Dominant Forces for Ocean Dynamics (section 7.1) Gravity ...

Dominant Forces for Ocean Dynamics (section 7.1)GravityPressure BuoyancyFriction wind, bottom, shear currentsInertial forces Coriolis, centrifugal force

Terminology for describing the flow in the ocean (section 7.3)• General Circulation.• Abyssal Deep (thermohaline circulation) • Wind-Driven Circulation • Gyres (mainly clockwise/anticlockwise in the

North/south hemisphere)• Boundary Currents : western and eastern currents

flowing parallel to coasts• Squirts or Jets are long narrow currents• Mesoscale Eddies (current instability, topographically

trapped)

Waves in the ocean (section 7.3)• Planetary Waves depend on the rotation of the earth for

a restoring force• Surface Waves sometimes called gravity waves and wind

waves. The restoring force is due to the large density contrast between air and water at the sea surface.

• Internal Waves are subsea wave similar in some respects to surface waves. The restoring force is due to change in density with depth.

• Tsunamis are surface waves with periods near 15 minutes generated by earthquakes.

• Tidal Currents (waves) are horizontal currents and currents associated with internal waves driven by the tidal potential.

• Edge Waves are surface waves with periods of a few minutes confined to shallow regions near shore. The amplitude of the waves drops off exponentially with distance from shore.

Pressure. The force is proportional to the area of the lateral facets of the fluid element and to the different of pressure acting on its opposite sides.

ො𝑥 − 𝛿𝐴𝛿𝑥𝜕𝑝

𝜕𝑥; ො𝑦 − 𝛿𝐴𝛿𝑦

𝜕𝑝

𝜕𝑦

Horizontal forces experienced by a fluid element while moving along a curved trajectory

Coriolis force is orthogonal to the fluid speed and deviates it to the right (left) in the north (south) hemisphere

ො𝑥 ρ𝛿𝑉𝑓𝑣 ; ො𝑦 − ρ𝛿𝑉𝑓𝑢 where 𝑓 = 22𝜋

𝑇𝑅𝑠𝑖𝑛𝜑

Here TR is the Earth rotation period and 𝜑 is latitude

Centrifugal force pushing the fluid element in the radial direction

ρ𝛿𝑉𝑢 2

𝑅where 𝑢 2 is the speed modulus and R the radius of curvature

Friction. It opposes to the motion, the expression is complex and depends of viscosity and turbulence. In simple cases and low speeds, it may be assumed to be proportional to the difference of speed between the fluid and the surrounding −𝐶𝑓ρ𝛿𝑉 𝑢 − 𝑢𝑏𝑎𝑐𝑘𝑔𝑟𝑜𝑢𝑛𝑑

Note: along the vertical the hydrostatic balance holds

we can write the equation of motion in the local y direction:

𝑑𝑣

𝑑𝑡= −

1

𝜌

𝑑𝑝

𝑑𝑦− 𝑓𝑢 +

𝑢 2

𝑅- friction

y

x

• If the motion is almost stationary: 1

𝑇𝑓≪ 1 (𝑇 is the time scale of the motion)

• The centrifugal force is small with respect to Coriolis : 휀𝑅 =𝑈

𝑓𝐿≪ 1 small Rossby number 휀𝑅 , where U and L are

the fluid speed and the horizontal scale of the motion• Friction is negligible

The pressure and Coriolis force must balance each other (geostrophic balance):𝜌𝑓𝑢 = −𝑑𝑝

𝑑𝑦

In general in the geostrophic balance:

If density is constant 𝜌 = 𝜌0 the geostrophic balance further simplifies to

1 current speed, 2 Coriolis 3 hydrostatic pressure

ො𝑥 𝜌𝑓𝑣 =𝑑𝑝

𝑑𝑥; ො𝑦 𝜌𝑓𝑢 = −

𝑑𝑝

𝑑𝑦

ො𝑥 𝑓𝑣 = 𝑔𝑑𝜂

𝑑𝑥; ො𝑦 𝑓𝑢 = −𝑔

𝑑𝜂

𝑑𝑦

That is a relation between current speed and sea surface elevation η which can be used for mapping ocean currents from satellite altimetry (figs.10.2, 10.3, 10.5)

Geostrophic balance (section 10.3)

Sea surface elevation measured by the NASA-CNES satellite Topex-Poseidon and processed at Jet Propulsion Laboratory in

Pasadena. The depressions and elevations associated with the currents shown in the previous fig.global_currents can be

identified. Arrows denote the direction and speed of currents. Image courtesy of NASA/JPL/Caltech

zhghgzp 212 )(

x

h

f

g

xf

g

x

h

f

g

x

h

f

gv

xf

gv

*

0

1

0

1

0

22

0

11

)(;

0

12*

gg

The hydrostatic relationship says that at any depth -z the pressure will simply be

given by the weight of the water above the level z. If there are two layers, layer 1

and 2 as in figure, then in layer 1 and 2 respectively,

So that the geostrophic balance gives

is called reduced gravity.

This implies that the difference of velocity (shear) between layer 1 and layer 2 depends on

the slope of the interface and on the difference of density between the two layers:

𝑣2 − 𝑣1 =𝑔∗

𝑓

𝜕ℎ

𝜕𝑥

In the figure above this relation is considered for the flow across a strait, such

as the Gibraltar or Otranto Strait

(Margules’ relation)

Margules’ relation (section 10.7)

Cross section of the Otranto Strait: temperature (left) and (salinity (right)

Levantine intermediate watter Inflow

Adriatic deep water outflow

Density drive coastal currents

The Margules’ relation can be applied to a coastal current such as that whose section is

shown in figure. In this case the speed of the fluid 2 is supposed to be nil (the ocean is at rest). The transport T of the coastal current is

where L is the width of the current, h its thickness (depending on x), H its thickness at the

coast. In this case the total transport depends on the density difference and on the

squared thickness of the current at the shore.

𝑇 = න0

𝐿

𝑣1ℎ𝑑𝑥 = න0

𝐿𝑔∗

𝑓

𝜕ℎ

𝜕𝑥ℎ𝑑𝑥 =

𝑔∗

2𝑓𝐻2

dx

the forces acting on the fluid are: • the downslope component of the reduced gravity 𝑔∗ sin 𝜃• the component of the Coriolis force associated to the

component of the earth rotation orthogonal to the slope 𝑣𝑓 cos 𝜃

• the friction which is proportional to the speed of the current, inversely proportional to its thickness h and proportional to friction coefficients describing the action on the current of the bottom rb and of the of the fluid at rest above the current rt :

Τ𝑟𝑡 + 𝑟𝑏 𝑣 ℎ

Downslope density driven currents

if α is the descent angle of the current (between the current and the horizontal direction), the balance of forces in thedirection orthogonal to the current and along the current are given as in figure

𝑔∗ sin 𝜃 cos 𝛼 =𝑣𝑓 cos 𝜃𝑔∗ sin 𝜃 sin 𝛼 = Τ𝑟𝑡 + 𝑟𝑏 𝑣 ℎ

So that the descent angle is given by the relation

tan𝛼 =𝑟𝑡 + 𝑟𝑏𝑓ℎ cos 𝜃

So that no friction implies no downslope motion (the current flows at a constant depth) and no rotation (f=0) implies an alongslope downward motion. The actual fluid descent results from two contrasting factors: the friction favoring it and the Coriolisforce preventing it.