Module 3 Climate Modeling Theory - 1

Module 3 Climate Modeling Theory - 1 William J. Gutowski, Jr. Iowa State University

Module 3 Climate Modeling Theory - 1 GOAL: Understand basis for modeling climate from (almost) first principles

Module 3 Climate Modeling Theory - 1 • OUTLINE (Part 1): • Symbolism • Conservation Laws • mass • thermodynamic energy • momentum • Equation of State • Water in the Atmosphere

Module 3 Climate Modeling Theory - 1 • OUTLINE (Part 1): • Symbolism

Some Symbolism t time x west-east coordinate y south-north coordinate z vertical coordinate f latitude l longitude horizontal wind u west-east component of v south-north component of w vertical wind

Module 3 Climate Modeling Theory - 1 • OUTLINE (Part 1): • Symbolism • Conservation Laws • mass • thermodynamic energy • momentum

Conservation of “M”

Conservation of Mass (Continuity Equation) = density [kg/m3] Source/sink = 0

Conservation of Mass

Conservation of Water Mass q = specific humidity [kg-(H2O)v/kg-air] s(q) = cond. - evap.

Conservation of Water Mass (column integral) E = sfc. evap.; P = precipitation

Conservation of W (Precipitable Water)

Conservation of General Constituent, i qi = amount of i [kg-(constituent i)/kg-air] e.g., CO2, O3, etc.

Conservation of Thermodynamic Energy ~ First Law of Thermodynamics ~ Heat input = D (internal energy) + (work done) = heating/mass [J-kg-1-s-1]

Conservation of Thermodynamic Energy ~ First Law of Thermodynamics ~

Conservation of Thermodynamic Energy

RNET FSH RNET

Conservation of Momentum ~ Newton’s Second Law ~

Conservation of Momentum ~ Newton’s Second Law ~ • Forces/mass: • gravity • pressure gradient • friction

Conservation of Momentum ~ Newton’s Second Law ~ Rotating Frame X

Conservation of Momentum ~ Newton’s Second Law ~ Rotating Frame

Conservation of Momentum ~ Newton’s Second Law ~ Sphere, Rotating Frame rotation of direction

Conservation of Momentum ~ Newton’s Second Law ~ Approximation: vertical

Conservation of Momentum ~ Newton’s Second Law ~ Approximation: vertical Hydrostatic Approximation Accurate to ~ 0.01% for weak vertical acceleration

Conservation of Momentum ~ Newton’s Second Law ~ Approximation: horizontal, extratropical

Conservation of Momentum ~ Newton’s Second Law ~ Approximation: horizontal, extratropical Geostrophic Approximation Accurate to ~ 20 - 30%

Module 3 Climate Modeling Theory - 1 BREAK

Module 3 Climate Modeling Theory - 1 • OUTLINE (Part 1): • Symbolism • Conservation Laws • mass • thermodynamic energy • momentum • Equation of State

Ideal Gas Law R = gas constant R = R(constituents) Common practice: R = Rd = 287 J-kg-1-s-1 T = Tv

Module 3 Climate Modeling Theory - 1 • OUTLINE (Part 1): • Symbolism • Conservation Laws • mass • thermodynamic energy • momentum • Equation of State • Water in the Atmosphere

q versus latitude & pressure [g-kg-1] Note: small part of atmosphere, but ...

… water • saturates • changes phase

Some Further Symbolism q specific humidity [kg-kg-1] mass (H2O)v/mass air e vapor pressure [Pa] partial pressure by water molecules m mixing ratio [kg-kg-1] mass (H2O)v/mass dry air RH relative humidity [%] ratio: m/msat

Water Cycle Q Q P P E E R

Water Cycle Heat released E Heat absorbed

Water is thus a primary • form of heat transport • heat absorbed when evaporates • released when water condenses • largest individual source of energy for the atmosphere

Water Cycle Radiation absorbed by water & re-emitted

Water is thus a primary • form of heat transport • heat absorbed when evaporates • released when water condenses • largest individual source of energy for the atmosphere • and greenhouse gas • ~ transparent to solar • absorbs/emits infrared

RH vs. latitude & pressure [%] RH 70 70

precipitation vs. latitude & longitude [dm-yr-1] [dm-yr-1] = [100 mm-yr-1] =[0.27 mm-d-1]

Lift Moist Parcel z 9.8 K/km T

Lift Moist Parcel z T z RH

Lift Moist Parcel z Lifting Condensation Level T z LCL RH 100 %

Stable Precipitation condensation collision coalescence

Module 3 Climate Modeling Theory - 1