CLIMATE/EARTH 401�Homework 1: Mathematical Manipulation

Richard B. Rood (Room 2525, CSRB)

rbrood@umich.edu

Cell: 301-526-8572

General Instructions

• The problems in this set are focused on mathematical manipulation and practice with definitions and skills you should have seen in earlier classes.
• 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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1.1) Homework problem

• Below the transformation of some scalar Φ from the vertical coordinate z to the vertical coordinate p, pressure, is expressed symbolically. Write expressions for the horizontal (x and y) derivatives of Φ and the vertical derivative (z) of Φ in the pressure coordinate system.

Φ(x, y, z, t) = Φ(x, y, p(x, y, z, t), t)

What happens if ∂p/∂z = 0?

1.2) Homework problem

In the absence of pressure forces and viscosity the equations of motion for purely horizontal motion are:

Calculate an equation for d(u²+v²)/dt.

1.3) Homework problem

The geostrophic wind is defined as. What is the divergence of geostrophic wind?

1.4) Homework problem

• In the equation below F is a force. K is a constant of proportionality. m is mass and r is distance. What are the units of K?

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• This is an equation for viscous force per unit mass. What are the units of μ?

F = Km²/r²