An example of vertical profiles of temperature, salinity and density

Geopotential Geopotential is defined as the amount of work done to

move a parcel of unit mass through a vertical distance dz

against gravity

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

is units of energy per unit mass (J/kg). However, it is call as “dynamic distance (D)” and sometime measured by the unit “dynamic meter” (1dyn m = 10 J/kg).

Note: Though named as a distance, dynamic height (D) is 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 z2) 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

Current Direction

• In the northern hemisphere, the current will be along the slope of a pressure surface in such a direction that the surface is higher on the right

• In the northern hemisphere, the current flows relative to the water just below it with the “lighter water on its right”

• Sverdrup et al. (The Oceans, 1946, p449) “In the northern hemisphere, the current at one depth relative to the current at a greater depth flows away from the reader if, on the average, the δ or t curves in a vertical section slope downward from left to right in the interval between two depths, and toward the reader if the curves slope downward from right to left”

€ 2Ωsinφv=1ρ∂p∂x€

2Ωsinφu=−1ρ∂p∂y

€ 2ΩsinφVH=1ρ∂p∂nH

Geostrophic Balance

€ VH=u2+v2Magnitude of the current

Horizontal pressure perpendicular to current direction

€ ∂p∂nH

€ ∂∂x ⎛ ⎝ ⎜ ⎞ ⎠ ⎟z=∂Φ∂x ⎛ ⎝ ⎜ ⎞ ⎠ ⎟p+∂Φ∂p ⎛ ⎝ ⎜ ⎞ ⎠ ⎟x,y∂p∂x ⎛ ⎝ ⎜ ⎞ ⎠ ⎟z=0€ ∂∂p=−1ρ€

∂p∂x ⎛ ⎝ ⎜ ⎞ ⎠ ⎟z=ρ∂Φ∂x ⎛ ⎝ ⎜ ⎞ ⎠ ⎟p€ ∂p∂y ⎛ ⎝ ⎜ ⎞ ⎠ ⎟z=ρ∂Φ∂y ⎛ ⎝ ⎜ ⎞ ⎠ ⎟p€

∂p∂nH ⎛ ⎝ ⎜ ⎞ ⎠ ⎟z=ρ∂Φ∂nH ⎛ ⎝ ⎜ ⎞ ⎠ ⎟p

€ 1=Φ2+ΔΦstd+ΔΦFor two levels,

€ 2Ωsinφv1=1ρ∂p1∂x ⎛ ⎝ ⎜ ⎞ ⎠ ⎟z=∂Φ1∂x ⎛ ⎝ ⎜ ⎞ ⎠ ⎟p=∂Φ2∂x+∂ΔΦ()∂x

€ 2Ωsinφv2=∂Φ2∂x€ 2Ωsinφv1−v2()=∂ΔΦ()∂x€

2Ωsinφu1−u2()=−∂ΔΦ()∂y€ 2ΩsinφV1−V2()=∂ΔΦ()∂nH

Integrating along a line, L, linking stations A and B

€ fV1−V2()=1L∂ΔΦ()∂nHdnH=1LΔΦB−ΔΦA( )AB∫

“Thermal Wind” Equation

xpfv∂∂=ρ

Differentiating with respect to z x

gzp

xfv

z ∂∂−=

∂∂

∂∂=

∂∂

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛⎟⎠⎞⎜

⎝⎛ ρρ

ygfu

z ∂∂=

∂∂

⎟⎠⎞⎜

⎝⎛ ρρ

Using Boussinesq approximation

yg

yg

zuf t

∂

∂≈

∂∂=

∂∂ ρρ

xg

xg

zvf t

∂

∂−≈

∂∂−=

∂∂ ρρ

Ory

g

y

gzuf

∂∂−=

∂∂−=

∂∂ δ

αα

α

x

g

x

gzvf

∂∂=

∂∂=

∂∂ δ

αα

α

Starting from geostrophic relation

Rule of thumb: light water on the right.

(for the upper 1000 meters)

€ ∂ρfr V ⎛ ⎝ ⎜ ⎞ ⎠ ⎟∂z=∇Hρ×r k

Deriving Absolute Velocities• Assume that there is a level or depth of no motion (reference

level) e.g., that V2=0 in deep water, and calculate V1 for various levels above this (the classical method)

• When there are station available across the full width of a strait or ocean, calculate the velocities and then apply the equation of continuity to see the resulting flow is responsible, i.e., complies with all facts already known about the flow and also satisfies conservation of heat and salt

• Use a “level of known motion”, e.g., if surface currents are known or if the currents have been measured at some depths.

• Note that the bottom of the sea cannot be used as a level of no motion or known motion even though its velocity is zero.

Dynamic Topography

at sea surface

relative to 1000dbar

(unit 0.1J/kgdynamic

centimeter)

Relations between isobaric and level surfaces

• A level of no motion is usually selected at about 1000 meter depth

• In the Pacific, the uniformity of properties in the deep water suggests that assuming a level of no motion at 1000m or so is reasonable, with very slow motion below this

• In the Atlantic, there is evidence of a level of no motion at 1000-2000 m (between the upper waters and the North Atlantic Deep Water) with significant currents above and below this depths

• That the deep ocean current is small does not mean the deep ocean transport is small

Level of No Motion (LNM)

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

Atlantic: A level of no motion at 1000-2000m above North Atlantic Deep Water

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?). The situation is possible in deep ocean where T and S change is small

Neglecting small current in deep ocean affects total volume transport

• Assume the ocean depth is 4000m, V=10 cm/s above 1000m based on zero current below gives a total volume transport for 4000m of 100m3/s

• If the current is 2cm/s below 1000m, the estimate of V above 1000m has an error of 20%

• The real total volume transport from surface to 4000m would be 180m3/s or 80% more than that assuming zero current below 1000m