Name:____________________________
SO335 Quantitative Methods: Assignment #3 (50 pts)
1. Gradient of a scalar.
a. (5 pts) For the following temperature field, T, calculate the temperature gradient in terms of x
and y:
T   3x 2 y   y
where  and  are dimensional constants that cause T to have units of K. Show all work.
b. (3 pts) Evaluate your answer in part (a) at the point (-2, 2).
For the following diagram showing isobars evenly spaced at 5 km intervals
N
p = 988 mb
p = 992 mb
p = 996 mb
p = 1000 mb
A
c. (3 pts) Draw the pressure gradient vector at point A.
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E
d. (5 pts) If the pressure at point A is 994 mb, and point A is located exactly between the two
isobars to its north and south, calculate the vector p at point A. Show all work.
2. Divergence of a vector.
a. (5 pts) For the following vector u
u   3x 2 yiˆ   yjˆ ,
where  and  are dimensional constants that cause u to have units of m s-1, find the
divergence. Show all work.
b. (2 pts) Evaluate your answer in part (a) at the point (-2, 2).
c. (3 pts) If the magnitudes of  and  are 1, is there divergence or convergence at (-2,2)?
Explain your answer.
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For the following diagram showing three evenly-spaced buoys 100 km apart,
N
A
B
C
E
d. (5 pts) If the current at point A is from the west at 1 m s-1 and the current at point C is from the
west at 2 m s-1, calculate  u at point B.
e. (2 pts) Is there divergence or convergence at point B? Explain your answer.
3. Vorticity of a vector.
a. (5 pts) For the following velocity vector u
u   3x 2 yiˆ   yjˆ ,
where  and  are dimensional constants that cause u to have units of m s-1, find the
vorticity. Show all work.
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b. (5 pts) For the following flow field, where the distance from buoys A and C is 100 km, the
distance from buoys B and D is also 100 km, and the fluid current speed at each buoy is 1 m s-1,
B
E
A
C
D
calculate k̂     u  at point E. Show all work.
c. (2 pts) What does the sign of the relative vorticity you calculated in part (b) tell you about the
spin of the fluid?
d. (5 pts) What if, instead of buoys in an ocean, the measurements in the diagram in part (b) were
mobile mesonets sampling the outer winds of a tornado. Re-calculate relative vorticity at point E
if the distance between A and C, and B and D, was 100 m and the wind speed a fast 50 m s-1 at
each point. Interpret your answer in terms of the difference in magnitude of vorticity between
parts (b) and (d).
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