Name:______________________________________________
Date:__________________
Worked with:_______________________________________________________________________
SO513 – Atmospheric and Oceanic Processes (Honors)
Lab 3: The Atmospheric Boundary Layer and Ekman Solutions (100 pts)
Brief Review of Lesson 5 Notes:
Consider an ocean with no interior flow and a horizontal wind field imposing a wind stress along
the surface ocean (𝜏𝑥 , 𝜏𝑦 ). Assuming a steady state, homogenous fluid with no pressure
gradient, the following set of equations represents the flow field in the surface Ekman layer:
𝜕2𝑢
−𝑓𝑣 = 𝜈𝑣 2
𝜕𝑧
𝜕2𝑣
𝑓𝑢 = 𝜈𝑣 2
𝜕𝑧
(Eq. 1)
Unless otherwise specified, this lab follows the notation used in the Lesson 5 notes. The upper
boundary forcing at z = 0 is from the wind stress:
𝜌𝑤 𝜈𝑣
𝜕𝑣
= 𝜏𝑦
𝜕𝑧
𝜌𝑤 𝜈𝑣
𝜕𝑢
= 𝜏𝑥
𝜕𝑧
(Eq. 2)
By substituting the upper boundary condition from Equation (2) into Equation (1) the surface
ocean Ekman Layer equations become:
−𝜌𝑤 𝑓𝑣 =
𝜌𝑤 𝑓𝑢 =
𝜕𝜏𝑥
𝜕𝑧
𝜕𝜏𝑦
𝜕𝑧
(Eq. 3)
With prescribed upper (Equation 2) and lower (u, v = 0 and z = -∞) boundary conditions, the
solution to Equation (3) in the Northern Hemisphere is:
𝑢=
𝛿𝐸 √2 𝛿 𝑧
𝜋
𝜋
𝑒 𝐸 [𝜏𝑥 cos (𝛿𝐸 𝑧 − ) − 𝜏𝑦 sin (𝛿𝐸 𝑧 − )]
𝜌𝑤 𝑓
4
4
𝑣=
𝛿𝐸 √2 𝛿 𝑧
𝜋
𝜋
𝑒 𝐸 [𝜏𝑥 sin (𝛿𝐸 𝑧 − ) + 𝜏𝑦 cos (𝛿𝐸 𝑧 − )]
𝜌𝑤 𝑓
4
4
(Eq. 4)
Part 1:
Show that for purely zonal winds, the Ekman in Equation (4) solution above matches the solution
in Equation (9) of the Lesson 5 class notes.
Part 2:
In the turbulent atmospheric boundary layer, the surface stress in the x and y directions are:
𝜏𝑥𝑧 = −𝜌𝑎 ̅̅̅̅̅̅̅̅̅
𝑈′𝑥 𝑈′𝑧
𝜏𝑦𝑧 = −𝜌𝑎 ̅̅̅̅̅̅̅̅̅
𝑈′𝑦 𝑈′𝑧
(Eq. 5)
where 𝜌𝑎 is the density of air and capital U and V denote atmospheric winds while lowercase u
and v denote ocean currents. Note that going forward, we will drop the tensor notation and
represent the Reynolds stresses as follows: 𝜏𝑥 = 𝜏𝑥𝑧 and 𝜏𝑦 = 𝜏𝑦𝑧 . However, obtaining
reliable measurement of turbulent fluxes in the marine atmospheric boundary layer (often on a
moving ship in heavy seas) has proven difficult. Instead, scientists use the bulk formula to relate
the momentum flux at the ocean surface to more easily measured variables via the dimensionless
drag coefficient, 𝐶𝑑 . The bulk formulae for the zonal and meridional wind components are:
𝜏𝑥 = 𝜌𝑎 𝑈∗ |𝑈∗ | = 𝜌𝑎 𝐶𝑑𝑥 (𝑈 − 𝑈𝑜 )|𝑈 − 𝑈𝑜 |
𝜏𝑥 = 𝜌𝑎 𝑉∗ |𝑉∗ | = 𝜌𝑎 𝐶𝑑𝑦 (𝑉 − 𝑉𝑜 )|𝑉 − 𝑉𝑜 |
(Eq. 6)
where 𝑈 and 𝑉 are the wind speeds at height (often taken at 10 meters above the surface) and
𝑈𝑜 and 𝑉𝑜 are the surface wind speeds, assumed to be zero for our application. The bars | |
denote absolute values. The flux scales 𝑈∗ and 𝑉∗ in Equation (6) are commonly referred to as
the “shear” or “friction” velocities and, assuming the processes are isotropic, they are related to
the covariance of the turbulent fluctuations in the Reynolds stress equation by:
̅̅̅̅̅̅̅̅̅
𝑈∗ |𝑈∗ | = −𝑈′
𝑥 𝑈′𝑧
′ 𝑈′
̅̅̅̅̅̅̅̅̅
𝑉∗ |𝑉∗ | = −𝑈
𝑦
𝑧
(Eq. 7)
You will find the functional form for the drag coefficient in the following paper. We will assume
a neutrally buoyant atmosphere. Hence use the neutral drag coefficient.
Kevin E. Trenberth, William G. Large, and Jerry G. Olson, 1989: The Effective
Drag Coefficient for Evaluating Wind Stress over the Oceans. J. Climate, 2,
1507–1516.
We will make two addition modifications to the neutral drag coefficient parameterization given
by Trenberth, Large, and Olson, (1989):

If the wind speed is zero, set the drag coefficient to zero (for simplicity).

Recent scientific studies have shown that at moderate wind speeds, the amplitude and
period of the wind driven, surface gravity waves augments the atmospheric boundary
layer flow by physically decoupling it from the wavy sea surface. This phenomenon
reduces the drag imposed by the sea surface on atmospheric boundary layer flow field by
altering the roughness length. Hence, for wind speeds greater than or equal to 20 m/s, the
drag coefficient saturates and is approximately constant, 𝐶𝑑 = 1.8x10-3.
Using your notes and the information provided here, complete the following tasks:
1. Make an array of wind speeds from 0.5 to 30 m/s. Using this array, calculate and plot the
drag coefficient and wind stress. You will find an appropriate estimated value for the
density of air in the text of Trenberth, Large, and Olson, (1989).
2. Based on your understanding of Trenberth, Large, and Olson, (1989), specifically the airsea temperature difference discussion surrounding Figure (1) in that paper, in which
months (generally) would you expect the drag coefficient off the Mid Atlantic coast to be
greater than and less than the neutral drag coefficient (i.e. the solid ∆𝑇 = 0 line in Figure
1)?
Using the code from Question 1, calculate the u and v Ekman spiral components for U =
10.0 m/s, V = 5.0 m/s and a latitude of 40 degrees North. Remember z is positive
upward. Plot the 3D Ekman spiral using the plot3 MATLAB function. Your vertical
scale should range z = 0 m to z = -500 m using a vertical grid spacing of 1 m. Rotate the
3D Ekman spiral to only show the “top” view or the hodograph view. Type help
plot3 into MATLAB for more information about that function. Does the spiral rotate
the correct direction for the Northern Hemisphere?
Furthermore, plot the vertical u and v component profiles with respect to 𝛿𝐸 𝑧. At what
depth to u and v go to zero? Constrain the vertical axis in the plot to this depth.
Part 3:
In equation (3), the derivate of the wind stress is only with respect to the vertical, z-direction.
Hence, the partial derivatives are effectively full derivatives allowing Equation (3) to be written:
−𝜌𝑤 𝑓𝑣 𝑑𝑧 = 𝑑𝜏𝑥
𝜌𝑤 𝑓𝑢 𝑑𝑧 = 𝑑𝜏𝑦
(Eq. 7)
In Equation (7), 𝜌𝑤 𝑓𝑢 𝑑𝑧 is the mass flowing per second in the x-direction through an area
defined by the vertical depth 𝑑𝑧 and one meter width area in the y-direction (similarly for mass
flowing in the y-direction). Hence, total mass following in the x- and y-directions (respectively)
from the level z to the surface is:
0
0
𝑓𝑀𝑥 = ∫ 𝜌𝑤 𝑓𝑢 𝑑𝑧 = ∫ 𝑑𝜏𝑦
𝑧
𝑧
0
0
𝑓𝑀𝑦 = − ∫ 𝜌𝑤 𝑓𝑣 𝑑𝑧 = − ∫ 𝑑𝜏𝑥
𝑧
𝑧
(Eq. 8)
In your Lesson 5 notes, you learned that the analytical solutions for Equation (8) are as follows,
where the mass transport is perpendicular to the surface wind stress:
𝑀𝑥 =
𝜏𝑦
𝑓
𝑀𝑦 = −
𝜏𝑥
𝑓
(Eq. 9)
Using Equation (9), calculate analytical the horizontal mass transport in the x and y directions for
U = 10.0 m/s, V = 5.0 m/s, and a latitude of 40 degrees North. Compare the analytical solution
with a numerical solution calculated using only the u and v Ekman spiral solutions in Equation
(6) with the trapz MATLAB numerical integration function. First, calculate the u and v
components of the Ekman spiral and numerically integrated these solutions using a vertical grid
spacing of 20m. Afterward, complete the numerical integration with 1m grid spacing. Did the
grid spacing improve numerical solution with respect to the analytical solution? Why (Hint: you
will ned to know how trapz work mathematically)?
Part 4:
1. Plot the zonal and meridional wind speeds from the Reanalysis data file which can be
access using the following MATLAB commands:
Ydrivepath = 'Y:\Mids Share\Oceanography\Data\Models\REANALYSIS\';
filename = 'NPac_AvgWinds_01_25_2015_to_01_30_2015.mat';
load(strcat(Ydrivepath,filename))
The wind is averaged over January 25-30, 2015. The gridded, zonal and meridional
components of the wind are found in the variables Uwind and Vwind, respectively.
Likewise, the gridded longitude and latitudes are found in windxgrid and
windygrid, respectively. Your plots should look like this:
2. Using the analytical solutions for the horizontal mass transport in Equation (9), plot the
resulting horizontal mass transport components from the Reanalysis wind field. Along
the Pacific Northwest coast, do you expect upwelling or downwelling? Why? Recalling
the physical-biological interactions you studied in SO272, what impact could this have on
local fisheries?