Figure 21. Hodographs of the wind vector in Ekman layers in the Northern and Southern Hemispheres

Here, G is the magnitude of the geostrophic wind; and,

positive angles counterclockwise from the x axis, the

turning angle α between the surface stress and the geo-

in this frame, the y component of the geostrophic wind

is no longer zero. Thus

strophic wind is given by

(

)

V ′(z)

2 1/ 2

tan α = lim

= -sgnf

G = Ug =

+ Vg′

′

Ug2

.

(131)

(133)

z→∞ U ′(z)

Figure 21 shows plots of eq 130 in both the Northern

or

α = 45 sgnf.

and Southern Hemispheres. The obvious feature in these

(134)

hodographs is the so-called Ekman spiral. The height of

The geostrophic wind makes a 45 angle with the sur-

the Ekman layer, hE, is commonly taken as

face stress in an Ekman layer.

hE = π δ

That 45 turning angle is easy to remember; remem-

(132)

bering the direction of the turning, however, is harder

since π δ is the lowest height at which the velocity vec-

since some of us deal with ABLs in both hemispheres.

tor has the same direction as the geostrophic wind (see

Thus, I find it helpful to bring some of the sophistica-

Fig. 21). From eq 128, we can estimate that hE is on the

tion of physics to boundary-layer meteorology through

order of 300 m.

the following right-hand rule.

We see in Figure 21 that the Ekman winds turn with

Figure 22 presents another view of the Ekman solu-

increasing height. In a Cartesian coordinate system with

tion. Notice, the longitudinal velocity component is dis-

A Right-Hand Rule for the Ekman Layer

1. Point the fingers of the right hand in the direction of the surface stress.

2. Curl the fingers in the direction of f. (f is up in the Northern Hemisphere; down

in the Southern Hemisphere.)

3. The right thumb then points in the direction that the wind will turn with increas-

ing height in an Ekman layer.

π

0.8

π

5

2

2π

Southern

--

Hemisphere

3

0.6

π

--

2

2π

45

0.4

π

--

3

π

--

0.2

3

4

2

π

π --

π

-- 12

--

18

6

--

z

--

z

36

δ

V'

--

0

U'/ G

G

0.6

0.8

1.0

0.2

0.4

π

--

π

18

π

--

0.2

--

6

12

4

36

π

--

π

3

--

45

0.4

2

π

H

Northern

H Southern

--

emisphere

2

0.6

Northern

2π

--

Hemisphere

0.5

1.0

0.5

0

0.5

1.0

0

3

π

0.8

V'/G

U'/G

Figure 21. Hodographs of the wind vector in

Figure 22. Nondimensional profiles of the longitudinal and

Ekman layers in the Northern and Southern Hemi-

transverse velocity components for Ekman layers in the North-

ern and Southern Hemispheres. The x (or U′) axis is aligned

spheres. The dots with numbers nearby mark non-

dimensional heights, z/δ. The straight line at 45 in

with the surface stress.

each hemisphere shows the geostrophic wind.

25