CLIMATE/EARTH 401�Homework 4: Basic definitions, scale analysis

Richard B. Rood (Room 2525, CSRB)

rbrood@umich.edu

Cell: 301-526-8572

General Instructions

• The problems in this set are focused on basic definitions and approximations, scale analysis, 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)

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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Scale factors for “large-scale” mid-latitude

4.1) Homework question

• What conservation law is represented? In the equation below what is the physical meaning of each of the terms? If the term is a force, then state whether it is a surface force, body force, or apparent force.

4.2) Homework Problem(note 4 parts in question)

In the next 2 slides, the equations of motion are provided using both z, height, as a vertical coordinate and p, pressure, as a vertical coordinate.

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Write out the material derivative in both coordinate systems. Show explicitly whether or not z or p is held constant when partial derivatives are taken (8 points). Identify the advection term. (4 points) What is the definition of vertical velocity in the two coordinate systems? (4 points) What are the units of the vertical velocity in the two coordinate systems? (4 points)

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)

4.3) Homework Problem

Given, frictional force is represented as

Question:

1. What are the units of K_d?
2. How do you represent the scaling of the operator?
3. What is the vertical scale length at which the friction is comparable to the Coriolis force for a given K_d ?

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You may assume that the vertical scale length is much smaller than the horizontal scale length, as in text and lecture.

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Showing only the dominate term of viscosity, re-write the equation in problem 1, adding viscosity to a geostrophicly balanced flow.

4.4) Homework Problem

This can be found in the text, but it is straightforward.

Just think about it and work from definitions. Hint: you do not need to use actual form of zonal mean.

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