Applications of the Basic Equations Chapter 3. Paul A. Ullrich

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1 Applicaions of he Basic Equaions Chaper 3 Paul A. Ullrich paullrich@ucdavis.edu

2 Par 1: Naural Coordinaes

Naural Coordinaes Quesion: Why do we need anoher coordinae sysem? Our goal is o simplify he equaions of moion. Someimes complicaed equaions are simple if looked a in he righ way.

3 Naural Coordinaes Quesion: Why do we need anoher coordinae sysem? Our goal is o simplify he equaions of moion. Someimes complicaed equaions are simple if looked a in he righ way. A large scales, he amosphere is in a sae of balance. A large scales, mass fields (ρ, p, Φ) balance wih wind fields (u). Bu mass fields are generally much easier o observe han wind. Balance provides a way o infer he wind from he observed pressure or geopoenial.

4 Geosrophic Balance Low Pressure High Pressure Flow iniiaed by pressure gradien Flow urned by Coriolis force

5 Geosrophic & Observed Wind Upper Tropo (300mb)

6 Describe he Previous Figure A upper levels (where fricion is negligible) he observed wind is parallel o geopoenial heigh conours (on a consan pressure surface). Wind is faser when heigh conours are close ogeher. Wind is slower when heigh conours are farher apar.

7 The Upper Troposphere Norh Geopoenial conours are depiced on a consan pressure surface. > Souh Wes Eas

8 The Upper Troposphere Norh Souh Geopoenial conours are depiced on a consan pressure surface. y > Wes Eas

9 The Upper Troposphere Norh Geopoenial conours are depiced on a consan pressure surface. = 0 ( 0 +2 )= 2 > y 0 +2 Souh 0 +3 Wes Eas

10 The Upper Troposphere Geopoenial conours are depiced on a consan pressure surface. > y Souh Wes Eas

11 Horizonal Momenum Assume no viscosiy du d p + fk u = r p du d dv d p p p p + fv fu Meridional gradien of geopoenial appears here

12 p p = fv g = fu g Meridional gradien of geopoenial appears here

13 The Upper Troposphere Norh Souh Geopoenial conours are depiced on a consan pressure surface. y fu g 2 y > Wes Eas

14 The Upper Troposphere Geopoenial conours are depiced on a consan pressure surface. > 0 0 Norh fu g 2 y fu g y 0 + Souh y y Wes Eas

15 The Upper Troposphere Think abou his a minue > 0 Norh Cold Warm Temperaure Gradien Pressure Gradien Coriolis Force y Souh Wes fu g y Eas

16 The Upper Troposphere Think abou his a minue We have derived a formula for he i (easward or x) componen of he geosrophic wind. We have esimaed he derivaives based on finie differences. Recall we also used finie differences in deriving he equaions of moion. There is a consisency: Direcion comes ou correcly (owards eas) The srengh of he wind is proporional o he srengh of he gradien.

17 The Upper Troposphere Think abou his a minue Wha abou he observed wind? Flow is parallel o geopoenial heigh lines Bu here is curvaure in he flow as well. IMPORTANT NOTE: This is no curvaure due o he Earh, bu curvaure on a consan pressure surface due o bends and wiggles in he flow.

18 Geosrophic & Observed Wind Upper Tropo (300mb)

19 The Upper Troposphere Wha abou he observed wind? Flow is parallel o geopoenial heigh lines Bu here is curvaure in he flow p p = fv g = fu g Quesion: Where is curvaure in hese equaions?

20 The Upper Troposphere Think abou he observed (upper level) wind: Flow is parallel o geopoenial heigh lines There is curvaure in he flow Geosrophic balance describes flow parallel o geopoenial heigh lines. BUT Geosrophic balance does no accoun for curvaure. Quesion: How do we include curvaure in our diagnosic equaions?

21 Naural Coordinaes Quesion: Why do we need anoher coordinae sysem? Our goal is o simplify he equaions of moion. Someimes complicaed equaions are simple if looked a in he righ way. A large scales, he amosphere is in a sae of balance. A large scales, mass fields (ρ, p, Φ) balance wih wind fields (u). Bu mass fields are generally much easier o observe han wind. We need o describe balance beween dominan erms: Pressure gradien, Coriolis and curvaure of he flow.

22 Naural Coordinaes A naural se of direcion vecors. When sanding a a poin, someimes he only indicaion of direcion is he direcion of he flow. Assumes no local changes in geopoenial heigh. Flow is along conours of consan geopoenial heigh. Assume horizonal flow only (on a consan pressure surface). An analogous mehod could be defined for heigh surfaces. Assume no fricion (no viscous erm) Analogous o a Lagrangian parcel approach.

23 The Upper Troposphere Norh Define one componen of hese coordinaes angen o he direcion of he wind. > Souh 0 +3 Wes Eas

24 The Upper Troposphere Define he oher componen of hese coordinaes normal o he direcion of he wind. > 0 0 Norh n n n Souh 0 +3 Wes Eas

25 Naural Coordinaes n Regardless of posiion: always poins in he direcion of he flow n always poins perpendicular o, o he lef of he flow n = k Righ-hand rule for vecors

26 Naural Coordinaes n Advanages: We can look a a geopoenial heigh (on a pressure surface) and esimae he winds. In general i is difficul o measure winds, so we can now esimae winds from geopoenial heigh (or pressure). Useful for diagnosics and inerpreaion.

27 Naural Coordinaes n However, for diagnosics and inerpreaion of flows, we need an equaion.

28 Naural Coordinaes Do you observe ha he normal arrows seem o poin a somehing in he disance? 0 Norh n n n Souh 0 +3 Wes Eas

29 Naural Coordinaes 0 Norh n n n Souh Wes Imagine ha he fluid is experiencing cenripeal acceleraion due o a force in he normal direcion. How would a fluid parcel reac? Eas

Naural Coordinaes 0 Norh n n n 0 + 0 +2 0 +3 Souh Wes R Definiion: The radius of curvaure of

30 Naural Coordinaes 0 Norh n n n Souh Wes R Definiion: The radius of curvaure of he flow is he radius of a circle wih angen vecor ha shares he same curvaure as he local flow. Eas

31 Naural Coordinaes Velociy in Naural Coordinaes u = V V = u Velociy Vecor Uni vecor angen o he flow Velociy Magniude Simplificaions: 1. Velociy is always in he direcion of 2. The value of u is always posiive

32 Naural Coordinaes Acceleraion in Naural Coordinaes Du D = D(V ) D = DV D + V D D Definiion of acceleraion Change in speed Change in direcion

33 Naural Coordinaes D D Quesion: How do we ge as a funcion of,? V R For simpliciy, consider a fluid parcel moving along a circular rajecory. Recall he use of circle geomery (from derivaion of Coriolis / cenrifugal force) 2 R 1 Final posiion of fluid parcel Iniial posiion of fluid parcel Radius of curvaure

34 Naural Coordinaes Beween he iniial and final posiions, he angen vecor changes by an amoun. Triangle 2 = 1 + Recall he use of circle geomery (from derivaion of Coriolis / cenrifugal force) R 1 Radius of curvaure

35 Naural Coordinaes Zoomed in Using geomery, his riangle has an inernal angle. 2 = 1 + Define angle R 1 Use he law of sines and he fac ha angen vecors have uni lengh: sin =sin( ) Since all angles are < 90 = n 1 = 2 2 For small displacemens, will poin in he same direcion as n 1 (= 90 o 1 )

36 Naural Coordinaes Zoomed in Using geomery, his riangle has an inernal angle. 2 = 1 + Observe ha for small displacemens (and using he fac ha angen vecors are uni lengh): Consequenly: R 1 n 1 n 1

37 Naural Coordinaes Zoomed in Using geomery, his riangle has an inernal angle. 2 = 1 + From he las slide: n 1 Disance raveled by fluid parcel s = R R 1 n 1 s R n 1

38 Naural Coordinaes Zoomed in Using geomery, his riangle has an inernal angle. From he las slide: 1 R s n 1 2 = 1 + R 1 Disance / Time = Velociy n 1 In he limi of! 0 D D = 1 R Ds D n 1 = V R n

39 Naural Coordinaes Remember our goal is o quanify acceleraion Du D = D(V ) D = DV D + V D D D D = V R n Du D = DV D + V 2 R n Change in speed?

experience a cenripeal force wih magniude V 2 /R owards he cener of he

40 Naural Coordinaes Recall from physics 101 cenripeal acceleraion: An objec raveling a velociy V forced o remain along a circular rajecory will experience a cenripeal force wih magniude V 2 /R owards he cener of he circle Du D = DV D + V 2 R n Change in speed Cenripeal acceleraion due o curvaure in he flow

41 Momenum Equaion Now ha we have an equaion for change in horizonal momenum in erms of angenal and normal vecors, we would like o derive a momenum equaion. The momenum equaion mus conain erms: Acceleraion Coriolis force Pressure gradien force

42 Momenum Equaion Coriolis Force Coriolis force always acs normal o he velociy, wih magniude f : F cor = fk u = fvn

43 Momenum Equaion Pressure Gradien Force Pressure gradien force acs in he opposing direcion of he pressure gradien. On a surface of consan pressure his leads o: F p = r +

44 Momenum Equaion Using he vecor form of he momenum equaion: Du D + fk u = r p Make all subsiuions: F p = r + F cor = fk u = fvn DV D + V 2 R n + +

45 Momenum Equaion DV D + V 2 R n + + In componen form: DV D = V 2 R + Along flow direcion () Across flow direcion (n)

46 Momenum Equaion Is his a simplificaion? Recall we are only considering flow along geopoenial heigh conours: 0 DV D = V 2 R + 0 Along flow direcion () Across flow direcion (n) By using naural coordinaes, we only require one diagnosic equaion o describe velociy.

Momenum Equaion One diagnosic equaion for curved flow: V 2 R + fv = @ @n Cenripeal acceleraion

47 Momenum Equaion One diagnosic equaion for curved flow: V 2 R + Cenripeal acceleraion Pressure gradien force Coriolis force Quesion: How does his generalize he geosrophic approximaion?

Physics 180A Fall 2008 Test points. Provide the best answer to the following questions and problems. Watch your sig figs.

Physics 180A Fall 2008 Test points. Provide the best answer to the following questions and problems. Watch your sig figs. Physics 180A Fall 2008 Tes 1-120 poins Name Provide he bes answer o he following quesions and problems. Wach your sig figs. 1) The number of meaningful digis in a number is called he number of. When numbers