Question

Problem Statement

Suppose you are a meteorologist who is writing a scientific article describing
how atmospheric flow is influenced by pressure. The article is for the general
scientific community, so your readers will be familiar with calculus but not at-
mospheric dynamics and may be rusty on vector calculus. Consider two cases
of pressure fields for all calculations.
Case 1 : p(x, y) = 990 + 9( x
L )2
Case 2 : p(x, y) = 1020 + 11 sin( πx
L ) sin( πy
L ),
where:
• p is the pressure, in hectopascals (hPa), where 1 hPa = 100 Pa
• L = 1000km is a constant
• x and y are in km
The domain of interest 0 < x < 2L and 0 < y < L, which is about the
size of a pressure system over the continental US. Weather maps often show
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pressure in millibars (mb), where 1 mb = 1 hPa = 100 Pa. Surface pressure
can vary between 990 mb and 1035 mb.
Fluid in Response to a Pressure Field
• 1. Sketch contour plots (using the learning objects of Module 5 online)
of p(x,y) by hand for both cases. You should be able to do this by think-
ing about what each of the factors does independently. Do not turn in
sketches with your project. The purpose is to help you explain the behav-
ior of these functions.
• 2. Explain Equation (2) in words. That is, how does fluid tend to acceler-
ate when it is exposed to differences in pressure? Given this explanation,
sketch by hand what the acceleration vector fields will look like for cases
1 and 2.
• 3. Calculate the acceleration vector fields using (2). Your solution should
be in m/s2. Do you think that observed winds in the atmosphere are
caused by acceleration due to pressure differences? Explain.
• 4. Numerically graph contours of p and the vector field −→a . Compare
them to your sketches. You should include the computer-generated plots
in your report, but not the sketches.
• 5. Calculate the curl of −→a for both cases
• 6. Is −→a a conservative vector field?
• 7. What is the potential function of −→a ?
• 8. The real fluid cannot accelerate indefinitely like in case 2. Explain why.
The Upper Atmosphere: Fluid in Response to Pressure Fields and Rotation
• 1. Rewrite −→
k × −→u in terms of u1 and u2. You may write −→u as u1
−→
i +
u2
−→
j + 0−→
k . Rewrite (4) as a system of two equations, one from the com-
ponent −→
i and one from the component −→
j . These are the equations of
geostrophic balance.
• 2. Calculate f, the Coriolis parameter, for Mesa’s latitude in radians/s.
• 3. Using parts 1 and 2 above calculate the vector field −→u for case 1 and
case 2. If units of meters and Pascals are used, your resulting velocity
will be in meters per second.
• 4. Numerically graph contours of p and the vector field−→u . Explain how
the fluid is moving with respect to the pressure.
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• 5. The vorticity, which is the −→
k -th component of the curl of the velocity
vector field, is a useful diagnostic tool in meteorology. Find the vorticity
for your two cases, ω = ∂u2
∂x − ∂u1
∂y . Plot contour and surface plots of the
vorticity and discuss how it relates to the pressure and velocity fields for
both cases.
• 6. Are the velocity vector fields for case 1 and case 2 conservative?
• 7. What do you expect the divergence to be, based on the vector field ?
Calculate the divergence. Did this confirm your expectation?
The Atmosphere Near the Surface: Fluid in Response to Pressure Fields, Rota-
tion, and Friction
Consider a different example of a velocity vector field.
Case 3 : −→u = [40 sin( πx
L ) cos( πy
L ) + 40 cos( πx
L ) sin( πy
L )]−→
i
+[−40 cos( πx
L ) sin( πy
L ) + 40 sin( πx
L ) cos( πy
L )]−→
j .
where: L = 1000km is a constant and x and y are in km.
Again the domain of interest 0 < x < 2L and 0 < y < L.
• 1. Graph the case 3 vector field numerically, and describe how the air is
moving.
• 2. Label where you would expect high and low pressure centers to occur
in your plot of case 3.
• 3. Calculate and plot the divergence of case 3, ∇•−→u . Flows where
∇•−→u > 0 are called divergent, while flows where ∇•−→u < 0 are con-
vergent. Explain how the air moves in convergent and divergent flows
and what type of pressure system each is associated with.

Answer 1

The student's question pertains to applying vector calculus to analyze atmospheric flow in response to pressure fields, involving calculations and graphical representations of fluid acceleration, the Coriolis effect, and divergence in meteorology.

Step-by-step explanation:

The student's question relates to vector calculus and its application in meteorology to describe atmospheric flow influenced by pressure fields.

The pressure, measured in hectopascals, varies as specified in two different cases. In case 1, pressure is a function of position squared, whereas in case 2, it is a sinusoidal function of both x and y coordinates.

Fluids respond to pressure gradients, and the atmospheric motion, or wind, is a result of such gradients. Fluid acceleration due to pressure differences can be calculated, and this will show whether atmospheric winds can result from such an acceleration.

Acceleration vector fields for both cases are calculated using the provided equations and the contours of the pressure fields and vector fields are graphed for comparison. The curl of the acceleration vector fields is also calculated to determine if the fields are conservative and to find potential functions.

Lastly, the impact of Earth's rotation (Coriolis effect) and friction on fluid motion at the surface level is considered. An additional vector field is introduced for this purpose, depicting fluid motion over the US and the corresponding divergence to indicate regions of low and high pressure.