Analysis of the Dynamical Equations Chapter 2. Paul A. Ullrich

Transcription

1 Analysis of the Dynamical Equations Chapte 2 Paul A. Ullich paullich@ucdavis.edu

2 Pat 1: Scale Analysis of the Momentum Equation

3 The Atmospheic Equations Du uv tan Dv + u2 tan Dw c p DT + uw = 1 cos + vw = u 2 + v 2 = 1 D = u Dp = J p = R d T Dq i = S +2 v sin 2 w cos + 2 u 2 u sin g +2 u cos + 2 v + 2 w Mateial deivative: + u Paul Ullich Analysis of the Dynamical Equations Mach 2014

4 Scale Analysis Question: What is the size (in time and space) of atmospheic phenomena that ae elevant fo lage-scale mid-latitude dynamics? Question: What ae the tems in the equations of motion that ae most elevant fo lage-scale mid-latitude dynamics? These questions ae closely connected: Scale analysis povides us a means to an answe. Paul Ullich Analysis of the Dynamical Equations Mach 2014

5 Scale Analysis Conside x and y components of the momentum equations: Du uv tan Dv + u2 tan + uw = 1 + vw = +2 vsin 2 w cos + 2 u 2 u sin + 2 Remembe the units each tem must have units of acceleation Define: L T Some chaacteistic distance Some chaacteistic time All tems should have units of L/T 2 Paul Ullich Analysis of the Dynamical Equations Mach 2014

6 Scale Analysis Question: How do we define a time scale? Idea: Take a typical tajectoy and ask how fa does a pacel go in a chaacteistic time? Paul Ullich Analysis of the Dynamical Equations Mach 2014

7 Scale Analysis We would like to define scales in tems of wind, pessue and density. Recall: hdistancei = hvelocityi htimei Define: U Some chaacteistic hoizontal velocity In tems of chaacteistic scales: L = U T So the chaacteistic time scale is given by T = L U Paul Ullich Analysis of the Dynamical Equations Mach 2014

8 Scale Analysis What do these scales mean? Conside, fo example, a typical topical cyclone: Fluid pacel Look at the oganization of the flow. The length scale is the diamete of the stom. The velocity scale is the maximum velocity of the flow. So the time scale is appoximately the time equied fo the pacel to move aound the stom. L Paul Ullich Analysis of the Dynamical Equations Mach 2014

9 Scale Analysis Definition: L U T H W P/ Hoizontal distance scale Hoizontal velocity scale Time scale (T = L/U) Vetical distance scale Vetical velocity scale Scale of hoizontal pessue fluctuations The mateial deivative D epesents change on the time-scale of motion. Hence, its scale is given by apple D = 1 T = U L Paul Ullich Analysis of the Dynamical Equations Mach 2014

10 Scale Analysis Typical scales associated with lage-scale mid-latitude stom systems: U 10 m s 1 W 0.01 m s 1 L 10 6 m H 10 4 m L/U 10 5 s P 10 hpa = 1000 Pa 1kgm 3 / 10 2 f s 1 a 10 7 m g 10 m s m 2 s 1 (Radius of Eath) (Gavity) (Kinematic Viscosity) Paul Ullich Analysis of the Dynamical Equations Mach 2014

11 Scale Analysis (Hoizontal Momentum) Hee is what each tem of the momentum equation looks like in tems of chaacteistic scales: Du uv tan Dv + u2 tan + uw = 1 + vw = +2 vsin 2 w cos + 2 u 2 u sin + 2 U U/L U U/a U W/a P U L Uf 0 Wf 0 H 2 Paul Ullich Analysis of the Dynamical Equations Mach 2014

12 (Hoizontal Momentum Equation) U 10 m s 1 W 0.01 m s 1 L 10 6 m H 10 4 m L/U 10 5 s Scales P 1000 Pa 1kgm 3 / 10 2 f s 1 a 10 7 m g 10 m s m 2 s 1 Du uv tan Dv + u2 tan + uw = 1 + vw = +2 vsin 2 w cos + 2 u 2 u sin + 2 U U/L U U/a U W/a ΔP/ρL Uf Wf νu/h Paul Ullich Analysis of the Dynamical Equations Mach 2014

13 (Hoizontal Momentum Equation) Lagest Tems Du uv tan Dv + u2 tan + uw = 1 + vw = +2 vsin 2 w cos + 2 u 2 u sin + 2 U U/L U U/a U W/ a ΔP/ρL Uf Wf νu/h Dominant Balance Paul Ullich Analysis of the Dynamical Equations Mach 2014

14 Only etain the lagest tems Pessue Gadient Foce Coiolis Foce 2 v sin =0 +2 u sin =0 Balance: If no time deivative tem is pesent the system is static. The emaining tems must balance each othe fo the equation to hold. Geostophic Balance Deceasing Pessue Low P Pessue Gadient Foce Coiolis Foce Path of ai pacel High P Paul Ullich Analysis of the Dynamical Equations Mach 2014

15 (Hoizontal Momentum Equation) Pessue Gadient Foce Coiolis Foce 2 v sin =0 +2 u sin =0 Low P Pessue Gadient Foce Coiolis Foce Path of ai pacel High P Definition: Fo lage-scale mid-latitudinal flows thee is an intinsic balance between pessue gadient foce and Coiolis foce. This balance is known as geostophic balance and leads to ai pacels taveling along lines of constant pessue. Paul Ullich Analysis of the Dynamical Equations Mach 2014

16 Scale Analysis (Vetical Momentum) Dw u 2 + v 2 = g +2 u cos + 2 w W U/L U U/a P H g Uf 0 W H 2 Paul Ullich Analysis of the Dynamical Equations Mach 2014

17 (Vetical Momentum Equation) U 10 m s 1 W 0.01 m s 1 L 10 6 m H 10 4 m L/U 10 5 s Scales P 1000 Pa 1kgm 3 / 10 2 f s 1 a 10 7 m g 10 m s m 2 s 1 Dw u 2 + v 2 = g +2 u cos + 2 w W U/L U U/a P sfc /ρh g Uf νw/h Paul Ullich Analysis of the Dynamical Equations Mach 2014

18 (Vetical Momentum Equation) Lagest Tems Dw u 2 + v 2 = g +2 u cos + 2 w W U/L U U/a P sfc /ρh g Uf νw/h Hydostatic Balance = g Paul Ullich Analysis of the Dynamical Equations Mach 2014

19 Scale Analysis Question: What ae the tems in the equations of motion that ae most elevant fo lage-scale mid-latitude dynamics? The lagest tems in the hoizontal and vetical momentum equations lead to two types of balance that dominate the obseved flow fo lage-scale mid-latitudinal stom systems: Geostophic Balance (Pessue gadient and Coiolis) Hydostatic Balance (Pessue gadient and gavity) Aside: Why is mid-latitudinal impotant? Paul Ullich Analysis of the Dynamical Equations Mach 2014

20 (Vetical Momentum Equation) Lagest Tems Dw u 2 + v 2 = g +2 u cos + 2 w W U/L U U/a P sfc /ρh g Uf νw/h The vetical acceleation Dw/ is 8 odes of magnitude smalle than hydostatic balance. The ability of the vetical momentum equation to estimate w is essentially nonexistent. Paul Ullich Analysis of the Dynamical Equations Mach 2014

21 Scale Analysis Scale analysis of the vetical momentum equation evealed that computing vetical velocity using this equation equies taking the diffeence of two tems which ae 8 odes of magnitude lage than the acceleation! Even tiny eos in computing the vetical pessue gadient will lead to lage eos in the vetical velocity. Motivates the next question Question: How can vetical velocity be computed? Paul Ullich Analysis of the Dynamical Equations Mach 2014

22 Vetical Velocity? Ve$cal mo$on is impotant: Rising mo$on leads to clouds and pecipita$on. The ve$cal accelea$on Dw/ is 8 odes of magnitude smalle than hydosta$c balance. The ability to use the ve$cal momentum equa$on to es$mate w is essen$ally nonexistent. Ve$cal velocity must be diagnosed fom some balance Note that small scales, thundestoms, tonadoes use vey diffeent chaacteis$c scales, so the ve$cal momentum equa$on can be employed in this egime. We will etun to this in a moment Paul Ullich Analysis of the Dynamical Equations Mach 2014