Turbulence and boundary layers

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Pierce Russell
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1 Trblence and bondary layers

2 Weather and trblence Big whorls hae little whorls which feed on the elocity; and little whorls hae lesser whorls and so on to iscosity Lewis Fry Richardson

3 Momentm eqations d dt f p x Fr x d dt f p y Fr y Away from the srface, we can ignore friction Led to geostrophic assmption By introdcing friction (drag) change the way we can balance the eqations of motion

4 Importance of Trblent Eddies Trblent eddies are important in the atmospheric bondary layer becase they can transport momentm, heat and moistre. As a reslt, the dynamical eqations that we hae discssed dring this semester mst be modified for se in the atmospheric bondary layer. We will now introdce a strategy for inclding the effects of trblence in the dynamical eqations. To do this, we will attempt to separate the trblent ariations in atmospheric properties from the large-scale ariations.

5 Vertical strctre in the ocean

6 Character of trblence The effects of trblence can be ignored in the free atmosphere, bt can not for flow near the srface (e.g., drag ensres flow is ageostrophic) Viscosity ensres wind speed is zero ery close to the srface Trblent transfer mch more efficient than moleclar effects (or thermal condition) Trblent eddies exist at all time and space scales between the limits of the bondary layer depth and the scale at which moleclar diffsion take oer (millimeter)

7 Sonic anemometer Measres ery small scale ariation in 3d wind field

8 Wind Speed (m/s) An Example of Trblence The effects of trblence are eident in this record of srface wind speed measred by an anemometer. The gsts and llls in the wind, which typically last for less than a minte, are indicatie of the passage of trblent eddies. Dring this gst, the wind speed increases by ~50%. stronger trblence weaker trblence Time

9 Trblence Eddies small, bt still mch larger than iscos scale Energy transferred to smaller scales, where ltimately dissipated by moleclar diffsion Small scale eddies generated by wind shear d V /dx and by boyancy (i.e., conection)

10 Eqations for trblence We wish to ealate the Fr terms in the momentm eqation (and eqialent terms in the heat and moistre eqations) Make se of random natre of trblent eddies in a statistical representation Trblent eddies small compared to synoptic scale motions (can ignore Coriolis acceleration associated with eddies) Also can make some frther simplifications of primitie eqations for near-srface conditions (Bossinesq assmption, i.e., density pretty constant)

11 Bossinesq approximation For limited depth region of the atmosphere (say, km), density changes are small compared to the mean density profile. Neglect density ariations except where they case boyant forces i.e., gien = 0 +D, se 0 eerywhere except for compting boyancy forces, where we se D A particlarly appropriate assmption for oceanic flow

12 Horizontal momentm Bossinesq eqations d dt d dt f f 0 0 p x p y Fr x Fr y Thermodynamic D d w dt dz d 0 Continity (mean density does not change) Vertical momentm (non-hydrostatic) x y w z 0 dw dt 0 p z g D 0 Fr z Note = 0 +D Where 0 is appropriate gien 0

13 Reynolds aeraging Define all qantities to be composed of a time mean and a deiation w w w At a gien point, this deiation from the time mean gies a measre of trblent eddies Compare this with spatial deiations we examined with waes

14 Wind Speed (m/s) An Example of Trblence The effects of trblence are eident in this record of srface wind speed measred by an anemometer. The gsts and llls in the wind, which typically last for less than a minte, are indicatie of the passage of trblent eddies. Dring this gst, the wind speed increases by ~50%. stronger trblence weaker trblence Time

15 Trblent ariations Instantaneos elocities can be decomposed into mean and trblent components: trblent elocity mean elocity instantaneos elocity Trblent elocities are the positie and negatie deiations of the instantaneos elocities abot the mean.

16 Stationary (topographically forced) waes NCEP Reanalysis Z500 Janary mean

17 NCEP Reanalysis Z500 Deiations from zonal mean Janary mean

18 Separate eddy ariations from backgrond A simple way to separate the trblent ariations from the large-scale ariations is to aerage or wind measrements oer a period of mintes.

19 Trblent ariations Instantaneos elocities can be decomposed into mean and trblent components: trblent elocity mean elocity instantaneos elocity Trblent elocities are the positie and negatie deiations of the instantaneos elocities abot the mean.

20 Rles of aeraging To inclde the effects of trblence in the primitie eqations, we need to hae some basic rles for dealing with the mathematics of aeraging. Compting the time aerage of a ariable A(x,y,z,t) that is a fnction of space and time: A( x, y,z ) N i0 N A( x, y,z,i ) discrete fnction A( x, y,z ) P P t0 A( x, y,z,t )dt continos fnction

21 Let A and B be two ariables that ary oer time and let c represent a constant. We will show that: B A B N A N B A N B A N B A i i i i i i i i N i i i 0 Compting the time aerage of a ariable A(x,y,z,t) that is a fnction of space and time: discrete continos B A B A B A B dt P Adt P B dt Adt P dt B A P B A P t P t P t P t P t Aeraging

22 More aeraging Throgh similar mathematical maniplation, we can derie the following aeraging rles: c ca A c ca A A AB B da dt A B A d A dt B Next we will apply these rles to ariables that are split into mean and trblent components.

23 E.g., momentm flx

24 Trblent goerning eqations z w y x x p f dt d 0 z w y x y p f dt d 0 D z w w y w x w g z p dt dw 0 0 D z w y x dz d w t 0 0 z w y x Note = (D)

25 Example: kinetic energy w w e E E w e E per nit mass w E w e w E Sbstitting for mean and deiations, Collecting terms, and taking time aerage, (notice mean of mean times deiation is zero) Mean kinetic energy Trblent kinetic energy

26 Dirnal TKE deelopment

27 TKE eqation Eoltion of TKE Use eddy form of goerning eqations to derie TKE eqation w e z p w z w e g w z w z w dt de ) ( ) ( 0 0 A B C D E F A: local change and adection by the mean flow B: mechanical prodction de to wind shear C: prodction by boyancy (conection) D: transport by eddies E: redistribtion by pressre (graity) waes F: dissipation, conersion of mechanical energy to heat

28 Flx Richardson nmber Ratio of boyant and shear eddy generation R f boyancy prodction mechanical prodction g w 0 w z w z A measre of the stability, or degree to which the atmosphere wants to resist deeloping trblence R f > stability dominates and flow is laminar (trblence tends to decay) R f < flow nstable and trblence deelops (eddies generated by strong wind shear) Can be sed to define top of the bondary layer, by looking for altitde where R f =

29 Richardson nmbers Flx Richardson nmber Gradient Richardson nmber (PR) Blk Richardson nmber All captre same essential physics: Can mechanical shear oercome stratification? 9

31 Field trip - assignment Measrements at East Bolder Commnity Park am - :5 Set p at 0- am. Some helpers wold be great! RTD: Depart Broadway and Eclid (~ 5 mintes): 5E (best/direct) 5 (then walk a bit to the soth) DASH (then walk a bit to the north) Bring: notebook to take field notes and log balloon theodolite angles. A watch will be sefl. 3

33 DASH

34 55 th street. Soth from Baseline Soccer fields Parking Meet here

35 Volnteers Few people ( or 3) to help with come with me at 9:30 to load p gear and drie to start set p. (Meet at stadim 070 lab 36) Others welcome to come early to get started on se p ~ 0am Can stay a little past :5 (need a few extra hands to help pack p) Car pool?s

Spring Semester 2011 April 5, 2011

Spring Semester 2011 April 5, 2011 METR 130: Lectre 4 - Reynolds Averaged Conservation Eqations - Trblent Flxes (Definition and typical ABL profiles, CBL and SBL) - Trblence Closre Problem & Parameterization Spring Semester 011 April 5,