CLIMATE/EARTH 401�Homework 7: Waves and Time constants

Richard B. Rood (Room 2525, CSRB)

rbrood@umich.edu

Cell: 301-526-8572

General Instructions

• The problems in this set are focused on vorticity as well as mathematical analysis.
• Please refer to earlier lectures and texts if you need more information.
• If these problems are taking too long, more than an hour, seek additional information (text, classmate, online, teacher)

Equations of motion�(z, height, as vertical coordinate)

tangential coordinate system on Earth’s surface

(x, y, z) = (+ east, + north, + local vertical)

Equations of motion�(p, pressure, as a vertical coordinate)

tangential coordinate system on Earth’s surface

(x, y, p) = (+ east, + north, pressure is vertical coordinate)

Vorticity equation

Vorticity equation in height coordinates (z) :

Variables and constants

• p is pressure
• ρ is density
• T is temperature
• α is specific volume
• u is velocity vector (ui, vj, wk)
• g is gravity force (assume constant)

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• R is the gas constant for dry air
• c_v is specific heat at constant volume
• c_p is specific heat at constant pressure = (c_v + R)
• a is radius of Earth
• Ω is angular velocity of Earth’s rotation
• ν is kinematic viscosity coefficient

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7.1) Homework question (20)

• Refer to Homework Problem 2.2 for reference

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Following an air parcel the time rate of change of temperature can be expressed as the difference between heating and cooling. If the heating is a constant, H, and the cooling is proportional to the temperature, then the conservation equation for temperature can be written as:

If T is in equilibrium, it does not change with time.

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Assume that the parcel is in equilibrium and the heating is turned off. It becomes zero. The Sun is extinguished!

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a. Calculate the time rate of change of the cooling of the parcel.

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b. How long does it take for the parcel to cool to ½ of its initial temperature.

7.2) Homework question (20)

In the derivation of an equation for the time rate of geopotential the following steps were used.

1. Start with the QG �vorticity equation

2. Use the definition of geostrophic wind.

3. Plug this in, and we immediately have an equation for the time rate of change of the geopotential height.

7.2) Homework question

• Prove Step 2 on the previous page.
• Assume that the divergence of the ageostrophic wind is zero (0) and that the meridional component of the geostrophic wind is zero (0). What is the local time rate of change of geopotential? You may assume a “wave-like” solution (or not).

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7.3 ) Homework question (40)

• Please refer to homework 4.4.
• Given the following slide from lecture 15.4.0, using the assumptions given in item 2, derive the wave equation given in item 3.

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Form a Perturbation Equation

1. Barotropic height tendency equation

2. Divide wind into mean and perturbation, assume no mean north/south wind

3. Plug perturbations in