atmospheric flow

8/16/2019 atmospheric flow

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Energy MeteorologyInstitute of Physics / ForWindCarl von Ossietzky Universität Oldenburg

Detlev Heinemann

National Cheng Kung University, Tainan, Taiwan – 10 September 2015

Basics of Atmospheric Flow Modeling

MODELING OF ATMOSPHERIC FLOW (I)

DAAD/NCKU Summer School – Lecture 3



CONTENTS OF LECTURES 3–5

I. Basics of Atmospheric Flow Modeling

! Dynamics of Horizontal Flow

! Atmospheric Boundary Layer

II. Atmospheric Flow Modeling: Overview & Large-scale! Overview

! Model Classes (linear, RANS, LES, ..)

! Scales of Atmospheric Motion

III. Atmospheric Flow Modeling: Meso- & Micro-scale! Meso-scale Wind Flow Modeling

! Small-scale Modeling: Large Eddy Simulation

2National Cheng Kung University, Tainan, Taiwan – 10 September 2015

MODELING OF ATMOSPHERIC FLOW





DYNAMICS OF ATMOSPHERIC FLOW



Newton’s Second Law (I)

in each of the three directions in the coordinate system, theacceleration a experienced by a body of mass m in response to a

resultant force "F is given by

This equation describes the motion of that body in an inertial (i.e.non-accelerating) frame of reference.



DYNAMICS OF ATMOSPHERIC FLOW



Newton’s Second Law (II)

If the coordinate system is accelerated, apparent forces are introducedto compensate for this acceleration of the coordinate system.

In an accelerated frame of reference two different apparent forces arerequired:

! a centrifugal force that is experienced by all bodies, irrespective oftheir motion,

!and a Coriolis force that depends on the relative velocity of the bodyin the plane perpendicular to the axis of rotation (i.e., in the planeparallel to the equatorial plane).




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HYDROSTATIC EQUATION & GEOPOTENTIAL (I)



Atmospheric pressure at any heightis due to the force per unit areaexerted by the weight of the airabove that height.

—> atmospheric pressure decreases with increasing height

Net upward force due to thedecrease in atmospheric pressurewith height: -$p

Net downward force due to gravi-

tational force acting on the slab:g%$z

+

If the net upward force on the slab equals the net downwardforce: Atmosphere is in hydrostatic balance.

Wallace & Hobbs (2006)



HYDROSTATIC EQUATION & GEOPOTENTIAL (II)



For an atmosphere in hydrostatic balance, thebalance of forces in the vertical requires that

or, with $z -> 0:

Integration then yields:

HydrostaticEquation

Note: $p is negative!

Balance of

- gravitational force

- vertical component of pressure gradient force



HYDROSTATIC EQUATION & GEOPOTENTIAL (III)

The geopotential & at any point in the Earth’s atmosphere isdefined as the work that must be done against the Earth’sgravitational field to raise a mass of 1 kg from sea level to thatpoint.In other words, & is the gravitational potential per unit mass.

Units of geopotential: Jkg-1 or m2s2.

d& = gdz = - 1/% dp

The geopotential &(z) at height z is thus given by

with &(z=0) = 0 at sea level.







PRESSURE GRADIENT FORCE

The x-component of the pressure gradient forceacting on a fluid element:

The horizontal components of the pressure gradientforce and acceleration, respectively, then are:



The pressuregradient force isdirected down thepressure gradient

'p from highertoward lowerpressure.



FRICTIONAL FORCE



Frictional force (per unit mass):

( represents the vertical component of the shear stress (i.e.,the rate of vertical exchange of horizontal momentum) in unitsof Nm-2 due to unresolved small-scale motion.

Free atmosphere (above theboundary layer):

Frictional force << pressuregradient force, Coriolis force

Within the boundary layer:

Frictional force ~ other terms in thehorizontal equation of motion



SHEAR STRESS



The shear stress "s at the Earth’s surface is oriented in theopposing direction to the surface wind vector vs.

Approximation by the empirical relationship:

"s = - % CD vs vs

where

% density of the airCD dimensionless drag coe)cient (varying with surface roughness and static stability)

vs surface wind vector

vs (scalar) surface wind speed





CORIOLIS EFFECTT h e a t m o s ph er e: A nI n t r o d u c t i on t om e t e or ol o g y /

L u t g en s ,T ar b u c k ,2 0 1 3

C omm onw e al t h of A u s t r al i a ,B ur e a u of M e t e or ol o g y ,2 0 0 6

Phenomenological description:

The Coriolis e+ ect describes an 'apparent' force that causes 'apparent' deflections.It increases with increasing latitude and wind speed, and alters the direction of thewind, but not its speed.

The Coriolis force can therefore balance the pressure force so that, in the northernhemisphere, the air will flow anticlockwise around a centre of low pressure andclockwise around a centre of high pressure.





MATHEMATICAL DESCRIPTION

CORIOLIS FORCE

1

2

3

4

5

Transformation of coordinates between the inertialreference frame (‘) and the reference frame rotatingwith the angular velocity of the earth ...

… and applied to the wind velocity vector

v=d‘r/dt …

…and substituting (1) in (2) addstwo new components: the Coriolisacceleration (2nd term) plus thecentripetal acceleration (3rd term)

The Coriolis force and acceleration in vectornotation …

… and the horizontal component only(with the geographical latitude ,).







EQUATION OF MOTION (I)

Horizontal motions of fluidelements in the atmosphere aregoverned by the acting forces:

Fh = Fp,h + Fc,h + Ffr,h

The individual acceleration offluid elements (air parcels) thusis:

dvh/dt = ah = ap,h + ac,h + afr,h

Then the horizontal equation ofmotion can be written:

dvh/dt = -1/% 'p - fk " v + F



In component form:

Notes:

1) dvh/dt is the Lagrangian time derivative

of the horizontal velocity component experienced by an air parcel as it moves

about in the atmosphere.

2 ) Acceleration is due to a change in velocityof the motion as well as due to a change in direction of the motion.










PRIMITIVE EQUATIONS (I)

The horizontal equation of motion is part of a completesystem of equations that governs the evolution of large-scaleatmospheric motions – the so-called primitive equations.

The other primitive equations are related to the verticalcomponent of the motion and to the time rates of change ofthe thermodynamic variables p, %, and T.

Equations containing time derivatives are called prognostic

equations.

Relationships between the variables that are not timedependent are called diagnostic equations.






PRIMITIVE EQUATIONS (II)

Based on conservation principles for momentum, mass & energy:

Five equations in five dependent variables: u, v, 2, &, and T.



v Horizontal equation of motion (momentum equation)

p Hydrostatic/hypsometric equation (using the ideal gas equation p=%RT)

T Thermodynamic energy equation (1st law, 3=0.286, 2=dp/dt)

% Continuity equation




BASICS OF ATMOSPHERIC FLOW MODELING (II)

ATMOSPHERIC BOUNDARY LAYER

! Characteristics of boundary layer flow

! Vertical structure of the Atmospheric Boundary Layer

! Atmospheric stability

! Vertical wind profile






CHARACTERISTICS OF BOUNDARY LAYER FLOW

Influence of surface

! friction, shear, turbulence

strong vertical gradients

vertical fluxes of momentum, heat, ..

Turbulence

! turbulent eddies are generated mechanically by strong shear asflow adjusts to condition at surface

! thermal generation of turbulence through buoyancy bydestabilized stratification (--> thermal stability)








GENERAL CHARACTERISTICS


Stability

! Structure of the Atmospheric Boundary Layer (ABL) is

influenced by underlying surface and by stability of the ABL! Unstable stratified ABL enhances turbulence production -> intensified exchange -> more uniform distribution of momentum, etc.

! Stable stratified ABL suppresses turbulence produced by

shear -> weak exchange -> weak coupling with surface







Diurnal pattern

! Strong mixing during daytime with upward heat flux from surface

! Strong turbulent mixing-> nearly uniform vertical profiles (‘mixed layer‘)

! top of ABL is capped by inversion

! inversion height rises quickly early in the morning with max. height of few km during daytime

! turbulence dies out in night time when vertical heat flux turns -> shallow stable layer near surface

! nocturnal boundary layer height is typically 50 to 200 m only



GENERAL CHARACTERISTICS






TURBULENCE


...in the laboratory, forwater flowing past a cylinder ...in the atmosphere, for acumulus-topped boundarylayer flowing past an island

Stull (2006)

Example: Karman vortex streets...




GENERATION OF TURBULENCE


! Generation can be mechanically, thermally, and inertially

! Mechanical turbulence, a.k.a. forced convection, results fromshear in the mean wind. It is caused by:

! frictional drag (slower winds near the ground than above)

! wake turbulence (swirling winds behind obstacles)! free shear (regions away from any solid surface)

! Thermal or convective turbulence, a.k.a. free convection—> plumes or thermals of warm air that rises and cold air thatsinks due to buoyancy forces

! Inertial turbulence: Generation of small eddies along the edgesof larger eddies (within the turbulent cascade). Special form ofshear turbulence (shear is generated by larger eddies)

Stull (2006)




ORIGIN OF TURBULENCE


! Turbulence is a natural response to instabilities in the flow andtends to reduce the instability.

! Example of thermal instability:Vertical turbulent motion of cold and warm air to reduce a large vertical temperature gradient

! Example of dynamic instability:Vertical shear in the horizontal wind:Turbulence mixes the faster and slower moving air

!Persistent mechanical turbulence in the atmosphere needs acontinuous destabilization by external forcings

Stull (2006)




ADIABATIC PROCESSES & ATMOSPHERIC

STABILITY

Stability regimes in dry air

Dry adiabatic lapse rate:





DRY ADIABATIC LAPSE RATE

For an adiabatic process, the first law of thermodynamics can be written as

For an atmosphere in hydrostatic equilibrium

Combining both equations (eliminatepressure):

with g: standard gravity %: density cp: specific heat at constant pressure

4: specific volume


Rate of temperature decreasewith height for a parcel of dryor unsaturated air rising underadiabatic conditions

„adiabatic“: no heat transfer intoor out of the parcel

When air rises (e.g., byconvection) it expands due to thelower pressure. As the air parcel

expands, it does work at theenvironment, but gains no heat.

-> It loses internal energy and itstemperature decreases.

The rate of temperature decreasewith height is 0.98 °C per 100 m.

c pdT − αdp = 0

dp = −ρgdz

Γd =−

dT

dz =

g

c p =−0.98 K/100m




VERTICAL STRUCTURE OF THE PLANETARY

BOUNDARY LAYER

geostrophic wind, ~1-2 km

~100 m

strong vertical wind shear,almost constant wind direction

decreasing turbulent e+ ects,

change in wind direction,small change in wind speed


Ekman layer and Surface layer are each characterized by di+ erent physicalconstraints.

35




! Definition of shear stress: ( = F/A [Nm-2] --> tangential frictional per unit area force acting on fluid

by shear

! Shear stress describes exchange of momentum (--> transport)

! Ex.: (xz describes vertical flux of horizontal momentum

! Laminar flow (i.e., no turbulence):

( = 6 7u/7z (Newton‘s law of friction)

with 6: dynamic viscosity [kgm -1s -1]

( = 8 % 7u/7z

with 8 = 6/% kinematic viscosity [m2s -1]

CHARACTERISTICS OF SHEAR FLOW





Turbulent flow

! Assumption: No shear at the top of the boundary layer --> (top = 0

! Assumption: Shear stress is proportional to vertical gradient of momentum (shear):

"xz = % Km (7u/7z)

generally: " = % Km $v

analogous to laminar flow with 8 exchanged by Km

with ...






! Km: turbulent exchange coe)cient for momentum (eddy di+ usivity, eddy viscosity)

Dimension: m2s-1

! Assumption: Km = const.

! approaching the top of the boundary layer--> decreasing shear stress

(top 9 0 for z 9 ztop

--> Ekman layer






EKMAN LAYER


Source: http://mathsci.ucd.ie/met/msc/fezzik/Phys-Met




EKMAN SPIRAL


The theoretical Ekman spiral describing the height dependence ofthe departure of the wind field in the boundary layer fromgeostrophic balance.

4 is the angular departure from the geostrophic wind vg.






SURFACE LAYER


! turbulent exchange coe)cient K can no longer be assumed to beconstant:

Km = Km(z)

(near the surface, the vertical eddy scale is limited by the distanceto the surface)

! shear stress ( varies only little with height: :( /( 9 0

! change of wind direction with height is very small

! --> (xz = % Km(z)#(7u/7z)

! Question: How to describe Km(z)?




SURFACE LAYER


! K depends on the characteristic size of turbulent eddies, which

increase in size with distance from surface, i.e. with height:

Km = f (z)

!Description of the characteristic length scale by mixing length l

with l = 3#z,

where 3 ! 0.4 (von Karman constant)

! Interpretation of mixing length l: length which is necessary for

changing the turbulent characteristics of (vertically) movingeddies

! Approach:

Km = l2 (7u/7z)




SURFACE LAYER


! Expressing the shear stress in units of a velocity:

u* = ! ( /% (friction velocity)

! This leads to:

( = % K m ("u/"z) (from slide 26)

= % l2 ("2u/"z2) <--- K m = l2 ("u/"z)

= % 32z2("2u/"z2) <-- l = 3#z

u* = 3z("u/"z) <-- u*2

= ( /%

! ("u/"z) = u* / 3z logarithmic wind profile

u(z) = u*/3 #ln (z/z0) integral form with integration constant z0




logarithmic wind profile inlogarithmic (left) and

linear (right) height scale

u(z) = u*/" ln (z/ z0)

3 ! 0.4: von Karman constantu*= "# / $: friction velocityz0: roughness length

here:z0 = 0.05 m

u* = 0.4 m/s

LOGARITHMIC WIND PROFILE





Validity:

! mean values

! horizontally homogeneous

! neutral stability






logarithmic wind profile inlogarithmic (left) and linear(right) height scale

u(z) = u*/k ln (z/z0)

k ! 0.4: von Karman

constant

u*= ;(/%: friction velocityz0: roughness length

here:z0 = 0.05 m

u* = 0.4 m/s






ROUGHNESS CLASSES &

ROUGHNESS LENGTHS TABLE

RoughnessClass

RoughnessLength [m]

Energy Index(per cent) Landscape Type

0 0.0002 100 Water surface

0.5 0.0024 73Completely open terrain with a smooth surface, e.g.concrete runwaysin airports, mowed grass, etc.

1 0.03 52 Open agricultural area without fences and hedgerows and veryscattered buildings. Only softly rounded hills

1.5 0.055 45 Agricultural land with some houses and 8 metre tall shelteringhedgerows with a distance of approx. 1250 metres

2 0.1 39 Agricultural land with some houses and 8 metre tall shelteringhedgerows with a distance of approx. 500 metres

2.5 0.2 31 Agricultural land with many houses, shrubs and plants, or 8 metre tallsheltering hedgerows with a distance of approx. 250 metres

3 0.4 24 Villages, small towns, agricultural land with many or tall shelteringhedgerows, forests and very rough and uneven terrain

3.5 0.8 18 Larger cities with tall buildings

4 1.6 13 Very large cities with tall buildings and skycrapers





INFLUENCE OF ROUGHNESS ON VERTICAL PROFILE

The vertical wind gradient for di+ erent values of surface roughness (built up area;scrub vegetation; open flat land or sea surface).Note that the wind power content of the wind at the same height (red line) variesconsiderably.

©

E W E A 2 0 0 2





INFLUENCE OF THERMAL STABILITY ON

VERTICAL WIND PROFILE

Wind speed variation with height in the surface layer for di+ erentstatic stabilities, plotted on linear (left) and semi-log graph (right)


Stull (2006)




GEOSTROPHIC DRAG LAW

Large-scale pressure di+ erences determine the fictitious geostrophic wind,which is representative for the wind speed driving the boundary layer (undercertain simplifying circumstances).

The geostrophic drag law relates this wind field to the surface layer wind.Using information about surface roughness and stability, it is then possible tocalculate the wind speed near the surface.

with

G: geostrophic wind speed u*: friction velocity 3: von Karman constant

f: Coriolis parameter z0: roughness length

A and B: dimensionless functions of stability (for neutral conditions: A=1.8, B=4.5)





SUMMARY: PBL CHARACTERISTICS


Surface layer(Prandtl layer):

! :(/( 9 0

! constant vertical flux! strong vertical gradient of vh

! small change in direction

! K(z) < const.

! logarithmic wind profile

Ekman layer:

! ( 9 0 for increasing z

! small vertical shear! strong chang in direction

! small change in direction

! K(z) = const.

Ekman spiral




REFERENCES

! J.R.Holton: An Introduction to Dynamic Meteorology (4th Edition,Academic Press, 200?)

! M. Jacobsen: Fundamentals of Atmospheric Modeling (2nd Ed.,Cambridge Univ. Press, 2005)

! E. Kalnay: Atmospheric Modeling, Data Assimilation and

Predictability (Cambridge Univ. Press, 2002)! D.P. Lalas, C.F. Ratto (Eds.): Modelling of Atmospheric Flow Fields

(World Scientific, 1996)

! R.B.Stull: An Introduction to Boundary Layer Meteorology (KluwerAcademic Press, 1988)

! H.A. Panofsky, J.A. Dutton: Atmospheric Turbulence. Models andMethods for Engineering Applications (Wiley, 1984)

! Pielke, R. A.: Mesoscale Meteorological Modelling (AcademicPress, 2nd Ed., 2002)

! ...


WIND ENERGY METEOROLOGY



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