Rossby waves (waves in vorticity)

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1 Rossb waes (waes in orticit)

2 Stationar (toograhicall orced) waes NCEP Reanalsis Z500 Janar mean 2

3 3 Vorticit eqation z w z w z w t 2 1 Change in relatie (ertical comonent o) orticit at a oint, Adection o relatie orticit Adection o lanetar orticit ( beta eect ) Conergence o absolte orticit Tilting o horizontal orte tbes Horizontal ariations in horizontal accelerations (baroclinic, or solenoidal term)

4 4 Vorticit eqation in ressre coordinates To obtain a ersion o the orticit eqation in ressre coordinates, we ollow the same rocedre as we sed to obtain the z-coordinate ersion: t t Using the -coordinate orm o the momentm eqations, this is: dt d which ields: Solenoidal term disaears a! momentm eqation momentm eqation d d d d dt d

5 For dr adiabatic low Thoght eeriment (otential temeratre/isentroic coordinates) The solenoid term disaears in ressre coordinates. Wh? Pressre ariations along ressre sraces is zero 1 2 The solidnoidal AND tilting term disaears in isentoic coordinates. Wh? w z w z Adiabatic condition, means no ertical motion so certainl no sheer in it! Here the ertical elocit is d w dt Onl need to worr abot diergence/stretching term! (bt this relies on assmtion o adiabatic low. i.e., a scaling)

6 Scaling qantities (midlatitde large-scale motions) U W L H dp 10 m/s 10-2 m/s 10 6 m 10 4 m 10 3 Pa 1 kg/m 3 d/ 10-2 T=L/U 10 5 s s -1 b m -1 s -1 6

7 Scaling orticit Relatie orticit ~ U L ~ s Absolte orticit, a = + ~ U L 1 0 Ro ~ 10 1 So, to the order o the Rossb nmber, a = + ~ We see immediatel the lanetar orticit las a er imortant role (which is a reminder that midlatitde sstems are close to being geostrohic) 7

8 8 Vorticit eqation scaling z w z w z w t ~ 10 ~ s L U 2 11 ~ 10 ~ s HL WU 2 10 ~10 ~ s Ub ~ ~ s L U 2 11 ~ 10 s HL WU ~ s L d d z w z w z w t 2 1

9 9 Scaled Vorticit Eqation t ) ( b So, or large scale weather sstems, the change o absolte orticit is aroimatel eqal to the rodction o orticit de to horizontal diergence (conergence) o orticit Scaling does not hold at smaller scales, where the ertical adection, tilting and baroclinic terms can become imortant For more accrac near ronts and cclones we need to retain the relatie orticit in the diergence term) t Local change in absolte orticit de to, 1) adection o absolte orticit 2) diergence o lanetar orticit

11 Vorticit limits or cclones and anticclones Cclones (low): d dt ( ) ( ) With conergence, orticit o a arcel will increase (more cclonic) As orticit increases, sbseqent increases will be greater Anticclones (high): With diergence, orticit o a arcel will decrease (more anticclonic) As relatie orticit becomes smaller, and seciicall close to, the diergence term will become zero. As sch, there is a limit to the amont o anticclonic orticit that can be obtained. So there is a limit to the rotational seed and size o anticclones. Recall we came to this same conclsion rom the gradient wind eqation this is the same essential hsics. 11

12 Vorticit in midlatitdes Geostrohic low, diergence is essentiall zero. (or, the ertical elocit is small: w~0) So the deth o the lid is constant. Simli rther to obtain a descrition o non-diergent barotroic orticit. d dt ( ) ( ) d dt 0 Ths, absolte orticit is consered ollowing the motion. 12

13 Conseration o orticit Momentm is consered Anglar momentm is consered (recall it was becase o this that we end with a Coriolis orce!) Similarl, absolte orticit is consered Absolte orticit is relatie orticit ls orticit o the lanet (i.e., local sin ls sin o lanet) The orticit o the lanet is jst the Coriolis arameter = 2 sin (!) So, ζ + = constant

14 Vorticit and Rossb waes Vorticit (ζ) is a measre o sin (either cratre or shear) Absolte orticit is relatie orticit ls the orticit o the lanet (sin o air ls sin o lanet) Absolte orticit is consered i.e., ζ + = constant Dislacement to the north, larger, so ζ smaller Anticclonic sin, trajector delects to the right Dislacement to the soth, smaller, so ζ larger Cclonic sin, trajector delects to the let (less delection to the right) So what is the orce that cases Rossb waes?

15 H cclogenesis L L L L ζ ζ L ζ ζ ζ ζ

16 Rossb waes Consider motion with constant absolte orticit Psh north, larger, so ζ smaller (anticclonic) Psh soth, smaller, so ζ large (cclonic) Psh north, Coriolis larger allows delection to the right Rossb waes moe westward

17 Rossb waes (ocean)

18 Rossb waes All waes need a restoring orce. For Rossb waes, comes abot de to changes in the Coriolis orce (the beta eect ) Imbalance between PGF and Coriolis roides restoring. Proagate westward (hase seed c = U b/k 2 ) Seed deends on size bigger is aster With some back grond (westerl) low Big waes roagate westward (against the mean low) Small waes roage eastward (with the mean low) This can be all seen with the (non-diergent) barotoic model (PS We did the 1d case. There is a generalization to 3d )

19 The irst nmerical weather orecast! Charne, J. G., Fjortot, R., and on Nemann, J., Nmerical integration o the barotroic orticit eqation. Tells, 2(4), 1950 From let to right, Harr Weler, John on Neman, M. H. Frankel, Jerome Namias, John Freeman, Ragnar Fjortot, Francis Reichelderer and Jle Charne in ront o ENIAC in 1950.

22 Problems Holton

Field tri - assignment Measrements at East Bolder Commnit Park 11am - 12:15 Set at 10-11 am. Some helers wold be great!

23 Field tri - assignment Measrements at East Bolder Commnit Park 11am - 12:15 Set at am. Some helers wold be great! RTD 203 direct Broadwa to cams. Leaing Aro eer 13 mintes to cams (Broadwa Eclid) Bring: notebook to take ield notes and log balloon theodolite angles. A watch will be sel. 23

25 55 th street. Soth rom Baseline Soccer ields Parking Meet here

The Vorticity Equation

The Vorticity Equation The Vorticit Eqation Potential orticit Circlation theorem is reall good Circlation theorem imlies a consered qantit dp dt 0 P g 2 PV or barotroic lid General orm o Ertel s otential orticit: P g const Consider