xgfv

ygfu

Remider on the geostrophic balance

Cartoon describing the current around a positive sea levela nomaly in the Northern hemisphere: the red (1), green (2) and blue (3) arrows represent the current, the Coriolis force and the hydrostatic pressure force, respectively. The scale is deliberately deformed to clearly represent a vortex of ocean dimensions. The height of the swelling is about one meter, while its horizontal extension is several thousand kilometers for an oceanic gyre, or hundreds of kilometers for a ring

The geostrophic balance is a diagnostic relationship between pressure and velocity distribution in a "rotating" fluid. It is valid if the frictions are negligible and Rossby's number

is small (it is usually 10-1) LfUR

Introduzione all’oceanografia fisica, Univ. di Lecce P.Lionello, A.A 2005-2006

gHx

p

gx

top

xa

1

Balance of forces in a fluid at rest,

The hydrostatic pressure is balanced by the gradient of atmospheric pressure and wind stress

Separating effects of atmospheric pressure and wind:

Effect of spatial variations of the atmospheric pressure: Inverse barometer effect: (approximately 1cm per 1hPa)

Wind set-up: a very small slope , order of 10-5 , increasing with the square of the wind speed and inversely proportional to the water depth. This is the main factor producing coastal floods when it acts over very long cross-shore distance (fetch)

gHx

top

x

x

p

xg a

Introduzione all’oceanografia fisica, Univ. di Lecce P.Lionello, A.A 2005-2006

negative surges

high pressure system, anti-cyclone

The inverse barometer effect

negative sea level anomaly

positive surges

low pressure system, cyclone

positive sea level anomaly

…. It is about 1cm per hPa

wind

Atmospheric pressure

Total level

Effect of wind

Effect of atmospheric pressure

Sea bottom

Cartoon, describing the combined effect of sea level pressure variation and wind stress blowing along the Adriatic Sea during a Sirocco storm

Ekman transport:

Ekman transport describes the stationary motion of a fluid in a rotating reference system subjected to the action of the wind (it similarly applies to the bottom friction) . In this case three forces are involved: drag due to the wind, Coriolis and friction exerted by the underlying fluid. The balance (steady conditions) can be achieved only if forces are disposed as in figure 9.2 and the surface currents deviates to the right (north hemisphere) with respect to the wind. The motion of the upper layer exerts similarly a drag on the underlying fluid, so that in turn the motion of the latter with further deviate to the right. The resulting vertical structure is called Ekman spiral (figure 9.3).

Note: Layer thickness = 100-200 meters, deep water condition, steady winds, absence of physical boundaries (coastlines).

HHxgfv

bot

x

top

x

HHygfu

bot

y

top

y

0

y

v

x

uH

The Ekman velocity 𝑢𝐸 is defined as deviation from geostrophy: 𝑢𝐸 = 𝑢 − 𝑢𝑔 , so that

HHfv

bot

x

top

xE

HHfu

bot

y

top

y

E 0

y

v

x

uH EE

A simple solution can be obtained for the total transport in a homogeneous layer of fluid. The shallow water equations give

Eucf

H

top

Eu

cf

H

bot

surface

bottom

Total Ekman transport and coastal upwelling/downwelling in the north hemisphere

For a uniform northerly wind blowing along the coast (aligned in the north-south direction) one has

bot

E

top

EE

E uux

uv

0;constant

top

Eu

bot

Eu

upwellingtop

Eu

bot

Eudownwelling

Figure 9.8 represents a sketch of the coastal circulation and the signature of cold water near the coast at the surface in caseof coastal upwellingWater in coastal areas with upwelling (downwelling) are cold(warm) and rich(poor) in nutrients,

ww.interactiveoceans.washington.edu/file/Coastal_Upwelling

Evidence of coastal upwelling in satellite chlorophyll observations

https://airsea-www.jpl.nasa.gov/cos/theory/upwelling_dynamics.html

Evidence of coastal upwelling in satellite Sea Surface Temperature observations

The Ekman SpiralThe Ekman transport was originally proposed by F.Nansen to explain why the icebergs where not moving in the direction of the wind. In this case there are three forces: Wind Stress, W; Friction of the underlying fluid F, which must be opposite the direction of the iceberg; Coriolis Force, C, which must be perpendicular to the velocityThe forces must balance for steady flow: W + F + C =0 (fig.9.2) The iceberg moves deviating to the right with respect to the wind (in the north hemisphere)

If a fluid is divided into layers each moving at a different speed, the surface layer is exactly in the same situation as the Nansen’s iceberg deviates to the right with respect to the wind.Each layer below it is set into motion by the drag of the layer above while the layer below exerts a friction opposing to the motion. Therefore the velocity of each layer is reduced and deviated to the right with respect to the fluid above it. The resulting variation of the speed with depth is the Ekman spiral (Fig.9.2and 9.3)

Mass balance and volume balance (if changes of density are neglected))• Bulk formulation for a marginal sea such as eq.(7.2)• Application to a column of fluid in “shallow water equation” • Local formulation with Euler formalism such as eq.(7.18) and (incompressible motion)

eq.(7.19)

Integral balance of volume for a fluid with uniform density and speed independent from the level

D

hB

η

If the fluid is incompressible, its volume does not change

x

x

2xxF 2xxF

0 HBHt uhD

0 HBHt uhD

0 HHt uD

Linearity 𝜂 ≪ 𝐷

Uniform water depth

Total thickness of the fluid:BhDH

Flux of volume: uFx

Net volume balance along the x-direction

tyuhDuhDxxBxxB 22

Local rate of change of volume tyxt

tyx

x

uhD B

Horizontal divergence of the flux ofvolumeLocal change of level of the surface Balance

simplifications

Mass M per unit surface 𝐴 = 𝑑𝑥 𝑑𝑦 is expressed as𝑀

𝐴= 𝐻−

η𝜌𝑑𝑧

Therefore : ∆𝑀

𝐴= 𝜌𝑆∆η + 𝐻−

η∆𝜌𝑑𝑧

∆η =∆𝑀

𝜌𝑆𝐴−න

−𝐻

η ∆𝜌

𝜌𝑆𝑑𝑧 =

∆𝑀

𝜌𝑆𝐴+න

−𝐻

η

𝛼∆𝑇 − 𝛽∆𝑆 𝑑𝑧

Mass variation

per unit surfaceSea level increase:

Addition/subtraction

of a layer of thickness ∆η

Changes pf density

with depth without

change of sea level

Sea level

variation

Mass

Addition/subtraction

Density

variations

Thermal

expansionHalosteric

contraction

By some algebra one obtains

Sea level variation at regional scale and steric effects

Previous formula shows that a local variation of sea level can be the result of two main factors

Mass addition/subtraction• Local scale redistribution of global mass addition (ice cap and continental glacier melting)• Space redistribution of mass caused by changes of both large and regional scale circulation • Space redistribution of mass caused global thermal expansion• Changes of mean atmospheric pressure at local scale (inverse barometer effect)• Change of mean wind intensity and direction (wind set-up)

Changes of density without change of mass• Local warming or cooling of water• Local salinization or freshening of water

These two factors act in general simultaneously