May 29, 2012 	  海面波によるVortex	  Force:鉛直幅で重み付け平均された運動方程式!

The	  Vortex	  Force	  in	  the	  thickness-­‐weighted	  mean	  momentum	  equations	  for	  surface	  gravity	  waves
Hidenori Aiki (JAMSTEC-RIGC)!
Richard J. Greatbatch (GEOMAR, Kiel, Germany)!

(wikipedia)

to be submitted	

Langmuir
Circulation

Craik and Leibovich momentum equations

Vortex ForceEquations in density-coordinates 	Andrews, 1983, JAS
Greatbatch, 1998, JPO

unweighted mean velocity (geostrophic flow, Ekman flow)
Bolus velocity (eddy-induced overturning circulation)

Rhines, 1982

ZONAL MEAN OF
GLOBAL OCEAN

Aiki and Richards, 2008, JPO

DRAKE PASSAGE

Aiki and Richards, 2008, JPO

May 29, 2012 	  海面波によるVortex	  Force:鉛直幅で重み付け平均された運動方程式!

The	  Vortex	  Force	  in	  the	  thickness-­‐weighted	  mean	  momentum	  equations	  for	  surface	  gravity	  waves
Hidenori Aiki (JAMSTEC-RIGC)!
Richard J. Greatbatch (GEOMAR, Kiel, Germany)!

(wikipedia)

to be submitted	

Langmuir
Circulation

Craik and Leibovich momentum equations

Vortex ForceStudies for
the Effect of Surface Waves on Circulation

Ver$cal	

Horizontal		  	  	  	  	  	  	  	  	  	  	  	  	  Radia$on	  Stress	  	  (Reynolds	  Stress	  +	  Form	  Stress)	

Vortex	  Force	

Integrated	

Eulerian	Longuet-­‐Higgins	  &	  Stewart	  (1964)	

GarreF	  (1976)	  	Eulerian	Eulerian	-­‐	Craik	  &	  Leibovich	  (1976)	  McWilliams	  et	  al.	  (2004)	

Lagrangian	 Lagrangian	-­‐	Andrews	  &	  McIntyre	  (1978)	  	  	  	  	  	  	  	  	  Leibovich	  (1980)	  	  	  	  	  	  Ardhuin	  et	  al.	  (2008)	

Lagrangian	 Eulerian	

Mellor	  (2003),	  Aiki	  &	  Greatbatch	  (2012)	

this	  study	3D Eulerian framework(Craik and Leibovich, 1976)
Ver$cal	

Horizontal		  	  	  	  	  	  	  	  	  	  	  Radia$on	  Stress	  	  (Reynolds	  Stress	  +	  Form	  Stress)	

Vortex	  Force	

Integrated	

Eulerian	Longuet-­‐Higgins	  &	  Stewart	  (1964)	

GarreF	  (1976)	  	Eulerian	Eulerian	-­‐	Craik	  &	  Leibovich	  (1976)	  McWilliams	  et	  al.	  (2004)	

Lagrangian	 Lagrangian	-­‐	Andrews	  &	  McIntyre	  (1978)	  	  	  	  	  	  	  	  	  Leibovich	  (1980)	  	  	  	  	  	  Ardhuin	  et	  al.	  (2008)	

Lagrangian	 Eulerian	

Mellor	  (2003),	  Aiki	  &	  Greatbatch	  (2012)	

this	  study	3D Eulerian framework (Craik & Leibovich, 1976)	
∂t U1′ = −∇p1′U1 = 0∂t U′2 + U1′ i∇U1′ = −∇p1′() ()∂t U′3 + U1′ i∇ U 2 + U′2 + U 2 + U′2 i∇U1′ = −∇p3′

Time scale of Langmuir Circulations is
two orders slower than wave period	()(∂T U 2 + ∇i U 2 U 2 + ∇i U′U′(

∇i U′U′∂T  α ∂ t2) = −∇ p4) = ∇i( U′U′ ) + ∇i( U′ U′ ) + ∇i( U′ U′ )41322314∂T  α ∂ tTime scale of Langmuir Circulations is
two orders slower than wave period	()(2∂T U 2 + ∇i U 2 U 2 + ∇i U′U′(∇i U′U′) = −∇ p44) = ∇i( U′U′ ) + ∇i( U′ U′ ) + ∇i( U′ U′ )14(3223)

1∂T U 2 + ∇ × U 2 × U 2 + ⎡⎣( ∇ × U′ ) × U′ ⎤⎦(4= −∇ p + U / 2⎡( ∇ × U ′ ) × U ′ ⎤⎣⎦4()()()

2())

4= ∇ × U1′ × U′3 + ∇ × U′2 × U′2 + ∇ × U′3 × U1′ = −∂t ∇ × U′3 × X1′

3D Eulerian framework (Craik & Leibovich, 1976)	
∂t U1′ = −∇p1′

U1 = 0∂t U′2 + U1′ i∇U1′ = −∇p1′() ()∂t U′3 + U1′ i∇ U 2 + U′2 + U 2 + U′2 i∇U1′ = −∇p3′∇ × U1′ = 0∇ × U′2 = 0() ()∂t ( ∇ × U′3 ) = −∇ × ⎡⎣ U1′ i∇ U 2 + U′2 + U 2 + U′2 i∇U1′ ⎤⎦∂T  α ∂ tTime scale of Langmuir Circulations is
two orders slower than wave period	

()(2∂T U 2 + ∇i U 2 U 2 + ∇i U′U′(∇i U′U′) = −∇ p44) = ∇i( U′U′ ) + ∇i( U′ U′ ) + ∇i( U′ U′ )14(3223)

1∂T U 2 + ∇ × U 2 × U 2 + ⎡⎣( ∇ × U′ ) × U′ ⎤⎦(4= −∇ p + U / 2⎡( ∇ × U ′ ) × U ′ ⎤⎣
⎦4()

()

()

2())

4= ∇ × U1′ × U′3 + ∇ × U′2 × U′2 + ∇ × U′3 × U1′ = −∂t ∇ × U′3 × X1′

t

X1′ = ∫ U1′dt
UStokes
2= X1′ ⋅ ∇U1′

Craik and Leibovich momentum equations	( ∂ + U i∇) U + (∇ × U ) × UT222

Vortex Force	

Stokes2= −∇π 4

3D Lagrangian framework(Leibovich, 1980)
Ver$cal	

Horizontal		  	  	  	  	  	  	  	  	  	  	  	  Radia$on	  Stress	  	  (Reynolds	  Stress	  +	  Form	  Stress)	

Vortex	  Force	

Integrated	

Eulerian	Longuet-­‐Higgins	  &	  Stewart	  (1964)	

GarreF	  (1976)	  	Eulerian	Eulerian	-­‐	Craik	  &	  Leibovich	  (1976)	  McWilliams	  et	  al.	  (2004)	

Lagrangian	 Lagrangian	-­‐	Andrews	  &	  McIntyre	  (1978)	  	  	  	  	  	  	  	  	  Leibovich	  (1980)	  	  	  	  	  	  Ardhuin	  et	  al.	  (2008)	

Lagrangian	 Eulerian	Mellor	  (2003),	  Aiki	  &	  Greatbatch	  (2012)	

this	  study	( x, y, z )Eulerian-Cartesian coordinates	 ( u, v, w ) =LLL
( a,b, c ) = ( x , y , z ) Lagrangian coordinates	 d ( x, y, z ) / dt
⎛ du / dt⎜⎜ dv / dt⎜ dw / dt⎝⎛ ∂x ⎞⎞
⎛ xa
⎜
⎟
⎟
⎜
⎟ = − ⎜ ∂ y ⎟ p = − ⎜ xb
⎜ ∂ ⎟⎟⎜ x⎠⎝ c⎝ z ⎠yaybycza ⎞⎟zb ⎟zc ⎟⎠−1⎛ ∂a ⎞⎜⎟⎜ ∂b ⎟ p⎜ ∂ ⎟⎝ c ⎠Original Expression	⎛ xa
⎜⎜ xb
⎜ x
⎝ cyaybycza ⎞ ⎛ du / dt⎟⎜zb ⎟ ⎜ dv / dtzc ⎟⎠ ⎜⎝ dw / dt⎞⎛ ∂a ⎞⎟⎜⎟
⎟ = − ⎜ ∂b ⎟ p⎟
⎜ ∂ ⎟⎠⎝ c ⎠Transformed Expression	

( x, y, z )Eulerian-Cartesian coordinates	
LL
L( a,b, c ) = ( x , y , z ) Lagrangian coordinates	
Chain Rule	⎛ xa
⎜x
⎜ b
⎜ x
⎝ cya
ybyc

⎛
⎞⎞⎛
⎞
∂za
∂
xa⎟ ⎜⎟⎜⎟zb ⎟ ⎜ ∂ y ⎟ = ⎜ ∂b ⎟⎜⎟⎟zc ⎠ ⎝ ∂ z ⎠ ⎜⎝ ∂ c ⎟⎠( x, y, z )Eulerian-Cartesian coordinates	 ( x ′, y′, z′ ) =( x, y, z ) − ( a,b, c )LLL( a,b, c ) = ( x , y , z ) Lagrangian coordinates	⎛ L ⎞⎛ xaud⎜ L ⎟⎜⎜ v ⎟ = − ⎜ xb
dt ⎜⎜ xL ⎟⎜⎝ w ⎟⎠⎝ c⎡⎛ L ⎞ ⎛− xa′⎢⎜ u ⎟d ⎢⎜L⎜ v ⎟ − ⎜ − xb′dt ⎢⎜L ⎟⎢⎜⎝ w ⎟⎠ ⎜⎝ − xc′⎣yayb
yc− ya′
− yb′
− yc′

za ⎞⎟zb ⎟zc ⎟⎠−1⎛ ∂a ⎞⎜⎟⎜ ∂b ⎟ p⎜ ∂ ⎟⎝ c ⎠L

Original Expression	

Transformed Expression	⎤⎞⎛⎞⎛ ∂a ⎞− za′u′⎥⎟⎜⎟ ⎥⎜⎟ L− zb′ ⎟ ⎜ v′ ⎟ + ..... = − ⎜ ∂b ⎟ p⎥⎜ ∂ ⎟− zc′ ⎟⎠ ⎜⎝ w′ ⎟⎠ ⎥⎝ c ⎠⎦LPseudomomentum

Eulerian mean velocity

VortexForcet

X1′ = ∫ U1′dt
UStokes
2= X1′ ⋅ ∇U1′

Craik and Leibovich momentum equations	( ∂ + U i∇) U + (∇ × U ) × U
T222

Vortex Force	

Stokes2= −∇π 4

Vertically Lagrangian
Horizontally Eulerianframework(this study)
Ver$cal	

Horizontal		  	  	  	  	  	  	  	  	  	  	  	  Radia$on	  Stress	  	  (Reynolds	  Stress	  +	  Form	  Stress)	

Vortex	  Force	

Integrated	

Eulerian	Longuet-­‐Higgins	  &	  Stewart	  (1964)	

GarreF	  (1976)	  	Eulerian	Eulerian	-­‐	Craik	  &	  Leibovich	  (1976)	  McWilliams	  et	  al.	  (2004)	

Lagrangian	 Lagrangian	-­‐	Andrews	  &	  McIntyre	  (1978)	  	  	  	  	  	  	  	  	  Leibovich	  (1980)	  	  	  	  	  	  Ardhuin	  et	  al.	  (2008)	

Lagrangian	 Eulerian	

Mellor	  (2003),	  Aiki	  &	  Greatbatch	  (2012)	

this	  study	Generalized Lagrangian	Andrews & McIntyre (1978, JFM)

Vertically Lagrangian Horizontally Eulerian
Mellor (2003, JPO)
Aiki & Greatbatch (2012, JPO)

Generalized Lagrangian	Vertically Lagrangian Horizontally Eulerian

Generalized Lagrangian	Vertically Lagrangian Horizontally Eulerian

Generalized Lagrangian	Vertically Lagrangian Horizontally Eulerian

Generalized Lagrangian	Andrews & McIntyre (1978, JFM)

Vertically Lagrangian Horizontally Eulerian
Mellor (2003, JPO)
Aiki & Greatbatch (2012, JPO)

Generalized Lagrangian	Vertically Lagrangian Horizontally Eulerian

Generalized Lagrangian	Vertically Lagrangian Horizontally Eulerian

Generalized Lagrangian	Vertically Lagrangian Horizontally EulerianEquations in VLHE coordinates	Aiki & Greatbatch (2012, JPO)	

No background vertical flow	

Blue line:
the vertical position of control cell	With background vertical flow （e.g. horizontal divergence of Langmuir Circulations）	

Blue line:
the vertical position of control cell
Red Arrow:the vertical position of water particle

Equations in VLHE coordinates	

Thickness Weighted Mean operator

Thickness Weighted Mean （TWM） Equations	

Total Transport Velocity

Aiki & Greatbatch (2012, JPO)	

Thickness Weighted Mean （TWM） Equations	Reynolds
StressFormStress	(RS ≡ ∇i z V′′A′′Acz)

FSV ≡ − zz′′′ ∇ ( p′′′ + η′′′ ) + pz′′′ ∇z′′′⎛⎞⎡⎤p′′′= − ⎢ z′′′ ∇ ( p′′′ + η′′′ ) ⎥ + ∇z z ′′′⎝⎠⎣⎦z( x, y, z )Eulerian-Cartesian coordinates	 ( x ′, y′, z′ ) =( x, y, z ) − ( a,b, c )LLL( a,b, c ) = ( x , y , z ) Lagrangian coordinates	⎛ L ⎞⎛ xaud⎜ L ⎟⎜⎜ v ⎟ = − ⎜ xbdt ⎜⎜ xL ⎟⎜⎝ w ⎟⎠⎝ c⎡⎛ L ⎞ ⎛− xa′⎢⎜ u ⎟d ⎢⎜L⎜ v ⎟ − ⎜ − xb′dt ⎢⎜L ⎟⎢⎜⎝ w ⎟⎠ ⎜⎝ − xc′⎣yaybyc− ya′− yb′− yc′za ⎞⎟zb ⎟zc ⎟⎠−1⎛ ∂a ⎞⎜⎟⎜ ∂b ⎟ p
⎜ ∂ ⎟⎝ c ⎠LOriginal Expression	

Transformed Expression	⎤⎞⎛⎞⎛ ∂a ⎞− za′u′
⎥
⎟⎜
⎟ ⎥
⎜
⎟ L− zb′ ⎟ ⎜ v′ ⎟ + ..... = − ⎜ ∂b ⎟ p⎥⎜ ∂ ⎟− zc′ ⎟⎠ ⎜⎝ w′ ⎟⎠ ⎥⎝ c ⎠⎦
L

Pseudomomentum

Eulerian mean velocity

VortexForceForm Stress Term	

Prognostic variable in the original equation system
Total Transport Velocity

Prognostic variable in the transformed equation system	

Prognostic variable in the original equation system
Total Transport Velocity= Lagrangian mean velocity	Prognostic variable in the transformed equation system	

Eulerian
Lagrangian=	 meanmean
velocity	
velocity	-	Stokes-drift
velocity	t

X1′ = ∫ U1′dt
U

Stokes2= X1′ ⋅ ∇U1′

Craik and Leibovich momentum equations	( ∂ + U i∇) U + (∇ × U ) × U
T222

Vortex Force	

Stokes2= −∇π 4

Summary
Ver$cal	

Horizontal	

Radia$on	  Stress	

Vortex	  Force	

Integrated	

Eulerian	Longuet-­‐Higgins	  &	  Stewart	  (1964)	

GarreF	  (1976)	  	-­‐	Craik	  &	  Leibovich	  (1976)	  Flux-­‐divergence	  form	  (Reynolds	  stress)	  Vector-­‐invariant	  form	  	  	  	  	  	  	  	  (3rd	  order	  vorVcity	  Eq.)	

-­‐	Leibovich	  (1980)	  Original	  expression	  (Form	  stress):	  	  	  	  PrognosVc	  variable:	  LM	  velocity	  Transformed	  expression	  	  	  	  	  PrognosVc	  variable:	  EM	  velocity	  	  	  	  	  =	  LM	  velocity	  -­‐	  psudomomentum	Eulerian	Eulerian	

Lagrangian	 Lagrangian	

Mellor	  (2003)	  Aiki	  &	  Greatbatch	  (2012)	  Lagrangian	 Eulerian	Horizontal	  Reynolds	  stress	  term	  is	  transformed	  as	  in	  CL76	  Form	  stress	  term	  is	  transformed	  as	  in	  	  	  	  	  Horizontal	  Reynolds	  stress	   Leibovich	  (1980)	  Form	  stress	  PrognosVc	  variable:	  EM	  velocity:	  PrognosVc	  variable:	  =Total	  Transport	  velocity	  	  	  	  	  Total	  Transport	  velocity		  	  -­‐	  pseudomomentum	  Studies for
the Effect of Surface Waves on Circulation

Surface	  Boundary	  Viscosity	  Term	  Spectrum	  Shape	◎	○	○	○	

Lagrangian	 Lagrangian	○	○	○	

◎	

Lagrangian	 Eulerian	◎	◎	◎	

○	

Ver$cal	

Horizontal	 Incompressibility	Eulerian	Eulerian	Generalized	  Lagrangian	  Mean	  theory	  	  (Andrews	  &	  McIntyre,	  1978)	

• PrognosVc	  variable	  in	  wave-­‐averaged	  momentum	  Eqs.	  	  	  	  Original	  expression:	  LM	  velocity	  	  	  	  Transformed	  expression:	  LM	  velocity	  -­‐	  pseudomomentum	  	  	  (Lamb,	  1932,	  page	  13;	  Lagrange,	  1788,	  page	  441-­‐447)	  • Hybrid	  Lagrangian-­‐Eulerian	  coordinates	  	  	  (Iwasaki,	  2001;	  Jacobson	  and	  Aiki,	  2006)	  • Lagrangian	  pseudomomentum	  Jacobson & Aiki, 2006, JPO	

Stokes-­‐dria	  velocity	  and	  its	  relaVves	
Stokes	  dria	  velocity	  	  Eulerian	  pseudomomentum	  	  (Longuet-­‐Higgins,	  1953)	  (Bretherton,	  1971;	  Buhler,	  2009)	  u x′ x ′ + uz′ z′
Quasi-­‐Stokes	  velocity	  	  (Andrews	  &	  McIntyre,	  1976;	  	  Iwasaki,	  2001;	  	  McDougall	  &	  McIntosh,	  2001)	  (u′z′ )+	−( uz′ − wx′ ) z′ / 2
Lagrangian	  pseudomomentum	  (Andrews	  &	  McIntyre,	  1978)	  − x x′ u ′ − zx′ w′
TWM	  pseudomomentum	  z=	(Aiki	  &	  Greatbatch,	  under	  revision)	

zz′u ′ − zx′ w′

References	
Aiki, H., and R. J. Greatbatch, to be submitted. http://www.jamstec.go.jp/frcgc/research/d1/aiki/!
The vortex force in the thickness-weighted-mean momentum equations for surface gravity waves!
Aiki, H., and R. J. Greatbatch, under revision, J. Phys. Oceanogr. http://dx.doi.org/10.1175/JPO-D-12-059.1!
The vertical structure of surface wave radiation stress on circulation over a sloping bottom from thickness weighted
mean theory!
Aiki, H., and R. J. Greatbatch, 2012, J. Phys. Oceanogr., 42, 725-747. http://dx.doi.org/10.1175/JPO-D-11-095.1
Thickness weighted mean theory for the effect of surface gravity waves on mean flows in the upper ocean"Aiki, H., and K. J. Richards, 2008, J. Phys. Oceanogr., 38, 1845-1869. http://dx.doi.org/10.1175/2008JPO3820.1!
Energetics of the global ocean: the role of layer-thickness form drag"!
Jacobson, T., and H. Aiki, 2006, J. Phys. Oceanogr., 36, 558-564. http://dx.doi.org/10.1175/JPO2872.1!
An exact energy for TRM theory!