Assignment #3 (50 pts)
Name: __________________________
1. (10 pts) For many atmospheric flows, rotation of the earth is important. The momentum
equation for inviscid flow in a frame of reference rotating at a constant rate  is:
u
1
  u   u   p  gkˆ  2 u   
t

For steady, two-dimensional horizontal flow with u =(u,v,0) and     z  , show that the
streamlines of the flow are parallel to constant pressure lines when the fluid parcel acceleration is
dominated by the Coriolis acceleration. [This seemingly strange result governs just about all
large-scale weather phenomena like hurricanes and other storms, and it allows weather forecasts
to be made based on surface pressure measurements alone!]
Hints:
1. When Coriolis accelerations dominate, u u 
u
 2  u  ,  
 t 
 2  u  , and
1

  p  .


2. Let y=Y(x) be the equation of a streamline contour. Then dY / dx  v / u is the streamline
slope.
3. Write out all 3 component equations of the momentum equation, and then build the ratio
v / u.
4. The pressure increment along a streamline is dp   p / x  dx   p / y  dy . Goal is to
show that dp = 0.
  x 
2. (5 pts) Explain effective gravity. Draw a force-body diagram for the resultant “equipotential
surface” to support your explanation.
3. (5 pts) A 2-D stream function is a scalar quantity that can be related to the velocity field by


and v 
. Given this relationship, show that the vertical vorticity is related to the
u
x
y
stream function as follows:
z 
v u
    2
x y
4. (8 pts) The results of problem (3) are useful for analysis of the Navier Stokes equations.
(a) Show how the full 2D Navier-Stokes equations can be reduced to the following linear forms:
u
1 p

  2 u
t
o x
v
1 p

  2 v
t
o y
(b) In (a) above,  is the kinematic viscosity and can be considered constant and density only


depends on the vertical,  o   o z  . Take
of equation (2) and subtract
of equation (1)
x
y
from it to show that
2


  2   2  or
    2 
t
t
5. (5 pts) Since
   o e i kx lyt 

  2 is a linear equation, we may assume a solution of the form:
t
Where  , k and l are constants and i   1 .

  2 to obtain an algebraic
t
relationship between  , k and l and the kinematic viscosity 
Substitute the linear solution,    o e i kx lyt  into
6. (5 pts) Substitute your solution for k, l  into  t    o e  it and explain what you observe
as lim  t  . Explain how this result shows you that the viscous term of the Navier-Stokes
t 
equation is a dissipative term.
7. (7 pts) Show how to derive the Reynolds-averaged, turbulent Navier-Stokes equations when
starting with the Navier-Stokes equations in the Boussinesq, rotating framework:
Start here:
Du
1 P
 fv  
  2 u
Dt
 o x
Dv
1 P
 fu  
  2 v
Dt
 o y
Dw
1 P 


g   2 w
Dt
 o z  o
End here:
Du
1 p



 fv
  2 u  u ' u '  u ' v'  u ' w'
Dt
 o x
x
y
z
Dv
1 p



 fu 
  2 v  v' u '  v' v'  v' w'
Dt
 o y
x
y
z
Dw
1 p 





g   2 w  w' u '  w' v'  w' w'
Dt
 o z  o
x
y
z