2. Conservation laws and basic equations

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1 2. Conservation laws and basic equations Equatorial region is mapped well by cylindrical (Mercator) projection: eastward, northward, upward (local Cartesian) coordinates:,, velocity vector:,,,, material (Lagrange) time derivative:, (t: time) (1),, Three conservation laws (four equations) (i) Equation of motion (Newton s 2 nd law; Angular momentum conservation law): 1 3 (ii) Equation of continuity (Mass conservation law): 1, 4 (iii) Equation (1 st law) of thermodynamics (Entropy conservation law): ln, Six variables:, (pressure), (temperature), (density), (specific humidity), (condensation ratio) (2) / potential temperature 5 Parameters: 0,0,, 2Ωsin (Coriolis; Ω: rotation angular. velocity; : latitude) ; 0, 0, : gravity acceleration and : gas constant and constant-pressure specific heat for dry air ; : Latent heat for water condensation External sources:,, ),, : momentum (friction), heat (net radiation), water vapor (evaporation)

2 Concept of continuum Actual matter = Σ molecule - Density = Σ molecular mass / unit volume = molecular mass number / unit volume - Pressure = Σ molecular momentum / unit time / unit area = molecular force / unit area - Temperature = Σ molecular kinetic energy / molecular number / Boltzmann constant (atmosphere) (ocean, ) (earth) Phase of matter: Gas, Liquid, Solid Continuum: Fluid, Plastic, Elastic, Rigid

3 Forces (interactions) working in the nature Intermolecular (electromagnetic) force saturation homogeneous nucleation - Kelvin s curvature effect supercooled tiny droplet - Raoult s solute effect vapor pressure / boiled point depression - Henry-Dalton s partial pressure law for a mixed gas Unsaturated surface of a droplet molecular diffusion evaporation Condensation at a solid surface (heterogeneous nucleation) large droplet Planetary gravitational force Density (hydrostatic) stratification Ocean - Photochemical / volcanic water vapor production - Gravitational separation / photodissociation Hydrogen escape / oxidation Ocean loss Precipitation (coalescence / sublimation) process - Gravity, radiation Equatorial tropopausal cold trap - Orography / sea-land heat contrast forced convection - Conditional instability moist convection (c) (d) Strong/weak nuclear & intermolecular electromagnetic forces (Israelachvili, 1985, 1992) Cloud/precipitation processes (Wallace & Hobbs, 1972, 2006) Planetary gravitation (Israelachvili, 1985, 1992)

4 Various fluid flows in the Earth System [(i) Momentum and (ii) continuity eqs. are common for any cases] Global atmosphere (Meteorology) Ocean (Oceanography) River (Hydrology) Compressible [+thermodynamics (iii)(iv)] Almost closed Zonal dominant Moisture effect [(v)] Almost free Incompressible [+thermal expansion] Almost closed Horizontal dominant Salinity effect Coastal effect Incompressible [+Level/stream change] Opened Almost one-dimensional Complex boundary

5 General principles governing planetary fluid Planetary (or geophysical) fluid: Gas/liquid under - sphericity: a=6370 km (for the earth) latitude φ, longitude λ, altitude z eastward displacement: dx = a cosφ dλ northward displacement: dy = a dφ - rotaion: Ω = 2π/86164s (for the earth) Coriolis parameter: f = 2Ω sinφ Rossby parameter: β= df/dy =2Ω cosφ - gravitation: g = 9.8 m/s 2 (for the earth) Variables: (6 for dry atmosphere) - wind (or stream) velocity components: (u, v, w) - thermodynamical state variables: (T, p, ρ) [ - humidity (or salinity): q ]

6 Centrifugal and Coriolis Forces Centrifugal force Ω must be balanced with pressure gradient etc. (: mass, : rotation radius, Ω: angular velocity, Ω : moving speed) CFF CFF Gravity If the body moves eastward by (relative to the earth) at latitude, Horizontal component of total centrifugal force cos Ω Ω sin 2 2Ω sin tan Coriolis parameter f sin

7 Exercise 2 (1) Explain the three terms in the last equation in the previous slide. (2) How different the Coriolis force in the northern and southern hemispheres? How about at the equator? (3) Do you think the motorcycle rider and the bathtub vortex must feel the Coriolis force of the earth s rotation? How about the Coriolis for the solar system, or of galaxy? Answers: (1) Ω cos sin 2Ω sin Coriolis force only considerable not separated from small gravity (neglectable) (2) Sin changes sign and direction of Coriolis force becomes opposite in northern/southern hemispheres. It becomes 0 and Coriolis force vanishes at the equator. (3) No, because the earth s rotation is in the time scale of 1 day. Similarly, we can neglect the earth s revolution around the sun with 365 days >> 1 day, as well as the solar revolution in our galaxy with 20 million years.

8 (a) A planet observed from space (b) A planet observed on itself velocity Sun planet (Earth) solar gravitation acceleration solar gravitation centrifugal force (c) A body (an air parcel) at rest on Earth observed from space normal force (pressure gradient) (d) An air parcel at rest observed on Earth pressure gradient (e) An air parcel moving with geostrophic westerly pressure gradient acceleration gravitation gravitation gravity centrifugal force due to Earth s rotation gravity Coriolis force

9 Traditional approximation Taking only the vertical component of rotation: cos Ω sin cos 1 2 Ω sin 2 2Ω sin tan the first term (maximum at 45º latitude, about 1/300 of g ellipticity of Earth) is involved in the gravity and excluded here Ω Ω a cos φ a φ Centrifugal Obtaining only the horizontal component of Coriolis and centrifugal forces: 2Ω cos sin φ Ω sin φ φ centrifugal x sin φ The other term on Ωcos is neglected even in the tropics (Phillips, 1966; Gill, 1982, 7.4).

10 Leonhard Euler (1707~1783) function : y = f (x) π, e, i (1748) Trigonometric expansion ( Fourier series) Newton s 2 nd law ( equation of motion ) (1736) F = m a Continuity equation for incompressible inviscid fluid (1757) u x w z 0. z w > 0 z z u > 0 u < 0 = + w < 0 x x x

11 2. Basic equations (cont.) Additional two laws (three equations; one variable: saturation water vapor pressure) (iv) Equation of ideal gas (Boyle-Charles law):, (v) Equation of water vapor saturation (Clapeyron-Clausius) :, 1 7 Parameters: and : molecular mass of dry air and water vapor Basic Eqs. (3)-(7) Horizontal mean Chapter 3 Vertical profiles (8)-(9) Zonal mean Chapter 4 Meridional distributions (23)-(27) All phenomena are governed by limited number of physical laws. Nonlinearity and complex sources produce complex phenomena. Chapter 5 Waves. (45)-(49) Chapter 6 Convections (69)-(74)

12 Julius Robert von Mayer ( ) James Prescott Joule ( ) /dc/pc/mayer.jpg (Roscoe, 1906; wiki/james_prescott_jou le) German scientist cruised in 1840 to East Jawa as a Dutch ship doctor, and noticed a concept called energy at present as exchangeable quantity between motion and heat. After returning to Germany in 1841, he submitted a paper to a journal of physics, but rejected. In 1842 his paper was accepted by a journal of chemistry, but was not so highly evaluated. In 1845 his second paper was rejected even by the chemical journal. After that he never submitted any papers to journals but published them by himself. In 1850 he became a farmer until his death. In 1854 von Helmholz recognized that Mayer was the first person discovering the energy. English brewer studied physics without any post at university or institute. He discovered the Joule s law and the mechanical equivalent of heat in early 1840s.

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