Important variables in meteorology Equation of state for

Important variables in meteorology Equation of state for dry and moist air SO 254 Introduction to Meteorology

Variables • Meteorology is, at its core, calculus and physics applied to air • That means that underlying every aspect of meteorology is math • That math has variables and units that accompany them • To set yourself up for success, it’s important to know the key variables • Table (next slide) summarizes some of the most-used variables in meteorology

Variable Letter or abbreviation Typical units Notes Temperature T (usually capitalized) Celsius, Kelvin, or Fahrenheit SO 345 (Atmospheric Thermodynamics) is an entire course devoted to temperatures! Pressure p (usually lower-case) Pascal (Pa) hectopascal (h. Pa) millibar (mb) kilopascal (k. Pa) One mb is equal to 1 h. Pa, and both are equal to 100 Pa Density ρ (Greek letter rho, lower-case) kg m-3 Mass divided by volume Mass m (typically lower-case) kg Distance x, y, z meters (m) Time t (usually lower-case) seconds (s) hours (h, sometimes hr) Water vapor w (mixing ratio) q (specific humidity) RH (relative humidity) Td (dew point temperature) w and q have units of kg kg-1. RH is unitless. Td has units of °C or °F Velocity u (east-west velocity component) v (north-south velocity component) w (vertical velocity component) u, v, and w all have units of m s-1 There are other measures of water vapor besides these. We will see more of them later in the course

Derivation: from an ideal gas to the equation of state for dry air • One of the most important relationships in all of meteorology, and one of the primitive equations that form the basis of all weather prediction models, is the equation of state • It is a very simple equation, p=ρRT, where R is the dry gas constant (R=287. 05 J kg-1 K-1), and J represents Joules • The equation of state for moist air is very similar, and takes into account water vapor in the atmosphere: p=ρRTv, where Tv is the virtual temperature: Tv = T + w/6, where Tv is the virtual air temperature in C, T is the air temperature in °C, and w is the mixing ratio of air, in g kg-1 • The equation of state for dry air comes from one of the basic equations in physics: the ideal gas equation

Derive an equation of state for dry air Classical physics has derived a simple equation governing behavior of ideal gases (what is an “ideal gas”? ) where p, V, T, R*, and n denote the pressure, volume, temperature, universal gas constant, and number of moles of a fixed collection of matter (i. e. , an air parcel).

Derive an equation of state for dry air It is not an easy manner to measure mass in the atmosphere (nor is it easy to measure volume). So we can use the assumptions of atmospheric composition (78% N 2, 21% O 2, etc) to transform n and R* into useful meteorological forms We express n as n=m/Md, where m and Md are the mass and molar weight of the dry air parcel Md = (0. 78*N 2 + 0. 21 * O 2 + 0. 01*other) = (0. 78*14*2 g/mol + 0. 21*16*2 g/mol + 0. 01*40 g/mol) = 28. 96 g/mol

Derive an equation of state for dry air Start with the ideal gas law, and put it in a form useful for atmospheric science n=m/Md R=R*/Md Move V/m to other side Density, ρ, = mass ÷ volume

Now, some basic properties of the atmosphere

How “tall” is the atmosphere? Depends on what you want • Weather occurs in about the lowest 15 km of the atmosphere Air extends up about 100 km *

• How does air pressure change with height? • Exponential decrease • How does density change with height? • Typically decreases exponentially, too At the surface: 1013 mb 1013 h. Pa 101, 300 Pa

Atmospheric pressure decreases exponentially with height 50% of earth’s air lies in the lowest 6 kilometers (3. 7 miles) Atmosphere officially extends up over 100 miles

• Pressure is basically the “weight” of the air above a location • If your location is at sea level, you will (generally) have a higher atmospheric pressure than other inland locations. • WHY? ? • Pressure decreases exponentially with height (more air molecules are found in air at the surface than in the same volume of air aloft)

How does air temperature change with height? • Not as straight-forward as pressure

Important definition: Inversion: a layer of air where the temperature change with height is opposite normal In troposphere, inversion is an INCREASE in temp with height