1

Equations of Equations of MotionMotion

BuoyancyBuoyancyEkman and Inertial Ekman and Inertial

MotionMotionSeptember 17September 17

2

0

cos21

sin21

sin21

z

w

y

v

x

u

Fguz

p

z

ww

y

wv

x

wu

t

w

Fuy

p

z

vw

y

vv

x

vu

t

v

Fvx

p

z

uw

y

uv

x

uu

t

u

z

y

x

Recall:

3

0

1

1

1

2

22

2

22

2

22

z

w

y

v

x

u

z

wAwAg

z

p

Dt

Dw

z

vAvAfu

y

p

Dt

Dv

z

uAuAfv

x

p

Dt

Du

zHH

zHH

zHH

Or:

2

2

2

22

sin2

yx

dtDt

D

f

H

u

where:

x-momentum

y-momentum

z-momentum

continuity

4

Figure 8.4 in Stewart

The buoyancy force acting on the displaced parcel is:

2gVF ‘

Buoyancy:

5

The acceleration of the displaced parcel is:

)( 2g

m

Fa

6

Stability EquationStability Equation

dz

dE

1

Stability is defined such that:

E > 0 stable

E = 0 neutral stability

E < 0 unstable

Influence of stability is expressed by a stability frequency N

(also known as Brunt-Vaisala frequency):gEN 2

7

Figure 8.6 in Stewart: Observed stratification frequency in the Pacific. Left: Stability of the deep thermocline east of the Kuroshio. Right: Stability of a shallow thermocline typical of the tropics. Note the change of scales.

8

Dynamic Stability and Dynamic Stability and Richardson's NumberRichardson's Number

If velocity changes with depth in a stable, stratified If velocity changes with depth in a stable, stratified flow, then the flow may become unstable if the change flow, then the flow may become unstable if the change in velocity with depth, the current shear , is large in velocity with depth, the current shear , is large enough. The simplest example is wind blowing over the enough. The simplest example is wind blowing over the ocean. In this case, stability is very large across the sea ocean. In this case, stability is very large across the sea surface. We might say it is infinite because there is a surface. We might say it is infinite because there is a step discontinuity in step discontinuity in , and N, and N2 is infinite. Yet, wind is infinite. Yet, wind blowing on the ocean creates waves, and if the wind is blowing on the ocean creates waves, and if the wind is strong enough, the surface becomes unstable and the strong enough, the surface becomes unstable and the waves break.waves break.

This is an example of This is an example of dynamic instabilitydynamic instability in which a in which a stable fluid is made unstable by velocity shear. Another stable fluid is made unstable by velocity shear. Another example of dynamic instability, the Kelvin-Helmholtz example of dynamic instability, the Kelvin-Helmholtz instability, occurs when the density contrast in a instability, occurs when the density contrast in a sheared flow is much less than at the sea surface, such sheared flow is much less than at the sea surface, such as in the thermocline or at the top of a stable, as in the thermocline or at the top of a stable, atmospheric boundary layer atmospheric boundary layer

9

Figure 8.7 in Stewart: Billow clouds showing a Kelvin-Helmholtz instability at the top of a stable atmospheric boundary layer. Some billows can become large enough that more dense air overlies less dense air, and then the billows collapse into turbulence. Photography copyright Brooks Martner, NOAA Environmental Technology Laboratory.

10

Richardson NumberRichardson Number

The relative importance of static stability The relative importance of static stability and dynamic instability is expressed by and dynamic instability is expressed by the Richardson Number:the Richardson Number:

2zu

gERi

Ri > 0.25 Stable

Ri < 0.25 Velocity shear enhances turbulence

11

Ekman FlowEkman Flow

12

Again:Again:

2

2

2

22

2

22

2

22

1

1

yx

z

vAvAfu

y

p

Dt

Dv

z

uAuAfv

x

p

Dt

Du

H

zHH

zHH

13

fvDt

Duu

fuDt

Dvv

0)(2

1 22 uvuvfvuDt

D0

22

Dt

Dc

vuc

Define c (current speed) as:

u and v change, but c stays constant: Coriolis force does no work!

002

1 2

Dt

Dc

Dt

Dc

14

Flow is in a circle: Inertial or Centripetal force = Coriolis force

fcr

c

2where

f

cr

Inertial radius

If

c ~ 0.1 m/s

f ~ 10-4

then

r ~ 1 km

15

Inertial Period is given by T where:

fT f

2

If

f ~ 10-4

then

Tf ~ 6.28x104sec ~ 17.4 hrs

16

r

fc

fc

fc

fc

cc

cc

c2/r

c2/r

c2/r

c2/r

17

Latitude () Ti (hr) D (km)

for V = 20 cm/s

90° 11.97 2.7

35° 20.87 4.8

10° 68.93 15.8

Table 9.1 in Stewart

Inertial Oscillations

Note: V is equivalent to c from previous slides, D is equal to the diameter or twice the radius, r

18

Figure 9.1 in Stewart

Inertial currents in the North Pacific in October 1987

19

Ekman flowEkman flow Fridtjof Nansen noticed that wind tended Fridtjof Nansen noticed that wind tended

to blow ice at an angle of 20to blow ice at an angle of 20°-40° to the °-40° to the right of the wind in the Articright of the wind in the Artic

Nansen hired Ekman (Bjerknes graduate Nansen hired Ekman (Bjerknes graduate student) to study the influence of the student) to study the influence of the Earth’s rotation on wind-driven currentsEarth’s rotation on wind-driven currents

Ekman presented the results in his thesis Ekman presented the results in his thesis and later expanded the study to include the and later expanded the study to include the influence of continents and differences of influence of continents and differences of density of water (Ekman, 1905)density of water (Ekman, 1905)

20

Figure 9.2 in Stewart

Balances of forces acting on an iceberg on a rotating earth

21

So again…So again…

2

2

2

22

2

22

2

22

1

1

yx

z

vAvAfu

y

p

dt

Dv

z

uAuAfv

x

p

Dt

Du

H

zHH

zHH

22

Can ignore all terms except Can ignore all terms except Coriolis and vertical eddy Coriolis and vertical eddy

viscocityviscocity

2

2

2

22

2

22

2

22

1

1

yx

z

vAvAfu

y

p

Dt

Dv

z

uAuAfv

x

p

Dt

Du

H

zHH

zHH

23

Balance in the surface boundary layer is between Balance in the surface boundary layer is between vertical friction (as expressed by the eddy viscocity) vertical friction (as expressed by the eddy viscocity) and Coriolis – all other terms are neglectedand Coriolis – all other terms are neglected

0

0

2

2

2

2

z

vAfu

z

uAfv

z

z

24

friction

u

c.f.

45°

At the surface (z=0)

25

If we assume the wind is blowing in the x-direction only, we can show:

4sin

4cos

)(

)(

D

ze

fAv

D

ze

fAu

Dz

z

x

Dz

z

x

26

Wind StressWind Stress Frictional force acting on the surface skinFrictional force acting on the surface skin

2WCDa ρa: density of air

CD: Drag coefficient – depends on atmospheric conditions, may depend on wind speed itself

W: wind speed - usually measured at “standard anemometer height” ~ 10m above the sea surface

2

3

3

/5.01.0

/1

1021

mNt

mkg

C

a

D

27

Ekman Depth (thickness of Ekman Ekman Depth (thickness of Ekman layer)layer)

f

AD z

2

For Mid-latitudes:

Av = 10

ρ = 103

f = 10-4

Plug these into the D equation and:

4520031010

1023

43

D meters

28

Figure 9.3 in Stewart

29

U10(m/s)Latitude

15° 45°

540m 30m

10 90m 50m

20 180m 110m

Typical Ekman Depths

Table 9.3 in Stewart

30

4sin

4cos

)(

)(

z

x

z

x

fAv

fAu

At z=0

y

x

v

u45°u

)(x

Remember, we have assumed that wind stress is in the x-direction only

31

4sin

4cos

)(

)(

efA

v

efA

u

z

x

z

x

At z=-D:

e-

y,v

x,u

v

u -π/4

u

)(x

-π

32

Ekman NumberEkman Number The depth of the Ekman layer is closely related to The depth of the Ekman layer is closely related to

the depth at which frictional force is equal to the the depth at which frictional force is equal to the Coriolis force in the momentum equation Coriolis force in the momentum equation

The ratio of the forces is known as Ekman depthThe ratio of the forces is known as Ekman depth

2fD

AE zz

Solving for d:

z

z

fE

AD

33

Ekman TransportEkman Transport

fvdzM

fudzM

xyE

yxE

)(0)(

)(0)(

In general, net transport in the Ekman Layer is 90° to the right of the wind stress in Northern Hemisphere

34

Ekman PumpingEkman Pumping

)()0(

0

0 00

0

Dwwy

M

x

M

dzz

wvdz

yudz

x

dzz

w

y

v

x

u

yxEE

D DD

D

but w(0) = 0

35

By definition, the Ekman velocities approach zero at the base of the Ekman layer, and the vertical velocity at the base of the layer wE (-d) due to divergence of the Ekman flow must be zero. Therefore:

)(

)(

DwM

Dwy

M

x

M

EEH

E

EE yx

vector mass transport due to Ekman flow

horizontal divergence operator

36

If we use the Ekman mass transports in we can relate Ekman pumping to the wind stress.

yxcurl

fcurlDw

fyfxDw

xyz

zE

xyE

)(

1)(

wind stress

37

ME Ek

pile up

of water

wE wE

Hi P

Lo P Lo P

anticyclonic