Time Scale for Cold-Air Pool Breakup by Turbulent Erosion

MeteorologischeZeitschrift, V ol. 12,No.4, 229-233(August 2003) c by Gebruder¨ Borntraeger 2003 Article

Time scale for cold-air pool breakupby turbulenterosion

SHIYUAN ZHONG1, XINDI BIAN2, and CHARLES DAVID WHITEMAN 3

1Departmentof Geosciences,University ofHouston, Houston, TX 2 USDAForest Service,East Central Research Station, East Lansing,MI 3Pacic NorthwestNational Laboratory,Richland, W A

(Manuscriptreceived November 15, 2002; in revisedform March 27, 2003; accepted May 2, 2003)

Abstract Turbulent erosion has been proposed asamajor mechanismfor removing wintertime cold-air pools (CAPs) from basins and valleys. The timescales involved in this erosion process, which areof great interest for winter weatherforecasting, have not been studied systematicallyin the past. In this short contribution, a semi-analyticalmodel isdeveloped to estimatethe timerequired for the dissipation of cold airpools from above by downward micro-scaleturbulent erosion for different wind speeds aloft, different staticstabilities inside the cold pool, and for different basin crosssections. The calculations show that micro-scaleturbulent erosion isa rather slow process and that the erosion ratedecreases rapidly with timeas staticstability increases in the capping inversion atthe top of the cold pool. The rateof erosion isdetermined mainly by wind speed above the CAPand the temperatureinversion strength inside the CAP; it is lesssensitive to the shape of the topography.Shallow CAPs of afewtens of metersin depth with aweakinversion maybe removed in a matterof hours if winds aloft aresuf ciently strong to initiate and maintain turbulent mixing. It isunlikely , however, that deeper CAPs with amoderateto strong inversion canbe destroyed by micro-scaleturbulent erosion unless combined with other regional and synoptic-scale processesthat produce larger-scaleturbulent mixing. Zusammenfassung Erosion durch Turbulenz ist schon lange alswesentlicher Prozess bei der Zerstorung¨ von winterlichen Kaltluftseen in Becken und Tälern betrachtet worden. DerZeitraum diesesErosionsprozesses, der beson- ders wichtig fur¨ Vorhersagen imWinter ist, ist bis jetztjedoch noch nicht systematischuntersucht worden. In diesemBeitrag wird ein semi-analytischesModell beschrieben, das die benotigte¨ Zeit fur¨ die Au osung¨ einesKaltluftsees durch mikroskalige turbulente Erosion von oben abschätzt.Dieses Modell berucksichtigt¨ die Windgeschwindigkeit oberhalb und die Stabilitätinnerhalb des Kaltluftseessowie den Querschnitt des Beckensbzw. Tales.Die Berechnungen zeigen, dassmikroskalige turbulente Erosion ein langsamerProzess ist und dasssich die Erosionsrate imLaufe der Zeit stark verlangsamt, wenn die statischeStabilit¨ at in der abgrenzenden Inversion oberhalb des Kaltluftseeszunimmt. DieErosionsrate wird vor allemvon der Windgeschwindigkeit oberhalb des Kaltluftseesund vom Temperaturgradienten innerhalb des Seesbestimmt. DerQuerschnitt des Beckensbzw. Talesspielt dabei eine weniger wichtige Rolle. Ein acherKaltluftsee von einigen Metern ( 50 m)mit einemschwachen T emperaturgradienten kann durch mikroskalige turbulente Erosion in einigen Stunden zerstort¨ werden, wenn die Windgeschwindigkeit oberhalb des Seesausreichend ist, um turbulente Durchmischung zu initiieren bzw. aufrechtzuhalten. Es ist unwahrscheinlich, dassein tiefer Kaltluftseemit einemm¨ aßigen bis starken Temperaturgradienten allein durch mikroskalige turbulente Ero- sion zerstort¨ werden kann, außer dass regionale oder synoptische Prozessevorhanden sind, die groß¨ ere Wirbel verursachen.

1 Introduction ceptablyhigh levels in CAPsthat last formany days (REDDY et al., 1995).CAPs canalso affect the valley Acoldair pool(CAP) is anaccumulation of cold air ina orbasin populationby producing hazardous episodes of basinor valley (relative tothe air abovethe valley) that persistent freezingrain, drizzle orfog,interfering with is characterized bya persistent temperature inversion. air andground transportation (S MITH et al., 1997).Un- CAPs are especially prevalentin the winter in basins or derstandingthe mechanisms leadingto CAPformation valleys havingpoorly developed along-valley wind sys- anddestruction is aproblemof practical as well as scien- tems. Highstatic stability inCAPs can trap air overnight tic interest. Several mechanisms havebeen proposed. or,in mid winter, for many days or evenweeks, allow- Inthe warmseason, diurnal CAPs formon near-calm ingpollutants, cloudsand moisture tobuild up. For ur- nightsby radiative cooling;they dissipate the nextmorn- banizedbasins, air pollutioncan accumulate to unac- ingwhen convection begins after sunrise (P ETKOVSEK , Correspondingauthor: C. DavidWhiteman, Paci c Northwest 1985; WHITEMAN and MCKEE,1982).During the cold NationalLaboratory, P .O.Box999, Richland, W A99352,USA, season,insolation alonemay be insuf cient to dissi- e-mail:[email protected]

0941-2948/03/0012-0231$ 02.25 DOI:10.1127/ 0941-2948/2003/0012-0231 c Gebruder¨ Borntraeger, Berlin, Stuttgart 2003 230 S.Zhonget al.:Time scale for cold-airpool breakup Meteorol.Z., 12, 2003 pate CAPsand several otherphysical processes have aloft continuesto increase, turbulenceproduction pro- beenlinked to their breakupsincluding cold air ad- ducedby vertical windshear mayeventually overcome vectionaloft (Z HONG et al., 2001),frontal passages the turbulenceconsumption caused by negative buoy- (WHITEMAN et al., 2001),air drainagefrom valleys ancyand dissipation. Asaresult, adownwardturbulent (ZAN¨ GL,2002),and turbulent erosion at the topof the sensible heat ux rcpq w is producedand the process of pool (PETKOVSEK ,1978,1985, 1992; V RHOVEC and turbulenterosion begins. When this downwardsensible HRABAR, 1996; RAKOVEC et al., 2002).Turbulent dis- heat uxbecomes equal to orgreater thanthe heat decit sipation orerosion was rst suggestedby P ETKOVSEK ina shallow layer ofcoldair at the topof the CAP,the (1985,1992) as amechanism forremoving CAPs in coldair layer dissipates byerosion. arelatively widebasin. Using an analytical approach, We use S todenote the horizontalarea at the topof heproposed two conditions that are necessary fortur- the CAP.Asturbulenterosion removes the air layer-by- bulenterosion to work: rst, astrongwind above the layer fromthe topof the coldpool, the topof the cold coldair poolis requiredto initiate the erosionprocess; pooldescends and the area S decreases (exceptin the second,this windspeed must increase continuouslyto special situation wherethe slope angle a is 90 ). The maintain the process.For typical Slovenianbasins he total sensible heat transporteddownward across the area foundthat awindspeed of 7 to9 ms 1 is required S duringa small time period Dt canbe calculated as forshear-generated turbulent mixing to start toerode the r c q w S Dt (2.1) temperature inversionfrom above. Because the capping p inversionat the topof the coldpool tends tostrengthen Theheat decit within ashallow layer nearthe topof duringthe erosionprocess, wind speeds abovethe cold the CAPcanbe estimated using poolhave to increase continuouslyso that the increase in DQ r cp Dq DV (2.2) shear productioncan compensate the increase in buoy- ancyconsumption and maintain turbulence.This re- where DV is the volumeof the layer. quirementfor a continuousincrease inwind speed above Forthis shallow layer ofcoldair tobe removed, the the CAPwas supportedby numerical simulations ofan turbulentsensible heat transporteddownward from the idealized basin byR AKOVEC et al. (2002).In their ideal- topmust equalor exceed the heat decit ofthis layer, ized simulations, the windspeed above the CAPstopped i.e., 1 increasing after reaching15 m s .Theerosion of the r c q w S Dt r c Dq DV (2.3) basin inversion,which started at awindspeed around p p 7 m s 1,ceased after the windspeed stopped increasing. Thetime requiredfor removing this shallow layer of Turbulentdissipation was also investigated bythe nu- coldair, therefore,is RHOVEC RABAR merical studyof V and H (1996)as one r cp Dq DV Dq DV ofthree mechanisms fordissipation ofdeep wintertime Dt (2.4) CAPs.Their study also conrmed that awindspeed in- r cpq w S q w S crease is necessary forturbulent erosion to continue. It Thedownward turbulent sensible heat ux q w can be also pointedout that CAPdissipation couldtake along describedusing K-theory (L UMLEY and PANOFSKY, time to complete (upto 11hours)even with windspeed 1964) as accelerations as large as 2.5m s 1 h 1. Theamount of time neededfor turbulent erosion to ¶q Dq q w Kh Kh (2.5) removea CAPis ofgreat interest forwinter weather ¶z Dz forecasting inbasins andvalleys. Inthis short contri- Substituting (2.5)into (2.4)yields asimple equationfor bution,we propose a methodfor estimating the time Dt, requiredto dissipate CAPsby turbulent mixing from Dq DV Dq DV DV Dz abovefor different windspeeds, stabilities, andbasin Dt (2.6) Dq K S cross sections. q w S Kh Dz S h Here, Dq cancels in the equationif weassume that the 2 Method depthof the layer removedis the same as the depthof the layer overwhich the turbulentsensible heat uxis cal- Asshownin Fig.1, we assume that aCAPhas anini- culated usingthe bulkapproximation based on K theory. tial depth h,apotential temperature difference qb qc The area S and volume DV are geometric parameters within the pool,and a shallow cappinginversion layer determinedby the shapeof the basin.For a circular basin (CIL)with a potential temperature difference q q . a b with aoorradius of r f lr anda constant slope angle a Wealso assume that windspeed above the topof the the horizontalarea at height z is CAP is u andwithin the CAPis zero.The wind speed differenceacross the topof the CAPproducesa verti- S z p r2 z (2.7) cal windshear that increases with increasing windspeed where aloft. Initially,the strongcapping inversion keeps the windaloft frommixing down into the CAP.Asthe wind r z r f lr z tan a (2.8) Meteorol.Z., 12, 2003 S.Zhonget al.: Time scale for cold-airpool breakup 231

If the windspeed aloft, the temperature difference across the cappinginversion, and the temperature inversion within the CAPare knownfor a circular basin with aslope angle a,Equations(2.6– 2.12) allow usto estimate the time interval requiredfor turbulent erosion to removea thinlayer nearthe topof the CAP.Theto- tal time periodfor the complete removalof the entire CAPis simply the summationof the time intervals, i.e. T å Dti i Onecan also estimate the minimumwind speed aloft requiredto initiate the process ofturbulent erosion based onthe critical Richardsonnumber. Turbulent erosion is enabledwhen the bulkRichardson number for the cappinginversion layer becomessub-critical, i.e., g Dq d Rb 0 25 (2.13) q Du 2 is the radius ofthe circular basin at height z. where d is the depthof the CIL(Fig.1). For simplicity, The volume DV is the volumeof the frustum ofa wehaveassumed calm conditionsinside the CAP,and coneand is givenby in suchcase, Du is equivalentto the windspeed aloft, u. FromEquation (2.13) we obtain p DV Dz r2 z Dz 2 3 g g Dq u 2 Dq d 2 d2 2 N d (2.14) r2 z Dz 2 q q d

r z Dz 2 r z Dz 2 (2.9) g Dq where N q d represents the Brunt-Vais¨¨ al¨a fre- Normallythe slope ofthe terrain a is muchsmaller quencyin the CIL. than 90 and both S and DV decrease as the topof the coldpool sinks. Theseterrain parameters becomein- dependentof height only when a 90 .Inthis case, 3 Results 2 2 r z r f lr S z p r f lr, and DV p r f lrDz and Equa- FromEquation (2.14), we canestimate the windspeed tion (2.6)then becomes aloft that will initiate turbulenterosion for different DV Dz Dz2 chosenstabilities andcapping inversion layer depths. Dt (2.10) K S K Results are shownin Fig.2. The minimum wind h h speedincreases with increasing CILdepths and Brunt- Theeddy diffusivity Kh neededin Equation (2.6) or Vais¨¨ al¨afrequenciesor capping inversion strengths. The its special form,Equation (2.10), is determinedfrom minimumwind speed is 7ms 1 for N 0.06 s 1 and BLACKADAR’s(1979)formulation Dz 50m,increasing tonearly 16 m s 1 for N 0.08 1 1 ¶u Ri 2 s and Dz 50 m. K l2 1 (2.11) Oncethe windspeed aloft exceedsthe threshold z Ri ¶ cr value,turbulent erosion begins. Assuming a CAPwith g ¶q ¶u 2 a500m depthand a cappinginversion layer of50 m where Ri q ¶z ¶z is the gradientRichardson number,and the critical Richardsonnumber Ricr is approxi- ina circular basin,we calculate, usingEquations (2.6– mately 0.25(T AYLOR, 1931; BUSINGER et al., 1971). 2.12),the time neededto destroythe CAPforsix sce- Thequantity l represents the turbulentmixing length narios with different values ofbasin geometry,stability , scale which,according to B LACKADAR (1962,1979), is Table 1: Parametersassumed in thecalculations for the six cases. kz 1 l kz 1 (2.12) case qa K qb K qc K u ms r f lr km a lo 1 295290 285 10 10 15 where k 0 35is the VonKarman constant, lo 2 295290 285 12 10 15 0 0063 u f 1 is the asymptotic valueof l forvery large z f is the Coriolis parameter,and u is the turbulentfric- 3 300290 285 15 10 15 tionvelocity, which can be parameterized as u cd u 4 300290 285 15 10 90 where cd is the dragcoef cient. D EARDORFF (1968) 5 300290 285 15 0 15 showed that cd is afunctionof Ri,anddecreases grad- ually from 1 0 07 u 1000 when Ri 0 to 0 when 6 300290 290 15 10 15 Ri Ricr 232 S.Zhonget al.:Time scale for cold-airpool breakup Meteorol.Z., 12, 2003

hancedwind shear. It is interesting to notethat changes inbasin geometry(Cases 3–5) result inrelatively small changesin the rates oferosion. The CAP topdescends at similar rates forthe circular basin with at oorand constant anglesidewalls (Case 3)and for the U-shaped (Case 4)basin,but is somewhatfaster forthe V-shaped basin (Case 5),indicating that if all otherconditions are the same, turbulenterosion is moreef cient fora V- shapedbasin. Finally, the fastest descent ofthe inversion topoccurs, as anticipated, forthe case ofneutral stability withinthe CAP(Case 6).In this case, forthe given strength ofthe cappinginversion (10 K) andthe wind speedaloft (15m s 1),turbulenterosion generated by the vertical shear ofhorizontal winds overpowers the in- versionto completely removea 500m deepCAP in a circular basin in slightly over30 hours.

4 Conclusion

Theresults indicate that micro-scale turbulenterosion ofcold air fromthe topof a CAPis arather slow process. Theerosionrate dependsmostly onthe windspeed aloft andthe inversionstrength andis less sensitive to the shapeof the topography.A shallow coldair pool ofa few10 s ofmeters with aweakinversion could beremoved in less thana dayif windsaloft weresuf- ciently strongto initiate andmaintain turbulentmix- andwind speed aloft. Theparameters usedfor the six ing.It would,however, take several daysto breaka CAP scenarios are summarized inT ab.1. Cases 1,2,3 and6 ofa fewhundred meters depthwith amoderateinver- assume acircular basin with aat oorof radius r and f lr sion.For CAPs that are deepor capped by a strongin- sidewalls ofangle 15 .Case 5assumes aV-shaped a version,it is unlikelythat micro-scale turbulenterosion basin where r 0andCase 4deals with aspecial case f lr alonecould destroy them unless the windsaloft were wherethe slope angleis 90degrees,which is close to a verystrong and increased rapidlywith time. Suchsitu- U-shapedbasin. In all butone case, apotential tempera- ations are normallyassociated with synoptic-scale dis- ture increase of5Kis assumed within the CAPwith po- turbancessuch as frontal passages. It is, therefore,con- tential temperature onthe basin oor 285 K and po- qc cludedthat micro-scale turbulenterosion can not break tential temperature at the base ofthe cappinginversion updeep wintertime coldair poolsunless combinedwith 290K. Case 6deals with the special case wherethe qb otherlarger-scale processes. It is ourhope that this the- stratication withinthe basin is neutral ( 290 qc qb oretical studyand our initial conclusionswill motivate K).Thecalculations start fromthe topof the CAPand further eld investigations ofthis phenomenon. progressdownward until the entire CAPis removed. Theresults ofthe calculations are givenin Fig. 3, whichshows the decrease ofthe CAPtopas afunction oftime forall six cases. Acomparisonof Cases 1and 2reveals the direct inuence of windspeed on the rate ofturbulenterosion. For the same stability andbasin geometry,the descent rate ofthe CAPtopor the erosion rate becomesconsiderably larger whenthe windspeed aloft increases from10 ms 1 inCase 1to12 m s 1 in Case 2.Consequently, the topof the CAPdropsabout 100m after 3daysfor Case 1,and nearly 400 m for Case 2.Asubstantial increase inthe sinkingrate ofthe CAPtopalso occurswhen wind speed increases from 12 m s 1 inCase 2to15 ms 1 inCase 3,but the dif- ferencebetween Cases 2and3 is smaller thanthat be- tweenCases 1and2. This is becausethe potential tem- peraturedifference across the CILdoubledfrom Case 2to Case 3,which partially cancels the effect ofen- Meteorol.Z., 12, 2003 S.Zhonget al.: Time scale for cold-airpool breakup 233

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