Ocean currents II
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: FD C D A a v 2 C D drag coefficient dimensionless factor for wind water interaction C D 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 9 Bft 7 Bft 10 Bft 8 Bft 11 Bft 12 Conversion from scale to wind velocity: v 0.836 B 3/ 2 m 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: FD C D A a v 2 a 1200 kg m3 v 45 km m 12.5 C D 0.001 h s 2 kg m FD 0.001 1200 3 A m 2 12.5 187.5 A N m s For a strong gale (B=12), v=35 m/s, the drag stress on the water will be: 2 kg m FD / A 0.0025 1200 3 35 3675 N m 2 m s kg m a 1200 3 v 35 C D 0.0025 m s or FD A 187.5 N m 2 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 D FD A C D a v 2 c Fc A 2m A u sin f u A m 2 sin f s 1 Coriolis parameter z A F / A z z z 1 f u D m/ A z z z force N 1 τ in vector terms f zˆ u D mass kg z zˆ u defines vector direction of transport f zˆ u τ D z 0 D f zˆ u dz f zˆ M Ek 0 M Ek u dz kg m s Ekman mass transport vector Ekman transport kg M Ek u dz m s assumption is a moreless linear increase of water density with depth 0 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: D z f 1 M Ek sin 900 f M Ek f 2 sin D z f u z depth vector M Ek 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? D z f M Ek f 2 sin 350 7.292 10 5 s 1 f 2 7.292 10 5 s 1 sin 350 4.183 10 5 s 1 kg 100m kg 10 m 20 , 000 m 3 60 s ms kg kg N 5 1 4.183 10 s 20,000 0.837 1 2 2 ms ms m M Ek u z 1200 D z f M Ek Weak force, done by winds of B≈2 with : kg 1 2 D m km m s v 0.913 3.3 kg CD s h 1200 3 0.001 m 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. D z f M Ek M Ek D z f CD a v 2 2 sin Calculate the mass transport MEk for a typical wind stress of D = 0.25 N/m2 at the southern latitude of 40oS. f 2 sin 400 7.292 10 5 s 1 f 2 7.292 10 5 s 1 sin 400 9.37 10 5 s 1 N 2 D 3 kg m 2 . 67 10 f 9.37 10 5 s 1 ms 0.25 M Ek 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? v2 f M Ek CD a v f M Ek CD a kg m s 0.29 m 1 km kg s h 0.002 1200 3 m About B=1-2 on the Beauford Scale 9.37 10 5 s 1 2.13 103 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 is more iscomplex because because of additional forces dueforces to friction and Reality more complex of additional due to the drag temperature which addatmospheric to the eddywind formation phenomenon! forceseffects, provided by the circulation 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 ref S , T , P z neglecting S , T , P P z Psurface g ref z Approximately a linear increase of pressure with depth – in contrast the surface z atmosphere displays an exponential Psurface 1 atm 105 Pa decrease of pressure with altitude . m kg z 100m P 100m 105 Pa 9.81 2 1000 3 100m 106 Pa s m z 1km P1km 107 Pa Flow at larger depth is directed by the pressure gradient and the z 4km P4km 4 107 Pa 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. Fc Fp f zˆ u 1 P 0 yielding a flow velocity u 1 zˆ P f usurface g f ref z L The pressure gradient is also affected along coastlines with upwards sloping ground level. 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 usurface g f ref z 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! usurface g f ref z L 26 kg kg kg 22 4 m3 m3 m3 f 2 sin 300 7.292 105 s 1 kg 3 1 m m usurface 3 10 kg s 7.292 105 s 1 1000 3 2,000,000m m 9.81 m s2 f 7.292 105 s 1 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. 1000 4.0 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. Both observables are defined by coastal boundaries, drag forces at the surface, density gradients at deep depths, and the Coriolis force. Geostrophic flow induced variations in ocean surface height The curvature of the flow and the horizontal velocity gradient across the flow causes a pressure gradient perpendicular to the flow direction, which translate into variations of the ocean surface height . m 10 s 1 10 m 0.1 f Lu s ; 1m 106 m g L 9.81 2 s m 4 1 4 1 Ocean Surface Height 6 for : u 0.1 s f 10 s ; L 1000km Example Gulf of Mexico Flow pattern http://www.aoml.noaa.gov/phod/dhos/altimetry.php Maps of sea level obtained from satellite altimetry measurements are used to derive surface ocean currents. Higher values of sea level (oranges and reds) are associated with gyres and warm eddies, while lower values are associated to colder features. Drifter trajectories illustrate circulation features. Sea height anomaly maps show the difference of sea level from average conditions, while sea height maps show absolute values of the sea level. Water height Altitude anomalies Changing average sea altitude levels Sea level trend between 1992 and 2009 with respect to a reference level , based on satellite altitude measurements. Yellow and red regions show rising sea level, whileon green and bluedata regions sea level. Based observational suchshow as tidefalling level measurements and satellite based altimetry
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