Ekman Flow Exercise

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Ekman Dynamics Exercise

Refer to Knauss section on Wind Stress – Ekman Transport

Typical vertical eddy viscosity, A z

= 0.01

Exercise One – Wind Stress

The wind acts on the ocean surface proportional to the square of the wind speed. In practice winds above 10 m/s (20 knots) are quite effective at moving the surface water and initiating upwelling.

𝑊𝑖𝑛𝑑 𝑆𝑡𝑟𝑒𝑠𝑠 = 𝜏 = 𝜌 𝑎

𝐶

𝐷

𝑊

2

Where 𝜌 𝑎

= 𝑑𝑒𝑛𝑠𝑖𝑡𝑦 𝑜𝑓 𝑎𝑖𝑟 = 1.3 𝑘𝑔 𝑚

−3

𝐶

𝐷

= 𝐷𝑟𝑎𝑔 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 = 1.4𝑥10

−3

𝑊 = 𝑤𝑖𝑛𝑑 𝑠𝑝𝑒𝑒𝑑 𝑖𝑛 𝑚/𝑠

Note – for more accurate calculations the drag coefficient varies with wind speed.

Make a plot of wind stress at wind speeds from 0 to 50 m/s

W=0:50;

W2=W.^2; tau=1.3*0.0014.*W2

What are the units of wind stress?

Exercise – Ekman Depth

The Ekman Depth is 𝐷

𝐸

= 𝜋√

2𝐴 𝑧

|𝑓|

so it depends on latitude in f.

Example script: lat=45

Az=0.01

De=pi*sqrt(2*Az/gsw_f(lat))

Plot Ekman depth with latitude from -90 to +90 degrees latitude.

Exercise – Maximum surface current due to wind speed

𝑉

0

= 𝑠𝑢𝑟𝑓𝑎𝑐𝑒 𝑐𝑢𝑟𝑟𝑒𝑛𝑡 = 𝜏 𝜌

√2𝜋

𝐷 𝑒

|𝑓|

V o

is the current at z=0, the surface. Calculate V o

for wind speeds of 30 m/s. Assume A is 0.01, f is calculated for 40N.

*******************STOP HERE*******************

Equations of motion with wind stress

If there are no horizontal pressure gradients, no acceleration and no internal fiction the Coriolis force is balance by friction represented by wind stress.

1 𝜌

𝜕𝜏 𝑥

=-fv

𝜕𝑧

1 𝜌

𝜕𝜏 𝑦

𝜕𝑧

=-fu

Exercise – Ekman Spiral

Solve for u and v with z going from 1 to 50 m . Calculate f for 40N (not the 45 they use)

This code from

Laure Zanna found Search https://courses.seas.harvard.edu/climate/eli/Courses/EPS131/2005fall_a/sectio ns/matlab-session.m clear all;

% grid in x y z xx=zeros(1,11); yy=zeros(1,11); % define zeros for the x-y direction zz=0:-4:-60; %define the depth

[x,y,z]=meshgrid(xx,yy,zz); clear xx yy zz

%parameters f=2*(7.3*10^(-5))*sin(45*pi/180); % coriolis parameter

A_v=10^-2; % friction coefficient delta=(f/(2*A_v))^(1/2); % 1/Ekman depth

% velocity field u=exp(delta*z).*cos(delta*z); v=exp(delta*z).*sin(delta*z); w=0*z; % define w=0 just for the plot figure; axes('position', [0.3,0.3,0.5,0.5]); %[left, bottom, width, height])

% use quiver3 to draw the Ekman spiral quiver3(x,y,z,u,v,w); xlabel('x','fontsize',13); ylabel('y','fontsize',13); zlabel('z','fontsize',13); view(-35,45) % set the angle of view (azimuth, vertical)

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