Level of No Motion (LNM)

Pacific:Deep water is uniform,current is weak below1000m.

Atlantic: A level of no motion at 1000-2000m

Current increases into the deep ocean, unlikely in the real ocean

“Slope current”: Relative geostrophic current is zero but absolute current is not. May occurs in deep ocean (barotropic?)

0=∇×∇ pρ p and ρ surfaces are parallel

Given a barotropic and hydrostatic conditions,

0=∂∂zVg

r

gVr

is geostrophic current.

xpfv

∂∂=

ρ1

xg

xp

zzxp

xp

zzvf

∂∂−

∂∂

∂∂−=

∂∂∂+

∂∂

∂∂−=

∂∂ ρ

ρρ

ρρρ

ρ 2

2

2111

For a barotropic flow, we have

dpdg

zp

dpd

zρρρρ −=

∂∂=

∂∂

and xp

dpd

x ∂∂=

∂∂ ρρ

Therefore, 012

=∂∂−

∂∂−−=

∂∂

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

xp

dpdg

xp

dpdg

zvf ρ

ρρρρ

And 012

=∂∂−

∂∂−=

∂∂−

∂∂

∂∂−=

∂∂

xg

zfv

xg

xp

zzvf ρρρ

ρρ

ρ

So x

gz

fv∂∂−=

∂∂ ρρ

Barotropic flow:

Since

(≈0 in Boussinesq approximation)

The slope of isopycnal is small and undetectable, for V=0.1 m/s, slope~10-6, i.e., 0.1m height change in 100 km

Baroclinic Flow: 0≠∇×∇ pρ and ),,( pTSρρ =

There is no simple relation between the isobars and isopycnals.

xpfv

∂∂=ρ slope of isobar is proportional to velocity

xg

zvf

∂∂−=

∂∂ ρρ slope of isopycnal is proportional to vertical wind shear.

With a barotropic of mass the water may be stationary but with a baroclinic field, having horizontal density gradients, such as situation is not possible

In the ocean, the barotropic case is most common in deep water while the baroclinic case is most common in the upper 1000 meters where most of the faster currents occur.

Relations between isobaric and isopycnal surfaces and currents

1½ layer flow

Simplest case of baroclinic flow:

Two layer flow of density ρ1 and ρ2.

The sea surface height is η=η(x,y) (In steady state, η=0). The depth of the upper layer is at z=d(x,y). The lower layer is at rest.

312 1 mkg≈−ρρ

)(1

zgp −= ηρ η∇×−= kfgVrr

1For z > d,

( ) gzgdgzdgdgp 212121 )()( ρρρηρρηρ −−+=−+−=

If we assume d∇−−=∇1

12ρ

ρρη

The slope of the interface between the two layers (isopycnal) =

100012

1 ≈−ρρρ times the slope of the surface (isobar).

The isopycnal slope is opposite in sign to the isobaric slope.

For z ≤ d,

02 =Vr

σt A B diff26.8 50m --- >50m27.0 130m 280m -150m27.7 579m 750m -180m

Isopycnals are nearly flat at 100m

Isobars ascend about 0.13m between A and B for upper 150m

Below 100m, isopycnals and isobars slope in opposite directions with 1000 times in size.

Example: sea surface height and thermocline

depth

Comments on the geostrophic equation

• If the ocean is in “real” geostrophy, there is no water parcel acceleration. No other forces acting on the parcel. Current should be steady

• Present calculation yields only relative currents and the selection of an appropriate level of no motion always presents a problem

• One is faced with a problem when the selected level of no motion reaches the ocean bottom as the stations get close to shore

• It only yields mean values between stations which are usually tens of kilometers apart

• Friction is ignored• Geostrophy breaks down near the equator• The calculated geostrophic currents will include any long-period

transient current

The β-spiralDetermining absolute velocity from density field

Assumptions: 1) Geostrophic

x

pfv

∂∂

=αy

pfu

∂∂

−= α

2) incompressible

3) steady state

0=dtdρ

0=∂∂tρ

0=∂∂

+∂∂

+∂∂

zw

yv

xu

ρρρ2) + 3)

Use the thermal wind equation with Boussinesq approximation

xg

z

vfo ∂

∂−=

∂∂ ρρ

yg

z

ufo ∂

∂=

∂∂ ρρ

0=∂∂

+∂∂

+∂∂

zw

yv

xu

ρρρTake into

zw

f

g

z

uv

z

vu

o ∂∂

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

∂∂

+∂∂

−ρ

ρ

Write u and v to polar format as

θcosVu =θsinVv =

⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂

=∂∂

2fV

g

zw

z oρ

ρθ

In the northern hemisphere, if w > 0, the current rotates to the right as we go upward (or to the left as we go downward)

( )yx

p

x

uf

x

fu

∂∂

∂−=

∂

∂=

∂

∂ 2

α

Take geostrophic equation

( )yx

p

y

fv

y

vf

y

fv

∂∂

∂−=

∂

∂+

∂

∂=

∂

∂ 2

α

0=∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

yf

vyv

xu

f

0=∂∂

+∂∂

+∂∂

zw

yv

xu

use

z

wfv

∂∂

=β dy

df=β

If v≠0, w changes with z and can not be zero everywhere. Thus the β effect makes the rotation of the geostrophic flow with depth likely, hence the term “β spiral”

Consider an isopycnal surface at ( )σ,, yxhhz o +−=If we go along this surface in the x-direction

0=∂∂

+∂∂

= dhz

dxx

dρρρ ⎥⎦

⎤⎢⎣⎡ ⎟

⎠⎞⎜

⎝⎛

∂∂⎟

⎠⎞⎜

⎝⎛

∂∂−=

∂∂

zxx

h ρρ

⎥⎦⎤

⎢⎣⎡ ⎟

⎠⎞⎜

⎝⎛

∂∂⎟

⎠⎞⎜

⎝⎛

∂∂−=

∂∂

zyy

h ρρ

Similarly

0=∂∂

+∂∂

+∂∂

zw

yv

xu

ρρρTake into

wy

hv

x

hu =

∂∂

+∂∂

f

v

z

w

y

hv

x

hu

z

β=

∂∂

=⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

∂∂

⎟⎠

⎞⎜⎝

⎛∂∂

∂∂

=∂∂

zx

h

f

g

z

v

o

ρ

ρ ⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

∂∂

−=∂∂

zy

h

f

g

z

u

o

ρ

ρ

Rewrite the thermal wind relation

0=∂∂

∂∂

+∂∂

∂∂

yh

zv

xh

zu

022

=⎟⎟⎠

⎞⎜⎜⎝

⎛−

∂∂∂

+∂∂

∂fzy

hv

zxh

uβ

02222

=⎟⎟⎠

⎞⎜⎜⎝

⎛−

∂∂∂′+

∂∂∂′+⎟⎟

⎠

⎞⎜⎜⎝

⎛−

∂∂∂

+∂∂

∂fzy

hv

zxh

ufzy

hv

zxh

u ooββ

Suppose we have derived u’ and v’ based on some reference level

If the observations should be error free, two levels would be sufficientConsidering the observation errors, particularly noise from time-dependent motions, this equation will not be exactly satisfied. Computationally, a least-square technique is used.