unit 7...from density current to bore to solitary waves during an ihop (internaonal h 2 o project)...

Unit 7 AtmosphericWavesand

Topographically-InducedFlowPhenomena

Reading Assignment

MR10Chap11:ThermallyForcedWindsinMountanousTerrain,pp.317-325Chap.12:MountainWavesandDownslopeWindstorms,pp.327-342Chap.13:BlockingoftheWindbyTerrain,pp.343-366

Atmospheric Waves Primer:

Ducted Gravity Waves, Solitary Waves, Density Currents, and Bores

h = A cos (kx – ωt) Oscillation in space and time. A = Amplitude ω = ct c = phase speed k = 2π / λx

= horizontal wavenumber λx = horizontal wavelength

Wave kinematics

h = 10 cos (kx – ωt)

h = 10 cos (ωt) At x=0, oscillation in time.

Wave kinematics

h = 10 cos (kx) At t=0, oscillation in space.

Wave kinematics

•  GravitywaveisawavedisturbanceinwhichbuoyancyactsastherestoringforceonparcelsdisplacedfromhydrostaNcequilibrium.

•  AlsoknownasbuoyancyoscillaNons.•  This buoyancy oscillation has a frequency related to stability,

known as the Brunt-Vaisala frequency (N)

Gravity Waves

Buoyancy - Driven by gravity through Archimedes’ Principle

Buoyancy

• GravitywavegeneraNonmechanisms:•  Topography

•  Airflowovermountains•  Changesinsurfaceroughness

•  ConvecNon•  ConvecNvepenetraNonintostablelayersaloQ•  Densitycurrentsimpingingonstableboundarylayer.

•  Shearinstability–Kelvin-Helmholtzinstability•  GeostrophicAdjustment

Gravity Waves

•  Also known as Kelvin-Helmholtz instability

•  Occurs when Ri < 0.25

(RAM 2002)

2

2

⎟⎠

⎞⎜⎝

⎛=

dzduNRi

N g ddz

=θ

θ

Gravity wave generation mechanisms Shear instability

Shear Instability – KH Waves

(NOAA)

12/16/2011 - BHM


•  When stability is present above or below, internal gravity waves may radiate away from the shear layer

(Scinocca and Ford 2000)

(CIMSS)

Gravity wave generation mechanisms Convection


(Bretherton and Smolarkiewicz 1989)

•  Contours of mixing ratio and potential temperature around simulated cumulus cloud


(Alexander 2002)

Gravity wave generation mechanisms Topography

•  Flow over “two-dimensional” mountains

(NASA)

ATS454/554Mesoscale 22

Exampleof“MountainWaves”•  PhototakenbyT.Lyzaduringthe

morninghoursof9June2012•  LocaNon:About20milessouthof

MilesCity,MT•  Viewingangle:towardtheNE•  RockyMountainsdistantbehind

picture


•  Geostrophic adjustment (unbalanced jet)

(Koch and O’Handley 1997)

Ducted gravity waves •  Superposition of two internal gravity waves

•  One moving downward, one moving upward

•  Reflected above by layer with decreasing m2 (vertical wavenumber) and below by the ground

•  m2 (below), is taken from the Taylor-Goldstein Eqn, and is dependent on:

§  Stability determined by the Brunt-Vaisala freq (remember, these are buoyancy oscillations!)

§  Change in vertical shear – particularly due to curvature.

•  Thus, ducts can be created by 1) thermal inversions and 2) significant change/curvature in the vertical wind shear.

•  If duct depth is ¼ of a vertical wavelength of the internal gravity waves, the waves constructively interfere.

m Nc U

d Udzc U

k22

2

2

22=

−+

−−

( ) ( )

Ducted gravity waves •  Superposition of two internal gravity waves

•  Here, vertical wavelength is 8 km, ¼ of vertical wavelength is 2 km – this is the duct.

Duct

Ducted gravity waves •  Flow in ducted gravity wave, u’ = A cos (kx – ωt) cos (mz)

•  u’ (and w’) sinusoidal in x and t

•  u’ maximized at surface, decreases with height

•  w’ maximized at top of duct, increases with height

Ducted gravity waves •  p’ and u’ correlated

•  Convergence and upward motion ahead of wave ridge

•  Divergence and downward motion ahead of wave trough

Ducted gravity waves •  Positive perturbation shear (du’/dz) in wave trough

•  Negative perturbation shear (du’/dz) in wave ridge

Ducted gravity waves

(Koch and O’Handley 1997)

•  Thermal ducts

Ducted gravity waves •  Wind ducts (speed and direction)


The Role of Gravity Waves in IniEaEng and Intensifying an EF5-Producing Supercell: 25 May 2008, Parkersburg, IA

(FromNWSDesMoinesServiceAssessment,2008)


•  TheParkersburgtornadowasadevastaNngEF5tornadothatoccurredduringthelateaQernoonhoursof25May2008

•  FirstgravitywavemayhaveplayedaroleinCI,althoughexistenceofprefrontaltroughlowersconfidenceinhowmuchofaroleitplayed

•  SecondgravitywavecoincidedwithrapidintensificaNonofsupercellandmesocyclongenesis

• Wewillcoverthelinkbetweengravitywavesandmesocyclones,QLCSmesovorNces,andtornadoesinUnit6


•  Layer of cooler, dense air overhead produces pressure rise, temperature drop and dewpoint rise at station as density current passes by.

•  Wind shift in direction of current motion

Density Currents

•  Often produced by cool thunderstorm downdrafts •  Typically have rear-to-front relative flow near surface •  Raised head near edge •  Vertical motion often produces condensation and lifts insects, dust. •  May produce CI

Density Currents

Seitter 1986

Density Currents

•  Often detected on radar due to condensation, insects, dust, etc.


Motion of a Density Current

•  Density current motion is tied to the depth of the cold pool and the negative buoyancy of the cold pool

•  The forward speed of a density current can be derived from forms of the hydrostatic equation and the x-equation of motion

•  Derivation time!

•  Flow suddenly changes from fast and shallow to slow and deep, maintaining constant flow rate.

•  Produces jump in height of flow

Hydraulic Jumps

•  A bore is a moving hydraulic jump in the atmosphere •  Depth of stable BL suddenly increases, flow in direction

opposite of bore motion decreases •  A wave phenomenon •  Often produced when density current impinges on stable BL,

often at night.

Atmospheric bores

(Simpson 2007)

•  Since wave phenomenon, often move faster than density current that produced it.

•  When a bore passes, result is steady or increasing temperature (due to mixing of BL), drop in dewpoint (due to mixing), abrupt “permanent” rise in pressure.

Atmospheric bores

Atmospheric bores •  Initial pressure rise often followed by smaller oscillations in pressure •  Result in destabilization of atmosphere due to mixing of stable BL.

In combo with lift along leading edge of bore, may cause CI.

Atmospheric bores

•  Bore strength determines whether bore is smooth, undular, or turbulent (energy imbalance at bore must be dissipated by waves/turbc)

Atmospheric bores

•  A single wave of elevation or depression in a stable layer of fluid (stable BL near surface)

•  A nonlinear form of wave between two fluids of different densities (similar to water waves)

•  Two types of dispersion occur •  Amplitude dispersion (larger waves move a little

faster, steepening wave crest) •  Wavelength dispersion (longer wavelengths move

faster)… similar process as shoaling of water waves

Solitary waves

•  Cause temporary increase in pressure (as opposed to bores that cause more permanent increase in pressure), temporary warming and drying.

Solitary waves

Currents and Bores and Waves, Oh My!

•  Densitycurrents,bores,andsolitarywavescanbeplacedonasortofone-direcNonspectrumofevoluNon

•  Knupp(2006)detailstheevoluNonfromdensitycurrenttoboretosolitarywavesduringanIHOP(InternaNonalH2OProject)IOPon21June2002


FromKnupp(2006)

Gravity Waves – Surface Trends •  DensityCurrent

•  Tempdecrease,pressureincrease,RHmayincrease.•  Bore

•  Tempsteadyorincrease•  sharppressureincrease(maybesemi-permanent)•  RHdecrease(mixing)

•  SolitaryWave–DuctedGravityWave•  Tempincreaseorsteady-Mixing•  TemporarypressureincreasewithU’windincrease•  PossibleRHdecreaseduetomixing

Overview

•  Topographycanimpactatmosphericflowsthroughavarietyofforcingsandscales

•  PVstretchinganditseffectsonRossbywaves(synopNcscale–DynamicsIandII)

•  ThermodynamicvariataNons(mesoscale)•  Slopeflows•  Valleyflows

•  Parceldisplacement(mesoscale)•  Mountainwaves•  Downslopewindstorms•  Terrainblocking

•  UniqueregionalclimatologicalfeaturesoQenassociatedwithtopography

•  DenverCyclone/DenverConvergenceVorNcityZone(DCVZ)•  CatalinaEddies

Slope Flow •  DrivenbybuoyancyvariaNonsalongamountainslope•  Atmosphereheatsfromthegroundupwardduringtheday,coolsfromthegroundupwardatnight

•  ChangeinelevaNonoflandsurface=horizontaltemperaturegradient•  BuoyancygradientduetohorizontaltemperaturegradientyieldshorizontalvorNcitygeneraNonandbothhydrostaNcandnonhydrostaNcp’(p’handp’nh)

•  Result:upslope(anabaNc)flowduringthedayanddownslope(katabaNc)flowatnight

•  AnabaNcflowp’•  Whentheslopeoftheterrainfeatureisgentle(small),p’hisdominant•  DepthofwarmsurfacelayertypicallyincreaseswithincreasingelevaNon,leadingtoanupslopePGF•  p’isconstrucNvetoupslopeflow

•  KatabaNcflowp’•  Whentheslopeoftheterrainfeatureisgentle(small),p’hisdominant•  CoolairnearsurfaceusuallybecomesdeeperwithincreasingelevaNon,leadingtoanupslopePGF•  p’isdestrucNvetodownslopeflow

•  ImplicaNon:anabaNc(upslope)windstypicallystrongerthankatabaNc(downslope)winds

•  Forsteepslope,p’nhoQendominates,leadingtop’overallacNngagainstbothanabaNcandkatabaNcflows

•  UpslopeflowtendstopeakafewhoursaQersunriseanddownslopeflowtendstopeakrightaroundsunset–peakdifferencebetweenmodifiedsurfacethermodynamicsalongslopesvs.valleytemperatures

•  AnabaNcflowstypically50-150mdeep,katabaNcflowstypically10-40mdeep

•  DepthofanabaNcflowincreaseswithincreasingelevaNon,whiledepthofkatabaNcflowincreaseswithdecreasingelevaNon

Fig.11.1fromMR10

UpwardPGFAnabaNcWind

AdaptedFig.11.3fromMR10Depthofwarmersfcairincreases

Slope Flow •  Convergencezoneforms

nearzoneofpeaktemperaturegradient

•  DMCcanformwithinthisconvergencezone,orDMCthatformsattopofmountaincanintensify

•  CommonformaNonmechanismforlong-livedPlainsMCSs/MCCs

•  WithoutDMC,erodedinversionwillleadtomixingdownofflowaloQanddestroyslopeflow(e.g.westerlyflowovertheRockies)

•  DuraNonandstrengthofslopeflowinverselyproporNonaltostrengthofflowaloQinthedownslopingdirecNon(westerlyflowaloQfortheRockies)

Valley Flow •  DrivenbythermodynamicvariaNonsalongtheaxisofavalley

orbetweenavalleyandaplainlocatedattheendofavalley

•  Diurnalwindsflowup-valleyduetoairinsidethevalleywarmingmorethantheairabovetheplain,whilenocturnalwindsflowdown-valleyduetoairinsidethevalleycoolingmorethantheplain

•  Down-valleywindscanbeamplifiedbydownslopeflow,leadingaphenomenonknownasadrainageflow

•  Valleyflowcanbeexplainedbythefirstlawofthermodynamics

𝑄= 𝜌𝑐𝑝𝑉𝑑𝑇/𝑑𝑡 

•  Keyvariable->volume(V)•  AssumingequalheaNngovervalleyandplain,slopededgesof

valleydecreasethevolumeofairimpactedbysameheatflux•  GivenconstantQ,ρ,andcp,Nmerateofchangeoftemperature

mustchange•  Result:greatertemperaturevariaNonwithinvalleythanacross

plain

•  ThevalleydoesNOTneedtoslopeforvalleyflowtooccur!•  Amodifiedslopeflowup/downvalleycanoccurinslopedvalleys

andcontributetovalleyflow

•  TopographicamplificaNonfactor(TAF)canquanNfyhowamplifiedthediurnalcyclebecomeswithinavalley

𝑇𝐴𝐹= 𝐴𝑥𝑧𝑝𝑙𝑎𝑖𝑛/𝐴𝑥𝑧𝑣𝑎𝑙𝑙𝑒𝑦 

•  Magnitudeofvalleyflowcanreach5-10ms-1Fig.11.8and11.9fromMR10

Slope and Valley Flows – Combined Summary

Sunrise•  Down-valleywindpersists

•  Upslopeflowbegins•  Valleycolderthanplain

Mid-morning(0900LST)•  ValleyflowtransiNoning

fromdowntoup•  Strongupslopeflow

•  ValleysameTasplain

Noon/earlyaQ.•  Up-valleywindisstrong•  Upslopeflowbeginsto

weaken•  Valleywarmerthanplain

LateaQernoon•  Up-valleyflowconNnues

•  Noslopeflow•  Valleywarmerthanplain

Evening•  Up-valleywindweakening

•  Downslopeflowbegins•  Valleybarelywarmerthanplain

Earlynight•  ValleyflowtransiNoningfromuptodown

•  Downslopeflowpeaks•  ValleysameTasplain

Middleofthenight•  Down-valleywindmature

•  DownslopeflowconNnues

•  Valleycolderthanplain

Latenighttomorning•  Down-valleywindfillsvalley

•  Noslopeflow•  Valleycolderthanplain

Fig.11.10fromMR10

Mountain Waves •  Internalgravitywavesforcedbyflowapproximately

perpendiculartoaridgeorseriesofridges•  MR10discussesthecomplexdynamicsofthese

waves–wewillnot!•  Reviewandexpansionofinternalgravitywavesfrom

Unit3•  Cause:mechanicaldisplacementofaparcelthat

maintainsconstantbuoyancyintoalayerofdifferingbuoyancycharacterisNcs

•  Thus,buoyancyservesasthe“restoringforce”,i.e.theforcethataimsto“restore”theparcelbacktoastateofrest

•  Foraninfiniteseriesofsinusoidalridges:•  DerivaNonofw’forwavesoveraseriesofridges,the

wavecrestNltsupstreamforarealverNcalwavenumber(m)

•  Ifmisimaginary,thenwavesbecomeincreasinglyevanescentwithheight

•  Forasingularridge:•  Evanescenceforimaginarym•  Differentwavelengthsnowsupportedbythe

topography•  ImplicaNon->WemustapplyaFouriertransform(ugh!)

tofindthewavenumberssupported•  k2<<m2forhydrostaNcwaves->theseareconfinedto

nearthemountainridge•  Downstreamwaves,ifany,arenon-hydrostaNc•  Unlesstrapped,energywillbetransportedupward

duetok2<<m2forhydrostaNcwaves(andfornon-hydrostaNcwavesforotherreasons)

Wavesoveraseriesofridges

Fig.12.3fromMR10

misreal

misimaginary

Mountain Waves Wavesformedoverasingularridge

Fig.12.4fromMR10

LenNcularcloud–causedbyliQingofparcelsbyupward-propagaNonofmountainwaves

Fig.12.2fromMR10

•  OfparNcularinterestaremountainwavesthatbecometrappedbyverNcalchangesinstabilityandshear

•  RecallfromUnit3:

𝑚2= 𝑁2/(𝑐−𝑢)2  − 𝜕2𝑢/𝜕𝑧2 /𝑐−𝑢  −𝑘2

𝑁2/(𝑐−𝑢)2  − 𝜕2𝑢/𝜕𝑧2 /𝑐−𝑢 =𝑙2 ->Scorerparameter

•  Becausek2<<m2formostmountainwaves,wavebehaviorisapproximatedwellbyanalyzingl2

•  SharpvariaNonsinstability(N2)andzonalwind(u)canleadtotwofluidlayersofdifferingScorerparameter,anupperlevel(lU)andalowerlevel(lL)

•  IflU<lL(ldecreasingwithheight),thenwaveswherelU<m<lLwillpropagateverNcallywithinthelayerwherel=lLbutwilldecayinthelayerwherel=lU(evanescentlayer)•  TheheightoftheinterfacebetweenlLandlUisdefinedasz=zr•  zrisaheightofperfectreflecNonofthewaveenergywithwavenumberm•  ReflecNonofwavesleadstoconstrucNveinterferenceamongwaveswithm=4zr(recallthatductdepth=¼verNcalwavelength,andzr=ductdepth)->thesewavesare“trapped”

Mountain Waves

Mountain Waves

Fig.12.1fromMR10

Visiblesatelliteimageryoftrappedmountainwaves

Fig.12.6fromMR10

Mountain Waves PicturebyT.LyzatakenSofMilesCity,MTonthemorningof9June2012,showingmountainwavecloudsoffoftheRockies

Fig.12.5fromMR10

LinearapproximaNon

Nonlinearapprox.–allowsforenhancementofwavesthroughnon-linearwaveinteracNons

Downslope Windstorms •  Intensewindstormsthatformonthedownslopesof

mountains

•  Associatedwithdeeplayersofairforcedoverterrain

•  Surfacelayeratthetopoftheterrainbarrierisusuallystronglystable

•  “CriNcallayer”presentabovestablesurfaceatcrest

•  NOTthermally-driven

•  Analogoustohydraulicjumps

•  CanbeexplainedbyusingaraNoknownastheFroudenumber

𝐹𝑟= 𝑢/𝑐 , where 𝑐= √𝑔𝐷 (shallow-waterwavespeed)

•  ForFr>1,flowissupercriNcaleverywhere(u>c)->flowslowsatopterrainpeakandacceleratesbacktooriginalvalueonthedownslope

•  ForFr<1,flowissubcriNcaleverywhere(u<c)->flowacceleratesoverpeakanddeceleratespastit

•  ForFr=1,flowisiniNallysubcriNcalbutacceleratestoasupercriNcalstate,peakingonthedownslopeoftheterrainfeatureunNlahydraulicjumprestoresflowtosubcriNcalstate

Fr>1

Fr<1

Fr=1

Fig.12.12fromMR10Fig.11.11fromMR10

•  ThreemostcommoncondiNonsfordownslopewindstorms1.  BreakingofwavesinverNcally-

deepcross-mountainflow2.  BreakingofwavesatacriNcal

level(shallowcross-mountainflow)

3.  Strongly-stableairatmountainpeakwithlessstableairabove(l2interface)

• Wavesamplifyduetowavebreaking

•  “SeparaNonstreamline”important(seeFig.12.13)

Downslope Windstorms

Fig.12.13fromMR10

Downslope Windstorms Fixedinterface(3km) Fixedmountainheight(500m)

Fig.12.14fromMR10 Fig.12.15fromMR10

unit 7...from density current to bore to solitary waves during an ihop (internaonal h 2 o project)...

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