Friction and Ocean Turbulence Part I

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1 Frcton and Ocean Trblence Part I L. Goodman General Physcal Oceanography MAR 555 School for Marne Scences and Technology Umass-Dartmoth

2 Frcton and Ocean Trblence Part I 3 Types of Flow Potental Flow No frcton; no vortcty Lamnar Flow Frcton bt no overtrnng Trblent Flow Frcton, hgh vortcty, overtrnng The Road to Trblence Trblent Flow Lamnar flow Who s ths? Homework problem: Gve one example of ocean flows whch are: (1) potental flow; (2) lamnar flow;(3) trblent flow

3 Nmercal Model of 3 D Trblence Drect Nmercal Smlaton (DNS) model of trblence

4 2 D Ocean Trblence Sea srface chlorophyll dstrbton derved from sea srface color n the western Sargasso Sea on May 27, 2007

5 Hgh Resolton Nmercal model of a Frontal Instablty Prodcng 2 D trblence

6 How do we measre ocean 3D trblence? Layered Organzaton of the Coastal Ocean (LOCO) Expermental Ste Monterey Bay CA SMAST LOCO Objectve: Stdy relatonshp of bologcal thn layers to trblence

7 SMAST T-REMUS Atonomos Underwater Vehcle Used for ocean trblence measrements

8 Green chlorophyll; Ble trblent dsspaton rate

9 Why s trblence mportant n oceanography? I. Ar Sea Interacton

10 II. Interor Mxng ( temperatre (heat) salt, momentm, PV, ntrents, plankton, larvae, bbbles, other consttents) Trblence prodced by an nderwater Internal wave beam Cortesy of M. Gregg

11 III. Bondary Frcton (Sde and bottom)

12 IV. Eddy (2 D Trblent) Mxng False Color Image of temperatre n Sea of Japan

13 Trblent Frctonal Effects: The Vertcal Reynolds Stress Ar Water d dt d dt dv dt No Trblence!!! p + ( f " ) = # # g + Fr! x or n component form 1 " p = # ( ) + f v+ Fr! " x 1 " p = # ( ) # f + Fr! " y dw 1 " p = # ( ) # g + Fr dt! " z x z y

14 Mean and Flctatng Qanttes ' = + ' ' mean + flctatng Three Types of Averages Ensemble Tme Space Ergodc Hypothess: Replace ensemble average by ether a space or tme average tme Notaton q! <q>

15 Mean Flow How does the trblence affect the mean flow? 3D trblence ', v ', w' Concept of Reynolds Stress! " ' w' = # < > < ' w' > < 0 w

16 Momentm Eqaton wth Trblence Moleclar Frcton Fr x, y, z d 1 " p # fv = # ( )+ Fr dt! " x dv 1 " p + f = # ( ) + Fr dt! " y dw 1 " p = # ( ) # g + Fr dt! " z where d " " " = + v + w dt " x " y " z Bt #! $ (dervatve of velocty shear)! = moleclar vscosty = 10 x z y "6 2 m sec d 1 " p # fv = # ( ) dt! " x dv dt + f = # 1 " p ( )! " y dw 1 " p = # ( ) # g dt! " z where d " " " = + v + w dt " x " y " z Bt = + ' x y w = + ' = v+v' = w+w'

17 Example Unform ndrectonal wnd blowng over ocean srface Dmensonal Analyss Bondary Layer Flow Gradent n x drecton smaller than n z drecton Mean velocty ndrectonal, no gradent n y drecton d 1 " p # fv = # ( ) dt! " x dv dt + f = # 1 " p ( )! " y dw 1 " p = # ( ) # g dt! " z where d " " " " = + + v + w dt " t " x " y " z " " " " 1 " p + + v + w # fv = # ( ) " t " x " y " z! " x " " " sng + v # w = 0 " x " y " z " " "( v) "( w) 1 " p + ( ) + # ) # fv = # ( ) " t " x 2 " y " z! " x

18 " ( ) 1 + " + " w # fv = # ( " p ) " t " x " z! " x Bt = + ' x y w = + ' = v+v' = w+w'!( + ')!( + ')( w + w') ( + ') +! x! z Now we average the momentm eqaton " ( ' w ' ) 1 + " + " < > # fv = # " p " t " x " z! " x # # 1 # p 1 #! + $ fv = $ + # t # x " # x " # z where! = $ " < ' w' >= " x"component of Reynolds Stress x x

19 General Case of Vertcal Trblent Frcton # #!! 1 # p 1 #! + + ( f $ ) = % + % g # t # x " # x " # z where! = % " < ' w' > for ' = ', ' = v ' j = 0 = 3 (z) x y Note that we sometmes se 1,2,3 n place x, y, z as sbscrpts Conventon: When we deal wth typcal mean eqatons we drop the mean Notaton! # #!! 1 # p 1 #! + + ( f $ ) = % + % g # t # x " # x " # z j j

20 Component form of Eqatons of Moton wth Trblent Vertcal Frcton # # # # 1 # p 1 #! + + v + w $ fv = $ ( )+ # t # x # y # z " # x " # z # v # v # v # v 1 # p 1 #! + + v + w + f = $ ( ) + dt # x # y # z " # y " # z # w # w # w # w 1 # p + + v + w = $ ( ) $ g # t # x # y # z " # z x y Note: n many cases the mean vertcal velocty s small and we can assme w = 0 whch leads to the hydrostatc approxmaton and # # # 1 # p 1 #! (1) + + v $ fv = $ ( )+ # t # x # y " # x " # z # v # v # v 1 # p 1 #! (2) + + v + f = $ ( ) + dt # x # y " # y " # z (3) # p = $ " g # z x y