Ρεύµατα παρουσία τριβής ανεµογενής κυκλοφορία

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Theory Serdrup theory for the wind-drien

1 Ρεύµατα παρουσία τριής ανεµογενής κυκλοφορία 6. Wind drien circulation Sarantis Sofianos Dept. of Physics, niersity of thens Ekman Theory Serdrup theory for the wind-drien circulation Stommel and Munk theory of the western boundary intensification

2 Global atmospheric circulation Coriolis Land/sea Polar Easterlies Westerlies Trades

Wind stress on the sea τ = c D ρ a u 10 u10 Weak winds where: ρ α air density (1.

Measurements showed that c D is related to the wind speed (not a constant).

Gale winds c D Eamples of c D used: = 14. 10 3 c c D D = 11. 10 3 = (0.

3 Wind stress on the sea τ = c D ρ a u 10 u10 Weak winds where: ρ α air density (1.3 kg/m 3 ) u 10 wind speed at10 m (m/sec) c D drag coefficient (unitless) Moderate winds Measurements showed that c D is related to the wind speed (not a constant). Neertheless, we usually use a constant alue for simplicity. Gale winds c D Eamples of c D used: = c c D D = = ( ασθενείς άνεµοι u) 10 3 ισχυροί άνεµοι c = ( u for 3m / sec u c D D = ( u for 6m / sec u u 6m / sec 10) )10 26m / sec 3

Eolution of wind-drien circulation theory: Fridtjof Nansen (1898) Vagn Walfrid Ekman (1902) Qualitatie theory, currents transport water at an angle to the wind.

4 Eolution of wind-drien circulation theory: Fridtjof Nansen (1898) Vagn Walfrid Ekman (1902) Qualitatie theory, currents transport water at an angle to the wind. Quantitatie theory for wind-drien transport at the sea surface. arald Serdrup (1947) Theory for wind-drien circulation in the eastern Pacific. enry Stommel (1948) Theory for westward intensification of wind-drien circulation (western boundary currents). Walter Munk (1950) Quantitatie theory for main features of the wind-drien circulation Kirk Bryan (1963) Numerical models of the oceanic circulation. The surface wind effect (Nansen theory): Based on obserations of iceberg moements W + F + C = 0

5 Ekman Theory R o = fl << 1 E k = Viscosity Terms Coriolis Term = L 1 f Θεωρώντας ότι το Η είναι το άθος επίδρασης του ανέµου << του ολικού άθους Και οι εξισώσεις κίνησης γίνονται: 2 fl f V = u << V 2 u 2 z 1 Ekman Layer τ u,υ f υ = 1 P ρ + V fu = 1 P ρ y + V 2 u z 2 2 υ z 2 () u g,υ g Geostrophic Layer (Interior) Η γεωστροφική ισορροπία συνεχίζει να ισχύει: f υ g = 1 P ρ fu g 1 = ρ P y (B) (Α) - (Β) f (υ υ g ) = V 2 u z 2 f (u u g ) = V 2 υ z 2 Ekman Equations

6 f ( υ υ g ) 2 u = V z (1) 2 f ( u u g ) 2 υ = V z (2) 2 Boundary conditions: For simplicity, only zonal wind u at z = 0 ρ z = τ υ ;ρ z = 0 at z u = u g ;υ = υ g Multiplying (2) by i = 1 and adding (1): d 2 V dz 2 if V = 0 (3) where V u u g + i( υ υ g ) B=0, since the solution can not become infinite with z becoming - sing the two surface boundary conditions: = τ δ(1 i) 2ρ Substituting in (3): τδ ( 1 i) u u g + i( υ υ g ) = 2ρ = τ ρ z " e $ f # δ 2 remember u g,υ g not z e ( 1+i )z δ = ( 1 i) τ ρ z 2 f 2 e δ 2 cos z δ sin z % δ ' i τ ρ z " e $ & f # δ 2 The solution is: ( V = e 1+i )z/δ + Be ( 1+i )z/δ " cos z δ + isin z % $ ' = # δ & 2 cos z δ 2 2 sin z % δ ' & cos π 4 cos z δ + sin π 4 sin z δ = cos " z δ + π % $ # 4 & ' sin π 4 cos z δ cos π 4 sin z δ = cos " z δ + π % $ # 4 ' & where δ = 2 f u = u g + τ ρ " e z δ cos z f δ + π % $ ' 4 # & υ = υ g τ ρ " e z δ sin z f δ + π % $ ' 4 # &

7 u = u g + τ ρ " e z δ cos z f δ + π % $ ' υ = υ 4 g τ ρ " e z δ sin z # & f δ + π % $ ' 4 # & ΙI. Surface Ekman elocity: ΙI. Vertical distribution of Ekman solution: at z = 0 u surface = u g + τ ρ! cos π $ # & f 4 " % υ surface = υ g τ ρ! sin π $ # & f 4 " % 45 o τ τ u surface Ekman Spiral Το άθος του στρώµατος Ekman (Ekman layer) ορίζεται ως το άθος όπου η ταχύτητα µειώνεται κατά ένα e (e-folding scale) u u g δ = 2 f D Ekman

ΙIΙ. The ertically integrated Ekman transport: Ολοκληρώνοντας την ταχύτητα λόγω επίδρασης του ανέµου επιφάνεια µέχρι το εσωτερικό του ωκεανού: = V = 0 ( u u g ) dz = 1 ρ f

8 ΙIΙ. The ertically integrated Ekman transport: Ολοκληρώνοντας την ταχύτητα λόγω επίδρασης του ανέµου επιφάνεια µέχρι το εσωτερικό του ωκεανού: = V = 0 ( u u g ) dz = 1 ρ f τ y - 0 ( υ υ g ) dz = 1 ρ f τ - Open sea upwelling/downwelling Consequences: Ekman Transport ( u u g ) από την 45 o 90 o Surface Current Coastal upwelling Ekman Transport

9 IV. Ekman pumping From continuity: 0 dw = δ 0 δ u + υ y + w z = 0 $ u + υ ' & ) dz % y ( $ τ y & % w(o) + w(δ) = 1 ρ f τ y ' ) ( t the bottom of the Ekman layer: w(δ) = 1 ρ f # τ y τ & % ( $ y ' Wind Stress Curl Coastal Ekman pumping w(δ) = L = τ y Lρ f We can not define the deriatie along the coastal zone

10 Serdrup theory for the wind-drien circulation Starting from the same equations of motion (with friction): f υ = 1 P ρ + V fu = 1 P ρ y + V 2 u z 2 2 υ z 2 (1) (2) (2) (1) y u f + f u f +υ y + f υ y = 1 2 P ρ y P ρ y υ ρ z 1 3 u 2 ρ y z 2 f " u + υ % $ '+ υ = 1 # y & ρ z 2 " υ u % $ ' # y & Integrating from the bottom (-Η) to the sea surface: ρ f 0 " u + υ % 0 $ ' dz + ρdz = # y & sing continuity and the fact that w(0)=w(-)=0 0 " τ y τ $ # y Meridional mass transport M y % ' & since at z = 0 ρ u z = τ ; Wind stress curl ρ υ z = τ y

11 M y = y τ τ y = wind stresscurl Depth integrated Vorticity equation for the wind-drien circulation - effect sing the depth integrated continuity equation Westerlies Polar Easterlies y τ 0 M y 1 τ = y M () = y East M y d = 1 2 τ y 2 ( East ) Trades East Δ( f +ζ ) +ζ τ = 0 Zonal elocity anishes at one boundary. Selection according to orticity conseration Eastern boundary: Δf ; ζ must be +; balanced by ζ τ Western boundary: Δf ; ζ must be -; does not balanced by ζ τ We need another orticity term

12 Connection of theories and dynamical features of the wind-drien flow: The wind-stress curl generates Ekman transport and the subsequent Ekman pumping. Ekman transport generates a conergence, which in turn generates geostrophic flow. The sum generates Serdrup transport. The flow center is deflected to the west by the -effect. What is the dynamical balance in the western boundary.

13 Ο Stommel πρότεινε µια πιο πλήρη σχέση για τη διατήρηση του στροιλισµού, που περιλαµάνει και όρο αποµάκρυνσης του σχετικού στροφιλισµού ζ, της µορφής -Rζ Interior + τ y = 0 τ Rζ + y = 0 WBL Rζ = 0 Ο Munk πρότεινε µια πιο πλήρη διατύπωση για τον όρο αποµάκρυνσης του σχετικού στροφιλισµού ζ, της µορφής: Interior 2 ζ 2 τ WBL ζ + = 0 + τ = 0 y 2 ζ = 0 y

14 Stommel Model: = Ο ζ R L L R R Munk Model: = Ο ζ L L We can estimate L or R/.

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