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Pressure and mass conservation 1 Today’s goals • • • • Some definitions: Gravity. What is a shallow atomsphere Coordinates. The concept of a “parcel” (or finite element in the fluid) • Vertical variations in pressure • Vertical variation in temperature (Seemingly obvious, but fascinating research topic) Some initial things to remember • Newton’s second law: “momentum concerned” • Ideal gas law: expression of mass for air • First law of thermodynamics “energy conserved” F = ma Gravity exists Earth is round There are days and nights (Earth is rotating about once per day) g a Alphabet soup is better in Greek p = rRT Forces? • Fg, Gravity • Fv, Viscosity • Fp, Pressure gradient F=ma All have the same units, N 4 Gravity • Newton’s law of gravity Me m r GM e m F r2 mass of the Earth mass of the parcel Distance between center of the Earth and air parcel So acceleration, Directed exactly toward the Earth’s center of gravity GM e F g0 * 2 m r 5 r m dme Me center of mass Integrating all mass elements of, it can be shown that the direction vector r points to the center of mass 6 Earth is big, atmosphere is small Radius of earth, a = 6371km Depth of atmosphere ~30km So we can use an approximation: r~ a Thus, 7 Exosphere Ionosphere Thermosphere Mesosphere Stratosphere Troposphere Salby, Fundamentals of Atmospheric Physics, page 11. Composition of the atmosphere • Air is a mixture of various gases • Particularly, nitrogen, oxygen, water vapor, carbon dioxide • We can assume air behaves like an ideal gas Definition: Lapse rate Pressure is mass • Pressure defined as force (imparted on parcel walls by molecules) per units area p=F/A • For horizontal surface the force is weight F = mg • So, mass: m = F/g = pA/g Define some coordinates • Position x,y,x But there are others! Horizontal: Longitude, latitude Vertical: Pressure, potential temperature,…. Velocity: u, v, w (Change in position with time) 13 The atmosphere is a fluid continuum • Can be described atmosphere by a set field variables that are continuous functions (in space and time) e.g. temperature, pressure, wind, mass, water content • It is convenient to consider these properties applying to a “small” parcel of air Volume dV = dx dy dz T, V, r, p, q x z Mass dm = r dV y •Total mass of the atmosphere M = ∫globe dm 14 Special look at mass….err pressure • What is pressure? NCEP Reanalysis: http://www.cdc.noaa.gov/ Pressure is mass • Pressure defined as force (imparted on parcel walls by molecules) per units area p=F/A • For horizontal surface the force is weight F = mg • So, mass: m = F/g = pA/g • What about change in forces on a parcel? Pressure and mass • Compute pressure at some height by integrating: p z p( z ) dp g rdz 0 • Compute sea level pressure: 0 ps g rdz If the surface pressure is 1000hPa, what is the pressure with half the atmospheric mass above? Half! (500hPa) Relationship between temperature, pressure and density The hydrostatic equation Exercises Practice question: • Holton 1.13, 1.16, 1.17, 1.18 Reading to next week • Holton2.6 • Holton 2.7 • Start on RY CH1