Analysis Methods in Atmospheric and Oceanic Science

Transcription

1 Analysis Methods in Atmospheric and Oceanic Science AOSC 652 Ordinary Differential Equations Week 12, Day 1 1

2 Differential Equations are central to Atmospheric and Ocean Sciences They provide quantitative descriptions of: motions of air and water circulation interaction of light and molecules photolysis frequencies and heating rates abundance of gases and particles biogeochemical cycling of carbon, interaction of all of above: nitrogen, sulfur, etc air quality ozone depletion and recovery clouds, aerosols, and precipitation 2

3 Differential Equations are central to Atmospheric and Ocean Sciences They provide quantitative descriptions of: motions of air and water circulation interaction of light and molecules photolysis frequencies and heating rates abundance of gases and particles biogeochemical cycling of carbon, nitrogen, sulfur, etc air quality ozone depletion and recovery clouds, aerosols, and precipitation interaction of all of above: climate change (impact of human activity on the physical, chemical, and thermodynamic state of the atmosphere and oceans) 3

4 Differential Equation: An equation that defines the relationship between an unknown quantity and one or more of its derivatives What is an ordinary differential equation? 4

5 Differential Equation: An equation that defines the relationship between an unknown quantity and one or more of its derivatives What is an ordinary differential equation? Equation contains functions of only one independent variable and one or more of its derivatives with respect to that variable 5

6 Differential Equation: An equation that defines the relationship between an unknown quantity and one or more of its derivatives What is an ordinary differential equation? Equation contains functions of only one independent variable and one or more of its derivatives with respect to that variable We ll now review examples of ordinary differential equations from classical physics and chemistry 6

7 Carbon- decay: d C 1 dt τ = C 7

8 Carbon- decay: d C 1 dt τ = C What is τ? 8

9 Carbon- decay: d C 1 dt τ = C What is τ? What must be the units of τ? 9

10 Carbon- decay: d C 1 dt τ = C Other ways of writing same equation: C = 1 τ 1 C = τ C C 10

11 Carbon- decay: d C 1 dt τ = C Other ways of writing same equation: C = 1 τ 1 C = τ C C Dot notation often used when independent variable is time 11

12 Newton s Second Law of Motion: F = m a : the net force on an object is equal to the mass of the object multiplied by its acceleration 12

13 Newton s Second Law of Motion: F = m a : the net force on an object is equal to the mass of the object multiplied by its acceleration How do we express this mathematically as a differential eqn? 13

14 Newton s Second Law of Motion: F = m a : the net force on an object is equal to the mass of the object multiplied by its acceleration How do we express this mathematically as a differential eqn? 2 d x F ( t) = m dt 2

15 Carbon- decay: d C 1 dt τ = C First Order Newton s Second Law of Motion: 2 d x F ( t) = m 2??? dt 15

16 Carbon- decay: d C 1 dt τ = C First Order Newton s Second Law of Motion: 2 d x F ( t) = m 2 Second Order dt 16

17 Heat Diffusion Equation: T 1 T = κ t ρ c z z What is T? What is t? What is ρ? What is c? What is κ? What is z? What is κ δt/ δz? 17

18 Heat Diffusion Equation: T 1 T = κ t ρ c z z What is T? Temperature What is t? time What is ρ? density What is c? specific heat What is κ? thermal conductivity What is z? distance conductive heat flux What is κ δt/ δz? 18

19 Heat Diffusion Equation: T 1 T = κ t ρ c z z Units What is T? Temperature (K) What is t? time (sec) What is ρ? density (kg/m 3 ) What is c? specific heat (J kg 1 K 1 ) What is κ? thermal conductivity (J m 1 s 1 K 1 ) What is z? distance (m) What is κ δt/ δz? conductive heat flux (J m 2 s 1 ) 19

20 Heat Diffusion Equation: T 1 T = κ t ρ c z z Sometimes written as: T t 2 T = α 2 z where α is a constant that describes the rate of heat diffusion 20

21 Solutions of Ordinary Differential Equations Carbon- Decay: d C 1 dt τ = C 21

22 Solutions of Ordinary Differential Equations d C 1 Carbon- Decay: = C dt τ Clearly, this equation can be rearranged and integrated to yield: C ( t) = C ( t = 0) e t / τ 22

23 Solutions of Ordinary Differential Equations d C 1 Carbon- Decay: = C dt τ Clearly, this equation can be rearranged and integrated to yield: C ( t) = C ( t = 0) e t / τ What is this called? 23

24 Solutions of Ordinary Differential Equations d C 1 Carbon- Decay: = C dt τ Clearly, this equation can be rearranged and integrated to yield: C ( t) = C ( t = 0) e t / τ Initial Condition If the independent variable were spatial rather than temporal, what would this term be called? 24

25 Solutions of Ordinary Differential Equations d C 1 Carbon- Decay: = C dt τ Clearly, this equation can be rearranged and integrated to yield: C ( t) = C ( t = 0) e t / τ What is this called? 25

26 Solutions of Ordinary Differential Equations d C 1 Carbon- Decay: = C dt τ Clearly, this equation can be rearranged and integrated to yield: C ( t) = C ( t = 0) e t / τ Half-life How much has the amount of C decayed, relative to the initial condition, when t = τ? 26

27 Solutions of Ordinary Differential Equations d C 1 Carbon- Decay: = C dt τ Clearly, this equation can be rearranged and integrated to yield: C ( t) = C ( t = 0) e t / τ Half-life How much has the amount of C decayed, relative to the initial condition, when t = τ? Times of t = τ, t = 2 τ, etc are often called one e-folding time, two e-folding time, etc. 27

28 Solutions of Ordinary Differential Equations d C 1 Carbon- Decay: = C dt τ Clearly, this equation can be rearranged and integrated to yield: C ( t) = C ( t = 0) e t / τ What is the half life of C? 28

29 Solutions of Ordinary Differential Equations d C 1 Carbon- Decay: = C dt τ Clearly, this equation can be rearranged and integrated to yield: C ( t) = C ( t = 0) e t / τ What is the half life of C? Given this half life and the fact that all C measurable in a paleo-climatic biological sample reflects the amount of C initially present in the sample at the time it was last living, minus isotopic decay, over what time horizon range can carbon- dating be applied? 29

30 Solutions of Ordinary Differential Equations d C 1 Carbon- Decay: = C dt τ Clearly, this equation can be rearranged and integrated to yield: C ( t) = C ( t = 0) e t / τ 30

31 Beer-Lambert Law: F(z,λ) = F TOA (λ) e τ(z, λ) (TOA : Top of Atmosphere) where: τ(z, λ) = m z σ λ [C] dz F : solar irradiance (photons/cm 2 /sec) σ λ : absorption cross section (cm 2 ) C : concentration of absorbing gas (molecules/cm 3 ) m : airmass: ratio of slant path to vertical path, equal to 1/cos(θ) for θ < 75 θ : solar zenith angle 31

32 Beer-Lambert Law: F(z,λ) = F TOA (λ) e τ(z, λ) (TOA : Top of Atmosphere) where: τ(z, λ) = m z σ λ [C] dz What is this τ called? F : solar irradiance (photons/cm 2 /sec) σ λ : absorption cross section (cm 2 ) C : concentration of absorbing gas (molecules/cm 3 ) m : airmass: ratio of slant path to vertical path, equal to 1/cos(θ) for θ < 75 θ : solar zenith angle 32

33 Beer-Lambert Law: F(z,λ) = F TOA (λ) e τ(z, λ) (TOA : Top of Atmosphere) where: τ(z, λ) = m z σ λ [C] dz If the absorbing species has a near constant mixing ratio wrt to altitude, we can write: [C] m.r. [Density] = m.r. [Density_surface] e z/h 33

34 Beer-Lambert Law: F(z,λ) = F TOA (λ) e τ(z, λ) (TOA : Top of Atmosphere) where: τ(z, λ) = m z σ λ [C] dz If the absorbing species has a near constant mixing ratio wrt to altitude, we can write: [C] m.r. [Density] = m.r. [Density_surface] e z/h What is H called? 34

35 Beer-Lambert Law: F(z,λ) = F TOA (λ) e τ(z, λ) (TOA : Top of Atmosphere) where: τ(z, λ) = m z σ λ [C] dz If the absorbing species has a near constant mixing ratio wrt to altitude, we can write: [C] m.r. [Density] = m.r. [Density_surface] e z/h What is H called? What does H equal? 35

36 Beer-Lambert Law: F(z,λ) = F TOA (λ) e τ(z, λ) (TOA : Top of Atmosphere) where: τ(z, λ) = m z σ λ [C] dz If the absorbing species has a near constant mixing ratio wrt to altitude, we can write: [C] m.r. [Density] = m.r. [Density_surface] e z/h What is H called? What does H equal? What approximation can we make that will allow this equation to be solved analytically? 36

37 F(z,λ) = F TOA (λ) e τ(z, λ) (TOA where: τ(z, λ) = m z σ λ : Top of Atmosphere) [C] dz Controls how atmosphere goes from this to this! to this? Top of Atmosphere From DeMore et al., Chemical Kinetics and Photochemical From Seinfeld and Pandis, Atmospheric Chemistry and Physics, Data for Use in Stratospheric Modeling, Evaluation No. 11, Copyright 20 University of Maryland. 37

38 Solar Thermal Curves of black-body energy versus wavelength for 5750 K (Sun s approximate temperature) and for 245 K (Earth s mean temperature). The curves are drawn with equal area since, to this? integrated over the entire Earth at the top of the atmosphere, the solar (downwelling) and terrestrial (upwelling) fluxes must be must be equal. From Houghton, Physics of Atmospheres,

39 Solar Thermal Curves of black-body energy versus wavelength for 5750 K (Sun s approximate temperature) and for 245 K (Earth s mean temperature). The curves are drawn with equal area since, to this? integrated over the entire Earth at the top of the atmosphere, the solar (downwelling) and terrestrial (upwelling) fluxes must be must be equal. From Houghton, Physics of Atmospheres, 1991 How do we model the thermal curve at the top of the atmosphere? 39

40 Solar Thermal Curves of black-body energy versus wavelength for 5750 K (Sun s approximate temperature) and for 245 K (Earth s mean temperature). The curves are drawn with equal area since, to this? integrated over the entire Earth at the top of the atmosphere, the solar (downwelling) and terrestrial (upwelling) fluxes must be must be equal. From Houghton, Physics of Atmospheres, 1991 How do we model the thermal curve at the top of the atmosphere? 40

41 Solutions of Ordinary Differential Equations Many simple equations simply can not be integrated to yield an analytic solution. For instance, the simple equation: dy y = 2 dt 1 + y can be rearranged and integrated to yield: 2 y ln y+ = t + C 2 which can be used to describe the evolution of y(t) as a function of t and C : i.e., can generate a family of plots of y versus t, for various values of C. 41

42 Solutions of Ordinary Differential Equations 2 Solution: y ln y+ = t + C 2 represented by a family of curves: 42

43 Solutions of Ordinary Differential Equations Many simple equations simply can not be integrated to yield an analytic solution. For instance, the simple equation: dy y = 2 dt 1 + y can be rearranged and integrated to yield: 2 y ln y+ = t + C 2 which can be used to describe the evolution of y(t) as a function of t and C : i.e., can generate a family of plots of y versus t, for various values of C. These plots, together with a measurement of y at particular time t, provide the solution. This type of solution is called an implicit solution 43

44 Solutions of Ordinary Differential Equations 2 Solution: y ln y+ = t + C 2 represented by a family of curves: One rather important solution is not yet represented 44

45 Solutions of Ordinary Differential Equations 2 Solution: y ln y+ = t + C 2 represented by a family of curves: The equilibrium solution, y = 0, is not represented by this figure 45

46 Readings for Wednesday: 25 pages from Storey 7 pages from Press 46