Class 8. Oceans

Figure: Ocean Depth (mean = 3.7 km)

Ship-measurements

Only a limited area covered

SST from buoys

-2.0 34.2 0C16.1

Ocean Surface Temperature from Remote Sensing

(NOAA)

source NOAA

http://www.osdpd.noaa.gov/ml/ocean/sst/sst_anim_full.html

- maximum insolation- albedo of water ~ 7%

warm

cold water sinks

cold water sinks

Ocean Surface Salinity

Evap>Prep

Prep>Evap

ARGO: profiling the interior of the ocean (up to z=-

2000 m)

ARGO: profiling the interior of the ocean (up to z=-

2000 m)

Data products:Temperature, salinity and density

Zonal average temperature in deep ocean

abyss z>1000 mhomogeneous mass of very cold water

warm salty stratified lens of fluid

Schematic of vertical structure

convection in the upper layer causes a vertically well mixed layer

strong vertical temperature gradient defines the thermocline

note: analogy to thermal inversion in the atmosphere

very cold water present below z<1000 m

Thermal expansion: Sea-level transgression scenarios for Bangladesh

Density (anomaly s), Temperature and Salinity

Fig. 9.2: Contours of seawater density anomalies ( = -s r rref in kg/m3)

rref = 1000 kg/m3

PSU = Practical Salinity Unit ≈ g/kg grams of salt per kg of solution

higher density

Simplified equation of state (defined with respect to

s0(T0,S0))

€

σ =σ0 + ρ ref −α T T − T0[ ] + βS S −S0[ ]( )

€

αT = −1

ρ ref

∂ρ

∂T

€

βS =1

ρ ref

∂ρ

∂S

Simplified equation of state (defined with respect to

s0(T0,S0))

€

σ =σ0 + ρ ref −α T T − T0[ ] + βS S −S0[ ]( )

a

s

s

Schematic of vertical structure

€

∂T

∂t= −

1

ρwcw

∂F

∂z

tendency due to radiative heating

T = temperatureF = heat flux (Wm-2)rw = density of watercw = heat capacity of water

μ

1000

dep

th (

m)

0

cold water -deep convection

cold waterdeep convection

upwelling

900S 900N 00 latitude

P>E P<E P<E P>E

Low salinity if precipitation (P) exceeds evaporation (E)

1000

dep

th (

m)

0

900S 900N 00 latitude

Thermohaline circulation

Sea level height

Which balances do apply in the ocean?

Hydrostatic balance -> yes

Geostrophic balance?

Thermal "wind"?

Ekman pumping/suction?

Rossby and Reynolds number in the ocean

Far away from the equator, e.g. latitude = 400,

North-South length scale L = 2000 km

(east-west larger)

Velocity scale U = 0.1

m/s

€

f = 2×2π

24 × 3600sin 400

( ) =1×10−4 s−1

Pressure in the ocean

€

p x,y,z( ) = ps + gρdzz

η

∫ = ps + g ρ η − z( )

high pressure low pressure

€

ρ =ρref =

ρdzz

η

∫η − z( )

mean densityin water column

Which sea level tilt is needed to explain U=0.1 m/s?

€

p x,y,z0( ) = ps + gρdzz0

η

∫ = ps + g ρ η(x,y) − z0[ ]

Example 1: assume density is constant

Geostrophic flow at depth

Example 2: assume density is NOT constant,

but varies in the x,y directions => r(x,y)=rref+s(x,y)

€

0 = −1

ρ

∂p

∂x+ fv

1000

de

pth

(m

)

0

900S 900N 00

latitude

2324

25

2626.5

27

Estimating the geostrophic wind from the density field:

The dynamic method

€

∂ug

∂z=

g

fρ ref

∂σ

∂y

€

ug z( ) − ug z1( ) =g

f

1

ρ ref

∂σ

∂ydz

z1

z

∫

This method allows for assessing geostrophic velocities relative to some reference level

One can assume that at a "sufficiently" deep height ug=0

Geostrophic flow at depth z

Example 3: I) assume density is NOT constant,

but varies in the x,y directions => r(x,y)=rref+s(x,y)

II) surface height is NOT constant

€

0 = −1

ρ

∂p

∂x+ fv

€

p x,y,z( ) = ps + gρdzz

η

∫ = ps + g ρ η(x,y) − z[ ]

€

v z( ) =g

fρ ref

η − z( )∂σ

∂x+

g

f

∂η

∂x

Geostrophic flow

Example 1: In the ocean geostrophic flow applies (not too close to equator)

Pressure induced by surface height variations η

Example 2: Horizontal density gradients cause a vertical change in the geostrophic flow velocity ("thermal" wind)

Example 3: In principle both height and density variations may apply

€

0 = −g∂η

∂x+ fvg

€

∂vg

∂z= −

g

fρ ref

∂σ

∂x

€

0 = −g∂η

∂y− fug

€

∂ug

∂z=

g

fρ ref

∂σ

∂y

€

v z( ) =g

fρ ref

η − z( )∂σ

∂x+

g

f

∂η

∂x

Determining the ocean flow from floating plastic ducks?

1000

dep

th (

m)

0

cold water -deep convection

cold waterdeep convection

upwelling

900S 900N 00 latitude

Ekman pumping/suction

Wind-driven ocean flow

Equations with wind-stress

€

x - component 0 = −1

ρ

∂p

∂x+ fv +

1

ρ ref

∂τx

∂z

€

y - component 0 = −1ρ

∂p∂y

−fu +1

ρ ref

∂τ y

∂z

Wind-driven ocean flow

Equations with wind-stress

€

x - component 0 = −1

ρ

∂p

∂x+ fv +

1

ρ ref

∂τx

∂z

Split velocity in geostrophic ('g') and ageostrophic parts ('ag')

€

v = vg + vag

€

vg =1

ρf

∂p

∂x

€

−fvag =1

ρ ref

∂τx

∂z

€

fuag =1

ρ ref

∂τy

∂z

e.g.

Ekman transport

€

MEk = ρ ref−δ

0

∫ uagdz

€

MEk = ρ ref uag,vag( )−δ

0

∫ dz =1

fτy,−τx( )

Ekman pumping (downwards)/suction

X wind into the screen

Ekman pumping (downwards)/suction

tropics

midlatitudes

elevated sea level heightin convergence area

Ekman pumping/suction due to wind stress

Ekman pumping/suction

Explanation

€

∂w∂z

= −∂u ag

∂x+

∂vag

∂y

⎛

⎝ ⎜

⎞

⎠ ⎟ mass conservation

€

w zsfc( ) −w zEkman,bottom( ) = −∂u ag

∂x+

∂vag

∂y

⎛

⎝ ⎜

⎞

⎠ ⎟dz

zEkman ,bottom

zsfc

∫

0

Ekman pumping/suction

Explanation

€

vag = −1

fρ ref

∂τ x

∂z

€

u ag =1

fρ ref

∂τ y

∂z€

w zEkman,bottom( ) =∂u ag

∂x+

∂vag

∂y

⎛

⎝ ⎜

⎞

⎠ ⎟dz

zEkman ,bottom

zsfc

∫

€

w zEkman,bottom( ) =1

ρ ref

1

f

∂τy,sfc

∂x−

∂

∂y

τx,sfc

f

⎛

⎝ ⎜

⎞

⎠ ⎟

1. we do not assume that f is constant, but f=f(y)2. variations in wind stress are much larger than in f

Ekman pumping/suction

Example

€

w zEkman,bottom( ) =1

ρ ref

∂∂x

τ y,sfc

f−

∂∂y

τ x,sfc

f

⎛

⎝ ⎜

⎞

⎠ ⎟

€

= 11000

2×10−1

2×106( ) 1×10−4

( )

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥=10−6 m /s

= 32 m/year

Ekman pumping/suction from wind stress climatology

€

w zEkman,bottom( ) =1

ρ ref

∂∂x

τ y,sfc

f−

∂∂y

τ x,sfc

f

⎛

⎝ ⎜

⎞

⎠ ⎟

downward

upward

f=0