Turbulent Mean Flow Effects: Inclusion of Rotation

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Turbulent Mean Flow Effects: Inclusion of Rotation Ocean Edd L U H Horizontal Equation of Motion Du 1 1 f u h p Dt z where D u u u ' w' kz Dt t z Added horizontal Friction (eddies) u u 1 v u u p u u u ( ) ( ) t z z i i i i i i i u w f u i kz A h i

Road to Geostroph: Diensional Analsis Acceleration Advection Coriolis Pressure Gradient Vertical Friction Horizontal Friction u u 1 v u u p u u u ( ) ( ) t z z i i i i i i i u w f u i kz A h 0 i U U U WU U ku z AU h = = fu? T L L H L H L U Ro fl Rossb Nuber Divide b fu If Ro 1, Ez 1, Eh 1 1 p Geostroph ( f u) i E 1? z fh i k A fl z h Eh Vertical EkanNuber Horizontal Ekan Nuber

Large Scale (Non Turbulent) Oceanic Characteristic Scales Open Ocean L U H L ~ 10 6 3 4 1-1 -4 5 sec 3 H ~ 10 f 10 sec U ~ 10 k= 10-10 A=10-10 U Ro fl E E z h k fh =10-3 -6 =(10 10 ) A fl -3-7 (10 10 ) sec sec Coastal Ocean L U H L ~ 10 4 4 1-1 -4 3 sec 1 H ~ 10 f 10 sec U ~ 10 k= 10-10 A=10-10 U Ro fl E E z h k fh =10-1 -4 =(10 10 ) A fl -1-3 (10 10 ) sec sec

Geostroph Northern Latitude

z z=d z=0 Geostrophic Equations p p p = fv = g z = B fu Barotropic Flow 0 v p A p B A >0 D-z p Hdrostatic Equation p = g( D p = g 1 p g v = f f 0 direction Flow into screen Flow out screen z

Eaple: The sea surface of a certain arginal sea of unifor densit and depth of 150 slopes to the north with a change in sea surface height of 1 c ever 10 k. Assue that it is in the northern heisphere. If the latitude of this sea is such that the Coriolis frequenc is given b f = 10-4 sec -1 (a) what is the agnitude and direction of the current at the surface, z =0? (b) at id depth, z =75? Eplain in words how our answer to (a) and (b) would change if this sea were located at a latitude lower and latitude higher than assued above. Assue the sae sea surface height and depth for this part. 10 k 1 c Z =75 North

z =0 10 k 1 c Z =75 North 10 6 10 4 10 6 9.8 10 g v sec.1 f 4 10 sec

Ekan Dnaics

Ekan Dnaics Du 1 p 1 fv ( )+ Dt z Dv 1 p 1 fu ( ) Dt z fu Eaple A: Stead state, no pressure gradient, wind blows in direction, constant densit 1 0 0 Y { } s q Y udz, where Y is length over which the stress occurs. f D z Note : D is defined b { } 0 0 q Volue Transport M q ass or Ekan transport zd 0

Ekan Transport { } s Ekan Laer q q Volue Transport Y udz Y f { } 0 s Y Y u X = ( *) f Note: X,Y are the characteristic lengths over which the wind stress occurs and transport occur.

Coastal Upwelling X Wind D Ekan Laer q w upwelling q X u w z w u ( q) ( u*) dz XY Xf

U 10 Upwelling Eaple Wind ( ) blows to the south off of Northern California just north of Montere Ba (latitude 37 o N) at stead rate of 10 /sec. If the coastal region being affected has diensions, X = 0 k, Y = 50 k, as shown in the figure below, estiate : (a) the Ekan volue transport, q, (b) upwelling velocit, w, and (c) how long it would take nutrients located at a depth of 100 to rise to the surface. q 100 Note: 1 5 1 f * sin( *sin(37* ) sec 8.810 sec kg 1.510 (10 ) ( u*) sec.510 ( ) kg water water 1000 sec 3 3 3 aircu D 10 4

Upwelling Eaple Continued q 100 4 3 Y 510 4 5 (a) q = ( u*) =.510 ( ) 1.410.14Sv 5 1 f 8.810 sec sec sec 4.510 ( ) q ( u*) sec 4 ( b) w 1.410 =1 XY Xf 4 5 1.0 10 8.810 sec sec da H 100 ( c) t 8 das w 1 da 3 6 1 Sv 10 sec

Ekan Dnaics: Upper Ocean Current Field (Assuptions: stead state, no surface slope, p=0, infinitel deep water, constant densit ) 1 fv z 1 fu z Edd Viscosit Assuption u k z v k z Surface Boundar Condition u s k z=0 = (u*) z Specif Direction of s 0 1 z b Note: z z f b k -1 Note: b is the "effective" Ekan depth where bh 1 1.37 u e U U (Deep water assuption)

Solution for { ( z)} = 0 { ( z)} = z0 z0 s s u U ep[ bz]cos( bz v U ep[ bz]sin( bz ( u*) U fk fk Note: z negative downward 45 o u Solution for { ( z)} = { ( z)} = 0 u U ep[ bz]sin( bz v U ep[ bz]cos( bz ( u*) U fk fk z0 s z0 Note: z negative downward 45 o s u

o Eaple: At a latitude of 45 N wind, U10,blows to the north at 10 /sec. - Observations show that the edd viscosit is. k =10 sec (a) What is the effective Ekan depth? (b) What is the agnitude and direction of the surface stress and surface current at z 0? ( c ) What is the agnitude and direction of the current and stress at z 5, 0. (d) Estiate the turbulent velocit at z 0, 5, 0. (e) How deep would ou have to go to be at a depth where the turbulence vanished? rad o 4 rad Note: f sin( sin(45 ) 10 4 sec sec (a) b 1 k 10 14 4 f 10 U (b) s kg N CD( U ) 1.510 (10 ).5 sec 3 10 3 N.5 o.5,45 to Northeast fk 3 kg 4 10 10 10 sec 3

( c ) z 5 5 5 u U ep[ bz]cos( bz.5ep[ ]cos( ) 14 14.5*.70*.91.16 sec 5 5 v U ep[ bz]sin( bz.5ep[ ]sin( ) 14 14.5*.70*.41.16.07 sec v 5 o tan( 0 45 5 u 14 u k = kub ep( bz)[cos( bz sin( bz ] z s ep( bz)[cos( bz sin( bz ].5 5 5 5 = ep( )[cos( sin( ] 14 14 14 N =.061

v k = kub ep( bz)[cos( bz sin( bz ] z s ep( bz)[cos( bz sin( bz ].5 5 5 5 = ep( )[cos( sin( ] 14 14 14 N =.163

( c ) at z 0 0 0 u U ep[ bz]cos( bz.5ep[ ]cos( ) 14 14.5*.3*.8.05 sec 0 0 v U ep[ bz]sin( bz.5ep[ ]sin( ) 14 14.5*.3*(.6).03 sec v 0 tan( 81 45 36 u 14 u k = kub ep( bz)[cos( bz sin( bz ] z s ep( bz)[cos( bz sin( bz ].5 0 0 0 = ep( )[cos( sin( ] 14 14 14 N =.059 o

v k = kub ep( bz)[cos( bz sin( bz ] z s ep( bz)[cos( bz sin( bz ].5 0 0 0 = ep( )[cos( sin( ] 14 14 14 N =.008 (d) u*=.5 at z=0,.5 u*=.016 1000 sec

at z=5,.061.168.18.18 u*=.013 1000 sec at z=0,.059.008.06.06 u*=.004 1000 sec N N ( e) z 1 b

v k z 45 o fu 1 z u k z 1 fv z Subsurface Stress Surface Stress Surface Current

Approach Using Cople Variables 1 u 1 v fv ; k fu ; k z z z z u fv k & fu k z Let u u iv i=sqrt(-1)=ep( i) u ifu k Solution u Aep( sz) z s v z ep( i) f if if f s = ep( i) (1 i) b k k k 4 k f f b = & s b k k

Case 1 ( z 0) 0; ( z 0) 0 (Wind to North) u k As ep( sz) z f Re( ) z0 0 Re( As) Re{ Aep( i) } 0 4 k A Uep( i) u Uep( i)ep( sz) u Re( u) v Iag( u) 4 4 u ep( bz) =( u*) z0 ep( bz) z ( u*) k U s ( u*) U fk z0

Case ( u*) k ( z 0) 0; ( z 0) 0 (Wind to East) u k As ep( sz) z f Iag( ) z0 0 Iag( As) Iag{ Aep( i) } 0 4 k A Uep( i) u Uep( i)ep( sz) u Re( u) v Iag( u) 4 4 u U s ep( bz) =( u*) z0 ep( bz) z U ( u*) z0 fk

Ekan Dnaics: Botto Laer North u g Sea surface slopes Upward to south 45 o p 0 Assuptions Stead State Upper Ocean Geostrophic u g 1 p 1 fv ( )+ z fu u =k z v =k z 1 p f 0 1 p 1 ( ) 0 Deep Ocean 0 0 Edd Viscosit z East

Case 1: Pressure Gradient to North Equations u fv k z v fu fug k z View Looking Down North u u u [1 ep( bz)cos( bz] g v u ep( bz)sin( bz g Solutions f b= bh>>1 k 45 o Note z positive upward b b u fk g u g East

Equations u fv fvg k z v fu k z Case : Pressure Gradient to West u v ep( bz)sin( bz g v v [1 ep( bz)cos( bz] g Solutions Solutions f b= bh>>1 k View Looking Down 45 o u v g North b b East v fk g

(a) Eaple: At a latitude of under no wind conditions the geostrophic current is observed to be.5 /sec to the east. Observations show that the edd viscosit is - k = 10 sec o 45 N (a) What is the agnitude and direction of the sea surface slope? (b) What is the effective Ekan depth? (c) What is the agnitude and direction of the stress and current on the botto, at z = 0, at = 0,at z = 40? (d) Estiate the turbulent velocit at z = 0, 5, 0. (e) How shallow would ou have to go to be at a depth where the turbulence vanished? p = gz ( p fu = g -4 1 10 sec.5 1 p fu - = sec 0 g 10 sec = - 510-6 (b) k 10 f 10 - -1 b = = =0-4

(c) b u fk g to the northeast 3 4 10.5 10 10.7 u v 0 at z =0 N u u [1 ep( bz)cos( bz] g v u ep( bz)sin( bz g u k kugb ep( bz)[cos( bz sin( bz)] z b ep( bz)[cos( bz sin( bz)] v k kugb ep( bz)[cos( bz sin( bz)] z b ep( bz)[cos( bz sin( bz)]

At z =0 0 u ug[1 ep( bz)cos( bz].5(1 ep( )cos(1).4 0 sec 0 v ug ep( bz)sin( bz.5(ep( )sin(1).15 0 sec b.7 ep( bz)[cos( bz sin( bz)] ep( 1)[cos(1 sin(1)] N =.5 b.7 ep( bz)[cos( bz sin( bz)] ep( 1)[cos(1 sin(1)] N -.05

At z =40 u ug[1 ep( bz)cos( bz].5[1 ep( )cos()].5 sec v ug ep( bz)sin( bz.5(ep( )sin().056 sec b.7 ep( bz)[cos( bz sin( bz)] ep( )[cos( sin()] N =.033 b.7 ep( bz)[cos( bz sin( bz)] ep( )[cos( sin()] N.09 At z =0,0, 40 N.5,.5,09 (d) u*.0,.0158,.0095 sec

Pressure Gradient Force Force Balance 45 o Geostroph fu g 1 p F p Botto Stress B u g kf B North U g U East fv fu Ekan Dnaics 1 z 1 p 1 ( )+ z Note: Forces are obtained b a vertical integral of above equations Coriolis Force F Co

Botto Transport Anoal (V) depth u g p depth u g East North North V ugb V -1 East 45 o z 0 ugb V u g -1 1 1 f ( u ug) fu p Let v=( u u ) g g 0 z 0 1 1 dz z 0 dz f ( u ug) z o z 0 k u fv V ug f o g b -1

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