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**Response of the Upper Ocean to Winds**

Chapter 9 Response of the Upper Ocean to Winds Physical oceanography Instructor: Dr. Cheng-Chien Liu Department of Earth Sciences National Cheng Kung University Last updated: 10 November 2003

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**Introduction Question Possible answer**

Same latitude but different climate? Charleston v.s. San Diego Norfolk v.s. San Francisco Possible answer Wind blows onshore cool, moist, marine, boundary layer a few hundred meters thick capped by much warmer, dry air San Diego Charleston warm, moist, marine, boundary layer that is much thicker Convection, which produces rain, is much easier on the east coast than on the west coast.

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**Inertial motion The equation of motion**

For a frictionless ocean at steady state (Eq. 9.1) No horizontal pressure gradient & w << u, v 2W cosf << g (Eq. 9.2) The Coriolis parameter f = 2W sinf The equation for the harmonic oscillator & Solution (Eq. 9.4) Inertia current or inertial oscillation (Fig 9.1) The free motion of parcels of water on a rotating plane The inertial period: Ti = (2p) / f = Tsd /(2 sinf ) (Table 9.1) Rapid change of wind inertia current Incoherent, transient, anti-cyclonic (clockwise in the northern)

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**Ekman layer Steady wind the Ekman layer**

Few-hundred meters thick in the upper ocean The doctoral thesis of Professor Walfrid Ekman Table 9.2: Contributions to the Theory of the Wind-Driven Circulation Nansen’s qualitative arguments Observation Wind tended to blow ice at an angle of 200 – 400 to the right of the wind in the Arctic Balance of three forces W + F + C = 0

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**Ekman’s solution The equation of motion Steady state**

A constant vertical eddy viscosity Molecular viscosity n << turbulent eddy viscosity Az

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**Ekman’s solution Solution Equation The Ekman spiral (Fig 9.3)**

Surface velocity Exponentially decays with depth Spiral Ekman’s constants Need two of three parameters (T, V0, Az) Bulk formula: T = rair CDU102 V0 = V0(U10, f) (Eq. 9.14)

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**Ekman’s solution (cont.)**

Ekman layer depth V(DE) is opposite to V0 DE = p / a (Eq and 9.16) Table 9.3: Typical Ekman Depth The Ekman number Ratio of the friction force and the Corilis force Eq. 9.17: Ez Eq. 9.18: at the Ekman depth DE, Ez = 1 / (2p2)

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**Bottom Ekman layer Solution Eq. 9.19**

u = U [1 - exp(-az)] cos az v = U exp(-az) sin az Boundary conditions: u(zb) = v(zb) = 0 The Ekman spiral (Fig 9.4) Direction: anti-cyclonic Amplitude

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**Bottom Ekman layer (cont.)**

Direction U(z = 0-) is in the direction of U(z > zb) // p U(z > zb) p U(z > zb) // p U(z = 0+) is 450 to the left of U(z > zb) U(z = 0-) is 450 to the right of U(z = 0+) Fig 9.5

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**Examining Ekman’s assumptions**

No boundaries Valid away from coasts Deep water Valid if depth >> 200 m f-plane Valid Steady state Valid if wind blows for longer than a pendulum day Time-dependent solution (Hasselmann 1970)

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**Examining Ekman’s assumptions (cont.)**

Az = fn(U102) fn(z) Not a good assumption If MLD < DE Az = fn(z) from MLD to DE Homogeneous density r fn(z) Probably good, except as it effects stability

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**Observations of flow near the sea surface**

Measurements Ekman's theory is remarkably good Results indicate: Inertial currents the largest component of the flow The flow is nearly independent of depth within the mixed layer for periods near the inertial period. Thus the mixed layer moves like a slab at the inertial period. Current shear is concentrated at the top of the thermocline The flow averaged over many inertial periods is almost exactly that calculated from Ekman's theory. The shear of the Ekman currents extends through the averaged mixed layer and into the thermocline. The Ekman-layer depth DE is almost exactly that proposed by Ekman (9.16), but the surface current V0 is half his value (9.14) The transport is 900 to the right of the wind in the northern hemisphere. The transport direction agrees well with Ekman's theory

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**Ekman mass transport Definition**

The integral of the UE, VE from the surface to a depth d below the Ekman layer (Fig 9.6)

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**Ekman mass transport (cont.)**

Volume transport Definition: The mass transport divided by the density of water and multiplied by the width perpendicular to the transport Y: the north-south distance across which the eastward transport Qx is calculated X: the east-west distance across which the northward transport Qy is calculated Unit: Sv

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**Application of Ekman theory**

Importance of upwelling Enhance biological productivity, which feeds fisheries Cold upwelled water alters local weather. Weather onshore of regions of upwelling tend to have fog, low stratus clouds, a stable stratified atmosphere, little convection, and little rain. Spatial variability of transports in the open ocean leads to upwelling and downwelling, which leads to redistribution of mass in the ocean, which leads to wind-driven geostrophic currents via Ekman pumping

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**Application of Ekman theory (cont.)**

Coastal upwelling Along California coast north wind offshore Ekman transport (Fig 9.7 left) upwelling (Fig 9.7 right) region of cold water (Fig 10.16) Upwelling nutrient productivity Important regions: Peru, California, Somalia, Morocco, and Namibia

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**Application of Ekman theory (cont.)**

Explain why the same latitude but different climate region of cold water cools the incoming air close to the sea a thin, cool atmospheric boundary layer cooling the city Hadley circulation in the atmosphere (Fig 4.3) downward velocity the warmer air above the boundary layer inhibits vertical convection rain is rare Rain forms only when winter storms coming ashore bring strong convection higher up in the atmosphere

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**Application of Ekman theory (cont.)**

Explain why the same latitude but different climate (cont.) Other reasons MLD tends to be thin on the eastern side of oceans upwelling can easily bring up cold water Currents along the eastern side of oceans at mid-latitudes bring colder water from higher latitudes All these processes are reversed offshore of east coasts

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**Ekman pumping Conservation of mass**

The spatial variability of the transports vertical velocities at the top of the Ekman layer ME: vector mass transport due to Ekman ﬂow in the upper ocean H: the horizontal divergence operator

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**Ekman pumping (cont.) Relate Ekman pumping to the wind stress**

The spatial variability of the transports vertical velocities at the top of the Ekman layer Substitute Into T: the vector wind stress ∵ no penetration ∴ w(0) = 0 need another vertical velocity to balance the geostrophic velocity wG(0) at the top of the interior flow in the ocean

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Important concepts Changes in wind stress produce transient oscillations in the ocean called inertial currents Inertial currents are very common in the ocean The period of the current is (2p)/f Steady winds produce a thin boundary layer, the Ekman layer, at the top of the ocean Ekman boundary layers also exist at the bottom of the ocean and the atmosphere The Ekman layer in the atmosphere above the sea surface is called the planetary boundary layer. The Ekman layer at the sea surface has the following characteristics: Direction: 450 to the right of the wind looking downwind in the Northern Hemisphere Surface Speed: 1–2.5% of wind speed depending on latitude Depth: approximately 40–300 m depending on latitude and wind velocity

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**Important concepts (cont.)**

Careful measurements of currents near the sea surface show that: Inertial oscillations are the largest component of the current in the mixed layer The flow is nearly independent of depth within the mixed layer for periods near the inertial period. Thus the mixed layer moves like a slab at the inertial period An Ekman layer exists in the atmosphere just above the sea (and land) surface Surface drifters tend to drift parallel to lines of constant atmospheric pressure at the sea surface. This is consistent with Ekman’s theory. The flow averaged over many inertial periods is almost exactly that calculated from Ekman’s theory Transport is 900 to the right of the wind in the northern hemisphere

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**Important concepts (cont)**

Spatial variability of Ekman transport, due to spatial variability of winds over distances of hundreds of kilometers and days, leads to convergence and divergence of the transport. (a) Winds blowing toward the equator along west coasts of continents produces upwelling along the coast. This leads to cold, productive waters within about 100 km of the shore. (b) Upwelled water along west coasts of continents modifies the weather along the west coasts. Ekman pumping, which is driven by spatial variability of winds, drives a vertical current, which drives the interior geostrophic circulation of the ocean.

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Momentum Equation

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Equation 9.1

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Equation 9.2

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Equation 9.4

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Ekman’s solution

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