In class, we derived the time rate of change of a variable in two ways: Eulerian and Lagrangian.
Write the time rate of change of temperature in both coordinate systems
Rewrite the two above equations for the case in which there is no vertical velocity.
Using the material derivative, solve the following problem:
The temperature does not vary in the west-east direction. The temperature at a point 50 km south of a station is 3 oC warmer than at the station. If the wind is blowing from the southwest at 20 m s-1 and the air everywhere is being heated by radiation at a rate of 1 oC h-1, what is the local temperature change at the station?
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Answer
Lagrangian
Eulerian
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Answer
Use the material derivative of temperature to solve this
Start by expanding the material derivative—we want the local change in temperature at the station
From the point ystation, you
are looking in the – (negative) y direction
Remember, you are looking at what is coming to you, not where you are going.
Or, which way is the T gradient pointing?
T+3
T
N
20 m/s
14.14 m/s
50 km
ystation
y
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Answer
Expand material derivative
Note that we only have a change in y
Solve for local change
Plug in the numbers
v = 14.14 m/s = 50.9 km/h
dy= 50 km, dT=3K dy is negative
DT/Dt=+1 deg C/hour
+
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Physically
At the station, you should be seeing warm air advection from the south.
The temperature you measure, will be the temperature from the south, + the heating from radiation.