tempting to compare the relative importance of the advective term and the Coriolis acceleration, which leads to (exercise) the Rossby number ju · ruj U Ro = ≈ ; (9) 2jΩ × uj 2ΩL named after a Swedish-born US meteorologist Carl-Gustav Arild Rossby (1898-1957). In this idealised case, the Coriolis force will only be significant for low Ro, so for U . 2ΩL, when the translational flow velocity is comparable or smaller to the local rotation-induced velocity. Example Consider ocean currents and weather cyclones, which are relatively steady rotating flows. With U ∼ 1 m/s (ocean) or U ∼ 10 m/s (wind), and the typical local angular velocity of 2Ω ≈ 10−4 Hz, we get Rossby numbers of Ro ≈ 0:01 (ocean) or Ro ≈ 0:1 (cyclones). We conclude that both are thus dominated by the Coriolis force, which plays the key role in their evolution. For a typical swimming human, with L ∼ 1 m and U ∼ 1 m/s, we find Ro ≈ 104, which justifies the complete neglect of the Coriolis force in this case. Interestingly, in systems where gravity is simulated by rotation (such as a space station), the Coriolis force will cause surprises when playing ball games (see Lautrup). For lower Reynolds numbers, it becomes important to compare the relative strength of viscous and Coriolis effects. The relevant dimensionless combination (exercise) is the Ekman number jνr2uj ν Ro Ek = = = ; (10) 2jΩ × uj 2ΩL2 Re named after a Swedish oceanographer Vagn Walfrid Ekman (1874– 1954). When the Ekman number is small, the viscous force is small compared to the Coriolis force. This is automatically the case for moderate Ro and large Re. In the sea and the atmosphere, the huge Reynolds number 4 makes the Ekman number negligible but, warned by our experience with boundary layers, we may expect the Ekman number to be larger close to boundaries. Indeed, it normally is of order unity close to bounding surfaces where viscosity dominates over advection, and competes with the Coriolis force. This gives rise to a particular boundary layer, called the Ekman layer, which we will discuss further. GEOSTROPHIC FLOW In most natural flow context, we may neglect both the Rossby ansd Ekman numbers, since Ro 1 and Ek 1. This means we are disregarding both viscous and advective terms in the Navier-Stokes equations. The flow is completely dominated by the Coriolis force and called geostrophic. For a steady flow, we end up with a particularly simple equation 1 − 2Ω × u − rP = 0; (11) ρ which has a nice hydrostatic analogy – the effective pressure gradient must always balance the Coriolis force. However, the Coriolis force has a rather special form, and this has far-reaching consequences. The most spectacular one is that geostrophic flow is essentially two-dimensional. To examine this, notice that u · rP = 0; (12) which implies that the effective pressure is constant on streamlines. Moreover, if the motion is horizontal, the gravitational force plays no role in the pressure, and effective pressure is the same as the hydrostatic pressure. This means that streamlines and isobars coincide in geostrophic flow. Thus, in a typical weather map, the direction of the wind can be readily read off from the isobars. Now, to account for the sign of the wind direction, we may use our knowledge of the Coriolis force direction, which makes winds in the Northern hemisphere turn anti-clockwise around the low pressure areas (cyclones) and clockwise around the high pressure areas (anti-cyclones). The wind velocity may be determined from the geostrophic equation (11) using Ω · u = 0 to find Ω × rP u = : (13) 2Ω2ρ In the absence of the Coriolis force, the air flow would happen along the pressure gradients only, from the high to low pressure regions. The 2D character of geostrophic flow may be seen as follows. Taking a scalar product of Eq. 11 with Ω, we find @P (Ω · r)P = Ω = 0; (14) @z 5 so the effective pressure is constant along the axis of rotation. In a constant gravitational field, using the definition of P and again disregarding the centrifugal contribution, we find the hydrostatic pressure as p = P (x; y) − ρgz: (15) The fact that the effective pressure is invariant under translations along the rotation axis extends to the whole flow. Taking the curl of Eq. (11) and using incompressibility, we find @u 0 = r × (Ω × u) = Ω(r · u) − (Ω · r)u = −Ω : (16) @z Thus the flow field u is a function of x and y only and is two-dimensional! This result is called the Taylor-Proudman theorem.
Recommended publications
  • Ekman Layer at the Sea Surface • Ekman Mass Transport • Application of Ekman Theory • Langmuir Circulation • Important Concepts
    Principle of Oceanography Gholamreza Mashayekhinia 2013_2014 Response of the Upper Ocean to Winds Contents • Inertial Motion • Ekman Layer at the Sea Surface • Ekman Mass Transport • Application of Ekman Theory • Langmuir Circulation • Important Concepts Inertial Motion The response of the ocean to an impulse that sets the water in motion. For example, the impulse can be a strong wind blowing for a few hours. The water then moves under the influence of coriolis force and gravity. No other forces act on the water. In physics, the Coriolis effect is a deflection of moving objects when they are viewed in a rotating reference frame. This low pressure system over Iceland spins counter-clockwise due to balance between the Coriolis force and the pressure gradient force. Such motion is said to be inertial. The mass of water continues to move due to its inertia. If the water were in space, it would move in a straight line according to Newton's second law. But on a rotating Earth, the motion is much different. The equations of motion for a frictionless ocean are: 푑푢 1 휕푝 = − + 2Ω푣 푠푖푛휑 (1a) 푑푡 휌 휕푥 푑푣 1 휕푝 = − − 2Ω푢 푠푖푛휑 (1b) 푑푡 휌 휕푦 푑휔 1 휕푝 = − + 2Ω푢 푐표푠휑 − 푔 (1c) 푑푡 휌 휕푥 Where p is pressure, Ω = 2 π/(sidereal day) = 7.292 × 10-5 rad/s is the rotation of the Earth in fixed coordinates, and φ is latitude. We have also used Fi = 0 because the fluid is frictionless. Let's now look for simple solutions to these equations. To do this we must simplify the momentum equations.
    [Show full text]
  • Small-Scale Processes in the Coastal Ocean
    or collective redistirbution of any portion of this article by photocopy machine, reposting, or other means is permitted only with the approval of The Oceanography Society. Send all correspondence to: [email protected] ofor Th e The to: [email protected] Oceanography approval Oceanography correspondence PO all portionthe Send Society. ofwith any permitted articleonly photocopy by Society, is of machine, reposting, this means or collective or other redistirbution article has This been published in SPECIAL ISSUE ON CoasTal Ocean PRocesses TuRBulence, TRansFER, anD EXCHanGE Oceanography BY JAMes N. MouM, JonaTHan D. NasH, , Volume 21, Number 4, a quarterly journal of The Oceanography Society. Copyright 2008 by The Oceanography Society. All rights reserved. Permission is granted to copy this article for use in teaching and research. research. for this and teaching article copy to use in reserved.by The 2008 rights is granted journal ofCopyright OceanographyAll The Permission 21, NumberOceanography 4, a quarterly Society. Society. , Volume anD JODY M. KLYMak Small-Scale Processes in the Coastal Ocean 20 depth (m) 40 B ox 1931, ox R ockville, R epublication, systemmatic reproduction, reproduction, systemmatic epublication, MD 20849-1931, USA. -200 -100 0 100 distance along ship track (m) 22 Oceanography Vol.21, No.4 ABSTRACT. Varied observations over Oregon’s continental shelf illustrate the beauty and complexity of geophysical flows in coastal waters. Rapid, creative, and sometimes fortuitous sampling from ships and moorings has allowed detailed looks at boundary layer processes, internal waves (some extremely nonlinear), and coastal currents, including how they interact. These processes drive turbulence and mixing in shallow coastal waters and encourage rapid biological responses, yet are poorly understood and parameterized.
    [Show full text]
  • On the Patterns of Wind-Power Input to the Ocean Circulation
    2328 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 41 On the Patterns of Wind-Power Input to the Ocean Circulation FABIEN ROQUET AND CARL WUNSCH Department of Earth, Atmosphere and Planetary Science, Massachusetts Institute of Technology, Cambridge, Massachusetts GURVAN MADEC Laboratoire d’Oce´anographie Dynamique et de Climatologie, Paris, France (Manuscript received 1 February 2011, in final form 12 July 2011) ABSTRACT Pathways of wind-power input into the ocean general circulation are analyzed using Ekman theory. Direct rates of wind work can be calculated through the wind stress acting on the surface geostrophic flow. However, because that energy is transported laterally in the Ekman layer, the injection into the geostrophic interior is actually controlled by Ekman pumping, with a pattern determined by the wind curl rather than the wind itself. Regions of power injection into the geostrophic interior are thus generally shifted poleward compared to regions of direct wind-power input, most notably in the Southern Ocean, where on average energy enters the interior 108 south of the Antarctic Circumpolar Current core. An interpretation of the wind-power input to the interior is proposed, expressed as a downward flux of pressure work. This energy flux is a measure of the work done by the Ekman pumping against the surface elevation pressure, helping to maintain the observed anomaly of sea surface height relative to the global-mean sea level. 1. Introduction or the input to surface gravity waves (601 TW; Wang and Huang 2004b), but they are not directly related to The wind stress, along with the secondary input from the interior flow.
    [Show full text]
  • Sea-Ice Melt Driven by Ice-Ocean Stresses on the Mesoscale
    manuscript submitted to JGR: Oceans 1 Sea-ice melt driven by ice-ocean stresses on the 2 mesoscale 1 1 2 1 3 Mukund Gupta , John Marshall , Hajoon Song , Jean-Michel Campin , 1 4 Gianluca Meneghello 1 5 Massachusetts Institute of Technology 2 6 Yonsei University 7 Key Points: 8 • Ice-ocean drag on the mesoscale generates Ekman pumping that brings warm wa- 9 ters up to the surface 10 • This melts sea-ice in winter and spring and reduces its mean thickness by 10 % 11 under compact ice regions 12 • Sea-ice formation (melt) in cyclones (anticyclones) produces deeper (shallower) 13 mixed layer depths. Corresponding author: Mukund Gupta, [email protected] {1{ manuscript submitted to JGR: Oceans 14 Abstract 15 The seasonal ice zone around both the Arctic and the Antarctic coasts is typically 16 characterized by warm and salty waters underlying a cold and fresh layer that insulates 17 sea-ice floating at the surface from vertical heat fluxes. Here we explore how a mesoscale 18 eddy field rubbing against ice at the surface can, through Ekman-induced vertical mo- 19 tion, bring warm waters up to the surface and partially melt the ice. We dub this the 20 `Eddy Ice Pumping' mechanism (EIP). When sea-ice is relatively motionless, underly- 21 ing mesoscale eddies experience a surface drag that generates Ekman upwelling in an- 22 ticyclones and downwelling in cyclones. An eddy composite analysis of a Southern Ocean 23 eddying channel model, capturing the interaction of the mesoscale with sea-ice, shows 24 that within the compact ice zone, the mixed layer depth in cyclones is very deep (∼ 500 25 m) due to brine rejection, and very shallow in anticyclones (∼ 20 m) due to sea-ice melt.
    [Show full text]
  • Effects of the Earth's Rotation
    Effects of the Earth’s Rotation C. Chen General Physical Oceanography MAR 555 School for Marine Sciences and Technology Umass-Dartmouth 1 One of the most important physical processes controlling the temporal and spatial variations of biological variables (nutrients, phytoplankton, zooplankton, etc) is the oceanic circulation. Since the circulation exists on the earth, it must be affected by the earth’s rotation. Question: How is the oceanic circulation affected by the earth’s rotation? The Coriolis force! Question: What is the Coriolis force? How is it defined? What is the difference between centrifugal and Coriolis forces? 2 Definition: • The Coriolis force is an apparent force that occurs when the fluid moves on a rotating frame. • The centrifugal force is an apparent force when an object is on a rotation frame. Based on these definitions, we learn that • The centrifugal force can occur when an object is at rest on a rotating frame; •The Coriolis force occurs only when an object is moving relative to the rotating frame. 3 Centrifugal Force Consider a ball of mass m attached to a string spinning around a circle of radius r at a constant angular velocity ω. r ω ω Conditions: 1) The speed of the ball is constant, but its direction is continuously changing; 2) The string acts like a force to pull the ball toward the axis of rotation. 4 Let us assume that the velocity of the ball: V at t V + !V " V = !V V + !V at t + !t ! V = V!" ! V !" d V d" d" r = V , limit !t # 0, = V = V ($ ) !t !t dt dt dt r !V "! V d" V = % r, and = %, dt V Therefore, d V "! = $& 2r dt ω r To keep the ball on the circle track, there must exist an additional force, which has the same magnitude as the centripetal acceleration but in an opposite direction.
    [Show full text]
  • Ekman Transport in Balanced Currents with Curvature
    MAY 2017 W E N E G R A T A N D T H O M A S 1189 Ekman Transport in Balanced Currents with Curvature JACOB O. WENEGRAT AND LEIF N. THOMAS Department of Earth System Science, Stanford University, Stanford, California (Manuscript received 24 October 2016, in final form 6 March 2017) ABSTRACT Ekman transport, the horizontal mass transport associated with a wind stress applied on the ocean surface, is modified by the vorticity of ocean currents, leading to what has been termed the nonlinear Ekman transport. This article extends earlier work on this topic by deriving solutions for the nonlinear Ekman transport valid in currents with curvature, such as a meandering jet or circular vortex, and for flows with the Rossby number approaching unity. Tilting of the horizontal vorticity of the Ekman flow by the balanced currents modifies the ocean response to surface forcing, such that, to leading order, winds parallel to the flow drive an Ekman transport that depends only on the shear vorticity component of the vertical relative vorticity, whereas across- flow winds drive transport dependent on the curvature vorticity. Curvature in the balanced flow field thus leads to an Ekman transport that differs from previous formulations derived under the assumption of straight flows. Notably, the theory also predicts a component of the transport aligned with the surface wind stress, contrary to classic Ekman theory. In the case of the circular vortex, the solutions given here can be used to calculate the vertical velocity to a higher order of accuracy than previous solutions, extending possible ap- plications of the theory to strong balanced flows.
    [Show full text]
  • A Coriolis Tutorial, Part 4
    1 a Coriolis tutorial, Part 4: 2 Wind-driven ocean circulation; the Sverdrup relation 3 James F. Price 4 Woods Hole Oceanographic Institution, 5 Woods Hole, Massachusetts, 02543 6 https://www2.whoi.edu/staff/jprice/ [email protected] th 7 Version7.5 25 Jan, 2021 at 08:28 Figure 1: The remarkable east-west asymmetry of upper ocean gyres is evident in this section through the thermocline of the subtropical North Atlantic: (upper), sea surface height (SSH) from satellite altimetry, and (lower) potential density from in situ observations. Within a narrow western boundary region, longi- tude -76 to -72, the SSH slope is positive and very large, and the inferred geostrophic current is northward and very fast: the Gulf Stream. Over the rest of the basin, the SSH slope is negative and much, much smaller. The inferred geostrophic flow is southward and correspondingly very slow. This southward flow is consistent with the overlying, negative wind stress curl and the Sverdrup relation, the central topic of this essay. 1 8 Abstract. This essay is the fourth of a four part introduction to Earth’s rotation and the fluid dynamics 9 of the atmosphere and ocean. The theme is wind-driven ocean circulation, and the motive is to develop 10 insight for several major features of the observed ocean circulation, viz. western intensification of upper 11 ocean gyres and the geography of the mean and the seasonal variability. 12 A key element of this insight is an understanding of the Sverdrup relation between wind stress curl 13 and meridional transport. To that end, shallow water models are solved for the circulation of a model 14 ocean that is started from rest and driven by a specified wind stress field: westerlies at mid-latitudes and 15 easterlies in subpolar and tropical regions.
    [Show full text]
  • 2.011 Motion of the Upper Ocean
    2.011 Motion of the Upper Ocean Prof. A. Techet April 20, 2006 Equations of Motion • Impart an impulsive force on the surface of the fluid to set it in motion (no other forces act on the fluid, except Coriolis). • Then, for a parcel of water moving with zero friction: Simplify the equations: • Assuming only Coriolis force is acting on the fluid then there will be no horizontal pressure gradients: • Assuming then the flow is only horizontal (w=0): • Coriolis Parameter: Solve these equations • Combine to solve for u: • Standard Diff. Eq. • Inertial Current Solution: (Inertial Oscillations) Inertial Current • Note this solution are equations for a circle Drifter buoys in the N. Atlantic • Circle Diameter: • Inertial Period: • Anti-cyclonic (clockwise) in N. Hemisphere; cyclonic (counterclockwise) in S. Hemi • Most common currents in the ocean! Courtesy of Prof. Robert Stewart. Used with permission. Source: Introduction to Physical Oceanography, http://oceanworld.tamu.edu/home/course_book.htm Ekman Layer • Steady winds on the surface generate a thin, horizontal boundary layer (i.e. Ekman Layer) • Thin = O (100 meters) thick • First noticed by Nansen that wind tended to blow ice at 20-40° angles to the right of the wind in the Arctic. Courtesy of Prof. Robert Stewart. Used with permission. Source: Introduction to Physical Oceanography, http://oceanworld.tamu.edu/home/course_book.htm Ekman’s Solution • Steady, homogeneous, horizontal flow with friction on a rotating earth • All horizontal and temporal derivatives are zero • Wind Stress in horizontal
    [Show full text]
  • Estimates of Wind Energy Input to the Ekman Layer in the Southern Ocean from Surface Drifter Data Shane Elipot1,2 and Sarah T
    JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 114, C06003, doi:10.1029/2008JC005170, 2009 Click Here for Full Article Estimates of wind energy input to the Ekman layer in the Southern Ocean from surface drifter data Shane Elipot1,2 and Sarah T. Gille1,3 Received 24 October 2008; revised 19 February 2009; accepted 24 March 2009; published 5 June 2009. [1] The energy input to the upper ocean Ekman layer is assessed for the Southern Ocean by examining the rotary cross spectrum between wind stress and surface velocity for frequencies between 0 and 2 cpd. The wind stress is taken from European Center for Medium-Range Weather Forecasts ERA-40 reanalysis, and drifter measurements from 15 m depth are used to represent surface velocities, with an adjustment to account for the vertical structure of the upper ocean. The energy input occurs mostly through the nonzero frequencies rather than the mean. Phenomenologically, the combination of a stronger anticyclonic wind stress forcing associated with a greater anticyclonic response makes the contribution from the anticyclonic frequencies dominate the wind energy input. The latitudinal and seasonal variations of the wind energy input to the Ekman layer are closely related to the variations of the wind stress, both for the mean and for the time-varying components. The contribution from the near-inertial band follows a different trend, increasing from 30°S to about 45°S and decreasing further south, possibly a consequence of the lack of variance in this band in the drifter and wind stress data. Citation: Elipot, S., and S. T. Gille (2009), Estimates of wind energy input to the Ekman layer in the Southern Ocean from surface drifter data, J.
    [Show full text]
  • Intermittent Reduction in Ocean Heat Transport Into the Getz Ice Shelf
    RESEARCH LETTER Intermittent Reduction in Ocean Heat Transport Into the 10.1029/2021GL093599 Getz Ice Shelf Cavity During Strong Wind Events Key Points: Nadine Steiger1,2 , Elin Darelius1,2 , Anna K. Wåhlin3 , and Karen M. Assmann4 • Observations at the western Getz Ice Shelf show eight intermittent events 1Geophysical Institute, University of Bergen, Bergen, Norway, 2Bjerknes Center for Climate Research, Bergen, Norway, of Winter Water deepening below 3Department of Marine Sciences, University of Gothenburg, Gothenburg, Sweden, 4Institute of Marine Research, 350 m depth during winter 2016 • The events are associated with Tromsø, Norway strong easterly winds and caused by non-local Ekman downwelling • The ocean heat transport into the Abstract The flow of warm water toward the western Getz Ice Shelf along the Siple Trough, West Getz Ice Shelf cavity is reduced by Antarctica, is intermittently disrupted during short events of Winter Water deepening. Here we show, 25% in the winter of 2016 due to the events using mooring records, that these 5–10 days-long events reduced the heat transport toward the ice shelf cavity by 25% in the winter of 2016. The events coincide with strong easterly winds and polynya opening in the region, but the Winter Water deepening is controlled by non-local coastal Ekman downwelling Supporting Information: Supporting Information may be found rather than polynya-related surface fluxes. The thermocline depth anomalies are forced by Ekman in the online version of this article. downwelling at the northern coast of Siple Island and propagate to the ice front as a coastal trapped wave. During the events, the flow at depth does no longer continue along isobaths into the ice shelf cavity but Correspondence to: aligns with the ice front.
    [Show full text]
  • Observations of Ekman Currents in the Southern Ocean
    768 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 39 Observations of Ekman Currents in the Southern Ocean YUENG-DJERN LENN School of Ocean Sciences, Bangor University, Menai Bridge, Wales, United Kingdom TERESA K. CHERESKIN Scripps Institution of Oceanography, University of California, San Diego, La Jolla, California (Manuscript received 29 October 2007, in final form 28 August 2008) ABSTRACT Largely zonal winds in the Southern Ocean drive an equatorward Ekman transport that constitutes the shallowest limb of the meridional overturning circulation of the Antarctic Circumpolar Current (ACC). Despite its importance, there have been no direct observations of the open ocean Ekman balance in the Southern Ocean until now. Using high-resolution repeat observations of upper-ocean velocity in Drake Passage, a mean Ekman spiral is resolved and Ekman transport is computed. The mean Ekman currents decay in amplitude and rotate anticyclonically with depth, penetrating to ;100-m depth, above the base of the annual mean mixed layer at 120 m. The rotation depth scale exceeds the e-folding scale of the speed by about a factor of 3, resulting in a current spiral that is compressed relative to predictions from Ekman theory. Transport estimated from the observed currents is mostly equatorward and in good agreement with the Ekman transport computed from four different gridded wind products. The mean temperature of the Ekman layer is not distinguishable from temperature at the surface. Turbulent eddy viscosities inferred from Ekman theory and a direct estimate of the time-averaged stress were O(102–103)cm2 s21. The latter calculation results in a profile of eddy viscosity that decreases in magnitude with depth and a time-averaged stress that is not parallel to the time-averaged vertical shear.
    [Show full text]
  • A Coriolis Tutorial, Part 4
    1 a Coriolis tutorial, Part 4: 2 wind-driven circulation and the Sverdrup relation 3 James F. Price 4 Woods Hole Oceanographic Institution, 5 Woods Hole, Massachusetts, 02543 6 https://www2.whoi.edu/staff/jprice/ [email protected] 7 Version5.6 May23,2019 Figure 1: A zonal section through the North Atlantic thermocline at 35oN, viewed toward the north. (up- per) Sea surface height (SSH) from satellite altimetry. (lower) Potential density from in situ observations. This section shows the remarkable zonal asymmetry that is characteristic of the major ocean gyres. Within a narrow western boundary region, the SSH slope is positive and comparatively very large. The inferred geostrophic current, the Gulf Stream, is northward and fast. Over the rest of the basin, the SSH slope is negative and much smaller, and the inferred geostrophic flow is southward and very slow. This broadly distributed southerly flow is wind-driven Sverdrup flow, mainly, and is the central topic of this essay. 1 8 Abstract: This essay is the fourth of a four part introduction to the effects of Earth’s rotation on the 9 fluid dynamics of the atmosphere and ocean. The theme is wind-driven ocean circulation, and the central 10 topic is the Sverdrup relation between meridional transport and the curl of the wind stress. The goal is to 11 develop some insight for several features of the observed ocean circulation, viz. western intensification of 12 upper ocean gyres and the geography of seasonal variability. To that end, a shallow water model is solved 13 for the circulation of a model ocean that is started from rest by a specified wind stress field that includes 14 westerlies at mid-latitudes and easterlies in the lower subtropics.
    [Show full text]