Basic dynamics

The equations of motion and continuity Scaling

Hydrostatic relation

Boussinesq approximation

Geostrophic balance in ocean’s interior

Newton’s second law in a rotating frame.(Navier-Stokes equation)

The Equation of Motion

FgVpdtVd

rrrrr++×Ω−∇−= 21

ρ

dtVdr

: Acceleration relative to axis fixed to the earth.

p∇−ρ1 : Pressure gradient force.

Vrr

×Ω−2 : Coriolis force, where sraddayrad 510292.724.365

112 −⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ ×=+=Ω π

⎟⎟⎠

⎞⎜⎜⎝

⎛ ×Ω×Ω−= Rggrrrrr

0: Effective (apparent) gravity.

VFrr

2∇≈ν : Friction. molecular kinematic viscosity.sm2610−≅ν

g0=9.80m/s2

1sidereal day =86164s

1solar day = 86400s

Gravity: Equal Potential Surfaces• g changes about 5%

9.78m/s2 at the equator (centrifugal acceleration 0.034m/s2, radius 22 km longer)

9.83m/s2 at the poles) • equal potential surface

normal to the gravitational vectorconstant potential energythe largest departure of the mean sea surface from the “level” surface is

about 2m (slope 10-5) • The mean ocean surface is not flat and smooth

earth is not homogeneous

In Cartesian Coordinates:

xFwvxp

dtdu +Ω−Ω+

∂∂−= ϕϕρ cos2sin21

yFuyp

dtdv +Ω−

∂∂−= ϕρ sin21

zFguzp

dtdw +−Ω+

∂∂−= ϕρ cos21

zuw

yuv

xuu

tu

dtdu

∂∂+

∂∂+

∂∂+

∂∂=

where

Accounting for the turbulence and averaging within T: ∫+

−

=2

2

)(1

),(

Tt

Tt

dttuT

Ttu

( ) ( )( ) ( ) vuvuvuvuvuvuvvuuuv ′′+=′′+′+′+=′+′+=

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

∂∂+

∂∂+

∂∂++

∂∂−=

∂∂+

∂∂+

∂∂+

∂∂

2

2

2

2

2

21

z

u

y

u

x

ufvxp

zuw

yuv

xuu

tu νρ

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

∂′′∂+

∂′′∂+

∂′′∂−=

∂′∂′−

∂′∂′−

∂′∂′−=

x

wu

x

vu

x

uu

x

uvx

uvx

uuxF

0=∂′∂+

∂′∂+

∂′∂

zw

yv

xu

Given the zonal momentum equation

If we assume the turbulent perturbation of density is small

ρρ ≅i.e.,

The mean zonal momentum equation is

Where Fx is the turbulent (eddy) dissipation

If the turbulent flow is incompressible, i.e.,

Eddy Dissipation

xuAuu xxx ∂∂=′′−=τ

yuAvu yxy ∂∂=′′−=τ

zuAwu zxz ∂∂=′′−=τ

Ax=Ay~102-105 m2/sAz ~10-4-10-2 m2/s

>> sm2610−≅ν

Reynolds stress tensor and eddy viscosity:

Where the turbulent viscosity coefficients are anisotropic.

,

2

2

2

2

2

2

z

uAy

uAx

uAzyx

Fzyx

xzxyxxx

∂∂+

∂∂+

∂∂=

∂∂+

∂∂+

∂∂= τττ

Then

yuAvu yxy ∂∂=′′−=τ

xvAuv xyx ∂

∂=′′−=τyxxy ττ ≠

(incompressible)

Reynolds stress has no symmetry:

A more general definition:

xvA

yuA xyxy ∂

∂+∂∂=τ yxxy ττ =

0=∂∂+

∂∂+

∂∂

zw

yv

xu

2

2

2

2

2

2

z

uAy

uAx

uAzyx

Fzyx

xzxyxxx

∂∂+

∂∂+

∂∂=

∂∂+

∂∂+

∂∂= τττ

if

Continuity Equation

Mass conservation law 0=⋅∇+ Vdtd r

ρρ

In Cartesian coordinates, we have

0=∂∂+∂

∂+∂∂+∂

∂+∂∂+∂

∂+∂∂

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

zw

yv

xu

zwyvxut ρρρρρ

( ) ( ) ( ) 0=∂

∂+∂

∂+∂

∂+∂∂

zw

yv

xu

tρρρρor

For incompressible fluid, 0=dtdρ

0=∂∂+∂

∂+∂∂=⋅∇ z

wyv

xuV

r

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

∂∂

∂∂≡∇

yxH ,),( vuVH ≡If we define and

0=∂∂+⋅∇zwVHH

r, the equation becomes

Scaling of the equation of motion• Consider mid-latitude (ϕ≈45o) open ocean

away from strong current and below sea surface. The basic scales and constants:

L=1000 km = 106 mH=103 mU= 0.1 m/sT=106 s (~ 10 days)2Ωsin45o=2Ωcos45o≈2x7.3x10-5x0.71=10-4s-1

g≈10 m/s2

ρ≈103 kg/m3

Ax=Ay=105 m2/sAz=10-1 m2/s

• Derived scale from the continuity equation

W=UH/L=10-4 m/s

0=∂∂+∂

∂+∂∂=⋅∇ z

wyv

xuV

r0~

HW

LU +

Scaling the vertical component of the equation of motion

111011101110105103101110111011101010 −+−+−+−−+Δ−=−+−+−+−H

Pz

2

2

2

2

2

2cos21

zwA

ywA

xwAgu

zp

zww

ywv

xwu

tw

zyx ∂∂+

∂∂+

∂∂+−Ω+

∂∂−=

∂∂+

∂∂+

∂∂+

∂∂ ϕρ

21

25

2543

2101010101010

HW

LW

LWU

HP

HW

LUW

LUW

TW z −−− +++−+Δ=+++

1010 3 =Δ−HPz PaHPz

74 1010 ==Δ

gzp ρ−=∂∂

Hydrostatic Equation

accuracy 1 part in 106

Boussinesq ApproximationConsider a hydrostatic and isentropic fluid

€ ∂p∂z=−ρg€

dpdρ=c2€ ∂ρ∂z=−ρgc2

€ ρz()=ρoexp−gd′ z c20z∫ ⎡ ⎣ ⎢ ⎤ ⎦ ⎥€

HS=c2g~200km>>H~1kmLocal scale height

€ dρdt=1c2dpdt=−ρgc2dzdt=−ρgc2w€

∂u∂x+∂v∂y+∂w∂z=gc2w€

O∂w∂z()Ogwc2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟=OWH()OWHS ⎛ ⎝ ⎜ ⎞ ⎠ ⎟=HSH>>1€ ∂u∂x+∂v∂y+∂w∂z=0

The motion has vertical scale small compared with the scale height

Boussinesq approximationDensity variations can be neglected for its effect

on mass but not on weight (or buoyancy).

ρρ ′>>oρρρ ′+= o ( ) pzpp o ′+=Assume that where , we have

gzp

oo ρ−=

∂∂

where gzp ρ′−=∂′∂

2

2

2

2

2

21zuA

yuA

xuAfv

xp

zuw

yuv

xuu

tu

zyxo ∂

∂+∂∂+∂

∂++∂′∂−=∂

∂+∂∂+∂

∂+∂∂

ρ

2

2

2

2

2

21zvA

yvA

xvAfu

yp

zvw

yvv

xvu

tv

zyxo ∂

∂+∂∂+∂

∂+−∂′∂−=∂

∂+∂∂+∂

∂+∂∂

ρ

0=∂∂+∂

∂+∂∂

zw

yv

xu

gzp ρ′−=∂′∂

ϕsin2Ω=f

Then the equations are

where

wϕcos2Ω−

(1)

(2)

(3)

(4)(The termis neglected in (1) for energy consideration.)

Geostrophic balance in ocean’s interior

Scaling of the horizontal components

2

2

2

2

2

2cos21

zuA

yuA

xuAwfv

xp

zuw

yuv

xuu

tu

zyx ∂∂+

∂∂+

∂∂+Ω−+

∂∂−=

∂∂+

∂∂+

∂∂+

∂∂ ϕρ

25

25

25443

222101010101010

L

U

L

U

L

UWUL

PL

UL

UL

UTU h −−−−−− +++−+

Δ−=+++

810810810810510910810810810710 −+−+−+−−−+Δ−=−+−+−+− Ph

3103103103101410310310310210 −+−+−+−−+Δ−=−+−+−+− Ph

Zero order (Geostrophic) balancePressure gradient force = Coriolis force

01 =+∂∂− fvxp

ρ

01 =−∂∂− fuyp

ρ yp

fu

∂∂−= ρ

1

xp

fv

∂∂=ρ

1

410=Δ Ph (accuracy, 1% ~ 1‰)

Re-scaling the vertical momentum equation

ρρρ ′+= o ( ) pzpp o ′+=

Since the density and pressure perturbation is not negligible in the vertical momentum equation, i.e.,

gzp

oo ρ−=

∂∂

gz

pg

z

p

z

p

z

p

z

p

z

p

z

p

z

p

z

p

z

p

z

p

oo

o

o

o

o

o

oo

o

oo

o

o

ρρ

ρ

ρρ

ρρρ

ρ

ρρρ

ρρρ

′−

∂′∂

−=

⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂′

−∂′∂

+∂∂

−≈⎟⎠

⎞⎜⎝

⎛∂′∂

+∂∂

⎟⎟⎠

⎞⎜⎜⎝

⎛ ′−−≈

⎟⎠

⎞⎜⎝

⎛∂′∂

+∂∂

⎟⎟⎠

⎞⎜⎜⎝

⎛ ′+

−=⎟⎠

⎞⎜⎝

⎛∂′∂

+∂∂

′+−=

∂∂

−

1

11

1

1

111

, , and

The vertical pressure gradient force becomes

Taking into the vertical momentum equation, we have

2

2

2

2

2

2cos21

zwA

ywA

xwAug

zp

zww

ywv

xwu

tw

zyx

o

o ∂∂+∂

∂+∂∂+Ω+′−∂

′∂−=∂∂+∂

∂+∂∂+∂

∂ ϕρρ

ρ

410~ =Δ′Δ PPz h

21

25

25423

2101010101010

HW

LW

LWU

HP

HW

LUW

LUW

TW z −−−− ++++−′Δ−=+++ δρ

H

P

z

p z ′Δ∂′∂~ δρρ ~′If we scale , and assume

1110111011105102102101110111011101010 −+−+−+−+−−−=−+−+−+− δρ

3/1 mkg=δρ

then

gzp ρ′−=∂′∂

and

(accuracy ~ 1‰)

Geopotential Geopotential is defined as the amount of work done to

move a parcel of unit mass through a vertical distance dz

against gravity is

dpgdzd α−==

(unit of : Joules/kg=m2/s2).

( ) ( ) ∫ ∫−=∫ ===−=2

1

2

1

2

1

12

z

z

p

pdpgdz

z

zdzzzz α

The geopotential difference between levels z1 and z2 (with pressure p1 and p2) is

Dynamic height

Given δαα += p,0,35 , we have

where ∫=Δ2

1

,0,35

p

pdppstd α is standard geopotential distance (function of p only)

∫=Δ2

1

p

pdpδ is geopotential anomaly. In general, 310~Δ

Δstd

( ) ( ) Δ−Δ−=∫ ∫−−==−=std

p

p

p

pdpdppppp p

2

1

2

1

,0,3512 δα

Δ is sometime measured by the unit “dynamic meter” (1dyn m = 10 J/kg). which is also called as “dynamic distance” (D)

Note: Though named as a distance, dynamic height (D) is still a measure of energy per unit mass.

∫=−=Δ2

112 10

1 p

pdpDDD δ Units: δ~m3/kg, p~Pa, D~ dyn m

Geopotential and isobaric surfacesGeopotential surface: constant , perpendicular to gravity, also referred to as

“level surface”

Isobaric surface: constant p. The pressure gradient force is perpendicular to the isobaric surface.

In a “stationary” state (u=v=w=0), isobaric surfaces must be level (parallel to geopotential surfaces).

In general, an isobaric surface (dashed line in the figure) is inclined to the level surface (full line).

In a “steady” state ( ),

the vertical balance of forces is

() ()()

()igi

iin

pinp tan

cos

sincos)sin( =∂

∂=∂∂

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛αα

0=∂∂=

∂∂=

∂∂

tw

tv

tu

np∂∂α

ginp =∂∂ )(cosα

The horizontal component of the pressure gradient force is

Geostrophic relationThe horizontal balance of force is

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛=Ω igV tansin21

ϕwhere tan(i) is the slope of the isobaric surface. tan (i) ≈ 10-5 (1m/100km) if V1=1 m/s at 45oN (Gulf Stream).

In principle, V1 can be determined by tan(i). In practice, tan(i) is hard to measure because

(1) p should be determined with the necessary accuracy

(2) the slope of sea surface (of magnitude <10-5) can not be directly measured (probably except for recent altimetry measurements from satellite.) (Sea surface is a isobaric surface but is not usually a level surface.)

Calculating geostrophic velocity using hydrographic data

⎟⎠⎞

⎜⎝⎛=Ω

11tansin2 igVϕ

⎟⎠⎞

⎜⎝⎛=Ω

22tansin2 igVϕ

The difference between the slopes (i1 and i2) at two levels (z1 and z1) can be determined from vertical profiles of density observations.

Level 1:

Level 2:

⎟⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛ −=−Ω

2121tantansin2 iigVVϕ

Difference:

⎟⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

⎟⎠⎞

⎜⎝⎛

−−−=

−=

−=

−=−Ω

4231

2121

2121

22

22

11

1121

sin2

zzzzL

g

AABBL

g

CCBBL

gCA

CB

CA

CBgVVϕi.e.,

because A1C1=A2C2=L and B1C1-B2C2=B1B2-C1C2

because C1C2=A1A2

Note that z is negative below sea surface.

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

⎟⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛ ∫−∫=−−− dpdp

Lzzzz

L

gp

p

A

p

p

B

2

1

2

1

4231

1 δδ

( ) dpdpzzgp

pA

p

pp

∫∫ +=−2

1

2

1

,0,3542 δα

( ) dpdpzzgp

pB

p

pp

∫∫ +=−2

1

2

1

,0,3531 δα

⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛ Δ−Δ

Ω=−

ABDD

LVV

ϕsin2

1021

Since

and

,

we have

The geostrophic equation becomes