CLIMATE/EARTH 401�Homework 3: Basic definitions, independent variables, and mathematical 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, 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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3.1 Homework problem

The material derivative is written below in two different coordinate systems. One uses pressure as a vertical coordinate; the other uses height.

What are the dependent and independent variables? When taking partial derivatives in one variable, what other parameters are held constant? Write down the definition of w and ω in terms of differentials of the independent variables. What are the units of ω?

3.2) Homework problem

1. Derivation. Starting from the �material (Lagrangian) form of �the continuity equation

Derive the Eulerian form. You �will need to use the chain rule �and the definition of the �material derivative.

2. What happens if there is no change in density following the motion?

3.3) Homework problem

The horizontal momentum equations in pressure coordinates can be written as:

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k is a constant which represents frictional dissipation.

Assume that the lines of constant geopotential are oriented in the east-west direction and that the flow is balanced. Derive equations that express the velocity components, u and v, in terms of pressure gradient, k and f. Draw a picture of this flow field, showing, quantitatively, the relation of the velocity field to the lines of geopotential.

3.4 Homework Problem

• In this figure, the large black arrow is the jet stream. If the pressure gradient force is balanced by the Coriolis force, mark the high and low pressure sides of the jet. Briefly, what is your reasoning?