SO513
Homework 4 (40 pts)
1. (5 pts) - In the equation, u F  
Name ________________________


e E z cos   E z   , identify all of the variables and
f v
4

1
give simple meanings for each.
2. (a) (10 pts) Show how to arrive at the following equations for flow in the atmospheric
Ekman Layer (boundary layer),
u  ug 1  e z cos   z  
v  ug e z sin   z 
from the general Ekman solution u  vi  Ae
conditions
1i  z
,
 Be1i  z  ug , using the following boundary
u  u g , v  vg at z  
u  v  0 at z  0
.
and the general Euler formula e  i  cos   i sin  .
Hint: Apply the far ( z   ) boundary conditions first to solve for coefficient A, then apply the surface boundary
conditions to solve for B, and finally apply the Euler formula remembering to set the respective real and imaginary
parts on the LHS and RHS equal to each other.
Figure 1: Ekman spiral for the atmospheric boundary layer. Note the values along the x and
y axes are ratios. Thus, at the point (1,0), the u component of the wind is exactly equal to its
geostropic value, and the v component of the wind is zero.
(b) (5 pts) Find the fractional values of u / u g and v / u g using the equations in part 5a for the
following heights, and plot the heights on Figure 1 above. Show your work below the table.
z
u / ug
v / ug

6
z

3
z

2
z
2
3
z


z
3
2
z
2

(c) (5 pts) Calculate the height (z) where the mean wind is nearly zonal and geostrophic (show
your work). This height is designated as the top of the Ekman Layer, and the layer is commonly
called the “boundary layer”, as it is the boundary between the free troposphere above and the
surface below. It is often marked by the presence of a strong temperature inversion, and if
substantial scalar tracers are present at the surface (including water vapor, pollution, and even
insects), these elements are often thoroughly mixed through the boundary layer.
1
 f 2
(d) (5 pts) The parameter  is defined as   
 , and K m is the vertical eddy viscosity
 2Km 
which is a measure of mixing in the boundary layer. It is itself defined as a ratio between
u
vertical turbulent flux of horizontal momentum ( uw ) and vertical shear of the mean wind (
):
z
uw
Km 
. Eddy diffusivities have been observed to range from 1  102 m2 s 1 . Calculate the
u / z
range of boundary layer depths (using your solution from part c) that correspond to this range in
eddy diffusivities. Assume a typical mid-latitude value for f. ( f  104 s 1 )
(e) (5 pts) What does this range (in boundary layer depth) imply about the role of turbulence and
vertical shear in determining the depth of the atmospheric boundary layer?
(f) (2 pts) At night, turbulent momentum transport typically decreases (due to loss of convective
heat transfer) but vertical shear increases (especially as near-surface winds go calm while flow
just above the surface remains relatively fast). What then can you infer about the typical height
of the atmospheric boundary layer at night?
(g) (3 pts) Finally, if the surface is characterized by many pollution sources, use K-theory (the
shorthand name given to this work by Prandtl and Ekman) to explain why pollution
concentrations are typically highest at sunrise and lowest in the late afternoon.