turbulence and atmospheric boundary layer lectures 10 and 11 · summary of lecture 9 1 mean winds...

Turbulence and atmospheric boundary layerLectures 10 and 11

Marta Wac lawczyk, S. P. Malinowski

Institute of Geophysics, Faculty of Physics,University of Warsaw

[email protected]

May 20, 2020

Marta Wac lawczyk, S. P. Malinowski (UW) Turbulence... May 20, 2020 1 / 59

Summary of lecture 9

1 Mean winds and temperature profiles in the surface layerMonin-Obukhov theory

2 Mean winds in the outer layerEkman spiral

3 Budget of turbulence kinetic energy in ABL


Surface energy balance in ABL

Recall the daily changes of the cler-sky ABL (diurnal cycle)



In this daily cycle, the solar radiation plays the role of an external forcingthat drives the turbulent motions.

Surface energy balance

F = Fs ↓ −Fs ↑ +FL ↓ −FL ↑

F - net radiation flux absorbed by the surfaceFs ↓ - the magnitude of the flux of shortwave solar radiation that reachesthe surface,Fs ↑ - the magnitude of the flux of shortwave radiation reflected by thesurface,FL ↓ - the magnitude of the flux of longwave radiation emitted byatmosphere,FL ↑ - the magnitude of the flux of longwave radiation emitted by thesurface,



Fs ↓ - incoming solar radiation isproportional to the sine of theelevation angle of the sunFs ↑ - the surface reflects someof the sunlight back upward,FL ↓ - depends on the airtemperature, which reaches itsmaximum in late afternoonbefore sunset and reaches itsminimum just after sunrise,FL ↑ - depends on surfacetemperature. Because of smallheat capacity, land surfacesrespond almost instantaneously:so FL ↑ is in phase with Fs ↓



The net heat flux F is next partitioned into

F = Fsh + Flh + Fgc

Fsh - sensible heat fluxFlh - latent heat flux,Fgc - conduction of heat downinto the ground,



Picture on the RHS presents an exemplary partition of the energy flux forfair weather conditions

Fsh - during daytime over a dry,unvegetated land most of thesuns energy goes into sensibleheat flux,Flh - during daytime, over moistlawns, forests etc. most of thesuns energy goes intoevaporation,Fgc - warms up the ground, thesurface skin temperature, isinversely proportional to theconductivity of the soil.



Turbulence in the ocean canquickly mix heat. So, oceansurfaces exhibit larger values ofFgc . Moreover, the specific heatof liquid water is also larger thanthat of soil. Hence, the oceanhas much larger heat capacitythan the land surface. The oceancan absorb and store solar energyduring the day and release it atnight. This results in nearlyconstant ocean surfacetemperatures through the diurnalcycle and small temperaturechanges through the annualcycle.

Figure: Author: Tiago Fioreze, CC BY-SA 3.0 source:

https://commons.wikimedia.org/wiki/File:Clouds over the Atlantic Ocean.jpg


BL approximation

b = gθv

θ0, θV = θ (1 + 0.61qV − ql )

where θV is the virtual potential temperature, ρ0 and θ0 are characteristicABL density and potential temperature, qV - water vapor mixing ratio, ql

- liquid water mixing ratio

Balance for the mean θ

∂θ

∂t+ uj

∂θ

∂xj=

∂

∂z

(κ∂θ

∂z− θ′w ′

)︸︷︷︸∼sensible heat flux

+Sθ

ρ0Cpκ∂θ/∂z - molecular sensible heat flux,ρ0Cpθ′w ′ - turbulent sensible heat flux,Sθ = 1/(ρ0cp)∂Fsh/∂z - source term


BL approximation

Balance for the mean qv

∂qv

∂t+ uj

∂qv

∂xj=

∂

∂z

(κ∂qv

∂z− q′vw

′)

︸︷︷︸∼latent heat flux

+Sq

ρ0Lκ∂qv/∂z - molecular latent heat flux,ρ0Lq′vw ′ - turbulent latent heat flux,Sq = 1/(ρ0L)∂Flh/∂z - source term


BL approximation + gradient diffusion hypothesis

Balance for the mean θ

∂θ

∂t+ uj

∂θ

∂xj=

∂

∂z

(κ∂θ

∂z+ κT

∂θ

∂z

)︸︷︷︸∼sensible heat flux

+Sθ

Balance for the mean qv

∂qv

∂t+ uj

∂qv

∂xj=

∂

∂z

(κ∂qv

∂z+ κT

∂qv

∂z

)︸︷︷︸∼latent heat flux

+Sq

where κT is the eddy diffusivity.As it is seen, the non-zero heat flux results in gradients of θ and q.


Bulk aerodynamic method to estimate the fluxes

Instead of solving the equations, we can parametrize the sensible heat fluxby the temperature difference between the surface and the air (θs − θair ).Let us assume, the temperature changes over a layer of height H. Thecharacteristic length scale of turbulence equals L and the characteristicvelocity scale |V | will be proportional to the mean wind speed at height H

Fsh = ρ0cpκT∂θ

∂xj∼ ρ0cpL|V |

(θs − θair )

H

Finally we assume

Fsh = ρ0cpCH |V |(θs − θair )

where CH is a dimensionless bulk transfer coefficient for heat.



Similarly, we can calculate the latent heat flux

Flh = ρ0LκT∂qv

∂xj∼ ρ0LL|V |

(qsat(θs)− qair )

H

Finally we assume

Flh = ρ0LCE |V |(qsat(θs)− qair )

where CE ≈ CH is the dimensionless bulk transfer coefficient for moisture.For momentum we have

u2τ = CD |V |2

where CD is the drag coefficient, uτ is the friction velocity, u2τ = −u′w ′



The coefficients CH , CE and CD are not constant, but depend on theexternal conditions.For example, CH ≈ 0.001− 0.005 in neutral stratification and flows overflat land. Exact value depends on the surface roughness. In the unstable,convective BL, CH becomes 2− 3 times larger. Conversely, as a stable BLdevelops, CH decreases.Turbulence is parametrized by the coefficient CH |V | - stronger windsmeans more shear = stronger turbulence production = stronger mixing.

The surface temperature θs should be treated as a response to to solarheating. E.g. given fixed Fsh, larger |V | results in smaller temperaturedifference (θs − θair ), hence, smaller θs . So, under windy conditions thesurface is cooler.


Values of drag coefficient for neutral stratification

Surface CD

Calm sea, desert, snow-covered flat plain 0.0014Beaches, ice, snow-covered fields 0.0028Grass prairie or farm fields, tundra 0.0047Cultivated area with low crops and single bushes 0.0075High crops, scattered obstacles (e.g. trees) 0.012Mixed farm fields and forest clumps, scattered buildings 0.018Regular coverage with large size obstacles, suburban houses 0.030villages, forestsCenters of large towns and cities, irregular forests 0.062


Vertical structure of ABL

Formation of the cappinginversion

The troposphere is staticallystable on average.

Turbulence generated byprocesses near the groundmixes surface air of lowvalues of potentialtemperature θ, with air ofhigher θ from upper region.


Vertical structure of ABL

The resulting mixture withinABL has relatively uniformpotential temperature.

This mixing has also createda temperature jump betweenthe ABL air and the warmerair aloft.

This is the cappinginversion.

So, turbulence creates thecapping inversion, but alsothe capping inversion trapsturbulence in ABL due to itsstrong stable stratification.


Daily changes of the mean profiles in ABL

Day time

Temperature and potentialtemperature near the groundincreases

Profile of θ is nearlyconstant within the outerlayer due to turbulentmixing (mixed layer)

Mixing ratio increases nearthe ground due toevaporation



Night time

Stable layer forms at nearthe ground in response tothe cooling of the air by theradiatively cooled surface.

Capping inversion formedduring day is sill present

Lower values of q near theground due to dew or frostformation.



Nocturnal jet

After the sunset turbulentfluxes across BL suddenlydisappear and residual layeris formed.

These turbulent fluxesreduced and turned the wind(Ekman spiral)

A sudden imbalance betweenthe main driving forces -Coriolis and the pressuregradient leads to theformation of the nocturnaljet.


Entrainment

During daytime, thermals rise across BL. Their inertia causes them toovershoot a small distance through the nocturnal inversion.This phenomenon is accompanied by intensive mixing and entrainment ofirrotational fluid into BL. As a result, BL grows rapidly.As it reaches the capping inversion, the entrainment zone becomes newcapping inversion.


ABL evolution


BL growth

Let us denote the height of the BL by h(t), the entrainment velocity by we

and the vertical velocity of the large-scale motion at the top of theboundary layer by wi (negative for subsidence).The entrainment velocity is proportional to the sensible heat flux Fsh butadversly proportional to the temperature jump across the inversion ∆θi .

we = βFsh

ρ0cp∆θi

where β is the Ball parameter

β = −Fshi

Fsh

and Fshi < 0 is the sensible heat flux across the capping inversion.


BL growth

So, the mixed layer is warmed up by the upward sensible heat flux fromthe surface and from the downward entrainment heat flux at the top

ρ0cpdθ

dt=

Fsh − Fshi

h(t)=

(1 + β)Fsh

h(t)

(we assume here that θ is approximately constant within the mixed layer)We can approximate the height of BL with the so-called encroachmentmethod which neglects penetration into inversion due to convection, thatis β = 0.


BL growth

Assume the stable temperatureprofile in the early morning

dθ

dz= γ or

dz

dθ=

1

γ

Find the value of the height h(t)which crosses the line z(θ), thatis

h(t) = z(θ)

Substitute to the equation fromthe previous page and integrate.

ρ0cp

∫z(θ)dθ =

∫Fshdt


BL growth

It means we calculate the amountof heat needed to increase the BLto the height h(t) from z(θ)profile and compare it with thethe amount of heat calculatedfrom the profile of the flux F

ρ0cp

∫z(θ)dθ =

∫Fshdt

The triangle

ρ0cp1

2γ(∆Θ)2 = Fsht


BL growth

Hence,

∆Θ =√

2γFsht/ρ0cp

and

henc (t) =√

2Fsht/(γρ0cp)

This describes the growth ofconvective ABL due to solarheating only (no entrainment).


BL growth

Let us now include entrainment,that is

ρ0cpdθ

dt=

(1 + β)Fsh

hentr (t)

Similarly as before we should findthe point where hentr (t) crosses aline

dθ

dz= γi or

dz

dθ=

1

γi

where we assume γi = const, soz = θ/γi , so hentr (t) = θ/γi


BL growth

We substitute hentr (t) = θ/γi

into the energy equation andintegrate (as we did before)

ρ0cp

∫ ∆Θ

0θdθ = γi (1+β)Fsh

∫ t

0dt

hence,

ρ0cp(∆Θ)2 = 2γi (1 + β)Fsht

Recall that for the case withoutentrainment we obtained

ρ0cp(∆Θ)2 = 2γFsht


BL growth

Comparing the two equalities weobtain

γi =γ

1 + β

Now, hentr = ∆Θ/γi , so

hentr (t) =√

2(1 + β)Fsht/(γiρ0cp)

or with γi = γ/(1 + β)

hentr (t) = (1+β)√

2Fsht/(γρ0cp).

Hence, finally

hentr (t) = (1 + β)henc (t).


Stable ABL


Stable BL

one of more difficult types ofBL to understand

BL tends to be 50-300 mdepth

start to form at dusk due toradiative cooling of theEarths surface

turbulence tends to beintermittent (localre-laminarisations)

gravity wave-like motions arepresent especially near thetop of BL

Figure: Smog over Almaty city, Kazakhstan, author: IgorsJefimovs, source: Wikipedia, CC BY-3.0


Stable BL

even the largest eddies donot span entire BL, sochemicals and aerosols havea tendency of layering

turbulence in SBL is formedmechanically, due to shear

the buoyant contribution ismuch smaller and it isgenerally a destructionprocess

since vertical motions aresuppressed turbulence isneither homogeneous norisotropic

Figure: Smog over Almaty city, Kazakhstan, author: IgorsJefimovs, source: Wikipedia, CC BY-3.0


External intermittency

The intermittency coefficient γ measures the fraction of turbulent fluidover the sampling space over which the statistics are taken.γ = 0 if the flow is purely laminar,γ = 1 if it is fully turbulent and

0 < γ < 1

if it is intermittent.

Figure: Vorticity in fully turbulent vs. intermittent flow [Ansorge & Mellado, JFluid Mech (805), 611-635, 2016]


Stable BL

Useful parameters to describe SBL

Brunt-Vaisala frequency

N =

√g

θv

dθv

dz

It is the maximum frequency of internal gravity waves. Waves within theSBL are trapped between the ground and the neutral layers above SBL,hence they propagate horizontally.

Flux Richardson number

Ri =

g

θvw ′θ′v

u′w ′ ∂u∂z + v ′w ′ ∂v

∂z

Ri increases with stratification, when Ri > Ric flow is assumed tore-laminarise. Ric ∼ 1.


Cloud-topped boundary layers


Cloud-Topped Boundary Layer over Land

On days when sufficient moisture is present in the BL, the liftingcondensation level (LCL) can be below h(t). In such case, the tops of thethermals that extend above the LCL are filled with cumulus clouds.

Figure: Cumulus humilis clouds in the foreground, taken at Swifts Creek, in the Great Alps of East Gippsland, Victoria,Australia. Author: Fir0002, Fir0002/Flagstaffotos, source: Wikipedia, CC BY-NC



short after sunrise BL toprises slowly while nocturnalinversion is burned off

but the LCL level risesrapidly due to sun’s warming(lowering of the relativehumidity)

as a result, BL top iscloudless in the morning



when the nocturnal inversiondisappears in the morningBL top rises rapidly

it jumps to the level∼ 1− 2km which can beabove LCL level

then fair-weather cumulusclouds are most likely toform

the top of these clouds doesnot penetrate beyond thecapping inversion



if there is sufficient verticalwind shear ∂u/∂z , ∂v/∂zhorizontal roll vortices canbe formed

These circulations organizethe stronger thermals intolines, oriented parallel to themean wind direction

Figure: Horizontal convective rolls, author: Daniel Tyndall,Departmant of Meteorology, University of Utah, source:https://pl.m.wikipedia.org/wiki/Plik:Convrolls.PNG (publicdomain)



if there is sufficient verticalwind shear ∂u/∂z , ∂v/∂zhorizontal roll vortices canbe formed

These circulations organizethe stronger thermals intolines, oriented parallel to themean wind direction

water vapor condenses onthe upward side of the roll,evaporation happens on thedownward side of the roll.

Figure: Horizontal convective rolls, author: Daniel Tyndall,Departmant of Meteorology, University of Utah, source:https://pl.m.wikipedia.org/wiki/Plik:Convrolls.PNG (publicdomain)


Cloud-Topped Boundary Layer

height of the rolls ∼ 1km,width ∼ 1− 10km

rolls form the cloud-streets

formation mechanism is notwell understood, but isrelated to the Ekman spiraland the presence of unstableconvective boundary layerbelow and stable inversionlayer above

Figure: Horizontal convective rolls, NASA image by JeffSchmaltz source:https://earthobservatory.nasa.gov/images/49254/winter-cloud-streets-north-atlantic (publicdomain)


Marine boundary layers

The cloud-topped boundary layerover the oceans are different fromtheir land counterparts due to

higher relative humidities

larger cloud cover

more important and complexrole of radiative transfer

significant role of drizzleplays in the boundary-layerheat and water balance

diurnal cycle governed bydifferent physics

Figure: Cellular convection, author: Imaleaper source:https://en.wikipedia.org/wiki/File:Open Cellular Convection.JPG,CC BY 3.0



Cloud cover reducesinsolation...

but also reduces outgoinglongwave radiation from theunderlying water.

Cloud cover also emits thelongwave radiation.

For thin BL (∼ 1km) cloudsare not much cooler thanthe Earth surface, so the neteffect is the radiative coolingof the BL.

This keeps relative humidityinside BL high. Figure: Cellular convection, author: Imaleaper source:

https://en.wikipedia.org/wiki/File:Open Cellular Convection.JPG,CC BY 3.0



As a result of net radiativecooling, the cappinginversion strengthens.

This in turn reducesentrainment of dry air intothe cloud layer

we = βFsh

ρ0cp∆θi

(larger ∆θi means smallerwe)

This contributes to themaintenance of the clouddeck.

Figure: Cellular convection, author: Imaleaper source:https://en.wikipedia.org/wiki/File:Open Cellular Convection.JPG,CC BY 3.0



Convection can be driven by

heating from below by thesurface buoyancy flux

cooling from above by theflux of longwave radiation atthe top of the cloud layer.

This induces formation of cell-likestructures (open or closed). Moreabout in the following lectures... Figure: Open-cell and closed-cell clouds off California,

Pacific Ocean, credit: Jacques Descloitres, MODIS RapidResponse Team, NASA/GSFC, source:https://visibleearth.nasa.gov/images/61786/open-cell-and-closed-cell-clouds-off-california-pacific-ocean


Boundary layer - terrain effects

Thermally driven circulations

During fair-weatherconditions, when mountainslopes are heated by the sun,the warm air rises along theslope forming an anabaticwind.

Figure: Modified picture of Glacier National Park inMontana, U.S., author: BorisFromStockdale source:https://commons.wikimedia.org/wiki/File:Glacier park1.jpg, CCBY-SA 3.0



Thermally driven circulations

After the sunset, when themountain slopes are cooledby longwave radiation, thecold, heavier air sinksdownslope as a coldkatabatic wind.

Figure: Modified picture of Glacier National Park inMontana, U.S., author: BorisFromStockdale source:https://commons.wikimedia.org/wiki/File:Glacier park1.jpg, CCBY-SA 3.0


Boundary layer - thermally driven convection

Sea and land breezes

Due to small heat capacityland surfaces react muchfaster to changes ofinsolation than the oceansurface - the air above landheats up and cools downfaster than the air overocean.

Lands are hotter in theafternoons and cool downfaster at night.

Figure: Schematic cross section through a sea breeze front,source: U.S. FAA AC 00-6A, Chapter 16, Figure 165. (publicdomain)


Boundary layer - thermally driven convection

Sea and land breezes

The resulting temperaturedifference between land andocean cause horizontalpressure gradients that driveshallow, diurnally varyingwinds: daytime sea breezesand land breezes atnighttime.

Figure: Schematic cross section through a sea breeze front,source: U.S. FAA AC 00-6A, Chapter 16, Figure 165. (publicdomain)



Consider a slope of angle α. Let us rotate the coordinate system such thatx and y axes are parallel to the slope (x is the downslope direction) and zis perpendicular to it.

Simplified momentum balance - x direction

∂u

∂t+ uj

∂u

∂xj= − 1

ρ0

∂p

∂x+ bsinα

b = g∆θv

θ0,

bsinα acts as a driving force.



Strong wind conditions

Stronger winds evolve due tothe synoptic-scale forcing.

They cause a number ofphenomena on smallerscales, including localaccelerations of the windover crests, blocking oflow-altitude winds, cloudformation, wake turbulence,Karman vortex streets,downslope wind storms, andhydraulic jumps.




local accelerations of thewind over crests

Recall mass flux across thestream US = const, where U isthe velocity and S is thecross-sectional area.




Von Karman vortex sheetsform in regions where fluidflow is disturbed by anobject. In the atmosphere,the resulting vortex streetscan be accompanied by thecharacteristic formations ofstratocumulus cloud sheets.

St =fL

U

where St is the Strouhal number,f is the shedding frequency, L isthe characteristic length.

Figure: Cloud vortices in the cloud layer off Heard Island,author: NASA / GSFC / Jeff Schmaltz / MODIS Land RapidResponse Team, (public domain)




hydraulic jumps are formedwhen high velocity fluidflows into a zone of lowervelocity

the fluid horizontal velocityrapidly decreases and thefluid ascends




Flow configuration whichcan occur can be predictedby the Froude number

Fr =U

NL

where N is the Brunt-Vaisalafrequency, L is the height of themountain. At small Fr , theairflow is forced to go around themountain and/or through gaps.At larger values of Fr more flowover the top of the mountainoccurs.


References

J.R. Garratt (1992)

The atmospheric boundary layer

Cambridge University Press

R. Stull (2005)

The atmospheric boundary layer In: Atmospheric Science (Eds. J. Wallace,P. V.Hobbs)

Elsevier


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turbulence and atmospheric boundary layer lectures 10 and 11 · summary of lecture 9 1 mean winds...

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