METR February 2002

Review State variables: p, ρ, T Pressure Temperature Equation of state: p = NkT/V = ρ R d T Virtual temperature T v = T ( r)

Hydrostatic balance Consider air between two horizontal surfaces of area A located at level z and level z + Δz. Then the downward force on the upper surface is -p(z + Δz)A and the upward force on the lower surface is p(z)A. An additional force acting on the air between the two surfaces is the downward weight force due to gravity -mg = -ρΔzAg z A p(z + Δz) p(z)

Hydrostatic balance Assuming the air is in equilibrium and experiencing no vertical acceleration, then the net force must be zero, so – p(z + Δz)A + p(z)A – ρΔzAg = 0 p(z + Δz) – p(z) = – ρgΔz (p(z + Δz) – p(z))/ Δz = – ρg Now, taking the limit Δz →0 gives Pressure in the atmosphere increases towards the surface due to the weight of the air above.

Hydrostatic balance Hydrostatic balance has the vertical pressure gradient of the air balance exactly by the weight due to gravity. It does not preclude vertical motion, but it does preclude vertical acceleration. Weather systems with strong vertical motion, such as thunderstorms, mountain waves or tornadoes, usually have strong non-hydrostatic vertical accelerations and hydrostatic balance does not hold. Synoptic-scale weather systems usually have weak vertical acceleration and hydrostatic balance holds quite well.

Hydrostatic balance Integrate the hydrostatic equation in the vertical For an isothermal atmosphere, T constant,

Hydrostatic balance Pressure decreases approximately exponentially with height, decreasing faster with height near the ground than higher up. For an isothermal atmosphere, the scale height H is the height over which the pressure decreases to 1/e of its original value. (e=2.72, 1/e=0.37) For T=288K, H~7.3 km

Zeroth Law Zeroth Law of Thermodynamics Two Systems individually in thermal equilibrium with a third system are in thermal equilibrium with each other. System - A collection of objects upon which attention is focused. Surroundings - Everything else

Zeroth Law Thermal Equilibrium occurs when there the net heat flow between 2 systems = 0. Heat flow ==> 0 when objects are at the same temperature. Consider the three boxes at temperatures T 1, T 2, and T 3 : 123 If T 1 = T 2 and T 2 = T 3, then boxes 1,2, and 3 are in thermal equilibrium satisfying the Zeroth law.

Conservation of Energy First Law of Thermodynamics – Conservation of Energy Energy can be exchanged between a system and its surroundings, but the total energy of the system and the surroundings is constant. “You can’t get something for nothing” (you can’t get more energy out of a system than you put into it) Experiments by James Joule (in the mid- to late-1800s) showed that heat and work are both forms of energy that can be transferred between a system and the surroundings

Conservation of Energy Energy Energy can neither be created nor destroyed. Energy can be converted among various forms, such as: Potential energy (e.g. gravity, PE = mgh) Kinetic energy (KE = ½ m v 2 ) Mass (E = mc 2 ) We can also define: Thermal energy – total of the kinetic energy of all molecules in a substance Internal energy – sum of the kinetic and potential energy of molecules and atoms from which a substance is made

Conservation of Energy Heat Heat is a measure of energy transfer by means of temperature differences. Heat, Q, was originally defined quantitatively as: “1 kilocalorie of heat = the amount of energy required to raise the temperature of 1 kilogram of water from 14.5 to 15.5°C” Substances differ considerably from one another in the quantity of heat needed to produce a given rise of temperature in a given mass.

Conservation of Energy Heat Capacity and Specific Heat Heat capacity, C = ΔQ/ΔT Heat capacity simply relates the amount of heat added to obtain a rise in the temperature of some unspecified amount in a substance. Specific heat, c = heat capacity = ΔQ mass m ΔT Specific heat of liquid water at 0°C = 4218 J K -1 kg -1 Specific heat of dry soil ~ J K -1 kg -1

Conservation of Energy Specific heat of air In order to uniquely define the specific heat of a gas, we must specify the conditions under which the heat ΔQ is added to the substance, e.g. constant pressure or constant volume For dry air, c p = 1004 J K -1 kg -1 and c v = 717 J K -1 kg -1

Conservation of Energy Latent heat is the energy given up or taken up by a system to cause a change of phase, such as water vapor condensing into liquid water. It is a key to understanding weather because latent heat is a major source of energy for thunderstorms and hurricanes. For evaporation, energy is transferred to liquid water molecules (from the soil or from solar radiation) so that they can speed up and change to water vapor. Since energy can’t be created, the substance that loses energy cools down. For condensation, energy is lost from the vapor molecules to the surrounding air as they condense to liquid, heating up the air. Latent heat of condensation, L c = 2,500 kJ kg -1