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Ocean and Climate Ocean currents II Wind-water interaction and drag forces Ekman transport, circular and geostrophic flow General ocean flow pattern Wind-Water surface interaction Water motion at the surface of the ocean (mixed layer) is driven by wind effects. Friction causes drag effects on the water, transferring momentum from the atmospheric winds to the ocean surface water. The drag force Wind generates vertical and horizontal motion in the water, triggering convective motion, causing turbulent mixing down to about 100m depth, which defines the isothermal mixed layer. The drag force FD on the water depends on wind velocity v: 2 FD CD Aa v CD drag coefficient dimensionless factor for wind water interaction CD 0.002, A cross sectional area depending on surface roughness, and particularly the emergence of waves! Katsushika Hokusai: The Great Wave off Kanagawa The Beaufort Scale is an empirical measure describing wind speed based on the observed sea conditions (1 knot = 0.514 m/s = 1.85 km/h)! For land and city people Bft 6 Bft 7 Bft 8 Bft 9 Bft 10 Bft 11 Bft 12 m Conversion from scale to wind velocity: v 0.836 B3/ 2 s A strong breeze of B=6 corresponds to wind speed of v=39 to 49 km/h at which long waves begin to form and white foam crests become frequent. The drag force can be calculated to: kg km m F C A v2 1200 v 45 12.5 C 0.001 D D a a m3 h s D 2 kg 2 m 2 FD 0.0011200 Am 12.5 187.5 A N or FD A 187.5 N m m3 s For a strong gale (B=12), v=35 m/s, the drag stress on the water will be: 2 kg m 2 FD / A 0.00251200 35 3675 N m m3 s kg m 1200 v 35 C 0.0025 a m3 s D Ekman transport The frictional drag force of wind with velocity v or wind stress x generating a water velocity u, is balanced by the Coriolis force, but drag decreases with depth z. m F A C v2 F A 2m Au sin f u D D D a c c A m 2sin f s1 Coriolis parameter z A F / A z z z 1 f u D m / A z z z 1 τ force N in vector terms f zˆ u D z mass kg zˆ u defines vector direction of transport τ f zˆ u D z 0 D f zˆ u dz f zˆ MEk 0 kg MEk u dz Ekman mass transport vector m s Ekman transport 0 kg MEk u dz m s assumption is a moreless linear increase of water density with depth M Ek u z Since the horizontal wind direction u, moving the water is perpendicular to the depth vector z, the direction of the frictional drag force D is perpendicular to both vectors and the magnitude is: vector depth 0 D z f 1 M Ek sin 90 f M Ek f 2sin D z f u z Example for Ekman transport What drag force (pressure) does it take at a latitude of 35oN to move water over a depth of 10 m within 1 minute by 100 m to the right? 0 5 1 D z f M Ek f 2sin 35 7.29210 s f 27.292105 s1 sin 350 4.183105 s1 kg 100m kg M u z 1200 10m 20,000 Ek m3 60s m s kg kg N z f M 4.183105 s1 20,000 0.837 1 D Ek m s m s2 m2 kg 1 2 m km Weak force, done by v D m s 0.913 3.3 C kg s h winds of B≈2 with : D 1200 0.001 m3 Typical surface wind stress conditions Annual mean wind stress on the ocean in units (N/m2). The green shade represents the magnitude of the stress. Typical wind stress values in the Westerlies reach ≈ 0.1 to 0.2 N/m2. The strongest stress component can be observed for the Roaring Forties, the weakest component is in the Doldrums. z C v2 z f M M D D a D Ek Ek f 2sin 2 Calculate the mass transport MEk for a typical wind stress of D = 0.25 N/m o at the southern latitude of 40 S. f 2sin 400 7.292105 s1 f 27.292105 s1 sin 400 9.37105 s1 N 0.25 2 kg M D m 2.67103 Ek f 9.37105 s1 m s About 2 tons of water are shifted within 1 sec by 1 meter to the left! Determine the wind velocity for a typical drag coefficient CD=0.002? kg 9.37105 s1 2.13103 f M f M m km v2 Ek v Ek m s 0.29 1 C C kg s h D a D a 0.0021200 m3 About B=1-2 on the Beauford Scale Impact on ocean currents The direction of Ekman transport depends on the hemisphere. In the northern hemisphere this transport is at a 90o angle to the right of the wind direction, and in the southern hemisphere it occurs at a 90o angle to the left of the wind direction. This generates gyres, circular motions in ocean basins limited by continental coasts. Reality Realityis more iscomplex more complex because becauseof additional of additional forces due forces to friction due toand the drag temperatureforces effects, provided which by the add atmospheric to the eddy windformation circulation phenomenon! and by the friction forces exerted by deeper water layers! Humboldt Current The cold Peruvian current (an eastern boundary current) flows towards the equator along the coast of Ecuador and Peru. It flows with a speed of 0.1 to 0.15[m/s]. In the absence of an El Niño, prevailing surface winds cause Ekman transport to the left or away from the coast, with subsequent upwelling of cold water. Kon Tiki, Heyerdahl’s thesis of populating Polynesia from the East rather than from the North-West by taking advantage of Humboldt current for sea travel. Pressure conditions Pressure gradient towards ocean depth can be expressed in terms of the salinity and temperature dependence of ocean water density dP g dz P g S,T, P z ref neglecting S,T, P P z P g z surface ref Approximately a linear increase of surface z pressure with depth – in contrast the P 1atm 105 Pa atmosphere displays an exponential surface decrease of pressure with altitude . m kg z 100m P100m 105 Pa 9.81 1000 100m 106 Pa s2 m3 z 1km P1km 107 Pa Flow at larger depth is directed z 4km P4km 4107 Pa by the pressure gradient and the Coriolis force, “geostrophic flow”. Geostrophic flow A geostrophic current is an oceanic flow in which the pressure gradient force is balanced by the Coriolis effect. The direction of geostrophic flow is parallel to the isobars, with the high pressure to the right of the flow in the Northern Hemisphere, and the high pressure to the left in the Southern Hemisphere. Fluid or gaseous media move from high pressure to low pressure regions. The force pushing the water is called the pressure gradient force Fp. In a geostrophic flow, water moves along the lines of equal pressure (isobars), instead of moving from a high pressure to low pressure region. This occurs due to Earth’s rotation that cause the Coriolis force Fc. The Coriolis force acts at right angles to the flow. When it balances the pressure gradient force (Fp=Fc), the resulting flow becomes the geostrophic flow. Flow velocity Variations of pressure conditions or isobars with depth are associated with temperature and salinity conditions and can cause horizontal flow. The pressure gradient is balanced by the Coriolis force. This allows an estimate of the flow speed. 1 F F f zˆ u P 0 c p yielding a flow velocity 1 u zˆ P f g z The pressure gradient is also affected along usurface coastlines with upwards sloping ground level. f ref L With f being the Coriolis parameter and g the earth acceleration. L represents the distance over which the salinity and temperature dependent density anomaly changes. Between 20oN and 40oN, L≈2000km. Geostrophic ocean flow Consider the gulf stream as a sample. The is a pressure or density with depth that in combination with the previously discussed Coriolis force affects the direction and determines the surface flow velocity g z usurface f ref L With f being the Coriolis parameter and g the earth acceleration. L represents the distance over which the salinity and temperature dependent density anomaly changes. Between 20oN and 40oN, L≈2000km. Estimate the gulf stream surface velocity usurface assuming a distance between 20oN and 40oN of L=2000 km for a depth of z=1000 m! g z kg kg kg usurface 26 3 22 3 4 3 f ref L m m m f 2 sin300 7.292 105 s1 f 7.292 105 s1 m kg 9.81 1000 4.0 2 3 m u s m 3101 surface kg 7.292 105 s1 1000 2,000,000m s m3 Overall agreement within the range of local speed variations. The maximum speed is observed at the western boundaries of the Gulf stream with v ≈ 1m/s, while in the interior of the gyre, the speed is much lower, v≈10cm/s. Gulf stream flow velocity Ocean current simulation for different temperature zones NASA/Goddard Space Flight Center Scientific Visualization Studio Single water drop flow The flow pattern is complex and the flow velocity varies greatly.
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